Xerox Project: Velocity Measurement & Correction System (VMCS)
Measurement of Velocity and Pressure in Supersonic Flow
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Transcript of Measurement of Velocity and Pressure in Supersonic Flow
Abstract
The study of supersonic fluid flow is one that has attracted so much
interest over the years. This is mainly as a result of the interesting behavior
of fluids when flowing at supersonic speeds, the corresponding changes in
their thermodynamic and mechanical properties; and most importantly, the
real‐world uses or applications such fluids can be put to. These real world
applications may be in the areas of Jet propulsion, machining and in some
cases fuel delivery systems.
Whenever an Engineer is confronted with a problem requiring the
application of fluid mechanics principles, it is often in the areas of optimized
fluid transportation and/or the use of a fluid to store, amplify or transmit
energy from one point to another. To be successful in his exploits, the
Engineer or scientist must be able to precisely measure and calculate
various fluid parameters like: temperature, pressure, density, velocity, Mach
number, etc; throughout the duration of the fluid flow processes.
This paper presents research data on present‐day means of measuring
fluid velocity and pressure during supersonic flow, the devices used in taking
these measurements, techniques employed and shortcomings encountered.
2
Acknowledgements
My profound gratitude goes first of all to God who has been the
source of my encouragement, inspiration and the strength with which I
move forward in life.
I also wish to acknowledge all the help provided by my family and my
friends. They are very instrumental to my success and it is my sincere wish
that they are successful in their various endeavors too.
3
Contents
Abstract………………………………………………………………………………………………….. 2
Acknowledgements………………………………………………………………………………… 3
Introduction……………………………………………………………………………………………. 5
Supersonic Fluid Flow Properties……………………………………………………………. 7
Categories of Flow Measuring Instruments…………………………………………….. 10
Pressure Measurement…………………………………………………………………………… 12
‐ Probing……………………………………………………………………………………. 12
Velocity Measurement……………………………………………………………………………. 15
‐ Laser Induced Thermal Acoustics…………………………………………….. 16
‐ Doppler Global Velocimetry…………………………………………………….. 17
‐ Planar Laser‐Induced Velocimetry……………………………………………. 18
‐ Rayleigh Scattering Velocimetry………………………………………………. 20
Emerging Technologies…………………………………………………………………………… 21
Constraints involved in Supersonic Flow Measurements………………………… 22
Conclusion………………………………………………………………………………………………. 25
References………………………………………………………………………………………………. 26
4
Introduction
Fluids are substances that deform continuously when subjected to a
shearing force. Liquid and Gaseous matter fall into the category of fluids;
and these fluids are further classified as compressible or incompressible
depending on whether the density of the fluid changes per unit time when it
is subjected to external pressures and imposed velocities.
Experimentally, it has been found that the speed of sound in any
medium depends on the Temperature, Density among other attributes as
regards that medium. Interestingly, it has also been found that the speed of
sound in any fluid medium has a relationship with the speed of the fluid.
This relationship is usually represented by the Mach number (symbol M)
named after the Austrian physicist and philosopher Ernst Mach. The Mach
number is a dimensionless parameter that denotes the ratio of the speed of
the fluid relative to its control boundary to the speed of sound in the same
medium.
Mathematically: M = u/a; in this equation, ‘u’ is the velocity of the
fluid and ‘a’ is the velocity of sound in the fluid.
5
Thus for M = 1, it means that the fluid is flowing with the same speed
as a sound wave would travel through it. For supersonic flow, M is larger
than 1 (M>1), which means that the fluid travels at a speed higher than that
characteristic of a sound wave travelling through the fluid in its present
state (this explains why the sound of certain aircraft‐engines trail behind the
craft during motion).
The study of supersonic flow is important as its understanding leads to
better utilization of fluid momentum in areas like: aircraft propulsion,
ballistics, fuel delivery and combustion in high speed Gas Turbines, etc. To
fully study flow phenomena, one must be able to accurately measure the
continuously changing fluid properties at various sections in the fluid flow
duct or path. It is as a result of this that researchers have devoted time to
develop probes, sensors, transducers, electronic modules, etc; all aimed at
fully observing and providing real‐time feedback on fluid flow properties in
order to properly monitor and regulate the particular device or process
affected by the fluid flow.
