Pressure Measurements

Material prepared by

Alessandro Talamelli, Antonio Segalini &

P. Henrik Alfredsson

Literature •  Springer Handbook of Experimental Fluid

Mechanics, Cameron Tropea, Alexander L. Yarin , John F. Foss (2007), ISBN-10: 3540251413

•  Measurement in Fluid Mechanics Stavros Tavoularis, Cambridge University Press (2009) ISBN-10: 0521138396

•  Fluid Mechanics Measurements, R. Goldstein, CRC Press; (1996) ISBN-10: 156032306X

•  Low-Speed Wind Tunnel Testing, Jewel B. Barlow , William H. Rae, Alan Pope, Wiley-Interscience; (1999) ISBN-10: 0471557749

Measurement techniques

•  Type of Measurements

–  Local measurements –  Integral measurements

–  Direct measurements –  Non-direct measurements

–  Field measurements –  Surface measurements

–  Time averaged measurements –  Time resolved measurements

Measurement chain

Sensor

Transducer

Acquisition data systems

Evaluation data systems

Sensor

•  Element which changes its status when in “contact” with the quantity to be measured

Sensor characteristics I

•  Spatial and temporal resolution

Spatial and temporal resolutions are coupled because it is not generally possible to distinguish if the quantity to be measured varies with time or if there are pseudo-temporal variations caused by the passage of spatial disturbances.

Sensor characteristics II •  Accuracy

•  Intrusivity: Non-intrusive methods are really non-intrusive (?)

•  Interference

•  Robustness

•  Calibration

•  Linearity

•  Cost

Transducer •  Element which transforms the changes

in the sensor’s status in an output “signal”

•  Typically it is an electrical signal

Ideal sensor/transducer

• Output signal is proportional to the magnitude of the physical quantity

• The physical quantity is measured at a point in space

• The output signal represents the input without frequency distortion

•  Low noise on output signal • Sensor does not interfere with the

physical process • Output is not influenced by other

variables

Output Signal

S(t)s

u

x

y

z

u x y z( , , )

t

s

! ! !s t( )! ! !u t( )

PhysicalQuantity TransducerandSensor

Basic Definitions I

In an internal point of a fluid the pressure can be defined as the mean of the three normal stress components acting over three surface elements orthogonal to each other in the point at rest with respect to the fluid. For a fluid in motion this value is called static pressure (definition by Aeronautical Research Council ).

Basic Definitions II If the fluid is brought to rest with an isentropic and adiabatic process, the pressure rises until a maximum that is called the total pressure. The stagnation pressure is the pressure measured when the velocity is zero. The dynamic pressure is the difference between the total and the static pressure.

pdyn = ptot - ps

Basic definitions III

2

21Vpp stot !+=

If the Bernoulli’s law is valid we have Otherwise: the kinetic pressure is

which give us

2

21 Vpkin ρ=

2

21 Vpdyn ρ=

Static pressure

Ideally the static pressure should be measured with a pressure probe that moves at the same velocity as the fluid particle. But unpractical!

Instead chose a stationary probe with respect to the laboratory (or airplane), choose a suitable shape and position the probe in a place where the pressure is equal to the static pressure of the undisturbed flow.

Static pressure probe STATIC PROBES

Presence of holes at a distance of 3D from the leading edge and 8-10D from the stem.

Sensitivity to the inclination of the asymptotic flow with respect to the probe axis.

s

s

Static pressure probe

s

ps-patm

•  WALL TAPPINGS •  On a wing or in pipes and ducts the static

pressure can be measured using holes in the surface (give attention to the sensitivity of the dimensions and the shape of the holes).

Static probes

•  The probe must be aligned with the flow (this effect can be reduced by using several holes)

•  Since the pressure is measured with holes, then the same problems of the wall tappings must be considered –  Effects of tip shape (geometry depends from flow

regime) –  Effects of probe blockage –  Effects of hole position –  Effects of the support

Blockage effect

•  Nose acceleration and probe support effects may compensate

Wall tappings •  The flow is very complex in proximity of the

tapping (only low Re simulations) –  Effects of orifice shape –  Effects of orifice orientation –  Effects of surface orientation –  Shape and position of the cavity (minimum depth) –  Compressibility –  Effects of the tapping orifice condition –  Effect of the distance from the measured point

Effect of d+

•  More problematic for high Re McKeon and Smits MST (2002)

Effect of d+

•  Less influence when d increases McKeon and Smits MST (2002)

Total pressure probe The stagnation pressure is obtained when the fluid is brought to rest through an isentropic and adiabatic process. In subsonic flow the Pitot tube measures the stagnation pressure (French hydraulic engineer 1695-1771)

Total pressure probe •  In supersonic flow there is a stagnation pressure

loss over the shock wave that is formed in front of the tube

•  Flat, hemisherical or elliptic head. In supersonic flow typically sharp wedge front.

Total pressure

•  Incoming flow direction •  Local Reynolds number (viscosity) •  Mach number •  Velocity gradient •  Wall proximity •  Turbulence

),,',',',,Re,(

21 2

2

2

2

2

2

2

0

dy

Uw

Uv

UuMf

U

ppC dm

p αϑρ

=−

=

Total pressure probe

•  Effects of finite dimensions –  Pressure measured in a finite region (not a

single streamline) -> spatial averaging –  This effect can be limited with small probes

(be careful! : robustness, time response) –  Blockage (d/L)

•  Directional sensitivity

Total pressure probe – direction sensitivity

Less sensitivity to the inclination of the flow in respect to the longitudinal axis than the static pressure probe.

