Measurement of the turbulent kinetic energy budget of a planar wakeflow in pressure gradients

Xiaofeng Liu, Flint O. Thomas

Abstract Turbulent kinetic energy (TKE) budget mea-surements were conducted for a symmetric turbulentplanar wake flow subjected to constant zero, favorable, andadverse pressure gradients. The purpose of this study is toclarify the flow physics issues underlying the demon-strated influence of pressure gradient on wake develop-ment, and provide experimental support for turbulencemodeling. To ensure the reliability of these notoriouslydifficult measurements, the experimental procedure wascarefully designed on the basis of an uncertainty analysis.Three different approaches were applied for the estimateof the dissipation term. An approach for the determinationof the pressure diffusion term together with correction ofthe bias error associated with the dissipation estimate isproposed and validated with the DNS results of Moser et al(J Fluid Mech (1998) 367:255–289). This paper presents theresults of the turbulent kinetic energy budget measure-ment and discusses their implications for the developmentof strained turbulent wakes.

1IntroductionThe response of a symmetric, turbulent plane near-wake toconstant favorable and adverse streamwise pressure gra-dients was the focus of an experimental investigation re-ported by Liu et al (2002). Their work was motivated by itsrelevance to high-lift for commercial transport aircraft. Insuch applications, the wake from upstream elements in amulti-element airfoil configuration develops in a strongpressure gradient environment. The nature of the wake’s

response to the imposed pressure field will significantlyaffect the overall aerodynamic performance of the high-liftsystem. The results presented by Liu et al demonstrate thatthe mean flow and turbulence quantities in the wake areextremely sensitive to the applied pressure gradient. Forexample, even a modest adverse pressure gradient wasfound to have a profound effect on increasing wakespreading and reducing the maximum velocity defect de-cay rate. Along with the enhanced wake widening, theadverse pressure gradient condition was found to sustainhigher levels of turbulent kinetic energy over largerstreamwise distances than in the corresponding zeropressure gradient wake. In contrast, the favorable pressuregradient case exhibited a reduced spreading rate, increaseddefect decay rate, and a more rapid streamwise decay ofturbulent kinetic energy relative to the zero pressure gra-dient case.

One of the most physically descriptive measures bywhich the evolution of a turbulent flow may be assessedis the turbulent kinetic energy per unit mass (TKE). Itsbudget, which examines the balance and contribution ofdifferent mechanisms such as convection, production,diffusion and dissipation in the TKE transport equation,provides insight into the physics of the flow and suggestsstrategies for turbulence modeling. Given the significanteffect that the imposed pressure gradient has upon theevolution of the wake turbulence quantities, as demon-strated by Liu et al and Carlson et al (2001), it is ofinterest to examine in detail the TKE budget for thestrained wake. Since direct numerical simulation (DNS)is limited to low turbulent Reynolds numbers, experi-ment is still the only feasible approach for obtaining theTKE budget in turbulent flows at high Reynoldsnumbers.

The measurement of the TKE budget in free shearflows has been the focus of several previous studies.These include Wygnanski and Fiedler (1969), Panchap-akesan and Lumley (1993), George and Hussein (1991),Hussein et al (1994), Heskestad (1965) and Bradbury(1965) in jet flows, Wygnanski and Fiedler (1970) in aplanar mixing layer, Raffoul et al (1995) and Browne et al(1987) in bluff body wakes, Patel and Sarda (1990) in aship wake, and Faure and Robert (1969) in the wake of aself-propelled body.

A series of turbulent kinetic energy (TKE) budgetmeasurements were conducted for the symmetric, turbu-lent planar wake flow subjected to constant zero, favorable,and adverse pressure gradients. This paper will focus onthe measurement procedure that was developed in order to

Experiments in Fluids 37 (2004) 469–482

DOI 10.1007/s00348-004-0813-3

469

Received: 24 November 2003 / Accepted: 9 March 2004Published online: 6 July 2004� Springer-Verlag 2004

X. Liu, F. O. Thomas (&)Center for Flow Physics and Control,University of Notre Dame, Notre Dame,Indiana 46556-5684, USAE-mail: Flint.O.Thomas.1@nd.edu

Present address: X. LiuDepartment of Mechanical Engineering,The Johns Hopkins University, Baltimore,Maryland 21218, USA

This research was supported by NASA Langley Research Center,Hampton, VA under Grant No. NASA NAG1-1987, monitored byBen Anders. The authors would also like to thank Michael M.Rogers of the NASA Ames Research Center for providing the DNSresults.

measure the strained wake TKE budget. Special consider-ation is given to the measurement of the dissipation term,and a comparison of three different methods is presented.The results are compared with the DNS wake simulationsat lower Reynolds number conducted by Moser et al(1998). To our knowledge, direct comparison of TKEbudget measurement with DNS simulation results has notbeen previously reported in the literature. The resultingwake TKE budgets are presented and their implications forthe development of strained turbulent wakes are dis-cussed.

2Experimental set-up

2.1Wind tunnelThe experiments were performed in an open-return sub-sonic wind tunnel facility located at the Center for FlowPhysics and Control at the University of Notre Dame. Thisfacility has been documented in detail in Liu (2001) as wellas in Figs. 1 and 2 of Liu et al (2002). Therefore, onlyessential aspects will be described here.

Ambient laboratory air is drawn into a square tunnelinlet contraction of dimension 2.74 m on a side with acontraction ratio of 20:1. Twelve turbulence reductionscreens at the tunnel inlet yield a very uniform test sectionvelocity profile, with a free stream fluctuation intensitylevel that is less than 0.1% (and less than 0.06% for fre-quencies greater than 10 Hz).

The reported experiments utilize two consecutive testsections. The upstream test section is 1.83 m in length,0.61 m in width and 0.36 m in height. This section

contains a wake-generating plate (described below) whilethe second forms a diffuser section which is used toproduce the desired constant adverse/favorable pressuregradient environment for wake development. The topand bottom walls of the diffuser are made of sheetmetal, and their contour is fully adjustable by means ofseven groups of turnbuckles in order to create thedesired constant streamwise pressure gradientenvironment.