These measuring techniques and the measuring devices applied are
reviewed and analyzed in this paper.
6
Supersonic Fluid Flow Properties
Every fluid in motion has certain properties which give an indication of
its present thermodynamic and kinetic state. These properties include:
‐ Density: the ratio of the fluid mass per unit volume. This is
denoted by the symbol ‘ρ’.
‐ Pressure: the force the fluid exerts per unit area perpendicular
to the direction of the force. This is denoted by the symbol ‘P’.
‐ Temperature: a measure of the degree of hotness or coldness of
the fluid. This is denoted by the symbol ‘T’.
‐ Velocity: the rate at which the fluid changes displacement with
respect to time. This is denoted by the symbol ‘U’.
‐ Enthalpy: a measure of the amount of heat energy the fluid
possesses. This is denoted by the symbol ‘h’.
‐ Entropy: a measure of the degree of disorderliness of the fluid
molecules. This is denoted by the symbol ‘e’.
‐ Internal energy: a measure of the amount of intrinsic or
chemical energy of a fluid. This is denoted by the symbol ‘E’.
7
During supersonic flow, the fluid is moving at speeds higher than the
characteristic speed of sound in the medium (i.e. M>1). This type of flow is
usually encountered in supersonic wind tunnels and at the diverging portion
of De Laval nozzles and is characterized by decreased fluid pressure and
reduced fluid density.
For a fluid undergoing steady flow without any work done, the
characteristic energies the fluid possesses are: pressure energy (due to the
force exerted by the fluid per unit area), kinetic energy (due to the velocity
and momentum of the fluid) and potential energy (due to the position of the
fluid above some certain datum level). These energies are represented by
the Energy equation: dp/ ρ + dv2/2 + gdz = const. (in differential form.)
Pressure kinetic potential
From the equation above, one can see that for a fluid flowing with a
fixed amount of energy assuming no energy losses are associated with the
flow; a decrease in either the pressure or potential energy will result in a
corresponding increase in the amount of kinetic energy (this is true for
conservation of momentum to hold). Although it is possible for these fluids
flowing at supersonic speeds to continue moving at such high speeds,
irregularities in the fluid flow path, higher external pressures and other
8
sources of flow friction result in the fluid flow suddenly breaking down,
shedding its high kinetic energy and assuming higher pressure energy. A
fluid breakdown of this nature results in the formation of shockwaves which
are pressure waves of finite and small thickness.
9
Categories of Flow Measuring Instruments
For supersonic flows, the standard method of measuring the velocity
with a hot‐wire anemometer, based on the temperature dependence of the
electric resistance of the hot wire, is impracticable due to the poor response
of the instrument with respect to the nature of supersonic fluid flows. Also,
due to the very dynamic nature and high sensitivity of supersonic fluid flow
to obstructions and undulations in the flow path, accurate measurement of
fluid properties is not an easy task as the measuring instrument must have a
way of interacting with the measurand in order to capture desired data
about the measurand. As a result of this and other factors, various
instruments have been designed with the aim of measuring fluid properties
to a high level of accuracy. These instruments can be broadly divided into
two categories (based on the technique employed) as follows: Intrusive and
Non‐intrusive instruments.
Intrusive Instruments
Intrusive instruments are based on an operating technique that requires the
transducer to be placed integral to the control volume thereby interfering
10
with the flow process. Instrumentation that falls under this category is
usually characterized by the placement of probes, sensors or creation of
ducts along the flow passage for the purpose of fluid sampling. The
disruption of the control volume is a cause of some errors as certain energy
losses due to friction and the formation of shock waves are encountered. An
example of a device that falls under this category is a pressure probe.
Non‐Intrusive Instruments
Non intrusive instruments employ a technique that allows the desired data
about the measurand to be captured without physical contact with the
measurand. Although these devices still interact with the molecules of the
measurand as a result of energy exchange on the atomic level, the effect of
this energy exchange process is not very consequential to the accuracy of
the captured data. Devices that fall under this category usually emit
electromagnetic pulses from one direction, pass it through the flowing fluid
and capture the transmitted, reflected or refracted pulses from the opposite
direction. Properties of the fluid are then established by interpolation of
data regarding the composition of the received pulses.
