Spalart1992 - One Equation Model

8/9/2019 Spalart1992 - One Equation Model

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AlAA 92 G439One Equatlon Turbulence Model

for Aerodynamic Flows

P R Spalart and S. R. AllmarasBoeing Commercial Airplane GroupSeattle, W 98214-2207

3 thAerospace SciencesMeeting ExhlbH

January 6-9. 992 Reno NV


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A ONE-EQUATION TURBULENCE MODELFOR AERODYNAMIC FLOWS

P. R. SpalartS. R. Allmaras-

Boeing Commercial Airplane GroupP.O. Box 3707, MS 7H-96Seattle, WA 98124-2207

Abstract

A traDSport equation for the turbulent viscosity wasusembled, using empiricism and arguments of dimen ional analysis, Galilean invarianee, and selective dependence on the molecular viscosity. It has similarities":'lith the models of Nee : Kovasznay, Secundov t .1.a n d l d w i n: Barth. The equation includes a destruction term that depends on the distance to the wall, related to the one in Secundov's model and to one dueto Hunt . Unlike early one-equation models the resulting turbulence model is local (i.e the equation at onepoint does not depend on the solution at other points),and therefore compatible with &rids of any structureand Navier-Stokes solvers in two or three dimensions.It is numerically forgiving, in terms o f near-wall resolution and stil"ness, and yields fairly rapid convergence tosteady state. The wall and freestream bonndary conditions are trivial. The model yields relatively smoothIaminar-tur rulent t _ i t i o n a t points epecified by theuser. It is powerful enough to be calibrated on 2-D mix-ing layers, wake&, and f l a t p l a ~bonndary layers, w h ~ hwe Itonsider to be the building blocks for aerodynanuclows. It yields satisfactory predictions of bonndarylayers in preaure gadients. Its Dumerical implementation in a 2 D steadY.fltate Navier-Stokes solver hasbeeR completed and is discussed. The cases presentedinclude .bock-induced eep&ration and a blunt trailingedge. The model locates Iihocks slightly farther forwardthan the JoImsoD-King model. I t perfonm well in thenear wake and appears to be a good candidate for morecomplex flows such as high-lift systems or wing-bodyjunctions. However, _ it is not clear whether steadysoIutioc' will or should be obtained.

Member AlAACopyricht eAmerican Institut. 1 _ t i a and AstJ'O"

_ics. In


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On the other hand, transport-equation models suchas It f and higher models [5] are usual lr local ' , although some have non-local near-wall terms, and havebeen a ~ a i l a b l eCor years. However, they are Car fromhaving shown a decisive advantage for the predictionofshock/boundary-Iayer interactions or eeparation Cromsmooth surfaces [6). They may be superior Cor massiveeeparation, but we do not have enough data, nor powerful enougb codes, to assess that. They are also muchmore difficult to use. Tbis is not so much because of tbeextra storage, but because they require finer grids neara wall, involve strong source terms that often degradethe convergence, and demand non-trivial upstream andfreestream conditions for tbe turbulence variables. Thenear-wall problema oCten lead to the use of wall functions (71, which are unwieldy and lose any justificationin the situation that matters most, namely, eeparation.

The new Baldwin-Barth model [8) is an attractive in

termediate.I t

hIlS only one equation and is local, except Cor the , dependence which they plan to disposeof in the long term. It is derived from thE; fro( model,through some Curther assumptions. Near the wall itdoes not require finer resolution than the velocity fieldit5eIC. Depending on the version, it predicts adversepressure-gradient cases and shock interactions betterthan Baldwin-Lomax, but not consistently as well asJohnson-King. Its accuracy will improve in time, and itis much more practical than two-equation models. Wemake more specific remarks on this model during thepresentation of the new one.

The present project was prompted by Baldwin .:Barth' s work, a nd b y the belief that generating a oneequation model as a simplified version of the k-c model

is not optimal. A one-equation model is simple enoughthat it can be generated from scratch , which may leadto better performance and certainly gives Culler controlover i ts mechanics. A case in point is tbe BaldwinBarth dift'usion term, which is constrained by the Iro(ancestry and the Curther assumptions made. We alsoallow a eemi-local near-wall term, as described below. Our calibration strategy was different. 'We expectto show that the new model has the same properties asthat of Baldwin : Barth in terms of compatibility witbunstructured grids and benign near-wall behavior, andis more accurate, especially away Crom the wall, as wellas slightly more robust. For instance, it accepts zerovalues in the freetit.eam.

The roster of one-equation models also includes tbose

o Bradshaw, Ferriss, : Atwell [9], Nee : Kovasznay[10J, Secundov and his co-workers [1 t], and Mitcheltree,Salas, : Hassan [121. Except for Secundov's and Baldwin : Bartb's, these models are not local, since theyuse length scales related to the boundary-layer thickness. Tbi s contribu tes to the common claim tbat oneequation models are not complete (i.e., they require acarefully-chosen length scale for each new flow' and tbat

two-equation models are the simplest complete models. The Nee-Kovasznay model was not followed uponpartly because it was not affordable at the time (1969).Tbe Secundo\' model is currently entered in the Collaborative Testing of Turbulence Models (C TTM, [13])and Prof. Bradsbaw was kind enough to pro\'ide theone-page description that was submitted. Dr. Secundov pro\'ided a few details in a personal communicationas well as a list of publications ranging from 19i to1986, but none in English. This model is presented asan evolution of the Nee-Ko\' Sznay model but is rich inilear-wall and compressibility corrections. In particula rwe reinvented their near-wall destruction term. I t isexpected for simple empirical models, developed underroughly the same constraints (invariance, and so on).t o exhibit strong similarities. However, the leeway islarge enough to produce models with widely dift'erentperCormance.

