-
Electronic Journal of Fluids Engineering, Transactions of the
ASME
Review of Hot-Wire Anemometry Techniques and the Range of
theirApplicability for Various Flows
P.C. StainbackDistingushied Research AssociateNASA Langley
Research Center
Hampton, VA 23681
and
K.A. Nagabushana†
Research AssociateOld Dominion University Research
Foundation
Norfolk, VA 23508
ABSTRACT
A review of hot-wire anemometry was made to present examples of
past work done inthe field and to describe some of the recent and
important developments in this extensive andever expanding field.
The review considered the flow regimes and flow fields in
whichmeasurements were made, including both mean flow and
fluctuating measurements.Examples of hot-wire measurements made in
the various flow regimes and flow fields arepresented. Comments are
made concerning the constant current and constant
temperatureanemometers generally in use and the recently developed
constant voltage anemometer.Examples of hot-wire data obtained to
substantiate theoretical results are presented. Someresults are
presented to compare hot-wire data with results obtained using
other techniques.The review was limited to wires mounted normal to
the flow in non-mixing gases.
† Currently Engineer at Computer Science Corporation, Laurel, MD
20707 Also, Consulting Research Engineer at Advanced Engineering,
Yorktown, VA 23693
NOMENCLATURE
a a1 4− constants in equation (47)a speed of soundA B, constants
in equation (24)A A1 3− constants in equation (18)b b1 3− order of
q in equation (47)B B1 5− constants in equation (19)
( )′A To Lkt′Aw overheat parameter, ( )12 ∂ ∂log logR Iw h( )′B
To 2 L k c rt p wπ
cp specific heat at constant pressure
cv specific heat at constant volume
cw specific heat of wired wire diameter of mesh( )d dt rate of
change of quantity ( ) with respect
to timedc diameter of cylinderdi diameter of jetdw diameter of
wire
′e instantaneous voltage across the wireE mean voltage across
the wireEout anemometer output voltage
′E finite-circuit parameter, ( ) ( )1 1 2− + ′ε εAwf
frequency
-
2Stainback, P.C. and Nagabushana, K.A.
F dimensionless frequencyF11 true one-dimensional spectral
densityFM measured one-dimensional spectral densityFtrf turbulence
reduction factor
Gr Grashof Numberh coefficient of heat transferhw height above
wire shock generator to probehw s, height above shock generator,
immediate
postshock valueI currentk thermal conductivity of air evaluated
at
subscript temperaturek1 wave number in the flow directionKn
Knudsen numberL characteristic lengthm mean mass flowmt ∂ µ ∂log
logt oTM Mach numberMm mesh sizen exponent for mass flow in
equations
(16) & (17)nt ∂ ∂log logk Tt oNu Nusselt number evaluated at
subscript
temperaturep mean static pressurepo mean total pressureP
electrical power to the hot-wirePr Prandtl numberq sensitivity
ratio, S Su Toq∞ dynamic pressureQ forced convective heat transferr
sensitivity ratio, S Sm Tord radial distance in cylinderical
polar
co-ordinaterp distance of virtual source of jet from origin
rw radius of wire′r r rp d−
R resistanceRe Reynolds number based on viscosity
evaluated at subscript temperature and wire diameter
RmTo mass flow - total temperature correlation
coefficient, ′ ′m T mTo oRuTo velocity - total temperature
correlation
coefficient, ′ ′u T uTo oR Toρ density - total temperature
correlation
coefficient, ′ ′ρ ρT To o
Ruρ velocity - density correlation coefficient,
′ ′u uρ ρRxx normalized auto-correlation functions sensitivity
ratio, S SToρS sensitivity of hot-wire to the subscript
variablet timeT temperatureTf ( )T Tw o+ 2u v w, , velocity in x
, y and z directions
respectivelyuτ frictional velocityx distance in the flow
directionxo virtual origin of the wakexw distance along the length
of wirey distance normal to the flow direction
α 1 12
21
+−
−γM
α1 linear temperature - resistance coefficient of wire
β ( )α γ − 1 2Mβ1 second degree temperature - resistance
coefficient of wireδ boundary layer thicknessδ* displacement
thickness for Blasius flowε finite circuit factor, ( )− ∂ ∂log logI
Rw w sε f finite circuit factor with fluid conditions
held constant while the hot-wire conditions change, ∂ ∂log logI
Rw w
ηy transformed co-ordinate distance normal to
bodyη recovery temperature ratio, T Tadw oθ temperature
parameter, T Tw oθ1 angle between plane sound wave and axis
of probeλ mean free pathµ absolute viscosityγ specific heat
ratio, c cp vρ densityτ time lagτw temperature loading parameter, (
)T T Tw adw o−τwr temperature parameter, ( )T T Tw adw adw−τwall
shear stress at the wall
′φ normalized fluctuation voltage ratio, ( )′e E STo
-
3Stainback, P.C. and Nagabushana, K.A.
Subscript
adw adiabatic wall conditionadw c, adiabatic wall temperature,
continuum flow
conditionadw f, adiabatic wall temperature, free molecular
flow conditionB due to buoyancy effectC constant current
anemometere edge conditioneff effective velocityf film conditionh M
and Ret are constant as Q variedo total conditionref reference
conditions electrical system untouched as M and Ret
varieds sound sourcesc settling chambert evaluated at total
temperatureT constant temperate anemometerw wire condition∞ free
stream or static condition
Superscript
' instantaneous ~ RMS mean
INTRODUCTION
Comte-Bellot1noted that the precise originof hot-wire anemometry
cannot be accuratelydetermined. One of the earlier studies of
heattransfer from a heated wire was made byBoussinesq2 in 1905. The
results obtained byBoussinesq were extended by King3 and
heattempted to experimentally verify his theoreticalresults. These
earlier investigations of hot-wireanemometry considered only the
mean heattransfer characteristics from heated wires. Thefirst
quantitative measurements of fluctuations insubsonic incompressible
flows were made in 1929by Dryden and Kuethe4 using constant
currentanemometry where the frequency response of thewire was
extended by the use of a compensatingamplifier. In 1934 Ziegler5
developed a constanttemperature anemometer for
measuringfluctuations by using a feedback amplifier to
maintain a constant wire temperature up to a givenfrequency.
In the 1950's, Kovasznay6,7 extended hot-wire anemometry to
compressible flows where itwas found experimentally that in
supersonic flowthe heated wire was sensitive only to mass flow
andtotal temperature. Kovasznay developed agraphical technique to
obtain these fluctuations,which is mostly used in supersonic flow.
Insubsonic compressible flows the heat transfer froma wire is a
function of velocity, density, totaltemperature, and wire
temperature. Because ofthis complexity, these flow regimes were
largelybypassed until the 1970's and 1980's whenattempts were made
to develop methodsapplicable8 for these flows. In recent years
therewere several new and promising developments inhot-wire
anemometry that can be attributed toadvances in electronics, data
acquisition/reductionmethods and new developments in
basicanemometry techniques.
Previous reviews, survey reports, andconference proceedings on
hot-wire anemometryare included in references 1,9-20.
Severalbooks21-24 have been published on hot-wireanemometry and
chapters25-30 have beenincluded in books where the general subject
matterwas related to anemometry.
This review considers the development ofhot-wire anemometry from
the earliestconsideration of heat transfer from heated wires tothe
present. Although mean flow measurementsare considered, the major
portion of the reviewaddresses the measurement of
fluctuationquantities. Examples of some of the moreimportant
studies are addressed for wires mountednormal to the flow in
non-mixing gases. Thepresent review attempts to bring the
development ofhot-wire anemometry up to date and note some ofthe
important, recent developments in thisextensive and ever expanding
field.
FLOW REGIMES AND FLOW FIELDS
Based on the applicable heat transfer lawsand suitable
approximations, hot-wire anemometry
-
4Stainback, P.C. and Nagabushana, K.A.
can be conveniently divided into the following flowregimes:
1. Subsonic incompressible flow2. Subsonic compressible,
transonic, and
low supersonic flows3. High supersonic and hypersonic flows
Within each of these major flow regimes are thefollowing
sub-regimes:
1. Continuum flow2. Slip flow3. Free molecular flow
In subsonic incompressible flow the heattransfer from a wire is
a function of mass flow, totaltemperature and wire temperature.
Since densityvariations are assumed to be zero, the mass
flowvariations reduce to velocity changes only. Thenon-dimensional
heat transfer parameter, theNusselt number, is usually assumed to
be afunction of Reynolds and Prandtl numbers andunder most flow
conditions the Prandtl number isconstant. Evidence exist which
indicate that Nut isalso a function of a temperature parameter6.
Insubsonic compressible, transonic and lowsupersonic flows the
effects of compressibilityinfluence the heat transfer from a wire.
For theseconditions the heat transfer from the wire is a
( )f u T To w, , ,ρ and ( )Nu f Re Mt t= , ,θ . In high
supersonicand hypersonic flows a strong shock occurs aheadof the
wire and the heat transfer from the wire isinfluenced by subsonic
flow downstream of theshock. Because of this, it was found
experimentallythat ( )Nu f Ret t= ,θ only, and the heat transfer
fromthe wire is again a function of mass flow, totaltemperature,
and wire temperature.
In continuum flow the mean free path of theparticles is very
much less than the diameter of thewire and conventional heat
transfer theories areapplicable. Where the diameter of the
wireapproaches a few mean free paths between theparticles, the flow
does not behave as a continuum,but exhibits some effects of the
finite spacingbetween the particles. These effects have
beenstudied31,32 by assuming a finite velocity and atemperature
jump at the surface of a body. Thisgas rarefaction regime was noted
as slip flow. In
free molecular flow the fluid is assumed to becomposed of
individual particles and the distancebetween the particles is
sufficiently large that theirimpact with and reflection from a body
is assumedto occur without interaction between the particles.Free
molecular flow is theoretically studied33 usingthe concepts of
kinetic theory34.