An example of a technique that is non intrusive is Laser‐Induced
Thermal Acoustics (LITA)
11
Techniques Employed in Flow Measurements
PRESSURE MEASUREMENT
PROBING
Probing involves the insertion of a small device that incorporates a
sampling tube and a sensor/transducer; in the flow channel or duct. Among
these devices, the designer needs to be careful about the disturbances they
introduce in the flow. In subsonic flows, one must sample at the same rate
as the local stream velocity; this is called “isokinetic” sampling. Smaller is
often the better, but while working in supersonic flows, designers are also
concerned by the shock that will develop at the tip of the probe. A detached
shock will ruin the measurements, because flow will spill around the tip and
species separation can occur. Thus it is desirable that the shock be attached.
During fluid flow, a small portion of the fluid is sucked into the
sampling tube or channel in the probe. An installed sensor/transducer in this
sampling channel measure the properties of the fluid in the channel before
it is fed to the exhaust. Electric signals from the transducer are then fed to a
digitizer in an external circuit. The digitizer amplifies and converts the
previously analogue signals to digital signals which are then fed to an
electronic module that interprets the signals and displays the results of the
12
measurement on a suitable display device which may be an Oscilloscope or a
Cathode‐Ray tube monitor. The electronic module may also feed the signals
to a computer for storage.
The diagram below shows an example of a probe used for
measurement. Fluid flows from the left into the probe where it is analyzed.
Image courtesy of: ‘A Sampling Probe for Fluctuating Concentration Measurement in Supersonic Flow’; thesis by Olivier Christian Xillo.
Pressure probes can be used measure the stagnation pressure, the static
pressure, and the flow angle within a fluid stream. Static pressure probes are
sometimes used to obtain static-pressure measurements in the flow rather
than at the boundaries. A static pressure probe may simply be a cylindrical
tube placed parallel to the flow with static tapings located on its body. A
variety of nose configurations can be used on the static tubes. Some of these
probes are known as wedge probes, cone probes, disc probes, and Prandtl
tube probes. If a half-open tube is placed so that its open end faces the
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14
VELOCITY MEASUREMENT
For supersonic flows, the standard method of measuring the velocity
with a hot‐wire anemometer, based on the temperature dependence of the
electric resistance of the hot wire, is impracticable. In wind tunnels the
velocity is more often calculated from several pressure measurements made
using a system of pressure gages installed along the channel length and over
the channel cross‐section. However, this system is fairly clumsy and requires
the determination of calibration coefficients since the measurements are
not direct.
The following table is an example of a table of calibration constants for
an aspirating probe:
X a m
0 4.19743 0.469548
0.2 2.80557 0.488318
0.4 2.24425 0.503715
0.6 2.04011 0.518208
0.8 2.10811 0.537137
1.0 2.42563 0.592056
15
A commonly used equation that relates pressure and velocity is the
steady flow equation written as follows:
(ρU)1A1= (ρU)2A2 =
((Pt2√γ)/(√T t2R)) * (A2M2) * ((1 + ( M2(γ2‐1)/2))^((1+γ)/2(γ+1))
As a result of this clumsiness involved in indirect determination of
velocity in supersonic flows, several other approaches have been made as
follows:
LASER INDUCED THERMAL ACOUSTICS (LITA)
In the Laser‐Induced thermal acoustics technique, two focused and
crossed 1.06‐μm laser beams from a Q‐switched Nd: YAG (150 mega‐joule /
pulse / beam) induce two counter‐propagating sound‐wave packets in the
sample volume defined by the crossing region. These sound waves
constitute gas density gratings in the fluid. The ~ 100 dB (re 20 μPa) sound
pressure level corresponds to a fractional density change of ~ 10‐4 and as a
result, the technique can be characterized as non‐intrusive. A third laser
beam (probe at 532‐nm) intercepts the sound wave packets. The sound
packets reflect a tiny fraction of the incident probe intensity to a detector
positioned at the Bragg‐scattering angle. Flow velocity and sound speed are
16
determined from distinct Doppler shifts of Bragg‐scattered light from the
third laser beam.
