hot-wire

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 εAw

f 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 o

Nu Nusselt number evaluated at subscript temperature

p mean static pressurepo mean total pressureP electrical power to the hot-wirePr Prandtl numberq sensitivity ratio, S Su To

q∞ dynamic pressureQ forced convective heat transferr sensitivity ratio, S Sm To

rd radial distance in cylinderical polarco-ordinate

rp 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

RmTomass flow - total temperature correlation

coefficient, ′ ′m T mTo o

RuTovelocity - total temperature correlation

coefficient, ′ ′u T uTo o

R 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 SToρ

S sensitivity of hot-wire to the subscript variable

t timeT temperatureTf ( )T Tw o+ 2

u v w, , velocity in x , y and z directions respectively

uτ 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 11

22

1+

−

−γM

α1 linear temperature - resistance coefficient of wire

β ( )α γ − 1 2M

β1 second degree temperature - resistance coefficient of wire

δ boundary layer thicknessδ* displacement thickness for Blasius flowε finite circuit factor, ( )− ∂ ∂log logI Rw w s

ε 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 o

θ temperature parameter, T Tw o

θ1 angle between plane sound wave and axisof probe

λ mean free pathµ absolute viscosityγ specific heat ratio, c cp v

ρ 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, ( )′eE STo


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


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 supersonic

and 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 from

the 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).


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 Qww = − (1)

( )dcdt

T I R Ld h T Tww 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


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.


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 τw

under 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 To

can 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


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, , , . For

incompressible 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 kt

where 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:


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:

( ) ( ) ( )ER R

A T B T mnw adw

T o T o

2

−= + (16)

where ( ) ( )A T

R A T

RT ow o

ref

=′

α 1

and ( ) ( )B T

R B T

RT ow o

ref

=′

α 1

.

For a CCA:

( ) ( ) ( )I RR R

A 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 conventional

form of ( )E f u= . The constant To and ρ version of

King'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 5

12

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


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 f

and he recommended the following equation:

( )Nu Ref f= +032 0 43052

. .. (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 the

mean 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:

( )[ ]Nue Pr Re

tt

=−

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


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 Baldwin35

Nut Ret , M∞ , τw

Nut 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 Mach

numbers ranging from 0.05 to 0.90 and Reynolds


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 was

experimentally21,112 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 , cp

and 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


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 and

S 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 . If

Ret → 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 RmTo

using 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


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 = 5 105 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


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.


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 for

the 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 the

condition where ( )Nu f Mt ≠ ∞ and S Su = ρ . Using the

variables 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 Nut

and 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 Nut

and 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 McDaid8

Stainback andJohnson85

E m , M∞ , To

E 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:


′ = − ′ − ′ + ′eE

Suu

S STTu T

o

ooρ

ρρ

(42)

where

( )SEu

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

ET

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


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 po

only and the Mach number, θ , and the totaltemperature must be held constant. On the otherhand, the evaluation of ( )∂ ∂

θlog log

,Nu Mt Ret

requires

that 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 MRet

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 , the

above 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 voltage

correlated in term of these primitive variables. Theuse of τw as an independent variable can cause


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, Nut

must 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 be

used 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 o

and 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.


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


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 Sρ

were 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 RmTo

but 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


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


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 o

and 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 cases where S Su = ρ

and S Su ≠ ρ . Examples of data obtained at high

supersonic and hypersonic Mach numbers werepresented. These results revealed that thefluctuation diagrams measured in the test section


were a straight line, indicating that thedisturbances in the test sections were due to soundmode.

Examples were also presented to illustratemeasurements made to substantiate turbulenttheories, to compare with other techniques, and toillustrate the concept of conditional sampling.

ACKNOWLEDGMENT

The authors acknowledge the support ofthis research by the National Aeronautics andSpace Administration under contracts NAS1-19320and NAS1-19672 respectively.

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204. Schlichting, H., 1987, "Boundary LayerTheory," McGraw-Hill Book Company, Co.

205. Saric, W.S., and Nayfeh, A.H., 1977,"Nonparallel Stability of Boundary Layer withPressure Gradient and Suction," AGARD-CP-224.

206. Kistler, A.L., and Vrebalovich, T., 1966, "GridTurbulence at Large Reynolds Numbers,"Journal of Fluid Mechanics, Vol. 26, No. 37,pp. 37-47.

207. Pao., Y.H., 1965, "Structure of TurbulentVelocity and Scale Fields at Large WaveNumber," The Physics of Fluids, Vol. 8, No. 6,pp. 1063-1075.