Figure 1 presents a plot of Mach numbervs. Reynolds number for
lines of constant Knudsennumber where dw = 0.00015 inch and for
flowconditions where 1.5 ≤ po , psia ≤ 150. Baldwinnoted35 that the
continuum flow regime existed forKn < 0.001 and slip flow
conditions existed for0.001 ≤ Kn ≤ 2.0. Other references suggest
thatslip flow conditions were attained only for Kn >0.01. Even
using the larger value of Kn for the slipflow boundary, (i.e., Kn
> 0.01) a 0.00015 inch wireoperated at low Mach numbers and at
atmosphericconditions is near the slip flow boundary. If thetotal
pressure is decreased or the wire diameterreduced, the value of Kn
would be shifted fartherinto the slip flow regime. Free molecular
flowconditions are generally assumed to exist for Kn >2.0.
Figure 1 can be used to delineate approximatevalues for M and Ret
for the various sub-regimes.
Various applications of hot-wireanemometry and the approximate
level of velocityfunctuations are:
Types of Flows Approximate29u u
1. Freestream of wind tunnels36,37 0.05%2. Down stream of
screens and grids25,38
0.20 - 2.00 %
3. Boundary layers39-42 3.0 - 20.0 %4. Wakes43,44 2.0 - 5.0 %5.
Jets45-47 Over 20.0 %6. Flow downstream of shocks48
7. Flight in Atmosphere49
8. Rotating Machinery50,51
9. Miscellaneous52-56
TYPES OF ANEMOMETERS
The two types of anemometers primarilyused are the constant
current anemometer (CCA)and the constant temperature anemometer
(CTA).
-
5Stainback, P.C. and Nagabushana, K.A.
A constant voltage anemometer (CVA) is presentlyunder
development57. Even though these threeanemometers are described as
maintaining a givenvariable "constant", none of these
strictlyaccomplish this. The degree of non-constancy forthe CCA is
determined by the finite impedance of itscircuit58. The constancy
of the mean wiretemperature for a CTA at high frequencies is
limitedby the rate at which the feedback amplifier candetect and
respond to rapid fluctuations in the flow.The CVA maintains the
voltage across the wire andleads constant rather than across the
wire57. Thenon-constancy effects in the CCA and the CVA canbe
accounted for by calibration of the CCA58 andby knowing the lead
resistance in the CVA.
The heat balance for an electrically heatedwire, neglecting
conduction and radiation is:
Heat Stored = Electrical Power In - AerodynamicHeat Transfer
Out
dcdt
T P Qw w = − (1)
( )dcdt
T I R Ld h T Tw w w w w adw= − −2 π (2)
If the heat storage term is properly compensated,then equation
(2) becomes:
( )I R Ld h T Tw w w adw2 = −π (3)
The measurement of fluctuations in a flowrequires a sensor, in
this case a wire, with a timeresponse up to a sufficiently high
frequency. Thetime constant of even small wires are limited andthe
amplitude response of these wires at higherfrequencies decreases
with frequency. Therefore,some type of compensation must be made
for thewire output. There are two methods foraccomplishing this.
Earlier approaches utilized aconstant current anemometer with a
compensatingamplifier that had an increase in gain as thefrequency
increased4. An example of the roll off inthe frequency of the wire,
the gain of the amplifierand the resulting signal is shown
schematically infigure 2. In principle, the output from the wire
canbe compensated to infinite frequencies. However,as the frequency
increases, the noise output fromthe compensating amplifier will
equal and
ultimately exceed the wire output, which limits thegain that can
be obtained. A schematic diagram ofa CCA is presented in figure
3.
The constant temperature anemometeruses a feedback amplifier to
maintain the averagewire temperature and wire resistance constant
{i.e.,dT dtw = 0 in equation (2)}, within the capability ofthe
amplifier. The practical upper frequency limitfor a CTA is the
frequency at which the feedbackamplifier becomes unstable. A
schematic diagramof a CTA is presented in figure 4. A
thirdanemometer, presently under development57, isthe constant
voltage anemometer. Thisanemometer is based on the alterations of
anoperational amplifier circuit and does not have abridge circuit.
A schematic diagram of a CVA ispresented in figure 5.
The upper frequency response of a CCA isgenerally accepted to be
higher than that of a CTA.There is some evidence that the frequency
responseof the CVA might equal or exceed that of the CCA.The
fluctuation diagram technique described byKovasznay is usually used
with a CCA to obtaindata at supersonic speeds. This technique
dependson the sensitivity of the wire being a function of
wiretemperature or overheat and the frequencyresponse of the wire
being assessable tocompensation to almost zero overheat.
Thistechnique has limited application for a CTA, since atlow
overheats, the frequency response of theanemometer approaches the
frequency response ofthe wire25.
An example of the difference between thefluctuation diagrams
obtained59 using a CTA and aCCA is presented in figure 6. The
intersection ofthe diagram with the vertical axis at S Su Toρ =
0
represents the total temperature fluctuation andthe data show
that the CTA cannot be used tomeasure these fluctuations. The
reason for this isillustrated in figure 7a where the total
temperaturespectra at low overheats for the two anemometersare
presented. In these cases the spectrumobtained with the CTA was
attenuated at afrequency that was about two orders of magnitudeless
than for the CCA. At high overheats the twomass flow spectra were
more nearly equivalent(figure 7b). However, in reference 60 the
output ofa laser was modulated and used to heat a wire to
-
6Stainback, P.C. and Nagabushana, K.A.
check the frequency response of a CTA. It wasshown that the
frequency response was essentiallyunchanged down to an overheat of
0.07.
The CTA can be used to makemeasurements in supersonic flows by
using twowires. For these flows the CTA is operated with twowires
having different but high overheats, digitizingthe voltages and
using two equations to obtain ′m ,
′To and ′ ′m To as a function of time61. Thenstatistical
techniques can be used to obtainquantities of interest. In general,
the CTA is moresuitable for measuring higher levels of
fluctuationsthan a CCA25. It remains to be determined howthe CVA
will compare with the CCA and CTA. Atpresent it appears that the
CVA has a higher signalto noise ratio than either CCA or CTA.
Additionaladvantages and disadvantages of the CCA versusthe CTA are
described in references 1, 29, 57, 62and 63.
At low speeds a linearizer is often used toconvert the
non-linear relationship between wirevoltage and velocity to a
linear relationship. Thereare two types of linearizers in use; the
logarithmicand the polynomial. A linearizer makes it possibleto
directly relate the measured voltage to thevelocity. However, the
linearization process doesnot result in better measured
quantities25.
LIMITATIONS OF HOT-WIRE ANEMOMETRY
Most of the data obtained using hot-wireanemometry is limited to
small perturbations.There are cases, however, where this
linearization ofthe anemometry equation is not accurate and
non-linear effects can influence both the mean47 andfluctuating25
voltages. Since high level fluctuationscan influence the mean
voltage measured acrossthe heated wire, it is important to
calibrate probesin flows with low levels of fluctuations.
Because of the mass associated with thewire supports, there can
be a significant amount ofheat loss from the wire due to conduction
to therelatively cold supports. This heat loss results in aspanwise
temperature distribution along the wirethat, in turn, causes a
variation of heat transferfrom the wire12,21,64 along its length.
In order tocompare the heat transfer results from one wire or
probe with another, the heat transfer rates must becorrected for
these losses. However, computation offluctuation quantities
requires that the uncorrectedvalues of the heat transfer rates be
used. Anexample of the temperature distribution along awire and its
mean temperature21 is shown in figure8. The finite length of the
wire and its attendanttemperature and heat transfer
distributioninfluences the level of the spectra (especially
athigher frequencies), correlations, and phaserelationships between
sensors25,65.
The spacial resolution of a wire is limited bythe length of the
wire and the size of the smallestscales of fluctuations in the
flow. If the length ofthe wire is larger than the smallest scale,
theresultant magnitude of the spectra will beattenuated at the
higher frequencies. The length ofthe wire with respect to the size
of turbulence canhave an effect on the measurements of
fluctuationintensity, space and time correlations, and
theturbulence scales and micro scales66-68.Additional spacial
resolution problems encounterednear walls were discussed in
references 69 and 70.Proximity to walls of wind tunnels or to
surfaces ofmodels can introduce errors in measurements dueto
increased heat transfer from the wire due toconduction to the
relatively cold walls21,25. Anexample of the effect of wire length
on normalizedspectra is presented in figure 9. The
spacialresolution of multi-wire probes is further limited bythe
distance between the wires. The hot-wire probeintrusion into the
flow can cause severedisturbance in certain flows. Examples are
flowswith large gradient such as boundary layers andvortices.
Because of the above, hot-wireanemometry has limited resolution in
space, time,and amplitude29.
A severe problem is encountered inhypersonic flows when the gas
is air. At higherMach numbers the total temperature must be
highenough to prevent liquefaction of air in the testsection. There
is a maximum recommendedoperating temperature for each wire
material.These two facts places severe limitations on themaximum
overheat at which wires can be operated.For example, the maximum
recommendedoperating temperature for Platinum-10% Rhodiumwire is
1842°R. For a M = 8 wind tunnel, the totaltemperature required can
be as high as 1360°R.
-
7Stainback, P.C. and Nagabushana, K.A.
Using a recovery temperature ratio of 0.96, themaximum values
for τw is 0.394 and θmax = 1.354.If gas rarefaction effects are
experienced and η isgreater than one, then the problem is even
moresevere. For η = 1.1 the maximum value for τwunder the above
conditions is 0.254. The abovevalues for τw are based on the
average temperaturefor the wire. For small L dw wires the
limitation onτw would be greater due to higher temperatures atthe
mid-portion of the wire. The total temperatureat low pressures
where η could be larger need notbe as high as those at higher
pressures, however,the constraint of constant total temperature
duringthe calibration process limits the amount that Tocan be
reduced. (Also see ref. 71-76).