In this paper x, y, and z denote the streamwise, lateral,and spanwise spatial coordinates, respectively.

2.2Wake-generating bodyThe wake-generating body is a Pexiglas plate (aligned withthe flow direction) with chord length of 1.22 m and athickness of 17.5 mm. The plate leading-edge consists of acircular arc with distributed roughness which gives rise to

Fig. 1. Experimentally-measured stream-wise pressure distributions for zero,adverse and favorable pressure gradientcases

Fig. 2. Twin X-wire probe configuration

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turbulent boundary layers that develop over the top andbottom surfaces of the plate. The last 0.2 m of the plateconsists of a 2.2� linear, symmetric taper down to a trailingedge of 1.6 mm thickness. The splitter plate model issidewall mounted in the test section, with endplates usedto minimize the influence of tunnel sidewall boundarylayers. The thickness of each endplate is 6 mm, and itspans from the leading edge to the 83% chord location ofthe wake-generating plate.

2.3Streamwise pressure gradientsThe streamwise pressure gradient is imposed on the wakeby means of fully adjustable top and bottom wall contoursof the diffuser test section. The flexible walls are iterativelyadjusted by means of seven groups of turnbuckles, untilthe desired constant streamwise pressure gradient dCp/dxis attained. The streamwise pressure distribution wasmeasured by means of a series of static pressure taps lo-cated on one flat sidewall of the diffuser test section at thesame lateral (y) location as the centerline of the wake. LDVmeasurements of the centerspan streamwise distributionof mean velocity, �U(x, y=0, z=0), were found to be fullyconsistent with the measured wall pressure variation,thereby confirming the suitability of the pressure tapplacement and its use in the characterization of thestreamwise pressure gradient imposed on the wake. Theimposed pressure will be expressed in terms of a pressurecoefficient,Cp= (P(x)–P¥)/q¥, where P(x) is the local staticpressure in the diffuser, and P¥ and q¥ are the static anddynamic pressures, respectively, upstream of the wakegenerating plate.

Three sets of experiments were conducted: 1) a zeropressure gradient (ZPG) base flow condition, dCp/dx=0.0 m–1; 2) a constant adverse pressure gradient(APG) condition with dCp/dx=0.338 m–1, and; 3) a con-stant favorable pressure gradient (FPG) condition withdCp/dx= –0.60 m–1. The zero pressure gradient wakeserved as an essential baseline case for comparison withthe nonzero pressure gradient wake development. Ineach case, a common zero pressure gradient zone occursimmediately downstream of the splitter plate trailingedge in order to ensure that the wake initial condition isidentical. The relative error in the imposed constantpressure gradient is never more than 1.7% to the 95%confidence level.

The measured streamwise pressure distributions cor-responding to these different experimental conditions areshown in Fig. 1. As indicated, the pressure gradients areinitially applied downstream of the plate trailing edge ata common location designated xp�40 h0 (where h0 is theinitial wake momentum thickness). In this manner, theinitial conditions at the trailing edge of the plate areidentical in each case. Also shown in this figure is alarger adverse pressure gradient case that was run butfound to give rise to intermittent, unsteady flow sepa-ration near the aft portion of the diffuser wall. For thisreason, measurements for this case will not be presented.It may be regarded as an effective upper limit on themagnitude of the constant adverse pressure gradient thatcan be produced by the diffuser without incurring

intermittent, unsteady flow separation effects. The dif-fuser wall coordinates corresponding to each pressuregradient case shown in Fig. 1 can be found in theAppendix of Liu et al (2002).

The quality of the flow field in the diffuser sectionwas carefully examined. These measurements revealedthat the mean flow remains spanwise uniform in thediffuser test section up to the last measurement station atx=1.52 m.

2.4Basic flow parametersThe experiments were performed at a Reynolds numberRe=2.4·106, based on the chord length of the plate and afree stream velocity of 30.0±0.2 m/s for all pressure gra-dient cases. The initial wake momentum thicknesswash0=7.2 mm, corresponding to a Reynolds numberbased on the initial wake momentum thicknessReh=1.5·104. The wind tunnel wall boundary layer thick-ness (99%Ue) is approximately 19 mm at the streamwiselocation corresponding to the trailing edge of the splitterplate.

2.5Flow field diagnosticsA multi-channel TSI IFA-100 constant temperature ane-mometer was utilized, together with a variety of hot-wireprobes, in order to acquire the required time-seriesvelocity fluctuation data. For measurements of thestreamwise and lateral or spanwise component velocity,Auspex type AHWX-100 miniature X-wire probes wereused. These probes utilize tungsten sensors with a nominaldiameter of 5 lm and a sensor length of approximately1.2 mm. In addition to the X-wire probes, a dual parallelsensor probe (Auspex type AHWG-100) was required forsome of the fluctuating derivative measurements in thedissipation estimate. The spacing between the dual sensorsof the parallel probe is 0.3 mm, and the sensor length wasapproximately 0.9 mm. In comparison, the estimatedKolmogorov length scale for the wake flow is approxi-mately 0.1 mm. The effect of the limited spatial resolutionof the probes used for fluctuating derivative measurementsis discussed in Sect. 4.4.

For the hot-wire measurements, the anemometeroutput was anti-alias filtered at 20 kHz and digitallysampled at 40 kHz. The 20 kHz Nyquist frequency waschosen to correspond approximately to the highestresolvable frequency of the hot-wire probes at themeasurement location for the TKE budget estimate,x=101.6 cm (x/h0=141). The total record length at eachmeasurement point is 13.1 s, which yielded fully con-verged turbulence statistics.

