lecture notes on wind tunnel testing
TRANSCRIPT
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C0NTENTS
Page
Introduction 1
Chapter - 1 Wind Tunnel 10
1.1 Introduction 10
1.2 Wind Tunnel Classification 10
1.2.1 The Type of Test Section 10
1.2.2 The Type of Return Circuit 11
1.2.3 The Speed of Flow in the Test Section 12
1.3 Types of Wind Tunnels 13
1.3.1 Subsonic Wind Tunnels 13
1.3.2 Transonic Tunnel 15
1.3.3 Supersonic Tunnel 16
1.3.4 Hypersonic Tunnel 17
1.3.5 Full Scale Tunnel 18
1.3.6 Compressed Air Tunnel 19
1.3.7 Other Tunnels 19
Chapter - 2 Wind Tunnel Intrumentation 20
2.1 Introduction 20
2.2 Pick-up or Transducer 21
2.2.1 Variable Resistance Transducer 21
2.2.1.1 The Wheatstone Bridge Principle 25
2.2.1.2 Summing Circuit 25
2.2.1.3 Differencing Circuit 28
2.2.2 Variable Capacitance Transducer 29
2.2.3 Variable Reluctance Transducer 30
2.2.4 Piezoelectric Transducer 31
2.3 Signal Conditioner 32
2.3.1 Signal conditioner for Variable Resistance Transducer 33
2.3.1.1 Excitation Supply 33
2.3.1.2 Bridge Balance 33
2.3.1.3 Shunt Calibration 34
2.3.1.4 Signal Amplification 35
2.3.2 Signal Conditioner for Variable Capacitance Transducer 36
2.3.3 Signal Conditioner for Variable Reluctance Transducer 37
2.3.4 Signal Conditioner for Piezoelectric Transducer 38
2.4 Data Acquisition System 39
2.4.1 Analog System 40
2.4.2 Digital System 41
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Chapter - 3 Tunnel Characteristics 43
3.1 Introduction 43
3.2 Air Speed Calibration 43
3.3 Determination of Velocity Variation in Test Section 47
3.4 Determination of Angular Flow Variation in Test Section 49
3.5 Turbulence Level 50
3.5.1 Drag Sphere 50
3.5.2 Pressure Sphere 51
Chapter - 4 Flow Visualisation 54
4.1 Introduction 54
4.2 Incompressible Flow Visualisation Techniques 54
4.2.1 Smoke Method 54
4.2.2 Tuft Method 56
4.2.3 Oil Flow Method 57
4.2.4 Evaporation Method 57
4.3 Compressible Flow Visualisation Techniques 58
4.3.1 Shadowgraph Method 58
4.3.2 Schlieren Method 58
4.3.3 Interferometer Method 59
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Chapter - 5 Pressure Measurement by Mechanical Device 60
5.1 Introduction 60
5.2 Measurement of Cp 61
5.2.1 Without Pre–Calibration of the Tunnel 61
5.2.2 With Pre–Calibration of the Tunnel 63
5.3 Pressure Distribution on Circular Cylinder Model 63
5.4 Pressure Distribution on Elliptical Cylinder Model 67
5.5 Pressure Distribution on Spherical Model 70
Chapter - 6 Force and Moment Measurement by Mechanical Balance 72
6.1 Introduction 72
6.2 Calibration 72
6.3 Measurements of Forces and Moments 78
6.4 Evaluation of the Tare and Interference Drag 80
6.4.1 Evaluation of the Tare and Interference Drag Separately 81
6.4.2 Evaluation of the Sum of the Tare and Interference Drag 82
Chapter - 7 Pressure Measurement by Transducer 84
7.1 Introduction 84
7.2 Time Response 86
7.3 Pressure Scanning 86
7.4 Measurement of Cp 89
7.4.1 With Pre–Calibration of Tunnel 89
7.4.2 Without Pre–Calibration of Tunnel 89
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Chapter - 8 Force and Moment Measurement by Internal (Sting) Balance 91
8.1 Introduction 91
8.2 Measurement of Lift 92
8.3 Measurement of Pitching Moment 95
8.4 Simultaneous Measurement of Lift and Pitching Moment 98
8.5 Other Forces and Moments 100
8.6 Interactions Effect 103
8.7 Factors Affecting the Accuracy of Measurement 104
8.7.1 Surface Preparation and Bonding of Strain Gauges 104
8.7.2 Noise Suppression 106
8.7.3 Thermal Effect 108
8.7.4 Optimising Excitation Level 111
Chapter - 9 Force and Moment Measurement by External Balance 114
9.1 Introduction 114
9.2 General Description 114
9.3 Operation 117
9.4 Calibration 118
9.5 Wind Tunnel Testing 123
Chapter - 10 Wind Tunnel Boundary Corrections (2D Flow) 124
10.1 Introduction 124
10.2 Horizontal Buoyancy 125
10.3 Solid Blocking 128
10.4 Wake Blocking 130
10.5 Streamline Curvature Effect 133
10.6 Summary of Two–Dimensional Boundary Corrections 135
Chapter - 11 Wind Tunnel Boundary Corrections (3D Flow) 138
11.1 Introduction 138
11.2 Horizontal Buoyancy 138
11.3 Solid Blocking 139
11.4 Wake Blocking 140
11.5 Streamline Curvature Effect 140
11.6 Downwash Effect 142
11.7 Summary of Three-Dimensional Boundary Corrections 143
Chapter - 12 Drag Measurement on 2D Circular Cylindrical Body 144
12.1 Introduction 144
12.2 Drag by Pressure Distribution on the Cylindrical Surface 145
12.3 Drag by Measuring Distribution in the wake of the Cylinder 149
12.4 Drag by Direct Weighing 153
Chapter - 13 Flow about an Aerofoil Section 156
13.1 Introduction 156
13.2 Formulation of the Problem 157
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13.3 Solution 159
13.3.1 Exact analytical Solution 159
13.3.2 Approximate Solution 159
13.3.3 Exact Numerical Solution 160
13.4 Lanearised Theory 160
13.4.1 Thickness Effect 161
13.4.2 Camber Effect 161
13.5 Exact Numerical Method (Panel Method) 163
13.6 Overall Aerodynamic Characteristics 166
13.6.1 Lift, Drag and Pitching Moment Coefficient 167
13.6.2 Location of Aerodynamic Centre 171
13.6.3 Location of Centre of Pressure 171
13.7 Wind Tunnel Testing 172
Chapter – 14 Measurement of Laminar Boundary Layer 177
14.1 Introduction 177
14.2 Boundary Layer Parameters 178
14.2.1 Displacement Thickness (s*) 179
14.2.2 Other Parameters 180
14.3 Laminar Boundary Layer in Zero Pressure Gradient 183
14.3.1 Theoretical Calculation 183
14.3.2 Wind Tunnel Testing 184
14.4 Laminar Boundary Layer in Favourable Pressure Gradient 188
14.4.1 Theoretical Calculation 189
14.4.2 Wind Tunnel Testing 190
14.5 Laminar Boundary Layer in Adverse Pressure Gradient 192
Chapter - 15 Measurement of Turbulent Boundary Layer 194
15.1 Introduction 194
15.2 Structure of Turbulent Boundary Layer 194
15.3 Log Law Relation 196
15.4 Power Law Relations 197
15.5 Wind Tunnel Testing 199
Chapter - 16 Flow about Rectangular and Swept Wings 202
16.1 Introduction 202
16.2 Theory 205
16.3 Prandtl’s Lifting Line Theory 206
16.4 Vortex Lattice Method 208
16.5 Wind Tunnel Testing 210
16.5.1 Measurement of Pressure Distribution 210
16.5.2 Measurement of Overall Forces and Moments Using Balance 215
Chapter - 17 Flow about a Slender Delta Wing 217
17.1 Introduction 217
17.2 Slender Wings in Attached Flow 217
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17.3 Slender Wings in Separated Flow 219
17.4 Wind Tunnel Testing 221
17.4.1 Measurement of Pressure Distribution 221
17.4.2 Measurement of Overall Forces and Moments 223
Chapter - 18 Flow about Composite Wings 224
18.1 Introduction 224
18.2 Straked Configuration 225
18.3 Canard Configuration 228
Chapter - 19 Drag Measurement of Sphere 231
19.1 Introduction 231
Chapter - 20 Supersonic Aerodynamics 235
20.1 Introduction 235
20.2 Shock Visualisation 237
20.3 Run Time of Tunnel 238
20.4 Determination of Mach Number 241
20.4.1 By using Area-Local Mach Number Relation 241
20.4.2 By Static Pressure Measurement on the Wall of the Test Section 242
20.4.3 By using Rayleigh-Pitot Formula 242
20.4.4 By using θ-β-M Relation (Shock Wave over a Wedge) 243
20.5 Variation of Mach Number along the Axis of Divergent Section of C-D
Nozzle 244
20.6 Variation of Mach Number along Diffuser Axis 245
20.7 Determination of the Exit Velocity 246
Appendix – 1 Notations 252
Appendix – 2 Note on Units 253
Appendix – 3 List of Facilities 255
Appendix – 4 2100 System : Strain Gage Conditioner and Amplifier System 256
References 281
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INTRODUCTION
The basic aim of aerodynamics is to obtain the flow quantities (especially, pressure
distribution and skin friction) about a body immersed in fluid. Very often, the interest is
limited only to obtain the overall forces and moments acting on the body.
There are two main ways these quantities can be found; theoretically and experimentally.
Both the procedures have their relative advantages and disadvantages and have acted and
are going to act as supplementary to each other in foreseeable future
The limitation of theoretical methods basically stems from the fact that the governing
equation of real fluid about a body – the Navier-Stokes equation can not, in general, be
solved theoretically. The theoretical methods are usually based on some simplified form
of this equation. With the assumption of inviscid (infinite Reynolds Number) and
incompressible (zero Mach number) flow, i.e., the ideal flow, the Navier Stokes equation
can simplified to Laplace’s equation. The solution of this ideal flow, because of the above
simplification, differs from the experimental results. Efforts are then made to employ
some ‘corrections’ due to the effects of viscosity and compressibility.
Even with simplification of inviscid incompressible flow, it is not easy to solve the
problem. For a few simple configurations, exact analytic solutions exist (Chap. 5).
Configurations of arbitrary shape are not amenable to analytic methods and demand
numerical solution. In the early days, a variety of approximate numerical methods were
developed. Examples are the different variants of linearised theory by Munk, Weber etc.
for aerofoil problems, Prandtl’s lifting line theory, Multhopp’s lifting surface theory,
Jone’s slender wing theory etc. for wing problems. With the advent of high speed digital
computers, more sophisticated exact numerical methods (Panel method) have been
developed. A variety of computer based theoretical schemes are also developed for
effecting the corrections due to viscosity and compressibility to these solutions.
Alternatively, attempts have been made to develop Euler as well as Navier-Stokes codes
with or without turbulence modeling.
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It is almost certainly the case that – however sophisticated these theoretical methods may
eventually become – the engineer will always wish to validate his design, prior to
manufacture, by means of physical experiment. In this respect, in aircraft industry, the
wind-tunnel experimentation will always play the superior role of the two.
Wind-tunnel testing, like the theoretical calculations, has its own deficiencies and
difficulties. Broadly speaking these are : the high capital and running cost associated with
a wind tunnel, the expenses, elapsed time and skill needed in manufacturing accurate
scale models, the difficulty in obtaining the adequate data (forces, pressure distribution
etc.), the difficulty of interrogating this data.
Students of aeronautical engineering are well aware of the fact that the forces and
moments etc. experienced in flight on an aircraft depends primarily on two non-
dimensional parameters : Reynolds number and Mach number. Reynolds number
expresses the relative contributions of inertia and friction forces in the motion of the
fluid. The Mach number is the ratio of the flight speed and the speed of sound. In general
it can be stated that only a full scale model operating at full scale speed can give a totally
correct simulation of a real aircraft in flight. However, because of power conservation
problem (specially for high-speed flow) the wind-tunnel model is generally constructed at
a much smaller scale than the real aircraft. This in itself presents numerous difficulties
associated with the acquisition of sufficiently detailed data on such a small model.
However a more serious problem arises in simultaneously recreating the Mach number
and Reynolds number experienced in flight.
If the working medium and its temperature are the same in the wind-tunnel as in full-
scale flight in the atmosphere, then proper matching of the Mach numbers requires the air
speeds to be the same in both cases. If this is not achieved, then at Mach numbers of
interest of most aircrafts, the effects of compressibility will be different between the
wind-tunnel and flight.
On the other hand, if the speeds are kept same for Mach number simulation, the Reynolds
number in the wind-tunnel will be reduced (proportional to the geometric scale of model)
relative to the real aircraft. Clearly, if the wind-tunnel speed is increased to approach
full-scale Reynolds number then the Mach number will be incorrectly simulated.
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Now many vital phenomena depend strongly on the Reynolds number and these include :
development of boundary layer, transition from laminar to turbulent boundary layer,
separation of boundary layer, vortex formation at high angle of attack etc. If the Reynolds
number is not matched properly, viscosity will be incorrectly simulated.
Numerous technological approaches have been proposed to overcome such difficulties.
One of these consists of modifying the properties of working medium and in particular
working at very low temperature or at high pressure. These approaches, in turn, present
other difficulties. However, since the present study is restricted to low speed regime
where compressibility effects are negligible, matching of both parameters is not
necessary and simulation of Reynolds number alone is sufficient.
The other difficulties associated with wind tunnel testing arise from the fact that the flow
conditions inside the tunnel are not exactly the same as those in the free air. Primarily, the
air in the tunnel is considered to be more turbulent than the free air this turbulence being
produced in the tunnel by propeller, vibrations of the tunnel walls etc. This consequently
increases the effective Reynolds number of the tunnel (Section 3.5). Excessive turbulence
makes the test data unreliable and difficult to interpret.
Secondly, the wind-tunnel model experiences spurious ‘constraint’ effects due to wind-
tunnel walls (chapter 10 and 11) which will be absent in free air. These extraneous forces
must be calculated and subtracted out. These forces arise from two sources. Due to
formation and growth of boundary layer in the test section, the effective area is
progressively reduced resulting in increase of velocity and decrease of static pressure
downstream. This variation of static pressure produces a drag force known as ‘horizontal
buoyancy’. Again, the presence of a model in the test section reduces the area through
which air flows. This ‘blockage’ caused by the model and its wake effectively increases
the average air speed in the vicinity of the model than they would be in free air, thereby
increasing all forces and moments at a given angle of attack.
Thirdly, the model in a tunnel is usually installed by some supports which in turn affect
the flow. The effect of this supports (the so-called ‘Tare’ and ‘Interference’ effects,
section 6.4) need to be calculated carefully and eliminated from observed values.
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The procedure involved in wind-tunnel testing may now be summarized. The prerequisite
of any experimental work is the calibration and evaluation of the tunnel (Chapter – 3)
itself. The wind-tunnel must be pre-calibrated to give the velocity of air flow during any
testing (since it is not practical to measure the velocity by pitot-static tube while the
model is in tunnel). The flow characteristics of the tunnel must be ascertained by
measuring the variation of velocity (static pressure) in the test section, flow angularity
and the turbulence level of the tunnel.
Wind-tunnel testing, then, involves model making, installation of model in the tunnel and
measuring forces, moments, pressure distribution etc. the forces and moments may be
obtained by any of the three methods :
a) Measuring the actual forces and moments with wind-tunnel balance
b) Measuring the effects that the model has on the airstream by wake survey (profile
drag, section 12.2)
c) Measuring the pressure distribution over the model by means of orifices
connected to manometer and integrating the pressure distribution over the model
surface.
The data acquired is then to be corrected for the tunnel boundary and support effects.
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Chapter 1
WIND-TUNNEL
1.1 Introduction :
The wind-tunnel is one of the most important facilities available for experimental work in
aerodynamics. Its purpose is to provide a region of controlled airflow into which models
can be inserted. This region is termed the working section or test section. For aeronautical
work, the flow in the test section should ideally be perfectly uniform in speed, direction
and vorticity. Such perfection can never be achieved in practice and the quality of a wind-
tunnel is related to the closeness to which the airflow in the test section approaches the
ideal.
1.2 Wind Tunnel Classification :
Wind-tunnels are usually classified according to the three main criteria :
i) the type of test section
ii) the type of return circuit
iii) the speed of flow in the test section
1.2.1 The type of test section:
The cross sectional form of a test section may be square, rectangle, octagonal, circular or
elliptic. Again, it can be closed or open. A closed test section is one which is completely
enclosed within solid walls, the airflow therefore being constrained by these walls. An
open test section is one which is not enclosed within solid walls (Fig. 1.1). Because the
flow is not constrained, it usually tends to expand, partly due to pressure difference and
partly due to mixing between the air in the test section and that outside. To allow for this
expansion, the downstream part of the tunnel is bell-mouthed.
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Figure 1.1 Open test section
Comparing these two types of test section, the closed type has the following advantages :
a) greater efficiency (i.e. reduced power losses)
b) better control of air flow
c) no loss of air
d) less noise
On the other hand, the open type of test section allows easier access to the model and
easier visual study of the flow.
1.2.2 The type of return circuit
A wind tunnel may either be open-circuit or closed-circuit tunnel. The open circuit tunnel
which is open at the both ends has no guided return of the air (Fig. 1.2). After the air
leaves the tunnel it circulates by devious paths back to intake.
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Figure 1.2 Open circuit tunnel
The closed circuit tunnel has, as the name implies, a continuous path for the air (Fig. 1.3).
The whole circuit, except possibly the test section, is enclosed.
1.2.3 The speed of flow in the test section:
Five categories of speed are usually recognized :
a) low speed (up to about 60 or 70 m/s)
b) high speed subsonic (but Mach number less than 0.9)
c) transonic (Mach number between 0.9 and 1.2)
d) supersonic (Mach number between 1.2 and 5)
e) hypersonic (Mach number greater than 5)
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Figure 1.3 Close circuit tunnel
The first two categories, low speed and high speed subsonic, are often taken together as
subsonic tunnels.
1.3 Types of Wind Tunnel :
1.3.1 Subsonic Wind Tunnel :
The simplest kind of subsonic tunnel consists of a tube, open at both ends, along which
the air is propelled. The propulsion is usually provided by a fan downstream of the test
section (a fan upstream would create excessive turbulence in the working section. Fig. 1.2
represents a tunnel of this type.
The following description relates to Fig. 1.2. The mouth is shaped to guide the air
smoothly into the tunnel; flow separation here would give excessive turbulence and non-
uniformity in velocity in the test section.
To make the flow parallel and more uniform in speed and top give a little time for
turbulence to decay, the mouth is followed by a settling chamber. The settling chamber
usually includes a honeycomb and wire-mesh screens.
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A honeycomb is a coarse mesh made of thin, broad plates set edgewise to the flow. It has
two purposes. First, it helps to guide the air to flow parallel to the tunnel axis. Second, if
there are any large eddies in the incoming flow, the honeycomb ‘cuts’ them into smaller
ones which can decay more rapidly than would the original larger ones.
The mesh-screens are fitted to reduce non-uniformities in flow speeds. A typical
installation might have one or two. The effects of screens on dynamic pressure variation
in the test section is shown in Fig. 1.4. The screen also serves to reduce the turbulence
level of the tunnel.
Figure 1.4 Effect of screen
The contraction followed by the settling chamber improves the quality of flow in the test
section. The air flows from the mouth of the tunnel at low speed into a comparatively
short settling chamber with a honeycomb and mesh screens. It is then accelerated rapidly
in the contraction. The contraction reduces turbulence and also non-uniformities in flow
speed and direction.
The test section is followed by a divergent duct, the diffuser. The divergence results in a
corresponding reduction in the flow speed, which has two principle effects. Firstly, it
enables an increased fan efficiency to be achieved. Secondly, the reduction in dynamic
pressure leads to reduced power losses at the exit from the tunnel in the laboratory.
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Leaving the diffuser, the air enters the laboratory, along which it flows slowly back to the
mouth of the tunnel.
A typical tunnel will have a working section of about 1 meter square and an overall
length of some 5 to 7 meters. The speed in the test section, will be controllable, upto
about 30 m/s.
1.3.2 Transonic Tunnel:
The main special feature of a transonic wind-tunnel is its test section. In this, test section
walls are neither open nor closed but a combination of both. The walls usually have
perforation or streamwise slots. The reason is as follows :
If, as an Fig. 1.5 an aerofoil is being tested in a transonic flow, shock waves occur. If the
walls were solid these shockwaves would be reflected from them and would impinge on
the model. The flow over the model would therefore be very different from that in free
flight and the test would be invalid.
If the test section were open, there would be a boundary between the jet and the
surrounding atmosphere; the shock (compression) waves would be reflected from this
boundary as expansion (rarefaction) waves. These would impinge on the model, so again
the test would be invalidated.
Figure 1.5 Reflection of shock wave
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If the walls are perforated or slotted (i.e., the test section is partly opened and partly
closed), the reflections are mixtures of compression waves and rarefaction waves and so,
depending on the degree of perforation, these tend to cancel each other out. The flow
over the model therefore approximates more closely to that in free flight.
1.3.3 Supersonic Wind Tunnel:
The simplest form of supersonic wind-tunnel is the blow-down type (Fig. 1.6). It consists
of a convergent-divergent duct whose upstream end is connected to a tank filled with
compressed air. The downstream end is usually open to the atmosphere. The air in the
tank then discharges through the duct. This means, of course, that the pressure in the tank
fall continuously, and therefore a reducing valve is fitted to maintain a constant pressure
at the inlet of the duct. The duration of each test run is necessarily limited with this type
of tunnel.
The blow-down type of tunnel is relatively cheap. In particular, a relatively low-powered
pump can be used to pressurize the tank taking, of course, a correspondingly long time to
do so. The power expanded in driving the tunnel during a test is many times greater than
the power of the pump.
The test section of this type of tunnel is followed by a convergent-divergent duct. It can
be shown that if the pressure ratio between the two ends of a convergent-divergent duct
exceeds 1.892, the flow is sonic (M=1) at the throat and supersonic downstream. A plane
downstream of the throat can therefore be used as a test section in which the flow is
supersonic.
Figure 1.6 Supersonic wind tunnel
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The Mach number at the test section will depend only on the cross-sectional areas at the
throat and the test section.
32
.
6
51
M
MA
A ST (1.1)
This shows that the test section Mach number is determined solely by the shape of the
tunnel (provided the pressure ratio is sufficient to maintain supersonic flow through the
test section). Because of this supersonic tunnels frequently consist of a basic ‘frame’ to
which various liners can be fitted. Each liner gives a unique area ratio and therefore a
unique Mach number in the test section. The shapes of some different liners for various
Mach number are illustrated in Fig. 1.7.
Figure 1.7 Shapes of liners
1.3.4 Hypersonic Wind Tunnel :
The main special feature of hypersonic wind tunnel is that provision must be made for
preheating the air before entering the tunnel.
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By suitable design of lines i.e. providing the large area ratio AT*S/ A* for generating high
Mach number, the Mach number in the test section of a supersonic wind-tunnel may be
increased to hypersonic regime. But another consequence of expanding air to high speed,
namely its change in temperature, becomes limiting criterion. The equation for the
temperature ratio along a streamline originating in a region where the flow is at rest with
temperature T0 and terminating where the temperature is T is given by
5
12
0 M
T
T (1.2)
For M = 10, this equation gives T =T0/21. Now if T0 be the absolute temperature 228k
then the wind temperature in the test section will be 13.5K. This is well below the
temperature where air becomes liquid. Thus a limiting Mach number in the test section
would be one at which air remains gaseous.
The obvious choice for increasing this limiting Mach number is not preheat the air to be
used in the tunnel to such an extent that the very low temperature in the test section is not
realized. Another choice is to use a gas which has very much lower condensation
temperature than air, e.g. helium. The majority of hypersonic tunnels, however use the
preheating method. The preheating of air may be done by heating the reservoir air or
alternatively to allow the air to pass through a heat exchanger as it leaves the reservoir to
enter the working section.
Apart from these wind tunnels, other types of wind tunnels are also designed and
fabricated. The effort to simulate both Mach number and Reynolds number of free flight
in wind-tunnel has resulted in development of two types of tunnels :
1. Full Scale Tunnel
2. Compressed Air Tunnel
1.3.5 Full Scale Tunnel :
The Full Scale Tunnel is capable of testing actual aircrafts of moderate size under near
flight condition. The wind tunnel, developed at Langley Field, U.S.A., attains wind
velocities up to 53m/s with an open jet 18m wide and 9m high. Apart from providing a
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total simulation of Mach number and Reynolds number, such wind tunnels also serve a
useful purpose in giving a correlation between flight and small model tests.
1.3.6 Compressed Air Tunnel :
The use of high pressure and therefore a high density in the test section can help to
achieve full scale Reynolds number with relatively small and low speeds. Some tunnels
are therefore completely enclosed in a large tank which can be pumped up to several
times atmospheric pressures. Such tunnels are termed compressed air tunnels.
It is worth mentioning that high pressure is no cure-all for getting a high Reynolds
number since model strength may be a limiting factor.
1.3.7 Other Tunnels :
There are also other types of tunnels built for various purposes. Some of these tunnels
are:
Smoke tunnel : For flow visualization
Spin Tunnel : For studying spin recovery
Low Turbulence tunnel : For testing at high Reynolds number
Stability Tunnel : For studying dynamic stability
Gust Tunnel : For studying effects of gust on models
V/STOL : For studying V/STOL configurations
Ice Tunnel : For studying formation and removal of ice on models
subjected to icing condition.
Automobile Wind Tunnel : For testing full scale automobiles.
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Chapter 2
WIND TUNNEL INSTRUMENTATION
2.1 Introduction
Instrumentation plays an important role in wind tunnel testing. The accuracy of
experimental results depends not only on the quality of the tunnel but also on the
performance of he measuring equipments.
The quantities which are frequently measured in wind tunnel testing are generally
pressure distribution and over all forces and moments acting on a model. Velocity, in
general, can be calculated from the pressure and hence need not be measured. However,
in some cases velocity itself (for example, fluctuating velocity components in turbulent
flow) may be of importance and need to be measured. Also, measurement of skin friction
may be necessary in some experiments.
Measuring instruments may, broadly, be classified as two types: mechanical and
electronic. Examples of mechanical type of instruments are the liquid-level manometers
for pressure measurement and wind-tunnel mechanical balances for measurement of
overall forces and moments. Such instruments lack the first response, capability of
measuring high and low values and amenability to automation required for unsteady or
short-duration high speed tunnel.
All these limitations may be overcome in electronic instrumentation system. An
electronic system usually consist of:
a) pick-up or transducer
b) signal conditioner
c) data acquisition system
The pick-up or transducer receives the physical quantity (pressure/force) under
measurement and delivers a proportional electrical signal to the signal conditioner. Here
the signal is amplified, filtered or otherwise modified to a format acceptable to the data
acquisition system. The data acquisition system may be a simple indicating meter, an
oscilloscope or a chart recorder for visual display. Alternatively, it may be a magnetic
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tape recorder for temporary or permanent storage of data or a digital computer for data
manipulation or process control.
2.2 Pick-up or Transducer:
A transducer may be defined as a device which provides an electrical output signal for a
physical quantity (pressure/force), whether or not auxiliary energy is required. Many
other physical parameters (such as heat, light, intensity, humidity) may also be converted
into electrical energy by means of transducers. Transducers used in wind tunnel testing
may be classified according to the electrical principles involved, as follows:
1) Variable resistance transducer (resistance strain gauge)
2) Variable capacitance transducer
3) Variable reluctance transducer
4) Piezoelectric transducer
Of all these transducers, resistance strain gauge, because of its unique set of operational
characteristics, has dominated in transducer field for the past twenty years.
2.2.1 Variable Resistance Transducer:
The strain gauge is an example of variable resistance transducer that converts a physical
quantity into a change of resistance. A strain gauge is a thin, wafer-like device that can be
attached (bonded) to a variety of materials. Metallic strain gauges are manufactured from
small diameter resistance wire such as constantan, or etched from thin foil sheets (Fig.
2.1). For simultaneous measurement of strain in more than one direction, two-element or
three-element rosettes are used. The resistance of the wire or metal foil changes with
length as the material to which the gauge is attached undergoes tension or compression.
In a gauge diaphragm pressure transducer, strain gauges are directly bonded on the
diaphragm while in a sting balance used for force measurement, strain gauges are bonded
on he sting (Fig. 2.2). While the load is applied, the resistances of the strain gauges
increase or decrease, depending on nature of stress (tensile or compression). The
sensitivity of a strain gauge is described in terms of characteristics called the gauge
factor, G, defined as the unit change in resistance per unit change in length
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Or, G = (RR) (L/L) (2.1)
where G = Gauge factor
R = Gauge resistance
R = change in gauge resistance
L = normal length (unstressed condition)
L = change in length.
The term L/L is the strain , so that equation (2.1) may be written as
G = (RR) (2.2)
Where = strain in the lateral direction.
Figure 2.1 Strain gauges (a: wire, b: foil)
The resistance R of a wire of length L can be calculated by using the expression for the
resistance of conductor of uniform cross-section.
2
4d
L
area
lengthR
(2.3)
Where = specific resistance of conductor material
L = length of the conductor
d diameter of the conductor
23
Figure 2.2 Sting balance
Tension on the conductor causes an increase L in its length and a decrease d in its
diameter. The resistance of the conductor then changes to
ddd
LLL
dd
LLRR
214
1.
4.
22
(2.4)
Equation (2.4) may be simplified by using Poisson’s ratio, , defined as a ratio of strain
in lateral direction to strain in axial direction. Therefore,
LLdd (2.5)
Substituting equation (2.5) in equation (2.4) gives
)21(
1
4 2 LL
LL
d
LRR
LLLLR 211
LLR 211 [neglecting higher order term]
The gauge factor can now be obtained as
21/ LLRRG (2.6)
24
Poisson’s ratio for most metals vary from 0.25 to 0.5 and the gauge factor is then of the
order of 1.5 to 2.0. For strain-gauge application, a high sensitivity is very desirable. A
large gauge factor means a relatively large resistance change which can be more easily
measured than a small resistance change. Semi-conductor gauges are now developed,
which have gauge factor of the order of 120.
The semi-conductor strain gauges are however neither so practical nor so widely used as
the conventional metallic gauges in general purpose, high accuracy transducers. It is
worth nothing that semi-conductor gauges were originally considered advantageous
because of their high output. This has less importance today because the same
semiconductor technology which created the type of gauge has also created smaller and
less expensive amplifiers high gain for use with conventional strain gauges.
Conventional metallic strain gauges are generally of four types : Constantan, Karma,
Isoelastic and platinum-tungsten. Constantan, a copper nickel alloy, of gauge factor 2.0 is
the most popular alloy for transducer gauges. It possesses an exceptional linearity over a
wide strain range and is readily manufactured. It is also easily solderable. Its primary
limitation in precision applications is a slow irreversible drift in grid resistance when
exposed to temperature above 75 C. Because the drift rate increases exponentially with
temperature, Constantan is not recommended for transducers operating continuously at
high temperature.
Karma (gauge factor 2.1) is a nickel-chromium alloy used in a variety of modified forms
for strain sensing. Like Constantan it displays extremely good linearity over a wide strain
range. It has greater resistivity than Constantan making higher grid resistance feasible. A
major advantage is its improved resistive stability, particularly at high temperature.
Isoelastic alloy offers exceptionally good fatigue life and a gauge factor 3.1, about 50%
higher than Constantan or Karma alloys. It has limited use in transducers because of its
poor zero stability with temperature variation. Because of its good fatigue life, it is
normally used for dynamic measurements.
Platinum-tungsten alloys, like Isoelastic, find their primary use in dynamic transducer
applications. With a gauge factor approximately two times greater than Constantan and
Karma, and with very good fatigue life, platinum-tungsten gauges are used almost
exclusively in ‘fatiguerated’ dynamic transducers.
25
2.2.1.1 The Wheatstone Bridge Principle :
The change in resistance due to applied load can be converted into a change in voltage by
the Wheatstone bridge circuit. Two types of Wheatstone bridge circuits are possible :
‘summing circuit’ and ‘differencing circuit’. Generally, in wind tunnel testing,
differencing circuit is used for measuring moment.
2.2.1.2 Summing Circuit :
In the summing circuit, resistance undergoing tension and compression are connected in
opposite sides of the Wheatstone bridge. Four unstressed strain gauges R1, R2, R3, R4 are
connected to form a Wheatstone bridge in summing circuit is shown in Fig. 2.3.
The current passing through the resistance R1 and R3 is I13 where
31
13RR
VI
(2.7)
Similarly, the current passing through resistances R4 and R2 is I42 where
24
42RR
VI
(2.8)
Figure 2.3 Summing circuit
The voltage at A is therefore,
26
1
31
113 .RRR
VVRIVVA
The voltage at B is,
4
24
442 .RRR
VVRIVVB
The voltage across A and B is,
4
21
1
31
RRR
VVR
RR
VVVVVV BAAB
31
1
4
4
2 RR
R
R
RV
4231
2143
RRRR
RRRRV
or, 4231
2143
RRRR
RRRRVV
(2.9)
Now, the output voltage V will be exactly zero, if
(1) 02143 RRRR or, 2
4
3
1
R
R
R
R
or, (2) RRRRR 4321 (say)
no matter what the input voltage V may be.
If any of the resistance changes due to applied load, the output voltage V will change.
Provision may be made to change only one resistance (quarter active bridge) or two
resistance (half active bridge) or three resistance (three-quarter bridge) or all four
resistances (fully-active bridge).
For the fully active bridge (Fig. 2.2), the output voltage due to applied load is calculated
in a simple manner. The resistance R1 and R2 are subjected to compression and will
therefore have a decrease in resistance value while resistance R4 and R3 will have a
increase in resistance.
The changed values of the resistances may be written as
27
RRR
RRR
RRR
3
2
1
RRR 4 (2.10)
RR , are the changes in resistances due to changes in strain at positions 1 and 2 (Fig.
2.2).
Substituting the values in equation (9) yields
RRRRRRRR
RRRRRRRR
V
V
)(
))((
224
22
RRR
RRRR
r
RRR
4
2
R
RR
2
(2.11)
If the strain gauges are bounded very close to each other, it can be assumed
RRR
and the equation (2.11) is reduced to
24
4
R
RR
V
V
or, R
R
V
V (2.12)
The equation shows a linear relationship. However, for quarter-bridge and half bridge a
non linearity appears in the expression for output voltage. For example, if only R4 is
active (quarter-bridge) and the other three resistance are passive (not bonded on the
sting), the expression for output voltage is
RR
R
V
V
4 (2.13)
For a half-bridge (taking only R4 and R3 active)
RR
R
V
V
2 (neglecting higher order terms) (2.14)
28
Similarly, for a three-quarter bridge (taking R4, R3 and R2 )
RR
R
V
V
4
3 (2.15)
Because of the linearity in relationship, fully-active bridge is usually used in
measurement techniques. It also has another advantage compared to others i.e. the
temperature compensation effect. In a fully active bridge, all resistances have same
temperature (neglecting the thermal e.m.f. effect) while in other bridges, the temperature
of active gauges may be different from those of the passive gauges which will cause a
change in resistance values resulting in further non-linearities.
2.2.1.3 Differencing Circuit :
The arrangement of resistance in the Wheatstone bridge in ‘differencing circuit’ is shown
in Fig. 2.4. Using the similar procedure, the output voltage V in this circuit is obtained
as
Figure 2.4 Differencing circuit
3421
3142
RRRR
RRRR
V
V
29
= RRRRRR
RRRRRRRR
2.2
(
224
2
RRR
RRR
24
2
R
RRR
[neglecting 2
RR with respect to 4R2]
R
RR
2
(2.16)
If the strain gauges are pasted close to each other, the output voltage will be virtually zero
since R will be almost equal to R.
2.2.2 Variable Capacitance Transducer :
The capacitance of parallel-plate capacitor is given by
)(.. 0 faradsd
AkC
Where A = area of each plate (m2)
d distance between the plates (m)
0 = 9.85 10 -12
(F/m)
k dielectric constant
Since the capacitance is inversely proportional to the spacing of the parallel plates, d, any
variation in d causes a corresponding variation in the capacitance. This principle is
applied in the variable capacitance pressure transducer (Fig. 2.5). A pressure, applied to a
diaphragm that functions as one plate of a simple capacitor changes the distance between
the diaphragm and the static plate. The resulting change in capacitance can be measured
with an AC bridge but it is usually measured with an oscillator circuit. The transducer, as
a part of the oscillatory circuit, causes a change in the frequency of the oscillator. This
change in frequency is a measure of the magnitude of the pressure applied.
