a ally simple turbulence closure model for attached and separated turbulence boundary layers

8/3/2019 A ally Simple Turbulence Closure Model for Attached and Separated Turbulence Boundary Layers

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1684 AIAA J O U R N A L V O L . 23, NO

A Mathematically Simple Turbulence Closure Modelfor Attached and Separated Turbulent Boundary LayersD. A. Johnson* and L. S. King*

NASA Ames Research Center, Moffett Field, CaliforniaA new turbulence closure model designed specifically to treat two-dimensional, turbulent boundary layers

with strong adverse pressure gradients and attendant separation is presented. The influence of history effects aremodeled by using an ordinary differential equation derived from the turbulent kinetic energy equation to describethe streamwise development of the maximum R eynolds shear stress in conjunction with an assumed eddy viscos-ity distribution that has as its velocity scale the maximum Reynolds shear stress. In the outer part of the bound-ary layer, the eddy viscosity is treated as a free parameter which is adjusted in order to satisfy the O DE for themaximum shear stress. Because of this, the model is not simply an eddy viscosity model, but contains features ofa Reynolds stress model. Comparisons w ith experiment are presented that clearly show the proposed m odel to besuperior to the Cebeci-Smith one in treating strongly retarded and separated flows. In contrast to two-e quation,eddy viscosity models, it requires only slightly more compu tational effort than simple models such as the Cebeci-Smith.

Nomenclature = inner part of boundary layeralA+cC di fC feC pD

gIfkLpq2uU Tvvxya76d*d *eKvv tpT t

modeling constant, u'v'm/kmva n Driest damping constantchord lengthturbulent diffusion modeling constantlocal skin-friction coefficient based on edgeconditionspressure coefficientnear- wal l damping term, D = (\ Qxp~y/A ),turbulent diffusion ratevariable changeboundary-layer shape factorturbulence kinetic energydissipation length scalepressureturbulent flow proper ty , w / 2 + t > / 2 + w / 2streamwise velocity componentwal l shear velocitynormal velocity componentturbulen t diffusion convection velocitycoordinate in streamwise directioncoordinate in normal directionangle of attackKlebanoff ' s intermit tency funct ionboundary-layer thicknessboundary-layer displacement thicknessincompressible boundary-layer displacementthicknessdissipation ratevon K ar m an ' s constantkinem atic viscosityturbulent eddy viscositydensityturbulent Reynolds shear stress

Subscriptse boundary-layer edge conditions

m = values of quant i ty where u'v' is a maximumo = outer par t of boundary layero o = freestream conditionsSuperscripts( _ )' = fluctuating quantity( ) = time-averaged qua ntity

IntroductionA CCU R A T E theoretical predictions of airfoil permance up to and beyond stall require that th e turbuboundary layer and wake be correctly described. For fidifference, viscous flow calculation methods, the accuracywhich this can be done depends on the validity of the clomodel used to describe th e turbulent Reynolds stresThus, th e turbulence closure model is fundamenta l to tmethods. Bradshaw 1 pu t this in another way: "A numerprocedure without a turbulence closure model stands insame relation to a complete calculation method as andoes to a bull."In this paper, a new turbulence closure model formulais presented. Based on comparisons with experiment, new model appears capable of treating two-dimensiosubsonic, pressure-driven separated flows and transoshock-induced separated flows. New modeling assumptar e applied that evolved from 1) recent measurements ofbulent flow properties in separated boundary layers (mpossible with the development of the laser velocimtechnique) and 2) insights gained from the comparisonthese experimental results with theoretical predictions baon existing turbulence closure models.A new approach is taken to model th e strong "histoeffects characteristic of turbulent boundary layers subjeto rapid changes in the streamwise pressure gradient . Andinary differential equation (ODE), derived from th ebulence kinetic energy equation, is used to describe streamwise development of the maximum Reynolds s