In the following section, a dissipation measurementtechnique based on the assumption of locally axisym-metric, homogeneous turbulence is described. Thisrequires the measurement of the mean-square fluctuating

derivative @v0

@z

� �2which cannot be obtained from a single

X-wire probe. For this measurement a twin X-wire probeconfiguration as shown in Fig. 2 was used. The spacingbetween the centers of the two X-wire probes is approxi-

471

mately 1.3 mm, as determined from an enlarged digitalimage of the twin X-wire configuration.

3Turbulent kinetic energy transport equationA generic form of the turbulent kinetic energy transportequation, valid for incompressible flow (see Hinze 1975) isgiven by

Dk

Dt¼ � @

@xiu0i

p0

qþ k0

� �� u0iu

0j

@Uj

@xiþ m

@

@xiu0j

@u0i@xjþ@u0j@xi

� �

� m@u0i@xjþ@u0j@xi

� �@u0j@xi

ð1Þ

where k ¼ 12 u0iu

0i is the mean turbulent kinetic energy per

unit mass, and k0 ¼ 12 u0iu

0i is the fluctuating turbulent

kinetic energy per unit mass. The left hand side representsthe material derivative of turbulent kinetic energy. Theterms on the right hand side are, respectively, the effectivediffusion of turbulent kinetic energy (by velocity fluctua-tions and pressure-velocity correlations), turbulence pro-duction, reversible viscous work, and turbulencedissipation to heat. Equation 1 may be written in theequivalent form,

Dk

Dt¼ � @

@xiu0i

p0

qþ k0

� �� u0iu

0j

@Uj

@xiþ m

@2k

@xi@xi� m

@u0j@xi

@u0j@xi

ð2Þ

It is important to point out that the last term in (2)is not equivalent to the dissipation term in (1). In fact,only if the turbulent flow is homogeneous does the lastterm on the right hand side of (2) become the properform for the dissipation. The difference lies in the cross-derivative correlation terms which to-date have not beenaccurately measured, although there was an attempt todo so by Browne et al (1987). Therefore, as in all otherpreviously-reported efforts to measure the turbulentkinetic energy budget in free shear flows, a concession ismade at the outset and utilize Eq. 2 is utilized as thebasis for our measurements. The use of the nine-termhomogeneous approximation for dissipation isreasonable given the fact that high Reynolds numberturbulent flows tend to approach a state of homogeneityat the smallest scales characteristic of the dissipativerange.

For the planar turbulent wake under considerationhere, we denote the streamwise, lateral and spanwisespatial coordinates as x1,x2 and x3 (which are equivalent tox, y, z), respectively. The corresponding mean velocitycomponents are denoted as �U1, �U2; and �U3(equivalent toU,V, and W) and the fluctuating velocity components asu¢1,u¢2, and u¢3 (equivalent to u, v, and w). For steady, 2-D

flow in the mean, we have @@t ðÞ ¼ 0; U3 ¼ 0 and @

@x3ðÞ ¼

0. Also, from the continuity equation, we have @U2

@x2¼ � @U1

@x1.

Therefore Eq. 2 can be simplified to a form appropriate forthe planar turbulent wake flow as follows:

0 ¼ �U1@k@x1� U2

@k@x2

Convection

� @@x1

u01p0

q � @@x2

u02p0

q

Pressure Diffusion

� @@x1

12 u031 þ u01u022 þ u01u023ð Þ � @

@x2

12 u021 u02 þ u032 þ u02u023ð Þ

Turbulence Diffusion

� u021 � u022

� �@U1

@x1� u01u02

@U1

@x2þ @U2

@x1

� �

Production

þm @2k@x2

1þ m @

2k@x2

2

Viscous Diffusion

�m @u01@x1

� �2þ @u01

@x2

� �2þ @u01

@x3

� �2�

þ @u02@x1

� �2þ @u02

@x2

� �2þ @u02

@x3

� �2

þ @u03@x1

� �2þ @u03

@x2

� �2þ @u03

@x3

� �2�

Dissipation ð3Þ

Equation 3 provides the framework to be used for themeasurement of the TKE budget in the wake. By mea-suring the individual terms in Eq. 3, the TKE budget forthe turbulent planar wake flow in pressure gradient can beconstructed. The approach utilized for the measurement ofeach term is briefly addressed below.

3.1Convection termThe convection term consists of two parts, the streamwiseconvection �U1

@k@x1

and the lateral convection �U2@k@x2

.Both can be measured directly. An X-wire probe is used toobtain both �U1 and �U2as well as the three normal-com-ponent stresses required for calculation of the cross-stream profiles of k. The streamwise spatial derivative @k

@x1is

obtained from the measurement of k at three adjacentstreamwise measurement stations via a finite differenceapproximation. Details of how streamwise derivatives arecomputed are discussed in Sect. 3.7 of the paper. Thelateral spatial derivative @k

@x2is obtained from differentiat-

ing an optimum fit to a high spatial resolution lateralprofile of k.

3.2Pressure diffusion termThe pressure diffusion term is not directly measurable. Inthe jet studies by Wygnanski and Fiedler (1969) andGutmark and Wygnanski (1976), this term was inferredfrom a forced balance of the turbulent kinetic energyequation. In a more recent axisymmetric jet study byPanchapakesan and Lumley (1993), the pressure transportterm was simply neglected. In the cylinder wake study byBrowne et al (1987), it was concluded that the pressuretransport term (obtained by forcing a balance of the tur-bulent kinetic energy equation) was approximately equalto zero. In the jet flow measurement conducted by Hussein

et al (1994), they ignored the termu01p0

q

� �and attempted to

472

estimateu02p0

q

� �by integrating the difference between the

so-called ‘‘transport dissipation’’ and the ‘‘homogeneousdissipation’’. In this study, the pressure diffusion term willbe inferred from the forced balance of the turbulent kineticenergy equation. This result will subsequently be comparedwith the DNS strained wake results of Moser et al (1998).