30
Figure 2.5 Variable capacitance transducer
2.2.3 Variable Reluctance Transducer :
Such transducers employ magnetic diaphragms as sensing element (Fig. 2.6). When a
differential pressure deflects the magnetic diaphragm, the air gaps (initially about 0.025
mm) also changes differentially and so does the reluctance. The two coils are connected
on a two-active arm bridge so that an output proportional to pressure is obtained.
Figure 2.6 Variable reluctance transducer
31
Another type of variable reluctance transducer is based on linear variable differential
transformer (LVDT). The LVDT is a three-coil device with a movable magnetic core
(Fig. 2.7). Two outer coils are connected in ‘opposition’ so that induced voltages are 180
out of phase with each other. When the armature is centered, these voltages are equal in
magnitude giving zero output. The pressure activates the diaphragm and when it moves
the magnetic fluxes are unbalanced to produce an output proportional to the pressure
applied.
Figure 2.7 Linear variable differential transducer
2.2.4 Piezoelectric Transducer:
The Greek word piezo means ‘to squeeze’. The piezoelectric effect is appropriately
described as generating electricity by squeezing crystals. This type of sensor is self-
generating, that is, it does not require external electrical power as do the variable
resistance or variable reluctance sensors.
A piezoelectric transducer is illustrated schematically in Fig. 2.8. The sensitivity can be
enhanced at the expense of resonant frequency by ‘stacking’ a series of elements together
with the appropriate electrical connection.
32
Figure 2.8 Schematic diagram of piezoelectric transducer
A variety of piezoelectric materials are used, with quartz being most popular. Although
piezo-electric transducers may be used for near static pressure measurements, they are
more frequently employed for transient measurement.
2.3 Signal Conditioner:
Signal originating from the transducer is fed to the signal conditioner in which it is
transformed into a form acceptable to the data acquisition system. Broadly speaking, the
signal conditioner provides circuitry for amplification, noise suppression, filtering,
excitation, zeroing, ranging, calibration and impedance matching. Because the operating
principles of the different transducers are different, a variety of signal conditioners have
been developed. The different types of signal conditioner for different transducers are
outlined below.
2.3.1 Signal conditioner for Variable Resistance Transducer :
The signal conditioner usually provides supporting circuitry for resistance strain gauge
transducer. Usually, the equipment is able to accept quarter-bridge, half-bridge and full-
bridge by providing appropriate dummy gauges. The circuitry usually provides excitation
power, balancing circuits, calibration elements, signal amplification etc.
2.3.1.1 Excitation Supply :
Normally DC excitation is used for resistance strain gauge transducer. Although AC
excitation can be used, the disadvantages outweigh the advantages. The accuracy of an
AC system is not as good as that of DC system. Also the noise rejection near the carrier
frequency is poor. Earlier DC amplifier circuit was based on the ‘chopper’ principle in
33
which the DC is first converted to AC and then amplified and later converted to DC.
Such a DC amplifier is fairly expensive. However, with the advent IC chips, DC
amplifiers are no longer more costly than AC system.
However, the DC power supplied must have high stability. To achieve this, the power
supply should be isolated from all other ‘common lines’ and from the AC power line. In
the other words, it should have a very low coupling to the power line and to the ground.
2.3.1.2 Bridge Balance :
The Wheatstone bridge circuit should ideally have zero voltage output under no load
condition, equation (2.9). However, because of normal gauge-to gauge resistance
variations and additional resistance changes during gauge installation, the bridge circuit is
usually in a resistively unbalanced state when first connected. It is advantageous to have
a balancing network to nullify any residual signal.
Figure 2.9 Parallel balance network
The most common arrangement uses a shunt on one side of the bridge as shown in Fig.
2.9, the fixed resistor in the potentiometer wiper lead being used to omit the loading
effect on the active arms of the bridge.
If all the resistance strain gauges are of exact equal values, the voltage at A and B will be
0.5 V and the output V will be zero. In this hypothetical case, the potentiometer wiper
34
lead will be at the center (position C) and the voltage there will also be 0.5 V and
therefore there will be no current through R4.
However, if due to any of reasons mentioned above, the output V is not zero, the voltage
at A is then either higher or lower than the voltage at B. in either case, bridge can be
balanced by moving the wiper lead downward (C2) or upward (C1) respectively.
The range of the balance network is given by
44R
R
V
V
if R4>>R
where V is the maximum out-of –balance (zero offset) that can be nullified. The range
can be extended by decreasing the value of R4. However, R4 can not be decreased
indefinitely because it will then have loading effect on the power supply. Usually, to limit
the loading effect, R4 is many times higher than R (of the order of 75 k to 100 k ).
2.3.1.3 Shunt Calibration :
Usually, in all signal conditioners, shunt resistors are provided across the arms connected
to balance network. The shunt resistor, when connected, can usually accommodate a
0.4% change of resistance of the arm shunted. This actually amounts to simulating 2000
strain on the arm shunted as shown in Fig. 2.10. From equation (2.2), = (R/R)/G. For
R = 120 , G = 2.0, R = 0.48, becomes 0.002 or 2000.
35
Figure 2.10 Shunt calibration
2.3.1.4 Signal Amplification :
Signal amplification is the major function of a signal conditioner. Usually, the output
voltage V (equations 2.12, 2.16) of a wheatstone bridge circuits is of the order of
microvolts since the change in resistance is usually of the order of 10 –5
to 10 –6
ohms.
Such a weak signal may not be accepted by the data indicator or recording system
(although microvoltmeters are now available) and therefore the signal originating from
transducer need to be amplified.
Signal requirements for amplifier are quite stringent. These include impedance matching
with the data indicator or recording device, high signal-to-noise ratio (SNR), low drift
(change in output voltage with time is called drift) etc.
With low impedance devices such as resistance strain gauges, no special problems arise
in the operational mode. A fairly conventional voltage amplifier with an input impedance
of 100k or greater in suitable for use with the data indicator system (such as DVM) or
36
C.R.O. For bridge circuits in which neither output terminal is grounded, a differential
amplifier is needed. Such amplifiers offer good common mode rejection characteristics.
The philosophy underlying noise cancellation is outlined in Fig. 2.11.
Figure 2.11 Noise cancellation by amplifier common-mode rejection
If the common mode rejection ratio is of the order of 105,
the noise that appears at the
output terminal is largely eliminated. Such transducers have the ability to handle direct
coupled signals, the D.C. drift being less than 10Vhour after allowing one hour warm-
up. Low drift rates are fairly difficult to achieve and the cost of D.C. amplifier with this
sort of performance is comparatively high.
2.3.2 Signal Conditioner for Variable Capacitance Transducer :
A number of signal conditioner is available based on the following schemes
i) D.C. polarization as the input circuit for an amplifier.
ii) An A.C. bridge circuit for use with and amplitude modulation system.
iii) A frequency modulating oscillator circuit.
iv) A pulse modulating circuit.
The D.C. polarization circuit, the simplest of these, is described here. It is effected by the
circuit shown in Fig. 2.12. in which C represents the capacitance of the transducer
together with that of the connecting cable and any stray parallel capacitance. The
polarizing voltage V is usually a few hundred volts. If it is assumed that the capacitance
C can be represented by a constant portion C0 plus a sinusoidally varying part C1 sinwt,
then
37
C = C0 + C1 sinwt
If C1<< C0 as will usually be the case with a transducer and the resistance R is made very
large then, it can be shown that
0
1
C
C
V
V
sin wt.
The output voltage thus follows the change in capacitance, which is dependent on the
pressure applied.
Figure 2.12 Simple D.C. polarization circuit
The limit of R is usually set by the leakage resistance in the circuit and the insulators in
the transducer must be stable and of high quality. With care, a ‘flat’ response down to a
few hertz can be obtained but a D.C. response is not possible with this circuit.
2.3.3 Signal Conditioner for Variable Reluctance Transducer :
Variable reluctance transducers require an external source of alternating current. The
output is an amplitude-modulated signal at the same frequency as the excitation and this
has to be ‘processed’ to recover the ‘pressure’ information. A block diagram of a typical
system is shown in Fig. 2.13.
The transducer is excited by an A.C. supply which is amplitude-modulated by the
transducer. A balance network is used to nullify the initial unbalance in the system and
the resistive component in the network is used to adjust the zero of the amplifier. The
band-pass filter, centered at the excitation frequency, removes ‘noise’. The signal is then
38
amplified by an A.C. amplifier and demodulated. It is then filtered to remove any ripple
from the carrier wave.
Figure 2.13 Carrier wave amplifier system
2.3.4 Signal Conditioner for Piezo-electric Transducer :
Piezo-electric transducers are self-generating; they do not require an external source of
energy. However, using them poses some special problem. In order to measure the charge
separation which occurs when the piezo-electric material is mechanically strained, a
measured circuit must be connected to it. The measuring circuit draws some current so
that the charge, Q, leaks away. To minimize this leakage, the input impedance of the
circuit must be made very large. Early approaches to this problem involved the use of
‘valve voltmeters’. The input impedance of such valves are very high so that negligible
current is drawn less than 10-12
Amp. Used in a simple voltage amplification circuit, Fig.
2.14, the output signal is function of cable capacitance CC, and any stray capacitance CS
between the input and ground as well as on the range- setting capacitor C1
39
Figure 2.14 Voltage amplifier
Thus, CS CCCC
mQV
10
This strong dependence on cable and stray capacitance is circumvented by using a
‘charge-amplifier’. This is an operational amplifier, in which the high input impedance is
retained but strong negative capacitive feedback is employed as shown in Fig. 2.15.
For such an arrangement, the output voltage V is given by
inF CmmC
QV
111
If the open loop gain –m, of the amplifier is very large (m > 50000), the output becomes
FC
QV
Thus a voltage proportional to charge Q is produced.
Figure 2.15 Charge amplifier
2.4 Data Acquisition System :
Data acquisition systems are used to measure, indicate and/or record signals originating
from transducers and signal conditioning process. Such systems can be categorized into
40
two major classes : analog system and digital system. The type of data acquisition
system, whether analog or digital, depends largely on the intended use of the recorded
input data. In general, analog systems are used when wide bandwidth is required or when
lower accuracy can be tolerated. Digital systems are used when the physical process
being monitored is slowly varying (narrow bandwidth) and when high accuracy and low
pre-channel cost is required. Digital data acquisition systems are in general more
complex than analog systems both in terms of instrumentation involved and the volume
and complexity of input data they can handle.
2.4.1 Analog System
An analog system may be defined as continuous function such as a plot of voltage versus
load (Fig. 2.16) or displacement versus pressure. Examples of the analog systems are the
analog panel meter, CRO, strip-chart recorder, X-y plotter etc.
Figure 2.16 Analog system
A complete analog instrumentation system used in wind tunnel testing may consist of
some or all of the following elements :
a) Transducers: for translating physical parameters into electric signal.
b) Signal Conditioners: for amplifying, modifying etc. of these signals.
41
c) Visual Display Devices: for continuous monitoring of the input signals. These
devices may include single or multi-channel CRO, storage CRO, panel meter,
numerical display and so on.
d) Graphic Recordings Instruments: for obtaining permanent records. These
instruments include strip chart recorder to provide continuous records on paper
charts, X-y plotter, ultraviolet recorders etc.
e) Magnetic Tape Instruments: for acquiring data, preserving their original
electrical form and reproducing them at a later data for more detailed analysis.
2.4.2 Digital System :
Digital systems handle information in digital form. A digital quantity may consist of a
number of discrete and discontinuous pulses (Fig. 2.17) which contains information about
the magnitude or nature of quantity. Digital system may consist of digital panel meter,
data-logger, computer etc. It is worth noting that if a digital system is employed, an
analog-to-digital (A/D) converter must be used before since the output signal from the
signal conditioner is in analog form.
Figure 2.17 Digital system
A complete digital instrumentation system may include some or all of the following
elements (Fig. 2.18).
a) Transducers: for translating physical parameters into electrical signals.
b) Signals Conditioners: for amplifying, modifying, etc. of these signal.
42
c) Scanner or Multiplexer: for sequentially connecting multiple analog signals to
one measuring/recording system.
d) Signal Converter: translates the analog signal to a form acceptable by analog-to-
digital converter. An example of signal converter is an amplifier for amplifying
log-level voltages generating by strain gauges.
e) Analog to Digital (A/D) converter: converts the analog voltage to its equivalent
digital form.
f) Digital Recorder: records digital information on punched cards, perforated
paper tape, magnetic tape, or a combination of these systems.
g) Auxiliary Equipment: this section contains instruments for system
programming functions and digital data processing. These functions may be
performed by individual instruments or by a digital computer.
Figure 2.18 Complete digital instrumentation system
Transd-
ucer
Signal
Condit-
ioner
Scanner/
Multiple-
xer
Signal
Conver
-ter
A/D
Conver
-ter
Digital
Record-
er
Auxiliary Equipment
and
System Programming
43
Chapter 3
TUNNEL CHARACTERISTICS
3.1 Introduction :
Once a wind tunnel is designed and constructed, the primary task is to calibrate and
evaluate the tunnel characteristics in terms of uniformity in wind speed and direction, and
also level of turbulence. A wind tunnel can be considered to have good characteristics if
the flow in the test section has uniform speed, no angular variation in direction and low
level of turbulence. Four tests are generally necessary for calibrating and evaluating a
tunnel. These are:
1. Air speed calibration.
2. Determination of velocity variation in the test section.
3. Determination of angular flow variation in the test section.
4. Determination of turbulence level.
3.2 Air Speed Calibration :
In any experiment, the wind tunnel flow speed (or dynamic pressure) must be known for
calculation of flow quantities. However, it is not desirable top insert a pitot-static tube in
the tunnel in the presence of a model. This is because of two reasons; firstly, the tube will
interfere with the model and secondly the tube will not read true owing to the effect of
model on it. It is therefore necessary to determine the airflow speed during an experiment
without using the pitot-static tube. This is possible by a prior calibration of a wind tunnel
manometer with respect to air speed.
The pitot-static tube (Fig.3.1) at station J is considered. If P0 be the total pressure, pj be
the static pressure and UJ be the oncoming flow speed at the test section, then from
Bernoulli’s equation
2
02
1JJ UpP
or, JJ PPU 02 (3.1)
44
Figure 3.1 Calibration of wind tunnel manometer
The pitot-static tube is connected to manometer M1 which shows a difference in water-
level of hJ , then
ghPP JwaterJ 0
The manometer M1 is inclined at an angle of 600,
gSinhPP JWaterJ 0
0 60 (3.2)
From equation (3.1) and (3.2)
gSinhU JWaterJ 0602 (3.3)
The air flow speed at test section can now be calculated from equation (3.3)
45
The calibration of flow speed UJ or dynamic pressure
2
2
1JJ Uq can now be
calibrated with the help of another manometer M2 . Applying Bernoulli’s equation at L
and S stations gives
22
2
1
2
1SSLL UpUp
or, SSLL qpqp where q is the dynamic pressure.
If the pressure drop between S and L stations due to friction is considered, total head at L
will be slightly smaller by an amount (say qSK1 where K1is he loss coefficient), then
1kqqpqp SSSLL
or, LSSL qkqpp 11
Applying equations of continuity between stations L and S
LLSS UAUA ; SLSL UAAU
Therefore, 2
11 LSSSL AAkqpp (3.4)
Applying equation of continuity between S and J
JJSS UAUA ; JSJS UAAU
or, JSJS qAAq2
Putting in equation (3.4)
2
1
21 LSJSJSL AAkqAApp
jqk2
or, 2kppq SLJ where k2 is a constant.
Now, if another manometer M2 is connected to stations L and S, then
ghpp LSwaterSL
or, 2kghq LSwaterJ
LSkh (3.5)
where k is a constant.
Equation (3.5) shows that the free stream dynamic pressure is linearly proportional to the
pressure difference in terms of manometer water level difference hLS. Free stream speed
46
(U) at station J Is also therefore directly related to pressure difference (in terms of hLS )
between two points L and S.
The lows peed wind tunnel (LSWT) in the department can be run at 11 different speed
setting. For 11 different speeds a table can be made concerning free stream speed C at
station J and hLS, as shown in Table 3.1.
Table 3.1. : Calibration of tunnel speed
No. of runs hJ (cm) qJ (N/m2) U at J (m/s) hLS(cm)
1.
2.
3.
-
-
11.
Calibration graphs (Fig. 3.2) can now be made in terms of q vs hLS and U vs hLS. Using
these graphs velocity or dynamic pressure in any subsequent experiment can be obtained
simply from hLS (without using pitot-static tube).
q U
(N/m2) (m/s)
hLS (cm) hLS (cm)
Figure 3.2 calibration graphs
47
3.3 Determination of Velocity variation in test section :
Velocity in the test section, even in the absence of model, is not uniform either in
horizontal or vertical direction. Owing to the effects of viscosity, the velocity near the
tunnel wall will be slower than the velocity on the centerline and velocity at downstream
will be greater than at upstream. To achieve uniformity of speed various means like using
guide vanes, breathers or screens are used.
To check uniformity of speed in vertical direction velocity at different vertical positions
(for example, points 1, 2, 3, 4, 5, in Fig. 3.3) can be measured by pitot-static tube.
Velocity at these points for a particular tunnel speed setting can be obtained from
ghU water 060sin2 (3.6)
Tunnel Roof
0 5 Test Section
Exit 0 4 Entrance
0 0 0 3 0 0
5
4 3
2
1
0 2
0 1
Tunnel Floor
Figure 3.3 Velocity measurement at five vertical and five horizontal positions
Velocity in the wind tunnel varies in longitudinal directions (i.e. along the axis of the test
section) because of viscous effects. As the flow progresses towards the exit, the boundary
layer is thickened resulting in an effective reduction of area, increase in velocity and
decrease in static pressure. Because of the decrease of static pressure there is tendency of
the model to be drawn downstream. This creates a drag force acting on the body, termed
horizontal buoyancy (chapter 10, 11), which is to be calculated and subtracted in any drag
measurement experiment.
Velocities (dynamic pressure) at different points along the tunnel center line (1,
2, 3,
4,
5 in Fig. 3.3) can be measured using the pitot-static tube as before. Subtraction of
48
dynamic pressure from total pressure (atmospheric pressure) will give static pressure at
these points.
A table can now made for calculation of velocity variation in vertical and horizontal
directions as shown below.
Table 3.2: Calculation of velocity at 9 points
Stations y cm h cm U m/s Stations x cm h cm U m/s p (N/m2)
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
U U (m/s)
(m/s) p (N/m2)
Height from floor, Distance along tunnel
y (cm) Centerline, x (cm)
Figure 3.4 Velocity variation in vertical and horizontal direction
49
Velocity variation with tunnel height (y) and velocity and static pressure variation with
distance along tunnel center line (x) can now be plotted (Fig. 3.4). Static pressure
gradient (p/x) should be calculated and noted.
3.4 Determination of Angular Flow Variation in the Test Section :
Due to defectiveness in design and construction, the flow in the test section may not be
horizontal. It is therefore necessary to know whether such angularity in flow exists and if
it exists then to measure it so as to allow compensations due to this angularity of flow.
The angular variation in the flow can be checked by using a spherical yawhead as shown
in Fig. 3.5. The yawhead has two smooth orifices usually 900 apart on the forward face of
a sphere. Obviously, if they are exactly placed, they will read equal pressure when the
flow is directed along the axis of the yawhead. If the pressure at the two points a and b
are not equal then it will indicate that the flow is inclined at an angle. This angle of yaw
may then be determined by simply rotating the yawhead till the pressures at these points
become equal. The angle of rotation of yawhead is then the angle of yaw of the flow. A
similar procedure can be adopted for measuring yaw in the horizontal plane by measuring
pressure at two other points a and b in the horizontal plane again 900 apart.
Figure 3.5 Spherical yawhead
An alternative way of measuring yaw angle is to fix yawhead in tunnel and to determine
the flow angularity by reading the pressure difference between two orifices and
comparing with a previous calibration of the yawhead.
It is believed that accurate testing can not be done if the variation in angle is greater than
50
5.0 degree. The larger angles of yaw distorts the span load excessively.
Unfortunately, the variation of flow angle across the jet may change with the tunnel
speed. If such a change is noted, a testing speed must be selected and the guide and anti-
twist vanes should be adjusted to give smooth flow at that speed.
3.5 Turbulence Level :
The flow conditions inside the wind tunnel are not exactly same as those in free air. The
flow inside the tunnel is more turbulent than the free air because of the effects of the
propeller, the guide vanes and the vibrations of tunnel walls. This discrepancy in the
turbulence level results in disagreement of tests made in the wind-tunnel and in the free
air at the same Reynolds number. By the same reasoning, tests made in different tunnels
at the same Reynolds number may not agree. A correction factor is therefore necessary
for compensating the turbulence created in the tunnel.
It is found that the flow pattern in the tunnel at a given Reynolds number corresponds
closely to the flow pattern in the free air at a higher Reynolds number. The increase ratio
is called the ‘turbulence factor’ and the effective Reynolds number RNe of the tunnel can
be obtained from the calculated Reynolds number using the turbulence factor of the
tunnel from
RNTFRN e (3.7)
The turbulence may be found with a sphere in two ways :
a) Drag sphere
b) Pressure sphere
3.5.1 Drag Sphere :
The drag coefficient of sphere is affected greatly by change in velocity. Contrary to the
layman’s guess, CD for a sphere decrease with increasing airspeed since the result of
earlier transition to turbulent flow is that the air sticks longer to the surface of the sphere.
This action decreases form or pressure drag, yielding a lower total drag coefficient.
Obviously, the Reynolds number at which the transition occurs at a given point on the
sphere is a function of the turbulence already present in the air and hence the drag
coefficient of a sphere can be used to measure turbulence . The method is to measure the
51
drag, D, for a small sphere 15 or 20 cm in diameter, at many tunnel speeds. After
subtracting the ‘horizontal buoyancy’ drag DB the drag coefficient may be computed
from
22 42
1
Ud
DDC B
D
(3.8)
Figure 3.6 Variation of CD with Reynolds Number
The sphere drag coefficient is then plotted against the calculated Reynolds number, RN
(Fig.3.6). The Reynolds number at which the drag coefficient equals 0.30 is noted and
termed the critical Reynolds number, RNC. The above particular value of the drag
coefficient occurs in free air at RN = 385000, so it follows that the turbulence factor may
be given by
TF = 385000/RNC (3.9)
Once the turbulence factor (TF) is obtained from equation (3.9), the effective Reynolds
number, RNe, can now be calculated from equation (3.7).
52
3.5.2 Pressure Sphere :
An alternative method (which will be used) of measuring turbulence makes use of
‘pressure sphere’. No force tests are necessary and the difficulty of finding the support
drag is eliminated. The pressure sphere has an orifice at the front stagnation point and
four more interconnected and equally spaced orifices at
0
2
122 from the theoretical rear
stagnation point (Fig.3.7).
Figure 3.7 Pressure Sphere
A lead from the front orifices is connected across a manometer to the lead from the four
rear orifices. After the pressure difference due to the static longitudinal pressure gradient
is subtracted, the resultant pressure difference, p for each Reynolds number is divided
by the dynamic pressure for the appropriate Reynolds number, and the quotient is plotted
against Reynolds number (Fig. 3.8). It has been found that the pressure difference p/q
is 1.22 when the sphere drag coefficient is 0.30 and hence this value of p/q determines
the critical Reynolds number RNC. Once the turbulence factor is determined, the
turbulence factor may then be determined, as before, from equation (3.9).
53
Figure 3.8 Variation of p/q with Reynolds number
This experiment is carried out on a sphere of diameter 20 cm. The following table may be
made for plotting p/q vs Reynolds number.
Table 3.3 : Experimental measurement of turbulence factor
No.of
Runs
hLS
(cm)
U from Fig.
1.2 b (m/sec)
q from
Fig.1.2 a
(N/m2)
hj
(cm)
p
= hjwg.sin600
(N/m2)
p/q RN
= UD/
1.
2.
3.
-
11.
Turbulence factor usually varies from 1.0 to 3.0. Values above 1.4 indicate that the tunnel
has too much turbulence for reliable testing. Low turbulence factor is necessary for the
test data to be reliable.
54
Chapter 4
FLOW VISUALISATION
4.1 Introduction :
Flow visualization techniques are a means of obtaining the qualitative pattern of the flow
about a body. Flows encountered in engineering application are often complex in nature.
Such techniques of flow visualization helps in obtaining a better understanding of the
flow characteristics. Many a times suitable mathematical methods have been developed
for a particular flow problem based on such qualitative studies.
Flow visualisation techniques can be classified as follows :
Flow visualisation techniques
Incompressible flow Compressible flow
Entire flow field Only on model Flow pattern Shock visualisation
1. Smoke 1. Tuft 1. Oil flow 1. Shadowgraph
2. Tuft on wire mesh 2. Oil flow 2. Interferometer
3. Evaporation 3. Schlieren
4.2 Incompressible Flow Visualisation Techniques :
4.2.1 Smoke Method :
Flow visualisation with smoke is achieved in a smoke tunnel with a facility to emit
cleaned smoke in streamer form (Fig. 4.1). Smoke is generated by burning kerosene or
paraffin. Particular care is needed in introducing the smoke in the tunnel by a blower
without disturbing the flow in the tunnel. This smoke follows the air flow and makes the
55
flow pattern visible. Smoke tunnels are usually low-velocity tunnels and most of them
have two dimensional test sections. Such tunnels are usually open circuit type to prevent
accumulation of smoke in the tunnel. The walls of test section are made of glass so that
the flow can observed (Fig. 4.2) and/or photographed.
Figure 4.1 Smoke Tunnel
Figure 4.2 Flow separation at high angle of incidence
56
4.2.2 Tuft Method :
Tufts are simplest and most often used. A large number of silk tuft are pasted at one end
on the surface of the wing. The length of each tuft is taken about 2 cm. The most rapid
method of installing the tufts is to attach them about every one inch to the tape and
pasting the tape on the model (Fig. 4.3). To obtain clear photography the model is usually
painted black while the tufts used are white. Since the open ended tufts align with the
flow the general direction of he tufts indicate the direction of the flow on the surface of
the body. Motion of tufts usually means that the flow in the boundary layer has become
turbulent. Violent motion or tendency a tendency to lift from the surface and point
upstream indicates separation.
If the tufts are to be used to examine the entire flow field they may be supported on wires
on a mesh installed inside the tunnel. Complete grids of wires normal to the flow can be
fixed in the tunnel behind a wing model. Tufts attached on one end on the mesh junctions
will align with the flow direction and show up trailing vortices.
Figure 4.3 Visualisation of flow over a straked wing by tuft method
57
4.2.3 Oil Flow Method :
In this method the model is pasted with a semi-liquid mixture of mobil oil and grease
and a dye. The dye taken for this purpose is a chemical known as Rhodamin B. When the
model is installed in the tunnel, the air flow spreads the mixture along the streamlines so
that after the tunnel has been stopped the flow pattern remains. The process requires
about 30 minutes of continuous air flow in the tunnel. The model is thereafter removed
from the tunnel and the flow pattern (Fig. 4.4) can be examined afterwards under
ultraviolet light.
An alternative approach is to mix mobil oil and titanium dioxide (dye) and paste on the
model. In this case the mixture gets dried up in a few minutes and the flow pattern can be
observed without using ultraviolet light. Care must be taken so that the oil does not
follow machining marks on the surface.
Figure 4.4 Visualisation of flow over a straked wing by oil flow method
4.2.4 Evaporation Method :
Napthaline may be dissolved in acetone and pasted on a model. When the tunnel runs
naphthalene evaporated quickly from the turbulent portion making that portion white. If
the model is painted black, transition from laminar to turbulent flow can be observed
easily.
Among the incompressible flow visualization techniques it may to be noted that tuft, oil
flow and evaporation method gives pattern of flow on the surface of the model only while
the smoke method (and tuft on mesh screen) gives the picture of the entire flow field.
58
Among the compressible flow visualization techniques, only the oil flow method,
described in section 4.2.3, can be used. Other methods are not suitable because of the
high speed involved.
4.3 Compressible Flow Visualisation Techniques :
4.3.1 Shadowgraph Method :
A parallel beam of light is produced by a point source. It is passed through a converging
lens and then through the working section. Since the flow in the working section is
compressible, refraction of light rays through the compressible medium will be different.
The screen will be illuminated where rays have converged. Shock waves then appear on
the screen as two adjacent bands, one dark and one light, corresponding to the sudden
increase in density gradient at the front of the shock and the sudden decrease in gradient
at the rear.
Figure 4.5 Shadowgraph picture of flow about a sphere
4.3.2 Schlieren Method :
Schlieren method is most widely used. It is sensitive to density changes whereas
shadowgraph method is sensitive to change in density gradient. The light rays passing
59
through the varying density area (test section) will be deflected. The screen will be
illuminated or darkened depending on the deflection of the light beam. This method is
described in details in chapter 20.
4.3.3 Interferometer Method :
A direct response to density changes is given by the interferometer which depends on the
interference fringes formed on the recombination of two light rays from the same source
which have taken different times to make the journey.
If the two path lengths are same, interference fringes may be produced. The light paths
are adjusted with no airflow disturbance to produce a uniform and parallel set of
interference fringes on screen giving uniform illumination. When the tunnel is run with
model installed, fringe spacing will change by an amount proportional to the phase
change by the disturbance at any point which is in turn proportional to the change of fluid
density integrated along the light path. If the interferometer is pre-calibrated, it will give
absolute values of density.
Figure 4.6 Schematic diagram of the interferometer system
60
Chapter 5
PRESSURE MEASUREMENT BY MECHANICAL DEVICE
5.1 Introduction :
Pressure, at different points on the surface of model, can be obtained by drilling holes on
the surface and connecting tubes from these points to a mechanical device like a multi-
tube liquid level manometer (Fig. 5.1). liquid levels, which are initially in the same level,
undergo changes in height proportional to the pressure applied and pressure at different
points in the surface can be calculated from the heights of the columns.
Figure 5.1 Liquid level manometer
Multi-tube, indicated schematically in Fig. 5.1 may be used in vertical position. For
increased sensitivity the manometer may be inclined at various angles in which readings
are multiplied by appropriate factors. Also, in stead of water, liquid of specific gravity
less than 1.0 may be used.
61
The reservoir for manometer liquid is usually mounted on a vertical rod at a height which
is adjustable. It is recommended that the reservoir be normally left open to atmospheric
pressures. Pressures p1, p2, p3,….are then gauge pressures i.e., pressures relative to
atmospheric datum. Pressure relative to some other chosen datum may be obtained by
connecting the reservoir and one manometer tube to the required datum.
Manometers are generally graduated so that height of liquid level may be read in cm and
the pressure is calculated from the height of the liquid column in the relevant tube. Some
manometers are graduated directly in N/m2 or in millibar (1mb = 100 N/m
2 ).
5.2 Measurement of Cp :
Pressure is usually expressed in non-dimensional form as pressure coefficient Cp . by
definition Cp is given by
2
2
1
U
ppC p
(5.1)
Using a liquid-level manometer as shown in Fig. 5.1, pressure coefficient Cp can be
obtained in two ways depending on whether the tunnel is precalibrated or not.
5.2.1 Without Pre-Calibration of the Tunnel :
If the tunnel is not pre-calibrated to give U, two holes are to be drilled on the walls of
the settling chamber and the test section and directly connected to the manometer in
addition to connecting pressure port of the configuration.
Now, by Bernoulli’s theorem,
SPUpP
2
02
1
where PS is the settling chamber pressure.
Or, ppU S
2
2
1
If the manometer is graduated in N/m2 ,(p - p) and (PS - p) can be obtained directly in
units of N/m2 and Cp can be obtained as the ratio of the two given by
pP
ppC
S
p (5.2)
62
Non-dimensional pressure coefficient is thus obtained simply as a ratio of pressure
differences and value of U is not needed. If U is needed (e.g., to calculate Reynolds
number) U can be obtained in a simple manner by assuming no frictional loss between
settling chamber and test section.
Under this assumption, U can be obtained as
pPU S2 (5.3)
If the manometer is graduated to give height of liquid column, Cp can be obtained as ratio
of column heights as shown below.
ghhpp liquid
and hhppU SliquidS 2
2
1
Where,
hS = height of column in the tube connected to settling chamber.
h = height of the water column in the tube connected to the pressure port on the
configuration where pressure is being measured.
h = height of the column in the tube connected to test section
This gives ,
pP
ppC
S
p
hhg
hhg
Sliquid
liquid
hh
hh
S
(5.4)
Cp is then obtained as ratio of height difference of liquid columns.
By assuming zero frictional loss between settling chamber and test section U can be
obtained as
pPU S2
ghhSliquid 2 (5.5)
63
5.2.2 With Pre-Calibration of the Tunnel :
If the tunnel is pre-calibrated to give U, pressure coefficient can be derived in terms of
U.
22
2
1
2
1
U
ghh
U
ppC
liquid
p
If the manometer is inclined at 600, then
2
0
2
1
60sin
U
ghhC
liquid
p
If the liquid is water, height is graduated in cm and density of air is taken as 1.225 kg/m3,
then
2225.12
1
866.081.91000
U
hhC p
270.138 Uhh (5.6)
Experimental measurement of pressure distribution on a few simple models are described
in the following sections. In all models several holes are drilled on the surface and
connected to the multi-tube manometer. Pressure distribution can then be obtained from
equation (5.4) or (5.6) depending on whether the tunnel is pre-calibrated or not. These
models include :
a) Circular cylinder model
b) Elliptical cylinder model
c) Spherical model
5.3 Pressure Distribution on Circular Cylinder Model :
Exact analytical solutions are available for limited cases of direct potential flow
problems. The problem of two dimensional flow about a cylindrical body is one of such
problems. For steady, inviscid, incompressible irrotational flow, for which the governing
equation is Laplace’s equation, the non lifting two dimensional flow about a cylindrical
64
body can be simulated by placing a doublet in uniform flow. The total velocity at any
point P (Fig. 5.2) is obtained as
sin2 Uqt (5.7)
Figure 5.2 Circular cylinder in uniform flow
The pressure distribution can be obtained from Bernoulli’s equation,
2
2
2
sin411
2
1
U
q
U
ppC t
P (5.8)
It may be noted that the expression for total velocity or pressure is independent of the
diameter of the cylinder.
The ‘ideal’ pressure distribution, given by equation (5.8), over the surface of the cylinder
will be symmetrical about the axis in the direction of the flow and about the plane normal
to it. Consequently, the net forces, lift and drag, are zero.
An experimental study can be undertaken to check how far the ‘real’ solution deviates
from the ‘ideal’ solution. For the case of uniform flow of real fluid, both the effects due
to compressibility and viscosity are to be taken into account. For the low speed test case
(0.1 Mach number) the effect due to compressibility may be justifiably ignored.
However, effect of viscosity alone will change the flow pattern considerably.
Primarily, the flow will be asymmetric about the axis normal to the uniform stream and
hence pressure distribution will also be asymmetric resulting in a net force (drag) acting
65
on the cylinder along the flow direction. However, the flow is still symmetrical about the
axis in the direction of the flow and hence no lift force acts on the cylinder.
Secondly, while the ‘ideal’ flow is always attached to the body surface, in real fluid, the
flow may separate under adverse pressure gradient. In the forward face of the cylinder (
between 00
to 900), the flow speed increases and pressure decreases, hence the flow is not
likely to separate in this region. In the backward face, ( between 900 to 180
0), the speed
decreases and pressure increases. Under the action of this increasing pressure (i.e.
adverse pressure gradient), the flow is likely to separate.
This separation is the so-called ‘boundary layer separation’. Since the flow velocity is
less in the boundary layer than in the free stream outside the boundary layer, the flow
separates in the boundary layer. The exact process of separation is yet little understood.
Generally speaking, at low speed the flow in the boundary layer is laminar and will be
attached to the body. Since the flow speed is less, kinetic energy associated with the flow
is also less, and the laminar flow is more susceptible to separation. As the flow speed is
increased, the boundary layer becomes turbulent. Transition for laminar to turbulent flow
is governed primarily by the Reynolds number of the flow.
The model chosen for experimental work is a circular cylinder of diameter 10.8 cm and
span 60.8 cm which extends from wall to wall (so that the flow is two dimensional).
Sixteen pressure holes are equally spaced at
0
2
122 apart (Fig. 5.3) on the surface of the
cylinder and are connected to a multi-tube manometer.
Advantage, however, can be taken for this circular cylinder model. Only one hole can be
drilled and pressure at different points on the circular section can be obtained by simply
rotating the model (chapter –12).
66
Figure 5.3 Pressure holes on cylinder surface
Both the theoretical and experimental Cp distribution can now be obtained from equation
(5.8) and equation (5.4) or (5.6) and plotted against . The difference is due to viscous
effects.