From Chue

Total pressure probe

•  Effects of viscosity (in high Reynolds number measurement Red can be low due to the small dimensions of the probe)

•  Viscous effect are negligible for ReD>100 •  For ReD>30

5.1Re101d

pC +=

Total pressure – velocity gradient

1) Indicated Pitot pressure > total pressure of the undisturbed flow if a Pitot tube is operated in a region where the total pressure varies in a direction ortoghonal to the asymptotic flow (e.g Boundary layer).

Total pressure – velocity gradient

1) Velocity gradient interference 2) With the presence of a flat wall parallel to the probe axis there could be a reflection effect with the consequent measured pressure higher than the total pressure. This effect is negligible for y>2d from the wall.

McKeon, Li, Jiang, Morrison and Smits MST (2003)

Total pressure – velocity gradient

•  Error 1) is normally corrected by changing the probe position rather than correcting the flow velocity

•  This is based on analytical displacement correction for a sphere in a velocity gradient

)(15.0, MacMillandy

==Δ

εε

)()(2

),17.01(18.0 2 ZagaroladydU

yUd

dy

cc

=−=Δ

ααα

)(),4tanh(15.0 McKeondy

α=Δ

Total pressure – velocity gradient

•  Wall correction

•  A new correction is proposed based on Preston probe pressure data

)(5.05.3exp015.0 MacMillandy

UU

⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛ −−=Δ

⎪⎩

⎪⎨

⎧

<<

<<

<

=+

+

+

1600110085.01108120.08150.0

dfordfordfor

dwδ

How important wall correction are ?

McKeon, Li, Jiang, Morrison and Smits MST (2003)

Velocity measurements with differential pressure probe: the Prandtl probe

•  Steady flow, low velocity, viscosity negligible: Bernoulli’s law holds

•  High velocity:

2

21Vpp stot !+=

( )pstot Vpp !" ++= 121 2

Corrective term f(M)

( )( )p

sttot ppV

!" +

#=

12

Measurement errors due to turbulence

A physical time-dependent quantity can generally be splitted in a mean part and in a fluctuating part

From Bernoulli’s law:

Taking the time-average:

Effect of the anisotropy

)(')()(')( tpPtptvVtv +=+=

( )22 ''221'' vVvVpPpP ststtottot ++++=+ !

( )22 '21 vVPP sttot ++= ρ Error due to

fluctuating velocity

Measurement errors due to turbulence

Another effect is linked to the radial gradient of the static pressure due to the fluctuations

This is of the opposite sign than the turbulence one. Therefore they compensate.

Velocity measurements - compressibility effects

εp Μ εp 0 0 0.1 0.0025 0.2 0.010 0.3 0.023 0.5 0.083 1.0 0.274

Note that the error in velocity is about half of εp !

Velocity measurements at a nozzle exit

•  Mass conservation equation:

•  Bernoulli’s law applied to a streamline passing on the reference section and to the nozzle exit:

exitexit AVAV =11

2211 2

121

exitexit VpVp !! +=+

V1 , A1 , p1 Vexit , Aexit , pexit

•  Combining the two relations:

•  If section 1 is characterized by a dimension much larger then the exit’s one, the corrective term can be neglected

!!"

#$$%

&'

(=

21

2

1

2

AAp

Vexit

exit

)

Velocity measurements at a nozzle exit

Pressure transducers

s

Absolute pressure transducer Measures the pressure relative to perfect vacuum pressure. (Example: barometer, used also for compressible flow) Gauge pressure transducer Measures the pressure relative to a given atmospheric pressure at a given location. (Example a tire pressure gauge). Vacuum pressure transducer This sensor is used to measure small pressures less than the atmospheric pressure. Differential pressure transducer This sensor measures the difference between two or more pressures introduced as inputs to the sensing unit.

Pressure transducers

s pd =ρ h sin(θ)

Pressure transducers

s

Betz manometer

Differential Pressure transducers

s

Capacitance principle

•  Very accurate •  Need to be calibrated (time to time) •  Expensive for multi point measurements

Pressure transducers

•  Need to be calibrate s

Pressure Scanners

•  Important in pipe/channel turbulence

Multi component velocity measurements

•  Five hole Pitot

•  Pressure distributions on the probe’s head is function of its geometry and of the flow direction

•  By sampling this distribution in five points is possible to determine the direction and magnitude of the velocity vector

Calibration

Multi component velocity measurements

•  Microphone: electromechanical transducer. The sensing element is a thin membrane that alters its shape under the pressure loading effect

•  High capacity to measure the pressure variations in the measurement point (the sensor measures variations up to at least 5 kHZ)

Time resolved pressure measurements

Time resolved pressure measurements

•  Capacitive type

•  Piezoelectric type

•  strain measurements

Time resolved pressure measurements

Pressure probe for measurements of pressure fluctuations inside the boundary layer (Tsuji et al 2007)

Time resolved pressure measurements

Frequency response for pressure probe of Tsuji et al. Frequenct is normalized with Helmholz resonator frequency.

Time resolved pressure measurements

Measured rms fluctuations of the pressure inside turbulent boundary layers at different Re. Lines are from numerical simulations.

Sensor characteristics •  Resolution √

•  Frequency response X

•  Accuracy √

•  Intrusivity X

•  Interference X

•  Robustness √

•  Calibration (X)

•  Linearity √

•  Cost √