208. Mack, L.M., 1977, "Transition Prediction andLinear Stability Theory," AGARD-CP-224.

209. Stetson, K.F., Thompson, E.R., Donaldson,J.C., and Sibr, L.G., 1983, "Laminar BoundaryLayer Stability Experiments on a Cone at Mach8. Part 1: Sharp Cone," AIAA-83-1761.

210. Demetriades, A., 1971, "Laminar BoundaryLayer Stability Measurements at Mach 7Including Wall Temperature Effects," AFOSR-TR-77-1311.

211. Billington, I.J., 1955, "The Hot-WireAnemometer and its use in Non-Steady Flow,"University of Toronto, Ins. Aero. Phys., Tech.Note No. 5.

212. Toy, N., and Thoma, S.H., 1987, "The TwinPulsed Wire Anemometer - A New Instrument forMeasurements in Complex Flows," Symposiumon Thermal Anemometry, ASME-FED Vol. 53,pp. 65-72.

213. Blackwelder, R.F., and McLean, 1987, "A Hot-Wire Probe for Velocity Measurement in ReverseFlows," Symposium on Thermal Anemometry,ASME-FED Vol. 53, pp. 73-78.


214. LaGraff, J.E., 1972, "Observation of HypersonicBoundary Layer Transition Using Hot-WireAnemometry," AIAA Journal, Vol. 10, No. 6, pp.762-769.

215. Sandborn, V.A., and Wisniewski, R.J., 1960,"Hot-Wire Exploration of Transition on Cone inSupersonic Flow," Proceedings of Heat Transferand Fluid Mechanics Institute Meeting,Stanford, Ohio, pp. 120-134.

216. Hussein, H.J., 1990, "Measurements ofTurbulent Flows with Flying Hot-WireAnemometry," The Heuristics of ThermalAnemometry, ASME-FED Vol. 97, pp. 77-80.

217. Spangler, J.G., and Wells, Jr., C.S., 1968,"Effects of Freestream Disturbances onBoundary Layer Transition," AIAA Journal, Vol.6, No. 3, pp. 543-554.

218. Obremski, H.J., and Fejer, A.A., 1967,"Transition in Oscillating Boundary LayerFlows," Journal of Fluid Mechanics, Vol. 29,part 1, pp. 93-111.

219. Mack, L.M., 1975, "A Numerical Method for thePrediction of High-Speed Boundary-LayerTransition Using Linear Theory," NASA-SP-347,Part 1, pp. 101-123.

220. Reshotko, E., 1976, "Boundary-Layer Stabilityand Transition," Annual Review of FluidMechanics, Vol. 8, pp. 311-349.

221. Laufer, J., and Vrebalovich, T., 1960, "Stabilityand Transition of a Supersonic LaminarBoundary Layer on an Insulated Flat Plate,"Journal of Fluid Mechanics, Vol. 9, part 2, pp.257-299.

222. Takagi, S., 1992, "Simple Method forDetermining the Laser-Velocimeter Focal Pointwith the Aid of the Hot-Wire Anemometer," AIAAJournal, Vol. 30, No. 6, pp. 1664-1665.

223. Blackwelder, R.F., and Kaplan, R.E., 1976, "Onthe Wall Structure of the Turbulent BoundaryLayer," Journal of Fluid Mechanics, Vol. 26,pp. 89-112.

224. Moin, P., and Kim, J., 1985, "The Structure ofthe Vorticity Field in Turbulent Channel Flow,Part I, Analysis of Instantaneous Fields andStatistical Correlations," Journal of FluidMechanics, Vol. 155, pp. 441-464 .

225. Perry, A.E., and Cheng, M.S., 1982, "On theMechanism of Wall Turbulence," Journal ofFluid Mechanics, Vol. 119, pp. 173-217.

226. Donovan, J.F., and Smits, A.J., 1987, "APreliminary Investigation of Large ScaleOrganized Motion in a Supersonic TurbulentBoundary Layer on a Curved Surface," AIAA-87-1285.

227. Rose, W.C., and Johnson, D.A., 1974, "Studyof Shock-Wave Turbulent Boundary LayerInteraction Using Laser Velocimeter and Hot-Wire Anemometer Techniques," AIAA-74-0095.

228. Polyakov, A.F., and Shindin, S.A., 1983, "SomeAspects of Measuring the Structure of Non-Isothermal Turbulence by SimultaneousApplication of DISA's LDA and Hot-WireAnemometer," DISA Information No. 28, pp.10-14.