3.3Turbulence diffusion termThe turbulence diffusion term is composed of the stream-wise turbulence diffusion � @

@x1

12 u031 þ u01u022 þ u01u023ð Þand

the lateral diffusion � @@x2

12 u021 u02 þ u032 þ u02u023ð Þ. In order to

determine the turbulence diffusion term, an X-wire probe

can be used to obtain u031 ; u01u022 ; u01u023 ; u021 u02 and u032 by

direct measurement. The remaining term u02u023 can beobtained indirectly from additional X-wire measurementsthrough application of a procedure developed by Townsend(1949) and described by Wygnanski and Fiedler (1969).Alternately, both Panchapakesan and Lumley (1993) and

Hussein et al (1994) simply assumed that u02u023 � u033 fortheir jet flow measurements, and demonstrated that theerror introduced by this assumption is less than 10%. In thisstudy, we will also use this approximation to estimate therequired term u02u023 .

3.4Turbulence production termThe shear production �u01u02

@ �U1

@x2þ @ �U2

@x1

� �and dilatational

turbulence production � u021 � u022

� �@ �U1

@x1can be measured

directly. Independent measurements of turbulenceproduction using both X-wire probes and two-componentlaser Doppler velocimetry (LDV) are presented in Liu et al(2002). Excellent agreement between the hot-wire and LDVmeasurements was obtained. These experiments showthat, despite the streamwise pressure gradients imposed,

the wake is shear dominated since � u021 � u022

� �@U1

@x1�

�u01u02@U1

@x1þ @U2

@x1

� �. Despite this, we include the dilatational

production term in the TKE budget. As indicated, themeasurement of local turbulence production requirescross-stream profiles of both local mean velocity andReynolds shear and normal stresses.

3.5Viscous diffusion termsAll previously cited investigations of the turbulent kineticenergy budget in free shear flows have ignored the viscousdiffusion terms. Wygnanski and Fiedler (1969) and Gut-mark and Wygnanski (1976) note that neglect of theseterms was based on the assertion of Laufer (1954) that theterm is comparatively small in the turbulent kinetic energyequation. Panchapakesan and Lumley (1993) explainedthat in free turbulent flows, away from walls, the viscouscontribution to the transport terms is negligible in com-parison with the turbulent contribution. In the wake underinvestigation here, this is substantiated by the directmeasurements of the local turbulent viscosity, as definedby the value of mt � �u01u02

@ �U1=@x2ð Þ. Results indicate

that mt=m � O 103ð Þ. Neglect of the viscous diffusion term isfurther validated from measured values of the second

derivative of the turbulence kinetic energy @2k@x2 which is

O(1.0 s–2). This leads to a corresponding value for theviscous diffusion O(10–5 m2/s3), which is only about10–7 times the peak value of the measured viscous dissi-pation term. It may be noticed that by neglecting theviscous diffusion term, the only difference betweenEq. 1and Eq. 2 is the expression for the dissipation term.

3.6Dissipation termA review of the cited literature reveals that the dissipationterm can be estimated in one of five ways. In this study,three of these approaches will be utilized to obtain pre-liminary dissipation estimates, and the results are com-pared. Each of the approaches is briefly described below.Ultimately, however, we will utilize a locally axisymmetricturbulence assumption for the dissipation estimate used inthe wake TKE budget.

3.6.1Isotropic turbulence assumptionIn high Reynolds number flows, the viscous dissipationtakes place at the smallest scales of motion. Due to theassumed loss of directional information during the energycascade to small scales, the turbulence may be approxi-mated as locally isotropic, in which case the dissipationterm can be radically simplified to (see Hinze 1975)

e ¼ 15m@u01@x1

� �2

ð4Þ

The required fluctuating spatial derivative can beobtained from the temporal derivative of u¢1 by invokingTaylor’s frozen field approximation,

@

@x� � 1

U1

@

@tð5Þ

This was the technique employed by Gutmark andWygnanski (1976) and Bradbury (1965) for their jet flowmeasurements.

3.6.2Locally axisymmetric homogeneous turbulenceassumptionUsing measurements in a round jet, and those of Browneet al (1987) in a cylinder wake, George and Hussein (1991)demonstrated that the mean-square derivatives of thefluctuating velocity are in good agreement with local axi-symmetric turbulence theory, the characteristic feature ofwhich is the invariance of statistical quantities with respectto rotation about a preferred direction. With theassumption of locally axisymmetric, homogeneous turbu-lence, the dissipation term can be estimated from either

e¼ m5

3

@u01@x1

� �2

þ2@u01@x3

� �2

þ2@u02@x1

� �2

þ8

3

@u02@x3

� �2" #

ð6Þ

or

e¼ m � @u01@x1

� �2

þ2@u01@x2

� �2

þ2@u02@x1

� �2

þ8@u02@x2

� �2" #

ð7Þ

473

In Eq. 6, the terms@u01@x1

� �2and

@u01@x3

� �2can be obtained

from a temporal derivative of the u¢1and u¢2time-series(obtained via the X-wire), respectively, combined with theuse of Taylor’s frozen field approximation. The term@u1

1

@x3

� �2can be obtained from a dual sensor, parallel probe

measurement. The estimate of the@u02@x3

� �2term requires a

twin X-wire probe configuration, which was shown inSect. 2.5.

3.6.3Semi-isotropic turbulence assumptionIn this approach, unmeasured fluctuating velocity deriva-tives in the homogeneous dissipation term are estimatedbased on measured fluctuating velocity derivatives. For

example, the streamwise derivatives@u01@x1

� �2,

@u02@x1

� �2, and

@u03@x1

� �2can each be estimated from temporal derivatives by

invoking Taylor’s hypothesis as described above. The

lateral and spanwise derivatives,@u01@x2

� �2and

@u01@x3

� �2can be

obtained by closely spaced parallel hot-wire probes separa-

ted in either the x2or x3 directions. The four remaining

derivatives@u02@x2

� �2;

@u02@x3

� �2;

@u03@x2

� �2and

@u03@x3

� �2in thedissi-

pation term can be subsequently estimated by invoking asemi-isotropy assumption, as described in detail by Wy-gnanski and Fiedler (1969), which assumes the nine spatialderivatives in the dissipation term observe the followingsemi-isotropy relationship:

ks@u01@x1

� �2¼ @u02

@x1

� �2¼ @u03

@x1

� �2

@u01@x2

� �2¼ ks

@u02@x2

� �2¼ @u03

@x2

� �2

@u01@x3

� �2¼ @u02

@x3

� �2¼ ks

@u03@x3

� �2

ð8Þ

where ks is the semi-isotropy coefficient. In the presentstudy, the coefficient ks will be determined from thestreamwise mean square derivative measurements.