PROBE PRE-CALIBRATION PROCEDURE
Once a probe is constructed, the followingprocedure should
ensure accurate and reliablemeasurements. First, the probe should
be operatedat the maximum q∞ and Tw that will be used duringthe
proposed test. This is done to pre-stress andpre-heat the wire to
ensure that no additional strainwill be imposed on the wire during
the test thatcould alter its resistance. For supersonic and highq∞
subsonic flows, the wires should also be checkedfor strain gaging,
that is, stresses generated in thewire due to its vibration. Note,
for testing in flowshaving high values of q∞ , the wires should
haveslack to reduce the stress in the wires and to helpeliminate
strain gaging. If strain gaging issignificant the wire should be
replaced. During thispre-testing many wires will fail due to faulty
wiresor manufacturing techniques, but it is better thatthe wires
fail in pre-testing rather than during anactual test.
A temperature-resistance relationship forwires is usually
requires to compute the heattransfer rate from the heated wires. It
is generallyrecommended that the following equation, which isa
second degree equation in ∆T , be used:
( ) ( )RR T T T Tref w ref w ref= + − + −12
1 1α β (4)
After the wires have been pre-stressed and pre-heated, they
should be placed in an "oven" and the
wires calibrated to determine the values for α1 andβ1. Once this
calibration has been completed, theprobes can be placed in a
facility for mean flowcalibration over the appropriate ranges of
velocity,density, total temperature and wire temperature.
STATISTICAL QUANTITIES
Data obtained using hot-wire anemometryare typically reduced to
statistical quantities. Overthe past few years the analysis of
random data hasbeen developed to a very high degree77-79. Thisplus
the rapid developments in electronics (i.e., theA/D converters and
high speed computers), havemade it possible to obtain almost any
statisticalquantity of interest within the error constraints ofthe
heated wire. Much of this is due to the fact thatthe digital
processing of data can be used to obtainmany quantities that are
difficult or impossible toobtain using analog data reduction
techniques.
Many types of single point and multi-pointstatistical quantities
can be obtained using hot-wireanemometry80-83. It is routine to
measure meanflow and RMS values, histograms and the higherorder
moments of skewness and kurtosis, autocorrelation, and one
dimensional spectra.Measurements of multi-point statistical
quantitiesinclude cross correlations, two-point histogramsand
higher order two-point moments, cross spectra,and coherence
functions. Attempts were made tomeasure higher moments up to eighth
order81.
These measurements can be used invarious ways to evaluate many
characteristics ofthe flow such as scales, decay rate, energy
contentetc25. The coherence function is a useful
statisticalquantity that can be used to evaluate variousproperties
of a flow84. It can often be used todetermine the predominant sound
propagatingangle and to determine the dominant mode presentin a
fluctuating flow field85,86.
A few examples of statistical quantities thatwere measured using
hot-wire anemometry arepresented in figures 10-13. Integral and
micro timeand length scales of a flow can be determined
fromautocorrelation functions such as the one presentedin figure
10. The higher moments of skewness andkurtosis (figure 11a-b) can
be used to determine if
-
8Stainback, P.C. and Nagabushana, K.A.
the fluctuations are Gaussian. For a Gaussiandistribution the
value of the skewness parameter iszero and for the kurtosis the
value is 3. Figure 11shows that both of these moments indicate that
themass flow and total temperature fluctuations areGaussian over
most of the thickness of theboundary layer. The value of third
order auto-correlation function, such as the one shown infigure 12,
can be used to support turbulent flowtheories. An example of
space-time correlationsmeasured in a turbulent boundary 59,87
ispresented in figure 13. The peak of thesecorrelations at t ≠ 0
indicate the presence ofconvection. The calculation of the
convectionvelocity, obtained by dividing the separationdistance by
the time at which the individual curvespeaks, indicates that there
was no significantvariation of the convective velocity over the
spacingsused. An example of normalized spectra measureddownstream
of a grid88 is presented in figure 14and show the increased
attenuation of highfrequency disturbances with increased
distancedownstream from the grid. (Also see ref. 89-92).
GENERAL HEAT TRANSFER RELATIONSHIPS
The heat transfer from a wire under thelimits of the present
report (i.e., the wires mountednormal to flow in non-mixing gases)
is 29:
Q f u c T Tp w adw= ( , , , , , )µ ρ (5)
if the fluid properties of µ, cp , and k are based on
To , then the above equation becomes:
Q f u T Tw o= ( , , , )ρ (6)
Since T Tadw o= η and ( ) ( )η ρ= =f Kn M f u To, , , .
Forincompressible continuum flows equation (6)reduces to:
Q f m T To w= ( , , ) (7)
Unless noted, the total temperature will be usedthroughout this
report to evaluate µ t , cp , and ktwhere as ρ will be based on T∞
.
For a wire with a given L dw the Nusseltnumber can be expressed
25 in terms of otherdimensionless parameters as:
( )Nu f Re Pr GrT T
Tu
c T Tt tw adw
o p w adw
=−
−
, , , ,
2(8)
and can be written as follows to show the effects
ofcompressibility:
Nu f Re Pr Gr MT T
Tt tw adw
o
=−
∞, , , , (9)
For relatively constant temperatures, Pr = constantand if Gr
Re< 3 , buoyancy effects will be small andGr can be neglected.
These approximations lead to:
( )Nu f Re Mt t w= ∞, ,τ (10)
MEAN FLOW MEASUREMENTS
SUBSONIC INCOMPRESSIBLE - CONTINUUMFLOW
Theoretical ConsiderationsThe functional relationship between
the
power to the wire or the heat transfer from the wireand the mean
flow variables are required todetermine the so called "static"
calibration of thewire from which the sensitivities to the various
flowvariables can be obtained in order to calculate
thefluctuations. Because of this the mean flow resultsand probe
mean flow calibration procedure areconsidered together.
The first attempt to obtain a theoreticalsolution for the heat
transfer from a heated wiremounted normal to the flow was carried
out byBoussinesq2. The equation that he obtained is:
( )( )Q L k c ur T Tt p w w adw= −2 π ρ (11)
Equation (11) can be expressed in terms of non-dimensional
quantities as follows:
-
9Stainback, P.C. and Nagabushana, K.A.
Nu PrRet t=2π
(12)
King re-analyzed the problem of heat transfer froma heated wire
and obtained the followingrelationship:
( )( )Q L k k c ur T Tt t p w w adw= + −2 π ρ (13)
or in terms of non-dimensional quantities:
Nu PrRet t= +1 2π π
(14)
From equation (11) and (13) it can be seenthat the only
difference between Boussinesq's andKing's results is the inclusion
of the additional termkt in King's result that attempts to account
for theeffects of natural convection. At "high" values ofReynolds
number the two results are essentiallyequal.
Using equation (3), equation (13) can beexpressed as:
( ) ( )[ ][ ]P A T B T m T To o w adw= ′ + ′ − (15)
where ( )′A To and ( )′B To are based on King's results.However,
the quantities ′A and ′B are usuallydetermined for a given wire by
direct mean flowcalibration. Often the exponent for the mass
flowterm is determined from a curve fit to the data. Thevalues for
the exponent can range25 from 0.45 to0.50.
For a CTA, equation (15) can be generalizedto:
( ) ( ) ( )E
R RA T B T mn
w adwT o T o
2
−= + (16)
where ( ) ( )A T R A TRT o
w o
ref
=′
α 1 and ( ) ( )B T R B T
RT ow o
ref
=′
α 1.
For a CCA:
( ) ( ) ( )I R
R RA T B T mnw
w adwC o C o
2
−= + (17)
where ( ) ( )A T A T RC o o ref= ′ α1 and ( ) ( )B T B T RC o o
ref= ′ α1 .Therefore, if wires operated with a CTA or a CCA
arecalibrated over a range of m and To , equations (16)and (17)
indicate that the calibration curves will bestraight lines if the
left hand side of the equation is
plotted as a function of mn . In general, the slopes,B , and
intercepts, A , will be functions of the totaltemperature. If the
total temperature is constant,A and B will be constants and if the
density isconstant, the mass flow term will reduce to velocity.An
example of voltage versus velocity for a wireoperated with a CTA is
presented in figure 15 forvarious values of total temperature.
(Also see ref.93-97).
Hot-wires have also been calibrated in theform of ( )u f E=
rather than the more conventionalform of ( )E f u= . The constant
To and ρ version ofKing's law for a CTA is E A B u2 = + and
whenexpressed as ( )u f E= gives:
u A A E A E= − +1 2 32 4 (18)
In this equation George et. al.,98 noted that A A1 3−are
functions of To . They proposed the followingequation for the
calibration of wires that isindependent of To for a limited
range:
Re B B Nu B Nu B Nu B Nut = + + + +1 2 3 4 512
32 2 (19)
where µ is evaluated at To and k evaluated at Tf .
Examples of DataA summary of heat transfer data from
cylinders in terms of Nut vs. Ret taken in thesubsonic continuum
flow regime was presented inreference 21. The results from these
experimentsare compared with the theoretical results ofBoussinesq
and King in figure 16. This figureshows that there is a relatively
good agreementbetween the measured results and King's theoryover a
wide range of Reynolds numbers. There is asubstantial difference
between Boussinesq's theory
-
10Stainback, P.C. and Nagabushana, K.A.
and King's theory and the measured results forReynolds numbers
less than about 100.
A large amount of heat transfer data wasalso presented by
McAdam99 in terms Nu f vs. Re fand he recommended the following
equation:
( )Nu Ref f= +032 0 43 052. . . (20)
In comparision, King's equation with Pr = 0.70 is:
( )Nu Ret t= +0 3183 0 6676 0 50. . . (21)
For Reynolds number equal to zero the twoequations give
essentially the same value for theNusselt number. At higher values
of Reynoldsnumber King's equation is about 40 percent higherthan
the values of Nusselt number presented byMcAdam. A film temperature
is often used as thetemperature at which k , ρ and µ are
evaluatedwhen correlating Nu versus Re. However, ρ issometimes
evaluated at the free stream statictemperatures. The use of the
film temperature forevaluating fluid properties has been questioned
inreference 100. However, in this reference thedensity in the
Reynolds number was evaluated atTf , where as, in reference 99 this
apparently was
not the case.