The following table may be made for plotting Cp vs. (Fig. 5.4).
Table 5.1 : Pressure distribution on circular surface
Tap
points
hLS U h h Cp
(Theoretical)
eq. (5.7)
Cp
(Experimental)
eq. (5.6)
1. 0
2. 22.50
3. 450
-
16. 337.50
67
Cp
-Ve
0 90 180 270 300 330 360
Figure 5.4 Pressure distribution on cylinder surface
5.4 Pressure Measurement on Elliptical Cylinder Model :
Exact analytical solution exists also for the case of potential flow about elliptical;
sections. Using conformal transformation, flow around a circular section can be
conformed into a flow around an elliptical section in such a way that the condition at
infinity is unaltered.
The flow at any point (r, ) on the surface of a circular section is given by equation (5.7)
sin2 uqt (5.7)
The flow past a circular section can be transformed into the flow past an elliptical section
by a conformal transformation (Fig. 5.4)
Z
bZ
2
(5.9)
68
Figure 5.5 Conformal Transformation
The velocities for corresponding points can be related by
d
dz
q
q
circle
ellipse (5.10)
Now, iea
b
z
bdzd 2
2
2
2
2
11 [since z = aei
]
2sin2cos12
2
ia
b
Therefore,
2
1
2
2
22
2
2
2sin2cos1
a
b
a
b
dz
d
2
1
4
4
2
2
2cos2
1
a
b
a
b
or, 2
1
4
4
2
2
2cos2
1
a
b
a
b
d
dz
Therefore,
69
2
1
4
4
2
2
2cos2
1
a
b
a
b
qq circle
ellipse
[from eq. (5.9)]
2
1
4
4
2
2
2cos2
1
sin2
a
b
a
b
U
(5.11)
The pressure distribution of the surface of the ellipse may be obtained from Bernoulli’s
equation,
2
1
U
qc
ellipse
p (5.12)
The pressure distribution on the ellipse can be experimentally determined by a elliptical
model extending from tunnel wall to wall so that two dimensional flow is obtained. The
major and minor axis of the elliptical model are 15.75 cm and 10.9 cm respectively.
Static pressure holes are made at sixteen points on the surface for measurement of
pressure. Cp at these sixteen points can be obtained from water level in the manometer.
A table can be made, as shown, for plotting of theoretical and experimental pressure
distribution vs .
The comparison of theoretical and experimental pressure distribution may be shown in a
similar manner as for circular cylinder.
Table 5.2 : Pressure distribution on elliptical cylinder surface
Tapping
points
hLS U h h cp
(Theoretical)
eq. (5.12)
cp
(Experimental)
eq. (5.6)
1. 0
2. 22.50
3. 450
-
-
16. 337.50
70
5.5 Pressure Measurement on Spherical Model :
The potential flow about a spherical body can be mathematically simulated by placing a
three-dimensional point doublet in a uniform stream. The flow about a sphere of radius
‘a’ can be shown to be generated by placing a doublet of strength (=2a3U) in
uniform stream U (Fig. 5.5).
The perturbation velocity components due to a doublet of strength , placed at origin, at
any point (x, y, z) are
5
5
22
5
4
3
4
3
)3(4
r
xzw
r
xyv
rxr
u
(5.13)
where,
222 zyxr
and = 2a3U
Taking doublet strength = 2a3U the perturbation velocity components on the surface
of the sphere at the center section are obtained as
Uu
2
1cos
2
3 2 [putting x = acos, y=0 and z = asin]
v = 0
Uw sincos2
3 (5.14)
The total velocity components due to the combined flow is
sincos2
3
12
1cos
2
3 2
UW
UU
The total velocity is given by
71
sin2
322
UWUqt (5.15)
The pressure distribution can be obtained from Bernoulli’s equation
2
1
U
qc t
p
2sin4
91 (5.16)
The spherical model undertaken for experimental work is drilled at 16 equally spaced
points for pressure measurement (Fig. 5.6)
Figure 5.6 Pressure holes on spherical surface
A table similar to that used for previous two experiments can be made and theoretical
(equation 5.16) and experimental (equation 5.6) pressure distribution can be plotted. The
theoretical pressure distribution is symmetrical over the surface of the sphere and hence
no force or moment acts on the sphere. The discrepancy with experimental results is due
to viscous effects.
72
Chapter 6
FORCE MEASUREMENT BY MECHANICAL BALANCE
6.1 Introduction :
A three component mechanical balance (Fig. 6.1) is basically designed to measure two
force components along mutually perpendicular axes (lift and drag) and a single moment
about an axis perpendicular to those of the forces (pitching moment). This type of
balance is usually a roof-top balance to be installed on the top wall of the tunnel. The
model is suspended from three vertical struts – two forward and one at the rear. Only
these vertical struts emerge in the tunnel.
The main lift beam, in conjunction with the pitching unit beam, gives the total lift. The
two front struts are connected to the main lift beam through hinges. The main lift beam
has two scale-pans for placing weights and two riders moving along a graduated scale.
The two front struts are to be attached to the main lifting surface of the model and they
transfer the lift force to the main lift beam. The weight placed in the beam scale pan
together with the rider displacements required to balance the torque imposed on the main
lift beam by the lift force gives a measure of the lift force provided the beam is properly
pre-calibrated. The rear strut also gives a part of the lift, which is measured in the same
way as in the main lift beam.
The pitching wheel, when rotated, guides a block along a threaded rod. As the block
travels up or down the attitude of the model in pitch is changed.
The drag beam comes under a torque due to the horizontal force on the front struts. This
horizontal force or drag is transferred from the front struts to the drag beam through
appropriate linkages. The moment on the drag beam is balanced by beam pan weights
and rider movement which gives a measure of the drag force encountered by the model.
73
Figure 6.1 Wind tunnel mechanical balance
6.2 Calibration :
The calibration of a balance require certain equipments and the idea is to make these
equipment as permanent as possible since calibration checks are needed many times
during the life of a balance.
The first equipment needed for calibration is a loading ‘TEE’ (Fig. 6.2). The tee
facilitates the application of static loads in order to simulate the lift and drag forces as
they arise from model tests in the tunnel. The tee is fitted to the struts of the balance, its
head to the two front struts and its tail to the rear strut.
74
Figure 6.2 Loading TEE for calibration
To simulate drag, a horizontal force is applied to the tee in the direction of the drag force
through a string attached to the tail of the tee. The string passes over a pulley and carries
drag weights in a scale pan attached to the free end of the string. Static lift forces are
simulated by dead weights placed in a weight pan hanging from the middle of the tee’s
head.
Riders are provided for minute adjustment in balancing. For the lift and pitching unit two
riders (left rider and right rider) are provided while for the drag beam one rider is
provided. Rider movements need to be calibrated. This is done for the lift unit in the
following way.
The step is initially balanced with no weight in either dead weight pan (which is hanging
from the tee via the pulley) or in the main lift beam pan (Fig. 6.3) and the rider positions
are recorded. Next, 50 gms of static lift is provided by placing this weight in the dead
weight pan. The set-up is again balanced first by moving the right rider (keeping left rider
stationary) and then by moving the left rider (keeping the right rider stationary) and these
displacements are then recorded.
75
Figure 6.3 Simulation of lift and drag by dead weight
Table 6.1 : Rider calibration
Load in dead weight pan Left Rider (mm) Right Rider (mm)
0 80 80
50gms 80 90
50 gms 70 80
The table shows that each rider movement of 1 mm is equivalent of 5 gms of load. It may
be noted that the riders move from left to right in balancing the static load. The combined
load from dead weight pan and rider movement is termed equivalent lift and this is
plotted against the load in beam pan.
Table 6.2 : Calibration of lift beam
Initial Positions : Left Rider = mm; Right Rider = mm
1 mm of rider displacement = 5 gms of static loa
76
Load (lift)
weight pan
(gms)
1
LR
mm
2
RR
mm
3
dLR
mm
4
dRR
mm
5
Load from
rider
(gms)
6
Load
In beam
weight pan
7
Equivalent
lift (gms)
(1)+(6)
1+6
169.62 60 100 0 0 0 16.81 169.62
250 60 65 0 35 165 34.64 415
450 60 65 0 0 0 50 450
750.8 60 85 0 -20 -100 78.23 650.8
Similarly, tables can be prepared for drag beam and pitching unit.
Table 6.3 : Calibration of drag beam
Initial positions : Rider = mm
1 mm of rider displacement = gms of static load
Load in dead
weight pan
(gms)
(1)
Rm
mm
(2)
dR
mm
(3)
Drag load
from rider
(gms)
(4)
Load in
beam weight
pan
(5)
Equivalent
drag (gms)
(1+4)
Table 6.4 : Calibration of pitching unit
Initial position : LR = mm ; RR = mm
1 mm of left rider displacement = gm of static lift load.
1 mm of right rider displacement = gm of static lift load.
77
Load (lift)
in dead
weight pan
(1)
LR
Mm
(2)
RR
Mm
(3)
dLR
mm
(4)
dRR
mm
(5)
Load from
rider (gms)
(6)
Load in beam
weight pan
(gms)
(7)
Equivalent
lift (gms)
(1)+(6)
Calibration charts may now be made for lift, drag and pitching unit as shown below.
Equivalent lift
Load in beam weight pan
Figure 6.4 Calibration of main lift beam
Equivalent drag
Load in beam weight pan
Figure 6.5 Calibration of drag beam
78
Equivalent lift
Load in beam weight pan
Figure 6.6 Calibration of pitching unit
6.3 Measurement of Forces and Moments :
Once the balance is calibrated, forces and moment acting on a model installed in the
tunnel can be computed easily from the beam weight pan load and calibration chart and
rider displacements. Tests are carried out on two models
1) a flat plate of dimensions 45.75 cm 30.5 cm
2) A rectangular wing of dimensions 61 cm 30.5 cm
(which spans the jet width) with NACA 0012 as its aerofoil section.
The interest is in finding the variations of CL, CD and CM with angle of incidence . The
tests with these models are carried out at the same speed for different angles of incidence
determined from the pitching screw settings.
Table 6.5 : Flat plate lift from main lift beam
Initial position: LR = mm, RR = mm
Incidence
(1)
LR
(2)
RR
(3)
dLR
(4)
dRR
(5)
Lift from
rider
(6)
Load in
beam
weight pan
(7)
Lift
from
Fig. 6.4
(8)
Total lift
L2 (gms)
(6) + (8)
79
Table 6.6 : Flat plate lift from pitching unit
Initial position : LR = mm, RR = mm
Incidence
(1)
LR
(2)
RR
(3)
dLR
(4)
dRR
(5)
Lift from
rider
(6)
Load in
beam
weight pan
(7)
Lift
from
Fig. 6.6
(8)
Total lift
L2 (gms)
(6) + (8)
Table 6.7 : Flat plate lift coefficient
Incidence
(degrees)
Total lift
(L1 + L2 )
gms
Total lift (L)
(L1 + L2 ) + 9.81
1000
Newton
½ ? 2
U (N/m2)
SU
LCL
2
2
1
Table 6.8 : Flat plate drag coefficient
Initial position : R = mm
Incid-
ence
(1)
R mm
(2)
dR
mm
(3)
Drag
load
from
rider
gms
(4)
Load in
drag
beam
pan gms
(5)
Drag
from
Fig.
6.5
gm
(6)
Total
drag D
gms
(4)+(6)
Total
drag =
(D9.8)/
1000
Newton
CD
221
U
D
80
Table 6.9 : Flat plate pitching moment coefficient
Incid-
ence
degree
LR
mm
dLR
mm
(4)
dRR
mm
(5)
Lift
due to
rider
gms
Load
in
beam
pan
Lift
from
Fig. 11.6
gms
Total
pitch-
ing
lift
M0.25C
N.m
Results can now be presented in graphical form. On the graph, the value of CLmax should
be noted.
Similar table can be created for the wing model.
6.4 Evaluation of the effects of the Support (Tare and Interference Drag) :
In any wind tunnel the model needs to be supported in some manner and the supports, in
turn, affects the drag measurement. Any strut connecting the model will add three
quantities to the forces read. The first is obvious drag of the exposed strut (tare) the
second is the effect of the strut’s presence on the free air flow about the model and the
third is the effect of the model on the free air flow about the strut. The last two items are
usually lumped together under the term “interference” and their existence should make
clear the impossibility of evaluation the total drag of the struts with the model out. This
procedure will primarily expose parts of the model support not ordinarily in the air stream
(although the extra length may be made removable) and will fail to record the
interference drag and will record only the tare drag.
The tare drag can be reduced by shielding a large part of the strut by fairings not attached
to the balance; only a minimum of strut bayonet is exposed to the air stream.
Theoretically the tare drag can be eliminated entirely by shielding the supports all the
way into the model (with adequate clearances inside), however the added size of shield
(and the presence of shield so close to model) will probably increase the interference drag
so much that no net gain will be achieved.
81
The tare and interference drag can be evaluated separately or jointly using the ‘mirror’ or
‘image’ technique. The separate evaluation of drag items is long and also unnecessary.
This separate evaluation approach is therefore rarely used. However, both the methods
are outlined below.
6.4.1 Evaluation of tare and Interference Drag Separately :
The model is first tested in the normal manner. Symbolically,
Dmeasured = DN = IUB/M + IM/UB + IUSW + TU (13.1)
where DN = drag of model in normal position.
IUB/M = interference of upper surface bayonets on model.
IM/UB = interference of on model on upper surface bayonets.
IUSW = Interference of upper support windshield.
TU = free air tare drag of upper bayonet.
Next, the model is supported from the tunnel floor by the ‘image’ or ‘mirror’ system. The
supports extend into the model but a small clearance is provided (Fig. 6.7) so that the
balance record only the drag of the exposed portion of the support (in the presence of the
model).
Figure 6.7 Arrangement for determining tare and interference drag separately
82
That is,
Dmeasured = IM/UB + TU (6.1)
For the interference run, the model is inverted and run with the mirror supports just
clearing the attachment points. This gives
Dmeasured = Dinverted + ILB/M + IUB/M + IM/UB + IUSW + ILSW + TL (6.2)
where Dinverted = drag of the model inverted (should equal that the drag of the model
normal, except for misalignment).
The upper supports are removed and a second inverted run is made giving
Dmeasured = Dinverted + ILB/M + IM/LB + ILSW + T1 (6.3)
The difference between the two inverted runs is the interference of the supports of the
upper surface. That is eq. (6.2) minus eq. (6.3) yields
IUB/M + IUSW (6.4)
By subtracting eq. (6.1) and (6.4) from the first run, the actual model drag is determined
if the balance is aligned.
6.4.2 Evaluation of the Sum of Tare and Interference Drag :
In this procedure, the sum of tare and interference drag can be found in three runs instead
of four in the previous method. In this case, the normal run is made, yielding
Dmeasured = DN + IU + TU (6.5)
where IU = IUB/M + IM/UB + IUSW
Next model is inverted and this gives
Dmeasured = Dinverted + IL + TL (6.6)
Then the dummy supports are installed. Instead of clearance being between the dummy
supports and the model, the exposed length of the support strut is attached to the model
and the clearance is in the dummy supports. This configuration yields (Fig. 6.8).
Dmeasured = Dinverted + IL + IU + TL + TU (6.7)
83
Figure 6.8 Arrangement for determining the sum of tare and interference drag
The differenc between eq. (6.7) and eq. (6.6) yields the sum of tare and interference TL
and IL .
A third method which is crude but simple is to minimize the interference drag by using
strut of aerofoil shape (and not using the windshield). The drag of the supports is than
essentially the tare drag which can be measured easily by the balance with the model out.
The drag of the suspension system can be measured this way and subtracted from the
drag measured of the two dimensional wing.
84
Chapter 7
PRESSURE MEASUREMENT BY TRANSDUCER
7.1 Introduction :
Pressure measurement in wind tunnel is of interest not only for determining pressure
distribution on aerodynamic shapes but also for determining test conditions in the wind
tunnel test sections. Up to about 15 to 20 years ago, the majority of wind tunnel pressures
were measured with liquid manometers and readings were taken manually. These
manometers , however, are not suitable for measuring very high or very low values of
pressure or for measurement of pressure in unsteady or short duration tunnel.
To overcome this limitations, these manometers have been replaced to a large extent by
pressure transducers in a scanning system with automated data recording system.
Pressures from 2 10-7
psia to approximately 1000,000 psia are successfully measured in
wind-tunnel with the aid of transducers whose electric output signal emanates from a
deflection or deformation caused by a pressure activated elastic sensing element. The
most common type of elastic sensing elements are the diaphragms. In order to produce an
electric signal, the elastic elements operates in conjunction with electrical sensing
element which provides an electrical change in response to the deflection or deformation
of the sensing element. The most frequently used electrical sensing elements include
metallic or semi-conductor strain gauges, variable capacitance device, variable reluctance
device and piezoelectric elements.
A summary of general performance of different pressure transducers as given in Table
7.1.
85
Table 7.1. Summary of performance of pressure transducers :
Type of transducers Range of pressure
measurement (psi)
Operating temperature
( 0F)
Resonant
frequency (KHz)
Variable resistance 10-4
to 100,000 - 430 to 300 0F Upto 1
Variable
capacitance
2 10-7
to 10,000 - 55 to 225 0F
Upto 200
Variable reluctance 3 10-5
to 10,000 - 63 to 250 0F Upto 25
Piezoelectric 5 10-4
to 100,000 - 400 to 500 0F Upto 500
In variable resistance transducer, a pressure change is converted into a change in
resistance caused by the strain in a strain gauge or gauges. Most strain gauge pressure
transducer incorporate four active strain gauge elements in a Wheatstone bridge circuit.
Variable reluctance transducer employs diaphragm as the sensing element. This
diaphragm is supported between two inductance core assemblies. A magnetic circuit with
core is completed. As the pressure is applied, reluctance changes.
In variable capacitance transducer pressure is applied on one plate of the capacitor. Since
the capacitance varies inversely proportional to the plate distance capacitance changes
due to applied pressure.
In piezoelectric transducer, voltage is generated when pressure is applied due to the
squeezing effect of the crystal. This type of transducer is self generating and does not
require any external power supply.
Variable resistance transducers are most widely used. In such transducers strain gauges
are bonded directly on diaphragm (Fig. 7.1). A diaphragm is essentially a thin circular
plate fastened around its periphery to a support shell. Stainless steel or beryllium copper
is generally used as the diaphragm material.
If the strain gauges are located as shown in Fig. 7.1, elements 1 and 2 will be in tension
and 3 and 4 will be compression. By electrically connecting these gauges a fully active
Wheatstone bridge is realized. Recent advances in the semi-conductor field have led to
the development of an integrated Wheatstone bridge consisting of four strain sensitive
resistive arms formed directly on the diaphragm.
86
Figure 7.1 Variable resistance pressure transducer
7.2 Time Response :
Response time of a transducer is not critical for a continuously running tunnel. When
measurements are required in short duration facility or when unsteady or transient data
are required, time response becomes primary in importance. In such cases transducer
must be flush mounted or be connected by very short tube lengths.
The response time, t, for a flush mounted transducer is
212
1
hft
where
f = transducer resonant frequency
h = damping ratio
7.3 Pressure Scanning :
Pressure distribution on models described in chapter 5 can also be obtained by use of
pressure transducer. However, measuring pressure at multiple numbers of ports on a body
poses a problem. Either it will require an equal number of pressure transducers, signal
87
conditioner and readout systems ( which will be expensive) or a single transducer will be
connected to all pressure holes one by one (which will be time consuming). This problem
can be overcome by a pressure scanning system. A number of tubes from various
pressure ports are routed to a common point and then applied individually to a single
transducer and readout system as shown on Fig. 7.2.
Figure 7.2 Pressure scanning system
The most important part of the pressure scanning system is the scanivalve (Fig 7.3). in a
scanivalve, the transducer is sequentially connected to the various pressure ports via a
radial hole in the rotor which terminates at the collector hole. As the rotor rotates, this
collector hole passes under the ports in the stator. Referring to the cutaway drawing (Fig.
7.3) the rotor is seen to be rigidly supported by a ball thrust bearing. The stator is
elastically connected to the block in a manner which allows the stator to follow the
surface of the rotor. Thus the pneumatic forces (pressure area) at each port which tends
88
to blow the rotor away from the stator are withstood by the ball thrust bearing. The stator
is epoxied into the block to prevent rotating.
Figure 7.3 Scanivalve
A scanivalve may be hand driven or solenoid driven. For a solenoid driven scanivalve,
further instruments are required. A solenoid controller is necessary for controlling the
stepping speed of the scanivalve. Also, interface controller is necessary for controlling
scanivalve port location.
89
7.4 Measurement of Cp :
The excitation voltage applied to a pressure transducer is generally 12 volts. A typical
value of full scale output of a PDCR23 transducer at this excitation is 17.5 mV for 1 psi
of pressures. If the reference side of diaphragm is connected to wall of test section (i.e.
datum pressure is free stream static pressure p ), the pressure coefficient at any
measurement location can be calculated from the output in two ways depending on
whether the tunnel is pre-calibrated or not.
7.4.1 With Pre-calibration of tunnel :
By definition, pressure coefficient Cp is given by
2
2
1
U
ppC p
If the tunnel is pre-calibrated, i.e., the free-stream speed U is known, Cp can be derived
in terms of U.
2
2
1
5.17/6.6894...
U
mVinreadoutC p
(1 psi = 6894.6 N/m2)
Since the transducer output is usually small, an amplifier is usually used. Taking the
amplifier gain into account Cp can be written as
5.17..225.12
1
6.6894....
2
gainamplifierU
mVinreadoutC p
)..(
)....(228.643
2 gainamplifierU
mVinreadout
(7.1)
7.4.2 Without pre-calibration of tunnel :
Cp can be derived in an alternative way if the tunnel is not pre-calibrated to give U. Cp
can be expressed as
2
2
1
U
ppC p
90
pP
pp
0
pP
pp
S
(PS = settling chamber pressure)
There are uaually 48 pressure port locations in a scanivalve. If the reference side is
connected to wall of test section to sense p and 48th
port location is connected to wall of
settling chamber then for pressure upto 47 points can be measured one by one by a
scanivalve as a simple ratio of digital outputs, given by equation (7.2).
91
Chapter 8
FORCE AND MOMENT MEASUREMENT BY ELECTRONIC
INTERNAL (STING) BALANCE
8.1 Introduction :
Aerodynamic forces and moments acting on a model in wind tunnel can be accurately
measured by variable resistance strain gauges. Usually a sting balance, also known as
internal balance or strain gauge balance, is used where strain gauges are bonded on the
sting (Fig. 2.2). These strain gauges are connected in Wheatstone bridge arrangement
(differencing circuits for forces and summing circuits for moments). When the tunnel is
started, the forces acting on the model change the resistance of strain gauges. The voltage
of the ‘unbalanced’ Wheatstone bridge is then a measure of the forces acting on the
model.
An aircraft model is subjected to three aerodynamic forces along three axes (lift, drag and
side force) and three moments (yawing, rolling and pitching) about the three axes. In
general, Wheatstone bridge circuit needs to be constructed for each of the forces and
moments. A general six component balance will then require six Wheatstone bridge
circuits (consisting of 24 strain gauges), a six channel signal conditioner, separate power
supply for each channel and appropriate data acquisition system. Philosophy underlying
bonding of strain gauges and Wheatstone bridge circuits needs to be studied for each
component separately.
It is worth nothing here that six components of forces and moments are measured in a
sting balance about the body axes (since the sting is attached to the body and moves with
the body) and not about wind axes. Hence body axes need to be converted to wind axes
which are more familiar.
92
8.2 Measurement of lift :
Generally, the symmetrically placed aircraft model will experience a lift L, drag force D
and pitching moment M at the aerodynamic center. The sting is like a cantilever beam on
which lift and pitching moment act. The sting is also subjected to axial stress due to drag
in addition to the bending stress due to lift (Fig. 8.1). it can be shown easily that the
output voltage will be proportional to lift force only if the ‘differencing’ circuit is used.
Figure 8.1 Differencing and summing circuits
The output voltage of the Wheatstone bridge circuit, from equation (2.16), is
R
RR
V
V
2
''' (8.1)
where
ab
ab
RRR
RRR
''
'
''
'
''' , bb RR = changes in resistance due to bending stress.
aR = changes in resistance due to axial stress.
Therefore,
R
RRRR
V
V abab
2
'''
R
RR bb
2
''' (8.2)
93
The above expression shows that the output voltage is independent of axial stress (i.e.,
drag forces).
Now the gauge factor G is defined as
RR
LL
RRG
and the change in resistance is expressed in terms of change in strain as
R = R G (8.3)
The longitudinal strains on the four strain gauges can be written as
2
23
22
11
hEI
ZZL
hEI
ZZL
hEI
ZZL
(8.4)
11
4
hEI
ZZL
The changed resistance value of four strain gauges will be
R1 = R + RG 1
R2 = R + RG 2 (8.5)
R3 = R + RG 3
R4 = R + RG 4
The output voltage is then
4321
3142
RRRR
RRRR
V
V
4331
3142
RGRGRGRG
RGRRGRRGRRGR
4321
3142
22
1111
GGGG
GGGG
21
2
2
2
1
2
12
24
2
G
G (since 3 = - 2 & 4 = - 1 )
122
G
(Neglecting higher order term)
94
hLEI
ZZG 12
2
LK1 (8.6)
where EI
ZZGhK
2
)( 121
= Constant
The output voltage is seen to be linearly proportional to lift force, L. It is to be noted that
the circuit can be made more sensitive by increasing the distance (Z2 – Z1) between the
strain gauges. It is worth noting here that the relationship will not remain linear except for
a fully active bridge.
The calibration constant K1 can be obtained by putting appropriate values of G, h, E, I
and (Z2 – Z1) in the expression K1 = Gh (Z2 – Z1)2EI. Alternatively, it can be obtained
by a calibration procedure as shown in Fig. 8.2. In this procedure, K1 is obtained by
putting known weight (W) at position (AA) on the string through a pulley and noting the
voltmeter readings. For an excitation voltage of 4.0 volts and amplifier gain 1000, typical
value of the constant K1 is of order of 0.028 mV/gm.
Sting
1 2 A
3 4 A
W
Figure 8.2 Calibration procedure for obtaining K1
The lift coefficient CL in any subsequent experiment can be directly related to the
millivolt output as
cos81.9
10002
1cos
2
1
SUK
VCC NL (8.7)
95
8.3 Measurement of pitching moment :
The pitching moment can be obtained from four strain gauges by constructing the
‘summing’ circuit (Fig. 8.1), which is constructed simply by interchanging R2 and R3 of
‘differencing’ circuit.
The output of the summing circuit can be shown to be independent of the drag force. The
output voltage can be written as
4231
2143
RRRR
RRRR
V
V
The changed values of the resistances may be written as
ab
ab
ab
ab
RRRR
RRRR
RRRR
RRRR
'
4
''
3
''
2
'
1
Substituting in previous equation yields
abbabb
abbabab
RRRRRRRR
RRRRRRRRRRR
V
V
2222
2222
2
bba
bababb
RRRR
RRRRRRRR
24
2
R
RRRR bb
(neglecting higher order term)
R
RR bb
2
(8.8)
Equation 8.8, analogous to equation (2.11), shows the output voltage V to be
independent of the axial stress due to the drag force.
The output of the summing circuit can now be shown to be proportional to pitching
moment. The output voltage is
2431
2134
RRRR
RRRR
V
V
96
Using equation (8.5),
2431
2134
22
RGRGRRGRGR
RGRRGRRGRRGR
V
V
21
2
2
22
1
2
21
24
22
GGG
GG
(since 3 = - 2 & 4 = - 1 )
2
21
G (neglecting higher order term)
11
112
2
2
ZZLEI
GhLK
GG
MKLK 21 (8.9)
where K2 = Gh / 2EI is a constant, M is the moment L (Z – Z1) due to L and K1 is the
constant defined earlier for lift.
The output voltage from equation (8.9) is seen to be dependent on both lift L and pitching
moment M. However, it is to be noted that K1, for this summing circuit, is made very
small to be negligible. This is because (Z2 – Z1) is made very small in summing circuit in
comparison to differencing circuit by fixing strain gauges very close to each other.
Equation (8.9) can be written as
MKV
V2
(8.10)
This equation shows that the output voltage in ‘summing’ circuit is proportional to
pitching moment only.
The value of the constant K2 can, in principle, be obtained theoretically by putting
appropriate values of G, h, E, I and (Z2 – Z1). However, it is desirable to determine its
value through static calibration. This can easily be done by using dummy weight (W) at
position AA, as before, to simulate lift and noting the change in output with increasing
load (Fig. 8.3).
97
1 2 A
A
3 4
W
Figure 8.3 Calibration procedure for obtaining K2
However, if an accurate estimate of both K1 and K2 are needed weights can be placed at
two positions, first at AA and then at BB, as shown in (Fig. 8.4) and the two values of
voltages are to be noted.
f
1 2 B A
3 4 B A
Figure 8.4 Calibration procedure for obtaining K1 and K2
From equation (8.9) the output voltages will be
MKLKV
V
BB
21
(8.11)
LfMKLKV
V
AA
21
(8.12)
From equations (8.11) and (8.12) K2 is obtained as
98
Lf
V
V
V
V
K BBAA
2 (8.13)
Once K2 is obtained from equation (8.11), K1 can be calculated from equation (8.11) or
(8.12).
The pitching moment M can be easily obtained from
LKV
VMK 12
or, 21 KLKV
VM
(8.14)
8.4 Simultaneous Measurement of Lift and Pitching Moment :
Lift and pitching moment cam be measured simultaneously by using eight strain gauges
to form two Wheatstone bridge circuits (both circuits as summing circuit) as shown in
Fig. 8.5. The output voltages of the two circuits will vary because of the variation in
moment with the distance f. the output voltages can be written from equation (8.9), as
121 MKLKV
V
AA
(8.15)
243 MKLKV
V
BB
(8.16)
A B
1 2 6 5
3 4 7 8
A B W
Figure 8.5 Calibration procedure for obtaining K2
Now, M2 = M1 + Lf (8.17)
99
Using equations (8.15), (8.16) and (8.17)
AAV
VMKLK
121 (8.18)
BBV
VMKLKfK
1434 (8.19)
In matrix form,
K1 K2 L AAV
V
=
K3 + fK4 K4 M BBV
V
On inverting,
L K1 K2 -1 AAV
V
= (8.20)
M K3 + fK4 K4 BBV
V
Equation (8.20) gives lift and pitching moment directly from the outputs of Wheatstone
bridge circuits provided, of course, four coefficient are measured from calibration.
It is however, to be noted that K1 and K3 are very small to be almost negligible. This is
because both Wheatstone bridge circuits are summing circuits where strain gauges are
pasted very near to each other. If K1 and K3 are assumed to be negligible, lift and
moment can be readily obtained as
100
4
2
2
1
K
V
V
M
K
V
V
M
BB
AA
and f
MML 12 (8.21)
8.5 Other Forces and Moments :
The mathematics underlying strain gauge instrumentation is described in details for lift
and pitching moment measurement in the previous sections. The same principles are
easily extended to measuring other forces and moments. In general, differencing circuit
(for forces) and summing circuits (for moments) are to be used. However, to measure
other components, cantilever or the sting needs to be specially machined for suitable
positioning of strain gauges for particular component.
Arrangement for measuring drag in a 3 component balance is shown in Fig. 8.6.
Figure 8.6 Three component balance
101
The normal forces (CN ) and axial force (CX ) obtained by the sting balance are converted
to lift (CL ) and drag (CD ) force by a simple conversion of axes (Fig. 8.7).
Figure 8.7 Axes system
CL = CN cos - CX sin (8.22)
CD = CN sin + CX cos (8.23)
The side force and yawing moment can be obtained using the same principle for
measuring lift and pitching moment. However, unlike in the previous case, the strain
gauges are to be bonded on the side surface of the sting as shown in Fig. 8.8.
To measure rolling moment, the sting to be machined such that the cross section is as
shown in Fig. 8.9 and strain gauges bonded are connected as a ‘summing’ circuit.
102
Figure 8.8 Measurement of side force and yawing moment
Figure 8.9 Measurement of rolling moment
103
8.6 Interactions Effect :
While it is desirable to design a strain gauge balance to make each bridge sensitive to
only one load component, it is not truly possible to eliminate completely interactions due
to other components. It may be therefore be necessary to take into account the presence
of non-linear interactions as well as the linear interactions.
As the most general case, a six-component strain gauge balance is considered. Such a
balance would measure six load components, i.e., three pure forces components (L, D, Y)
and three moments (M, R, N).
Each bridge indicator reading, as a consequences of interactions, is function of all six
components. If only linear interactions assumed it can be written in general case
NKRKMKYKDKLKR NRMYDL (8.24)
Where R is the indicator reading corresponding to , being any one of L, D, Y, M, R,
N.
There will be six equations of this type, one for each reading where KL ,KD, etc. are
constant coefficients ( the so-called calibration coefficients). A total of 36 coefficient (
first order coefficients) are to be calculated.
All these coefficients can be calculated by loading the balance with each component
independently. Repeating this procedure 36 time will give the 36 balance coefficients.
If single load component is applied to the balance, where , like , is one of the
component is present, from Eq. (8.24), R is given by
KR (8.25)
By plotting R against for several values of , K can be obtained as the slope of the
curve. If the plot is linear, the slope at once gives K = R . If plot is non-linear, the
effect of non-linear interactions is also to be taken into account.
If non-linear interaction takes place, equation can be written in a polynomial form as
LDKNKRKMKYKDKLKR LDNRMYDL )(
RNKLYK RNLY )()( .... (8.26)
104
To obtain the second order calibration coefficients, it will be necessary to load the
balance in combination of various pairs of components.
If two load components, and (say) are applied to the balance simultaneously, then the
six readings are determined by relations, each of the form
)(KKKR (8.27)
If one of the applied loads, , is maintained constant and the other, i.e., is varied then a
plot may be made of R against . Measurement of the slope at = 0 gives
)(
0
KKR
(8.28)
Comparing this with the values of (R ) = 0 when = 0 will show whether or not
K( ) is negligible. If K( ) is not negligible, then R should be determined for a
few different values of . The slope of a graph of R against gives K( ).
In this way, all the calibration coefficients in the six equations (8.24) may be determined.
8.7 Factors Affecting the Accuracy of Measurement :
To obtain high accuracy in strain gauge instrumentation, careful attention must be made
on different aspects, in general, these include :
i) Surface preparation and bonding of strain gauges.
ii) Noise suppression.
iii) Thermal effect.
iv) Optimizing excitation level.
8.7.1 Surface Preparation and Bonding of Strain Gauges :
Strain gauges can be bonded satisfactorily to almost any solid material if the material
surface is prepared properly. Concept of surface preparation is based on the
understanding of cleanliness and contamination. Negligence to surface preparation may
yield most unsatisfactorily gauge installation and hence erroneous result.
The system of surface preparation includes five basic operations :
105
- Solvent degreasing
- Abrading
- Application of gauge layout lines
- Conditioning
- Neutralising
Degreasing is performed to remove oils, greasing, organic contaminations and soluble
chemical residues. Degreasing is done to avoid having subsequent abrading operation
drive surface contaminants into the surface material.
Degreasing can be accomplished using a hot vapour degreaser or aerosol spray cans of
chlorothene SM or Freon. Spray applicators of cleaning solvent are always preferable
because dissolved contaminants can not be carried back to the parent solvent. If possible,
entire test piece should be degreased. Otherwise, for large objects, area sufficiently larger
than the gauge area should be cleaned.
Surface abrading is done to remove any loosely bonded adherents (rust, paint etc.) and to
develop a surface texture suitable for bonding. The abrading operation can be performed
in a variety of ways, depending upon the initial condition of the surface and the desired
finish for gauge installation. For rough surface it may be necessary to start with a file. For
moderately smooth surface, abrading can be done by silicon-carbide paper of appropriate
grit.
The normal method of accurately locating and orienting a strain gauge on the test surface
is to first mark the surface with a pair of crossed reference line at the point where the
strain measurement is to be made. The reference or layout lines should be made with a
tool which burnishes rather than scribes. A scribed line usually raises a burr or creates a
stress concentration. On aluminium and most other alloys a 4H drafting pencil is a
satisfactory and convenient burnishing tool.
After the gauge layout lines are marked, final surface preparation is accomplished by
water based cleaners. To dislodge and remove oxides and mechanically bound
contaminants, a mild phosphoric acid compound is used for conditioning the surface.
This is immediately followed by the neutralizing any chemical reaction introduced by the
acidic conditioner to produce optimum surface conditions for strain gauge bonding.