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N O V E M B E R 1985 TURBULENCE CLOSURE MODEL FOR B O U N D A R Y LAYERS 1viscosity model , but also contains features of a Reynoldsstress model. Since only an ODE needs to be solved with thismode l rather than one or more partial differential equat ions ,the proposed closure model requires only slightly more com-putat ional effort than algebraic models such as the Cebeci-Smith. 2The performance of th is closure model will be i l lustratedth ro ugh comparisons with experiment fo r three difficult flowcases: 1) a transonic shock-induced separated f low,3 2 ) alow-speed diffuser flow with separat ion, 4 and 3) a super-critical airfoil at transonic conditions.5 The first tw o flowcases contain large regions of separated flow; the third con-tains only a small separated zone near the trailing edge of theairfoil . Included in these comparisons are results obtainedwith th e popular Cebeci-Smith model.

DiscussionClosure Model Formulation

In recent years , considerable effort has been directedto ward incorporat ing two-equation (e.g., k e and k co2 ),eddy viscosity formulations into numerica l predic t ionmethods with th e expectation of improved predictive ac-curacy for strong inviscid-viscous interaction cases. Weighedagainst the added numerical complexi ty and computat ionalt ime required to solve tw o additional partial differentialequat ions, the improvements in predictive accuracy overalgebraic models, such as the Cebeci-Smith, have no t beendramatic. In general , the two-equation formula t ions do bet-te r than th e algebraic formulat ions for separated flow cases,in that the rapid skin-friction recovery observed experimen-ta l ly downstream of reattachment is better predicted and thecalculated mean velocity profi les within th e separated zoneare in bet ter qual i ta t ive agreement with experiment. Yet fortransonic cases with massive separat ion, the prediction ofsurface pressure has been less than satisfactory (e.g., Ref . 6).For cases in which th e flow remains attached, results haveeven been obtained that could be const rued as inferior tothose obtained with th e Cebeci-Smith mo de l (e.g., Ref . 7).The development of the present closure model was basedon a completely different philosophy from that used indeveloping the two-equat ion formula t ions . The goal was todevelop an adequate closure model for a limited class offlows rather than a universal model. Th e following premisemot ivated the development . I f the Cebeci -Smith model canperform as well as it does with the known invalid assumptionthat the turbulent shear stress depends only on local proper-ties of the mean f low, then it should be possible to develop abetter closure model fo r boundary- layer flows without a t -tempt ing to predict , throughout the entire viscous layer, th eproduc t ion , dissipation, and diffusion of both the turbulencekinetic energy and the dissipation rate, as is done in two-equation formulations. For rapidly varying flows, it ispostulated that a proper accounting of convection effectsw as essential, with diffusion effects playing a less dominantrole.To account for convection and diffusion effects on theReynolds shear stress development , an ODE is proposed toprescr ibe the maximum Reynolds shear stress development inthe streamwise direction. More precisely, the ODE describesth e development of -u'v'm along the pa th of m a x i m u mshear stress and is derived from the turbulence kinetic energyequa t ion . For convenience, in th is paper -u'v' will bereferred to as the Reynolds shear stress r,, although it is reallygiven by rt/p. Along with this ODE for the maximum

wher e vti an d vto can be thought of as describing the eviscosity in the inner and outer parts of the bo un da ry laNotice in Eq. (1) that vt vti when vtivto.Thus, Eq. (1) provides a smooth blendingtween the regions where vt**vti and where vt~vto. Itmakes v t functionally dependent on vto across most of bo un da ry layer. This is a critical element of the propoclosure model since vtot as will be described shortly, is uto control the rate of growth of u'v'm. The inner eviscosity v ti is assumed to be given by the followrelationship:

where D is the same damping t e rm used in th e Cebeci-Smformulat ion. Typically, in zero-equation, eddy viscomodels, the eddy viscosity in the inner part of the boundlayer is represented by D2K 2y2 \du/dy\. For zero-pressgradient boundary layers , Eq. (2) reduces to that fohowever , for nonzero-pressure-gradient flows, it is beliethat Eq. (2) is more broadly valid. This will be illustrawhen the results obtained with th e proposed closure moar e compared with those obtained with the Cebeci-Smmodel. An adjustment in the damping term D is requireEq. (2 ) to accoun t for a y dependency instead of adependency in the near-wall region. This is done by simusing a value of 15 fo r A + rather than the usual value ofused in the Cebeci-Smith model .Th e outer eddy viscosity vto is assumed to have the dibut ion