Doppler Global Velocimetry (DGV)
Velocimetry has to do with the measurement of fluid speed or the
speed of sound in a fluid. The Doppler Global Velocimetry technique or DGV
for short invented by Komine (in 1990) applies the Doppler Effect (‐ the
change in the appearance of light waves or other types of waves from an
object that is moving away or towards an observer) in the measurement of
supersonic flow velocity. As part of the Doppler Effect, when the object
moves towards the observer, reflected light from the object appears to be
shifted towards the blue or shorter wavelength of visible light more than it
would have been if it were stationary; when the object moves farther, the
opposite effect occurs and the light appears to move towards the red or
longer wavelength – this shifting is referred to as the Doppler Shift.
Instruments based on this technique can measure the velocity of the
fluid because the Doppler Shift is proportional to the fluid speed relative to
the light sensing device. Basically, these instruments usually consist of a light
emitter, a light sensor (or Receiver), auxiliary components to tune the
emitted and absorbed rays, a transducer to convert the absorbed signals to
17
electric signals and a computer to compute, display and/or store output
data. The light rays are emitted at a predefined wavelength at a section of
the supersonic tunnel or flow‐tube; reflected rays are picked up by a sensor,
and result computations are done using optical geometry and Doppler
equations.
The diagrams below show typical velocity maps obtained using
Doppler Global Velocimetry.
Planar Laser‐Induced Velocimetry (PLIF)
When certain materials absorb various kinds of energy, some of the
energy may be emitted as light. This process involves two steps: (1) the
incidental energy causes the electrons of the atoms of the absorbing
material to become excited and jump from the inner orbits of the atoms to
the outer orbits; (2) when the electrons fall back to their original state, a
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18
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19
The measurements show that the free stream flow is diverging (velocity
increasing away from the centre of the image, which is the axis of
symmetry). This flow divergence is caused by using a conical instead of a
contoured nozzle. A conical shock wave is clearly visible and the maximum
velocity is achieved in outermost part of the shock layer, where the velocity
has a significant radial component even before it passes through the shock.
Rayleigh Scattering Velocimetry
Rayleigh scattering Velocimetry is another measuring technique that
applies Rayleigh scattering to the measurement of fluid velocity.
Rayleigh scattering is the scattering of electromagnetic radiation into
different wavelengths by very small particles of matter. It is responsible for
red sunrises and sunsets as well as the blue of the daytime; and this is as a
result of the scattering of different wavelengths of the sunlight based on the
position of the sun relative to the observer.
20
Emerging Technologies
Holographic Interferometry
Naturally electromagnetic waves can interfere with each other. This
Interference occurs when two or more waves overlap or intersect. For
example, interfering light waves are responsible for the colors occasionally
seen in soap bubbles ‐ the light waves that reflect off the inner surface of
the bubble interfere with light waves of the same wavelength that reflect off
the outer surface of the bubble. Some of the wavelengths interfere
constructively, and other wavelengths interfere destructively. Since different
wavelengths of light correspond to different colors, the light reflecting off
the soap bubble appears colored.
Holographic Interferometry involves the exploitation of the
phenomenon of interference in the study of fluid flow and the
representation of captured images as 3‐Dimensional images using a
photographic plate and lasers.
Experiments have investigated the usefulness of holographic
Interferometry as a flow visualization technique for studying supersonic flow
in air inlets.
21
Constraints encountered in Supersonic Flow Measurements
Despite the current advances in technology, science and related
disciplines, accurate measurements have always been a problem. This is
majorly due to the fact that a majority of instruments are not very dynamic
and as a result they can hardly provide real‐time feedback (their response
time is > 0).
The following are few of the common challenges involved in the
measurement of supersonic flow properties (limited to pressure and velocity
measurements):
• The standard method of measuring the velocity of a particular
flow process with a hot‐wire anemometer, based on the temperature
dependence of the electric resistance of the hot wire, is impracticable due
to the very low sensitivity and low response time of the instrument relative
to the speed of the flowing fluid.
• Also, the determination of the composition of a mixture by the
conventional sampling methods is not an easy task. Devices used for
sampling are often bulky and never designed to be used inside a small wind
tunnel. Consequently, gas analyzers are usually located outside the tunnel,
22
and samples taken in the tunnel test‐section are conveyed to the analyzer
via a set of tubing. This kind of system usually has the problem of very low
frequency response and thus in situ sampling is preferred.