Presentation and CaIibration of the Mogd

Overvjew

In this section we present four nested versions of themodel Crom the simplest, applicable to Cree shear flows.to the most complete, applicable to viscous flows pastsolid bodies and with laminar regions. As each additional physical effect is considered. new terms are addedand calibrated. They are identified by a common le ttersubscript in the constants and functions involved (e . ..constant cu function f ~ 2 ;note that tbe constants aienormalized so tbat the functions are of order 1). Thenew terms are passive in all tbe lower versions of tbemodel, so that the calibration proceeds in order. Thispresentation may seem heavy, but should be instructiwas it allows the reader to criticize the tbeory or the calibration layer by layer and to test the rele\-ant versionin tbe situation he or she chooses. It should also helppreserve some clarity in later alterations of tbe model.The Appendix gives a compendium of the equations forthe complete model.

Constitutive Relation

The \:entral quantity is tbe eddy viscosity v . TIJf'Reynolds stresses are given by - IT; iij = 2V,Sij whet{S,j lJU,/fJz j + lJUj/lJzj /2 is the strain-rate tensor. Compared witb a two-equation model we naturallymiss tbe Ir term (turbulence kinetic energy). This is nota major effect in thin sbear flows, and the addition of2k/3 to the diagonal elements of the stress tensor is approximate in any case. Note that even in two-equationmodels there bas been an erosion of the meaning of kit5elf [14J, and also that tbe equation II, = k ~If i,.dearly not satisfied in the log la:rer with the true and


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c. oeare basing this criticism on experimental anddirect-simulation results [15].

In a one-equation model, or in other models (e.g.,zero-equation) which produce f but not k, we could obtain a rough approximation of k as proportional to thestreues given by V,Sij, and introduce it on the diagonal.It would certainly be consistent to make it large enoughfor the Reynolds-stress tensor to be positive-


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We break our convention for 0 . which belongs to theC6 series , because of the traditional notation of Prandtlnumbers.

The diffusion te rm of (2) conserves the integral of thequantity v c. . Recall the lack of a destruction term.This lack w u responsible for a mild inconsistency iniIotropic turbulence. It could also inva1idate the modelin the c . . . or Ihear flowl in which v, decreues (neptive Dv,/Dt luch a t an axilymmetric wake . However,the difruaion term can eaaily brine down the centerlinevalue or v and the true constraint is that under (2) theintesra1 or v:+ c u cannot decrease. With tbe cl .- icalexponents of the aelf-.imilar axisymmetric wake (lengthscale oc 11/3, velocity oc t- 2/ 3 ), we find that the intecralincreases provided tbat cn 1. Even iCthe calibrationdoes not include the axisymmetric wake, it is preferableto satisfy this constraint.

Another cODitraint may be obtained from the behavior or a turbulent front. The diffusion term admits thefollowing (weak) one-dimensional solution:

/I,( ,t) = max 0. A [Z+ A I ; (62) tD, (3)for any cODitant A. This is a linear ramp propagatingat the velocity -A(I +(62)/0 f cn > - I it propa,atesinto the non-turbulent region, which is physically correct. The equivalent of cn is 0 in the Secundov model(the diffusion term is conservative) . It equaled - 2 inthe original Baldwin-Barth model [16) and is somewhatbelow 1 in the published version [8), so that under thediffusion term alone the turbulent front recedes. Webelieve this effect is to blame for the sensitivity of thatmodel to the freestream value of /I (or RT). Note thatBaldwin & Barth are constrained in their choice of e62

by the connection with the k-l model, in tbe originalversion, and by their calibration in the log layer, in allversiODI. We avoid this constrai nt thank s to a near-wallterm as explained below. The weak solution shown in(3) i. of , reat interest in practice, as it indeed gives the.tructure of tbe aoIution at the edge of a turbulent re,ion. Tbis occurs because the diffusion term dominatesthere _ Fig. 6, below).

An approximate analysis of the turbulent front, coupling the eddy viscosity and the shear rate, indicatesthat the ratio (1 + en) 0 also deserves attention. If it islarger than 2 the eddy-viscosity front, whicb is a ramp,is slightly ahead of the shear front, making the velocityprofile smootber. f it is lower than 2 the two fronts coincide. f t equals 1 tbe velocity profile of a mixing layeris exactly triangular, i.e., IU.I exhibits a step which isunphysical . Although these considerations border ontbe cosmetic they indicate that (1 + en)/O' should belarger than 1 and suggest that 2 m a ~ be a favorablevalue.

1I\'eak solutions such as (3), and raises the possibility ofnon-unique solutions. Indeed if tbe initial condition isIzl we have a weak solution in which II, behaves l ihz 1/(2+< ,,) near z = 0, and a smooth solution with

/I > 0 at % = O The difference is confined to aboundary layer near = O In a numerical setting withstraightforward second-order centered differencing. theweak solution will be obtained if the diffusion term iswritten /I V2 /1,+( 1+cu)(V 11,)2, but the smooth solutionwill be obtained if it is written as in (2). Other formstbat give the smooth solution are V2(vl/2)+Ct2(V/l 1)2and (l+eu)V(/I, VV,)-Ct2/1r V2/11 The later addition ofa term proportional to the molecular viscosity formallyresolves this non-uniqueness and leads to the smoothsolution with /I, > O Howe\'er, particularly at highReynolds numbers, it is desirable to use a favorable formof the diffusion term.

Outside a turbu lent shear flow the Reynolds stresses.particularly the diagonal components, do not exactlyvanish. However, they are induced by pressure fluctuations and bear little relationship to the local straintensor S i j For that reason, it is as well to have the eddyviscosity be zero outside the turbulent region. and thisis the value we recommend in the freestream . In addition, tbe model is essentially insensitive to non-zerovalues (wbich may help some numerical solvers). provided that they are much smaller than the values in theturb ulent region. Thi s is due to the dominance cf theturbulent region (/I, > 0) over the non-turbulent one, asillustrated by the ramp solution (3) . This feature addsto the black box charact er of the new model and represents a substantial advantage over the Baldwin-Barthmodel and many two-equation models, some of whichare highly sensitive to freestream values-that of thetime scale for instance.