3.6.4Direct measurement of all nine fluctuating derivativetermsOf course, the most sophisticated method for obtaining thedissipation is to measure all nine fluctuating derivativeterms by use of twin X-wires, as described by Browne et al(1987). Their bluff body wake study indicated that thelocal isotropy assumption is not valid for a cylinder wakein the self-preserving region with relatively low Reynoldsnumber. However, the requirement of twin X-wiresseverely limits the spatial resolution of the fluctuatingderivative measurements. In the present study we will notuse this approach since the spatial resolution of the twinX-wire probe configuration was deemed too large to obtaina reliable measurement of the required derivatives.

3.6.5Forced balance of the TKE equationThe easiest way to evaluate the dissipation term is from aforced balance of the turbulent kinetic energy equation.

However, this approach assumes that the pressuretransport term is negligible, which we will subsequentlydemonstrate is not the case. Therefore, this approach willnot be used in this study.

3.7Measurement of streamwise derivatives of meanquantitiesFor the finite-difference approximation of streamwisederivatives of mean quantities, the selection of the distanceDx between adjacent streamwise stations will greatly affectthe measurement uncertainty. In the measurementsreported here, Dx was optimized using an uncertaintyanalysis. With profiles of a given mean turbulence quantityobtained at three consecutive streamwise measurementstations, a natural approach for taking the spatial deriva-tive of a given function f(x) would be to use a central-difference approximation with xi=x, (xi–xi–1)=Dx and(xi+1–xi)=Dx. This gives,

df

dx¼ fiþ1 � fi�1

2Dxþ O Dxð Þ2

� �ð9Þ

It may be shown (Gerald and Wheatley 1994) that thenumerical differentiation based on evenly-spaced quadraticLagrangian polynomial interpolation is identical to thecentral difference scheme. If there is no positioning errorassociated with the probes, the uncertainty of the estimateof df

dx is solely determined by truncation, which is basically abias error due to the use of the central-difference scheme.This will obviously decrease as Dx decreases, which wouldsuggest that we want the spacing between the adjacentmeasurement stations to be as close as possible. However,in reality, there are unavoidable positioning errors associ-ated with both the streamwise and lateral locations of theprobe. With this positioning error, the behavior of theuncertainty of df

dx will be totally different. Assuming dx anddy positioning errors associated with the x and y locationsof the probe, respectively, the propagation of these errors tothe finite difference representation of df

dx was investigated.This reveals that uncertainty in df

dx due to probe positioninguncertainty actually increases as Dx decreases. Therefore,the total uncertainty in df

dx is comprised of two parts, that dueto positioning error, which decreases with Dx, and that dueto truncation error, which increases with Dx. This aspect isclearly shown in Fig. 3, which compares the variation ofposition, truncation and total uncertainties of dk/dx withDx. Note that the two competing trends give rise to anoptimal Dx separation for the measurements. In this study,the TKE budget measurements were obtained at thestreamwise location x=101.6 cm (x/h0=141) for the ZPG,APG, and FPG cases. Based on an uncertainty analysis likethat described above, the optimal streamwise separation ofthe measurement stations was selected as Dx=12.7 cm.Therefore multiple traverses at streamwise locationsxi–1=88.9 cm, xi= 101.6 cm and xi+1=114.3 cm wereobtained as described in Liu (2001).

4Results for the zero pressure gradient wakeIn this section we separately present each of the measuredterms in Eq. 3 for the ZPG turbulent wake case. These

474

results were obtained by the methods outlined in theprevious section.

4.1Convection termThe lateral distribution of the streamwise convection��U1

@k@x1

, the lateral convection ��U2@k@x2

, and their sum forthe symmetric wake in ZPG at x/h0=141 are presented inFig. 4. In this figure, both convection terms are non-di-mensionalized by using the local wake half-width d(x) asthe reference length scale, and the local maximum velocitydefect Ud(y) as the reference velocity scale. Here d isdefined as the lateral distance from the centerline of thewake to the position at which the local velocity defectdrops to half of Ud. From Fig. 4, it can be seen that for thesymmetric wake in ZPG, the streamwise convectiondominates the total convection distribution.

4.2Production termFor the ZPG wake, the dilatational production

� u021 � u022

� �@ �U1

@x1is zero. Figure 5 presents the measured

shear production term (simplified as �u01u02@ �U1

@x2since

@ �U2

@x1� 0) as obtained for the ZPG wake at x/h0=141. The

production has been appropriately scaled by local valuesof d(x) and Ud(x). Peak turbulence production is sym-metric across the wake, and occurs near y/d=±0.9, which isassociated with the lateral location of maximum meanstrain rate @ �U1

@x2. Very similar results were obtained from a

separate flow field survey of the symmetric wake usingLDV, as presented in Liu et al (2002).