vto= const -7where 7 is Klebanoff ' s intermittency function, y = l+ 5.5(y/5)6]. The constant in Eq. (3) is determined byvalue of u 'v 'm prescr ibed by the ODE mentioned earThat is, the constant is adjusted as needed such thatfollowing relationship is satisfied:

-UV(du/8y)\m (du/dy)\ f

lengthscale velocityscale

wher e v t is given by Eq. (1). The subscript m denotes thatquantity is evaluated where -u'v' is a m a x i m u m .strain rate in Eq. (4), (du/dy)\m, like d an d < 5 * , is a deminable property of the mean velocity profile. Forvience, th e boundary- layer approximat ion for the strainha s been used. To indicate that th e eddy viscosity at the ltion of - u'v'm can be represented as the product of a lescale and a velocity scale, the right-hand side of Eq. (4shown in two forms. The length scale is given(-u'v!n)Y2/(du/dy)\m and the velocity scale by (-u'v'm)For the inner eddy viscosity representat ion [see Eq. (2)] ,velocity scale is also (-w't^)1/ 2, but the length scale isThe use of (-u'v'm)V2 as the velocity scale is supportedth e work of Perry and Schofield.8 For boundary-layer fwith adverse pressure gradients they found that this quanprovided a much bet ter velocity defect correlation thanth e wall shear velocity ur.The adjus tment of vto in order to satisfy the ODE u'v'm and the use of (-u'v'm)yi as the velocity scale for


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1686 D. A. JOHNSON AND L. S. KING AIAA JOURNA

To complete the formulation, an equation is needed for-u'v'm. This equation is developed from the turbulencekinetic energy equat ion, using similar assumptions to thoseused by Bradshaw et al.9 in the development of their approx-imate Reynolds stress model. Along th e path 5 of maximumkinetic energy /c , the turbulence kinetic energy equation fo ran incompressible boundary layer reduces to

d* -= -u'v' dyproduction diffusion dissipation

(5)The approximations have been made that x and s are nearlycoincident and that y is nearly normal to 5. The m denotesthat the quant i ty is evaluated where k or u'v' is a . max-i m u m . This equation is also valid for compressible (subsonican d transonic) boundary layers, if mass-averaged variablesar e employed.10To simplify this equation further , a dissipation lengthscale Lm is defined as

and the assumption (-ufv'm/km) =al=const is in t roduced.Upon rearrangement, Eq. (5) takes the formdud y

-> &(-u'v'm (6)contributiondu e todiffusion

contributiondu e toconvection

For convenience, the turbulent diffusion term is representedinEq. (6) by > m .This ordinary differential equat ion has been employed inthe development of integral boundary-layer methods byMcDonald 11 and Green et al. 12 In the former method, i t isused to obtain an expression for the shear stress integral (theintegrated value of the Reynolds shear stress across theboundary layer), whereas in the latter method it is used toobtain an expression for the entrainment coefficient.The dissipation length scale Lm is assumed to scale with yin the inner part of the b o u n d a r y layer and with th eboundary-layer thickness 6 in the oute r part of the boundarylayer. The functional form assumed fo r Lm is.225

Lm=0.096 ym/d> 0.225 (7)In Eq . (6), Lm(du/dy)\m can be interpreted as the squareroot of the turbulent shear stress that would result if convec-tion an d diffusion effects were negligibly small. This term isthus replaced by the quant i ty (-u'v'm e q ) ' / 2 which is as-sumed to be determined from th e following equilibrium eddyviscosity distr ibution:

tt eq ,)] (8a)(8b)

quantity is approximated by the following expression:

aid[0.1-(y/d)m]( " t o V*Veq /

where Cdif is a modeling constant. The formulat ion of thexpression is given in the Appendix .To solve Eq . (6), it is advantageous to make the followichange of variables:g=(-u'v'm) and (1