The amplification of the eddy viscosity by the production term is of interest. Note that the linearizationof (2) for small /I preserves only the production term .Consider the steady flow at a velocity Uoo past a bodyof size L, with thin shear- or boundary layers of thickness 6; these are orders of magnitude . Outside the th inshear layers the deformation tensor is of order Uoc / L sothat, irrespective of the exact definition of S, the logarithm of the amplification ratio will be on the ord erof Cu (a growth rate on the order of euS over a residence time on the order of L/ U oo ) . Thus, small valueswill remain small. In contrast in the tbin shear layerstbe logarithm ofthe amplification ratio under the effectof the production term alone would be on the order ofeuL/6, and therefore large in the usual situation si'lceL ; 6. Thus small values of /I,. whether inherited fromthe freestream , or resulting from numerical errors. or introduced intentionally at the trip as described later.will cause transition in the thin shear layers o ~ By

The fact that the dependent variable v, itself is the transition we mean growth to such levels that the difdiffusion coefficien t is responsible for the existence of fusion terms . which are nonlinear. b ecome actin . Th e

4


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Ii

destruction term introduced l ater is also nonlinear . The structure at the edge of the turbulent r egion. as seen insequence of exponential srowth , followed by saturat ion the mixing-layer case in FiS. 2 . The centered-differenceat levels on the order of U 6 , is consistently observed in solution cannot faithfully reproduce a weak solution atpractice . Note also that even though the contribution of the front. but the disturbance d >es not propagate . Time

the eu term is pOlitive the diffusion terms cannot cause steps in excess of the stability limit were revealed firstrunaway IfOwth. This is so because their contribution by short oscillations near the centerline .is neptive at a local maximum of II I Recall also thec:outraint on the intesral of 11:+' . The eddy-viscOlity o . 5 T - ~ - - - - - r - - - . . , . . - - . . . . . , . - . . , - ' _bud,et in a weJl..developed solution always includes aIliJeable c:oatribution (rom the production term .

We DOW calibrate the free--shear-ftow version o( themodel by requiring correct levels of shear stress iD twodimensional mixing layers and wakes . Fair values forthe peak shear stress are 0.01 4U)2 in the mixiDg layeraDd 0.06 4U)2 iD the wake, where U is the peakvelocity difference [17]. This lives two conditions forthree free cOllltants C.1> 11 , and C.2 and leaves a onedimensional family of solutions which is .hown inFiS . 1 parametri%ed by the Prandtl number 11' , the eas

iest quantity to interpret. The ranse of values of 11' weconsider plausible is at most [0 .6,1]. The correspondingvalues of cn are between 0 .6 and 0 .7 ana satisfy our

guidelines (-1 < cn ::; 1 with a margin. The ratio(1+c.,)/O varies from about 2.7 to 1.7 and also satisfiesits suideline (> 1).

1.70

1.115~

. \ Cb2 II1.10I

1.55

'..

.50 I / U h C w 2U 5

UO

' ----. , '10Cll1 r ---- :------- .30 -

S0.1 0.7 0.' 0.8 1.0

11'

Fisure 1: Calibrated model constants . - locus of solu -

O , " t - - - t - - ~ t - - - h f 4 - f : , . o ' ~ - - - -

0 3 + - - - - + - - - - f - - i l ' i : * - + - - - + - - - + - - - -

0.2 t - t - - J J - - t - - - t-..... . . . ,"" -- - ---;

I ~ = ~ I 5000.1 ; . + - - - t - - . . . , ~ - l . . - i l l

I50

0.02 0.04 0.06y

Fisure 2: Profiles in a time-developing mixing layer .Normalized with velocit y difference and time. Velocityprofiles adjusted to the same slope at y = o

Based primarily on the edge beha\ior . we favor afairly diffusive member of our plausibl e range , namely11' = /3 , C61 = .1355 , Cl2 = 0.622 , (l+c62)/O ::::: 2.4 . Inthe mixing layer it gives a velocity profile close to thatlUIIIociated with identical rollers of uniform vorticity(see Fig . 2). Very low values of 11' would be needed tobrins it close to the hyperbolic-tangent profile, which isa common approximation . In the wake the peak eddyviscosity is 0.46M where AI is the momentum of thewake, in sood asreement ith experimen t [18]. We didnot attempt to match any axisymmetric flow, partlybecause they are not pre\'8lent in our applications andpartly because , for most models, these flows conflictwith the 2-D flows . The model is not intended to beuniversal.

tiOlll; +, x , point selected for the calculations. ~ a r w a l lRegion. high Reynolds Number

The solutions were obtained numerically using astretched srid, centered second-order finite differences ,Runse-Kutta fourth -order time intesration , and zerovalues in the freestream . This artless treatment wouldrapidly reveal poor numerical properties in the model .The srowth of the layers was followed unti l a self-similarItate was attained. The solutions exhibit the ramp

5

In a boundary layer the blocking effect of a wall isfelt at a distance through the pressure term , which act sas the main des t ruction term for tbe Reynolds shearstress . This susgests a destruction term in the transportequation for the eddy viscosity. Dimensional analysi sleads to a combination VJ I IIr/d)2 , witb d the distan ceto th e wall . Th e subscri p t It st ands f or ' walr . Thi s


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term will be passive in free shear flows (d > oc) andtherefore does not interfere with our calibration up tothis point. The Secundov model includes this ty pe ofterm (however their te rm differs from ours in the viscous

and the oute r regions). The idea of a near-wall, but notViICOUl , "blocking" 'erm is also in Hunt [19]. It is alsorelated to the algebraic models, which take the smallerof two eddy viscOllities. The outer eddy viscosity scaleswith the boundary-layer thicknell, and the inner eddyvilcOllity is pven by the mixing length, I oc d.