4.3Turbulence diffusion termFigure 6 presents measured profiles of the streamwise

turbulence diffusion � @@x1

12 u031 þ u01u021 þ u01u023ð Þand the

Fig. 3. Uncertainty analysis of dk/dx for ZPG at x=101.6 cm,y=0.0 cm

Fig. 4. Convection in the ZPG symmetricwake at x/h0=141

Fig. 5. Turbulent production in the ZPG symmetric wake atx/h0=141

475

lateral turbulence diffusion � @@x2

12 u021 u02 þ u032 þ u02u023ð Þfor

the symmetric wake in ZPG at x/h0=141. The diffusionterms have been scaled appropriately by local values ofd and Ud. It is apparent from this figure that for the ZPGturbulent wake, the lateral turbulent diffusion is thedominant diffusion mechanism. By comparison, thestreamwise turbulence diffusion is negligible. Sincestreamwise turbulence diffusion is not significant and thelateral diffusion serves only to locally redistribute turbu-lence kinetic energy, we expect that cross-stream integra-tion should give,Z þ1

�1� @

@x2

1

2u021 u02 þ u032 þ u02u023ð Þ

� �dx2 ¼ 0

In order to gauge the accuracy of the measurement ofthe lateral diffusion term, the profile of the total turbulencediffusion shown in Fig. 6 was numerically integratedacross the wake, and the result was indeed found to bezero (within experimental uncertainty).

4.4Dissipation termAmong all of the terms in the turbulence kinetic energyequation, the measured dissipation term is most likely topossess significant bias error. There are two primary errorsources associated with the dissipation estimate. First, asdescribed in Sect. 3, since we neglect the cross-derivativecorrelation terms and resort to the homogeneousapproximation for dissipation, this will give rise to a biaserror due to mathematical modeling. Second, the limitedspatial resolution of the hot wire probes required for themean-square derivative measurements will give rise to abias error due to spatial resolution.

For the mean-square derivatives there are tworequirements for a reliable measurement. First, the spatialresolution of the probe should resolve scales on the orderof the Kolmogorov length scale; second, the temporalresolution of the velocity fluctuation time-series recordshould capture the highest frequencies associated with theconvection of dissipative scales past the sensor(s). There islittle difficulty in fulfilling the temporal resolution

requirement based on the available probe size. Therefore,the Nyquist frequency of the data record provides a suf-ficient match to the temporal resolution requirement.

Regarding the spatial resolution requirement, unfortu-nately, as described in Sect. 2.5, the dimensions of the hot-wire sensors and their spacing in multi-sensor configura-tions are all greater than the Kolmogorov length scale,which is approximately 0.1 mm near the centerline of thewake. However, probe spatial resolution is critical for areliable mean-square derivative estimate, as described indetail by Wallace and Foss (1995). Through an investiga-tion of the effect of the finite-difference spacing on themean-square derivative estimate obtained from DNS data,Wallace and Foss demonstrated that the estimate of themean-square derivative is attenuated dramatically as finitedifference spacing is increased, which is equivalent to theissue of probe spatial resolution in the measurement.

The effect of spatial resolution on fluctuating derivativeestimates can be clearly seen by comparing the samefluctuating derivative as measured by the dual sensorparallel probe with that from the X-wire. Consider, for

example, the mean square derivative term@u01@x1

� �2, which

can be obtained from measured time-series data byinvoking Taylor’s frozen field hypothesis. All results

obtained for@u01@x1

� �2using both X-wire and parallel probes

under ZPG conditions are shown in Fig. 7. From this fig-ure, it can be seen that although the cross-stream profileshapes are the same, the parallel probe gives higher peak

values for the quantity@u01@x1

� �2than the X-wire probe does.

This disparity can be attributed to the difference betweenthe effective measurement volume of the parallel andX-wire probes. In particular, Fig. 7 clearly illustrates thatthe larger effective measurement volume of the X-wireresults in a lower mean-square derivative measurement

due to effective spatial low-pass filtering. Therefore@u01@x1

� �2,

as measured by the parallel probe will be closer to the truevalue than the corresponding X-wire measurement,although it too will be biased by some degree due toinsufficient spatial resolution.

In this study, the bias error associated with the dissi-pation measurement was minimized via a two-step pro-cedure. First, for those fluctuating derivatives that can bemeasured by both the parallel probe and X-wire (or X-wirepair), the degree by which the magnitude of the fluctuatingderivative is reduced (relative to the parallel probe) due toprobe spatial resolution limitations was quantified. In eachof these comparisons, the cross-stream profiles of fluctu-ating derivatives had identical shapes but the magnitudeswere reduced below those measured by the parallel probeconfiguration. This allowed the determination of correc-tion factors to be applied to those fluctuating derivativesthat could only be measured by the X-wire (or X-wireprobe pair). This partially compensated the magnitude ofthe fluctuating derivative to an equivalent effective reso-lution of the parallel probe. The only assumption requiredis that the scaling factor would be the same for thosederivatives in which we have no corresponding parallel

Fig. 6. Turbulence diffusion in the ZPG symmetric wake atx/h0=141

476

probe measurement. Once the derivatives were partiallycompensated in this manner, preliminary estimates of thedissipation term were made by making the local isotropy,locally axisymmetric turbulence, and quasi-isotropic tur-bulence approximations, as outlined in Sect. 3.6. Thesepreliminary dissipation estimates are compared in Fig. 8,where it can be seen that significant disparities occur be-tween estimates. Note that the dissipation term based onthe local isotropy assumption is much smaller in magni-tude than for the other two methods.

Each of the preliminary dissipation estimates wasincorporated into the wake TKE budget Eq. 3 (along withthe other measured terms) and the pressure diffusion termwas extracted from a forced balance. Since the pressurediffusion term serves primarily to locally redistributeturbulent kinetic energy, one expects that a cross-streamintegration of this term should be very close to zero (aswas previously demonstrated for the measured turbulencediffusion). In fact, the accuracy of each preliminary dis-sipation estimate was assessed by checking the lateralintegration character of the resulting pressure diffusion

term in each case. It was found that the dissipation esti-mate based on the locally axisymmetric turbulenceassumption leads to a result in which cross-stream inte-gration of the pressure diffusion is closest to zero. The ideaof using the zero cross-stream integration character ofturbulence diffusion to assess the accuracy of dissipationis not unique to our study. In his measurement of the TKEbudget of a turbulent planar jet, Bradbury (1965) assumedlocal isotropy in his measurement of dissipation, and thiswas subsequently corrected by requiring that the sum ofthe pressure and turbulence diffusion terms (extractedfrom a forced balance) should exhibit zero integrationacross the jet.