106
Once the surface material is prepared, strain gauges are to be bonded on the surface
properly. Because a strain gauge can perform no better than the adhesive with which it is
bonded to the test piece, the adhesive is a vitally important component in every strain
gauge installation. Ideally, the adhesive would cause the strain gauge to act as an integral
and inseparable part of the surface material – without adding influence of its own.
One adhesive which is widely used for routine measurement in stress applications under
environmental conditions the cyanoacrylate adhesive. This adhesive transforms from a
liquid to solid when pressed into a thin film between the gauge and the mounting surface.
This adhesive is very easy to handle and cures almost instantly to produce an essentially
creep-free, fatigue-resistant bond.
Other types of adhesive include mainly the epoxy adhesives. The epoxies form a large
class of adhesives used for strain gauge bonding because of the wide range of
characteristics available. Some of the epoxies are single-component and others are two-
component. Epoxy-phenolic adhesive are used for higher operating temperature.
8.7.2 Noise Suppression :
Strain measurements are often made in the presence of electric and/ or magnetic field
which can superimpose electrical noise on the measurement signals. If not controlled, the
noise can lead to inaccurate results and incorrect interpretation of the strain signals
altogether. It is therefore necessary to apply noise-reduction measures top any strain
gauge experimentation.
Virtually every electrical device which generates, consumes or transmits power is
potential source for causing noise in strain gauge circuits. In general, the higher the
voltage or current level, and the closer the strain gauge circuit to the electrical device the
greater will be the electrical noise. The common sources of electrical noise include : AC
power lines, motor starters, transformers, relays, generators, rotating and reciprocating
machinery, are welders, vibrators, fluorescent lamps, radio transmitters etc.
Electrical noise from these sources can be categorized into two basic types: electrostatic
and electromagnetic. The two types of noise are fundamentally different and thus require
different noise-reduction measures. Unfortunately, most of the common noise sources
107
listed above produce combinations of the two noise types, which can complicate the
noise-reduction problem.
Electrostatic fields are generated by the presence of voltage – with or without current
flow. Alternating electrical fields inject noise into strain gauge systems through the
phenomenon of capacitive coupling. Fluorescent lighting is one of the more common
sources of electrostatic noise.
The simplest and most effective barrier against electrostatic noise is conductive shield. It
functions by capturing the charges that would otherwise reach the signal wiring. Once
collected, these charges must be drained off to a satisfactory ground (Fig. 8.10). If not
provided with a low resistance drainage path, the charges can be coupled into signal
conductors through the shield-to-cable capacitance.
Figure 8.10 Electrostatic shielding
Another source of electrostatic noise is leakage to ground through the strain gauges. This
leakage, if excessive, can cause noise transfer from the test piece to the gauge circuit.
Any strain gauge installation on a conductive specimen forms a classic capacitor which
can couple noise from the test piece to the gauge. It is therefore essential to make certain
that the test piece is properly grounded and the leakage between gauge circuit and test
piece is well within bounds.
Electromagnetic fields are ordinarily created either by the flow of electric current or by
the presence of permanent magnet. In order for noise voltage to be developed in a
conductor, magnetic lines of flux must be ‘cut’ by the conductor. Signal conductors in the
vicinity of moving or rotating machinery are generally subjected to noise voltages from
108
this source since moving machine member (made of iron and steel which are
ferromagnetic) redirect existing lines of flux.
The most effective approach to minimizing electromagnetic noise is not to attempt
magnetic shielding of the sensitive conductors but to ensure that noise voltages are
induced equally in both side of the amplifier input. Achievement of noise cancellation
by this approach is discussed in section 2.3.1.4 (Fig. 2.11).
The noise, electrostatic or electromagnetic, can be effectively assessed by the signal
conditioner by a simple but significant feature- a control for removing excitation from the
Wheatstone bridge. With such a control, the instrument output can be easily checked for
noise, independently of any strain signal.
A simple but effective way of reducing noise is to reduce amplifier gain and compensate
by increasing bridge excitation voltage.
8.7.3 Thermal Effect :
Ideally, a strain gauge bonded to a test piece would respond only to the applied strain in
the material and be unaffected by other variable in the environment. Unfortunately, the
resistance strain gauge is somewhat less than perfect. The electrical resistance of the
strain gauge varies not only with strain but with temperature as well. In addition, the
relationship between strain and resistance change (i.e., the gauge factor C) itself varies
with temperature. These deviations from ideal behavior can be important under certain
circumstances and can cause significant error if not properly accounted for.
Once an installed strain gauge is connected to a strain indicator and the instrument
balanced, a subsequent change in the temperature of the gauge installation will generally
produce a resistance change in the gauge.
However, because this change in resistance due to the thermal effect will be registered by
the strain indicator as strain, the indication is usually referred to as ‘temperature-induced
apparent strain’ or ‘apparent strain’ in the test material.
The net apparent strain is caused by two concurrent algebraically additive effects in the
strain gauge installation. First, the electrical resistivity of the strain gauge is temperature
dependent and any resistance change due to this effect appears as strain to a strain
indicator. The second contribution to apparent strain is caused by the differential thermal
109
expansion between the strain gauge and the test material to which it is bonded. With
temperature change, the test piece expands or contacts, and since the strain gauge is
firmly bonded to the test material, the gauge grid is forced to undergo the same expansion
or contraction. To the extent that the thermal expansion coefficient of the grid differs
from that of the test material, the grid is mechanically strained in conforming to the free
expansion or contraction of the test material. Since the grid is, be design, strain sensitive,
the resultant resistance change appears to the strain indicator as strain in test material.
The net ‘apparent strain’ can be expressed as the sum of resistivity and differential
expansions effects:
TG
GT
G
(8.29)
where
BG = Thermal coefficient of resistance of grid material.
G = Gauge factor.
T - G = Difference in thermal expansion coefficients between test
piece and grid respectively.
T = Temperature change.
It should not be assumed from the form of equation (8.27) that the apparent strain is
linear with temperature because the coefficient within the bracket are themselves
functions of temperature. The equation clearly demonstrates, however, that the apparent
strain exhibited with temperature change depends not only upon the strain gauge but also
on the material on which the gauge is bonded.
The first part of apparent strain, i.e., the strain due to thermal expansion of grid can be
eliminated by compensating gauges. For an quarter-arm bridge, an identical
compensating or ‘dummy’ gauge connected on an adjacent arm of the Wheatstone bridge
is mounted on an unstrained specimen for the identical material as the test piece and
subjected always to the same temperature as the active gauge.
Under these hypothetical conditions, the apparent strains in the active and dummy strain
gauges should be identical. Since identical resistance change in adjacent arms of the
Wheatstone bridge does not unbalance the circuit, the apparent strains in the active and
dummy gauges should cancel exactly. This part of the apparent strain can be cancelled by
110
same philosophy by using half-bridge (for example, two active gauges on the two sides of
a thin bending beam will have same temperature and cancel the apparent strain if
connected in adjacent arms of the Wheatstone bridge) and full bridge.
The second part of apparent strain which is due to difference in thermal expansion of
strain gauge and test material can be eliminated by the concept of self-temperature-
compensation. The metallurgical properties of certain gauge alloys – in particular,
constantan and modified Karma – are such that these alloys can be processed to minimize
the apparent strain over a wide temperature range when bonded to test materials with
thermal expansion coefficients for which they are intended. Strain gauges employing
these specially processed alloys are referred to as ‘self-temperature-compensated’.
Fig. 8.11 illustrates the apparent strain characteristics of the self-temperature-
compensated by this figure, the gauges are designed to minimize the apparent strain over
the temperature range from about 00 F to 400
0 F. When the self-temperature-compensated
gauge is bonded to a test material having the thermal expansion coefficient for which the
gauge is intended and when operated within the temperature range of effective
compensation, strain measurements can usually be made without the necessity of
correcting for apparent strain.
Table 1 shows a number of common materials and gives the thermal expansion
coefficients for each, along with the recommended S-T-C number. For apparent strain
cancellation, strain gauges of appropriate S-T-C number should be bonded to the test
material.
111
Figure 8.11 Variation of apparent strain with temperature
Table 8.1 : S-T-C number of different materials
Test material Thermal expansion coefficient
per degree Fahrenheit
Recommended S-T-C
number
Aluminium 12.9 13
Brass 11.1 13
Bronze phosphor 10.2 09
Copper 9.3 09
Molybdenum 2.2 03
Steel 6.0 (average) 06
Stainless steel 9.0 (average) 09
Tilonium 4.9 05
8.7.4 Optimising Excitation Level :
The excitation voltage applied to a strain gauge bridge creates a power loss in each arm,
all of which must be dissipated in the form of heat. Only a negligible fraction of the
power input is available in the output circuit. This causes the sensing grid of every strain
gauge to operate at a higher temperature than the test material to which it is bonded. It
can be considered that heat generated within a strain gauge must be transferred by
112
conduction to the mounting surface. The heat flow through the specimen causes a
temperature rise in the test material, which is a function of its heat-sink capacity and
gauge power level.
Consequently, both sensing grid and test material operate at temperatures higher than
ambient. When the temperature rise is excessive, gauge performance will be affected in a
number of ways. Firstly, a loss of self-temperature-compensation (S-T-C) occurs when
the grid temperature is considerably above the specimen temperature. Secondly,
hysteresis and creep effects are magnified since these are dependent on backing and glue-
line temperature. Thirdly, zero (no-load) stability is strongly affected by excessive
excitation.
One of the simple but effective way of determining the optimum excitation level is to
gradually increase the bridge excitation under zero-load condition until a definite zero
instability is observed. The excitation should then be reduced until the zero reading
becomes stable again without a significant offset from the low-excitation zero reading.
For most applications, this value of bridge voltage is the highest that can be safely used
without significant performance degradation.
Optimum strain gauge excitation level can also be determined on the basis of heat sink
property of test material and gauge size and resistance. Heavy sections of high thermal
conductivity metals such as copper or aluminium are excellent heat sinks. Thin section of
low-thermal-conductivity metals such as stainless steel or titanium are poor heat sinks.
Higher excitation level is permissible for test material having good heat-sink properties.
Similarly, higher strain gauge resistances permit higher excitation level.
Power dissipated in grid (watts) may be given by
R
VPG
4
2
(8.30)
while power density in grid (watts / m2) may be given by
APP GG / (8.31)
113
where
R = Gauge resistance in ohms
A = Grid area (Active gauges length gauge area)
V = Bridge excitation in volts
When grid area (A), gauge resistance ( R ) and grid power density ( PG ) are known :
APRV G 2 (8.32)
Table 2 provides the values of power density of various metals.
Table 8.2 : Heat sink conditions
Accuracy
requirement
Excellent
Aluminium or Copper
Good
Thick Steel
Fair
Thin Steel
Poor
Plastic
High 2 – 5
3.1 – 7.8
1 – 2
1.6 – 3.1
0.5 – 1
0.78 – 1.6
0.1 - 0.2
0.16 - 0.31
Moderate 5 -10
7.8 - 16
2 – 5
3.1 – 7.8
1 – 2
1.6 – 3.1
0.2 – 0.5
0.31 – 0.78
Low
10 – 20
16 - 31
5 – 10
7.8 - 16
2 – 5
3.1 – 7.8
0.5 – 1
0.78 – 1.6
114
Chapter 9
FORCE AND MOMENT MEASUREMENT BY
ELECTRONIC EXTERNAL BALANCE
9.1 Introduction :
Measurements of forces and moments on a model in wind tunnel are made either
mechanically or electronically. The basic advantages of electronic measuring system, i.e.,
fast response, high and low values capability and amenability to automation are outlined
in Chapter 2. In an electronic system pick up or transducer converts the physical quantity
under measurement into electrical signal.
Internal electronic balance or sting balance (where strain gauge is used as pick-up or
transducer) for measurement of forces and moments are discussed in details in Chapter 8.
in this chapter an external electronic balance (3-component) is described. The advantage
of this system is that, unlike in sting balance, it is kept outside the tunnel and hence flow
is not disturbed by it.
In this balance, aft lift, fore lift and drag are measured by three load cells are obtained
from three digital voltmeters. Pitching moment is obtained by simple manipulation.
9.2 General description :
The general arrangement of the external balance is shown in Fig. 9.1. it is mounted on the
side wall of the working section outside the tunnel and is designed for airflows from right
to left when balance is viewed from front.
115
Figure 9.1 Three component external balance
The balance is constructed mainly of aluminium alloy and its main frame work comprises
a mounting plate which is secured to the tunnel test section and carries a triangular force
116
plate. The force plate and mounting plate are connected by three supporting legs,
disposed at the corners of the force plate. Each leg is attached to the force plate and
mounting plate by spherical universal joints. The effect of this is to constrain the force
plate to move in a plane parallel to the mounting plate, while leaving it free to rotate
about a horizontal axis. The necessary three degrees of freedom are thus provided.
Models for use with the balance are provided with a 12 mm diameter mounting stem and
this is inserted in the bore of model support and secured by a collet tightened by model
clamp. The model support is graduated on the periphery and is free to rotate in the force
plate for adjustment of the angle of attack of the model, while its position may be locked
by an incidence clamp.
The force plate may be locked in position by two centering clamps. It is to be noted that
“ this plate should always be tightened when balance is not in use or when changing
models”.
The forces acting on the force plate are transmitted by way of flexible cables to strain
gauge load cells which measure respectively the fore and aft lift forces and the drag force.
The drag cable which lies horizontally, acts on a line through the center of model support
while the two lift cables act vertically through points disposed equidistantly from the
center of the model support and in the same horizontal plane as the support. The distance
between the fore and aft lift tapes is 0.127 m (5.0 inch) and sum of the force in these
tapes thus gives the lift on the model while the difference when multiplied by 0.127 gives
the pitching moment in Newton meters. A drag balance spring acts on the force plate to
apply preload to the drag load cell.
The output from each load cell is taken to a strain gauge amplifier carried on the
mounting plate and then via a flexible cable to a display unit comprising a set of three
voltmeters shoeing the output from the respective load cell circuits. Lift and drag forces
are then calculated directly from the load cell outputs by using the calibration factors.
When calibrating the balance there is possibility of slight friction in the force plate
supports. To overcome this, small vibration is provided. The motor which requires a 12
volt DC supply is carried on the mounting plate and controlled by the vibrator push
button. It is not usually necessary to use the vibrator when using the balance for force
117
measurements on the model as their usually sufficient vibration present to overcome any
friction in the mechanism.
9.3 Operation :
To fit a model, centering clamps are tightened, model is set at zero incidence and
incidence clamp is tightened. Model supporting stem is slided into model support and
model clamp is tightened. Centering clamps are released to ensure that model moves
freely without touching tunnel walls. It is to be noted that under “ no circumstances
model clamp is tightened in the absence of model otherwise the collet will be damaged”.
After switching on the supply it is desirable to allow a warm up time of 15 minutes for
the load cells before taking any readings. Once centering clamps are released the display
unit will indicate values corresponding to the zero readings of lift and drag. Vibrator is to
be separated before recording the zero reading of fore and aft lift and drag.
It will generally be found most convenient to set the incidence of the aerofoil models to
give a lift force acting downwards, thus giving positive values of lift load cell read-out.
To measure the aerodynamic forces tunnel speed is set to a desired value and hold display
button is pressed on the display unit. Reading of digital voltmeter are then recorded.
When it is desired to make a series of measurement of lift and drag over the range of
model angles of incidence this angle may be set by releasing the incidence clamp,
rotating the model support to the desired angle and retightening the clamp. “the centering
clamps must be locked before releasing the incidence clamp or handling the force plate
in any way. Otherwise there is risk of damaging the load cells”.
At the end o the test sequence, zero readings of load cells are taken for recheck.
Range of loading as per manufacturer’s specifications are as follows :
Lift Force 0 – 100 N
Drag Force 0 – 50 N
Pitching Moment 0 – 2.5 Nm
118
Initial calibration factors given by manufacturer are :
Fore Lift 7.570 N/volt
Aft Lift 7.418 N/volt
Drag 7.496 N/Volt
If is model angle of incidence, a, f and d are aft load cell, fore load cell and drag load
cell read out respectively, a0, f0 and d0 being the zero readings then
Aft lift (A) : A = a1(a – a0)
Fore lift (F) : F = f1 (f – f0 )
Drag (D) : D = d1 (d – d0 )
Total Lift (L) : L = A + F
Moment (M) : M = 0.127 (F – A)
Where a1 , f1 and d1 are the aft lift, fore lift and drag calibration factors respectively.
9.3.1 Setting up load cells :
At times it may be necessary to readjust the cables connecting the force plate to the load
cells. It is essential that this is done correctly otherwise there is possibility of overloading
the cells.
To readjust, the centering clamps are first tightened. Forces are transmitted from lift and
drag cables to the load cells by way of a conical nipple brazed to the cable and an
adjusting screw secured by locknut which contacts the nipple and transmits the load to
the cell. To adjust, locknut is loosened and adjusting screw is turned anti-clockwise until
the cable is just tight. After that it is turned clockwise by one half of a revolution and
locknut is tightened. There will then be a play of approximately 0.25 mm between cable
and load cell when centering clamps are locked.
9.4 Calibration :
It is desirable to calibrate the balance periodically. Essentially the calibration procedure
involves the application of known lift and drag forces using dead weights. Fig 9.2 shows
119
the set up for calibrating in the open circuit. The balance is usually mounted on a frame
attached to the wind tunnel. However, a separate fixture also can be prepared for
calibration where balance can be mounted.
Figure 9.2 Schematic arrangement off calibration rig
The balance is supplied with calibrating arm having a 12 mm diameter stem which could
be secured in the model clamp. A pivoted link is fitted to the arm in one of the three
positions, either on the axis of the arm or at the points displaced 63.5 mm on either side
of the axis.
The calibrating arm is locked in the model clamp with the arm cross member lying
horizontally and the projection of the arm from the balance set so that arm loading point
lies approximately on the axis of the wind tunnel.
To calibrate lift load cells the pivoted link is fitted at the location point on axis of the
calibrating arm and dead weights are applied to the loading link using a suitable hook.
120
Since the lift load cells are disposed symmetrically on each side of the balance axis, it
may be assumed that dead loads so applied are divided equally between the two cells.
To calibrate the drag load cell, a horizontal force is applied to the calibrating arm by way
of loading link using dead weights, a nylon chord and a pulley (Fig. 9.2).
If desired, the individual fore and aft lift load cells may be calibrated by applying dead
weights using the loading link, at the locating points at each end of the transverse
member of the calibrating arm.
The calibrating procedure is as follows :
(a) The power supply to the balance is switched on and left for twenty minutes for
warm up.
(b) The centering clamps are released and the zero readings of load cells are
recorded.
(c) The dead weights are applied and the load cell outputs are recorded.
(d) The dead weights are removed and the zero readings are noted again.
(e) The procedure is repeated ten times thus collecting ten set of readings.
(f) Average values of load cell outputs are calculated.
Suitable loads for calibration are 100 N and 50 N for lift load cell calibration and 50 N
and 25 N for drag cell. It is usually desirable to carry out calibration at two set of weights
: one at rated load and the other at half rated load to confirm linearity of the relationship
between load cell output and the load.
It will be observed that in amplifier box there are three holes labeled ‘set zero’ and three
labeled ‘set bridge volts’. In each case the hole gives access to an adjusting screw that
may be reached by a small screwdriver.
It should not normally be necessary to make any adjustment to these settings, if they are
changed they will need recalibration.
The set zero adjustment is made with the force plate clamped in which condition none of
the load cells is subjected to any loading. The output from each cell as shown on the
display unit should then be approximately zero, although this setting is not critical.
121
To check the bridge voltage use is made of the calibration cable provided with the
balance. The cable has a male and female termination and may be inserted between each
load cell in turn and the input plug to the amplifier. The calibration cable has two free
leads which can be connected to a high grade digital voltmeter.
After warming up the bridge supply voltage should be set to 10.000 volts 0.005 volts on
all three circuits.
Table 9.1 : Calibration of drag cell
(a) Weight = 0.5 kg
Serial No. Reading with load
(x)
Reading with no load
(y)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
2.060
2.081
2.078
2.078
2.074
2.081
2.067
2.081
2.076
2.066
1.416
1.430
1.456
1.443
1.445
1.453
1.460
1.459
1.455
1.460
Mean 2.0741 1.4511
d = 2.0741 d0 = 1.4511
Calibration factor = (0.5 9.81) (2.0741 – 1.4511) = 7.865 Nv
122
Table 9.2 : Calibration of lift load cells
(a) Weight = 1000gms
Serial
No.
Fore lift reading
With load Without load
Aft lift reading
With load Without load
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
3.777 3.123
3.802 3.132
3.804 3.140
3.805 3.142
3.800 3.140
3.816 3.135
3.806 3.139
3.814 3.146
3.813 3.142
3.814 3.144
3.201 2.703
3.279 2.701
3.283 2.704
3.287 2.710
3.283 2.708
3.207 2.713
3.292 2.711
3.290 2.706
3.288 2.715
3.289 2.712
Mean 3.8051 3.1383 3.2859 2.7083
Calibration factor: 7.349 N/v 8.483 N/v
(b) Weight = 2035 gms
Serial
No.
Fore lift reading
With load Without load
Aft lift reading
With load Without load
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
4.473 3.124
4.493 3.131
4.522 3.126
4.495 3.116
4.494 3.104
4.466 3.105
4.475 3.112
4.486 3.104
4.473 3.102
4.498 3.094
3.937 2.692
3.920 2.692
3.924 2.697
3.933 2.699
3.925 2.706
3.925 2.702
3.914 2.706
3.936 2.697
3.920 2.701
3.946 2.713
Mean 4.4875 3.1118 3.928 2.7005
Calibration Factor: 7.248 N/v 8.123 N/v
123
9.5 Wind Tunnel Testing :
Wind tunnel testing is carried out on two-dimensional wings : one with NACA 0012
section and the other a supercritical aerofoil. Both the models are 30.48 cm (1ft) in chord
and 61 cm (2 ft) in span.
After the models are installed in tunnel zero reading are recorded before starting the
tunnel. After the tunnel is started readings may be taken from the three digital voltmeters
(if necessary, by pressing the push button to hold the display).
Fore and aft lift and drag may now be determined by subtracting the respective zero
readings and using the calibration charts. Total lift is obtained by summing the fore and
aft lift and pitching moment at the holding point can be determined by multiplying 0.127
with the difference of the two lifts. Pitching at any other point (e.g. leading edge or ¼ c)
can be derived by moment transfer theorem.
124
Chapter 10
WIND TUNNEL BOUNDARY CORRECTIONS (2D FLOW)
10.1 Introduction :
The conditions under which a model is tested in a wind tunnel are not the same as those
in free air. The closed (or open) boundaries of test section in most cases produce
extraneous forces. This must be subtracted out in order for the results to be comparable
with those in free air.
The presence of test section boundary walls produces :
i) A variation in static pressure along the test section due to formation and
subsequent thickening of boundary layer downstream. The effective area is
reduced progressively downstream resulting in an increase of velocity and
decrease of pressure downstream. The change in pressure upstream and
downstream of the model produce a drag force known as ‘horizontal buoyancy’.
ii) A lateral constraint to the flow pattern about a body, known as ‘solid blocking’.
In a closed wind tunnel, solid blocking is the same as an increase of speed,
increasing all forces and moments at a given angle of attack. It is usually
negligible with an open test section, since the airstream is then free to expand in
a normal manner.
iii) A lateral constraint to the flow pattern about the wake known as ‘wake
blocking’. The effect increases with an increase of wake size and in a closed test
section increases the drag of the model. Wake blocking is usually negligible
with an open test section since the airstream is then free to expand in normal
manner.
iv) An alternation to the local angle of attack along the span. In a closed test section
the angle of attack near the wingtip of a model with large span is increased
excessively, making the tip stall early. The effect of an open jet is just the
opposite (tips unstalled). In both cases the effect is diminished to the point of
negligibility by keeping model span less than 0.8 times the tunnel test section
span.
125
v) An alternation to the normal curvature of the flow about a wing. The wing
moment coefficient, wing lift and angle of attack are increased in a closed wind
tunnel and are decreased with an open jet.
vi) An alternation to the normal downwash so that the measured lift and drag are in
error. The closed jet makes the lift too large and the drag too small. An open jet
has just the opposite effect.
It is to be noted that the additional effects resulting from the customary failings of wind
tunnels – local variations in velocity, angularity of flow, tare and interference etc. – are
extraneous to the basic wall corrections and it is assumed that the errors due to these
effects have already been removed before wall effects are considered. Methods governing
their removal are discussed in Chapter 6.
Since the manner in which the two and three dimensional walls affect the model and are
simulated is quite different they will be considered individually. Wall corrections for
two-dimensional testing are given here. Wall corrections for three-dimensional testing are
discussed in next chapter, Chapter 11.
In order to study effects primarily concerned with two dimensional flow, it is customary
to build models of constant chord which completely span the test section from wall to
wall. The trailing vortices are then practically eliminated. Consequently, corrections due
to downwash and spanwise variation of local angle of attack are not needed. These
corrections, needed for three dimensional testing, are given in Chapter 11.
Corrections for tests under two-dimensional flow conditions include :
i) horizontal buoyancy.
ii) Solid Blocking
iii) Wake blocking
iv) Streamline curvature effect
10.2 Horizontal Buoyancy :
Almost all wind tunnels with closed throats have a variation in static pressure along the
axis of the test section resulting from the thickening of the boundary layer as it progresses
toward the exit and to the resultant effective decrease of the jet area. It follows that the
pressure is usually progressively reduced as the exit is approached and there is a tendency
126
of the model to be ‘drawn’ downstream. The static pressure variation along a jet is
usually as shown in Fig, 10.1.
Static pressure
(N/m2 )
distance along tunnel center line
Figure 10.1 Variation of static pressure along tunnel center line
The variation of cross-sectional area of the model (NACA 0012 model of Chapter 9) is
shown in Fig. 10.2. it is seen that the variation of static pressure from, say station 2 to
station 3, is (p2 - p3). This pressure difference acts on the average area (S2 + S3)/2. The
resulting force for that segment of the model is therefore
2
32
32
SSppDB
Figure 10.2 Variation of cross-sectional area of aerofoil model
This equation is simply solved by plotting local static pressure against body section area,
the buoyancy then becoming the area under the curve. However, for the case where the
127
longitudinal static pressure gradient is a straight line (as shown in Fig. 10.1), the equation
gives horizontal buoyancy as
dsdsdpSD XB
where SX = model cross-section area at station x
S = distance from model nose
dp/ds = slope of longitudinal static pressure gradient
Now, SXds = body volume, horizontal buoyancy can be obtained as
DB = -(dp/ds) (body volume) (10.1)
Aerofoil body volume can be obtained approximately by the expression
Model volume = 0.7 model thickness model span.
In deriving eq.(10.1) only the pressure gradient effect is taken into account. But the
existence of failing static pressure gradient not only implies that the test section is getting
effectively smaller but also that the streamlines are getting squeezed. This squeezing
effect should also be incorporated in the calculation of horizontal buoyancy. More
accurate formulae incorporating this squeezing effect have been derived by Glauert and
also by Allen and Vincenti.
The expression for horizontal buoyancy by Glauert is
dsdptDB
2
12
(10.2)
where t = body thickness
1 = body-shape factor (about 4.2 for NACA 0012 aerofoil)
The expression for horizontal buoyancy derived by Allen and Vincenti is
dsdph
dsdpcDB
2
22
2 68
(10.3)
where h = tunnel height
c = model chord
22
48
h
c (10.4)
cxddxdzpcz2
11
0
2 1116
(10.5)
128
= 0.24 for NACA 0012 aerofoil (x, z are the aerofoil coordinates), c its
chord and p its no chamber (basic) pressure distribution.
The amount of ‘horizontal buoyancy’ (DB) is then subtracted from the observed values of
drag in order for the result to be comparable with free air condition. This is usually small
for wings, but large for fuselages and nacelles.
10.3 Solid Blocking :
The presence of a model in the test section reduces the area through which air must flow,
and hence by Bernoulli’s principle increases the velocity of air as it flows over the model.
The increase of velocity, which may be considered constant over the model for
customary model sizes, is called ‘solid blocking’ (Fig. 10.3). its effect is a function of
model thickness, thickness distribution, and model size, and is independent of the
camber. The velocity increment at the model due to solid blocking can not simply be
obtained by direct area reduction. It is much less than the increment one obtains from the
direct area reduction since it is the streamlines far away from the model that are most
displaced.
To understand the mathematical approach, solid blockage for a circular cylinder in a two-
dimensional tunnel is considered. The cylinder in an open free stream may be
mathematically simulated by placing a doublet of strength = 2a2U in a uniform
stream where ‘a’ is the radius of the cylinder and U is the free stream speed.
Next, the presence of tunnel roof and floor is to be mathematically simulated. It is well
known that any boundary wall near a source, sink, doublet or vortex can be simulated
easily by the addition of a second source, sink, doublet or vortex above the boundary
wall. A solid boundary is formed by the addition of such an image system which
produces a zero streamline matching the solid boundary. An open boundary, on the other
hand, requires an image system that produces a zero velocity potential line which
matches the open boundary. After the image system, as shown in Fig. 10.3, is established,
its effect on the model is the same as that of boundary it represents.
The image system is developed in the following way. A single doublet of strength at A
is considered, which is to be contained within the solid walls 1 and 2. To simulate wall 1,
a doublet is needed at B and for wall 2,another doublet is needed at C. Now, doublet B
129
needs a doublet B’ to balance it from wall 2 and doublet C needs a doublet C’ to balance
it from wall 1 and so on out to infinity. The image system for a closed rectangular test
section thus becomes a doubly infinite system of doublets.
Figure 10.3 Mathematical simulation of solid body between tunnel roof and floor
The axial velocity in the tunnel centerline due to first doublet at B is ,
222 222 hUahU
or, 2
2
h
a
U
U
Since velocity by a doublet varies inversely with the square of the distance from the
doublet, the doubly infinite doublet series may be summed as
130
2
2
12
12
h
a
nU
U
total
Sb
2
22
3 h
a (10.6)
It is seen that a 0.25 m radius cylinder in a tunnel 2.5 m high act as though the clear jet
speed (U) were increased by 3.3 percent.
Now, the blockage due to a given aerofoil of thickness t may be represented on that due
to “ equivalent ” cylinder of diameter t(1)1/2
. with this approach the solid blocking for a
two-dimensional aerofoil may be found from simple doublet summation. Glauert wrote
this solid blocking velocity increment as
2
2
12
2
1
2
822.043 h
t
h
tSb
(10.7)
where 1 = 4.2 for NACA 0012 aerofoil.
Allen and Vincenti obtained their expression by rewriting eq.(10.7). introducing as in
eq.(10.4) and using 2 = 4 1 t2/ c
2, solid blocking correction is obtained as
Sb = 2 (2 = .24 for NACA 0012 aerofoil) (10.8)
A simple form of solid blocking correction is given by Thom as
23
1 )..(mod
C
volumeelKSb (10.9)
where K1 = 0.74 for a wing spanning the tunnel breadth
C = tunnel test section area
10.4 Wake Blocking :
Any real body without suction-type boundary layer control will have a wake behind it ,
and this wake will have a mean velocity lower than the free stream (Fig. 10.4). According
to the law of continuity, the velocity outside the wake in a closed tunnel must be higher
than free stream in order that a constant volume of fluid may pass through the test
section. The higher velocity in the mainstream has, by Bernoulli’s principle, a lowered
pressure and this lowered pressure, arising on the boundary layer (which later becomes
the wake) grows on the model and puts the model in a pressure gradient resulting in a
velocity increment at the model.
131
Figure 10.4 Velocity characteristics of wake
To compute this wake effect, the wake and tunnel boundaries are to be mathematically
simulated. The wake simulation is fairly simple. In the two-dimensional case a line
source at the wing trailing edge emitting, say ‘blue’, fluid will result in a ‘blue’ region
simulating the wake. In order to preserve continuity a sink of same strength should be
added far downstream.
The simulated wake may be ‘contained’ within the floor and ceiling by an infinite vertical
row of source sink combination (Fig. 10.5).
132
Figure 10.5 Mathematical simulation of wake of a body contained between tunnel roof
and floor.
The axial velocity induced at the model is
2
hU
(10.10)
where = strength of source-sink.
h = tunnel height.
The increment in horizontal velocity due to wake blocking can be written as
Dwb Chc
U
U
4
(10.11)
Maskell suggests that the correction be
Dwb Chc
U
U
2
(10.12)
133
10.5 Streamline Curvature Effect :
The presence of ceiling and floor prevents the normal curvature of the free air that occurs
about any lifting body and – relative to the straightened flow – the body appears to have
more camber (around 1% for customary sizes) than it actually has. Accordingly, the
aerofoil in a closed wind tunnel has too much lift (and moment about quarter-chord) at a
given angle of attack and, indeed, the angle of attack is too large as well. This effect is
not limited to cambered aerofoils only, since, using the vortex analogy, any lifting body
produces curvature in the airstream.
Streamline curvature effect may be estimated by assuming that the aerofoil in question is
small and may be approximated by a single vortex at its quarter-chord point. The image
system necessary to contain this vortex between floor and ceiling consists of vertical row
of vortices above and below the real vortex (Fig. 10.6). the image system extends to
infinity both above and below and has alternating signs.
Figure 10.6 Mathematical simulation of streamline curvature
134
The first image pair may be considered first. It is apparent they induce no horizontal
velocity since the horizontal components cancel, but as will be seen, the vertical
components add.
From simple vortex theory, the vertical velocity at a distance x from the lifting line will
be
222 xh
xw
(10.13)
Substitution of reasonable values for x and h in the above equation reveals that the
boundary induced upwash angle varies almost linearly along the chord, and hence the
stream curvature is essentially circular.
The chordwise load for an aerofoil with circular camber may be considered to be a flat
plate loading plus an elliptically shaped loading.
Considering the flat plate loading first, the upwash induced at half chord by the two
images closest to the aerofoil, by eq. (10.13), may be given by
22 4
4
22
ch
cw
Since UcCL2
1 the angular correction needed for the nearest image becomes
LCch
c
U
w22
2
48
1
Assuming that (c/4)2 is smaller to h
2 and again using
22
48
h
c
the angular correction is obtained as
LC
3
6
The second pair of vortices induces a upwash velocity
22
42
4
22
ch
cw
and an angular correction , LC3
6
4
1
135
Adding for all the infinite pairs of images, angular correction may be obtained as
LSC C
........
16
1
9
1
4
11
63
radianC
C
L
L
.......2
1
.12
6 2
3
LC.2
3.57 degrees (10.14)
The additive lift correction is
scLscC ..2
LC..2
1.2
LC. (10.15)
and the additive moment correction is
LsccM
CC
4
4
1 (10.16)
10.6 Summary of Two-dimensional Boundary Correction :
The data concerned for the NACA 0012 aerofoil model in Chapter 9 are the following :
Free stream speed = U m/s
Free stream dynamic pressure = q N/m2
Reynolds number = Re
Angle of incidence =
Drag = D
Lift = L
Applying the wind tunnel boundary corrections the corrected values can be obtained as
summarized below.
Sb is given by eq.(10.6) or (10.8). To get wb from eq.(10.11) or (10.12), CD needs to be
corrected first.
Considering horizontal buoyancy into account, CD may be corrected by using
136
SU
DDC B
D2
2
1
where DB is given by eq.(10.1), (10.2) or (10.3). wb can then be obtained from eq. (10.11)
or (10.12).
Corrected value of free stream speed U may be obtained from
)1()1( UUU wbSbC
Corrected value of dynamic pressure q may be obtained from
2112
qqq C
and the Reynolds number from
1RRC
Lift coefficient CL is found from
SU
LCL
2
2
1
The corrected lift coefficient taking blockage effect can be obtained from
SU
LCLc
22 12
1
or, 21 LLc CC
Taking both blockage effect and streamline curvature effect into account
21LLc CC [from eq.(10.15)]
Where can be obtained from eq.(10.4).
Corrected value for incidence is
LC C.2
3.57 [from eq. (10.14)]
Corrected drag coefficient CD (taking both solid and wake blocking into account) may be
obtained as
wbSbDDC CC 231
137
Corrected moment coefficient CM(1/4)C may be given as
4
.21
4
1
4
1L
CMCM
CCC
Both the uncorrected and corrected values can be put tabular form as shown in Table
10.1.
Table 10.1 : Uncorrected and corrected values of different parameters
Uncorrected Corrected
U R CL CD CM(1/4)C U R CL CD CM(1/4)C
138
Chapter 11
WIND TUNNEL BOUNDARY CORRECTIONS (3D FLOW)
11.1 Introduction :
Wind tunnel boundary corrections for three dimensional testing follow the same
reasoning used for two dimensional testing (Chapter 10). The correction factors are,
however, different since both vertical and horizontal wall effects are now taken into
account. Also, an additional correction is needed for the wall effects on downwash by the
trailing vortices issuing from the trailing edges of the wing models.