In a claaical log layer with friction velocity u.. wehave S = U.,f(IC4) and "I = u. lCd. Equilibrium betweenthe production and diffusion terms (all positive) and thedeltruction term is pOIIIible providedC ,I = U/,,2 + (1 + cu)/tT .

1.0

0.1

I ,0.6

II

I

iI1I

/F _ Vhlar

7i~ ,

j

HIe, 1'--j'LOQ iFPG \ I layer I[II I

/ iIIi

uIIr i,.rtolBL

'Ielta show that the model, when equipped with the 0.2/ II/V I Ieltruction term, can produce an accurate log layer.

This rel is OIl th' treatment of the vilcous regiOll, d ~ 0.0 ,

ICribed below. On the other hand it produces too low 0.0 0 0.8 1.2ra Rin-friction coefficient in a flat-plate boundary layer .This shows that the destruction term 8. formulated de- Figure 3: I , function involved in the destruction term.cays too slowly in the outer region of the boundary layer . see (4-6).To address this deficiency and allow a new calibrationwe multiply it by a non-dimensional function ItIJ whichequals 1 in the log layer. Note that C ,I is not negotiable fact that S may vanish . The region r > 1 is exercised(i.e., we would not adjust the C, at the expense of the only in adverse pressure gradients, and then rarely belog-law constants), and also that \II e were not able to yond r = .1. Having 1 ,(0) = is not essential , becaus eobtain an accurate skin friction just by using the free- in free shear f l o ~ s~ h edestru.ction term vanishes on acdom left by the free-shear-flow calibration (Fig. I). The count o ~the cP. m Its d e n ~ l I l D a t o r .A reasonable valuemodel becomes for C ,3 IS 2. \ \e then cahbrate C to match the skin-

friction coefficient in a flat-plate boundary layer. WeDVI =CtlSVI+. . [V.(V I VVI)+Ct2(VVI)2]-C",l/w [ .i)2 adopt the value o f t h ~CTTM .' namely C, = 0.00262Dt u d at R4 = 104 [13] which requires C ,2 = 0.3 . All the

T (4) boundary-layer tests relied on a code writ ten by Mr . D.Note that Secundov et .1. did not foUow the I , route. Darmofal, ofM.I.T . during a short s tay at Boeing.

The choice of an adequate argument for I , was in -spired by algebraic models, in which the mixing lengthplays a major role near the wall. This length can b edefined by I ; VI/tiS and we use the square of I / d asa convenient nOll-dimensional argument:

V,r ; : Sf 2 d2. (5)

Both r and ItIJ equal 1 in the log layer, and decrease inthe outer region. Note that any dimensionally conectfunction oC(v d,S) that reduces to - C t I J l , , : l U ~in a loglayer would be as eligible as the one we are choOlling (.f).A satisfactory I , function is

[

1 + /I ] 1/6I.,(r)

=,6 C'6

3, ,

= +C.,2 (r G - r) . (6)

9 +c ,3

This function is shown in Fig. 3. The results aremost sensitive to the slope of Iv. at r = 1, which iscontrolled by C ,2. The step from 9 to /", is merely alimiter that prevents large values of I which could upset the code and give an undeserved importance to the

6

Figure 4 shows the velocity, eddy-viscosity. and shearstress profiles in a flat plate boundary layer at R, :::: 104 .At 104 , C, = 0.00262 and H = 1.31. The Clausershape factor G ; : J2/C,(H - 1)/H is settled at 6.6 .and the shape of the profile is satisfactory . Noticeagain the ramp structure of V, at the edge of the shearflow. The peak value of V, is O.021U oo f. compared witl.0.0168Uoo 6 in the Cebeci-Smith model [2]. Conversely .the Cebeci-Smith eddy viscosity is higher near 11/6 = 1.The .hear-stress profile approaches the wall with a finite slope, rapidly turning to zero slope at the wall a5diICusaed in the context of direct-simulation results [15].

Figure 5 shows the velocity profile in wall coordinates, illustrating the log layer and the smooth depar

ture in the wake. Again, the shape of the outer regionappears good, showing that the destruction term andthe '" function are fair approximations, at least in thi 5flow. The arrival at the freestream velocity is a little tooabrupt, as it 1\'as in the mixing layer , in Fig . 2. This b e-havior cannot be corrected except by making th e mod elvery diffusive (low u) .


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4 I

y/S

3

2

r r ~ f f .

\

o '\""-

-1 /

7fo t

I-4

o

II,

i

r---- ~ r I,I > < - \ i.--- ;

i

I

IiII,I

4 8

yWFigure 4: Profiles in a flat-plate boundary layer at Figure 6: Eddy-viscosity budget in a flat-plate bound-R, s:t: 10, outer coordinates. U normalized with Uoo r ary layer. Normalized with r IVa l / .with rO l , and v, with 0 025U oo 6 .

"

Jj"" I,I iV 'I

,i i

i,I

'I i I

I,

I; Ii I:U+ Ii i

I I, IIi ,I III :I ,II IIi I

1000 10000y+

Figure 5: Velocity profile in a flat-plate boundary layerat R, s:t: 10", inner coordinates. - model; - - -log law.

The budget of Vt is shown in Fig. 6. The sum (i.e.,Dv'; Vi is positive throughout. It is 0 at the wall,then roughly follows a ramp up to the edge of the turbulent region. Its outer part is representative of theouter part of either one of the free shear flows, including the vanishing contribut ion of the destructionterm. The production is equal to the shear stress.In the outer part the dift'usion is primarily responsi.

ble for the advance of the turbulent front. in qualitative agreement with the budgets of "legitimate" turbulence quantities such as the kinetic energy. Near thewall the dift'usion again makes a strong positive contribution, balanced by the destruction. The ideal nearwall budget is. in the units of the figure. production= , dift'usion = ,2(1 + e.2)/u /eu 3. destruction =-1 - K2 1 + et2)/u /eu -4 . Note that it is not maintained far up into the layer at all; correspondingly. Vtdoes not follow its ideal linear log-layer behavior ( Yu T )far up either (see Fig. 4). However this does not preventa log layer from forming.