229. Logan, P., McKenzie, R.L., and Bershader, D.,1988, "Hot-Wire Accuracy in SupersonicTurbulence from Comparisons with Laser-Induced Fluorescence," AIAA Journal, Vol. 26,No. 3, pp. 316-322.

230. McKenzie, R.L., and Fletcher, D.G., 1992,"Laser-Spectroscopic Measurement Techniquesfor Hypersonic, Turbulent Wind Tunnel Flows,"NASA-TM-103928.


0.1

1

10

0.1 1 10 100 1000

P , psiao 1.5 3. 5 9 12 15 30 60 90120 150

Free-Molecular

Kn 2 1 0.1 0.01 0.001

FlowContinuum

Slip flowflow

M

Ret

Figure 1. Mach number vs. Reynolds number forlines of Constant Total Pressure andConstant Knudsen Number;d inchw = 0 00015. ; T Fo = °560 .

f

E

Amplifier

Resultant Signal

Wire

Figure 2. Frequency Compensation for CCA.

Compensaor

Amplifier Eout

Hot-wire

Square WaveCalibrator

ConstantCurrentSupply

Figure 3. Schematic Representation of ConstantCurrent Hot-Wire System; (ref. 62).

Amplifier Eout

Hot-wire

"Variable"CurrentSupply

AmplifierFeedback

Figure 4. Schematic Representation of ConstantTemperature Hot-Wire System; (ref. 62).

-

+

Eout

R R

wire leadw l+

+

Figure 5. Schematic Representation of ConstantVoltage Hot-Wire System; (ref. 57).

eSTo

S Su Toρρ

0.1

0.2

0 1 2

CCACTA

Figure 6. Modal Analysis that Compares the CTAand CCA Systems; y δ = 0 23. ; (ref. 59).


f Hz,

′′T f

T fo

o o

2

2b gb g%

CCA 0.042CTA 0.005 Noise

102 103 104 105

100

10 1

10 2

10 3

−−

−−

−−

%T To o

fo = 102

(a) Low Overheat;Total Temperature Fluctuations

Figure 7. Normalized Power Spectra that Comparethe Constant Temperature Anemometer(CTA) and Constant CurrentAnemometer (CCA) Systems; y δ = 0 23. ;(ref. 59).

ρρ

ρρ

u f

u fo

′′2

2

f Hz,

CCA 0.135CTA 0.12Noise

102 103 104 105

100

10 1

10 2

10 3

−−

−−

−−

~

fo == 102

ρρ ρρu u~

(b) High Overheat;Mass-Flow Fluctuations

Figure 7. Concluded.

290

295

300

305

310

315

-0.12 -0.08 -0.04 0 0.04 0.08 0.12

Tungsten Wire

d cmL cmT K

I Amps

w

o

====== °°

==

−−

−−

5 100 25296

2 5 10

4

2 3 2

x.

. x

T Kw ,°°

x cmw ,

T Kw == °°310 8.

Figure 8. Typical Temperature Distribution Along aConvection Controlled Hot-Wire; (ref. 21).

1

.35

.01

0

0

0.5

1

0.01 0.1 1 10

F kF k

M 1

11 1

k L1

ηηk L

ηηk = Kolmogorov scaleL = Length of wire

Figure 9. Hot-Wire Spectral Transfer Function; (ref.143).


-0.2

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

25.4 mm 5.3 15.70 m/s 25,00050.8 mm 5.3 7.70 m/s 25,000

M M d u Rem m w Mm

Rxx ττ

x Mm == 48

u Mmττ

Figure 10. Auto-Correlation Coefficient (( ))Rxx ττ ; (ref.

25).

y δδ

′′e

e

3

3

-1.5

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1 1.2

MassFlowTotal Temperature

(a) Third Moment; SkewnessFigure 11. Distribution of the Mass-Flow and Total

Temperature Fluctuations Across theBoundary Layer; (ref. 59).

y δδ

′′e

e

4

4

2

4

6

8

0 0.2 0.4 0.6 0.8 1 1.2

MassFlowTotal Temperature

(b) Fourth Moment; FlatnessFigure 11. Concluded.

25.4 mm 5.3 15.70 m/s 25,00050.8 mm 5.3 7.70 m/s 25,000

0.02

0.04

0.06

0.08

0 0.8 1.6 2.4 3.2

RE

3

ττ

u Mmττ

M M d u Rem m w Mm

x Mm == 48

Ru u t u u t

uE

312

2 2

3τ

τ τ τ τ=

′ ′ + − ′ ′ +

Figure 12. Composite Third-Order Auto-Correlation

Coefficient (( ))R uE

3ττ ; (ref. 25).