The corrections for three dimensional testing include :
i) horizontal buoyancy
ii) solid blocking
iii) wake blocking
iv) streamline curvature effect
v) downwash effect
11.2 Horizontal Buoyancy :
The philosophy behind the buoyancy correction has been described in Chapter 10. for the
three dimensional case, the correction for pressure gradient effect only may be written, as
before, as
ds
dpDB (body volume) (11.1)
The total correction for both pressure gradient and streamline squeezing effect has been
given by Glauert as
ds
dptDB
3
34
(11.2)
where 3 = body shape factor for three-dimensional bodies
= 4.2 for NACA 0012 wing
and t = maximum body thickness
= .05856m for the case of 1.6 aspect ratio rectangular wing
139
11.3 Solid Blocking :
The solid blocking correction for three dimensional flow follows the same philosophy
described earlier for two dimensions. The body can be represented by a source-sink
distribution is now contained by the walls of the tunnel. The simulation of the tunnel
walls for the three-dimensional case requires image system for horizontal boundary walls
(floor and ceiling ) as well as for side vertical walls as shown in Fig. 11.1. The image
system as before extends to infinity on all sides.
Figure 11.1 Mathematical simulation of solid body between horizontal as well as lateral
boundaries of the tunnel.
Summing the effect of images, velocity increment due to solid blocking for a wing may
be given by
23
)...(11
C
volumewingK
U
USb
(11.3)
where,
K1 = body shape factor (1.008 for NACA 0012 wing)
1 = factor depending on the tunnel test section shape and model span to tunnel
width ratio
C = tunnel test section area (61 cm 61 cm for low speed tunnel)
Thom’s short-form equation for solid blocking for a three dimensional body is
23
)...(mod
C
volumeelK
U
USb
(11.4)
where K = 0.9 for a three-dimensional wing.
140
11.4 Wake Blocking :
The correction for wake blocking follows the logic of the two dimensional case in that
the wake is simulated by a source of strength Q at the trailing edge which is matched for
continuity by adding a downstream sink of same strength Q. The image system consists
of a doubly infinite source-sink system spaced at a tunnel height (h) apart vertically and a
tunnel width (B) apart horizontally as shown in Fig. 11.1.
The axial velocity induced by the image system is
BHQU 2
The incremental velocity is
Dwb CC
S
U
U
4
(11.5)
where,
S = model wing area
C = tunnel test section
CD = drag coefficient of the wing
The increase of drag due to pressure gradient may be subtracted by removing the wing
wake pressure drag
DD CC
volumewingKC
23
11 )...( (11.6)
where K1, C and 1 are as defined for eq. (11.3).
11.5 Streamline Curvature Effect :
The correction for streamline curvature for three dimensional testing follow the same
philosophy as those for the two dimensional case in that they are concerned with the
variation of the boundary induced upwash along the chord. But for the three dimensional
system is shown in Fig. 11.2. Basically it consists of the real wing with its bound vortex
CD and trailing vortices C and D. The vertical boundaries are simulated by the infinite
system of horse-shoe vortices and the horizontal boundaries are simulated by the infinite
lateral system . Linking the two systems is the infinite diagonal system.
141
Figure 11.2 Mathematical simulation of streamline curvature effect
The effect of the image system can be summed up and without going into the details of
the formulation the correction factors can be written as
3.57..12 LSC CCS (11.7)
where,
2 = factor representing the increase of boundary induced upwash at a point p
behind the wing quarter-chord in terms of the amount at the quarter-chord.
= 0.195 for the present model and tunnel.
= a factor which is function of the span load distribution, ratio of model
span to tunnel width, the shape of the test section and whether or not the
model is on the tunnel centerline.
= 0.137 for the present case.
S = model surface area.
C = tunnel test section area.
The additive lift correction is
142
aC SCLS (11.8)
where a = wing lift-curve slope
= 0.088 per degree for a 3D wing
The additive correction for the moment coefficient is
LSCMSC CC .25.0 (11.9)
11.6 Downwash Effect :
The downwash induced by the trailing vortex system needs to be corrected for the tunnel
wall effects. Through elementary vortex theory the correction factor for the tunnel
boundary induced downwash can be developed. The only mathematical tools needed are
the expression for the induced velocity w due to a vortex of strength at a distance r
r
w4
(11.10)
and the relation between lift and circulation for a uniformly loaded wing of span b
LCbSU 2 (11.11)
combining the two gives
LCrb
SUw
8 (11.12)
Now r represents the vortex spacing in the image system which may be expressed as
some constant times a tunnel dimension, e.g. the tunnel height h and the model wing span
may be expressed in terms of the tunnel width B.
The induced angle at the centerline of the test section is then
LC
hbBbK
s
U
w
81
for any one image.
Summing the whole field and setting B/8Kb = and noting that hB is the test section
area C, for the complete system is obtained as
)3.57./1 Li CCS (11.13)
Now, the induced drag coefficient may be written as
LiDi CC where i = induced angle
143
Therefore, the change in induced drag caused by the boundary induced downwash
becomes
2
1 LLiDi CCSCC (11.14)
11.7 Summary of Three-Dimensional Boundary Corrections :
Data (, CL, CD, U) obtained from testing of a wing model in a closed three dimensional
tunnel may be corrected to free air conditions according to the following relations :
The corrected value of wind speed is
21 UU C (11.15)
where wbSb (11.16)
The dynamic pressure is
22 12
1 Uq C
212
1 2 U
21 q (11.17)
The Reynolds number is
1RRC (11.18)
The lift coefficient is (from eq.{(11.7) & (11.8))
aCSCC LLC ).3.57).((21 12 (11.19)
The angle of attack is (from eq. (11.7) & (11.3))
21 13.57 LC CCS (11.20)
The drag coefficient is (from eq.(11.6) & (11.14))
2
1. .21 LDCD CCSCC (11.21)
144
Chapter 12
DRAG MEASUREMENT ON CYLINDRICAL BODY
12.1 Introduction :
The resistance experienced by a body as it moves through a fluid is what is commonly
known as drag. Total drag of a body may be separated into a number of items each
contributing to the total. As a first step it may be divided into “pressure drag” and
“friction drag”. The pressure drag may itself be considered as the sum of three items :
1) boundary layer pressure drag
2) trailing vortex drag or induced drag
3) wave drag.
Some of these items depend on viscosity, others may exist in inviscid fluid.
Schematically,
Total drag
Friction drag Pressure drag
(depends on viscosity)
Boundary layer normal Trailing vortex drag Wave drag
pressure drag (does not depend (does not depend
(depends on viscosity) on viscosity) on viscosity)
Trailing vortex drag can exist only in the case of flow about a three dimensional lifting
body and depends on the lift generated. The wave drag is associated with the formation
of shock waves in high speed flight. For the particular case of low speed two-dimensional
flow about a circular cylinder, both the items can be eliminated. The drag components
acting on the body are the friction drag and boundary layer normal pressure drag. The
summation of these two components is the profile drag.
Flow around a circular cylinder (Fig. 12.1) is considered. To analyze the drag force it is
convenient to assume the cylinder t be moving in a stationary fluid. To an observer
moving with the cylinder the fluid will appear to be approaching as a uniform stream. At
145
any point A on the surface of the cylinder, the effect of fluid may conveniently be
resolved into two components, pressure (p) normal to the surface and shear stress(
)along the surface. The combined effect of pressure and shear stress (skin friction) in the
direction of oncoming fluid gives rise to the drag force (profile drag).
Figure 12.1 Uniform flow past a circular cylinder
It is worth noting that in the case of ‘ideal’ fluid, shear stress is zero and the pressure
distribution (given by cp = 1 – 4sin2 ) is symmetrical over the forward and backward
face of the cylinder which cancel out exactly giving zero drag force. For a real fluid shear
stress exists and the pressure distribution is no longer symmetrical resulting in an overall
rearward force. This force is so called boundary layer normal pressure drag.
There are three methods of measuring the drag force :
1) by measuring pressure distribution on the surface on the cylinder,
2) by measuring pressure distribution in the wake of the cylinder,
3) by direct weighing.
12.2 Drag by Measuring Pressure Distribution on the Cylinder Surface :
Consider an element s on the surface of the cylinder at a point where the normal is
inclined at an angle to the direction of U as shown in Fig. 11.2.
The element of drag d per unit cylinder length due to p and is
dspD .sincos
146
and integrating round the whole parameter yields
dspD .sincos
Figure 12.2 Elemental drag due to p and
Drag may be expressed in non-dimension form as
SU
DCD
2
2
1
where S is the area. For a bluff body like circular cylinder S represents the frontal
projected area normal to U. For a cylinder of diameter d and length unity S becomes
(d x l) d. the characteristics dimension is the diameter d i.e. the width measured across the
cylinder normal to the flow. This is in contrast to the concept of wetted surface area used
for streamlined body like aerofoil section where the characteristics dimension used is the
length of the body along the direction of flow or the ‘chord’ of the aerofoil.
ds
UU
p
ddU
DCD .sin
2
1cos
2
1
1
2
1 222
or, dsCCd
C fpD .sincos1 (12.1)
where,
Cf = skin friction coefficient
CD = drag coefficient
147
Cp = pressure coefficient
This equation shows that the drag of cylinder may be found by measuring p and over
the surface. Now it is easy to measure the distribution of p over a cylinder merely by
drilling fine holes into its surface, but measurement of is a much more difficult task. For
the case of a circular cylinder, however, the contribution on drag from shear stress (the
skin friction drag) is found to be very much smaller than from pressure (boundary layer
pressure drag) and may safely be neglected.
Making this assumption and writing
ddRs .2/.. simplifies equation (12.1) to
ddCd
C pD .2cos1
2
0
2
0
.cos2
1dC p (12.2)
Using equation (12.2) CD can be calculated from the measured distribution over the
cylinder surface.
The circular cylinder model is provided with a fine pressure tapping at one point on its
surface. A protractor is attached to the cylinder and the pressure taping is connected to
the manometer. By rotating the cylinder about its axis to successive angular positions (00
, 50, 10
0, …, 360
0 ) the complete pressure distribution round the whole surface may be
recorded.
Pressure taping at three points are connected to the manometer for measuring inlet total
pressure Po(i.e. the pressure in the settling chamber), inlet static pressure p and static
pressure on the cylinder P (Fig. 12.3). The dynamic pressure of the oncoming flow q is
pPUq O
2
2
1 (12.3)
Manometer reads directly in terms of millibar, written mbar (1 mbar =10-3
bar = 100
N/m2 ). A table may now be prepared.
148
Figure 12.3: Pressure distribution on the surface of the cylinder
Table 12.1 : Pressure distribution on the surface of the cylinder
degrees
h0 h h (P0 - p)
= h0 - h
p - p
= h - h
Cp Cp Cos
0
5
10
-
360
Cp can be obtained simply as the ratio of (h - h ) and (h0 - h). Two graphs of Cp and
Cp cos as function of can now be plotted [Fig. 12.4(a), 12.4(b)]. CD May be obtained
from Fig. 12.4(b) by use of planimeter to measure the area beneath the curve. Usually
2
0
02.2cospc
149
From equation (12.2),
2
0
01.1.cos.2
1dcC pD
Figure 12.4 Variation of Cp and Cp cos with
To obtain Reynolds number of the flow, value of U is needed. This can be obtained
from equation (12.3) written in the form
pPU 02
Reynolds number is obtained from
dUR
12.3 Drag by Measuring Distribution in the Wake of the Cylinder :
The second method of determining the drag is based on the application of momentum
equation to the air flow. The flow of a fluid along a duct of width 2h past a cylindrical
body (Fig. 12.5) is considered. The velocity is U and the pressure is p at upstream.
Downstream of the cylinder the velocity the velocity is no longer uniform; let the velocity
be u at distance y from the duct center line. The pressure across the downstream section
is assumed to be uniform and has the value pe. It is convenient to refer to the space
bounded by the upstream section, downstream section and duct walls as the control
volume and the surface formed by these boundaries as the control surface.
150
Figure 12.5 Application of the Momentum equation
The forces in the direction acting on the fluid in the control, volume are, per unit length
of cylinder :-
at the upstream section 2h p
at the downstream section -2h pe
at the cylinder -D
It is to be noted that the force exerted by the cylinder on the fluid (which has a minus
sign) is equal and opposite to the force exerted by the fluid on the cylinder. Forces due to
shear stress on the walls of the duct and due to the fluid weight are neglected.
The momentum flux per unit width over the downstream section =
h
h
dyu 2.
The momentum flux per unit width over the upstream section = dyU
h
h
2.
Equating the net force in the x-direction to the momentum flux out of the control volume
h
h
h
h
e dyUdyuDhphp 22.22
151
Rearranging and making non-dimensional gives the result
dyU
u
dU
pp
d
h
dU
DC
h
h
e
D
2
2
22
12
2
1
2
2
1
The integral may also be made non-dimensional by the substitution y = h
So that dU
uhdy
U
uh
h
.1.1
1
1
2
2
2
2
And then final result is
dU
u
d
h
U
pp
d
hC e
D .12
2
1
21
1
2
2
2
(12.4)
Equation (12.4) provides a means to calculate CD from the pressure drop along the duct
and the velocity distribution in the wake. It is to be noted that the derivation does not
restrict the result to pressure drag only; the contributions of both pressure and skin
friction forces are contained in the force which comes into the momentum equation. The
ski friction drag on the wall also contributes to the momentum change and is therefore
included in D. it is also worth mentioning that equation (12.4) applies only to the case of
flow along a duct where the flow is confined between parallel walls.
diameter of the cylinder d = 48 mm = .048 m
half width of working section h = 50 mm = .05 m
152
Table 12.2 : Velocity traverse in wake.
y
(mm)
(= y/h)
P0 - p
2
2
1 U
N/m2
U
m/s
Pe - pe
2.2
1u
N/m2
u
m/s
u / U 221 Uu
0 0.0
2 0.04
4 0.08
50 1.0
0 0.0
-2 -0.04
-4 -0.08
-50 -1.0
Readings are recorded at successive values of the distance y from the center line, made
dimensionless by dividing by h in the next column. The third column indicates the Pitot
pressure Pe ( pe =0, atmospheric datum) and hence represents the local dynamic pressure
2.2
1u at a point in the exit section. This is also made dimensionless by dividing by
2
2
1U . Next two columns show u / U and 221 Uu which can be plotted as shown
in Fig. 12.6.
In stead of determining u and U individually to calculate 221 Uu ,an alternative
approach is to obtain u / U directly from the formula
pPpPUu ee 0
The drag coefficient may now be obtained by use of the curve 221 Uu in Fig. 12.6.
The area beneath the curve is usually found to be –0.074. CD may now be calculated from
equation (12.4).
153
Figure 12.6 Velocity traverse in the wake of the cylinder
12.4 Drag by Direct Weighing
Fig. 12.7 shows the essential components of the working section in which drag may be
measured by direct weighing. The body is mounted on an arm which extends through a
hole in one wall of the working section and which is supported on a flexible link so as to
form a balance. Now the drag experienced by the body in the air flow may be directly
measured by balancing the setup with weights in the scale pan. It is recommended that
exact balance is found by suitably trimming the wind speed rather than making small
adjustments to weights in the scale pan. At each wind speed the total pressure P0 and
static pressure p at inlet are recorded.
Diameter of the cylinder d = 12.5 mm = .0125 m
Frontal projected area of the cylinder S = .0125 .048 sq. m
154
Figure 12.7 Drag measurement by a mechanical balance
Table 12.3 : Drag by direct weighing
No. of
runs
P0
p
P0 - p
2?.2
1 U
N/ m2
D
gm
D N CD
SUD 2.2
1
1.
Three methods of drag measurement should yield almost identical values of CD.
However, it is to be noted that both wake traverse and direct weighing include pressure as
155
well as skin friction component whereas surface pressure measurement method takes
only pressure drag into account.
This particular procedure can be repeated for
1) flat plate
2) aerofoil section
3) square cylinder section.
156
Chapter 13
FLOW ABOUT AN AEROFOIL SECTION
13.1 Introduction :
The problem is to obtain pressure distribution on the surface of an aerofoil in two-
dimensional steady incompressible flow and derive the overall aerodynamic
characteristics of the aerofoil by integrating the pressure distribution.
Cartesian coordinates Oxz are taken with the origin coinciding the leading edge of the
aerofoil. The free stream velocity U is inclined at the angle of incidence to the Ox axis
as shown in Fig. 13.1.
Figure 13.1 Cartesian coordinate system
The equation of the aerofoil profile relative to the axis system is denoted by
z = fu(x) on upper surface
= f 1(x) on lower surface (13.1)
In the case of symmetrical profile fu(x) = - f 1(x) (13.2)
The perturbation velocity induced due to the presence of the body in the flow may be
assumed to be u and w in x and z directions respectively.
The total velocity components at any point on the surface of the aerofoil are
157
wUW
uUU
sin
cos
and the total velocity and pressure are
22 WUqt (13.3)
21 tp qC
13.2 Formulation of the Problem :
In theoretical analysis, the unknowns that are to be computed are the two components of
perturbation velocity (u, w) and pressure (p). These unknowns can, in principle, be
calculated from principles of conservation of mass and momentum.
For steady inviscid, incompressible flow, these equations are :
Eq. of continuity : 0z
W
x
U
(13.4)
Euler’s eq. of momentum : x
p
z
UW
x
UU
.
1
: z
p
z
WW
x
UU
1 (13.5)
The situation is simplified if the flow is considered to be potential (irrotational).
Condition of irrotationality is
0x
W
z
U
(13.6)
The velocity field, under the assumption of irrotational flow, can now be expressed as the
gradient of a scalar potential such that
z
Wx
U
,..... (13.7)
Using (13.4) and (13.70, Laplace’s equation is obtained.
02 (13.8)
The simplicity of potential flow derives from the fact that the velocity field is determined
from Laplace’s equation, eq. (13.8), which contains equation of continuity, eq. (13.4),
and condition of irrotationality, eq. (13.6). The equation of momentum, eq. (13.5), is not
used and the velocity is determined independent of pressure. Once the velocity field is
158
obtained, pressure can be known by integrating equation (13.5). Equation (13.5) can be
integrated to give one of the forms of Bernoulli’s equation. For steady incompressible
flow, Bernoulli’s equation becomes (in the simplest form)
2.2
1tqp Constant
Using this expression, pressure distribution Cp can be obtained as
2
2
1
2
1
U
q
U
ppC t
p
(13.9)
where p and U are the pressure and velocity at infinity.
Since onset flow U always satisfies the Laplace’s equation, eq. (13.8) can be further
simplified by assuming
where = total potential
= perturbation potential
= potential due to onset flow U
Since 02 , equation (13.8) can be written as
02 (13.10)
Laplace’s equation in perturbation potential is a second order linear differential equation
and requires two boundary conditions for solution, one on the body surface and other at
infinity.
The boundary conditions are :
i) flow at the body surface must be tangential, i.e., the normal component must
be zero,
qn = 0 on body surface (13.11)
ii) the perturbation velocities must tend to zero at infinity i.e,
u, w 0 at infinity (13.12)
The problem of calculating inviscid, incompressible, irrotational flow about a body
finally reduces to solving equation (13.10) subject to the boundary conditions, equations
(13.11) and (13.12).
159
13.3 Solutions :
Several methods have been developed for solution of the problem formulated above.
These methods may broadly be classified as
Solution of Laplace’s equation
Approximate solution Exact solution
Analytic Numerical
13.3.1 Exact Analytic Solution :
Exact analytic solution of Laplace’s equation can be obtained only for an extremely
limited class of simple body surface (e.g. flow past a half-body, Rankine oval, circular
body etc.). However, in two-dimensional flow problems, advantage can be taken of the
fact that in two dimensions the problem of solving Laplace’s equation can be replaced by
the problem of finding a suitable conformal transformation of the boundary. The use of
this technique has resulted in a number of useful potential flow solutions. For example,
using Joukowski’s transformation flow past a circular section can be mapped onto flow
past a flat plate, elliptical section or Joukowski aerofoil. Nevertheless, these solutions
comprise a restricted class.
13.3.2 Approximate Solution :
The solution of Laplace’s equation presents difficulty because of the nonlinear boundary
condition, equation (13.11). The governing equation, the Laplace’s equation is linear and
requires no simplification. The non-linearity enters in the problem through the boundary
condition. Because exact analytic solutions are limited and because exact numerical
methods are beyond the capability of hand calculation, approximate methods were
developed in the past. The earliest theory developed for solving this problem, the so-
called linearised theory, is based on a through –going linearisation of the boundary
condition. The boundary condition is linearised under the assumption of small
160
perturbations. These simplifying assumptions obviously place a limit on the accuracy of
the solution. Usual assumption are :
a) the body is slender with small local surface slope
b) the perturbation velocity components due to body are small with respect to onset
flow.
13.3.3 Exact Numerical Solution :
With the advent of high speed digital computer, exact numerical methods have become
feasible. These methods do not use any simplifying assumptions in the formulation and
are applicable to a variety of body surfaces. However, since the solution is achieved
numerically, numerical inaccuracies enter into the solution.
A distinction must be made between approximate solution and numerically exact
solution. In the later, the analytic formulation, including all equations, is exact and
numerical approximations are introduced for the purpose of numerical calculation. Exact
numerical methods have the property that the errors in the calculated solutions can be
made as small as desired, by sufficiently refining the numerical calculations. In contrast,
approximate solutions introduce analytic approximations into the formulation itself and
thus place a limit on the accuracy that can be obtained in a given case regardless of the
numerical procedure used.
13.4 Linearised Theory :
Boundary condition of flow tangency, eq. (13.11) can be written as
uU
wU
dx
dz
cos
sin (13.13)
In linearised theory (or thin aerofoil theory) the perturbation velocities u, w and angle of
incidence are assumed to be small, so that
cos = 1 , sin =
u,w << U
Equation (13.13) is then reduced to
U
w
U
wU
dx
dz
(13.14)
161
The equation (13.14) can now be split in two parts, for thickness and camber distribution.
Taking tC zzz
and tC www
Equation (13.14) becomes
dx
dz
U
w tt
thickness effect (13.15)
dx
dz
U
w CC
camber effect (13.16)
13.4.1 Thickness effect : Symmetrical aerofoil at zero incidence
This problem is solved by distributing source distribution on the chord of the aerofoil and
satisfying the boundary condition, equation (13.15) on the chord of the aerofoil.
dx
dzUx t
2)( (13.17)
and the perturbation velocity due to the source distribution placed along the chord is
C
t
t dx
ddzUxu
0
)0,(
(13.18)
Total velocity induced at any point qt is
tt uUq
C
t dx
ddzU
0
11
(13.19)
13.4.2 Camber (and incidence) effect : Cambered aerofoil at incidence
Effects due to camber is simulated by placing a vorticity distribution of strength (x) o the
chord of the aerofoil. The velocity induced wC at any point due to the distribution of
vorticity is
C
Cx
dw
02
.
From equation (13.16)
162
dx
dzU
x
d C
C
02
.
the solution of which can be expressed as
1
0 sin2
1cot2 nAAU n (13.20)
where cos12 cx
The first term 2
1cot0A represents the vorticity which occurs with a straight line flat
plate aerofoil. The coefficient of sine series An depends on the shape of the aerofoil.
The lift and pitching moment coefficient for the wing of span can now be obtained as
cU
dxU
cU
U
cU
LCL
222
2
1
.
2
1
2
1
22 1
0
AA (13.21)
2222
..
..
2
1
.
2
1cU
xdxU
cU
MC EL
ELM
22
210 AAA
(13.22)
where
0
0
1d
dx
dzA C
and
dndx
dzA C
n .cos2
0
(13.23)
The coefficient A0, An can be obtained by integration if the aerofoil surface is expressed
by the equation (13.1). For arbitrary aerofoil section, body surface is, however, described
by a set of points and hence these coefficients are to be obtained numerically.
The aerofoil section chosen for present study is NACA 0012 aerofoil. The equation of
this aerofoil contour is given by
4
5
3
4
2
3215 xaxaxaxaxatz
163
where t = 0.12, a1 = 0.2969, a2 = -0.126, a3 = -0.3516, a4 = 0.2843, a5 = -0.1015
The velocity distribution on this aerofoil of chord unity at zero incidence may be obtained
from equation (13.19) as
U
uUq t
t 1
where
2
5434451 43243
2
1
3
4{
1
1ln
2[
5xaxaaxaaa
x
x
x
at
U
ut
}]1
ln432 3
5
2
432x
xxaxaxaa
(13.24)
and the pressure distribution
2
1
U
qC t
p (13.25)
Lift and pitching moment coefficient can be obtained from equations (13.21), (13.22)
and (13.23) as
2LC (13.26)
2... ELMC (13.27)
13.5 Exact Numerical Method (Panel method) :
Panel method involves singularity distribution on the surface of the configuration and
unknown strength of the singularity distribution is obtained numerically by satisfying the
boundary condition of zero normal flow on the surface of the configuration. Singularities
used are sources, doubles, vortices, dipoles etc. These singularities are well known
solutions of Laplace’s equation, eq. (13.10). Since Laplace’s equation is linear, any
combination of the singularities can be added to provide new solutions. Also, all these
singularities automatically satisfy the condition at infinity, equation (13.12). So the whole
problem is reduced to obtaining the strength of the singularity distribution by satisfying
the boundary condition of zero normal flow, equation (13.11), on the boundary surface.
Satisfaction of boundary condition on boundary surface results in an integral equation
over the surface.
164
For bodies of arbitrary shapes, this integral equation can not be solved numerically. The
numerical solution is the central problem of all panel methods. To solve this integral
equation numerically, the body must first be approximated by a number of flat panels.
The body is specified to the computer by a set of point which lie exactly on the body
surface. The input order is such that the body surface is approximated by a number of
straight line panels (Fig. 13.2). On each panel, one point (normally taken to be the mid-
point) is selected where the boundary condition (this integral equation) is satisfied.
Satisfying this integral equation at all mid-points results in a set of linear algebraic
equations for the values of source density on the panels. Once these are solved, flow
velocities can be computed at the mid-points.
Figure 13.2 Approximation of the body surface into number of panels
Numerous versions of panel methods have been developed using various combinations of
singularities distributed inside and/or on the surface of the body. Some of the versions for
two-dimensional flow are :
a) Source-only singularity model
b) Vorticity-only singularity model
c) Composite singularity model
d) Internal singularity model
In what follows a brief description of the source-only singularity model developed by A.
M. O. Smith and J. L. Hess is described.
In this method the aerofoil surface is discretised into N number of flat panels (Fig. 13.3)
on each of which are placed uniform source and uniform vorticity distribution. The
165
strength of the source distribution is assumed to be constant for each panel but varying
from panel to panel. The strength of the vorticity distribution is assumed to be constant
for all panels. The strength of the singularity distribution is obtained by satisfying the
boundary condition of zero normal flow at the mid-point of all N panels resulting in n
number of linear algebraic equations.
Figure 13.3 Discretised model
Boundary condition of flow tangency can be written, as before, as
xfuU
wU
dx
dz
cos
sin (13.13)
or, sincos UUxfxfuw
Normalising with respect to c and U gives
sincos xfxfuw
where j = 1, 2, ……….,N (13.28)
On each panel is placed a uniform source distribution i, i = 1,….,N which varies from
panel and a uniform vorticity distribution which is the same for all panels. Thus, there are
(N+1) unknown singularity strengths. There are n linear equations arising from satisfying
equation (13.14) at N panel mid-points together with the Kutta condition, Usually taken
to be equal velocity (pressure) at the mid-point of trailing edge panels 1 and N.
166
Normalised perturbation velocities due to distribution of sources and vorticity can be
expressed as
N
i
ji
N
i
ijij BAu11
.
(13.29)
N
i
N
i
jiijij DCw1 1
.
where uj, wj are the normalized perturbation velocity components at the mid-point of j-th
panel induced by the distribution of sources and vorticity on the aerofoil surface and Aji
, B ji are the appropriate influence coefficients. These coefficients Aji , B ji depend only on
the coordinates of the i-th and j-th panel.
Using equations (13.14) and (13.15), the boundary condition of zero normal flow can be
expressed as
sincos11
xfBxfDAxfC j
N
i
jijjii
N
i
jijji
where j = 1, 2,…….., N (13.30)
The solution for the unknown variable i and is obtained from the n linear equations,
equation (13.16) together the Kutta condition of equal pressure at the mid-points of panel
1 and N. once the solution is obtained, the perturbation velocity components are obtained
from equation (13.15).
The total velocity is given by
22sincos jjj wuq (13.31)
and the pressure coefficient
21 jpj qC (13.32)
Pressure distribution, obtained by this method, is given in the tables for comparison with
experimental values for NACA 0012 aerofoil model.
13.6 Overall Aerodynamic Characteristics :
Once the pressure distribution are obtained, following quantities can be calculated :
1) Lift, drag and pitching moment coefficients
167
2) Location of aerodynamic center
3) Location of center of pressure.
13.6.1 Lift, Drag and Pitching Moment Coefficients :
Fig. 13.4 represents an aerofoil section at an incidence to the fluid stream, which is
assumed to be moving from left to right at a speed of U. Through the nose of the
aerofoil are drawn axes Ox and Oz parallel and perpendicular to the chord line
respectively. The chord of the aerofoil denoted by c. the ordinates of the highest and
lowest points of the section are z2 and z1 respectively.
Figure 13.4 Aerofoil section at incidence
Taking a slice of the aerofoil of unit spanwise length, the forces acting on a small element
of length s, of the surface may be considered. The normal forces on the element is
composed of ps inwards and artificial ps outward leaving a net inwards force (p -
p)s. This force may be resolved into components z and x acting parallel to the Oz
and Ox axis respectively. Then
cossppz
xpp per unit span (since xs cos )
for an element on upper surface. For an element on lower surface it becomes
xppz per unit span.
168
If this is low integrated with respect to x between the limits x = 0 and x = c, the total
force Z is obtained as
dxppdxppZ
CC
00
upper surface lower surface
Using subscripts u and 1 for the upper and lower surface respectively
C
udxppppZ
0
1 (13.33)
Thus, given the variation of pressure p along the chord of the aerofoil it is possible to
calculate the lift. It is to be noted that a fictitious value of pressure inside the aerofoil is
assumed (p). It is assumed to be p for the purpose of non-dimensionalising. The actual
value is quite immaterial.
Equation (13.33) is easily put into coefficient form as follows.
Defining CZ by CZ =
SU
Z
..2
1 2
Considering unit span, the area S is equal to c and therefore
CZ =
cU
Z
..2
1 2
= dxpppp
cU
c
u.
..2
1
1
0
12
Now 1/ c(dx) = d(x/ c). This gives
CZ =
1
0
1 )/(. cxdCC ppu
( Since
pC
U
pp
2..2
1
by definition )
=
1
0
)/(. cxdC p (13.34)
A similar argument may be used to give the following relations
169
δX = sin..)( spp
zs .sin..
leading finally to CX = cZ
cZ
p czdC
/
/
2
1
)/(. (13.35)
The pitching moment may also be calculated from the pressure distribution. For
simplicity, it will be found about the origin of Ox and Oz axes.
δZ = xpppp u ..)(1 per unit span and therefore the contribution to the
pitching moment due to this element of Z-force is
xdppppMuEL 1..
when the total pitching moment due to z-force is
cxdcxCCC ppuELMZ
1
0
1..
1
0
cxdcxC p (13.36)
since, 222
2
1
2
1cU
M
cU
MCM
and S = c.
Similarly, the contribution to CM due to the X-force may be obtained as
czdczCC
CZ
CZ
pELMX 2
1
... (13.37)
Four graphs can now be plotted giving Cp Vs x/c, Cp vs z/c, Cp.(x/c) vs x/c and
Cp.(z/c) vs z/c and the integrals given by eqs. (13.20), (13.21), (13.22) and (13.23) may
be graphically evaluated.
The force coefficients CX and CZ are parallel and perpendicular to the chord line, whereas
the more usual coefficients CL and CD are referred to the air direction. The conversion
from one pair to the other may performed by reference to Fig. 13.5, in which CR, the
coefficient of resultant aerodynamic force, acts at an angle to CZ. CR is resultant to both
of CX and CZ , and of CL and CD, and therefore from Fig. 13.5.
170
Figure 13.5 Conversion of axes
sinsincoscos)cos( RRRL CCCC
Now, ZR CC cos and XR CC sin
Therefore,
sincos XZL CCC (13.38)
Similarly,
)sin( RD CC
cossin XZ CC (13.39)
The total pitching moment coefficient is
........ ELMXELMZELM CCC (13.40)
The experimental values of CL and CM L.E. can be compared with the solutions of
linearised theory, e.g., equations (13. ) and (13. ). The agreement between two results
is expected to be good despite the approximations of linearised theory. This is because in
deriving equations (13. ) and (13. ), both the effects due thickness and viscosity are
neglected and these effects tend to cancel each other. The thickness effect tends to
increase the CL while the effects due to viscosity tends to decrease the CL. In wind tunnel
testing, both the effects are present and they cancel each other. Lastly, it is to be noted
that CD can not be obtained by potential flow theory.
171
13.6.2 Location of Aerodynamic centre :
Aerodynamic centre is, by definition, the point on the chord of an aerofoil for which CM
is virtually constant, independent of the lift coefficient, i.e.,
0... caM
L
CdC
d (13.41)
Taking moment about the leading edge gives
accaEL xDLMM sincos..
or, sincos.... DLaccMaEML CCcxCC
Since CD is usually much less than CL
cxCCC acLcaMEML .... (assuming cos = 1)
Differentiating with respect to CL gives
c
xC
dC
dC
dC
d ac
caM
L
ELM
L
....
By definition of aerodynamic center, ,0... L
caM
dC
dC
Therefore, ... ELm
L
ac CdC
d
c
x (13.42)
Aerodynamic center can now be calculated from equation (13.28) as the slope of CM L.E.
vs CL curve.
13.6.3 Location of Centre of Pressure :
The center of pressure is, by definition, the point on the aerofoil section where pitching
moment is zero, i.e., the aerodynamic forces acting on the aerofoil section may be
represented by the lift and drag alone acting on the point.
Unlike aerodynamic center (which is a fixed point lying within the profile), the center of
pressure moves with change of lift coefficient and is not necessarily within the aerofoil
profile.
From Fig. 13.6 ML.E can be written as
cpEL xDLM sincos..
172
Figure 13.6 Location of center of pressure
Dividing by 22
2
1cU gives
cxCCC cpDLELM sincos...
Assuming CD << CL and cos = 1,
cxCC cpLELM ...
or, L
ELMcp
C
C
c
x... (13.43)
13.7 Wind Tunnel Testing :
The model is of span 61 cm and it extends from wall to wall (so that the flow is two-
dimensional). At the center section, pressure holes are made at 29 Points (Fig. 13.7)
round the aerofoil contour. These are connected to the multi-tube manometer.
173
Figure 13.7 Experimental model
Tables can now be made for Cp distribution for multiple number of angles of incidences,
50, 10
0, 15
0 and 20
0. This range will cover the stall. Tables, for 0
0 and 10
0 incidences,
giving results of exact numerical method, are shown
174
Table 13.1 : Flow at 00 incidence
Tapping
points
x/c z/c hLS
cm
U
m/s
h
Cp
Experime-
ntal
Cp
Linearised
theory
Cp
Exact
theory
1 0.0 1.0
2,27 .015 -.054
3,28 .03 -.271
4,27 6 -.375
5,26 .09 -.406
6,25 .15 -.41
7,24 .2 -.383
8,23 .3 -.340
9,22 .4 -.291
10,21 .6 -.160
11,20 .8 -.049
12,19 .9 +.031
13,18 .95 +.101
14,17 .975 .161
15,16 .99 .28
175
Table 13.2 : Flow at 100 incidence :
Tapping
points
x/c z/c hLS
cm
U
m/s
h
Cp
Experime-ntal
Cp-Linearised
theory
Cp Exact
theory
1 .0 -5.05
2 .015 -3.65
3 .03 -2.70
4 .06 -2.21
5 .09 -1.74
6 .15 -1.63
7 .2 -1.09
8 .3 -.75
9 .4 -.48
10 .6 -.182
11 .8 -.005
12 .9 +.007
13 .95 +.152
14 .975 +.652
15 .99 +.68
16 .99 +.22
17 .975 +.19
18 .95 +.153
19 .9 +.142
20 .8 +.16
21 .6 +.22
22 .4 +.305
23 .3 +.402
24 .2 +.504
25 .15 +.722
26 .09 +.86
27 .06 +.93
28 .03 +.810
29 .015 +.810
176
Graphs can now be plotted for Cp vs x/c for all values of angles of incidence showing
comparison of numerical solution by panel method with experimental results. Difference
is again due to effects of viscosity.