This completes the calibration of the model save forviscous eft'ects. \\' e later examine the performance of themodel in what is probably the most sensitive situation.the outer region of a boundary layer in adverse pressuregradient.

Near-wall Region. finite Reynolds Number

In the buft'er layer and viscous sublayer. additionalnotation is needed. Besides the wall units, y+ and soon, we introduce v which will equal v, except in theviscous region, and X == v/v This is in analogy withMellor .: Herring's notation [20], because from the wall

to the log layer we have X = Y+ .We follow Baldwin .\: Barth [8] in choosing a transported quantity vwhich behaves linearly near the wall.This is beneficial for numerical solutions: ;; is actuallyeasier to resolve than U itself, in contrast with (, forinstance. Therefore. the model will not require a finergrid than an algebraic model would. To arrive at this


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we cODIider the c1aaical log layer and de\'ise near-walldamping f u n c t i o n s ~that are compatible with known

results. Th ese functions are disti nct from the I nearwall inv lCid destruction term .

The eddy viKOCIity /I, equals yu. in the los layer,but not in the buffer layer. We define v 0 that it equals,, ' , . all the way to the wall. This leads to

X31 ~ 1= . , . . - : -r .

~ + t;;)(7)

The 1 1 fWKtion is borrowed from Mellor k Herring.The subKript tI stands for viscous . We prefer thevalue c,,1 = 7.1 to Mellor Herring's 6.9, which wehelieve yields a low intercept for the log law. Other ~fuJldioas could be used, for instance the one compatiblewith the Van-Drieat damping is I,,) = ( 1 + 41+2 -1)/2X, where t+ =X[I - exp(-xllCA+)] and A+ is thefamiliar 2G [2]. We have not encountered cases b whichthe choice of I .) mllde a difference . Note that there isDO basis for (7) to apply at the edge of the turbulentregion, where X is lao of order 1 and smaller. However,the eddy ViK06ity has little inftuence there, because ofthe absence of steep gradients.

The production term also needs atten tion. III it S isreplaced with S, given by

X~ 2= 1 - 1 + xl,,) (8)

The function / , , 2 is constructed, just like /01> 110 thatS would maintain its log-layer behavior S = .. (KYall the way to the wall. S is singular at the wall, buti i is 0 there, 10 that the production is well-behaved 'Note that there is a range of X in which S is _ thanS and may become negative. This should DOt upsetthe Dumerial salven. Other quantities involved in the

inviKid model are redefined in terms of if iDst ead of/I for instance r ii/ SK 2tP).

We f iwly add a viscous diffusion term, cOMistentwith a Dirichlet boundary condition at the wall, if =O.This term too is based on an analogy, rather tbaa a rigoroua equation. In addit ion behaves linearly, 110 thatits Laplacian will be small in an established dut ion .Accordin&ly, we insert the molecular viscOCIity ia a convenient place and pay little attention to a factor of tT .The transport equation has become

Dv -_ 1 [ ]Dt = c 61 S/ I+ ; V./I+V)VV)+C.2(VV)

-c",t tIJ [ ~ r (9)

Tbis equation now yields equilibrium (D VIDt = ) allthe way to d = 0 in a classical law-of-the-wan situation. Furthermore, the numerical evidence shows thattbis clusital solution is a stable IOlution of the system

8

made of the momentum equation and (9), as it has beenobtained starting with a wide range of initial conditions .This includes the results of Figs . 4 and 5 ; in particularFig. 5 displays the viscous and buffer layers.

Fine effects of pressure gradients or transpiration arelikely to strain the accuracy o f the model , although weexpect the trends to be correct and the database is far .from definitive. Note that the traditional tools for inserting these effects into the algebraic models are ruledout, because they are not local.

Lamjnar &CioD aDd Trjppjnc

The final set of terms provides control over the laminar resions of the sbear layers, a control which hastwo aspect s: keeping the 8011' laminar where desired .and obtai ning transition where desired. Navier-StokescodeiJ with &Igebraic models usual ly have crude off-on devices or short ramps hued on the grid index along thewall . These do not help the convergence of the codes . Inaddition we require a device that is useable on unstru ctured grids. The subscript t will stand for trip . Wei.Jse this word to mean that the transition point is imp06ed by an actual trip, or natural but obtained from aseparate method. 0 0 scco. t d o , d tAt t . rh l t c fmollel 6( tM/std to prellict tAt transition location. Tbisis true of all the models we kno,, . Some models . inc1uding Ir-(, predict relaminarization. In situations tbatshould induce relaminarization, this one tends to dropthe eddy viscOCIity to low levels, but without snappingto O. We may be able to improve on tbis using the f'2function below.

We delCribed how transition was expected only inthin shear layers. The linearized version of (9) for small

v only contains the production term: DvlDt = cuSii ;therefore , v = 0 is an unstable IOlution of (9) (goingin the direction of DIDt). In a boundary-layer codethe uro solution is easily maintained, but in a Na\'ierStokes code exactly -aero values are rarely preserved, sothat the model is primed by numerical errors upstream of the trip . It then transitions at a rate thatdepends on numerical details and has little to do wit hthe boundary layer's true propensity to transition, ascontrolled by pressure gradient, suction, and so on. W everified this behavior, and it is not acceptable .

A solution is to alter the production term so thatv= 0 is a stable solution, albeit with a small basin ofattraction. For this we take the convention that in thelaminar region v s of order /I at ID06t, and recall th eargument x. Note that if V ::; / I , then /II


10/23

C 4 = 2. In any case c,3 must be larger than 1. Asfor Ct. it could be decreased several-fold , if a code stillyielded premature transition . The cross-over point of1- / 12 i.e., the bound of the basin of attraction), is atX = og CI3 C,.. However, 2 was small enough in ourcode . Values much smaller than 1 would start affectingthe results in the turbu lent region .