∆∆x δδ

y δδ == 0 25.

Correlation

0

0.2

0.4

0.6

0.8

Time Delay, msec

Coefficient

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6

11.52242

y δδ == 0 50.

Figure 13. Examples of Filtered Space-Time CorrelationCoefficient - 4 kHz; (ref. 87).


2.0

0.4

0.5

0.6

0.8

1.0

1.2

1.6

0.3 m

0.01

0.1

1

1 10 100 1000

x

u m sM mm

Rem

Mm

====

==

1025

16 700,

′′′′

==

uu

x

2

2

0 3.

Wave number k, 1

Figure 14. Decay of Energy of the SpectralComponents; (ref. 88).

40

50

60

70

0 2 4 6 8 10

291304307313

Over Heat Ratio = 1.8

T Ko

u

E2

Figure 15. Voltage vs. Velocity for a Wire Operatedwith CTA; (ref. 93).

10 2 100 10 2 104−−

10 3

10 1

10 1−−

Nut

Boussinesq

Ret

King

Figure 16. Summary of Heat Loss from CircularCylinders Over a Wide Range ofReynolds Number in Continuum Flow;(ref. 21).

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5

Evaluation of physical variables

N Evaluated at mean film temperatureR Evaluated at mean film temperature Evaluated at ambient temperatureGr Evaluated at ambient temperature

κκµµρρµρµρ

Forced convection

Free convection

Mixed Flow

Nuf

Re f

T T Grw o

1 1 3 1 10

1 32 9 8 10

1 64 1 88 10

2 0 3 08 10

6

6

5

5

. . x

. . x

. . x

. . x

−−

−−

−−

−−

d cmw == 0 00090.

Figure 17. Interaction of Free and ForcedConvection; (ref. 104).

Ret

Nut

Figure 18a. Nusselt Number Correlation forCylinders in Subsonic Slip Flow;T Fo = °80 ; T Fw = °584 ; (ref. 35).

1

10

1 10 100

0.050.10.20.40.60.80.90.95

1.051.251.51.751.9

M

Nut

Ret

Figure 18b. Spangenberg's Reported Heat LossMeasurements from ElectricallyHeated Wires in Air; (ref. 171).


ττw

NuNu

w

∞∞

∞∞ ==ττ .221

0.8

0.9

1M = 0.05

M = 0.20

M = 0.90

M = 0.40

M = 1.25

M = 0.05

M = 0.20

M = 0.90

M = 0.40

M = 1.25

0.8

0.9

1

0 0.5 1 1.5 2

ρρλλ

====

0 001259

3. gm cmdw

ρρλλ

====

0 000420

3. gm cmdw

Figure 19. Normalized Nusselt Number Variationwith Overheat for 0.00015 inchDiameter Platinum Wire Containing10% Rhodium; (ref. 110).

0.1

1

0.1 1 10

Constant Kn KnConstant M

M

0.04

0.4

2.0

.05

.1 .2

.3.4 .5 .6 .7 .8

Nut

Ret

Figure 20. Predicted Nusselt Number Correlationfrom Approximate Slip-Flow Theory;(ref. 35).

0.01

0.1

1

10

100

0.01 0.1 1 10 100 1000

Nut

Ret

BoussinesqKing

Figure 21. Summary of Supersonic Heat Transferfrom Transverse Cylinders in RarefiedAir Flow; (ref. 21).

0.1

0.01 0.1 1

4.73.52.11.050.54Constant MConstant Kn

Kn

M .1 .2 .3

0.5

50.01

.6 .9

Nut

Ret

Figure 22. Comparison of Heat-Transfer Data withFree-Molecule-Flow Theory Usingα = 0 57. ; (ref. 33).


0.050

M

0.30.20.1

10.8

0.1

10

1000

0.001 0.1 10 1000 100000

Nut

Ret

Figure 23. Correlation of Convective Heat Transferfrom Transverse Cylinders; (ref. 10).

0

0.2

0.4

0.6

0.8

1

0.01 0.1 1 10

ηη *

Kn

ηη* ,

, ,

==−−−−

T TT T

adw adw c

adw f adw c

Figure 24. Normalized Recovery Temperature Ratio vs.Free Stream Knudsen Number; (ref. 21).

ηη

M

0.96

0.98

1

1.02

1.04

1.06

0 0.2 0.4 0.6 0.8 1

Free Molecular Flow1.000.800.600.400.200.100.080.06Continuum Flow

Kn

Figure 25. Recovery Temperature Ratio vs. MachNumber for Constant Values ofKnudsen Number.