In the current study, the pressure diffusion termobtained from the forced balance of the TKE equationincludes not only the pressure diffusion itself, but also anerror term. The error term can be further decomposed intobias and random error components. The random errorcomponent would not be expected to exhibit a systematicvariation across the wake, and consequently its effect islikely to be canceled upon cross-wake integration.

Fig. 8. Comparison of dissipation estimatesfor the ZPG symmetric wake at x/h0=141

Fig. 7. Comparison of@u01@x1

� �2measured by

X-wire and parallel probes in the ZPG symmetricwake at x/h0=141

477

However, the bias error will clearly remain. If we make theplausible assumption that the dissipation, including boththe measured mean-square derivative terms and theunmeasured cross-derivative correlation terms, is thedominant source of this bias error, then reducing the biaserror associated with the dissipation should bring thecross-wake integration of the pressure diffusion term tozero.

The attribute of zero lateral integration of the pressurediffusion term can be utilized as a constraint to correct thebias error associated with the axisymmetric turbulencedissipation estimate. More specifically, with the pressurediffusion term obtained from the forced balance of theTKE equation, we can use a shooting method to iterativelyadjust a constant scaling factor to be applied to the dis-sipation term until we get a zero lateral integration of thepressure diffusion. The constant scaling factor serves tocompensate the bias error due to insufficient spatial res-olution of measurement probes, as well as the bias errordue to the mathematical modeling of the dissipation term.

4.5ZPG turbulent kinetic energy budgetFigure 9 presents cross-stream profiles of the terms inEq. 3 for the ZPG planar wake at x/h0=141. The viscousdiffusion term is assumed to be negligible. All terms exceptpressure diffusion have been obtained from direct mea-surement. Error bars associated with the measured termsare also shown in this figure. Note that these error barsreflect only the uncertainty associated with measurementand data analysis. The uncertainty in the dissipationassociated with mathematical modeling is not included.The pressure diffusion profile shown in Fig. 9 is obtainedby forcing a balance of the TKE equation, and therefore it

actually consists of both the true pressure diffusion andthe (minimized) total error of the measurement. All termsin Fig. 9 have been scaled in a consistent manner withlocal values of dand Ud. Positive values indicate a localgain in TKE while negative values indicate a loss. Note forexample, that turbulence dissipation is always negativeand production positive. This figure presents the turbu-lence dissipation corrected such that cross-wake integra-tion of the pressure diffusion term is zero.

The double peaks of the production term correspondapproximately to the locations of the maximum meanstrain rate in the upper and lower shear layers of the wake.At the center or near the edges of the wake, the mean shearis zero or asymptotically approaches zero, and there is noproduction.

Note that both turbulence diffusion and pressure dif-fusion terms have similar profile shapes. The diffusionterms respond to the lateral gradient in turbulent kineticenergy associated with newly-generated turbulenceresulting from the production term. Both terms clearlyserve to transport turbulence laterally away from regionsof high mean strain, where it is produced, and towardthose locations with low production (like the wake cen-terline and outer edges). Note also that while turbulencediffusion is greater than pressure diffusion, the latter termis certainly not negligible as has often been assumed. Inaddition, there is no evidence in Fig. 9 that there is a so-called counter-gradient transport mechanism for thepressure diffusion term, as suggested by Demuren et al(1996).

As for the dissipation term, it can be seen from Fig. 9that the greatest dissipation occurs across the centralregion of the wake, where the turbulence level is mostintense.

Fig. 9. Turbulent kineticenergy budget for the planarwake in ZPG at x/h0=141

478

4.6Comparison with DNS resultsMoser et al (1998) investigated the TKE budget of a tem-porally evolving planar turbulent wake using DNS. Theyapplied forcing to the initial wake, and then investigatedthe influence of the forcing on the far wake development.Their unforced wake corresponds to the ZPG conditions ofour wake study, with three basic differences: (1) theyobtained the TKE budget in the far wake similarity regionwhile ours is obtained in the near-wake region; (2) theirwake develops in the temporal domain while ours obvi-ously develops spatially; and, (3) their mass-flux Reynoldsnumber, which is equivalent to the momentum-thicknessReynolds number in spatially developing wakes, is only2000, an order of magnitude smaller than ours(Reh=15000).

For the temporally-developing wake flow in DNS, theonly non-zero mean velocity component is �U1, and due tohomogeneity in the streamwise and spanwise directions,derivatives of averaged quantities with respect to x1 and x3

are zero. For the experiment, the spatially-developingwake flow is homogeneous in time t and spanwise direc-tion x3. Therefore, in the Reynolds stress transport equa-tion labeled as A1 in Moser et al (1998), there is nostreamwise or lateral convection term for the Reynoldsstress transport. However, the temporal derivative term inDNS can be transformed into the streamwise convectiveterm in the spatial domain, and vice versa, through thefollowing relationship:

@

@t¼ Ue

@

@xð10Þ

where Ue is the external free stream velocity outside of thewake in the spatial domain. Figure 12a in Moser et al(1998) provides the budget of the quantity of q2(= 2k) forthe temporally-developing wake simulated by DNS. Byusing the transformation specified by (10), we can makedirect comparisons of our experimental data with the DNSresults. The time derivative term in DNS matches thestreamwise convection term in the experiment, and pro-duction, turbulence diffusion, pressure diffusion, anddissipation in DNS match the corresponding terms in theexperiment. The only term left unmatched is the lateralconvection term in the experiment. More specifically, inorder to make a fair comparison with the DNS results, theexperimentally-measured terms in Eq. 3 need to be scaled

byU3

d

4dU1

Ue

� �. In this manner, the streamwise convection

term in the experiment will have the same scale as the time

derivative of q2shown in Fig. 12a of Moser et al (1998).Figures 10, 11, 12, 13, and 14 present comparisons

between the experimental and DNS profiles of the dissi-pation, production, streamwise convection, turbulencediffusion, and the pressure diffusion, respectively. In thesefigures, open circles represent the experimental results andthe solid line represents the DNS simulation. Consideringthe different Reynolds numbers and stages of wakedevelopment for the experiments and simulations, theagreement between the experimental and DNS results isquite encouraging. In particular, the agreement betweenthe measured and DNS-based turbulent diffusion term is

quite remarkable. Even the comparison of the pressurediffusion terms shows good general agreement. Note thatthe scatter of the DNS data for the pressure diffusion term