Overall aerodynamic characteristics can now be calculated using linearised theory and
experimental values and shown as in Table 13.3.
Table 13.3 : Overall aerodynamic characteristics
Linearised Theory Experimental Values
CL CM1/4 C xac xcp CL CM1/4 C CD xac xcp
00
50
100
150
200
177
Chapter 14
MEASUREMENT OF LAMINAR BOUNDARY LAYER
14.1 Introduction :
It is well established by experiment that when a fluid moves over a solid surface there is
no slip at the surface. The fluid in immediate contact with the surface is at rest relative to
the surface. The relative velocity increases from zero at the surface to the velocity in the
free stream through a layer of fluid which is called the boundary layer.
Figure 14.1 General characteristics of boundary layer over a flat plate
Steady flow over a smooth flat plate is shown in Fig. 14.1 where the streaming velocity
U is constant over the length of plate. It is found that the thickness of the boundary layer
grows along the length of the plate as indicated on the figure. The motion in the boundary
layer is laminar at the start, but if the plate is sufficiently long, a transition to turbulence
is observed. This transition is produced by small disturbances which, beyond a certain
distance grow rapidly and merge to produce the apparently random fluctuations of
178
velocity which are characteristics of turbulent motions. The parameter which
characterizes the position of transition is the Reynolds number Rx based on distance x
from the leading edge:
xURX
(14.1)
The nature of the process of transition depends on factors such as turbulence in the free
stream and surface roughness of the boundary. It is not possible to give a single value of
Rx at which transition will occur, but is usually found in the range 1 105 to 5 10
5.
The concept of boundary layer is a mathematical one, which divides the flow region in
two parts – one outside the boundary layer and other inside the boundary layer. The rate
of change of velocity with distance from the surface ( u/y ) is large in the boundary
layer, outside it is small. The viscous stress which is related to the velocity gradient
( = u/y ) is therefore large only in the boundary layer, elsewhere it is small.
Consequently, flow inside the boundary layer is highly viscous whereas the flow outside
it can be treated as inviscid. The governing equation of flow outside the boundary layer is
therefore Laplace’s equation while that for inside the boundary layer is the Navier- stokes
equation.
However in studying the boundary layer flow, Navier Stokes equation is not solved.
Using an ‘order of magnitude’ study Prandtl has simplified the equation to what is known
as ‘Boundary Layer Equation’. Either the boundary layer equation or an integrated form
of this equation, known as von karman’s Momentum integral equation is used for
obtaining boundary layer characteristics.
In boundary layer calculation interest is rarely in the calculation of velocity profiles or the
thickness of boundary layer. The interest is often limited to calculating certain boundary
layer parameters e.g., displacements thickness , momentum thickness , shape factor H,
local and overall skin friction coefficients ( Cf and CF ).
14.2 Boundary Layer Parameters :
14.2.1 Displacement thickness () :
179
A little consideration will show that the boundary layer thickness , shown in Fig. 14.1 as
the thickness where the velocity reaches the free stream value is not a entirely satisfactory
concept. The velocity in the boundary layer increases towards U in an asymptotic
manner, so that distance y at which the velocity may be considered to have reached U
will depend upon the accuracy of the measurement. A much more useful concept of
thickness is the so called displacement thickness . This is defined as the thickness by
which fluid outside the layer is displaced away from the boundary by the existence of the
layer, as indicated schematically in Fig. 14.2, by the stream line approaching B. In Fig.
14.2 the distribution of velocity u within the layer is shown as a function of distance y
from the boundary as the curve OA. If there were no boundary layer the free stream
velocity U would persist right down to the boundary as shown by the line CA. The
reduction volume flow rate (per unit width normal to the diagram) due to reduction of
velocity in the boundary layer is therefore
h
dyuUQ0
(14.2)
Figure 14.2 Velocity distribution and displacement thickness of boundary layer
Which is the shaded area OAC in the figure, the dimension h being chosen so that u = U
for any value of y greater than h. if the volume flow rate is now considered to be restored
by displacement of the streamline at AA away from the surface to a position BB through
a distance , the volume flow rate between AA and BB is also Q and this is seen to be
180
Q = U (14.3)
In the other words, flow over a solid surface having a boundary layer of thickness , is
equivalent to flow with no boundary layer over a solid surface of thickness (Fig. 14.3).
Equating the results of equation (14.2) and (14.3) gives
hh
dyU
udyuU
U00
11
Figure 14.3 Mass defects due to boundary layer
Now h is any arbitrary value which satisfies the condition
u = U
or, 1 – u/U = 0
for all values of y greater than h. the value of h may therefore be increased indefinitely
without affecting the value of integral. So h may be allowed to increase towards infinity,
viz h and the result obtained is
dyUu
0
1 (14.4)
in he practical measurement of from a measured velocity distribution the infinite upper
limit presents no difficulty.
14.2.2 Other parameters :
181
Other parameters are developed by considering momentum effects within the boundary
layer. A control volume of length x, height h (greater than boundary layer thickness )
and unit thickness normal to the plane of diagrams as shown in Fig. 14.4 is considered.
Figure 14.4 Mass and momentum flux in boundary layer
The rate of mass inflow is m at the left hand and the rate of mass outflow at the right
hand end is xdxmdm . Consideration of continuity then shows the outflow through
the upper surface to be - xdxmd . The momentum equation may now be written.
The net rate of flux of x-component of momentum M from the control volume is the
sum of
xdxMdM at the right hand end
- M at the left hand end
and xdxmdU at the upper surface
if the surface shear stress is w acting in the direction shown in the diagram, the
momentum equation is then
xdxMdUMxdxMdMxw
This may be simplified to give
182
dxMddxmdUw
or, MmUdx
dw
(14.5)
Now
h
udym0
and
h
dyuM0
2
Substituting in Eq. (14.5) gives
h
w dyuuUdx
d
0
2
or,
h
w dyU
u
U
u
dx
dU
0
2 1
Since u = U for all values of y greater than h, the arbitrary upper limit may be replaced
by infinity, giving
0
2 1 dyU
u
U
u
dx
dUw (14.6)
It is convenient to express w in dimensions form of local skin friction coefficient Cf as
2
2
1
U
C wf
(14.7)
Eq. (14.6) then becomes
0
12 dyU
u
U
u
dx
dC f (14.8)
The momentum thickness of he boundary layer, , may be defined, based on the
momentum defect in the boundary layer as
dyU
u
U
u
0
1 (14.9)
Equation (14.8) then gives local skin friction coefficient as
dxdc f 2 (14.10)
The total skin friction force per unit width of a plate of length L is
183
L
wf dxD0
L
f dxcU0
2
2
1 [from eq. (14.7) ]
dxdx
dU
L
0
2 22
1
Taking = 0 at x = 0 and writing L as the momentum thickness at distance L from the
leading edge Df can be obtained as
Lf UD 22
1 2
(14.11)
The skin friction force Df is now written in terms of dimensionless overall skin friction
coefficient CF where
LU
DC
f
F2
2
1
L
L2 (14.12)
This equation gives the overall skin friction coefficient on a flat plate very simply in
terms of the momentum thickness at the trailing edge and the length of the plate.
It is frequently useful to refer to the ratio of displacement thickness to momentum
thickness and this is called the shape factor H :
H = / (14.13)
All the boundary layer parameters described above are to be measured and compared
with the theoretical solutions for three cases :
a) Laminar boundary layer in zero pressure gradient
b) Laminar boundary layer in favourable pressure gradient
c) Laminar boundary layer in adverse pressure gradient
14.3 Laminar Boundary Layer in Zero Pressure Gradient
14.3.1 Theoretical calculation
184
All the boundary layer parameters can be obtained from the solution of momentum
Integral equation which is given as
2
21
U
Hdx
dU
Udx
d w
(14.14)
For the case of a flat plate at zero incidence 0 dxUd and the momentum integral
equation is simply
2
udx
d w
(14.15)
An approximate calculation of boundary layer equations is now possible by assuming any
velocity profile (linear, cubic etc.) in the boundary layer. Exact solution for this problem
using the condition of similarity has been obtained by Blasius. The different solutions are
given below in tabular form :
Table 14.1 : Solutions of boundary layer characteristics
Form of
u / U
/ x / x / x Cf / x CF / 2 H
Linear
y / XR
464.3
XR
732.1
XR
577.0
XR
577.0
R
155.1
3.00
Cubic
3
2
1
2
3
yy
XR
64.4
XR
740.1
XR
646.0
XR
646.0
R
292.1
2.70
Exact
Blasius
solution
-- XR
721.1
XR
664.0
XR
644.0
R
328.1
2.59
14.3.2 Wind Tunnel Testing :
Fig. 14.5 shows the arrangement of test section attached to the outlet of the contraction of
the air flow bench. A flat plate is placed at mid height in the section with a sharpened
edge facing the oncoming flow. One side of the plate is smooth and other is rough so that
by turning plate over, results may be obtained on both types of surface. The length of the
flat plate is 0.265 m and boundary layer can be studied at four stations.
185
A fine pitot tube may be traversed through the boundary layer at a particular station (x) of
the plate. The end of the tube is flattened to that it presents a narrow slit opening to the
flow. The traversing mechanism is spring loaded to prevent backlash and a micrometer
reading is used to indicate the displacement of the pitot tube. The thickness of tube (2t) is
0.40 mm. The setting of micrometer can be adjusted by assuming displacement tube 0.20
mm when it touches the surface.
Figure 14.5 Arrangement of test section
An experimental study can now be undertaken to obtain the edge of the boundary layer
on a flat plate and measure the boundary layer parameters experimentally.
To obtain the edge of boundary layer, the pitot tube is set at about 10 mm distance from
the surface. The difference in total pressure P0 and the static pressure p should be of the
order of 61 N/m2, which will give a uniform speed ( U ) of about 10 m/s. At this speed,
the boundary layer will (possibly) remain laminar throughout. Reading of the total
pressure P measured by the Pitot tube are then observed over a range of setting of
micrometer as the tube is traversed towards the plate. At first the reading will be
substantially constant (P0), indicating that the traverse has started in the free stream; if
186
this is not the case, measurement may be started with an initial setting further from the
plate. As the pitot tube readings begins to fall ( P< P0 ) indicating that the pitot tube has
entered the boundary layer, the micrometer setting is recorded. This setting indicates of
the boundary. A table can now be set for boundary layer thickness at four stations.
Table 14.2: Boundary layer thickness
No. Station
(x) m
P0
N/m2
p
N/m2
2
2
1U
N/m2
U
m/s
RX Micrometer
Reading
mm
1.
2.
3.
4. .265
Figure 14.6 Development of boundary layer in downstream direction
Edge of the boundary layer can now be obtained by plotting x vs (Fig. 14.6). To
measure the boundary layer parameters, attempt may now be made to find out the
velocity profile, u(y) within the boundary layer at each of these four stations. This can be
found by traversing the pitot tube through the boundary layer till it touches the surface.
187
Since the pitot-tube has certain thickness (2t = 0.40 mm), it can not measure pressure (P)
exactly at the surface but at 0.2 mm away from the surface.
Values of u / U are found from
pP
pP
U
u
0
where P = total pressure measured by the pitot-tube
P0 = total pressure in the air – box
p = static pressure in the air – box
Table 14.3 : Velocity profile (Stations : 1, 2, 3, 4 )
Station
(x) m
P0 p U
pP
pP
0
Micrometer
reading mm
(y)
mm
u / U 1- u / U
Graphs can now be plotted for u / U vs y and u / U (1- u / U ) vs y as shown in Fig.
14.7.
188
Figure 14.7 Velocity profile in boundary layer
The appropriate areas under the curves can be measured by planimeter, which will give
and and in turn H (
/ ) and Cf . These experimental results (,
, , H, Cf) can
now be compared (as shown in Table 14.4) with the theoretical solutions given in the
Table 14.1. This procedure will be repeated for other stations. Measurement of at the
last station (x = .265 m) will yield overall skin friction CF (= 2L / L).
Table 14.4 : Comparison of boundary layer parameters (Stations : 1, 2, 3, 4)
Cf CF /2 H
Linear
Cubic
Experimental
The entire procedure can now be repeated for the rough surface (by turning over the
smooth surface) and effect of roughness on boundary layer growth may be studied.
Roughness of plate serves to increase the rate of growth of boundary layer.
14.4 Laminar Boundary Layer in Favourable Pressure Gradient :
14.4.1 Theoretical Calculations :
189
The preceding section is related to boundary layer development along a smooth flat plate
with uniform flow in the free stream, i.e., in conditions of zero pressure gradient along
the plate. If the free stream is accelerating, substantial changes takes place in the
boundary layer development.
If the flow accelerates, by Bernoulli’s theory, pressure falls in the direction of flow. From
Bernoulli’s theory
2
02
1 UpP
Differentiating with respect to x gives
020
dx
dUU
dx
dp
dx
dP
or, dx
dUU
dx
dP
2 (14.16)
The failing pressure in the direction of flow is considered to be favourable since it does
not try to obstruct the motion (and the flow never separates). However, the boundary
layer under this favourable (negative) pressure gradient grows less rapidly than in zero
pressure gradient and transition to turbulence is inhibited.
The governing equation of boundary layer flow is the so-called momentum integral
equation expressed in the form
2
1
1
1
21
UH
dx
dU
Udx
d w
(14.17)
In stead of U, U1 is used to show that free stream velocity changes with x.
For the case of zero pressure gradient the second term vanishes (since d U1/dx is zero).
For this case exact solution (Blasius solution) is available. However no exact solution
exists for the case of non-zero pressure gradient. This has led to the development of
approximate methods. Usually such approximate methods have been developed with the
limited objective of predicting reliable over-all characteristics of the boundary layer, e.g.
momentum thickness, displacement thickness etc., rather than details of boundary layer
flow (i.e. velocity profile in the boundary layer). The momentum integral equation,
equation (14.17) provides the basis for such methods. However, assumptions are made in
solving this equation. All these methods (i.e., Pohlhausen’s method, Thwaite’s method,
190
Young’s method) are based on the simplifying assumption that laminar boundary layer
velocity profiles can be regarded as a uni-parametric family. The assumption is
reasonably valid when external pressure gradient is favourable or weakly adverse but
when the gradient is strongly adverse this assumption losses validity and these methods
fail.
For favourable or a weak adverse pressure gradient, Thwaites method predicts the
momentum thickness as
dxUU
X
X
X
1
0
5
1
1
6
1
1
2 45.0 (14.18)
The suffix x1 indicates that the quantities are evaluated at x = x1.
14.4.2 Wind Tunnel Testing :
In the experimental set-up, liners can be placed on the walls of the working section to
produce the accelerating flow (Fig. 14.8) along the length of the plate. The thickness of
the liners is 2.5 mm at the beginning and 15 mm at the end and the length is about 22 cm.
The experimental determination of the boundary layer parameters can now be made with
the liners fitted to get the characteristics of accelerating flow. The procedure is the same
as described before and is not presented here.
191
Figure 14.8 Accelerating flow by use of liners
The boundary layer parameters determined experimentally can be compared with the
solutions obtained by Thwaites method. To obtain the parameters by Thwaits method it is
necessary to calculate U1. This can be done in the following way.
If the thickness of linear is assumed to linearly increasing, at any station x, the thickness
is
xt
220
5.2155.2
The liners reduce the effective span by 2t.
The area of the clear jet is 58 mm 100 mm and the area of effective jet with liners
attached is
58 mm
x
220
5.2155.22100
From equation of continuity, the free stream speed U1 then can be obtained as
U1 =
xU
1136.095
100 (14.19)
Where x is in mm.
192
Inserting equation (14.19) in equation (14.18), the momentum thickness can now be
obtained by Thwaites method at any station x1 for the accelerating fluid.
14.5 Laminar Boundary Layer in Adverse Pressure Gradient :
For a decelerating free stream the reverse effects are observed. The boundary layer grows
more rapidly and the shape factor (H) increases in the downstream direction. The
pressure rises in the direction of flow and this pressure rise tends to retard the fluid in the
boundary layer more severely than in the main stream since it (boundary layer) is moving
less quickly and the flow separates. Due to the same reason (i.e. for having less energy)
laminar boundary layer separates earlier than the turbulent boundary layer. In turbulent
boundary layer there is mixing of flow (and energy) between the boundary layer and the
free stream. Energy diffuses from the free stream through the outer part of the boundary
layer down towards the surface to maintain the forward movement against the rising
pressure. However, if the pressure gradients are sufficiently steep, the diffusion is
insufficient to sustain the forward movement and the flow along the surface reverses
forcing the main stream to separate. It is this separation, or stall as it is sometimes called
which leads to the main component of drag on bluff bodies and to the collapse of lift
force on an aerofoil when the angle of incidence is very high.
In the experimental set-up, decelerating flow can be produced by simply reversing the
liners. The procedure of measuring the boundary layer parameters is the same as
described before for zero pressure gradient case and is not presented here.
Finally, the entire experimental results can be presented for various cases as shown in
Table 14.5.
193
Table 14.5 : Boundary layer Characteristics
Smooth Plate Rough Plate
Zero
pressure
gradient
Favourable
pressure
gradient
Adverse
pressure
gradient
Zero
pressure
gradient
Favourable
pressure
gradient
Adverse
pressure
gradient
H
194
Chapter 15
MEASUREMENT OF TURBULENT BOUNDARY LAYER
15.1 Introduction :
It is stated earlier that the case with which the laminar flow in a boundary layer changes
to turbulence is dependent on Reynolds number. However, proper understanding of the
physical mechanism of transition to turbulent flow is not yet achieved despite much
efforts devoted to this problem. It would be seen from experimental evidence that the
onset of turbulence requires (firstly) the presence of disturbances of some definite size
and (secondly) the Reynolds number must be sufficiently high. In the absence of
extraneous sources of such disturbances, e.g. external turbulence or surface imperfections
which shed eddies into the flow, they may be the result of amplification of small
disturbances in conditions of instability in the laminar boundary layer. Where conditions
are such as to favour instability e.g. adverse pressure gradient, surface imperfections,
external turbulence, high Reynolds number etc., transition take place earlier and
transition region is also short. Likewise, in favourable pressure e gradient etc., either
transition will not take place at all or else will take place gradually.
15.2 Structure of Turbulent Boundary Layer :
The structure of turbulent boundary layer is as shown in Fig. 15.1. The turbulent
boundary layer can be divided into three regions : outer region, inner region and the
laminar sub-layer. Over the outer part of the layer there is a region predominantly large
scale eddies. In this region, the turbulence is not continuous but intermittent. This region
extends from 1.2 down to 0.46, where is the boundary layer thickness. The
intermittent nature of eddy formation in this region and the large size of eddies, which are
of the order of , result in the instantaneous edge of the boundary layer.
195
Figure 15.1 Structure of turbulence boundary layer
This region, referred to as outer region is one of the relatively uniform mean velocity. It
is also a region of relatively low shear stress.
Below this region of large scale intermittent eddies is the fully turbulent region (the inner
region) extending from about 0.4 down to the laminar sublayer. In this region the shear
stress is dominated by the turbulent contribution (Reynolds stress or shear stress
vu ), which is much greater than the viscous stress (u /y).
Below this inner region is the laminar sub-layer. This laminar sub-layer is a very narrow
region of flow adjacent to the wall. Turbulent fluctuations and hence Reynolds stress
become small in this region. The dominant shear stress is purely viscous one, (u /y),
which is constant in the sub-layer.
The governing equation of turbulent boundary layer flow is the momentum integral
equation and for the case of flow over a flat plate at zero incidence (i.e. zero pressure
gradient) this equation can be written as
2
Udx
d W
(15.1)
for the case of laminar boundary layer flow, this equation was solved by assuming a
velocity profile (linear, cubic etc.) in the boundary layer. For the case of turbulent
196
boundary layer, two types of velocity profiles are usually assumed, the so-called ‘Log’
law relations and ‘power’ law relations.
15.3 Log Law Relations :
According to this law, the velocity profile in the boundary layer is described by a
logarithmic expression. Three expressions are used for the three regions of the boundary
layer structure.
The velocity profile in the laminar sub-layer is given by
yu
u
u (15.2)
It is suggested that the laminar sub-layer is in the region defined by
8.71 yu (the edge of the laminar sub-layer is at y = y1)
For inner region, the inner velocity law has been developed by Prandtl based on mixing
length theory. The inner velocity law can be written as
Byu
Au
u
ln (15.3)
where A and B are constants given by A = 2.5 and B = 5.5 and u is the friction velocity
defined as
Wu
The above equation (15.3) is found to be reasonably valid for the outer region also. In
such case, to avoid any difficulty that may arise from u tending to minus infinity as y
tends to zero, a slightly modified form is used for the outer region
yuCA
u
u1ln (15.4)
where C is a constant (C 9.0).
However, in view of the wide applicability of equation (15.3) von Karman has suggested
that full boundary layer profile (except for laminar sub-layer) be given as
yyu
Au
u
10log
197
A simplified form of the above expression is obtained by simply assuming a turbulent
boundary layer profile of the logarithmic type
Byu
Au
u
10log (15.5)
where A and B are constants (A = 2.5ln10, B = 5.5).
For purpose of comparison with experimental data, the velocity distribution in the
turbulent boundary layer may be assumed as
By
Au
u
10log (15.6)
15.4 Power Law Relations :
It is experimentally found that the velocity distribution measured in pipes can be given by
nayU
u 1
1
(15.7)
where U1 is velocity on the pipe axis, y is the distance from the wall, a is the radius and n
is the number which depends on Reynolds number.
For turbulent boundary layer over a flat plate, a power law of the type represented by eq.
(15.7) can be made consistent with the inner region velocity law, eq. (15.2) by writing
nyu
Cu
u1
1
(15.8)
where C1 is a constant, whose value, like that of n, depends on the Reynolds number. The
following table lists the values of C1 and n for various values of Reynolds number for
flow in pipes (also valid for flow over a flat plate) :
n C1 Re
7 8.74 5.5 104
8 9.71 2.4 105
9 10.60 6.3 105
10 11.50 1.6 105
198
It may be easily verified that for a power law with index 1/n the displacement and
momentum thickness of the boundary layer on a flat plate are given by
n
nH
nn
n
n
2
21
1
1
(15.9)
Like the velocity distribution law, eq. (15.3), these power laws do not fit the laminar sub-
layer nor do they satisfy the condition that u/y = 0 at the outer edge of boundary layer.
Using eq. (15.8), velocity at the edge can be obtained as
nU
Cu
U1
1
(15.10)
Eqs. (15.8) and (15.10) gives
1
1
11
n
n
n UC
u
U
(15.11)
The momentum integral equation, eq. (15.1) can now be solved using the above velocity
distribution.
The momentum integral equation is
2
2
2
U
u
Udx
d W
Substituting for from eq. (15.9) and U u from eq. (15.10) in the momentum integral
equation gives an equation in .
1
2
1
2
121
n
n
n UC
dx
d
nn
n
On integration (taking = 0 at x = 0)
32
2
n
XRCx
(15.12)
199
where 3
1
12
12
32
n
n
nnC
n
nnC
The local skin friction coefficient is
)3(2
322
22
n
X
w
f RCU
u
Uc
(15.13)
where )1(2
2
)1(2
13 2
nnn
CCC
3222 nnnC
The momentum thickness is given by
)3(2
4
n
XRCx
(15.14)
where 24
21C
nn
nC
The displacement thickness is given by
)3(2
4
2
n
XRCn
n
xH
(15.15)
The overall skin friction coefficient for both sides of the flat is
c
n
fF RCn
ndxCcC
0
32
31
312
3
2
42
nRC (15.16)
where
cUR
For n = 7 and C1 = 8.74 this gives
5/1
5/1
51
0722.0
0576.0
37.0
RC
RC
Rx
F
Xf
X
(15.17)
15.5 Wind Tunnel Testing :
The first apart of the experiment is to check whether the experimental observation fits the
expression for velocity profile, eq. (15.6) and then to obtain values of A and B. This work
should be carried at a uniform speed (U) of about 30 m/s. Velocity profile at any station
200
(except possibly the first station where the flow may be laminar) by simply traversing the
pitot-tube. The details of the procedure is outlined in Experiment. No. 14 and are not
repeated here. A table can be made giving u/ U against y. is, as before, taken as that
value of y for which pitot pressure P equals P0 .
A graph may be plotted for u / U vs log10(y/) and values of the constants A and B can
be obtained. This experiment can be repeated for other stations also.
Table 15.1 : Velocity profile (Stations : 1, 2, 3, 4)
Station P0
N/m2
p
N/m2
Pitot
pressure
pP
pP
U
u
0
Micrometer
reading
y / log10(y/)
The second part of the experimental work is to obtain the different boundary layer
characteristics experimentally by procedure outlined in Experiment No. 14. u/U and
u/U (1- u/ U) may be plotted against y /. On the first graph 1/7 th power law profile
may also be plotted corresponding to
7
1
y
U
uto check the validity of the power law relations.
The appropriate areas under these curves will give and and hence H. The
experimental boundary layer parameters can be compared with theoretical results
obtained with power law relations using n = 7 as shown in Table 15.2.
Table 15.2 : Comparison of boundary layer parameters
Station Experimental Theoretical(1/7–th power law)
H Cf CF
H Cf CF
1
2
3
4
201
The values of and using 1/7 power law are usually noticeably higher than the
experimental values. This indicates that over initial part of the length of the surface the
boundary layer is laminar.
The third part of the experimental work is to observe the effect of pressure gradient on
the growth of the boundary layer. This can be done by using the liners to provide
accelerating and decelerating flow. The boundary layer is supposed to grow appreciably
thicker in the rising pressure gradient.
The fourth and final part of the work is to study the effects of roughness by using the
rough side of the plate. The roughness of surface serves to increase the rate of growth of
boundary layer.
All the experimental results can be presented and can be shown in tabular form for the
last station (L = 0.265m) :
Table : U = ; Re = U c/ =
Station Smooth surface Rough surface
L=0.265 m Zero
pressure
gradient
Favourable
pressure
gradient
Adverse
pressure
gradient
Zero
pressure
gradient
Favourable
pressure
gradient
Adverse
pressure
gradient
H
Cf
CF
202
Chapter 16
FLOW ABOUT RECTANGULAR AND SWEPT WINGS
16.1 Introduction :
The characteristics of flow about a three-dimensional wing of finite span effects from
that about a wing of infinite span in a number of ways. For the flow about a wing of
infinite span, the flow characteristics do not change in spanwise direction and hence the
flow is essentially two-dimensional. However, this is not the case for a wing of finite
span.
Let the lift or more precisely, circulation at the center section of wing (station 1) be 1 .
now, the air can not sustain any load and therefore load and the circulation at the wing
tips, must be zero. Therefore, there must be a variation of circulation (from 1 at center
section to zero at the tip) along the span, whatever may be the form of variation. If
therefore, 1 be the circulation at station 1, then the circulation at any other station 2 will
be 2 which is different from 1 . This difference in circulation ( 2 -1) must therefore
appear as vorticity shed in wake (Fig. 16.1).
This shedding of vorticity in the wake is the most important characteristics of three
dimensional flow. It may be worth noting that shedding of vorticity does not take place in
two dimensional flow except for unsteady cases. In general, the shedding is continuous as
spanwise circulation varies continuously from center section to tips. However, intensity
of shed vorticity become more at the neighborhood of tips. Also, these shed vortices roll
up at the tips. This is why shed vorticity along the trailing edge is so often referred to as
the ‘tip vortices’.
203
Figure 16.1 Two and three dimensional flow characteristics
The flow at any section of a three-dimensional wing differs from the flow which would
occur round the section of a two-dimensional wing owing to the influence of the trailing
vortex system. The velocity induced by this vortex system is normal to the span of the
wing and to the direction of motion and is directed downwards is general. The normal
induced velocity at a point on the wing will be denoted w (downwash). It has two very
important consequences which alter the aerodynamic characteristics of three dimensional
wings.
Firstly, the induced velocity w effectively reduces the angle of incidence by a small angle
w/ U (Fig. 16.2). If is the geometric angle of incidence of the wing, the effective
angle of incidence will be
U
we (16.1)
In other words, the wing center section then behaves the same as if it is part of an wing of
infinite span at an angle of incidence e and gives the lift coefficient at this angle of
incidence.
204
Figure 16.2 Effects of downwash
Secondly, the lift force is, however, inclined backwards at an angle w/ U and therefore
gives a component in the direction of drag force. This component is called induced drag
since it is caused by the induced velocity of the trailing vortices. The induced drag is
LU
wDi
(16.2)
The total drag of the wing is obtained by adding the profile drag (boundary layer normal
pressure drag and skin fraction drag) and the induced drag. If the profile drag coefficient
is now written as CD0 and the induced drag coefficient as CDi, total drag coefficient of a
three-dimensional wing can be written as
DiDD CCC 0
LD CUwC 0 (16.3)
The governing equation of steady, incompressible, inviscid, irrotational flow about a
three-dimensional wing is the Laplace’s equation :
02
2
2
2
2
22
zyx
(16.4)
where is perturbation potential.
The boundary conditions are the same as for the two dimensional case :
i) flow at the wing surface must be tangential,
qn = 0 on wing surface (16.5)
iii) the perturbation velocities must tend to zero at infinity,
205
u, v, w 0 at infinity (16.6)
16.2 Theory :
Various solutions of Laplace equation, equation (16.4), have been developed, which can
be broadly classified as
Solutions of Laplace’s equation
Approximate solution Exact solution
Linearised Theory
Analytic Numerical
(flow past a sphere)
Panel Method
The two-dimensional linearised theory have been extended to the three-dimensional flow.
Effects of thickness and camber are separated in exactly the same manner using small
perturbation assumption. The thickness effect gives the pressure distribution at zero lift
and camber effect (which includes incidence ) gives the lifting characteristics.
Weber, Holme and Hjelte developed approximate methods for the thickness problem
using the linearised boundary condition.
The classical theories which deal with camber and incidence effects are :
i) Prandtl’s lifting line theory (applicable for large aspect ratio rectangular wings)
ii) Jone’s slender wing theory (applicable for low aspect ratio delta wing)
iii) Multhopp’s lifting surface theory (applicable for wings of arbitrary planform).
iv) Faulkner’s vortex lattice theory (applicable for wings of arbitrary planform).
However, all these methods (apart from restriction with regards to planform size and
shape) suffer from the fact that they can predict only the overall forces and moments and
not the details of pressure (or velocity) distribution on the wing surface.
Numerically-exact numerical solutions were obtained later with the development of
‘Panel Methods’. These methods are capable of predicting pressure distribution also.
206
In the present context, lifting line theory and lattice theory are presented for comparison
with experimental work involving measurement of overall forces and moments. Also,
pressure distribution obtained by a version of panel technique is quoted in the tables for
comparison with experimental pressure distribution.
16.3 Prandtl’s Lifting Line Theory :
In Prandtl’s lifting line theory the wing is replaced by a single line. The procedure is to
find normal induced velocity w and the effective angle of incidence e at each point of
the span to calculate the corresponding elementary lift and drag forces and finally to
integrate across the span of the wing. The first stage is therefore the determination pf the
normal induced velocity at a point on the span in terms of the strength of the trailing
vortices.
The circulation round the aerofoil varies across the span, being symmetrical about the
center and failing to zero at the tips. Between the points y and (y+dy) of the span of
aerofoil the circulation falls by –(d/dy) dy and hence a trailing vortex of this strength
springs from element dy of the span as shown in Fig. 16.3. There is therefore a sheet of
trailing vortices extending across the span of the wing and the normal induced velocity at
any point of the span must be obtained as the sum of the effects of all the trailing vortices
of this sheet. The normal velocity induced at any point (y1 ) of the wing is
S
Syy
dydy
d
yw1
14
1
(16.7)
Figure 16.3 Trailing vortex sheet
207
Another equation is required connecting the circulation and the normal induced
velocity w.
The second equation is
UcCL2
1
Uac e02
1 (where a0 is the two-dimensional lift curve slope)
U
U
wca 0
2
1
wUca 02
1 (16.8)
For solution of circulation , Prandtl chooses the variation of across the span as
1
sin4 nAsU n (16.9)
Substitution of this form of in eqs. (16.7) and (16.8) gives
sinsinsin nnAn (where = a0c/8s) (16.10)
The solution of the fundamental equation, eq. (16.10), is obtained numerically. Once the
solution is obtained, the lift and induced drag are determined very simply in terms of
coefficients An of the series for the circulation.
The lift and induced drag coefficients are given by
1AACL (16.11)
21
LDi CA
C
(16.12)
where A is aspect ratio. is a small positive number given by .12
1
2
A
nAn
The total drag coefficient is given by (Fig. 16.4)
2
0
1LDD C
ACC
(16.13)
208
Figure 16.4 Total drag coefficient
The lift curve slope for three dimensional wings can be given in terms of the slope for
two-dimensional wings as
Ad
dC
d
dC
d
dC
D
L
D
L
D
L
2
2
31
(16.14)
16.4 Vortex Lattice Method :
In vortex lattice method, wing chordal surface plane (z = 0) is divided into a large
number of quadrilateral panels as shown in Fig. 16.5. On each panel is placed a horse-
shoe vortex consisting of a bound vortex and two trailing vortices extending to infinity
downstream. The strength of horse-shoe vortex is assumed to be constant for a particular
panel but varies from panel to panel. The unknown horse-shoe vortex strength can be
obtained by satisfying the boundary condition of zero normal flow at collocation points.
209
The collocation points are taken to be ¾ panel chordwise portion mid-way between
chordal lines.
The problem is to determine the strength of horse-shoe vortices i, i=1,….,NM. This is
done by satisfying boundary condition of zero normal flow at NM panel collocation
points. This results in a system of NM linear algebraic equations. In matrix form
UA iji (16.15)
where Aji is the influence coefficient matrix, i is the strength of the horse shoe vortex
strength on i – th panel and is the angle of incidence.
Figure 16.5 Vortex lattice model
The bound vortices only carry lift. Loading on each bound vortex is given by
12 yyUF iZ
Total lift can be obtained by summation of load carried by all panel as
MN
i
MN
i
iZL yyU
SU
F
SU
C1 1
1222
2
1
1
2
1
1
12
2yy
SUi (16.16)
210
The pitching moment coefficient about any axis xm is given by
miimiZM xxyyScU
xxF
ScU
C 122
2
2
1
1
(16.17)
Sectional lift coefficient and spanwise loading can be obtained by similar numerical
integration.
16.5 Wind Tunnel Testing :
16.5.1 Measurement of pressure distribution :
The experimental work is conducted on a rectangular model of aspect ratio 2.4 having
NACA 0012 aerofoil section. 45 pressure holes are drilled on the wing surface (Fig. 16.6)
round the section at two spanwise stations (at 10% semi-span and at 90% semi-span).
Figure 16.6 NACA 0012 wing model
211
Results can be obtained for angles of incidence of 00, 6
0, 15
0 and 20
0 . Exact numerical
solutions for two cases of = 00 and = 6
0 are given in the following tables for
comparison.
Table 16.1 : Pressure distribution at = 00 (10% semi-span)
Tapping
points
x/c z/c hLS
(cm)
U
m/s
h
(cm)
Cp (experimental) Cp (panel
method)
1
2,29
3,28
4,27
5,26
6,25
7,24
8,23
9,22
10,21
11,20
12,19
13,18
14,17
15,16
0
0.015
0.03
0.06
.09
.15
.2
.3
.4
.6
.8
.9
.95
.975
.99
1.0
-.044
-.25
-.35
-.312
-.262
-.150
-.035
+.03
+.09
+.148
+.26
1.0
-.01
-.219
212
Table 16.2 : Pressure distribution at = 00 (90% semi-span)
Tapping
points
x/c z/c hLS
(cm)
U
m/s
h
(cm)
Cp
(experimental)
Cp (panel
method)
30
31,58
32,57
33,56
34,55
35,54
36,53
37,52
38,51
39,50
40,49
42,47
43,46
44,45
0
.015
.03
.06
.09
.15
.2
.3
.4
.6
.0
.95
.975
.99
1.0
-.01
-.219
-.314
-.334
-.322
-.288
-.23
-.19
-.11
-.028
+.072
+.11
+.24
213
Table 16.3 : Pressure distribution at = 60
(10% semi-span)
Mapping points x/c z/c hLS U h Cp (experimental) Cp (exact theory)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
0
.015
.03
.06
.09
.15
.2
.3
.4
.6
.8
.9
.95
.975
.99
.99
.975
.95
.9
.8
.6
.4
.3
.2
.15
.09
.06
.03
.015
-
-1.396
-1.268
-1.1
-.932
-.782
-.685
-.535
-.42
-.223
-.074
+.016
+.098
+.166
.25
.27
+.19
+.13
+.065
+.01
-.056
-1.65
-1.08
-.078
-.029
.095
.21
.48
0.76
214
Table 16.4 : Pressure distribution at = 60
(90% semi-span)
Mapping points x/c z/c hLS U h Cp (experimental) Cp (exact theory)
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
0
.015
.03
.06
.09
.5
.2
.3
.4
.6
.8
.9
.95
.975
.99
.99
.975
.95
.9
.8
.6
.4
.3
.2
.15
.09
.06
.03
.015
-
-.918
-.88
-.755
-.67
-.529
-.45
-.338
-.255
-.126
-.032
+.03
+.09
+.153
+.258
+.26
+.159
+.061
+.042
-.011
-.073
-.125
-.141
-.138
-.108
-.03
+.08
+.28
+.588
215
Experimental pressure distribution may now be graphically plotted and compared with
the exact numerical solution given in the tables.