0.5

1

0 0.5 1 1.5 2 2.5 3 3.5 4

51.22599

f, Hz

r wττ

φφ∆∆ f

Figure 26. Fluctuation-Diagram of the FilteredSignals (Single-Wire); (ref. 97).


0.050.10.20.3

M

Ref. 120

10 2

10 3

10 4

−−

−−

−−

105 106 107

uu

Re ft∞∞−−, 1

Figure 27. Velocity Fluctuations Measured in theTest Section of LaRC Low TurbulencePressure Tunnel; (ref. 37).

1

2

3

4

5

6

7

10 15 20 25 30 35 40

4

8

12

16

20

24

28

uusc

x Mm/

′′

′′

uu

vu

sc

sc

2

2′′u

usc

2

′′vusc

2

uusc

Figure 28. Effect of 16:1 Contraction onTurbulence Generated by 2-inchSquare Mesh Grid; RMm

= 3710 ; (ref.

118).

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

u'v'

u'v'

} Turbulence reduction for onemanipulator only. Incomingturbulence from free stream

} Turbulence reduction for onemanipulator only. Incomingturbulence from upstream honeycomb

22

−−++

KK

11

11

++

++

K

K11++ −−++ ++

a aKa K

where

a Kp

qK== ==

++1 11

. ;∆∆

∆∆ p q

Ftrf

1 4 20 8 28 36HC M M M Mm m m m 42Mm

Figure 29. Measured Turbulence Reduction Factorfor Various Screens and Combinationsof Screens with Honeycomb ComparedWith Different Theories; (ref. 120).

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Re u x

uu

δ

τ

δρµ

= ≅

≅

7 5 104

0 037

.

.

y δδ

uu

wu

vu

(a) Zero Pressure GradientFigure 30. Relative Turbulence Intensities in a

Boundary Layer Along a Smooth Wall;(ref. 131).


0

0.02

0.04

0.06

0.08

0.1

0.12

0.005 0.01 0.015 0.02 0.025

0.2

0.3

0.4

Re xuu

δ

τ

≅

≅

7 5 104

0 037

.

.

uu

wu

uu

vu

y δδ

(b) Constant-Stress LayerFigure 30. Concluded.

0.4

0.8

1.2

1.6

2

2.4

2.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

500,00050,000

Ret

uuττ

′′r rp

(a) Across the pipeFigure 31. Distribution of velocity fluctuation in a

pipe; (ref. 40).

0.8

1.6

2.4

0 0.02 0.04 0.06 0.08 0.1

500,00050,000

Ret

uuττ

′′r rp

(b) Near the wallFigure 31. Concluded.

2

4

6

8

0 2 4 6 8 10

F x 10 = 0.6Broadband - Klebanoff's Mode

4

0

1

Magnified 20x

u = 11.6 m/sec

√√0.63 cm GRID

} TS-frequency

TS packets

≈≈

Re = 717

ηηy

,%uu

uue

u ue

Figure 32. Types of Fluctuation SimultaneouslyPresent in Boundary Layer; (ref. 145).


ξξ2 == ++r x rd p

T Cx r

d

Re betweenand

o

p

j

d

== °°++

==

170

20

17 00030 000

,,

uulocal

,

TT

o

olocal

Relative Intensity

0.1

0.2

0.3

0.4

0.5

0 0.04 0.08 0.12 0.16

,max

TT

o

o

max

uu

Figure 33. Distribution of Relative Intensity of TurbulentVelocity and Temperature Fluctuations in aHot Round Free Jet; (ref. 157).

′′ −−uu

x xd

o

c

2

2

0

0.02

0.04

0.06

0.08

0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

500650800950

x/dc

ξξ == −−y d x xc o

Figure 34. Relative Turbulence Intensity (u-component) in the Wake of aCircular Cylinder; Redw

= 1360 ; (ref.

43).

Figure 35. General Fluctuation diagram for SubsonicCompressible Flow; (ref. 168).

0.01

0.02

0.03

0 0.25 0.5 0.75

Sound

Flow

Measurement

X

r

,%φφ

(a) Disturbances from Single Sound SourceFigure 36. Fluctuation Diagram; M = 0 76. ; (ref. 167).


0.2

0.4

0.6

0 0.2 0.4 0.6

Flow

MeasurementX

Wire

r

,%φφ

(b) Combination of Vorticity andEntropy Modes Downstream of aCylinder


1

10

0.0001 0.001

0.80.60.50.20.10.05

M ∞∞ Pt Rh Wired inL mmT K

T TR R

w

o

o

Ref

−=== °==

100 000150 683031 6530 7403

%. ..

