i

Abstract:

A supersonic wind tunnel is an experimental apparatus that simulates the flow field at

supersonic speeds (1.2 < M < 5). It is a test bed for examining the fluid mechanics and

associated fluid phenomena for air traveling around flying objects faster than the speed of

sound (M > 1). The supersonic nozzle is the core component of the supersonic wind tunnel.

The Mach number and flow properties are determined by the nozzle geometry. The present

research reports a design scheme for designing a convergent-divergent nozzle to produce a

flow that have a speed greater than the speed of sound (M > 1). The design scheme is based

on the method of characteristics.

ii

Table of Contents:

Abstract: ................................................................................................................................................... i

1 Introduction: .................................................................................................................................... 1

2 De Laval nozzle: Theoretical Background & Operating Conditions: ............................................. 1

3 Method of Characteristics ............................................................................................................... 3

4 Calculation of Flow field: Method of Characteristics ..................................................................... 5

5 Conclusion: ..................................................................................................................................... 8

6 References ....................................................................................................................................... 9

List of Figures:

Figure ‎2-1: Schematic of a de Laval Nozzle .............................................................................. 1

Figure ‎3-1: Characteristic lines downstream of a supersonic throat .......................................... 4

Figure ‎3-2: Notation for point-to-point method of characteristics calculations ........................ 4

Figure ‎4-1: Schematic point numbering [4, pp.529] .................................................................. 5

Figure ‎4-2: Flowfield along center line ..................................................................................... 7

Figure ‎4-3: Supersonic contour for maximum divergence angle 75o ........................................ 8

1

1 Introduction:

A two-dimensional nozzle can be divided into the following regions [1]:

1. The contraction, where flow is subsonic

2. The throat, where flow achieves sonic conditions

3. An initial expansion region, where the slope of the contour increases up to its maximum

value

4. The straightening region, where cross sectional area increases, while slope decreases to

zero

5. The test section with uniform and parallel walls.

In this project, the considered de Laval Nozzle is given by A/A* = 1+2.2(x-1.5) 2

which is

purely a converging – diverging nozzle. The nozzle does not have any straightening section.

That’s to say, when 0<x<1.5, the nozzle is converging, and when 1.5<x<3, the nozzle is

diverging and in expansion region.

2 De Laval nozzle: Theoretical Background & Operating Conditions:

A de Laval nozzle or convergent-divergent nozzle is a tube that is pinched in the middle as

shown in Figure 2-1. It was invented by Gustav de Laval in 1888 for use in steam turbines.

This nozzle heavily relies on the properties of supersonic flow to accelerate gases beyond

Mach 1. It is used to accelerate a pressurized gas passing through it to a supersonic speed,

and upon expansion, to shape the exhaust flow so that the pressure energy propelling the flow

is converted into directed kinetic energy. Because of this, the nozzle is widely used in some

types of steam turbines, and is used as a rocket engine nozzle. It is also used in supersonic jet

engines.

Figure ‎2-1: Schematic of a de Laval Nozzle

To accelerate gasses beyond Mach 1, there must be a chocked condition at the throat of the

nozzle

a. Chocked flow occurs when the exhaust velocity at the throat is Mach 1

2

b. In chocked conditions, a further increase of pressure in the combustion chamber will

not accelerate gases in throat beyond Mach 1

c. Acceleration beyond Mach 1 is caused by a drop in ambient pressure, or backpressure

d. Temperature and pressure drop as Mach number of exhaust gases increases

e. Higher Mach number of exhaust gases means greater thrust

The phenomena of this converging-diverging nozzle can be explained using the

following formula [2]

)1()1( 2 u

duM

A

dA

If M2 is less than 1, ΔA must be negative for Δu to be positive. That is to say:

f. An increase in velocity is related to a decrease in area for 0 ≤ M ≤ 1

g. On the contrary, an increase in Δu is related to an increase in ΔA for M > 1.

h. For supersonic flow, the velocity increases in diverging duct and decreases in the

converging duct.