When applying in situ sampling, one is confronted with the
disturbances they introduce in the flow as is usually the case with pressure
probes. In subsonic flows, one must sample at the same rate as the local
stream velocity; this is called “isokinetic” sampling. Smaller sizes of
instruments are often the better, but when working in supersonic flows, one
is also concerned about the shock that will develop at the tip of the probe. A
detached shock will ruin the measurements, because flow will spill around
the tip and species separation can occur. Thus the design has to be such that
the shock is attached. Unfortunately, there are no devices that satisfy these
constraints and are also capable of determining the composition of arbitrary
mixtures of several gases.
The diagram above shows a detached and an attached shock at the tip of a pressure probe.
23
• Laser velocity meters use either the Doppler Effect or scattered‐
light modulation on a system of interference bands. Such measurement
schemes are fairly complicated and to increase the scattered‐light intensity
a large number of aerosol particles of a fixed size are introduced into the
flow. This raises the question of whether the velocity of the scattering
particles is equal to that of the flow and this method of introducing of large
quantities of aerosol, cannot be used in the case of combustible mixtures.
• One challenge usually encountered in the application of Non‐
Intrusive methods in measurement is the presence of noise due to flow
luminosity, scattered laser light and the operating frequencies of certain
equipment. Equipment that involves Lasers and laser beams in
measurement are usually susceptible to this type of problem and presently
the signal to noise ratio is improved by using certain types of filters (e.g. UG‐
5 filters).
24
Conclusion
Taking any measurement in a supersonic flow process is a tedious task
that requires a great deal of care and meticulous handling. These
measurements involve the use of probes, light and electromagnetic energy
emitting devices, sensors. Electronic modules, computers, etc; and the
fundamental principle behind the operation of these devices lies in the
transduction process that involves the interaction of an instrument with the
measurand. Changes in the state of the instrument as a result of this
interaction are used to calibrate the instrument within acceptable limits of
errors but in spite of all the recent technological advancements, several
constraints exist which are taken care of with the application of correction
factors, sub‐equations, etc.
However the case may be, measurement in supersonic flow remains a
core and basic part of engineering and the future looks bright due to the
current trends in optimized engineering analysis.
25
References
[1.] J. ‐F. DEVILLERS, G. B. DIEP “Hot‐wire Measurements of Gas
Mixture concentrations in a Supersonic Flow” DISA INFO. 14, 1973 (pp. 29‐
36)
[2.] Forkey, J.N. (1996): Development and Demonstration of Filtered
Rayleigh scattering – A Laser Based Flow Diagnostic for Planar Measurement
of Velocity, Temperature and Pressure.
[3.] Princeton University Department of Mechanical and Aerospace
Engineering Technical Report2067, 1996.
[4.] Meyers, James F. (1992): Doppler global velocimetry – the next
generation? AIAA 17th Aerospace Ground Testing Conference, Nashville, TN,
paper no. AIAA‐92‐3897, July 6‐8, 1992.
[5.] Meyers, J. F. (2005): Doppler Global Velocimetry Measurements
of Supersonic Flow Fields. VKILS 2005‐01, Advanced Measuring Techniques
for Supersonic Flows.
26
[6.] R. C. Hart, G. C. Herring, & R. J. Balla, “Pressure Measurement in
Supersonic Air Flow by Differential Absorptive Laser‐Induced Thermal
Acoustics,” Optics Letters, Vol. 32, No. 12, 2007, pp. 1689‐1691.
[7.] R. C. Hart, G. C. Herring, & R. J. Balla, “Common‐Path
Heterodyne Laser‐Induced Thermal Acoustics for Seedless Laser
Velocimetry,” Optics Letters, Vol. 27, No. 9, 2002, pp. 710‐712.
[8.] Miles, RB and Lempert W (1990). Two‐dimensional
measurement of density, velocity, and temperature in turbulent high‐speed
air flows by UV Rayleigh scattering. Appl. Phys. B. 51: 1‐7.
[9.] Hiller B, Hanson RK (1988), Simultaneous planar measurements
of velocity and pressure fields in gas flows using laser‐induced fluorescence,
Appl. Opt., Vol. 27, 33‐48; Paul PH, Lee MP, Hanson RK (1989), Molecular
velocity imaging of supersonic flows using pulsed planar laser‐induced
fluorescence of NO, Opt. Lett., Vol. 14, 417‐419.
[10.] http://www‐g.eng.cam.ac.uk/whittle/
[11.] http://www.ntis.gov/