..

∆∆Nu∞∞

ρρ ≈≈ 1 Kn

Figure 37. Nusselt Number as a Function of ρ ; (ref. 110).

0

0.1

0.2

0.2 0.4 0.6 0.8

0.00040.00060.00080.0010.0012

Density; gm/cm 3

M

Su

(a) Velocity Fluctuations; T Ko = °303 ; τw = 0 672. .Figure 38. Spangenberg Data; (ref. 163).

0

0.1

0.2

0.3

0.0004 0.0006 0.0008 0.001 0.0012

0.050.10.20.40.60.8

ρρ; /gm cm3

SρρM

(b) Density Fluctuations; T Ko = °303 ;τw = 0 672. .

Figure 38. Continued.

-2.5

-2

-1.5

-1

-0.5

00 1 2

0.050.10.20.40.60.8

STo

M

ττW

(c) Total Temperature FluctuationsT Ko = °303 ; ρ = 0 0008 3. gm cm .



0.40.450.50.550.60.650.70.750.80.826

CHOCKEDCONDITION}

,%uu

105 106 107

Re ft, −−1

M ∞∞

101

100

10 1−−

(a) Velocity FluctuationsFigure 39. Disturbances Measured in the LaRC 8'

Transonic Pressure Tunnel using aThree Element Hot-Wire Probe; (ref. 86).

0.40.450.50.550.60.650.70.750.80.826

CHOCKEDCONDITION}

105 106 107

,%ρρρρ

Re ft, −−1

M ∞∞

100

10 1

10 2

−−

−−

(b) Density FluctuationsFigure 39. Continued.

0.40.450.50.550.60.650.70.750.8 CHOCKED CONDITION

105 106 107

,%TT

o

o

Re ft, −−1

M ∞∞

10 1

10 2

−−

−−

(c) Total Temperature FluctuationsFigure 39. Continued.

0.40.450.50.550.60.70.750.80.826

CHOCKEDCONDITION}

105 106 107

,%mm

Re ft, −−1

100

10 1

10 2

−−

−−

M ∞∞

(d) Mass Flow FluctuationsFigure 39. Concluded.


0.02

0.04

0.06

0.08

0 0.2 0.4 0.6 0.8 1

}

}

u u

v u

w u

ρ ρ

y δδ

Figure 40. Normalized RMS Velocity and DensityFluctuation Distribution Across theBoundary Layer; (ref. 170).

-50

-40

-30

-20

-10

0

1 10 100 1000 10000

-5/3

FREQUENCY Hz,

PSD dBE Hz

,2

T

ut

ρ~

Figure 41. Spectra of Mass Flux and TotalTemperature Fluctuations; 3.6 km;M = 0 57. ; Feb. 79; (ref. 49).

Probe 1Probe 2

King's Law

d , in.

0.000500.000200.00015

0.001

0.01

0.1

1

10

0 0.1 0.2 0.3 0.4 0.5

w

%

M∞∞

u u

T Tm m

o o

ρ ρ

(a) Measured FluctuationsFigure 42. Fluctuations Measured in the LaRC Low

Turbulence Pressure Tunnel;p psiao = 15 ; (ref. 172).

0.01

0.1

1

0 0.1 0.2 0.3 0.4

"Y" Wire Probe (3 wire Results) Parallel Wire Probe (3 wire Results)King's Law

M∞∞

,%mm

m m or u u

(b) Mass Flow FluctuationsFigure 42. Concluded.


rr00

φφmm

TT

o

o

Figure 43. Generalized Fluctuation Diagram forSupersonic Flow.

rrββ

ppppσσ ωω

−α−α 00

φφ

-r-r

TT

o

o

EntropyEntropy

mm

SoundSound(Finite Source Velocity)(Finite Source Velocity)SoundSound(Zero Source Velocity)(Zero Source Velocity)VorticityVorticity

Figure 44. Generalized Mode Diagram forSupersonic Flow.

4

8

12

16

20

24

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

r

MRet

====

5 00250 000

.,

xφφ 103

Figure 45. Mode Diagram Measured in Test Sectionfor M = 5 0. ; (ref. 185).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 2 3 4 5

,%mm

M

Re inRe in

t

t

/ . ,/ . ,

≈≈≈≈

90 000330 000

(a) Mass Flow FluctuationsFigure 46. Variation with Tunnel Mach Number;

(ref. 185).