The linear velocity of the exiting exhaust gases from the de Laval Nozzle can be calculated

using the following equation [3]:

)2(11

21

P

P

M

TRv e

e

Where,

Ve = Exhaust velocity at the nozzle exit, m/s

T = absolute temperature of the inlet gas, K

R = Universal gas constant = 8314.5 J/(kmol.K)

M= the molecular mass of the gas, kg/kmol

Pe = absolute pressure of exhaust gas at the nozzle exit, Pa

P = absolute pressure of inlet gas, Pa

In addition, the area-Mach relations equation given by [2, pp.5, 204] is following which is

extremely important to calculate the Mach number based on the area ratio:

11

2

2

2

* 2

11

1

21

M

MA

A

3

3 Method of Characteristics

In supersonic flow, there are lines along which pressure waves are propagated are called

characteristics lines in gas dynamics. Method of characteristics is a standard approach to the

calculation of the supersonic region of a converging-diverging nozzle; however it cannot be

used for the subsonic or transonic regions [2, pp. 454]. Hence, the Method of Characteristics

is a numerical procedure suitable to solve two-dimensional compressible flow field problems.

By using this technique, flow properties such as direction and velocity, can be calculated at

any distinct point throughout a flow field.

Three distinct properties of the method of characteristics in gas dynamics are given below

described by [4]:

1. Property: A characteristic in a two-dimensional supersonic flow is a curve or line

along which physical disturbances are propagated at the local speed of sound relative

to the gas.

It dictates that when a small disturbance is introduced into a subsonic flow, the effect of the

disturbance is felt throughout the entire flow field. However, when a small disturbance is

introduced into a supersonic flow, the effect of the disturbance is confined to that portion of

the flow field contained within a Mach cone defined by the Mach lines or waves (the zone of

influence).

In short, Mach lines = characteristics: lines of disturbance propagation (depends on domain of

dependence and region of influence)

2. Property: A characteristic is a curve across which flow properties are continuous,

although they may have discontinuous first derivatives, and along which the

derivatives are indeterminate.

It states that a characteristic line can be thought of as an infinitesimally thin interface between

two smooth and uniform, but different regions. The line is a boundary between two

continuous flows. Though the streamlines passing through a field of these Mach waves are

continuous, the derivative of the velocity and other properties are discontinuous. If there is a

discontinuity in the derivative of a variable, but not in the variable itself, the discontinuity is

said to be weak. Thus a Mach line is a weak discontinuity.

3. Property: A characteristic is a curve along which the governing partial differential

equations(s) may be manipulated into an ordinary differential equation(s).

It states that a characteristic is a curve along which the governing PDE becomes an ODE.

This property is significant because ODEs are often easier to solve than PDEs.

In a nutshell, method of characteristics is to calculate the slope of the characteristics, and to

calculate the change of flow variables along the characteristics: thus resulting expression is

the compatibility equation, and finally to solve the pair of ordinary differential equations. A

schematic view of characteristic lines is given in Fig. 3-1.

4

Figure ‎3-1: Characteristic lines downstream of a supersonic throat

For notations, the point which connects to P by a right-running characteristic line is

considered A, and the point connecting with a left-running line is considered point B, as

shown in Figure 3.2. Right-running characteristics are considered as C_. Similarly, left-

running characteristics are considered to as C+.

Figure ‎3-2: Notation for point-to-point method of characteristics calculations

Besides, a two–dimensional irrotational flow field can be written following using Cramer’s

rule [2,pp.11, 387-90]:

dydx

dydxa

v

a

uv

a

u

dydv

dudxa

v

a

u

xy

0

0

12

1

0

0

101

2

2

22

2

2

2

2

2

5

Setting Denominator = 0, we get the following characteristic equations:

)tan(

chardx

dy

Setting Numerator = 0, we get the following compatibility equations:

)()( sticcharacterialongtheCKconstM

)()( sticcharacterialongtheCKconstM

4 Calculation of Flow field: Method of Characteristics

The de Laval contour shape is given by the following equation:

A/A* = 1+2.2(x-1.5) 2. Based on the contour shape, the maximum divergence angle has been

calculated which is 75o. Since the given nozzle is symmetric about x-axis, the total

divergence angle has been divided by 2, and the value 37.5o has been fed into the program

with 5 points on the initial line. The schematic point numbering on the contour is given in