Bradshaw 27 notes that there is a differenceof opinion
throughout the hot-wire anemometrycommunity about the usefulness of
a universalcorrelation based on variables evaluated at a
filmtemperature. These correlations provide a usefulguide for
plotting results and comparing mean flowresults obtained by
different investigators.However, if good accuracy is to be obtained
for thefluctuations, individual calibration of probes
isrequired.
Often attempts are made to measure meanvelocities using hot-wire
anemometry. It can beshown using equation (13) that the voltage
across awire is a ( )f u T To w, , ,ρ . Therefore, to measure
themean velocity, the other variables must be heldconstant or a
method must be used to correct thedata for any variation in
variables other thanvelocity65,71,101. Because of this, the
hot-wireanemometer is not a very good mean flow
measuring device, even if one utilizes some of thecorrections
that have been developed. However, forlimited variations in the
independent variables,corrected velocity measurements were reported
inreference 101.
Very Low VelocitiesAt very low velocities the heated wire
can
cause a relatively significant vertical movement of afluid due
to buoyancy effects on the lower densityfluid adjacent to the wire.
This results in a changein the effective velocity around the wire.
Effortswere made to calibrate and use hot-wireanemometry at very
low velocities102-107 wherenatural convection effects were present.
Theinfluences of natural convection are parameterizedby the Grashof
number. Experimental evidence108
indicated that if Gr Re< 3 then free convection effectswere
negligible. An example of the effect of lowvelocities on the
Nusselt number is presented infigure 17.
King also provided an equation suitable forlow speed flows:
( )( )P
Lk T T
b rt w adw
w
=−2π
log(22)
where b k e c ut p=−1 γ ρ and γ = Euler's Constant =
0.57721. In terms of non-dimensional quantities,equation (22)
becomes:
( )[ ]Nu e Pr Ret t= −2
2 1log γ(23)
Equation (22) is valid for udw < 0.0187 where u isin cm/sec
and dw is in cm. Equation (13) is validfor udw > 0.0187.
For velocities as low as 1.0 cm/sec, Hawand Foss102 attempted to
correlate their data usingKing's equation in the form:
E A Bun2 = + (24)
A deviation of their data from a fitted curve wasobserved at u ≈
30 cm/sec. The diameter of the wireused in their experiment102 was
not noted.However, if one assumes a value of 0.00015 inch or
-
11Stainback, P.C. and Nagabushana, K.A.
0.00020 inch, the limits for the application ofequations (13)
and (22) indicate velocities of 49cm/sec or 37 cm/sec, which are
not too differentfrom 30 cm/sec. The use of equation (22) wouldnot
improve the correlation presented in reference102 since it can be
shown that as u → 0 in equation(22) Ew → 0. The data of reference
102 indicatesthat at u = 0 the intercept of the curve is
greaterthan the value indicated by the intercept inequation (24).
Correlations obtained using theresults of a theory based on
Oseen108 flow wouldnot improve the correlation since this
approachgives results that are similar to those obtainedusing
equation (22). For a heated wire tested inhorizontal wind tunnels,
ueff cannot reach zero since
the effective velocity is:
u u ueff B2 2 2= + (25)
and for a heated wire uB ≠ 0 .
SUBSONIC SLIP FLOW AND TRANSONIC FLOW
Theoretical ConsiderationsThese two flow regimes will be
treated
together since the experimental results are similar.Kovasznay6
extended hot-wire anemometry resultsto compressible flows and
showed that there was asignificant difference between the heat
transfer incompressible and incompressible flows.
Severalexperimenters obtained heat transfermeasurements at low
speeds and found anapparent compressibility or Mach
numbereffect35,64,109 at Mach numbers as low as 0.1.Spangenberg110
conducted extensive tests over awide range of variables and
determined that theapparent compressible flow effects at Mach
numberas low as 0.05 was really due to gas rarefaction(e.g., slip
flow).
In this flow regime the heat transfer fromthe heated wire is
generally given as:
( )Q P Lk T T Nut w o t= = −π η (26)
In transonic flow and subsonic slip flowsthe Nusselt number is
no longer only a function ofReynolds number and Kings' law is no
longerapplicable. The most common functional
relationship for the Nusselt number in these flowregimes
is58:
( )Nu f M Ret t= , ,θ (27)
since it was found that Nut is also a function of atemperature
parameter. Another functionalrelationship that was used to analyze
gasrarefaction effects is35:
( )Nu f M Knt w= , ,τ (28)
In subsonic compressible flows the recoverytemperature of the
wire can change and functionalrelationships for η are:
( )η = f M Ret, or ( )η = f M Kn, (29)
DependentVariable
IndependentVariable
Reference
Nut Ret , M∞ , θ Morkovin58
Nut Kn, M∞ , τw Baldwin35Nut Ret , M∞ , τwNut Kn, M∞ , θη Ret ,
M∞ Morkovin58
η Kn, M∞ Vrebalovich111
Table I. Functional Relationships for Nut and ηη .
Morkovin chose ( )Nu f M Ret t= ∞ , ,θ and( )η = ∞f M Ret, for
the development of his equations.
In order to emphasize the gas rarefaction effects,Baldwin chose
( )Nu f M Knt w= ∞ , ,τ and ( )η = ∞f M Kn, .Independent variables
that might be used to relateNut and η to the dependent variables
are presentedin Table 1. Although Morkovin and Baldwin chosethe
variables of and , one could just as wellhave chosen the variables
noted in or . It willbe shown later that the variables in might be
themost efficient group to use.
Examples of DataBaldwin35 and Spangenberg110
investigated the heat transfer from wires over awide range of
Ret , M, and Tw in the slip flow andtransonic flow regimes. Their
results, presented infigures 18a-b, shows that ( )Nu f Re Mt t= ,
for Machnumbers ranging from 0.05 to 0.90 and Reynolds
-
12Stainback, P.C. and Nagabushana, K.A.
numbers ranging from 1 to about 100. The effectsof wire overheat
on the values of Nut were alsodetermined by Baldwin and Spangenberg
andexamples of these effects are shown in figure 19.The values of
Nut can increase or decrease withincreased overheat depending on
the Mach numberand Knudsen number.
Results from theoretical calculations madefor the effects of
slip flow on heat transfer fromwires were reported in reference 31.
An example ofthese results is presented in figure 20. The levels
ofthe calculated Nusselt number do not agree withmeasured results,
however, the trends of thetheoretical results agree with the
experimentaltrends shown in figure 18.
SUPERSONIC CONTINUUM FLOW
General results for compressible flow showsthat ( )Nu f M Ret t=
, ,θ . However, it wasexperimentally21,112 determined that ( )Nu f
Mt ≠for Mach numbers greater than about 1.4. Typicalheat transfer
data for supersonic flow is presentedin figure 21 to illustrate the
approximate invarianceof Nut with M. At higher Mach number
andrelatively low total pressures, there is a highprobability that
much of the data presented forsupersonic and hypersonic flows are
in the slip flowregime.
FREE MOLECULAR FLOW
Standler, Goodwin and Creager33
computed the heat transfer from wires for freemolecular flow and
an example of their resultsalong with measurements are presented in
figure22. A combination of continuum flow, slip flow andfree
molecular flow results10 are shown in figure23. From this figure it
can be seen that forcontinuum flow at large Reynolds numberNu Ret
t≈
½ . For free molecular flow Nu Ret t≈ andslip flow results
smoothly connect the two regimes.Therefore, for slip flows, Nut
varies with exponent ofReynolds number which range from ½ to 1.
RECOVERY TEMPERATURE RATIO
The recovery temperature ratio must beknown to compute the heat
transfer from heatedwires. In general, the recovery temperature
ratio isa function of Mach and Reynolds numbers or Machand Knudsen
numbers. However, for Machnumber greater than about 1.4 the
recoverytemperature ratio is not a function of Mach numberfor
continuum flow. A "universal" curve presentedby Vrebalovich111
(figure 24) correlated thetemperature recovery ratio with Knudsen
numberfor all Mach numbers. Using the results presentedin figure
24, the temperature recovery ratio forcontinuum flow and free
molecular flow, curves ofη vs. M and Kn can be calculated. An
example ofthese calculations is presented in figure 25.
FLUCTUATION MEASUREMENTS
SUBSONIC INCOMPRESSIBLE FLOW
Theoretical Considerationsa. Constant Temperature Anemometer
For a constant temperature anemometer,King's3 equation can be
expressed as:
ER
L k k c ud T Tw
t t p w w o
22= + −π ρ η (30)
where Rw and Tw are constants.
If one assumes that the changes in kt , cpand η can be
neglected, the change in E will be afunction of ρu and To as given
by the followingequation for small perturbations:
′ = ′ + ′eE
Smm
STTm T
o
oo
(31)
where
[ ]SRe Pr
Re Prm
t
t
=+
14
2
1 2
π
π and ST
wo
= −12
ητ
(32)
From the above equation it can be seen thatSm → 0 as Ret → 0 and
Sm → ¼ as Ret → ∞ . For the
-
13Stainback, P.C. and Nagabushana, K.A.
temperature sensitivity, STo → − ∞ as τw → 0 and
STo → 0 as τw → ∞ . Equation (31) shows that
( )E f m To= , where S E mm = ∂ ∂log log andS E TT oo = ∂ ∂log
log .
Since equation (31) shows that ( )E f m To= , ,the fluctuation
of mass flow and total temperaturecan be measured using a CTA
113-115. This canbest be done by using two wires operated
atdifferent, but high overheats, digitizing the data,and solving
two equations for ′m , ′To and ′ ′m To asfunctions of time.
If the total temperature and the Machnumber varies
significantly, then kt and cp must be
differentiated with respect to To and ηdifferentiated with
respect to Mach number. Underthese conditions it would be more
appropriate touse the equation obtained by Rose and McDaid8
with the assumption that ( )Nu f Mt ≠ .