Fig. 10. Comparison of experimental and DNS (Moser et al 1998)dissipation profiles for the symmetric wake in ZPG

Fig. 11. Comparison of experimental and DNS (Moser et al 1998)production profiles for the symmetric wake in ZPG

Fig. 12. Comparison of experimental and DNS (Moser et al 1998)convection profiles for the symmetric wake in ZPG

479

is likely due to an insufficient period for the time-aver-aging. Note also that the experimental pressure diffusionterm contains not only the pressure diffusion itself, butalso the total measurement error of the TKE budget.Therefore, the comparison of the pressure diffusion termscan also be viewed as a measure indicating the overallaccuracy and reliability of the TKE budget measurement.Observed disparities between the convection, productionand dissipation terms can be attributed to different Rey-nolds numbers and different stages of developmentbetween the experimental and the DNS data. Moreover, thedisparity between the convection terms of the experi-mental and DNS data may also be attributed largely to theabsence of lateral convection for the DNS simulation,which evolves temporally as a strictly parallel flow.

5Effect of the pressure gradient on the planar wake TKEbudgetTo investigate the influence of the pressure gradient on thewake TKE budget, terms for the ZPG, APG, and FPG caseswere normalized by using the local wake half-width d, and

the square root of the local maximum turbulent kineticenergy

ffiffiffiffiffiffiffiffiffikmax

p, as the reference length and velocity scales,

respectively. The comparisons between the normalizedTKE budget terms for different pressure gradient cases arepresented in Figs. 15, 16, 17, and 18.

As reported in Liu et al (2002), when the adversepressure gradient is imposed, the wake widening rate isenhanced, the velocity defect decay rate is reduced, and theturbulence intensity and the Reynolds stress are bothamplified. In contrast, when the wake develops in afavorable pressure gradient, the wake widening rate isreduced, the velocity defect decay rate is increased, and theturbulence intensity and Reynolds stress are bothdecreased in relation to corresponding zero pressure gra-dient values. The wakes studied in this paper are all shear-dominated, despite the imposed streamwise straining.However, as noted in Liu et al (2002), the dilatationalproduction term is found to play an important role inaugmenting and suppressing the turbulence for the APGand FPG cases, respectively. Acting as a trigger, this term

Fig. 13. Comparison of experimental and DNS (Moser et al 1998)turbulence diffusion profiles for the symmetric wake in ZPG

Fig. 14. Comparison of experimental and DNS (Moser et al 1998)pressure diffusion profiles for the symmetric wake in ZPG

Fig. 15. Comparison of the convection profiles for the symmetricwake in ZPG, APG and FPG

Fig. 16. Comparison of the turbulence diffusion profiles for thesymmetric wake in ZPG, APG and FPG

480

gives rise to an initial disparity in turbulence levels afterimposition of the pressure gradients, and subsequentlyalters the shear production term through modification of�u0v0. Measurements of the Reynolds stress correlationsuggest no significant modification in the phase relation-ship between u¢ and v¢ due to the imposed pressure gra-dients. Cross-stream profiles of �u0v0

k obtained at

various streamwise locations exhibit collapse for eachpressure gradient case investigated.

Consistent with this scenario, Fig. 17 clearly showslocal turbulence production for the APG case exceeds thatfor ZPG. In contrast, production for the FPG case is sup-pressed below that for ZPG. These differences are directlyassociated with the dilatational production term. Theeffect of the imposed pressure gradient is also significantfor the convection term, as shown in Fig. 15, since thisterm is directly related to the mean motion of the flowfield. As expected, streamwise convection is greatest forthe accelerated FPG case and less so for the APG. Given theeffect that the pressure gradient has on the turbulenceproduction term (as shown in Fig. 17), it is not surprising

that the turbulence diffusion exhibits comparable dispar-ities among the imposed pressure gradient cases, as shownin Fig. 16. In contrast, Fig. 18 indicates that the influenceof pressure gradient on dissipation is minimal comparedto the other terms. These comparisons suggest that thefundamental TKE transport mechanism is not altered bythe imposed pressure gradients. Rather, Figs. 15, 16, 17,and 18 suggest that the imposed pressure gradient exertsits influence on the turbulence field primarily through themean flow and largest scale energy-containing motionsrather than the fine-scale turbulence.

6ConclusionsA series of turbulent kinetic energy (TKE) budget mea-surements were conducted for a symmetric, turbulentplanar wake flow subjected to constant zero, favorable, andadverse pressure gradients. Special consideration wasgiven to the dissipation estimate. On the basis of experi-mental evidence supporting similar profile shapes for themeasured mean-square derivatives, and requiring zerocross-stream integration of the pressure diffusion term(obtained from the forced balance of the TKE equation), adissipation bias error correction method was proposedand implemented in the experiments. More specifically, ascaling factor was determined using a shooting method,and applied to the dissipation estimate to compensate thebias errors due to the limited spatial probe resolution andmodeling of the dissipation term. This approach is vali-dated through the comparison of the experimental TKEbudget with the DNS results obtained by Moser et al(1998). Although the stage of wake development andReynolds numbers are different for the experiments andDNS simulations, good general agreement is observed.

Comparison of the different terms in the TKE budgetsof the wake subjected to the imposed adverse, zero, andfavorable pressure gradients indicates that the funda-mental TKE transport mechanism is not altered by theimposed pressure gradients. The imposed pressure gradi-ent exerts its influence on the turbulence field primarilythrough the mean flow and largest scale energy-containingmotions rather than the fine-scale dissipative turbulence.

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