Fig. 4-1. Hence, points 1 to 5 have been assigned the theta values 37.5o, 30

o, 22.5

o, 15, 7.5

o,

and 0o respectively. Then, the Prandtl-Meyer angle ν (M) was calculated using the following

Prandtl-Meyer function based on initial Mach number = 1.22

Figure ‎4-1: Schematic point numbering [4, pp.529]

After this, K- , K+, and μ have been calculated using the following Equations:

)()( sticcharacterialongtheCKconstM

)()( sticcharacterialongtheCKconstM

The y-coordinate of point 1 in the example is arbitrarily chosen to be 1 “unit” [4, pp.528]

6

(a physical dimension corresponding to the throat half-height), and therefore the x-coordinate

From these points, the radius of the initial value line is determined by the following equation:

The coordinates of the other first five points have then been calculated using

and to form a curved sonic line. After the first five initial points, θ and ν of

downstream points have been calculated using the following equations of compatibility using

the initial points' flow field:

and

For non-boundary points, the slopes of the characteristic lines leading to the point in question

have been calculated using the following equations because it renders to reasonably better

answers [4, pp. 525]

and

From the above equations, the location of P is given by the following equations:

and

Solving this pair of equations simultaneously yields , the location of point P:

Equation (a) is used to calculate C_ for points on the centerline, while Equation (b) is used to

calculate C+ for points along the contour. The slopes of type right running characteristics of

contour points were calculated using Equation (c). Likewise, the slopes of left running

characteristics for centerline points were calculated using Equation (d)

and

After getting the Mach number at each point on the center line, the following isentropic

equations [2, pp.3, 80] have been used to calculate

,

, and

:

1

2

2

11

M

T

T

o

12

2

11

M

P

P

o

7

1

1

2

2

11

M

o

The calculated flow field is given in the Fig. 4-2 and the contour shape is given in Fig. 4-3.

Figure ‎4-2: Flowfield along center line

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

2

4

6Mach Variation across centerline

M

Position (m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.5

1Pressure Variation across centerline

P/P

0

Position (m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.4

0.6

0.8

1Temperature Variation across centerline

T/T

0

Position (m)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.5

1

/0 Variation across centerline

/ 0

Position (m)

8

Figure ‎4-3: Supersonic contour for maximum divergence angle 75o

5 Conclusion:

A supersonic wind tunnel with A/A* = 1+2.2(x-1.5) 2

has been designed in Matlab using

Method of characteristics based on the source codes taken from [5-7]. The flow field has been

calculated based on the maximum divergence angle provided from the area ratio. For the

given geometry, the design Mach number is 4.6588 , all other calculations are seemingly

feasible provided the entry points for initial value line and the number of points along the

contour in the expansion region be same.

0.1 0.2 0.3 0.4 0.5 0.6 0.7

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Supersonic Contour Image for Mach 4.6588

9

6 References

[1] Lu, F. K., Wilson, D. R., & Matsumoto, J. Rapid valve opening technique for supersonic

blow-down tunnel. Experimental Thermal and Fluid Science, Vol. 33, No. 3, 2009, pp. 551-

554.

[2] Anderson, J. D. Modern compressible flow: with historical perspective. 3rd

ed., McGraw-

Hill, Singapore, 2004.

[3] Sutton, G. P. Rocket propulsion elements - An introduction to the engineering of rockets.

6th

ed., Wiley-Interscience, New York, 1992, pp. 646

[4] John, J. E., & Keith, T. G. Gas Dynamics: Third Edition. Upper Saddle River, NJ:

Pearson Prentice Hall, 2006, pp. 494 – 551.

[5] Retrieved on 15 DEC 2012 from

http://www.mathworks.com/matlabcentral/fileexchange/14682-2-d-nozzle-design

[6] Retrieved on 15 DEC 2012 from

http://www.mathworks.com.au/matlabcentral/fileexchange

[7] Retrieved on 15 DEC 2012 from http://www.wpi.edu/Pubs/E-project/Available/E-project-

102809-221348