Instead of using King's equation, considerequation (15) for
measuring mass flow and totaltemperature fluctuation. For a CTA
equation (15)becomes:
( )( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )( )
d EnB T mn
A T B T mnd m
A TA T
T
A T B T mn
mnB TB T
T
A T B T mnd T
o
o o
oo
o
o o
oo
o
o ow
o
log log
loglog
loglog
log
=′
′ + ′
+
′′
′ + ′
+′
′
′ + ′
−
2
12
33
∂∂
∂∂ η
τ
b. Constant Current AnemometerFor the CCA anemometer, Kings'
equation
becomes:
EI L k k c ud T Tt t p w w o= + −2π ρ η (34)
Again assume that kt , cp and η are constant, the
change in E is given by the following equation:
′ = ′ + ′eE
Smm
STTm T
o
oo
(35)
where
( )( )[ ]
( )( )( ) ( )S
Re Pr
Re PrSm
f
t
t
Tf
o=
−
− += −
−
− −1
4
2
1 2
1
236
ε
ε ε
π
π
ε
ε εη
θ ηand
If d Ilog = 0 then
[ ]SRe Pr
Re Prm
f
t
t
=+
14
2
1 2ε
π
πand [ ]ST fo
= −−
12ε
ηθ η
(37)
Equation (36) and (37) shows that ( )S fm w= τ . IfRet → 0 then
Sm → 0 , but if Ret → ∞ then
[ ]Sk
kmt w
t w
=−
ττ θ2
. On the other hand, if τw → 0 then
Sm → 0 and if τw → ∞ then [ ]SRe Pr
Re Prm
t
t
→+
12
2
1 2
π
π. If
τw → 0 then Sk
Tt
o→
ηθ
and if τw → ∞ then STo → 0 .
Again, it is possible to measure both m , To and RmTousing a CCA
113 and the fluctuation diagramdeveloped by Kovasznay6. An example
offluctuation diagrams for two discrete frequenciesmeasured with a
CCA is presented in figure 26.Again if the total temperature and
the Machnumber varies significantly, then it would be
moreappropriate to use Morkovin's equation with theassumption that
( )Nu f Mt ≠ .
Similarly, to measure mass flow and totaltemperature
fluctuation,equation (15) for CCAbecomes:
( )( )
( )( ) ( )
( )( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )( )
d EnB T mn
A T B T mnd m
A TA T
T
A T B T mn
mn B TB T
T
A T B T mnd T
f
o
o o
f
oo
o
o o
oo
o
o o
o
log log
loglog
loglog
log
=−
−
′
′ + ′
+
−
−
′′
′ + ′
+′
′
′ + ′
1
2
1
238
ε
ε ε
ε
ε ε
∂∂
∂∂
Mass flow fluctuations measured insubsonic flows can be very
misleading where thereis a significant amount of far-field sound.
Themean square value of mass flow fluctuation is:
′
=′
+
+
′
mm
uu
Ruuu
2 22
2
ρρρ
ρρ
~ ~(39)
The magnitude of the mass flow fluctuationdepends on u , ρ , and
Ruρ where − ≤ ≤1 1Ruρ . As an
-
14Stainback, P.C. and Nagabushana, K.A.
example, assume Ruρ = 1, indicating downstream
moving sound, and u u = ρ ρ. Under theseassumption the mass flow
fluctuation equals twicethe velocity or density fluctuation.
However, ifRuρ = −1, indicating upstream moving sound, and
u u = ρ ρ; the mass flow fluctuation are zero.
Examples of DataMost of the measurements made using hot-
wire anemometry were and still are being made inthe subsonic,
incompressible, continuum flowregime. An extensive amount of data
wasaccumulated over the years in various flow fields25.Some of
these data will be presented in thefollowing section.
a. FreestreamSome of the first fluctuation measurements
made using hot-wire anemometers were obtained inthe freestream
of wind tunnels to help evaluate theeffects of turbulence on the
transition of laminarboundary layers116. The purpose of this effort
wasan attempt to extend wind tunnel transition data toflight
conditions in order that the on-set oftransition might be predicted
on full scale aircraft.Measurements in the freestream are also
requiredto study the effect of freestream disturbances onlaminar
boundary layer receptivity. An example ofmeasurements made in the
freestream is presentedin figure 27 for the Low Turbulence
PressureTunnel located at the NASA Langley ResearchCenter37. The
filled symbols represent data takenin the facility during 1940117
and the curvesrepresent measurements made in 1980. Theagreement
between the two sets of data is very goodwhen it is noted that the
data taken in 1940 wasobtained at po = 4 atmospheres and the low
datumpoint at Re xt = 5 10
5 is for M ≈ 0 02. .
Fluctuation measurements were also madein various location
within wind tunnel circuits,predominantly in the settling
chamber.Anemometry was used to evaluate the efficiency
ofcontractions in reducing vorticity levels in the testsection118.
An example of the results obtainedthrough a contraction is
presented in figure 28.The absolute value of the velocity
fluctuation in thedirection of the flow was reduced through
thecontraction but the relative values were greatlyreduced
depending on the area ratio of the
contraction. For example in figure 28 the velocityfluctuation
downstream of the contraction areratioed to the mean flow in the
large section of thecontraction where the local velocity is low. If
thesedownstream velocity fluctuations were ratioed tothe local mean
velocity, these normalizedfluctuations would be substantially
smaller, i.e.,
.u usc = 2 6 vs. .u u = 0 16.
b. GridsIt was found that screens or grids can
effectively reduce vorticity fluctuations. Because ofthis
attenuation, screens have been extensivelyinvestigated25,38,119
using hot-wire anemometryto optimize their characteristics for use
in windtunnels to reduce the vorticity levels in the testsection.
An example of these measurements ispresented in figure 29 where the
turbulentreduction factor is given as a function of the ∆ p
q∞across the screens120. Therefore, the use ofscreens in the
settling chamber along with acontraction of adequate area ratio,
cansubstantially reduce velocity fluctuations in the testsection
due to voriticity121. (Also see ref. 122-130).
c. Boundary LayersHot-wire measurements were made in
turbulent boundary layers 42 to measure theReynolds stresses and
other fluctuation quantitiesto furnish data for the development of
turbulentboundary layer theories. An example ofmeasurements made in
the boundary layer on a flatplate131 is presented in figure 30a-b.
Figure 30ashows the significant variation of the
velocityfluctuation across the boundary layer while figure30b shows
an example of the local streamwisevelocity fluctuation ratioed to
the local velocity.Forming the ratio in this latter manner
indicatesthat the velocity fluctuations can exceed 40 percent,a
value that is too large for an accurate assumptionof small
perturbation.
Extensive measurements were made ofturbulent flows in pipes40,41
to comparetheoretical and measured results. From thesemeasurements
many statistical quantities wereobtained including Reynolds
stresses; triple andquadruple correlations; energy spectra; rates
ofturbulent energy production, dissipation, anddiffusion; and
turbulent energy balance. An
-
15Stainback, P.C. and Nagabushana, K.A.
example of the streamwise velocity fluctuationsacross a pipe is
presented in figure 31a-b.
Hot-wire anemometry has been extensivelyused to investigate the
characteristic of variousboundary layer flow manipulators such as
LargeEddy Break-up devices (LEBUS) 132, Riblets133
and roughness elements134. Laminar boundarylayer transition due
to T-S waves135, cross flow136
and Gortler vortices137 was extensively studiedusing the
hot-wire techniques. Also the effects ofheat addition138, sound139
and vorticity140 onboundary layer characteristics have
beeninvestigated.
The hot-wire anemometer with a single wirecannot determine the
direction of flow. However, atechnique using a multi-wire, "ladder
probe" wasdeveloped141 to study the separated boundarylayer where a
significant amount of reverse flowoccurred. This technique was used
to determinethe location of the zero average velocity in asubsonic
turbulent boundary layer. (Also see ref.142-144).
d. Laminar Boundary Layer ReceptivityOne of the major
impediments to a through
understanding of laminar boundary layer transitionis the ability
to predict the process by whichfreestream disturbances are
assimilated into theboundary layer. These free stream
disturbancescan be either vorticity, entropy, sound or acombination
of these fluctuations.
The effect of freestream fluctuations on thestability of the
laminar boundary can beinvestigated by making measurements in
thefreestream and in the boundary layer to evaluatethe receptivity
of the boundary layer tofluctuations145 in the freestream. An
example offluctuations measured in a subsonic boundarylayer on a
flat plate for various frequency bands ispresented in figure 32.
Kendall145 presented threetypes of measurements made in a
laminarboundary layer due to velocity fluctuations from thefree
stream. The first type is illustrated by the x'sand consisted of
broadband velocity fluctuationwhere the peak level occurs towards
the inner partof the boundary layer. This type of measurement
isnoted as the Klebanoff's mode and is representedby the solid
line. The results obtained when the
data were filtered at the Tollmein-Schlichting (T-S)frequency
are represented by circles. Although thefrequencies were identical
to those of T-S wavesthey were not T-S waves since the
convectionvelocity was equal to the free stream value. Themaximum
level of these fluctuations occurred at theouter part of the
boundary layer. The third type offluctuation is represented by the
dotted line. Thesewere true T-S waves which occurred in packets
andhad a convective velocity of 0.35 to 0.4 of the freestream
velocity. These peak fluctuation levelsoccurred near the wall.
(Also see ref. 146-156).
e. JetsHot-wire measurements were obtained in
jets45-47 to measure the Reynolds stressesassociated with free
shear layers and to helpevaluate the RMS levels and frequencies
associatedwith jet noise. An example of the velocity andtemperature
fluctuations measured157 across aheated jet is presented in figure
33. The two typesof fluctuations were normalized by the maximumand
the local mean values, respectively. (Also seeref. 158-160).
f. WakesVarious statistical quantities were
measured downstream of a heated cylinder byTownsend43 to obtain
experimental results to helpimprove turbulent theories applicable
to this type offlow. Some of the quantities obtained
includedturbulent intensities, sheer stress, velocity-temperature
correlation, triple velocity correlation,diffusion rate and energy
dissipation.Measurements were made from 500 to 950diameter
downstream of the cylinder wheredynamical similarity was assumed to
exist. Anexample of the mean curve fitted to the u-component of the
velocity fluctuation is presentedin figure 34 and shows good
similarity. Uberoi andFreymuth44 made extensive
spectralmeasurements downstream of a cylinder and theirdata
indicated that only the spectra of large-scaleturbulence were
dynamically similar.