Sectional lift, drag and pitching moment coefficients can also be obtained at these two
spanwise stations (10% and 90%) by numerical or graphical integration of the pressure
distribution using the procedure outlined for two-dimensional case in Chapter 13. These
results can be presented in tabular form as shown below.
Table 16.5 : Sectional characteristics
Station Sectional CL Sectional CD Sectional pitching moment
CM L.E.
10% semi-span
90% semi-span
16.5.2 Measurement of overall forces and moments using balance :
The model chosen for this work is a rectangular one of aspect ratio 1.6 having the same
NACA 0012 aerofoil section. The procedure for measurement of forces and moments is
outlined in Chapter 9 and is not repeated here. It is advised that the 3-component balance
should be calibrated again since balance characteristics change from time to time.
The value of CL, CD, CM(1/4)C can be obtained over a range of incidence ( = 00, 5
0, 10
0,
150 and 20
0 ) to cover the stall. dCL /d and CLmax can be obtained from the above values.
CD can be plotted against C2
L (Fig. 16.4). CD0 is the value of drag coefficient obtained at
= 00. These results can then be compared with the two-dimensional results to observe
the difference between three dimensional and two dimensional cases (Table 16.6).
Table 16.6 : Comparison of 2D and 3D cases
2D 3D
CD CD CM(1/4)C
d
dCL CL CD CM.15c
d
dCL
216
These experimental results can be compared with the solution obtained either by the
lifting line theory or the vortex lattice theory as shown in Table 16.7.
Table 16.7 : Comparison of experimental and numerical values :
Experimental Linearised/Vortex lattice theory
CD CD CM(1/4)C
d
dCL CL CD CM.15c
d
dCL
The whole experimental procedure can be repeated for a swept wing of aspect ratio
having 450 leading and trailing sweep. However, for this wing, linearised theory will fail
and comparison of experimental results can be made with vortex lattice theory.
217
Chapter 17
FLOW ABOUT A SLENDER DELTA WING
17.1 Introduction :
The flow about a slender delta wing with sharp leading edge differs vastly from that
about a rectangular wing. Primarily, expressions for lift coefficient, drag coefficient,
wing lift-curve slope etc., are different for slender delta wings indicating different values
of lift and drag. Secondly, flow over slender delta wing begins to separate quite early at
small angle of incidence (as low as 30 as compared to 14
0 - 18
0 for rectangular wings).
The character of this flow separation is also entirely different from that on a rectangular
wing which causes stall. Because of the adverse pressure gradient, flow over a delta wing
of large sweep angle separates along the leading edges. This leading edge flow separation
results in an increase of lift rather than a decrease of lift as is the case for rectangular
wing at high angle of incidence.
Theories of slender wings with or without leading edge separation are described in the
following sections.
17.2 Slender Wings in Attached Flow :
For this case. The theoretical method was developed by Munk and later extended by
Jones. The basic idea of this theory (slender wing theory ) is that for a elongated wing at
a small angle of attack the flow pattern in any transverse plane, i.e. a plane substantially
normal to the main stream direction, approximate near the wing to that of two-
dimensional flow.
The governing equation of incompressible potential flow is the Laplace’s equation,
02
2
2
2
2
22
zyx
(17.1)
The slender wing theory, in its classical presentation, begins by neglecting the term
22 x in eq. (17.1) leaving the approximate equation
02
2
2
2
zy
(17.2)
218
The justification for this approximation is, at least, intuitively the slender elongated form
of the wing under consideration. It seems clear that since the geometrical properties of
the wing vary only slowly in the x-direction, the derivative 22 x ,which is the rate of
change of the perturbation velocity in the same direction must also be small, at least in
the neighbourhood of wing.
Eq. (17.2) is exactly the equation for the perturbation potential in two-dimensional
incompressible flow, i.e., the flow pattern in any transverse plane normal to the
mainstream, is two-dimensional. Hence transformation techniques which are used for
solution of two-dimensional cases can be employed here.
Using a transformation of the type
xsZZ 222 (17.3)
where s(x) is the local semispan, (Fig. 17.1).
The solution is obtained as
21
22 yxsU (17.4)
Figure 17.1 Flow past slender delta wing
The loading or the pressure difference between upper and lower surface is given by
1PPup CCC
219
dx
xds
yxs
xs )(
)(
4
2
122
(17.5)
For the case of a wing of triangular planform it follows from eq.(17.5) that the loading is
constant along any straight line (for example, along OA in Fig. 17.10 through the vertex.
This is because ds(x)/dx is constant for triangular planform and 22 )()( yxsxs is
constant for such a line. This is the essence of conical flow. The spanwise loading for
such low aspect ratio delta wings is always elliptic.
The lift coefficient of the wing of aspect ratio A is given by
2ACL (17.6)
Hence 2AddCL (17.7)
The induced drag coefficient is given by
21
LDi CA
C
(from eq. (16.12))
21LC
A ( = 0 since spanwise loading is elliptic)
2LC (17.8)
17.3 Slender Wings in Separated Flow :
At small angle of incidence the flow begins to separate and the equations (17.6), (17.7)
and (17.8) for lift, drag etc. remain no longer valid. The separation takes the form of a
vortex sheet which springs from each leading edge. Each sheet then rolls up into a
streamwise vortex some distance above the wing surface (Fig. 17.2) and the flow beneath
these vortices retains the full stream total head. Indeed, the vortices tend to increase the
flow velocity on the upper surface and in consequence they increase the lift above the
value to be expected without leading edge separation.
220
Figure 17.2 Flow separation along leading edges
The problem of flow separation is essentially non-linear in which strength of the
separated vortices as well as their positions are unknown. The calculation procedure
should therefore be an iterative one. However, early workers have developed non-
iterative schemes by assuming positions of separation vortices.
Lift of a slender delta wing is given by Mangler as
2
22
ACL (17.9)
while the expression given by Weber is
2
3
22
ACL (17.10)
Later, attempt has been made to develop mathematical models incorporating the leading
edge separation by Brown and Michael, Mangler, Smith etc. They developed simplified
models in which the rolled up vortices are represented by a single pair of concentrated
vortices as shown in Fig. 17.3.
More accurate methods for this non-linear leading edge separation problem have been
developed later within the framework of vortex lattice theory as well as panel method by
a number of workers. These methods essentially make use of electronic computer for
computation.
221
Figure 17.3 Simplified flow model
In recent years, attempts have been made to obtain the details of flow separation using
Euler and Navier-Stokes codes.
17.4 Wind Tunnel Testing :
17.4.1 Measurement of pressure distribution :
To obtain spanwise loading, 23 pressure holes are drilled on a delta wing model of aspect
ratio 1.0 at five chordwise station (Fig. 17.40 given by
RCx / .73, .786, .841, .90, .956
222
Figure 17.4 Slender delta wing model
Loading at all these points can be measured and compared with theoretical solution given
by eq.(17.5). Theoretical and experimental load distribution can be given in tabular form
as shown in Table 17.1.
Also, for all chordwise stations, spanwise load variation can be obtained by plotting Cp vs
y/s(x).
Table 17.1 : Load distribution at = 20 (unseparated flow)
U =
Tapping
point
x/CR y/s(x) hLS h Cpu Cp1 Cp1
experimental
Cp
theoretical
(eq. 17.5)
1
2
-
23
223
Experimental work for separated flow case is carried out on the same model having
aspect ratio 1.0 at moderate to large incidence. Loading distribution can be obtained at
these points and plotted against y/s(x) as shown in Fig. 17.5. This pressure distribution
shows peak around 90% semi-span because of the separated flow along the leading edge.
Figure 17.5 Spanwise load distribution
17.4.2 Measurement of overall forces and moments :
Using the 3-component balance, lift, drag and pitching moment can be obtained for the
same delta wing model of aspect ratio 1.0 covering a wide range of incidence upto 400 in
steps of 40. the experimental results can be plotted as CL, CD, CM vs. .
Experimental results can be compared with the empirical relations given by equations
(17.9) and (19.10). The liner lift given by equation (17.2) should also be superimposed on
this graph to indicate the additional non-linear lift due to leading edge flow separation at
moderate to high incidence.
224
Chapter 18
FLOW ABOUT COMPOSITE WINGS
18.1 Introduction :
Aerodynamic characteristics of a wings of rectangular and slender delta planform have
been studied in the preceding experiments. Both the planforms have their advantages as
well as disadvantages. Rectangular wings show a high lift-curve slope (little less than
2) but suffer from a increase of drag and decrease of lift (stall) at high angle of attack
(around 180) due to separation of flow on the upper surface of the wing. Slender delta
wings, on the other hand, show a low lift –curve slope together with high lift-dependent
drag. However, such wings exhibit a leading edge separation even at moderate angle of
attack, a phenomenon, which helps in increasing the lift rather than decreasing it.
It is then immediately realized that considerable benefit may therefore be gained by a
composite planform. In such composite planforms, a vortex generating surface (i.e. a
slender wing with sharp leading edge) is placed ahead of a wing. Lifting capability is
significantly enhanced, particularly at moderate to high angles of attack. This is due to
favourable interaction between the leading edge separated flow on the front wing with the
flow on the mainwing. Because of this favourable interaction, separation from the main
wing surface is also delayed resulting in reduction of drag at a given angle of attack.
The surface used to generate the interacting vortex may take any of the several forms :
the most commonly used are the strake and the canard. The canard has the additional
advantage of providing an additional control surface. The aerodynamic characteristics of
such composite configurations are essentially chracterised by the combination of vortex
flows and the classical attached aerofoil flow on the same wing (rear wing). The main
advantage of the combined flow is observed in a dramatic extension of the lift curve
through the stall. The main disadvantage is the existence of the ‘free’ vortices quite close
to the main wing surface.
225
18.2 Straked Configuration :
A strake is a thin sharp edged highly swept delta wing which is added at the leading edge
of the basic wing. Usually it is added to the inboard portion of a rounded edged delta (or
swept) wing (Fig. 18.1). It may be worth noting here that the configuration is called
cranked wing if the leading edge of the entire planform is rounded. On the other hand, if
the leading edge of the entire planform is sharp, it is called a double delta configuration.
Figure 18.1 Strake wing configuration
The remarkable feature of the straked wing is the dramatic extension of the lift curve
through the stall (Fig. 18.2) and indeed often the stall does no occur at all in the wind-
tunnel investigation. There is a little change in the lift-curve slope upto about 12 – 140
for wings with and without strakes. However, lift coefficient increases significantly
above this angle for the straked wing. This increase is not that simply obtained by adding
the nonlinear lift from the additional slender wing of the strake surface; there is obviously
considerable interaction taking place between the two lifting surfaces.
226
Figure 18.2 Variation of lift with incidence
However, this benefit of increased lift coefficient of a straked wing is to be counter-
balanced against two adverse features. Firstly, the drag at low incidence is quite high.
Secondly, the increase of lift coefficient is usually accompanied by a decrease in static
longitudinal stability.
The theoretical solution of this problem involves calculation of separated flow along the
leading edge of the strake, calculation of attached flow on main wing surface and suitable
way of effecting the influence of separated flow on basic wing. While a number of
methods are available for calculating either the separated flow (on the stake) or attached
flow (on the basic wing), the main difficulty is to evaluate the effect of the vortex field
from the strake as it passes over the main wing. Simple approximate methods have been
developed based on the assumption that the separated vortices pass over the main wing
parallel to the basic wing planform. A more exact method for this non-linear flow
problem has been developed based on panel method by Rubbert.
227
Th emodel chosen for experimental work is a double-delta wing planform of aspect ratio
1.34. The sweep angle of the strake is 770 (Fig. 18.3)
Figure 18.3 Strake wing model
To obtain spanwise pressure distribution at different chordwise stations, wing surface is
grooved in the spanwise direction. Thirty-four pressure holes are distributed on nine
chordwise positions given by
x/CR =0.19, 0.26, 0.33, 0.40, 0.473, 0.544, 0.61, 0.78, 0.88
where CR is the root-chord.
Load distribution (Cp) can be obtained at these points experimentally for various
incidence and plotted against y/s(x) where s(x) is the local semi-span (Fig. 18.4).
228
To obtain the overall lift, drag and pitching moment coefficients, the three-component
balance can be used for the same model and can be plotted against angle of attack
.dCL/d can be obtained from the graph.
Figure 18.4 Spanwise load distribution
18.3 Canard Configuration :
Unlike in strake configuration, there is a horizontal gap between the front vortex
generating surface and the rear main wing, i.e. it is a two wing configuration. It is often
described as a tail-first configuration (Fig. 18.5).
229
Figure 18.5 Canard wing model
The main advantages of canard configurations are observed in an enhancement of lifting
capability, particularly at high angles of attack, and in an extension of the lift curve
through the small compared to the canard-off case. This extension of the linear
characteristics is due to a delay of the bubble-type separation from the wing surface. This
results in a reduction of separation drag and an improved effectiveness of the aft located
control surface.
A significant disadvantage for canard configurations is associated with problem arising
from the presence of free vortices quite close to the configuration surface. This may pose
considerable difficulties particularly for the sideslipping motion due to the problem of
vortex breakdown at high angle of attack, which effectively puts an operational limit on
such configurations.
Theoretical models for canard configurations are not so common. The difficulty seems to
get the canard induced vortex field represented properly as it passes over the rear wing
since the strength, shape and position of separation vortices are all unknowns. However,
flow models using iterative scheme have been developed within the framework of vortex-
230
lattice theory as well as panel method. Also, Euler and Navier-Stokes code have been
developed in recent years for this problems.
For wind tunnel testing of canard configuration, a model is chosen in which the basic
wing is a swept wing of aspect ratio 3.2 having a leading edge sweep of 29.30. The canard
is a delta wing of aspect ratio 1.0 (Fig. 18.5).
For pressure measurement 48 pressure holes are set on surface of the basic wing. These
holes are set along the 58% chord line of t he basic wing, being distributed along the
upper surface of the starboard wing and along the lower surface of the port wing.
Pressure can be measured by a single pressure transducer using a scanivalve system.
The overall forces and moments can be measured on the same canard model using either
the sting balance or the three component external balance.
231
Chapter 19
DRAG MEASUREMENT OF SPHERE
19.1 Introduction :
Study of drag of a spherical body is interesting because it depicts the characteristics of
boundary layer. First, its drag is related to Reynolds number. Second, laminar flow
having less energy with which to surmount roughness or corners separates from a surface
more easily than does turbulent flow. Third, the maintenance of a laminar boundary layer
becomes more difficult as the Reynolds number (the speed or length) increases. Fourth,
laminar flow is encouraged by a pressure gradient falling in the direction of flow and
separation easily by a pressure gradient rising in the direction of flow.
In the light of the above discussion drag characteristics under conditions of changing
Reynolds number may be analysed. For simplicity flow about an aerofoil (Fig. 19.1) may
be considered first.
In Fig. 19.1(a) the flow is a predominantly laminar and hence drag should have been
lower. However, as mentioned earlier, laminar boundary layer has less energy and fails to
negotiate the downstream curve (after maximum thickness point) of the aerofoil and the
flow separates. Hence the drag becomes high because of excessive separation (point A in
Fig. 19.2).
As the flow speed increases, transition point moves ahead (fig. 19.1b) according to the
third feature mentioned earlier. Since the turbulent flow is more robust the flow sticks to
the surface of the aerofoil and does not separate. This results in a net decrease of drag
coefficient (point B Fig. 19.2).
Still higher Reynolds number results in a increase of drag (Point C in Fig. 19.2) and the
transition point has moved t0 the furthest point (the minimum pressure). Its further
motion is resisted by the favourable pressure gradient from the leading edge to that point.
232
Figure 19.1 Characteristics of flow about an aerofoil
Figure 19.2 Variation of CD with Reynolds number
The above flow pattern is perfectly valid for a spherical body also. The added feature is
that the decrease in CD in the region A to B is so rapid that even the total drag
D (= DSCU 2
2
1 ) decreases despite the increase in free stream velocity U .
233
The above characteristics can be studied experimentally by measuring the drag of a
sphere using the three-component wind-tunnel balance. A sphere (of radius 15 cm) is
attached to the balance and the drag of the sphere is measured.
The drag coefficient based on the projected frontal area S (= 42d ), where d is the
diameter of the sphere and is given as
SU
DCD
2
2
1
the Reynolds number is determined with the diameter of the sphere as the characteristics
dimension,
dURe
The tunnel can be run at various speed setting and the drag of the sphere may be
recorded. The results can be presented in Table 19.1.
Table 19.1 : Drag of sphere
No. of Runs hLS Drag of sphere (D) U CD Re
1
2
3
4
-
-
10
The variation of both drag and drag coefficient may be plotted against the Reynolds
number. This experiment highlights the effects of Reynolds number (i.e. scale effect) on
drag.
It may be noted here that Reynolds number affects the lift also. The variations in lift
curve slope of an aerofoil caused by increasing the Reynolds number are very small, but
234
in general the lift curve will be straightened up, the slope will increase slightly and the
stall will become more abrupt (Fig. 19.3). It follow that CLmax and the angle at which it
occurs are increased.
Figure 19.3 Variation of lift coefficient with incidence
235
Chapter 20
EVALUATION OF A SUPERSONIC WIND TUNNEL
20.1 Introduction :
The supersonic wind tunnel system is usually developed as a blow down facility where
stored high pressure dry air is discharged through the working section. It consists of two
basic units, the compressor system with reservoir storage and the wind tunnel section
with its associated ducting system.
The compressor system is a reciprocating compressor providing air at a final pressure of
5-6 bar. The air is stored in a reservoir system of total volume sixteen cubic meter. The
wind tunnel system is connected to the reservoir system by means of a manually
controlled valve. The valve is used to control the pressure in the settling chamber
upstream of the working section.
Such supersonic tunnels are short-duration tunnels and usually require special measuring
devices for measurements. Different experiments that are conducted in the supersonic
tunnel are :
1. Shock visualisation
2. Determination of Run time of a tunnel
3. Determination of Mach Number in the Test Section
4. Variation of Mach number along the axis of divergent section of C-D
nozzle
5. Variation of Mach number along diffuser axis
6. Determination of the exit velocity
236
237
20.2 Shock Visualisation :
In order for shock pattern to be visualized, various types of optical systems are available.
These systems are :
a) Shadowgraph technique
b) Interferometer technique
c) Schlieren technique
In the present facility a Schlieren system has been used. The basic philosophy of the
Schlieren system is as follows.
With the flow disturbances around a model, localized changes in pressure and hence
density occur. If parallel light is shone into the working section, then the light emitted
from the section will not be wholly parallel but will have zones of increased and
decreased intensity due to the diffractive effects of the density changes. By using an edge
and other lenses and mirrors it is then possible to produce a sharp image of the flow
disturbances.
Fig. 20.1 shows schematically the Schlieren system. Light from a source is focused by
means of a lens onto a variable aperture unit which is used to vary the total illumination
through the remaining optical path. This aperture is placed at the focus of lens 3 such that
parallel light passes into the wind tunnel working section.
Te light issuing from the working section is focused onto a Schlieren edge. The edge can
take two distinct forms. For black and white Schlieren patterns, the edge is basically a
glass slide with 50% of the area blacked out giving a sharp straight dividing line between
the clear and obscured areas.
For colour Schileren, a three colour edge is used with the central colour as a narrow
straight edged band.
238
After the Schlieren edge, the light is focusssed onto a screen where an image is formed.
When the system is set up, the Schlieren edge position is adjusted so that the illumination
on the screen is about 50% of that without the edge interrupting the optical path (for
black and white Schlieren only). With the tunnel operating, the flow disturbances cause
the focused image of the working section to be changed as mentioned previously. The
Schlieren edge effect is to chop off those optical path deflections which pass into the
obscured area on the edge. This produces an image of the model complete with the flow
pattern shown as lighter and darker zones.
20.3 Run Time of Tunnel :
For a single run of the tunnel, air is prescribed and stored in reservoir. By running the
compressor for about 20-30 minutes, air can be stored in the reservoir at about 5 kg/cm2
(about 70 psi). by opening the valve continuously air can then be released from the
reservoir at a fixed settling chamber pressure (say 1.2 kg/ cm2 , about 20 psi).
The aim of the experiment is to determine duration of the flow in the tunnel at a constant
settling chamber pressure. It is usually of the order of 30 seconds. The run-time can be
calculated theoretically in the following manner.
Let, t = run time
V = volume of the reservoir
i = initial density of air in reservoir
f = final density of air in reservoir
Pi = initial pressure of air in reservoir
Pf = final pressure of air in reservoir
Total mass flow from reservoir in time t is
fif VM
ifiV 1
Now, nifif PP
1
For polytropic expansion, n = polytropic index (1 for isothermal expansion)
Then, ifif PPVM 1
239
ifii PPTRV 1 (20.1)
Now, total mass flow through the test section in time t can be written as
UAtM f (20.2)
where A = area of the test section
= density of air in the test section
U = velocity of air in the test section
For isentropic flow,
1
1
2
02
11
M
where 0 is the total density and M is the Mach number.
Assuming, = 1.4
2
52
0 2.01
M
2
52
00 2.01
MRTP (since P = RT) (20.3)
Now, velocity U in the test section can be written as
U = M a where a is speed of sound
Or, U = M RT
2
0
2
11 M
RTM
(since 2
02
11 MTT
)
2
0 2.014.1 MTRM (20.4)
Using equations (2), (3) and (4)
tAM
RTMM
RT
PM f ..
2.01
4.12.01
2
02
52
0
0
(20.5)
For convenience, flow at the throat may be considered. In that case M = 1, A = A. Then
Eq. (5) is reduced to
tT
APM f .
0165.0
0
0
(20.6)
Equating equations (20.1) and (20.6)
240
VP
P
RT
Pt
T
AP
i
f
i
i
1.0165.0
0
0
or,
00
01053.0
P
P
P
P
T
T
A
Vt
fi
i
(20.7)
Pi, Pf, P0 are to be observed from the dial gages and then to be converted to absolute
pressure by adding the atmospheric pressure.
The dimensions of the tank and tunnel are as follows :
Tank diameter (d) = 6 ft = 1.8288 m
Tanks height (h) = 20 ft = 6.096 m
Tank volume = .4 d2 .h = 16.0127 m
3
Throat area = A = 4 sq. inch =.00258 m
2
T0 = Ti = Tatm = 220C = 532.5
0 K.
With all these data, it is now possible to calculate the run-time theoretically for a
particular settling chamber pressure P0 using equation (20.7). Run time can be obtained
experimentally by a stop watch. For various settling chamber pressures (P0 ) different
run-time will be obtained theoretically and experimentally. Calculated and experimental
values can be put in tabular and graphical forms as shown below.
Table 20.1 : Run-time calculation
No. Pi
(absolute)
Pf
(absolute)
P0
(absolute)
Run-time
(theoretical)
Run-time
(experimental)
1
2
3
241
t (sec)
Gauge pressure (P0 )
Figure 20.2 Variation of Run-time with settling chamber pressure
20.4 Determination of Mach Number :
Mach number in the test section can be determined by four techniques. These are done as
follows:
20.4.1 By using Area-Local Mach Number Relation :
There is a relation between area at any cross section and the Mach number given as,
)1(2
1
2
2
11
1
21
M
MA
A (20.8)
where is the adiabatic index of air (=1.4) and A is the area of the throat. At the test
section, one can calculate the area of cross-section (A) and also the throat area (A).
Using eq. (20.8), one can determine the value of M.
A A
………………..
M =1
M
Figure 20.3 Convergent -Divergent Nozzle
242
Example: 947.28.3
2.11
A
A
Using this value in the eq. (20.8), M = 2.6334.
20.4.2 By Static Pressure Measurement on the Wall of the Test Section :
Relation between static (p) and stagnant pressure (P0) is given as,
1
20
2
11
M
p
P
11
21
02
p
PM (20.9)
Now, observed valve pressure is 24.7 psi and hence,
P0 = 24.7 psi + 1 atm = 38.7 psi = 266836.5 N/m2
A mercury tube is used. A height difference of 1 cm in the tube will mean a pressure
difference of 1.3366 103 N/ m
2. Now, static pressure can be written as,
P = Patm + 1.3366 103 (h)
From the mercury tube h = -26.1 and therefore P = 66439.74 N/ m2
Putting these values in the eq. (20.9), M = 1.5615.
20.4.3 By using Rayleigh-Pitot Formula :
Rayleigh-Pitot formula is given as
121
124
1 21
2
22
1
02
M
M
M
P
P (20.10)
where P02 is the total pressure at station 2 and P1 is static pressure at station 1.
243
P1, M1 M2
P02
Pitot-tube
P02 : Total pressure at
P1 : Static pressure at
Figure. 20. 4 Rayleigh-Pitot Formula
Same mercury tube is used here also.
P02 = Patm + 1.3366 103 (h1)
P1 = Patm + 1.3366 103 (h2)
Example : From the mercury tube h1 = 71.2 and h2 = -57.7
Putting these values in the eq. (20.10), M = 2.4342.
20.4.4 By using M Relation (Shock Wave over a Wedge) :
A shock wave is formed when supersonic flow passes over a wedge. This strategy can be
used to determine the Mach number of the flow
shock
wedge
M
Figure 20.5 Shock wave over the wedge
2 1
1
2
244
There is a relation between a and b, shown in the Fig.20. 5.
2)2cos(
1)(sin).cot(2)tan(
2
22
M
M (20.11)
where is the adiabatic index of air (= 1.4). so if one finds out the value of a and b
experimentally, M can be easily determined from the eq. (20.11).
From one experiment = 12.50 and = 40
0. Then putting these values in the eq. (20.11)
will give value of M = 2.108.
20.5 Variation of Mach number along the axis of divergent section of C-D nozzle
Variation of Mach number ahead of the test section due to liners can be obtained from
measurement of pressure at different points as shown in Fig 20.6.
Figure 20.6 Pressure ports along liners
If P0 and p be the absolute total and static pressure at any point and M be the Mach
number of flow then,
1
20
2
11
M
p
P
2
722.01 M (assuming = 1.4)
or,
15.0
7
2
0
p
PM (20.12)
The total pressure (P0 ) is constant throughout the test section in the absence of shocks in
the flow and is known from dial gauge of settling chamber. The dial gauge pressure
should again be converted into absolute pressure by adding the atmospheric pressure. The
static pressure at different points can obtained from mercury manometer readings
connected to several static pressure holes. Results can be given in tabular and graphical
forms.
245
Table 20.2 : Variation of Mach Number
P0 =
Stations Distance from settling
chamber
Manometer height
(h)
Static pressure
(p) absolute
M
1
2
.
.
8
9
10
2.4
2.0
Mach no. 1.6
1.2
0.4
0
0 4 8 12 16 20 24
Distance along nozzle axis from throat
Figure 20.7 Variation of Mach number along nozzle axis
20.6 Variation of Mach number along diffuser axis
This is done in the same manner as in the preceding section.
246
20.7 Determination of the exit velocity :
The manometer used to calculate the total and static pressure at the exit has pressure
difference of 0.2 mPa for 1 small division. Difference between the total and static
pressure at the exit is given as,
50 exitexit PP small divisions
= 5 0.2 100 = 100 N/m2
Therefore, uexit
= 12.65 m/s.
247
Appendix – 1
NOTATIONS
a = speed of sound
A = cross-sectional area at any section
A = throat area
b = wing span
CD = drag coefficient
CL = lift coefficient
CF = overall skin friction coefficient
CM = pitching moment
Cp = pressure coefficient
Cf = local skin friction coefficient
c = wing chord
D = drag
d = diameter
h = height of water column
L = lift
M = Mach number
Mf = Mass flow
P0 = total pressure
P = static pressure
p = static pressure at infinity
qt = tangential velocity
qn = radial velocity
q = free stream dynamic pressure
248
Appendix – 2
NOTE ON UNITS
Throughout this book the International System of Units (SI) is used. In SI units, the units
of mass, length and time are
Kilogram kg
Meter m
Second s
In this system the unit of force, defined as that force which when applied to a unit mass
of 1 kg produces an acceleration of 1m/s2 is the
Newton N
The Newton may be expressed in term of units of mass, length and time by the equation
1 N = 1 kg m/ s2
Pressure p at a joint in a fluid is determined in terms of normal force acting on an element
of a plane surface through the point the pressure is the ratio of force to area. The units are
therefore
Pressure p N/m2 or kg/ms
2
The SI units provides for the use of particular names for certain derived units and the
Pascal, represented by the symbol Pa, is used to represent a pressure of 1 N/m2 , viz.
1 Pa = 1 N/m2
For many purposes this unit is rather small, and another unit, the bar is used. This is
defined by
249
1 bar = 105 Pa = 10
5 N/m
2
The millibar, written mbar (or simply mb) is a further convenient unit.
1 mb = 10-3
bar = 100 Pa = 100 N/m2
In British system of units the fundamental units of mass, length and time are
Pound lb
Foot ft
Second s
These are related to SI units by the conversion factors
1 lb = 0.45359237
1 ft = 0.3048 m
Properties of Air : Sea-level conditions
Temperature T = 150 = 288.16 k
Pressure p0 = 101325 n/m2
Density = 1.2256 kg/m3
Absolute coefficient of viscosity = 1.783 10-5
kg/ms
Kinematic coefficient of viscosity = 1.455 10-5
m2 /s
Speed of sound in air a (= RT) = 340 m/s
Ratio of specific heat =1.4
Gas constant R =287.2 J/kg K
250
Appendix – 3
List of Facilities
Serial
No.
Tunnel Test section
dimension
Maximum
speed
Type of Tunnel Maximum
power
1. Low speed
Tunnel
61cm 61cm 34 m/sec Suction type 15 HP
2. Smoke
Tunnel
39cm 5cm 5-8 m/sec Blow down
type
Small fan
3. Air Flow
Bench
10cm 5cm 33 m/sec Blow down
type
0.75 HP
4. Gust Tunnel 61cm 91cm 35 m/sec Blow down
type
50 HP
5. Industrial
Wind Tunnel
226cm 145cm 16 m/sec Blow down
type
75 HP
6. Calibration
Tunnel
21cm 21cm 30 m/sec Blow down
type
10 HP
7. Cascade
Tunnel
48cm 10cm Mach No.
= 0.5
Blow down
type
75 HP
8. Supersonic
Tunnel
5cm 10cm Mach No.
= 2.2
Blow down
type
75 HP
251
Appendix – 4
2100 System : Strain Gage Conditioner and Amplifier System
1. DESCRIPTION
1.1 General
The series 2100 modules comprise a multi-channel system for generating conditioned
high-level signals from strain gage inputs for display or recording on external equipment.
A system would be comprised of :
(a) One or more two-channel 2120B Strain Gage Conditioners.
(b) One or more 2110B Power Supplies (each Power Supply will handle up to ten
channels; i.e., five 2120B Conditioner / Amplifiers).
Optionally, one or more 2111 DC- Operated Power Supplies (each 2111 module
is capable of powering up to eight channels; i.e., four 2120B Conditioners/
Amplifiers; or up to ten channels when maximum bridge voltage and output
current are not required).
(c) One or more rack adapters or cabinets, complete with wiring, to accept the above
modules.
1.2 Significant Features
The principal features of the system include :
Independently variable and regulated for each channel (0.5 to 12 Vdc).
Fully adjustable calibrated gain from 1 to 2100.
Bridge-completion components to accept quarter – (120Ω, 350Ω and 1000Ω),
half-and full-bridge inputs to each channel.
LED null indicators on each channel – always active.
100 mA output.
All supplies and outputs short-circuit proof with current limiting.
Compact packaging – ten channels in 5.25 ×19 in (133 × 483 mm) rack space.
252
2. SPECIFICATION
All specifications are nominal or typical at +230C unless noted. Performance may be
degraded in the presence of high-level electromagnetic fields.
Operational Environment : Temperature : 140 to 122
0F (10
0 to 50
0C). Humidity : Up to
90% RH, non-condensing.
2.1 Model 2120B Strain Gage Conditioner
Note : These specifications apply for each of two independent channels per module.
Inputs :
Inputs Impedence : > 100 MΩ (balance limit resistor disconnected).
Source Current : ±10 nA typical; ±40 nA max.
Configuration : Two- to seven-wire to accept quarter-, half-, or full –bridge strain
gage or transducer inputs. Internal half bridge, dummy 350Ω and dummy120Ω
completion gages and three-wire calibration capability provided. Sec 4.3c for
dummy1000Ω provision.
Protection : Input is protected from damage of inputs up to ±50V differential or ±25V
common mode.
Amplifier :
Gain : 1 to 2100; continuously adjustable; direct reading.
Gain steps X2, X20, X200; with ten-turn counting knob, X0.5 to X10.50 ±1% typical.
Frequency Response (min) :
Normal Range : dc to 15 kHz: - 3 dB at all gain settings and full output.
dc to 5 kHz: - 0.5 dB at all gain settings and full output.
Extended Range : (Configured by internal jumper – see 6.1 c ).
dc to 50 kHz: -3 dB at all gain settings and full output.
Dc to 17 kHz: -0.5 dB at all gain settings and full output.
Noise RTI * : (350 Ω source impedance) 1 μV p-p at 0.1 Hz to 10 Hz; 2 μVrms at 0.1
Hz to 50 kHz.
Noise RTO**
: 50 μV p-p at 0.1 Hz to 10 Hz; 80 μV p-p at 0.1 Hz to 100 Hz; 100
μVrms at 0.1 Hz to 15 kHz; 200 μVrms at 0.1 Hz to 50 kHz.
* Referred to input
** Referred to output.
253
Temperature Coefficient of Zero : ±1 μV/ 0C RTI, ±210 μV/
0C RTO; -10
0 to +60
0C
(after 30 minute warm-up).
Common-Mode Voltage : ±10V
Common-Mode Rejection : (dc to 60 Hz)
Gain Multiplier CMR (dB)
X2 67
X20 87
X200 100
Output Range : ±10V (min) at ±100mA; current limited at ≤140 mA.
Capacitive Loading : Up to 0.15 μF.
Excitation :
Type : Constant voltage.
Range : 0.5 to 12 Vdc (continuously adjustable for each channel) with 120 Ω full-
bridge load.
Short-Circuit Current : Less than 40 mA max.
Noise : 2 mV p-p dc to 20 kHz.
Load Regulation : ±0.2 % no-load to 120 Ω load (±10% line change).
Balance :
Method : Potentiometric.
Range : ±2000 με, ±4000 με or ±6000 με (quarter, half or 350 Ω full bridge) ranges
selected or disabled by internal jumpers.
Calibration :
Controls : Two-position (center off) toggle switch.
Standard Factory-Installed Resistors : (174.8 KΩ ±0.1%) simulate ±1000 με at GF
= 2.
Optional Calibration Relays : Provides remote operation of excitation (off/ on) and
shunt calibration.
Relays are powered by user-supplied voltage source, and must be specified when
ordering rack adapters or cabinets.
Size : 5.25 H × 2.94 W × 10.97 D in (133 × 75 × 279 mm).
Weight : 2.2 lb (1.0 kg).
254
2.2 Model 2110B Power Supply (AC – Operated)
Outputs : ±15 Vdc and + 17.5 Vdc; protected against continuous circuits.
Input : 107, 115, 214, 230 Vac ±10% (selected internally); 50 – 60 Hz.
Power : 40 W typical, 100 W max.
Meter : 0 to 12 Vdc (with switch) to read bridge excitation.
Also ac input and dc output go/no-go monitor.
Size : 5.25 H × 2.44 W × 12.34 D in (133 × 62 × 313 mm).
Weight : 6.7 lb (3.1 Kg).