Introducing this change in var iables and the modeled eqution fo r 33W results in Eq. (6) taking the form used in the iplementation of this closure model,

convectivechange respectively.The closure model that has been developed can be sumar ized as being composed of three parts: 1) a noequil ibr ium eddy viscosity distr ibution [Eqs. (1-4)] , 2rate equation [Eq. (11)] for the s treamwise development u'v'm, and 3) an equilibrium (production = dissipatioeddy viscosity distribution [Eqs. (8)]. The equil ibr ium edviscosity model of Eq. (8) plays several impor tant rolFirst, it is used to determine geq an d v t0t eq in Eq. (11). It ais the resultant eddy viscosity distribution when dg/dx~Finally, it is used as the initial eddy viscosity distributionstarting th e calculations.With th e proposed closure model , initial conditionsEq . (11) are needed in addition to a mean streamwise velocprofile and a value for ~u'v'm to start a calculation. Tmost straightforward approach is to assume that th e flowin equilibrium at the start of the calculations,

(t =vttcq and =g e q i . e . ,In this case, a value of u'v'mt eq is needed. This quantican be an experimental value or a value determined fromequilibrium eddy viscosity model such as the Cebeci-SmiAlternatively, the wall shear velocity ur, as determined eithfrom experiment or from calculations, can be used. Tstarting mean velcoity profi le ca n either be a calculated or exper imental prof i le . Obviously , th e calculations should nbe started wh ere th e shear stress is changing rapidly in tstreamwise direction, such as near the point where turbuletransit ion is initiated.In the next section, th e p e r fo rman ce of this closure mois illustrated through compar ison with experiment. All of calculations to be presented were obtained for a value


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NOVEMBER 1985 TURBULENCE CLOSUREMODELFORBOUNDARYLAYERS 16

Closure Model PerformanceThe closure model just described has been incorporatedinto th e finite difference (direct/inverse) boundary-layer pro-gram of Carter . 1 3 The boundary-layer equat ions are nottruly predictive, since either the pressure gradient (directmode) or a mass flux parameter peued* (inverse mode) mustbe prescribed as par t of the boundary condit ions. Never-

theless, the boundary-layer equations provide a means ofperforming a consistency check on a turbulence closuremodel, the premise being that if all remaining properties ofthe flow are predicted by specifying peue8* (e.g., surfacepressure, skin friction, an d mean velocity profiles), then on ecan assume that the turbulence closure model is valid. Con-versely, if the remaining flow proper t ies are not predicted ac-cura te ly , then the closure model must be invalid. Used in thisfashion the inverse boundary- layer method is a powerful toolfo r evaluating turbulence closure models.Some may question the appropr iateness of applying th eboundary-layer equations to flows with massive separation.However , from experience they have been found to performmuch better than expected. This w as demonstrated, for ex-ample , in Ref. 14 th rough direct comparison with solutionsto the Reynolds-averaged, Navier-Stokes equations for ashock-induced separated f low. The compar isons were m a d eby employing the results from the Navier-Stokes solutions asboundary conditions for the boundary-layer calculations.Similar pressure recoveries an d mean velocity profiles wereobtained for a flow with quite extensive separation (the flowcase was the shock-induced, separated flow considered inthis section).The boundary- layer method of Carter solves theboundary-layer equations by relaxation. At each iteration,Eq . (11) w as solved by the implicit Euler method and vto inEq. (4) was determined by Newton ' s method . The number ofiterations to convergence was nearly identical to that re -quired by the Cebeci-Smith model. Interestingly, with th eCebeci-Smith model this convergence rate required the use ofNewto n linearization, whereas with th e proposed closuremodel this was unnecessary.Th e first test case to be presented is the axisymmetric,transonic, shock-wave/turbulent boundary-layer experimentof Bachalo and Johnson.3 Of the three test cases consideredin this paper, this one represented the severest challenge tothe proposed closure model. A sketch of the flow model ispresented in Fig. 1 along with the boundary-layer displace-ment thickness distribution that results from the strong in-viscid/viscous interaction that develops on this model when7^=0.875. In Fig. Ib, x is the distance from th e leadingedge of the circular arc section and c is the chord length ofth e circular arc section. For the approach Mach number of0.875, a shock wave results at x/c-0.65 that is sufficientlystrong to produce an extensive region of separated f low.Separation occurs just downstream of the shock wave(x/c-0.10) and the flow does not reat tach unti l downstreamof the circular arc section (x/c\.15).The measured surface pressure distribution fo r this in -teraction and those predicted by the proposed non-equilibrium turbulence closure model (NTM) and the Cebeci-Smith model (CEBSM) are compared in Fig. 2. The NTMcalculations were started using the CEBSM, with the NTMturned on well ahead of the leading edge of the circular arcsection. The calculations downstream of this point were per-formed in the inverse mode. As is evident from Fig. 2, theNTM predicts a surface pressure distribution tha t is in muchbetter agreement with experiment than that predicted by the