-
16Stainback, P.C. and Nagabushana, K.A.
SUBSONIC SLIP FLOW AND TRANSONIC FLOW
Theoretical ConsiderationsIn compressible flows the heat
transfer
from a wire is usually described by the followingequation:
( )Q Lk T T Nut w o t= −π η (40)
Differentiating the above equation for the casewhere Q P=
gives:
( )d P d d Nu d d T d kw
w tw w
o tlog log log log log log− = − − +θ
ττ
ητ
ηη
τ41
The terms on the right hand side of theabove equation depend
only on the functional formsassumed for Nut and η (Table I) and the
chosenindependent variables (Table II). Ultimately theseterms
depend on the variation of Nut and η withthe flow variables along
with the aerodynamic andthermodynamic properties of the flow. The
finalform for the left hand side of the equation dependson the type
of anemometer used.
It was shown in reference 161 that, for awire mounted normal to
the flow, ( )E f u T To w= , , ,ρ .Morkovin58 and Baldwin 35
related Nut and η tothe non-dimensional variables noted in Table
I.However, recent results presented by Barreet.al.,162 suggested
advantages from using thefollowing variables: ( )E f m M T To w= ∞,
, , and
( )E f p m T To w= , , , . In order to obtain the equation
forthe CCA, they transformed Morkovin's equationsinto their
variables. This transformation is notnecessary, since once the
variables are chosen, theequations can be derived directly using a
methodsimilar to the one described by Anders161.
The equations obtained using the variablesof from table II gives
the same results as those of
, Morkovin58, namely ( )E f m To= , under thecondition where (
)Nu f Mt ≠ ∞ and S Su = ρ . Using thevariables in , Barre
et.al.,162 applied theresultant equations to measurements made in
aturbulent boundary layer under the assumptionthat p = 0 without
assuming that S Su = ρ . They also
extended the equation to the case of supersonic
flow in the test section of wind tunnels by assumingthat all the
fluctuations were sound, again notassuming S Su = ρ .
Once the independent variables are chosen,it is not necessary to
derive the equations using Nutand η . The "primitive" variables, u
, ρ , To etc.,greatly simplifies the manipulation of thecalibration
data and can be used to correlate E asa function of u , ρ , To
etc., without evaluating Nutand η . This technique might have
advantages inthe calibration of wires and the ease of operation
ofcalibration facilities.
The possible sets of variables based on theabove discussion is
presented in Table II.
DependentVariable
IndependentVariable
Reference
E , Nut , η u , ρ , To Baldwin35,Morkovin58
E , Nut , η m , M∞ , To Barr, Quine andDussauge162
E , Nut , η p∞, m , To Barr, Quine andDussauge162
E u , ρ , To Rose and McDaid8Stainback and
Johnson85
E m , M∞ , ToE p∞, m , To
Table II. Various Independent Variables toDerive the Hot-Wire
AnemometryEquations
Therefore, there are many forms for the hot-wireequations
depending on the variables chosen andthe anemometer used. One
should choose thevariables that are most convenient for the
flowsituation under investigation.
a. Constant Current AnemometerUsing the heat transfer equation
(40) and
the functional relationship from equation (27) and(29), the
change in voltage across a wire can berelated to the changes in u ,
ρ , and To . An exampleof a set of equations obtained for a
constant currentanemometer was given by Morkovin58 as:
-
17Stainback, P.C. and Nagabushana, K.A.
′ = − ′ − ′ + ′eE
Suu
S STTu T
o
ooρ
ρρ
(42)
where
( )S Eu
E A NuRe
NuM M Reu
w
wrwr
t
t
t
t
= =′ ′
+
− +
∂∂ τ
τ∂∂ α
∂∂ α
∂ η∂
∂ η∂
loglog
loglog
loglog
loglog
loglog
1 143
SE E A Nu
Re Rew
wrwr
t
t tρ
∂∂ ρ τ
τ∂∂
∂ η∂
= =′ ′
−
loglog
loglog
loglog
(44)
( )( )S E
T
E Ak A A n m
Nu
Re
Nu
M
AM
mRe
To
w
wr
t wr w w wr t tt
t
t
w tt
o= =
′ ′+ ′ + ′ − − + +
− ′ +
∂∂ τ
τ τ∂
∂ α
∂
∂
α∂ η
∂∂ η
∂
loglog
log
log
log
log
loglog
loglog
1 11
2
12
45
For a large range of Reynolds numbers andMach numbers, Su and Sρ
in equation (42) are
unequal35. Following Kovasznay's technique forsupersonic flow,
dividing equation (42) by STo ,
squaring and forming the mean gives:
( )′ = ′
+′
+
′
+ − −φ
ρρ
ρρ
ρρρ ρ
2 22
22 2
2 2 2 46quu
sTT
qsRuu
qRuTuT
sRTT
o
ou uT
o
oT
o
oo o
~~ ~ ~ ~ ~
This is a general equation for a wire mountednormal to the flow
in compressible flows whereS Su ≠ ρ . This is a single equation
with six
unknowns. In principal, this equation can besolved by operating
a single wire at six overheatsand solving six equations to obtain
the threefluctuating quantities and their correlations. In thepast,
it was generally stated that the calibration ofthe wire cannot be
made sufficiently accurate or thevelocity and density sensitivities
cannot be madesufficiently different to obtain a suitable
solutionusing this technique. Demetriades163 noted thatthe
coefficient in equation (46) must occur to atleast the fifth
degree. This constraint, however,appears to be too restrictive. For
example, assumethat s is a function of q as follows:
s a a q a q a qb b b= + + +1 2 3 41 2 3 (47)
It can be shown that b1 , b2 and b3 can have anyvalue provided
that the substitution of therelationship for s into equation (46)
results in an
equation having at least six terms. An analysis ofdata obtained
at transonic Mach number bySpangenberg indicates that s can be
non-linearlyrelated to q and suggest that a solution to
equation(46) is possible. A more detailed discussion of thiscan be
found in ref. 164.
If solutions for equation (46) are possible,what is the form of
the fluctuation diagram? Inequation (46), φ is a function of q and
s ,therefore, the fluctuation diagram exists on a three-dimensional
surface, a hyperboloid, rather than aplane as for the case when S
Su = ρ . However, the
important information, fluctuation quantities, existsin the φ −
q and φ − s planes. For example whens = 0 , equation (46) reduces
to an equation for ahyperbola in the φ − q plane, where the
asymptotegives the velocity fluctuations. If q = 0 , equation(46)
reduces to an equation for a hyperbola in theφ − s plane and the
asymptote represents thedensity fluctuations. When q and s both are
zero,the intercept on the φ axis gives the totaltemperature
fluctuation. In planes parallel to theq s− plane, the locus of
points of the fluctuationdiagram is governed by the velocity and
densityfluctuations and their correlation. The crossproduct term,
qs, requires a rotation of the axisbefore the characteristics of
the locus can beidentified. The locus of points on the surface of
thehyperboloid will depend on the relative changes inq and s as the
overheat of the wire is changed.
Although the fluctuation diagram exists onthe surface of a
hyperboloid, the fluctuations can bedetermined from the
intersection of the hyperboloidwith the φ − q and φ − s planes.
Because of this, thefluctuation and mode diagrams were defined as
thetraces of these intersections in the notedplanes165,166. A
general schematic representationof the fluctuation diagram for
equation (46) ispresented in figure 35.
Even though there is much evidence thatS Su ≠ ρ over a large
range of Ret and M∞ in subsonic
compressible flow, some experimenters haveconducted tests167-169
under the condition whereS Su = ρ . When S Su = ρ for subsonic
compressible
flow, the fluctuation and mode diagram techniquedeveloped by
Kovasznay can be used to obtain the
-
18Stainback, P.C. and Nagabushana, K.A.
mass flow and total temperature fluctuations. Thegeneral
fluctuation diagram is identical to the onefor supersonic flow,
namely, a hyperbola. Also, themode diagrams for entropy and
vorticity modes areidentical. The sound mode is, however,
different.For supersonic flow the angle that plane soundwaves makes
with respect to the axis of a probe canhave only two values. If the
sound source is fixedthen cosθ1 1= − M . If the sound source has a
finite
velocity then cosθ1 1= − −
ua
ua
s . However, for
subsonic flows the values of θ1 can range from 0º to360º. An
example of fluctuation diagramsmeasured in subsonic compressible
flows under thecondition where S Su = ρ is presented in figure
36a-b.
b. Constant Temperature AnemometerThe hot-wire equation for a
CTA that
corresponds to equation (42) for a CCA is:
′ = ′ + ′ + ′eE
Suu
S STTu T
o
ooρ
ρρ
(48)
and for S Su ≠ ρ is a single equation with three
unknowns. Hinze indicated that a CTA cannot beused to obtain
fluctuations using the mode diagramtechnique since the frequency
response of theanemometer approaches the frequency response ofthe
wire at low overheats. Because of theseproblems, it was suggested
that a three wire probebe used and the voltage from the
anemometerdigitized at a suitable rate85. Then three equationsare
obtained that can be solved for ′u , ′ρ , ′To as afunction of time.
Statistical techniques can then beused to obtain statistical
quantities of interest.
To insure that the solution to the threeequations are
sufficiently accurate, the three wiresare operated at different and
high overheats tomake Su and Sρ as different as possible. This
will
insure that the condition number for the solutionmatrix is
reasonably low. A complete description ofthis technique is given in
references 85 and 86.
Some experimenters have conductedtests8,170 where S Su = ρ in
transonic flows. For
these tests a set of equations similar to those givenby Morkovin
for a CCA were derived for a CTA.
The functional relationship for the voltageacross a wire for
subsonic slip flows is identical tothe functional relationship for
transonic flow.Therefore, the three wire probe technique
underdevelopment for transonic flows should beapplicable for both
regimes. Since slip flows areidentified by the Knudsen number, the
possibility ofobtaining useful measurements using a three wireprobe
might be improved by using differentdiameter wires in addition to
using differentoverheats.