0

2

4

6

8

10

1 2 3 4 5

M

Re in

Re int

t

/ . ,

/ . ,

≈≈

≈≈

90 000

340 000

xp

Mγγ ρρ 2 104

(b) Pressure FluctuationFigure 46. Concluded.


0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5

M

Re inRe in

t

t

/ . ,/ . ,

≈≈≈≈

90 000340 000

uu

s

Figure 47. Variation of Source Velocity with TunnelMach Number; (ref. 185).

0

1

2

3

-10%

+10%

0.1

0 200 400 600 800 1000 1200 1400

,%mm

,%TT

o

o

p N cmo / 2

Figure 48. Root-Mean-Square Mass-Flow and Total-Temperature Fluctuations in the LaRC22" Hypersonic Helium Tunnel; (ref. 181).

2

4

6

8

0 2 4 6 8 10 12 14 x 106

,%pp∞∞

p N cmo, /∞∞2

Figure 49. Pressure Fluctuation in the LaRC 22"Hypersonic Helium Tunne; (ref. 181).

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14 x 106

Uncertainty

uu

s

p N cmo, /∞∞2

Figure 50. Relative Source Velocities in the LaRC22" Hypersonic Helium Tunnel; (ref.181).


0.5

1

1.5

2

2.5

0 0.5 1 1.5

ρρττ

′′u

wall

2

S S

S S

Klebanoff s dataRange of data

u

u

≠≠

==ρρ

ρρ

'

y δδ

Figure 51. Velocity Fluctuation Measurements inBoundary Layer; Influence of theSensitivity Coefficient to MachNumber; Shaded Zone - Range of theAvailable Velocity Fluctuation Data inSupersonic Boundary Layers,Including Hot-Wire and Laser-DopplerMeasurements; (ref. 162).

-1

-0.9

-0.8

-0.7

-0.6

-0.5

0.4 0.8 1.2 1.6

RuTo

S S

S Su

u

≠

=ρ

ρ

M

Figure 52. Variation of RuTo with Mach Number

Through Boundary Layer; (ref. 177).

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 1 2 3 4 5 6 7 0 0.1 0.2 0.3

y in.

.m T Wire noA

o

13 16 32 5

−−−

Figure 53. Variation of Wide-Band RMS Mass-Fluxand Total-Temperature FluctuationsAcross the Boundary Layer. Kistler'sResults at M = 4 76. (Solid Line) areIncluded for Comparison; (ref. 188).

0.02

0.04

0.06

0.08

0.1

0.12

0 1 2 3 4 5 6 0 0.1 0.2 0.3

y in.

.u T Wire noA

o

13 16 3

−−

Figure 54. Variation of Wide-Band RMS Velocity andStatic Temperature Fluctuations Acrossthe Boundary Layer; ′ ≠p 0 ; (ref. 188).


0.05

0.1

0.15

0.2

0 1 2 3 4 5 6 7

Range of wall p' from Kistler

Expected p'

Laufer's datalevel from

0 0.1 0.2 0.3

y in.

.p Wire noA

ρ13 16 3

−−

Figure 55. Variation of Wide-Band RMS Pressureand Density Fluctuations Across theBoundary Layer; ′ ≠p 0 ; (ref. 188).

0

1

2

3

4

-0.1 0 0.1 0.2 0.3 0.4

Peak(y = 1.68 cm)

Center line(y = 0)

αα for M

7 8.4

Location m m T TPeakCenter line

o o

. % . %

. % . %10 9 0 42

6 2 0 26

φφ

r

Figure 56. Mode Diagrams in Wake of 15° Half-Angle Wedge; (ref. 181).

r

eV e∆∆ 1

0.1

0.2

0 0.1 0.2 0.3 0.4 0.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0

0.1

0.2Shock Generation h h

Free StreamWedgeWedgeWedge

cm Wirecm Wire

w w s,

.

..

. .

. .

10 0 910 0 52 0 90 01 0 90 01 0 9

12

°°°

Figure 57. Fluctuation Mode Plots; M ≈ 21;p x N mo = 12 41 106 2. ; (ref. 48).

Strazisar et.alSchubauer & SkramstadRoss et.al.WortmannKachanov et.al.

60

120

180

240

300

360

420

200 400 600 800 1000 1200

x 106

Parallel

Saric & Nayfeh

uρρδδ µµ*

f

Figure 58. Neutral Disturbance in Studies ofLaminar Boundary Layer Stability;(ref. 205).