2.3 Model 2111 DC-Operated Power Supply
Outputs : : ±15 Vdc and + 17.5 Vdc; protected against continuous circuits.
Input : 12 Vdc nominal (9 to 18 Vdc range). Power 60 W max; 78 % efficiency at full
load.
Reverse Polarity Protection : Internal shunt diode.
Meter : 0 to 12 Vdc (with switch) to read bridge excitation. Dc output go/no-go monitor.
Size : 5.25 H × 2.44 W × 12.34 D in (133 × 62 × 313 mm).
Weight : 3.0 lb (1.4 kg ).
2.4 Model 2150 Rack Adapter
Application : Fits standard 19-in (483 – mm) electronic equipment rack.
Accepts one Power Supply and one to five Strain Gage Conditioners.
Completely wired.
Power : 2-ft (0.6 – m) three-wire line cord; 10-ft (3-m) extension available.
Size : 5.25 H × 19 W × 14.17 D in overall (133 × 483 × 360 mm).
Weight : 6.6 lb (3.0 kg ).
2.5 Model 2160 Portable Enclosure
Description : Completely self-contained adapter and cabinet with all wiring for two or
four channels.
Accepts one power supply and one or two Strain Gage Conditioners.
Power : 8-ft (2.4 – m) detachable three-wire cord.
255
Size : 5.55 H × 8.75 W × 13.80 D in (141 × 222 × 350 mm).
Weight : 5.2 lb (2.4 kg ).
3. CONTROLS
3.1 Model 2110B/2111 Power Supplies
Bridge Volts Meter : Displays the voltage on each input bridge (as selected by
CHANNEL selector). Also used to monitor ac line and dc outputs of Power Supply .
Channel Selector : Positions 1 to 10 select channel to display bridge excitation on Meter
(“1” is Channel farthest to left in cabinet, etc).
The Dc position monitors a mixed output from the +15, -15, and +17.5V power supplies
and should always read on the “DC” line at “10” on the Meter. The AC position (2110 B
only) monitors the peak-to-peak ac line input (at a fixed transformer tap). A reading
anywhere in the band from 9 to 11 on the Meter indicates that the input voltage is proper
for the selected transformer tap (see 4.1e). No reading indicates the equipment is
ungrounded.
External Meter Jacks : Supplies Meter voltage to an external meter if desired for more
precise adjustment of bridge supply voltages.
Power Switch: The central power switch for this supply and all Conditioners connected to
it. (The pilot lamp may take several seconds to extinguish when the power is turned off).
3.2 Model 2120B Strain Gage Conditioner
(one channel described; both identical and independent)
Output Lamps : LED indicators always monitoring amplifier output. Primarily used to
adjust AMP ZERO and Bridge BALANCE. (Fully lit with 0.07V output).
Balance Control : A ten-turn potentiometer to adjust bridge balance. Normal range
±2000 με. (See 4.12 a to extend range) EXCIT must be ON to set bridge balance.
Gain Controls :
Multiplier Switch : Provides gain steps of X2, X20 And X200.
Potentiometer : Ten-turn with counting knob provides multiplier of 0.50 to 10.50.
Total amplifier gain is the product of the multiplier switch and potentiometer
settings.
256
Bridge Excit Control : A 25-turn trimmer to adjust bridge excitation from 0.50 to 12
Vdc. The actual setting is monitored on the Meter and the External Meter Jacks on the
Power Supply (the proper channel must be selected).
AMP ZERO Control : A 25-turn trimmer used to set the electrical “Null” of the input
amplifier zero. (EXCIT should be OFF and the input circuit connected when this is done).
EXCIT Switch : A toggle switch controlling the excitation to the input bridge. (Any
amplifier output with EXCIT at OFF is dc amplifier offset, thermal EMF from the bridge
or ac pickup in the wiring).
CAL Switch : A two-position (with center off) toggle switch to shunt-calibrate the input
bridge. As delivered, “A” simulates ±1000 με, and “B” simulates -1000 με by shunting
the internal 350Ω half bridge. Other shunt calibration configurations are possible by
internal resistor and jumper changes (see 5.0 Shunt Calibration).
INPUT Receptacle (Rear Panel) : A ten-pin quarter-turn connector to connect input
gage(s). (Quarter, half and full bridges can be accepted simply by using the appropriate
pins. Sec 4.3c for details.) Mating connector supplied.
OUTPUT Receptacle : A standard BNC connector delivers the amplifier output (±10V
at ±100 mA).
257
4. OPERATING PROCEDURE
4.1 SETUP and AC Power
4.1a: the individual Conditioner and Power Supply modules are not stand-alone
instruments. They are designed to plug into a prewired cabinet or rack adapter which (1)
supplies ac line power (fused) to the Power Supply, (2) distributes dc regulated voltages
to all Conditioners and (3) connects the bridge voltage monitoring meter in the Power
Supply to the various channels.
If one or more 2150 ten-channel Rack Adapters are used, these should be mounted in a
standard 19-in (483-mm) equipment rack; 5.25 in (133 mm) vertical height is required for
each ten channels.
4.1b : Before installing a 2110B Power Supply module in each cabinet or rack adapter,
check that each 2110B module is set for the proper ac line voltage:
Slide the right-hand side cover almost all the way back to expose the two toggle
switches on the printed-circuit board. One switch, as marked, sets for nominal 115 or
230V; the other sets for NORM line (115/230V ±10%) or LOW line (107/214V
±10%). Replace side cover.
The POWER switch on the front panel should be at OFF. Install the Power Supply in the
right-hand position of the cabinet; push in to engage the input/output plug, and secure the
retaining screws.
4.1c : Install 2120B Conditioners in the remaining positions in the cabinet. Push the
modules in to engage the power-input plugs, and secure the thumb screws. (Blank covers
are available for unused positions).
4.1d : Plug the line cord into an ac receptacle, making certain that the third pin goes to a
good ground. The equipment must be grounded for safety and best performance.
Note : If the line plug must be replaced with a different type, observe this color code
when wiring the new plug :
Brown : High line voltage.
Blue : Low Line voltage (i.e., “neutral” or common)
Green/ Yellow : Ground.
258
4.1e : Check ac power. On each Power Supply, turn the CHANNEL selector to “AC”.
Turn the POWER switch on (up). The red pilot lamp should light and the meter should
read between 9 and 11. if not, observe meter reading:
Pegs at full scale. Turn power off immediately. This indicates that the input voltage is
much too high for the internal switch settings (probably a 230V input with switches set
for 115V; see 4.1b).
Reads low (between 8 and 9-1/4). The ac line voltage is significantly below 115V (or
230V). Remove Power Supply and reset internal switch for LOW line.
Reads around 5. This indicates that the internal switches are probably set for 230V
input, whereas the voltage is actually 115V. Turn POWER OFF, remove module, and set
switches (see 4.1b).
Reads 0 (no reading). Red pilot lamp not lit: The ac receptacle has no power or the fuse
(at the rear of the instrument) is open. Pilot lamp lit: Equipment is not properly grounded.
Either the third pin was not used, or the receptacle used is not properly grounded.
4.1f : Check dc power. On the Power Supply, turn the CHANNEL selector to “DC”. The
meter should read very near the line at 10. if not, this indicates that either (1) there is an
internal short in one of the Conditioner modules (remove them one at a time), or (21) one
or more of the regulated power supply circuits is defective (see 6.4 Internal
Adjustments).
4.1g : Check bridge excitation regulators. Scan the CHANNEL switch through positions
1 to 10; all positions should read some voltage between 0.5 and 12V. (However, switch
positions corresponding to channels not installed will, of course, read zero.)
4.1h : The system is now ready for ruse. If it is planned to use the system immediately, it
is suggested that the POWER be left on (for warm-up); otherwise turn all POWER
switches to OFF.
Caution : Prior to removing or installing the 2110B Power Supply into a rack adapter or
cabinet, the ac power cord must first be unplugged. Refer all servicing to qualified
technicians.
4.2 DC Power
259
4.2a : The 2111 module is capable of powering up to eight channels (four Model 2120B
modules) at maximum rated bridge voltage and output current or up to ten channels when
maximum bridge voltage and output current are not required. The 2111 functions
similarly to the 2110B Power Supply, with the exception of the 12 Vdc nominal input,
which supports battery operation only.
4.2b : Remove the line cord, which is not used when a 2111 module is installed, from the
ac receptacle of the cabinet.
Set POWER switch on the front panel to OFF. Install the 2111 module in the right-hand
position of the cabinet; push to engage the input/output plug, and secure with thumb
screws.
4.2c : 12 Vdc power is supplied through the four-conductor recessed male connector on
the 2111 rear panel. Connections are made to the mating female connector (TRW/Cinch-
Jones S2404-CCT; Vishay Micro-Measurements P/N 12X300606) with #16 AWG (1.3 –
mm dia.), or larger, wire. Assure that the operating voltage at the input connector will be
maintained within 9 to 18 Vdc. Make connections as above.
4.2d : Turn POWER on and check for proper operation as described in 4.1f through 4.1h
for the 2110B.
4.3 INPUT Connections
4.3a : It is suggested that the system be turned on and allowed to stabilize while
preparing the input connectors; Power consumptions is negligible. To prevent powering
any input circuits at this time, turn the EXCIT toggle switches OFF on all channels.
4.3b : each channel uses a separate (and interchangeable) input plug. Two loose plugs are
supplied with each 2120B Conditioner (one per channel). If additional plugs are desired,
they are available from Vishay Micro-Measurements or through electronic parts
distributors.
Vishay Micro-Measurements 12X300515
ITT/Cannon KPT06B12-10P
Bendix PT06A-12-10P(SR)
4.3c : Connect the input to each channel, using the connectors supplied, in accordance
with Figs. 1a and 1b.
260
Note : Except when using an external full bridge, there must be a jumper in the plug
connecting pins D and E; this connects the midpoint of the internal 350Ω half bridge
to the S+ amplifier input, thus completing the necessary full bridge for proper
amplifier operation.
Generally, no modifications or jumpers are required inside the 2120B Conditioner
regardless of the external bridge configuration used. (However, there are provisions for
accepting 1000Ω quarter-bridge inputs and for changing the shunt calibration circuit –
see Note on the following page and 5.0 Shunt Calibration, respectively.)
Note : 2120A/B Strain Gage Conditioners, with serial numbers above 85000, provide
the capability for 1000Ω quarter-bridge operation. For this mode, the 120Ω dummy
terminal (pin H of input plug) is converted to a 1000Ω dummy terminal by removing
a shunt from a factory installed Vishay 880Ω precision resistor in series with the
internal 120Ω dummy gage. To make this conversion, the user must desolder a solder
pad located on the circuit side of the PC board. Figure 2 shows the location of the
880Ω resistor (component side) and the solder pad.
4.4 Wiring Considerations
The precautions given below provide important considerations for correct wiring
technique. Both precautions may be applicable if you are testing to measure both static or
dynamic data. For additional information on electrical noise, please consult Vishay
Micro-Measurements Tech Note TN-501, Noise Control in Strain Gage Measurements.
Note : The system or the signal input wiring could be subjected to high-level EMI
(electromagnetic interference) or ESD (electrostatic discharge); therefore, shielded
cabling should be used to preserve data integrity as well as prevent permanent
damage to the instrument. External bridge sensor(s) should also be shielded. The
shield should be connected to the hood of the INPUT connector. The system chassis
should be connected to a good “earth grounded”, generally through the line cord.
Dynamic Data: It is extremely important to minimize the electrical noise that the gages
and lead wires pick up from the test environment; this noise is usually related to the 50 or
60 Hz line power in the area:
261
a) Always use twisted multi-conductor wire (never parallel conductor wire);
shielded wire is greatly preferred, although it may prove unnecessary in some
cases using short leads.
b) Shields should be grounded at one (and only one) end; normally the shield is
grounded to the INPUT connector hood and left disconnected (and insulated
against accidental grounding) at the gage end. Do not use the shield as a
conductor (that is, do not use coaxial cable as a two-conductor wire).
c) The specimen or test structure (if metal) should be electrically connected to a
good ground.
d) Keep all wiring well clear of magnetic fields (shields do not protect against
them) such as transformers, motors, relays and heavy power wiring.
e) With long leadwires, a completely symmetrical circuit will yield less noise (e.g,
a half bridge on or near the specimen will usually show less noise then a true
quarter-bridge connection).
Static Data : Precise symmetry in leadwire resistance is highly desirable to minimize the
effects of changes in ambient temperature on leadwires.
a) In the quarter-bridge circuit,. Always use the three-lead-wire circuit shown in
Fig. 1a, rather than the more obvious two-wire circuit.
b) If possible, group all leadwires to the same channel in a bundle to minimize
temperature differentials between leads.
c) If long leads are involved, calculate the leadwire desensitization caused by the
lead resistance. If excessive in view of data accuracy required, adjust effective
gage factor, increase wire size, or increase gage resistance – or all three, as best
suits the situation.
4.5 MILLIVOLT Inputs
The 2120B Conditioner can accept low-level dc inputs, (using pins A and D), provided
two requirements are observed:
a) The common-mode voltage should not exceed ±10V in normal operation, and
must never exceed a peak voltage of ±25V.
262
b) The input cannot be completely floating: there must be a ground return
(generally less than 10MΩ ), either external or within the 2120B. In the case of
thermocouples welded to a nominally grounded structure, this return is adequate.
A ground return exists within the standard 2120B due to the presence of the
bridge-balance circuit. However, if the external signal is adequately grounded,
this resistance can be removed (remove jumpers P and N – see 4.12a)
The user is also cautioned regarding two sources of possibly significant error:
a) Bias Current : Each input (pins A and D) requires an input current of
approximately 10 nA; this current will flow through the input wiring to the
ground return, which must exist. With a floating input (in which case the
balance resistor must remain installed), the bias required at pin D will flow
directly from the balance resistor, but the bias for pin A will flow through the
entire input circuit; with low source impedances that is insignificant and can be
offset with the AMPZERO control. High source impedances can result in
measurable offsets (with a 5000Ω source impedance the offset may approach
0.2 mV RTI).
b) Any nonsymmetry in the ground returns of the inputs will reduce the CMR of
the amplifier to some degree.
4.6 OUTPUT Connections
Cautions : If it is possible in way to damage the indicator or recorder connected to the
OUTPUT with inputs of 15V or 140 mA, the OUTPUT should not be connected until the
channels have been balanced (see 4.11 and 4.12).
4.6a : The model 2120B (Serial Numbers above 134733) uses a standard BNC connector
for each output. The shells are connected directly to the chassis which is tied to earth
ground. For protection against high-level EMI (Electromagnetic Interference), ESD
(Electrostatic discharge), and permanent damage, it is highly recommended that standard
shielded BNC cables be used. The Shield should be connected directly to the body of the
connector. Assembled cables with a BNC connector on each end are available from local
electronic parts distributors and from the Vishay Micro-Measurements using the part
numbers listed below:
263
BNC Cable Length MG Accessory No.
2 ft (0.6 m) 2100-A62
4 ft (1.2 m) 2100-A81
10 ft (3.0 m) 2100-A49
4.6b : It should be noted that the OUTPUT indicator lamps on the front of the 2120B at
all times monitor the voltage on the OUTPUT connector. If both lamps are extinguished,
the output voltage is zero (typically within 2 mV maximum circuit offset). Full brilliance
of either lamp indicates a voltage in excess of 70 mV (possibly as high as 15 V).
4.7 OUTPUT Current Limits
The output is capable of ±10V into a load of 100Ω or higher. With a load of 100Ω or
lower, the output will deliver up to ±100 mA, but in no case greater than 140 mA.
The maximum output can readily be limited to less than 140 mA by increasing the value
of two resistors per channel (R34 and R37, normally 6.2Ω ±5%).
The desired value of R34 and R37 can be calculated with the following formula :
MAX
CI
R870
1 (Eq. 1)
where, RC1 = current limit resistor in ohms (R34, R37).
IMAX = maximum current output in mA.
4.8 OUTPUT Voltage Limits
The open circuit output voltage can exceed ± 13.5V which can cause problems with
certain sensitive recording devices. The printed circuit board has provisions on each
channel for a voltage divider to reduce the output voltage. A series and a shunt resistor
(R38 and R39, respectively) can be added for this purpose. In the standard unit, r38 is
shunted by a closed “Z” solder pad and R39 is open.
To limit the output voltage, calculate the desired resistor values using Ohm’s law. Then,
install the appropriate resistors and open the “Z” shunt (by cutting the small shorting
trace). For noise reduction, it is desirable to keep the total amplifier load resistance low
but it should be above 100 ohms to avoid current limiting. If additional assistance is
264
needed, contact assistance is needed, contact the Vishay Micro-Measurements
Applications Engineering Department.
4.9 Operation
4.9a : On each channel make certain that the EXCIT switches are OFF (thus removing
excitation to all gage circuits) and the CAL switches are in the center (OFF) position.
4.9b : If it is possible to damage or overload the indicators or recorders connected to the
OUTPUTS with 15 Vdc (or 140 mA for low-resistance devices), the OUTPUT plugs
should not be connected at this time.
4.9c : On the Power Supply module, turn the POWER switch on. The red pilot lamp
should light.
4.9d : On the Power Supply (2110B only) module, set the CHANNEL selector at AC;
the meter should read within the AC band.
4.9e : Turn the CHANNEL selector to DC; the meter should read on the DC check line.
4.10 EXCITATION
4.10a : Set desired excitation on each channel; turn the CHANNEL selector to channel 1;
adjust BRIDGE EXCIT (using a small screwdriver) to read the desired BRIDGE Volts on
the Power Supply Meter.
Note : If greater accuracy is desired than can be achieved with the built-in meter.
Connect an external meter to the EXTERNAL METER banana jacks on the Power
Supply (the minus jack is nominally chassis-ground).
Turn CHANNEL selector to channel 2 and repeat the above procedure adjusting
BRIDGE EXCIT on the next channel; continue until all installed channels are
satisfactorily adjusted.
Note : To achieve best stability and lowest noise at the out-put, it is desirable to
use the maximum excitation that the input to each channel can accept. Small or low-
resistance gages or gages bonded to a poor thermal conductor (such as most plastics
or composites) must necessarily use low excitation. For additional information on
excitation selection, please consult Vishay Micro-Measurements Tech Note TN-
502, Optimizing Strain Gage Excitation Levels.
265
4.10b : Connect gage INPUT plugs (if not already connected).
4.11 Amplifier Zero
Adjust the AMP ZERO for each Channel. (To some extent the amplifier balance is
affected by symmetry of the source impedances seen by the amplifier inputs).
Using a small screwdriver, adjust each AMP ZERO until both OUTPUT lamps are off.
(If the “+” lamp is lit, turn the adjustment counterclockwise, etc.)
If, at best null, both lamps are lit, this is an indication of excessive noise (probably 50 or
60 Hz) at the input. Check wire shielding, etc. Refer to 4.4 Wiring Considerations for
further discussion.
4.12 Bridge Balance
4.12a : Adjust balance. For each channel, turn the EXCIT switch to ON; then turn the
BALANCE control to extinguish the OUTPUT lamps.
Note : As delivered, the BALANCE controls can correct for approximately ±2000
με unbalance in a quarter, half or 350Ω full bridge. With full-bridge inputs other
than 350Ω , the balance range will be reduced for lower bridge resistance and
increased for higher resistance. For example, with a 120Ω full bridge, the balance
range is reduced to under ±700 με. If the balance range proves inadequate for the
gages or transducers in use, the “balance limit resistor” can be changed from
75000Ω to 37000Ω by moving the jumper from area P to area N, thus doubling the
balance range. Also, jumpers can be located at both area P and N, thus tripling the
original balance range. An Extension of the balance range will produce a reduction
in the setability of the balance control. This is especially noticeable for strain gages
with resistance of 350Ω and higher. Bridge balance may be disabled by removing
both P and N jumpers. Spare or unused jumpers can be stored on the pins next to
the right channel gain switch.
4.12b : Connect OUTPUT plugs for each channel (unless already connected).
4.13 GAIN
4.13a : Adjust GAIN for each channel. There are two general methods of settings the
GAIN control:
266
a) Mathematical : In many cases it is possible to predict and preset the amplifier gain
required. For example, assume the input is one active gage with GF = 2 (this will
produce 0.5 μV per με per volt of excitation) and bridge excitation has been set at 5
Vdc. Further assume that the desired output from the 2120B is 2V for 500 με. At
500 με the bridge will deliver 1.25 mV (500 με × 5 V × 0.5 μV/V/ με = 1.25 mV).
To achieve 2 V output from the amplifier will require a gain of 1600 (2V/1.25 mV
= 1600 ). Set the GAIN control at 8.00, and the multiplier switch at X200.
b) Empirical : Without regard to bridge excitation or amplifier output voltage,
assume that the desired output is 25-mm deflection on a recorder for a 500 με input.
Using shunt calibration (such as the 1000 με built into the 2120B Conditioner),
adjust the GAIN as required to achieve the desired deflection – for example, a 50-
mm deflection should occur when the 1000 με shunt calibration resistor in the
2120B is selected (assuming GF=2, for which the calibration resistors are
calculated).
In practice, even though the mathematical approach is possible in many situations,
the shunt-calibration method should also be used as the final exact adjustment.
The user is cautioned to consider the effects of leadwire resistance and the
calibration circuit actually in use when calculating the strain simulated by shunt
calibration. See 5.0 hunt Calibration.
4.13b : All controls are now set. However, just before taking data, it is advisable to check
balance on each channel:
a) Briefly turn EXCIT to OFF; if the OUTPUT lamps are not at null (both
extinguished) adjust AMP ZERO as necessary. This can be done at any time during
a test – and should be done occasionally on an extended test.
b) Under no-load conditions (and with CAL at OFF and EXCIT at ON) the
OUTPUT lamps should indicate null; if not, adjust the BALANCE control.
Note : In both steps above, it may be desirable to observe the output recorder rather
than the OUTPUT lamps. First, there may be a very small offset (5 to 10 mV)
between true zero output and the zero indicated by the lamps and, second, it may be
necessary to compensate for a small mechanical or electrical zero offset in the
recording device.
267
4.13c : Once the GAIN and BALANCE control settings have been finalized, it is
recommended that the knobs be locked in position to prevent accidental rotation.
Counting knobs utilize a lever which must first be pulled away from the panel and then
rotated clockwise (towards the bottom of the panel). The knob can be unlocked simply by
rotating the lever back to the counterclockwise stop.
4.14 NOISE
Before taking dynamic data, it is highly desirable to document the output noise
attributable to wiring and other sources vs. the total dynamic output which includes this
noise plus the dynamic strain signals:
Momentarily turn EXCIT to OFF. Any output observed now is NOT caused by strain
(whether a dynamic strain is being generated or not). “White” noise (full spectrum) is due
to the amplifier and cannot be reduced except by reducing GAIN – it should not exceed
several microvolts rms referred to the input (that is, the observed signal divided by
amplifier gain). A recurrent waveform (usually 50 or 60 Hz or multiplies of this
frequency) indicates electrical pickup at he gages or in the wiring to the gages; if
excessive, the source should be located and corrective measures taken. For additional
information on electrical noise, please consult Vishay Micro-Measurements Tech Note
TN-501, Noise Control in Strain Gage Measurements.
5. SHUNT CALIBRATION
Note : It should be emphasized that the purpose of shunt calibration is to determine the
performance of the circuit into which the gage(s) is wired, and in no way does it verify
the ability of the gage itself to measure strain or the characteristics of its performance.
5.1 Equations
Shunt calibration can be achieved by shunting any one of the four arms of the input
bridge [which includes an active gage(s) and the bridge completion resistors within the
2120B Conditioner]. The 2120B provides for shunting any of these arms. No matter
which arm is shunted, the same equation applies:
268
610
ACAL
ACAL
RRK
R (Eq. 2)
where, CAL = Strain simulated (microstrain).
AR = precise effective resistance of arm shunted (ohms).
K = effective gage factor of strain gage.
CALR = resistance of calibration resistor (ohms).
K may be the actual package gage factor of the strain gage in use, or it may be adjusted
for leadwire desensitization:
LG
G
RR
RKK
(Eq. 3)
where, K = package gage factor of active gage.
GR = resistance of strain gage (ohms).
LR = resistance of leadwire (s) in series with active gage (usually the
resistance of one leadwire) (ohms)
When shunting either bridge arm to which the balance limit resistor is connected, I is
theoretically necessary to correct for this shunting effect in determining AR . While the
exact value depends on the position of the balance potentiometer, a good approximation
(which assumes the pot is at mid position) is:
BLPA
BLPAA
RRR
RRRR
42
4
(Eq. 8)
where, AR = resistance of resistor or gage in arm.
PR = resistance of balance potentiometer.
BLR = resistance of balance limit resistor.
It should be noted that, for the 2120B Conditioner as shipped (where shunt calibration is
across the 350Ω dummy half bridge), this correction is only 0.2%.
5.2 Shunting Internal Half Bridge (350Ω) :
Use : Quarter and half bridge (full bridge with reduced accuracy).
Advantages : Same resistors regardless of active gage resistance. No special wiring
required. Can simulate tension and/or compression.
269
Disadvantage : Leadwire desensitization may be significant (use Eq. 7 in Eq. 6).
Location of resistors and jumpers on the printed circuit board is shown in Figs. 3, 4 and 5.
(Note that separate resistors are used for CAL A and CAL B, so that these may be
different values; to calculate strain use AR = 349.3Ω).
5.3 Shunting Internal Dummy Gage (120 or 350Ω) :
Use : Quarter bridge only.
Advantage : Automatically corrects for leadwire desensitization (using three-wire
circuit). No special wiring. Accuracy independent of precise gage resistance.
Disadvantage : Only usable if internal dummy gages are in use. Simulates tension only.
Location of resistors and jumpers on the printed circuit board is shown in Fig. 6. AR is
resistance of dummy resistor. K = K.
5.4 Shunting Active Gages
While there is no electrical problem in shunting active gages at the specimen (they must
be accessible), accomplishing this at the Conditioner with only the usual three-lead
connection will introduce serious errors if the leadwires have measurable resistance. The
reason is that one signal lead, which is supposed to be only a remote voltage-sensing
lead, now carries current (to the calibration resistor); the error thus introduced is
approximately four times that which would be expected by normal “leadwire
desensitization” equations.
The above problem applies equally to active (or compensating) gages in stress analysis
and to all transducer applications.
As rough guide, 1% error will be introduced if the resistance of each lead is :
For 120Ω gages : 0.3Ω (7 ft AWG 26, 30 ft AWG 20)
(2.1 m × 0.4 mm dia; 9 m, 0.8 mm dia)
For 350Ω gages : 0.9Ω (20 ft AWG 26, 85 ft AWG 20)
(6 m, 0.4 mm dia; 25 m, 0.8 mm dia)
5.4a : To properly shunt-calibrate active gages or transducers, the accepted technique is
to provide two additional leads dedicated to the calibration circuit; for quarter-bridge
operation this is customarily called the “five-leadwire circuit”.
270
Pins J and K on the input connector are used for this application. Figures 7 & 8 show a
half bridge, but the calibration wiring also applies to full bridges and transducers; for true
quarter bridges, Fig.7, compression only, applies.
The added external leads should be connected directly to the strain gage terminals. AR =
actual gage resistance. K = K. (In a transducer, the connection should be made at the
connector on the transducer. AR would be the effective resistance of the shunted
transducer arm.)
MODEL 2130 DIGITAL READOUT
AND
MODEL 2131 PEAK READING DIGITAL READOUT
GENERAL
The 2130 module provides an LED digital display plus a channel selector in a 2100
System-compatible package. The 2131 also includes peak reading capability. These units
simply slide into either the 2150 Rack Adapter or the 2160 Enclosure. An additional line
connection is not required as power is derived from the 2110B or 2111 Power Supply
through the rack adapter or enclosure. Standard cables (two supplied) make the necessary
signal connection between the 2130/2131 and each of the 2120B Strain Gage Conditioner
channels to be used in the display mode. The 2130/2131 will accept and switch up to ten
inputs. Additionally, front-panel jacks are provided for utility inputs such as measuring
bridge excitation via the 2110B or 2111 EXERNAL METER jacks. An external
monitoring device, such as an oscilloscope, can also be connected to the rear-panel output
connector to give simultaneous indications for a given selected input. All input and
output connections are single-ended.
SPECIFICATIONS
All specifications nominal or typical at +230C unless noted. Performance may be
degraded in the presence of high-level electromagnetic fields.
271
2130/2131 Common Specifications
Input Capacity : 10 channels, BNC (rear panel); 1 Channel, banana jacks (front panel).
Switched Output : Not attenuated, BNC (rear panel)
Attenuator Accuracy : ±0.1 % or better.
Update Rate : 3 readings/second, nominal.
Digital Display : 3-1/2 digit LED, ±1999 counts.
Display Height : 0.3 in (7.6 mm).
Operational Environment :
Temperature : 140 to 122
0F (-10
0 to 50
0C).
Humidity : Up to 90 % RH, non-condensing.
Power : 2110B/2111 Power Supply.
Size : 5.25 H 2.94 W ×10.97 D in (133 ×75 × 279 mm)
Weight : 2 lb (0.8 kg)
2130 Specifications
Input Voltage Range : ± 1999 mV (X1 range);
± 19 990 mV (X10 range)
Input Impedance : 100 KΩ.
Accuracy : ± 0.05% ± 1 count
2131 Specifications
Input Voltage Range : ± 1999 mV (X1 range); ± 10 V (X10 range);
Input Impedance : Greater than 1 MΩ.
Accuracy :
Step Input : ± 0.1 % ± 5 counts for repetitive step inputs of greater than 10 milliseconds
duration.
Repetitive Step Input : ± 0.2 % ± 5 counts for repetitive step inputs of greater than 500
μsec duration. Number of steps
required ≥ DurationPulse
ondsmilli
.
sec.10
272
Repetitive sine Wave Input : ± 5.0 % ± 5 counts for repetitive sine wave of frequency
less than 1000 Hz.
± 1.0% ± 5 counts for repetitive sine wave input of frequency less than 200 Hz.
Storage Stability : ± 3 counts/ minute maximum at +750F (+23C).
Peak Modes : MAX (usually positive) excursion and MIN (usually negative) excursion.
Peak Reset : Manual or Automatic.
CONTROLS
OUTPUT Display : Provides a digital reading of the input as selected by CHANNEL
selector. Typically used to monitor strain or bridge voltage.
CHANNEL Selector : Positions 1 to 10 select the input channel for display. (Generally,
position 1 is channel farthest to the left in rack, etc.)
The EXTERNAL position selects the input that is connected to the adjacent front-panel
jacks.
EXTERNAL Jacks : provides ability to accept a front-panel input, typically bridge
voltage from 2110B/2111 EXTERNAL METER jacks.
ATTEN Switch : X1 position gives ±2 volt range.
X10 position gives ±20 volt range (±10V for 2131).
POWER Switch : This switches the 17.5 Vdc power supply. The pilot lamp LED
indicates when power is on.
SIG OUT (Rear Panel) : BNC receptacle used to monitor the input signal on an external
indicating instrument such as an oscilloscope. The desired channel is selected with the
front-panel CHANNEL selector switch.
Input Connectors (Rear Panel) : 10 BNC receptacles (typ. Connected to 2120A
OUTPUT receptacles).
RESET Switches (2131) : AUTO (Toggle Switch) – When set to AUTO, the stored peak
reading is periodically reset to the existing input level. Automatic reset will occur
approximately every 5 to 10 seconds.
MAN (push-button Switch)- When pressed, resets peak reading to the existing input
level; the button should be held 1 second or more for complete reset.
273
PEAK MODE Switches (2131) : ON – Display reads the stored peak reading. When off
(down), display reads the existing input level.
MAX – The most positive (algebraic) input is stored.
MIN – The most negative (algebraic) input is stored.
SETUP
Install the 2130/2131 into the rack or enclosure as discussed in 4.0 Operating Procedures;
the 2130/2131 is installed in exactly the same manner as the 2120B Conditioners, filling
one of the rack or enclosure slots.
INPUT CONNECTIONS
Connect the OUTPUT connector of each 2120B channel to be displayed to the
appropriate 2130/2131 input, preferably using standard BNC cables. These can be
purchased locally or from Vishay Micro-Measurements. See Section 4.6a for a listing.
If desired, bridge voltage can be displayed by connecting banana plug jumpers between
the 2110B/2111 EXTERNAL METER jacks and the 2130/2131 EXTERNAL jacks
(connect red to red and black to black).
274
OPERATION
To prevent damage to small gages or sensitive galvanometers, before turning on power to
the 2110B/2111 and 2130/2131, complete all steps in 4.0 Operating Procedures through
4.9e.
Turn POWER on to both the 2110B/2111 and 2130/2131. SET the 2131 PEAK MODE
and RESET-AUTO switches to off. Continue on 4.10 utilizing the CHANNEL selector to
choose the desired channel for display. Observe the OUTPUT DISPLAY when adjusting
the 2120B balance and gain controls as well as when taking data. For convenience, the
OUTPUT DISPLAY may be set up to read directly in engineering units.
The ATTEN switch is normally used in the X1 position but the X10 psition is required
when the reading goes over 1999 counts and the display flashes (indicating overrange). In
the X1 mode, the 2130/2131 reads directly in millivolts (tens of millivolts in the X10
mode).
To use the 2131 without utilizing the peak reading feature, keep the PEAK MODE and
RESET toggle switches set to the off (down) position. To take peak readings using the
2131, achieve desired calibration as discussed in the above paragraphs and proceed as
follows depending upon type of input signal:
Non-Recurring Peaks
Set RESET-AUTO to off (down position).
Set PEAK MODE rotary switch to MAX and the toggle switch to ON.
Press RESET-MAN firmly (approximately 1 second).
Load specimen or structure through the peak value of interest.
Read OUTPUT DISPLAY.
If the MIN peak is of interest, turn PEAK MODE to MIN and press RESET-MAN again.
Notes
In both MAX and MIN modes, a peak reading can have either a positive or negative sign.
For example, if RESET results in a 1500 count reading (Static load offset), a – 440 count
input excursion from the offset level will result in a reading of 1060 in the MIN mode or
an unchanged reading of 1500 in the MAX mode. If, instead, the excursion were 200, the
reading would have been an unchanged 1500 in the MIN mode or 1700 in the MAX
mode.
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A very slow display change can be due to peak storage drift that is not necessarily due to
change in the strain amplitude. Typically, the MAX peak storage can drift in either
direction, whereas MIN peak storage tends to drift in the positive direction.
In the presence of 50/60 Hz pickup, the display will read slightly higher in peak reading
mode because the pickup appears to the 2131 as a normal (although small) dynamic
signal. Therefore, this pickup should be minimized by using twisted and shielded strain
gage input wiring.
Recurring Peaks
Set PEAK MODE toggle switch to off (down position).
Establish static load (if required) and cyclic load.
Set PEAK MODE to ON.
Switch RESET to AUTO.
Press RESET-MAN firmly (approximately 1 second) or wait for automatic reset
to reset display.
MAX or MIN should now be displayed according to the position of the PEAK
MODE control.
To read the opposite peak, set PEAK MODE rotary switch accordingly and
repeat prior two steps.
To determine the peak-to-peak amplitude, algebraically subtract the MIN
reading from the MAX reading.
A decreased cyclic strain amplitude will be reflected in the display after reset
occurs.
SERVICE
A schematic of the 2130 and the 2131 can be found on the next page. Replacement parts
can be obtained from the factory.
There is an internal adjustment in the 2130/2131 for span sensitivity. This can be
trimmed by applying exactly ±1.900V to the input and adjusting the potentiometer at rear
of meter until readout displays ±1.900. A five minute warm-up is recommended.
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REFERENCES
1. Alan Pope & Low Speed Wind Tunnel Testing.
J. J. Harper John Wiley and Sons, 1966.
2. E. Markland A First Course in Air Flow, Techquipment Limited, 1976.
3. R. C. Pankhurst & Wind Tunnel Techniques, Pitman Publishers, 1968.
D. W. Holder
4. E. L. Houghton & Aerodymics for Engineering Students.
A. E. Brock Edward Arnold Ltd.
5. L. Bernstein Force Measurement in Short-duration Hypersonic
Facilities.
AGARD – AG214, November 1975.
6. L. Bernstein Lecture notes on Experimental Methods in Mechanics of
Fluid.
7. Strain Gauges and Instrumentation. Technical Training
Programme, C –25. Measurement Group, U.S.A.
8. Modern Strain Gauge Transducers : Their Design and
Construction. Part I – IX, October 1984.
9. T. A. Cook A Notebook on the Calibration of Strain Gauge Balance
for Wind Tunnel Models. RAE Technical Note. No. Aero
2631, December 1959.
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