In order to still balance the budget near the wall weoffset the change in the production term with an opposite change in the destruction term, involving 12, Againwe take an empirical approach , and have numerical evidence that it yields a stable system . A user that is doingboundary-layer calculations can leave the /'2 term out(i.e., l e t C'3 = ).

The task ill now to obtain transition. We refer totransition points, in 2-D, and transition lines, in 3D .We address only boundary-layer transition, but a gen

eralization to free shear 60ws will be easy. Usually each2-D body u two transition points. We wish to initiate transition near these points in a smooth manner,and compatible with any grid . For this a source termis added that will be nonzero only in a small domain ofinfluence. This domain should not extend outside theboundary layer. Not wanting to find this edge, nor violate in\'Ilriance principles, we invoke the quantities I Uand Wt. Il U is the norm of the difference between thevelocity at the trip (i.e., usually zero since the wall isnot moving) and that at the field point we are considering . Wt is the vorticity at the 1\'all at the trip point .Upstream ofthe trip in a boundary-layer code, it is fairto take 1 1 at the wall at the current station, since 1 1at the trip ill not available yet . The thickness of the

boundary layer is on the order ofU./WI

whereU.

isthe edge velocity. Recall that it is still laminar. Wealso introduce d the distance from the field point tothe tr ip point or line.

Storing, or repeatedly computing, d t for esch fieldpoint ill a penalty, but it would not be difficult to keepa list of the points that are within a r ~ s o n b l edistanceof the trip and to compute the trip ter m only for thOlie.The Uler may also watch for the following peculiar situation: the body could be so thin that the trip on oneside causes transition on the other side . A definition ofd l as the length of the shortest line that links the fieldpoint to the trip while not crossing the body would solvethe problem, but in an expensive manner. t is simplerjust to override d, (Le., set it to a large value) for fieldpoints and trips that are known to be on oppOllitt: sidesof the body. The angle between the line from the tripto the field point and the normal to the wall may beuseful.

Dimensional analysis points to AU 2 as a properscale for the source term . and we arrive at

9

D V [1 f J - 1 [ ... ( - - ]=CbJ - 12 11+;; \ . (1I+1I)\ V)+Cb 2 ( \ v f

- [cu,du - ~ 112] [ ~ r+ III AU 2. (11)

with

111 =CII gl exp (-CI2 : . J2 [d2 + 9 ~ d ; 1, 12)and 9, min(O. I, taU w,AZ) ",here Il.z is the grid spacing along the wall at the trip . This equation specifiesthe two I terms, and the trip term is the last in (11).The Gaussian in confines the domain of influence ofthe trip terms as needed ; it is roughly a semi-ellipse .The magnitude is adjusted so that the integrated contribution for a particle crossing the domain of influencl"is on the order or U.6 6 the boundary-layer thickness ,as is ensured by typical algebrai c models [2]. The oddfactor 9 is passive in a situation with an extremely finegrid, but is quite active il.nd necessary in practice. Thisis because the domain ofinfluence Dfthe tri p scales withthe boundary layer thickness , which is very small in thelaminar part. As a result, that domain easily falls between two streamwise grid points, so that the trip is notfelt at all. The 9, factor guarantees that the trip termwill be nonzero over a few streamwise sta(ions .

The value C 2 = 2 reflects typical values of 6 dU.in laminar boundary layers and is not a candidate formuch adjustment. Tests indicate a range of at I' asttwo decades for CII between values so low that transition miscarries, and values so high that; ; Ilnd the skinfriction overshoot. The value Ctl = is well within thatrange; successful transition was obtained with 0.1 andwith 10. It is possible that at very low Reynolds llumbers CII would require more attention . In any case . werecommend that the user check for transition on bothsurfaces . To start with, the two trips should bracket the.tagnation point. Except at very low Reynolds numbersthe skin-friction coefficient is enough of a criterion. Notethat the trips are often near the leading edge, in a region where the skin friction has violent variations du eto the pressure gradients. Therefore, it is advisable tocheck the skin-friction coefficient a little downstream. ina weak pressure gradient.

Note that the grov.th of ;; to nonlinear levels under the effect of the production term occurs in a fewboundary-layer thicknesses (C6I being roughly 0.13) .This is consistent with the idea that the trip term mimics the secondary instabilities invoked by recent transition theories , which have growth rates on the order of1/6 . However since the streamwise gr id is often muchcoarser than 6, transition will still appear very steep tothat grid . The contribution of the trip source term israpidly overwhelmed by the exponen t ial amplificationdue to production . Thus . we ha ve a formal ad\ an tag'


11/23

over the "off-on" models in that transition is a smoothprocess, but in practice it is debatable. Naturally, anadaptive gid will focus points at transition , and approach the ideal situation .

InjtW apd fmstream C9nditjons

Fair rsuIts have been obtained by initially letting iiuniformly to iw freeetream value. The turbulent viscatity emanatea at the tripe and IIpreads without noticeably dqrading the cODvergeDCe of the code. In thefreestre m the ideal value ill zer9 . Some solvers mayhave trouble with thill, either because of round-olf errorsor 80IJle cODvergence teat s dividing by ii. Fr_t ream val_ ofi l up 0 roughly ~ I Oare easily t9lerable with theCUrreDt CI3 and C'4 COIUItants. Recall that the ,,1factorin (7) theD maltea ~ f much Ilmal1er than v 119 that thelaminar boundary layers are not disturbed .