,SHOCK

00 / /SEPARATED //FLOW0 1(a ) \

.06

.05

.04.03

.02.01

0

00o o

SHOCK LOCATION/ 0 0

1 ~ (b) oo 1 1 1 1 1 1 1 1 1 1

.5 .6 .7 .8 .9 1.0 1.1, 1.2 1.3 1.4 1.5x/c

Fig. 1 Shock-induced separated f low, M^ =0.875: a) flow mob) boundary-layer displacement thickness distribution.

.9 1- 1x / c 1.3 1.5

Fig. 2 Shock-indu ced separated flow, M00= 0.875: a) surfpressure distribution; b) skin-friction distribution; c) shape-fadistribution.

incorporated into a t ruly predictive method (i.e., a coupinviscid/viscous calculation method or a Navier-Stokes stion method). Included in Fig. 2 are the predicted distrtions for skin friction C fe an d shape factor H . Skin fric


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1688 D. A. JOHNSON AND L. S. K I N G A I A A J O U R N

0.938

0 .2 .4 .6 0 .2 .4 .6 0 .2 .4 .6 0 .2 .4 0 .2 .4 .6 .8 1.0 1.2

Fig. 3 Shock-induced separated f low,= 0.875: mean velocity profile comparis

16r

12

02.4

2.0

NT M CEBSM

(a )

*.1.29 = 9eq ^-EQUILIBRIUM

CONDITIONS

(b ) I____i_____i_____[_.6 .7 .8 .9 1.0

x/c1.1 1.2 1.3

Fig. 4 Shock-indu ced separated flow, M00= 0.875, "history ef -fects": a) maximum Reynolds shear stress development; b) non-equilibrium state of flow.

FLOW

BOUNDARY L A YER ED G E

10 -

O00

O

-1.2

-.8 CEBSM

DIRECT .INVERSE(a )

5r

x 29>

O

(b )

8r

O O


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NOVEMBER 1985 TURBULENCE CLOSURE MODEL FOR BOUNDARY LAYERS 1

in much better quali tat ive agreement with the experimentthan are those predicted by the CEBSM. The smallerboundary-layer edge velocities predicted by the CEBSM are aresult of the overprediction of pressure recovery. All of theprofiles predicted by the CEBSM exhibit a positive curvaturein the inner part of the boundary layer that is not observedin the exper iment. This departure between experiment andthe CEBSM results will also be seen to be present for theother flow cases considered. At x/c-l.Q, the separatedmean velocity profile predicted by the CEBSM takes on thecharacteristic*form so often seen in the literature when thisclosure model is applied to separated flows. The curvature ofthe profile at this station is of opposite sign to that observedin the experiment and that predicted with the NTM. The