Hot-Wire CalibrationEvaluating the required partial
derivatives
requires care when carrying out the mean flowcalibration. For
example, consider Morkovin'sequations (42-45). The evaluation of(
)∂ ∂ θlog log ,Nu Ret t M must be obtained by varying poonly and
the Mach number, θ , and the totaltemperature must be held
constant. On the otherhand, the evaluation of ( )∂ ∂ θlog log ,Nu
Mt Ret requiresthat the total pressure be changed when the
Machnumber is varied in order to maintain Ret constant.Similar
constraints also must be observed when( )∂ η ∂log log Ret M and (
)∂ η ∂log log M Ret are evaluated.Similar care must also be taken
in evaluating thepartial derivatives when using other dependent
andindependent variables. Some of these variationsand their
constraints makes the operation of windtunnels very time consuming
if an accurate meanflow calibration is to be obtained.
If Nut is assumed to be a ( )f M Ret∞ , ,θ , theabove described
constraints applied, and theoperational envelope of the facility
considered, thereis a skewing of the Nut vs. Ret curves for
constantMach numbers because ( )Re f ut = , ρ (figure
18b)171.Because of this, the region over which the
partialderivation can be evaluated is reduced due to thisskewing.
If Nut is assumed to be a ( )f M Kn w∞ , ,τ ,plots of Nut vs. Kn
for constant Mach number is notas skewed and a more complete set of
derivativescan be evaluated from a given number of datapoints
(figure 37). This efficient use of data can alsobe obtained for (
)E f u To= , ,ρ with the wire voltagecorrelated in term of these
primitive variables. Theuse of τw as an independent variable can
cause
-
19Stainback, P.C. and Nagabushana, K.A.
extra complications in the calibration of wires whenη varies
with M or Ret . Under these conditions thewire temperature must be
changed when η variesto hold τw constant110.
Based on past experience85,172-175, thefollowing method appears
to provide a reasonabletechnique for correlating data to obtain the
requiredsensitivities. Consider the correlation of Nusseltnumber
for the situation where ( )Nu f M Knt w= , ,τ .First the measured
variation of Nut with oneindependent variable must be obtained with
theother independent variables held constant. Thevariation of Nut
with the remaining independentvariables must be obtained under the
sameconstraints. After these data are obtained, Nutmust be curve
fit to one of the independentvariables, say Kn, for all constant
values of M andτw. The curve fitting process will, in general,
resultin ( )∂ ∂ τlog log , ,Nu Kn f M Knt w= . This method must
beused to obtain other partials ie.,
( )∂ ∂ τlog log , ,Nu M f M Knt w= and( )∂ ∂ τ τlog log , ,Nu f
M Knt w w= . A similar technique
should be used to obtain ∂ η ∂log log Kn and∂ η ∂log log M .
After the partial derivatives areobtained, the sensitivities, i.e.,
Su , Sρ and STo can be
determined. Each of these sensitivities will, ingeneral, be
functions of all the independentvariables. Spangenberg published
the only dataknown to the authors which were obtained underthe
above constraints. He presented the Nusseltnumber as a function of
M, ρ , and τ but for agiven diameter wire ρ and Kn are reciprocals
ofeach others. An example of the sensitivitiesobtained for one set
of Spangenberg data ispresented in figures 38a-c. These figures
showsthat each sensitivity, is in general, a function of thethree
independent variables.
Examples of Dataa. Freestream
Mean flow measurements were made in thetransonic and slip flow
regime in the 1950's35,110.It was only during the 1970's and 1980's
thatattempts were made to measure8,49,85,86,170
fluctuations in these regimes because of thecomplexity of the
response of the heated wire tovelocity, density and total
temperature.
In order to evaluate the effect of free streamfluctuations on
boundary layer transition, hot-wiremeasurements were made in the
Langley ResearchCenter 8' Transonic Pressure Tunnel during
theLaminar Flow Control experiments86. For theseflow conditions S
Su ≠ ρ with S Su < ρ . Because of this
situation a three wire probe was used to measureu u, ρ ρ, T To o
and m m and examples of thesemeasurements are presented in figure
39a-d. (Alsosee ref. 176-178).
b. Boundary LayerHorstman and Rose170 made
measurements at transonic speeds where, for theirflow condition,
it was found that S Su ≈ ρ . For this
condition the transonic hot-wire problemdegenerated to the
supersonic flow problem whereonly m m , T To o and RmTo could be
measured. From
their measurement of m m , the velocity and densityfluctuations
were computed by assuming that T To oand p p were zero. An example
of these results ispresented in figure 40. In this figure Horstman
andRose's hot-wire results, represented by the circles,are compared
with the thin film results obtained byMikulla170.
c. Flight in AtmosphereAny attempts to extrapolate the effect
of
wind tunnel disturbances on laminar boundarylayer transition to
flight conditions requires someknowledge of the disturbance levels
in theatmosphere. Much of the fluctuation data obtainedin the
atmosphere was measured using sonicanemometers on towers179. There
was a limitedamount of data obtained in the atmosphere
usinghot-wire anemometry on flight vehicles49,180.Otten et. al.,49
expanded the methods devised byRose and McDaid by using a two wire
probe. Onewire was operated by a CCA at a low over heat tomeasure
To. The other wire was operated with aCTA that was sensitive to m
and To. The resultsfrom these two wires were used to measure m
andTo in the atmosphere. An example of spectral data
obtained in the atmosphere is presented in figure41 and reveals
the expected −5 3 slope, for m andTo.
-
20Stainback, P.C. and Nagabushana, K.A.
d. Subsonic Slip FlowFor this regime ( )Nu f M Ret t w= ∞ , ,τ
and
S Su ≠ ρ . These results are identical to those in the
transonic flow regime and attempts have beenmade to apply the
three wire technique developedfor transonic flows to subsonic slip
flows. For testsin subsonic slip flows the three wires were
ofdifferent diameters in addition to being operated atdifferent
overheats. Some very preliminary dataobtained using this technique
in the Langley LTPTtunnel is presented in figure 42a-b
wherecomparison with results obtained using King'sequation are
made.
HIGH SUPERSONIC AND HYPERSONIC FLOW
Theoretical Considerationa. Constant Current Anemometer
In the 1950's and 1960's hot-wireanemometry was extended into
the high supersonicand hypersonic flow regime6,7,181,182. For
highsupersonic flows it was found experimentally thatS Su = ρ and
equation (42) becomes:
′ = − ′ + ′eE
Smm
STTm T
o
oo
(49)
Dividing equation (49) by the total temperaturesensitivity,
squaring, and then taking the meanresults in the following
equation:
′ =′
−
+
′
φ 2 2
22
2r
mm
rRmm
TT
TTmT
o
o
o
oo
~ ~(50)
This equation was derived by Kovasznay6 and usedto generate
fluctuation diagrams for supersonicflows. This equation was also
used in references167 and 168 for subsonic compressible flows.
Thegeneral form of equation (50) is a hyperbola wherethe intercept
on the φ -axis represents the totaltemperature fluctuation and the
asymptotesrepresent the mass flow fluctuation7.
Kovasznay demonstrated that the basiclinear perturbation in
compressible flows consistsof vorticity, entropy and sound. He
termed thesebasic fluctuations as "modes". If the
fluctuationdiagram is assumed to consist of a single mode
thediagrams were termed "mode diagrams". An
example of a general fluctuation diagram and thevarious mode
diagrams for supersonic flow arepresented in figures 43 and 44.
b. Constant Temperature AnemometerFor this case equation (42)
becomes:
′ = ′ + ′eE
Smm
STTm T
o
oo
(51)
This is a single equation in two unknowns and atwo wire probe
can be used to obtain ′m , ′To and
′ ′m To similar to the compressible subsonic
flowcase61,183,184.
Examples of Dataa. Freestream
In order to evaluate the relative "goodness"of supersonic wind
tunnels and to relate the levelsof disturbances in the test section
to laminarboundary layer transition on models, a largeamount of
hot-wire measurements were made inthe test sections of supersonic
and hypersonic windtunnels. In 1961 Laufer185 presentedmeasurements
made in the test section of the JetPropulsion Laboratory 18 x 20
inch supersonicwind tunnel over a Mach number range from 1.6 to5.0
using CCA. An example of the fluctuationdiagrams obtained by Laufer
is presented in figure45. From these diagrams the mass flow and
totaltemperature fluctuations were obtained. Examplesof the mass
flow fluctuations are presented in figure46a. There was a
significant increase of m m withMach number ranging from 0.07% at M
= 1 6. toabout 1.0 to 1.35% at M = 5 0. , depending onReynolds
number. All of the fluctuation diagramswere straight lines and
Laufer demonstrated thatthese results indicated that the
fluctuations werepredominantly pressure fluctuations due to
sound.Examples of the calculated pressure fluctuationsare presented
in figure 46b. Laufer concluded thatthe pressure fluctuations
originated at theturbulent boundary on the wall of the tunnel
andbecause of the finite value of the temperaturefluctuations the
sound source had a finite velocity.An example of the sound source
velocities ispresented in figure 47.
A large amount of hot-wire data was takenin the freestream of
various facilities to measure
-
21Stainback, P.C. and Nagabushana, K.A.
disturbance levels in efforts to develop quietsupersonic wind
tunnels. A review of this effort wasreported in reference 186.
Measurements in the freestream of theLangley Research Center
Mach 20 High Reynoldsnumber Helium Tunnel were performed by
Wagnerand Weinstein181. All of their fluctuation diagramswere
straight lines similar to the results obtained insupersonic flows.