Numerica Solution Procedure

We expand 9n the approach of Baldwin Barth (8]for solution of the turbulence transport model within aNavier-Stokea solver. The model is advanced in time using an implicit IOlution procedure deaigned to achieve apatitive turbu lence field for all transient IOlution sta tes,as weD as a fast cODvergence ratl' to steady-state. \\'ehave incorporated the turbulence-model solution module into a modified version of Martinelli .: Jameson'sFLOI03 (21], where the updates of the velocity field andthe turbulence are decoupled at each time step. Theturbulence is updated at the start of each multistageRunge-Kutta time IItep on the finest grid of the multi

g id cycle ; on coarser grids the turbulence is frozen.We begin by rewriting the one-equation model (11)

in a form more convenient for numerical analysis,

':: = M V)v+ P ii)iI- D ii)i i+ T (13)where M V)ii is the c9mbined advection/diffusionterlDli,

Note that T and D v) are aiways positive or zero andP(V } is positive as long as S is positive.

Care should be used in interpreting our notation. Forexample, the complete production 9perator is P ii)Vrather than P ii) alone . The n9tation is intended tosimplify the foll9wing matrix theory analysis.

The turbulence transport equati"n is discretized on agrid with v the vector of unknowns at all gid points .The solution is integrated in time using an implicitbackward-Euler scheme of the form,

[1- At ( d(v") + rr(v") - tr(v")} Av n = (18)At [M(v") + P( v n) - D(vn)] v n + AtT

where the nonlinear matrix operat9rs M, P, D and Tare the discrete analogues of AI, P, D and T, respectively; d, V, J ) are implicit matrices; At is the timestep; and Av is the solution change ,

(19)

The exact solution of the turbulence transport equation cannot become negative . It can be shown that ifii = 0 at some location and the surrounding values arenon-negative, then 8v/8t ~ O. An underlying goal oftbe solver is to reproduce this analytic behavior-a nonnegative turbulence field-throughout the solution process (i.e. ii 0 at all grid points and at all time steps).

We athieve this goal ofa non-negative turbulence fieldthrough the use of positive discret .. operators and M-type matrices. A positive oper ator . when applied to avector of non-negative elements , will produce a vectorwith n9n-negative elements . An M-t.ype matrix is di

agonally dominant with positive diagonal elements andneptive (or zero) off-diagonal elements. A key property of an M-type matrix is that its inverse containsonly non-negative elements; hence, the inverse of an ~ f -type matrix is a positive operator . Our goal can beaccomplished by careful discretization and constructionof the implicit operators d, rr and Ii .

Rearrangement of (18) gives v n + directly a ~ a f U ~ -tiOD ofv n ,

M V)i i= - i i .V)v+; ' [V. [ v+ V)Vii] + c.,(VV)2] , [1_ At{fd + rr -IJ V,,+I = (20)(14) [1+ At M - Jd) + (P - Ji ) - (D - J)} n + AlT.

the production IOUfCe term is

P V)V = cull - 1121S V,

the wall destruction source term is_ eu v[

_]D V)v = [coud. - 2/t2] d '

and the trip function is

T = IIAU 2

(15)

(16)

(17)

10

Assuming v n ill non-negative , then non-Ilegativity ofv n +1 is guaranteed i f the right-hand-siele operator is

positive and the left-hand-side operator forms an M-type matrix. Thus, sufficient c o n s t r i n t ~on the discret eoperators are given by,

- d is M-type- rr is M-type

+ J) is M-type

(218)

(21 b

(2Ic)


12/23

[M - K J v ~ 0[p - JS] v ~ 0[D -I>] v:S 0

for all v ~ 0

for all v?.:

f o r a l l v ~ O(22a)

(22b)

(22c)

Baldwin k Barth show, for their model, that similarconstraints on the implicit and explicit operators in thebackward-Euler scheme guarantee positivity of the discrete solution. They also show that these constraintsgive unconditional stability of the numerical solutionprocedure.

Description of the discretization and construction ofimplicit matrices will be given in detail for I-D, wherethe notation is simpler. Extension to 2-D and 3-D isstraightforward because there are no cross derivativesin the one-equation model.

The model is discretized using a cell-centered finitedifference scheme.

Advection Operator s

The advection terms are discretized using f:rst-orderaccurate upwinding.

[M(v)v]ll)= - [ut ( ii~ ; : - l )+ U.- C ' i ~ ;iii)] ,(23)

where the advection velocities U and U- are defined.

ut = ( U i+ IUil). U i- = _(U i - lUi/). (24)The Jacobian of the advection operator is.

M ~ L l= -- ( - U n / A X i ,K ( ~ /= -. +ut - Un/AXi, (25)f V ~ i ~ 1= - (+Un/AXi.

Note that _}i;fl) is M-type and M - IVI)(1lv = 0,satisfying the positivity constraints.

Diffusion Operators

We have found that positive operators are more easilyconstructed if the diffusion terms are rearranged into theform,

where liberties have been taken with differentiation ofthe molecular viscosity v. The form of (26) avoids discretization of the term (V'ii)2, which does not easily lenditself to positive discrete operators, The diffusion terms

are discretized using second-order-accurate central dif-ferencing.

[M(V)V]\2) 1 + Cb2 1 [( ; ;\ ii. 1 - V.=

v+ vIHl/2- -' -- ':'CT Ax. A.r.+I/2

_ (v + i)i-l/2 ii - Vi_I] , (27)I.JI. Xi-I/2

[M(V)VW ) cn 1:: - - v + i i ) i - -CT Ax.

[iii+ - Vi iii - Vi-I]AXi+I/2 - AXi-l/2 .

(28)

The diffusion coefficients (V+V)i+I/2 and (v+ ii).-1/2 atcell faces are taken as averages of adjacent cell-centeredvalues,

(v + ii)i:H/2 = ~ [(v + I). + (v + ii).:i:Il, 29)

and AXi+I/2 and Ari_l /2 are distances between cellcenters.

Th . J b' ,2) d 'M3) Ie approXimate aco lans M an are 0 )-tained by freezing the diffusion coefficients.