CN 8

01.21.0

-_! -8

NTM CEBSM

=* EQUILIBRIUM CONDITIONS

(b )

Fig. 7 Low-speed diffuser f low, "history" effects: a) maximumReynolds shear stress development; b) nonequilibrium state of flow.

misprediction in profile shape is a result of the eddy viscorelationship used in the inner part of the boundary la(i.e., vti=D2K2y2 \du/dy\). It will be seen in the other fcases to also result in mispredictions of profile shape, ewell ahead of where separation occurs.Whereas th e inner eddy viscosity distribution used in CEBSM results in poor mean velocity profile predictionsis not the primary cause for the overprediction of pressrecovery. This is instead a result of the CEBSM predicto o rapid a rise in turbulent shear stress in the vicinity of shock wave. On the other hand, through Eq. (11), the Nlimits the rate at which the shear stress can grow at shock. As a result, the eddy viscosity is reduced atherefore, th e inner part of the boundary layer loses mmomentum at the shock, which in turn reduces its abilitnegotiate pressure gradients farther downstream. In Figthe development of -u'v'm with streamwise distance is cpared for the two closure models. Included in this figurthe quanti ty geq/g predicted by the NTM. This quantwhich is also equal to (v t/v t> eq l w ) 1 / 2 , is a measure of thefluence Eq. (11) has on the shear stress development. Itcan easily be shown to equal the ratio of dissipation to duction. If equilibrium were assumed, i t would have value of unity . Notice that even though the shock has greatest influence on this quanti ty , even downstream of shock, nonequi l ibrium effects are predicted to persist. Inrecovery zone (i.e., th e region where 6* is decreasing), gbecome greater than unity because of the slow decay-u'v'm predicted by Eq. (11) in this region. It is this sdecay in -u'v'm that caused the skin friction to be higthan that predicted by the CEBSM in the region downstrof reattachment.The next flow case to be considered is the low-speedfuser flow studied by Simpson et al.4 A sketch of the fmodel is shown in Fig. 5. Th e u p p e r surface of the diffusecontoured to produce strong adverse pressure gradients. lower-surface boundary layer, which undergoes masseparation because of the imposed pressure gradients ,studied in the exper iment. Exper imental profiles weretained from upstream of the adverse pressure gradient regto a final downstream station where the separat ion becso massive that three-dimensional influences on the fb ecam e excessive. The boundary- layer disp lacemthickness dist r ibut ion fo r this experiment is included in5.

16

I8

x = 3.01 m 16

O EXP NTM

CEBSM

32

24

16

x = 3.972 m


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1690 D. A. JOHNSON AND L. S. K I N G AIAA JOURN

.02

* .01

CDSMA671 AIRFOIL SECTION O

oO

o

O

.6x/c 1.0

Fig. 9 Supercritical airfoil flow, M00=0.72 and a = 4.32 deg:boundary-layer displacement thickness distribution.

Th e calculations were started from an experimentallydetermined m ean velocity profile and a measured value fo rU T at Jt = 0.8 m . With these data available, it was notnecessary to start th e calculations with the CEBSM. Directmode calculations were performed up to x = 2.2 m ; then,from that station on, the calculations were performed in theinverse mode. Presented in Fig. 6 are the exper imental andpredicted surface pressure, skin-friction, and shape-factordistributions. As in the previous test case, the CEBSMpredicts too large a pressure recovery, whereas th e pressurerecovery predicted with the NTM is in good agreement withexper iment. In Fig. 7, the developments of -u'v'm predictedby the two closure models are presented along with the quan-tity geq/g predicted by the NTM. These results indicate thatnonequil ibr ium effects ar e also of importance fo r this low-speed flow. Mean velocity profiles at a few select stations ar epresented in Fig. 8. The trends illustrated in this figure ar esimilar to those observed for the shock-induced separatedflow. The CEBSM is seen to predict the wrong profile shape,even upstream of separation (the x=3A2 station is justupstream of the measured mean separat ion point) .The last test case to be considered is the supercritical air-foil exper iment of Johnson an d Spaid.5 This experiment wasperformed at conditions intended to simulate cruise. Accord-ingly, th e shock which developed at x/c=0.40, on thisDSMA 671 airfoil section was relatively weak, but strong af tloading resulted in a rapid growth in the boundary-layerdisplacement thickness along the upper surface near thetrail ing-edge region, as shown in Fig. 9. The boundary layerw as sufficiently retarded that a small separation bubbledeveloped at */c^0.98.The NTM calculations fo r this case were started withCEBSM. At x/c = 0.22 the NTM was turned on (a transitionstrip was located on the model at */c = 0.17). For bothclosure models, th e calculations were performed in the directmode up to the shock and, from th e point downstream, th ecalculations were per formed in the inverse mode. The result-