1

2

3

4

5

6

0 10 20 30 40 50 60

x 103

x Mm

u

v

2

2′Biplane square mesh grid

M M d umm m s

Pressure 4 BarRe

m m

,

.

−

≈

172 5 4 61

500λ

uu

2

2′

Figure 59. Decay of u

u

2

2′ and

u

v

2

2′ Downstream of a

Grid; (ref. 206).

0.001

0.01

0.1

1

10

0.01 0.1 1 10

X (cm)4080160

POWER SPECTRAL DENSITY

Wave Number, cm- 1

−−53

slope

(a) Poser Spectral Density of u2;u m s= 10 ; (ref. 88)

Figure 60. One-Dimensional Spectrum.

0.01

1

0.0001 0.01 1

POWER APECTRAL DENSITY

SPECTRAL ENERGY

Normalized Wave Number

100

10000

100000

E k

E k1 1

2 1

xM

xkk

m

d

=

−

45

1855

1 7 1

53

.

M mmM du m s

ReRe

m

m

M

1705 4

61 00

2 4 10500

6

..

. x

λλ ≈≈

(b) Spectral Energy; (ref. 206)Figure 60. Concluded.

Heated JetHeated GridTheory according to Pao

0.01

1

0.00001 0.001 0.1 10

POWER APECTRAL DENSITY

SPECTRAL ENERGY

Normalized Wave Number

0.001

100

10000

Figure 61. One-Dimensional TemperatureSpectrum; (ref. 207).


0

1

2

3

4

5

0.01 0.1 1 10

Schubauer-SkramstadHall-HislopDrydenBailey-Wright

x 10- 6

,%u u

Ret

Theory, xuλλ ρρ µµ == 4 104

Figure 62. Effect of Free-Stream Turbulence onBoundary Layer Transition; (ref. 208).

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Stetson et. al. ; R = 1728, M = 6.8Kendall; R = 1740, M = 7.7Demetriades; R = 1740, M = 7.0Theory, Mack; R = 1730 Equiv.

e

ee

x 103

Second Mode; ψψ = 0°

First Mode; ψψ = 55°

F x 104

−− αα

Figure 63. Amplification Rate vs. Frequency,Comparison with Others; (ref. 209).

0 0.1 0.2 0.3 0.4 0.5

0.0

- 2.0

2.0

4.0

- 4.0

0.0

- 2.0

2.0

4.0

- 4.0

Channel 2

Channel 1

ρρρρ

uu rms

′′

TIME seconds x 10 3−−

Figure 64. Typical Fluctuating Mass Flow Datafrom Two Hot-Wires IndicatingSimilarity Between the Signals; (ref.226).

-4

-3

-2

-1

0

1

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

UpstreamDownstream-1

0

1

2

yδδ

== 0 9.

yδδ

== 0 1.

TIME seconds x 10 3−−

ρρρρ

uu rms

′′

Figure 65. Ensemble Averaged Positive Events inthe Weak Perturbation Case; (ref. 226).


X = 5.375X = 8.375X = 9.375

Open Symbols = Laser VelocimeterClosed Symbols = Hot-Wire Anemometer

0.04

0.08

0.12

0.16

0 0.5 1 1.5 2 2.5

M = 2.06

y cm,

uu

Figure 66. Comparision of Streamwise VelocityFluctuations Measured by Hot-Wiresand Laser Velocimeter; (ref. 227).

y δδ

FLUCTUATION AMPLITUDE

0.025

0.05

0.075

0 0.5 1 1.5

Single High OverheatMultiple Overheats

LASER-INDUCED FLUORESCENCE

M = 2.06

T Tρ ρ

Hot Wire T To o− ≈ ρ ρ

Figure 67. A Comparison of the Distribution ofRMS Fluctuation Amplitudes in StaticTemperature and Density within theBoundary-Layer obtained using a Hot-Wire at Single and Multiple OverheatRatios with Direct MeasurementsObtained using Laser-InducedFluorescence (LIF). The LIF data havebeen Corrected for Instrument Noise;(ref. 229).

RMS Fluctuation Amplitude

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.02 0.04 0.06 0.08

NO-LIF

NO-LIF

O2-LIF/Raman

O2+N2 Raman

Hot-Wire

yδδ

T T

T T

T T

ρ ρ

ρ ρ

Figure 68. Comparisons of Temperature andDensity RMS FluctuationAmplitudes Obtained with LIFTechniques or Implied by Hot-WireAnemometer Data. Flow Conditionsare the Same as in Figure 66. ForEach Variable, the FluctuationAmplitude is Normalized by theLocal Time-Averaged Value; (ref.230).

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