Examples of their measuredmass flow and total temperature
fluctuations arepresented in figure 48. The mass flow
fluctuationswere substantially higher than the values measuredby
Laufer at M = 5 0. . Pressure fluctuationmeasurements presented in
figure 49 indicate thatat low total pressures the boundary layer on
thenozzle wall was probably transitional at the acousticorigin of
the sound source. Relative sound sourcevelocities are presented in
figure 50. The sourcevelocities for the Mach 20 tunnel at the
higherpressures are significantly higher than thosemeasured by
Laufer at Mach numbers up to 5.(Also see ref. 187).
b. Boundary LayerMeasurements were made in supersonic
and hypersonic turbulent boundary layers toextend the range of
Reynolds stress measurementsneeded in the development of turbulent
boundarylayer theories. Barre et. al.,162 conducted hot-wiretests
in a supersonic boundary layer wheretransonic effects were
accounted for by using atransformation of equation (42-45) from u ,
ρ , To top , m , and To . Using the assumption that
( )E f p m To= , , and p p ≈ 0, reduced their equation to( )E f
m To= , . Under these condition the fluctuation
diagram developed by Kovasznay was used toobtain m m , T To o
and RmTo without the assumption
that S Su = ρ .
Examples of their results are presented infigures 51 and 52.
Figure 51 shows that thequantity ρ τ′u w
2 is greatly underestimated if the
assumption is made that S Su = ρ when the velocity
in the boundary is transonic and S Su ≠ ρ . Figure 52
show the variation of RuTo with the local Mach
number through the boundary layer. The expectedvalue for RuTo is
-0.85 and the data obtained for
S Su ≠ ρ agrees well with this value. However, data
evaluated where Su was assumed to be equal to Sρwere
substantially higher at the lower transonicMach numbers.
Fluctuations in a hypersonic boundarylayer were made by Laderman
and Demitrades182
and reported in reference 188. An example of themass flow and
total temperature fluctuationsmeasured across the boundary layer is
presented infigure 53. The velocity, density, static
temperature,and pressure fluctuations were calculated using themass
flow and total temperature fluctuations andvarious assumptions. An
example of thesemeasurements is presented in figures 54 and 55.
Additional measurements were made inhypersonic boundary layers
by Laderman189 andon an ogive cylinder by Owen and Horstman59 atM ≈
7 0. . The measurements made by Owen andHorstman included not only
data for m , To and RmTobut integral scales and microscales,
probabilitydensity distributions, skewness, kurtosis
andintermittancy distribution across the boundarylayer. A summary
paper by Owen 190 presentsadditional data which included
space-timecorrelation, convective velocities,
disturbanceinclination angle, and turbulence life
timedistributions. (Also see ref. 87, 191-196).
c. WakeAn example of fluctuation diagrams
measured in the wake behind a 15º half anglewedge at M = 15 5.
is presented in figure 56. Fromthese results Wagner and Weinstein
concluded thatthe predominant fluctuation in this wake wasentropy
since the fluctuation diagrams werestraight lines that intersected
the r -axis atapproximately −α .
d. Downstream of ShockIn supersonic and hypersonic flows the
disturbances measured in the freestream of the testsection are
not necessarily the disturbances thatcan affect the transition of
the laminar boundarylayer on a body. The passage of sound
wavesthrough shocks will result in a combination ofvorticity,
entropy, and sound downstream of theshock197. Because of this the
fluctuation diagramwill no longer be a straight line but a
general
-
22Stainback, P.C. and Nagabushana, K.A.
hyperbola48. An example of this result ispresented in figure 57.
(Also see ref. 198-201).
CONFIRMATION OF THEORETICAL RESULTS
Hot-wire anemometry was used extensivelyto validate or confirm
theoretical results. Someexamples of these efforts are presented
below.
Theoretical studies of the stability oflaminar boundary layers
to small disturbanceswere initially performed by Tollmein202,203
andSchlichting204. These calculations indicated thatdisturbances of
a given frequency could decrease,remain constant or be amplified
depending on thefrequency chosen and the Reynolds number. Thefirst
experimental verification of the theory wasmade by Schubauer and
Skramstad135. Anexample of recent experimental and
theoreticalresults for the determination of the neutral
stabilityboundary205 in a laminar boundary layer ispresented in
figure 58.
Hot-wire measurements were madedownstream of "grids" to evaluate
the theory for thedecay of turbulence. Tests conducted by
Kistlerand Vrebalovich206 to evaluate the "linear" decaylaw is
presented in figure 59 and confirm this lawfor large values of the
Reynolds number. At lowerReynolds number25 the exponent can be
closer to1.20 - 1.25. Measurements of spectra for
velocityfluctuations downstream of grids was also madeand compared
with theory. Two examples of thesespectra are presented206,207 in
figure 60a-b. Thespectra in figure 60a had an insignificant amount
ofenergy in the expected inertial sub-range indicatedby a slope of
−5 3. This result was attributed to thelow Reynolds number of the
flow. The spectrapresented in figure 60b was measured in a
highReynolds number flow and revealed a significantamount of energy
in the inertial sub-range.Theoretical calculations were made for
thetemperature spectra in a heated jet anddownstream of a heated
grid. An example of thetheory and measurements207 is presented in
figure61. Attempts have been made to predict theinfluence of
measured freestream fluctuations onlaminar boundary transition. An
example of thisefforts208 is presented in figures 62.
A considerable amount of data wasobtained by Stetson et. al.,209
in hypersonic flow tostudy the stability of laminar boundary layer.
Anexample of these results are presented in figure 63and indicates
the existence of first and secondmode instabilities in the laminar
boundary layer.These results were in agreement with thoseobtained
by Kendall155 and Demetriades210 andwas in qualitative agreement
with the theoreticalresults obtained by Mack156.
OTHER APPLICATIONS
Hot-wire anemometry was used in shocktubes in an attempt to
check the frequencyresponse of probes and to trigger
otherevents52,211.
Theoretical results indicated that therewould be "temperature
fronts or steps" in cryogenicwind tunnels due to the injection of
liquid nitrogeninto the circuit. Measurements were made using
ahot-wire anemometer to determine the possibleoccurrences of these
thermal steps53.
A pulsed hot-wire was used to measure thevelocities and flow
angle in low speed flows212. Awire, which was operated by an
anemometer, wasplace in the wake of a second wire which could
bealternately heated. The time lag between heatingthe forward wire,
this pulse being detected by thesecond wire and the distance
between the wireswas used to compute the velocity of the flow.
Inreference 213 a somewhat different technique wasdescribed that
used two CCA's and a CTA tomeasure the velocity and flow angle.
Hot-wire anemometry was used to obtainthe location of transition
in a laminar boundary inaddition to obtaining some information on
thefluctuations in the laminar, transitional andturbulent boundary
layers 214,215. In some flowswhere the fluctuation levels are high,
such as a jet,a moving hot-wire probe was found to improve
theaccuracy of the results216. This technique isusually noted as
flying hot-wire anemometry. (Alsosee ref. 217-221).
Hot-wire anemometry was used222 tomeasure the focal points of
the laser beams for
-
23Stainback, P.C. and Nagabushana, K.A.
Laser Transit anemometry (LTA) . Using atraversing mechanism and
a CTA, the distancebetween the two beams was determined bymeasuring
the difference between the twomaximum voltage outputs from the
anemometer. Itwas noted that additional information could
beobtained such as the mean value of the beamintensity intersected
by the wire, laser beam power,beam separation, beam diameter, beam
divergence,cross-sectional beam-intensity distribution andrelative
beam intensity.
CONDITIONAL SAMPLING
Organized motion or structures in aturbulent boundary layer has
been extensivelystudied20,223-225 using the concept of
conditionalsampling. The flow of a turbulent boundary layerover a
concave surface was studied in reference226 to search for organized
motion in the boundarylayer. The possible existence of organized
motion isillustrated in figure 64 from measurements madewith two
hot-wire probes located 0.1δ apart. Theconditional sampling
technique was used todetermine the characteristic shape of the mass
flowsignal during the passage of an organized motion.An example of
these results are presented in figure64. Figure 65 shows that the
measured event at anupstream and a downstream station had the
samegeneral characteristic shape.
COMPARISON OF HOT-WIRE MEASUREMENTSWITH OTHER TECHNIQUES
In the past the hot-wire anemometer, withall its limitations,
was the only instrument availablethat was capable of measuring
fluctuations withadequate fidelity12. To some extent, this is
nolonger the case as other techniques such as LV,LIF, CARS, and
Raman scattering are now availablefor measuring various mean flow
and fluctuatingquantities. In inviscid flow where the
fluctuationscan be low, the anemometer is presently the
onlyreliable instrument available. Compared to othertechniques the
anemometer is still relatively simpleto operate and relatively
inexpensive. Because of itslong history, the results obtained
usinganemometry is still often used as a standard forevaluating
measurements obtained using other
techniques. The extent to which the anemometercan maintain these
advantages depends on thecontinued development of the other
techniques.
Tests were conducted in turbulentboundary layers to compare
hot-wire results withother techniques to validate
hot-wire/laservelocimeter (LV)227,228 and hot-wire/laser-induced
fluorescence (LIF)229 techniques. Anexample of velocity
fluctuations obtained using hot-wire anemometer and a LV system is
presented infigure 66 for streamwise velocity fluctuations.
Theagreement between the two sets of data is verygood. Measured
density and static temperaturefluctuations measured with a hot-wire
and a LIFsystem in a turbulent supersonic boundary layer
ispresented in figure 67. Again, except for two valuesfor the
density fluctuation obtained with the LIFsystem, the agreement
between the two sets of datais very good. Various experiments were
made usingLIF and LIF/RAMAN techniques to measure T To oand ρ ρ
where results were compared with hot-wiremeasurements228. An
example230 of these resultsis presented in figure 68. The only
disagreementbetween the hot-wire and the other results
wasattributed to a shock that apparently did not crossthe hot-wire
probe.
CONCLUDING REMARKS
A review was made to illustrate theversatility of hot-wire
anemometry in addition tonoting some of its limitations. The review
includedexamples of results obtained in the various flowregimes and
various types of flow fields. Examplesof data were presented for
the subsonicincompressible flow regime that were used toevaluate
the flow quality in the test section of windtunnels, to obtain
measurements in turbulentboundary layer, and to substantiate or
validatevarious theories of turbulence.
Recently attempts to extend hot-wireanemometry into the
transonic and subsonic slipflow regimes were presented for ca