4 r

(0

EX P N T M

.2 1.0x/c

Fig. 10 Supercritical airfoil flow, M00=0.72 and a = 4.32 desurface pressure distribution; b) skin friction distribution; c) shfactor distribution.

mance. On the other hand, the NTM formulat ion predicpressure plateau similar to that observed experimentwhen separat ion occurs. The results shown in Fig. 10 sugthat the NTM should do substantially better in predicairfoil performance when it is incorporated into an aiprediction method. Mean velocity compar isons at repretative stations ar e presented in Fig. 11 and the developmof -u'v'm and the quanti ty g eq/g ar e presented in Fig.


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NOVEMBER 1985 T U R B U L E N C E C L O S U R E M O D E L F O R B O U N D A R Y LAYERS 1

.04

.03

y/ c .02

.01

0 .2 .4 .6 0 .2 .4 0 .2 .4 .6 .8 1.0u/ue

Fig. 11 Supercritcal airfoil f low, ^^=0.72 and a = 4.32 deg:mean velocity profile comparisons.

NTM

o1.41.21.0

(a )

JFI- *

EQUILIBRIUMCONDITIONS

(b )1. 0

x/cFig. 12 Supercritical airfoil f low, M00=0.72 and a = 4.32 deg"history" effects: a) maximum Reynolds shear stress development;b) nonequilibrium state of f low.

ConclusionsA new turbulence closure model has been formulatedspecifically fo r two-dimensional , turbulent boundary-layerflows subjected to severe adverse pressure gradients. Basedon compar isons with exper iment, this closure model appearscapable of treating subsonic, pressure-driven separated flowsand shock-induced separated flows. From the standpoint ofengineering practicality, the proposed closure model has

cluded in this paper ; for example, the performance bouary layers with zero and favorable pressure gradients th e solution sensitivities to the values prescribed for modeling constants.The next obvious step will be to incorporate the propoclosure model into a truly predictive method. Ideally,predictive method should be based on the Reynolaveraged, Navier-Stokes equations, since this woeliminate the possibility of falsely at t r ibut ing errors cauby approximations to the mean flow equations to the closmodel. Once this is done, it will be interesting to see how proposed model will perform against two-equation, eviscosity and full Reynolds stress closure models.

AppendixIn the early stages of this study, th e diffusion term in (6) w as assumed to be negligibly small relative to the convtive term in that same equation. Calculations performed wthe diffusion term neglected produced excellent resultsretarded flow regions [i.e. , regions where d(-ufv^/dx wpositive an d consequently , where vto was less than * > /0 eq aresult of the convective term in Eq. (6)]. However, in

downstream recovery regions the neglect of turbulent difsion resulted in too slow a decay in -u'v'm with th e streawise distance. Moreover , in the case where massive sepation occurred upstream, the neglect of the turbulent difsion term resulted in the outer eddy viscosities downstrebecoming so large that converged solutions could not be tained. It became obvious that turbulent diffusion couldbe neglected in flow recovery regions. O ne solution to problem would be to assume that the convective and difsion terms in Eq. (6) exactly cancel each other in regionflow recovery (i.e. , that v to never exceeds vto> eq ) . Howevthis strong assumption appears to result in too rapid a dein -u'v'm and also in ah underpredict ion of skin-frictrecovery.To improve the prediction of the streamwise decay-u'v'm in flow recovery regions, a model for the diffusterm in Eq. (6) was formulated. The diffusion modelBradshaw et al.9 was not adopted since it was deemed esstial that the eddy viscosity distribution reduce to equilibrium eddy viscosity distribution of Eq. (8) in regiwhere the convective term in Eq. (6) was negligibly small.accomplish this, th e turbulen t diffusion term as descrilater is related to the nonequilibrium state of flow in outer part of the boundary layer.Similarly to Bradshaw et al. , th e "bulk" convecthypothesis of Townsend 16 is used as a starting point,

(

whe re v is a lateral convection velocity representing thebulent t ranspor t by the larger and more energetic eddymotions. Along the path of maximum kinetic energy,(Al) takes the fo rm:(

where the relationship -ufv'm/km=al ha s been assumAn approximate model for the variation of v . with y/5shown in Fig. Al is proposed based on the q2v fsurements obtained by Bradshaw 17 for two adverse press


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1692 D. A. JOHNSON AND L. S. K I N G A I A A J O U R N

i.o

y /6

(LOCATION OF -u' v'm)

Y, A R B I T R A R Y SCALEFig. Al Qualitative model for v as a function of y/b.

In Eq. (A3), *0 is the maximum lateral convection velocitythat is hypothesized to occur at 7/5= 0.7. This quant i ty is inturn assumed to be propor t ional to

1- dissipationproduction (A4)

The term inside the absolute value sign represents the im-balance between production and dissipation in the outer par tof th e boundary layer, which can be expressed in terms ofeddy viscosities,

1- dissipationproduction 1- u'v')dii/dy

"to, eq(A5)

The scaling of v0 to (-u'v Y2 in Eq. (A4) is consistentwith the arguments of Townsend. The other te rm in Eq.(A4) has been added to make the t u rbulen t diffusion func-t ionally dependent on the nonequilibrium state of the flow.In regions where the flow is being retarded, the diffusionterm in Eq. (6) is of the same sign as the convective term andthus can result in even larger depar tures from equi l ibrium. Inregions where the flow is recovering, it opposes the convec-tive t e rm, hence causing a more rapid recovery to equilib-r ium conditions. The intent in the development of an expres-sion for the turbulent diffusion term was to allow a morerapid return to equi l ibr ium in flow recovery regions with aminimal change in the closure model performance in retardedflow regions. Th e expression given by Eq. (A4)has thesepropert ies . Combining Eqs. (A3-A5) and in t roducing th emodeling constant , Cdif, Eq. (9) is obtained,

1-to, eq

(A6)

This model for the turbulent diffusion is acknowledgedbe very approximate . As more experience is gained withturbulence closure approach proposed, a mo re refined mofo r th e turbulent diffusion should evolve.

AcknowledgmentsThe authors gratefully acknowledge the kindness of DrE. Carter of United Technologies Research Center fo r min g avai lable to us h is boundary- layer program wdocumentat ion and his assistance during the initial stagesapplying this program.

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sonic Turbulent Separated Flow," AIAA Paper 81-1241, 1981.14Johnson, D. A., H o r s t m a n n , C. C., and Bachalo, W."Comparison between Exper imen t and Predict ion for a TransTurbu lent Separated Flow," AIAA Journal, Vol. 2 0 , J u n e 198737-744.15Johnson, D. A. and King , L. S., "A New Turbulence CloApproach fo r Two-Dimensional Boundary Layer s with SeparatiNASA, to be published.16 Townsend , A. A., T he Structure of Turbulent Shear FCambridge Univers i ty Press, Cambridge, England, 1976.17 B r a d s h a w , P., "The Turbu lent St ructure of Equi l ibr ium Boar y Layers," Journal of Fluid Mechanics , Vol.29, Pt . 4 , 1967625-645.

a ally simple turbulence closure model for attached and separated turbulence boundary layers

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