. . . ,{2) 1 + C62 _ 1Mi,i_1 = + - - - V + V ) i _ l / 2 A

CT ~ x . ~ r _ l / 2

: - :{2) 1 + Cb2 1M, = - - - v + ii)i-I/2 ~

' , ' CT Xi x . _ I / ~ 30)1 + Cb _ 1

- - - - - (v + 11 ) . + 1 / 2 - - : - ' - -CT ~ . r ~ X + 1 / 2

{ 2 ) 1 + Cb2 1M i .i+ 1 = + - - - v + i ) '+I /2 ' , , - . -

CT . . . r ...... rHln

,3 ) cb2 IM ; I : : - - V + i i ) i .t t f ~ Z i ~ J t - 1 / 2

...,{.3,l C6 1Mi,. = + ::'(v+ i i ) i - - ' : - -

CT A X i ~ . r i _ l / (31)Cb2

+ - v + ii)i ,CT AXiA.r.+I/2

{ 3 ) C62 1Mi,i+l = - - v + ii)i .

CT Ax.Axi+l/2

These operators satisfy constraint (22a), since

[M - M ] ( 2 ) v = 0, [ i 71 (3 )M M J v : : O , 32): - : { 2 ) : - : { 3 )

When negated, the sum M + M forms an \I-typt

matrix. This can be shown by considering individualmatrix elements. For example, the sum of the matrixelements above the diagonal becomes.

Kf '2 ) + Kr 3 ) _ *',i+1 ' ,HI - C T ~ X i A . r + 1 / 2

[ 1+ Cb2)(V + ii),+1/2 - Cb2( II + ii .] 3:3)


13/23


14/23

With the _ r e e term6 taken ""ether. only two posit h'ity coutramts are required .

1) - P' 0, (41a)

[1>-D]-lP'-P] = [1>-P}-[D-P] ~ O.(4Ib)Eumin J the four poaibJe s ip combinations for ~ DPJ and [D - P}. the.e poritivity coostraints are always.w.&ed by (40). Furthermore. aaalytic caoeellation ofproductioD and delltructioD is relec&ed in (40). resultinsin nperior O D ~ p D C ecompared to (38).

We evaluat e the Jacobians P ' ar.d D' analytically usinS the chain rule. The differentiation is s tmpt forward. except for the wall delltructioo term I . . Sinceevaluation and difl'erentiatioo of I involves terr.lS liker - s and r12 care must be taken for r small or larse.lest ODe'. computer besina to complain. Tbe form of(6) ciws acceptable numerical behavior for r - O. Forlarse r, the functiOD ItIl asymptotes to [1 + c 31 1/t1 , so

eome cutoff' (say r = 10) . the derh-ative (I ,)' is8et to aero.

Approximate FI&Wiation

With the implicit and explicit operators properly defined. the backward Euler solution procedure (18) suarantees a positive turbulence field at each update . Itdoes. h o w e ~ e r .require expensive inversion of a larsesparse matrix in 2-D or 3-D. Approrimate factorization of the implicit operator will reduce the cost of anupdate, but unconditioaal positivity is lost . To see thisetrect consider a 2-D flow with zero IOUrce terms. Theuusplit implicit system is,

[1- ~ t (R( + R.,)] y = t (M( + M,,) y". (42)where M ( and M are finite ditrerences in tbe ( and"coordiDate directions , respectively , and U( and U"satis 'y tbe positivity coutrainu (21a) and (22a). I f heimplicit operator is approximately factorized. tben thesystem becomes.

[1 ~ t U ( ](1 - ~ t U , , )~ y = t ( M (+ M.,)y".43)

R.earraqing this system to i80Iate y .. l pves.

( 1 - ~ t U d( 1 - ~ t U . , ] y + 1= 1 + ~ t M - U dt ( M . ,- U.,) + t 2 U ( U I j )v . (44)

The splittins error results in the term ~ t 2 J J ' ( t ; J . , y ,which may DOt be positive. For small enou,h ~ t theapliuinS error wiD DOt ruin positivity, but the constraintequatioo on ~ t is a nonline"r matrix equation in it&elfand is quite difficult to solve.

We conclude that there appearll to be DO approximatefactorization of(18) that retainll t unconditional positivity of the orisinal unllplit ,. . d . We propose u s i n ~

13

subiterations of an app roximate factorization scheme ateach time step .-hich will r ~ c o v e rthe update of theoriginal unsplit I ~ ' s t e mat con\'ersenee (of the subiterat ion process). We use th'e lub ite rat ion proces of

Steinthorsson .: Shih [22]. where the splitt n,; error issimply Iaged one subiteraticlD. The objeeth'e of theapproximate factorization is then to minimize splittin gerror for rapid conversence. alld to minimize the number of .ubiterations where th. candid ate updat ed fieldcontains neptive points .

In 2-D we have tried conventional ADI .-ith aD additional factor due to the SOUtel terms ,

[1 - ~ t U d[1 - ~ t t ; J J[1 - ~ t ( F-1i)J.t\ ,"= t R Y .(45)

where R is the residual or risht-band-side operator in(18) . We have aI.o tried the approximate factorizationdeveloped by Shih .: Ch} 'u [23J for finite-rate chemistry.

[N -~ t U d

N-I [N - .1tU ] = ..1tR," , (46a)N : : 1 ~ t ( p '-1). 46b)

As disc.-ed in {23]. this second form bas Iml'er splittint:error and con"erses much faster to the unsplit updatethan conventional ADI. H o w e ~ ' e r .even this second formrequires several subiterations at start up to achie"e anon-nesative upd ated turbulence field: for example . approximately 30 subiterations are needed for a time stepof t = 10 (baaed on chord and freestream density andpreaure) to eliminate all nepth 'e updated points.

\ \ e use an approximate factorization that outperforn be '.ll (45) and (46) . Prior to splittin,;. we d i ~ i d etJtrousb by the diqonal mment of tbe implicit operato r . then use a conventional approxi mate factorization.

Denotins the diqonal elements of the implicita d , ' ~ -

tion/ditrusion operators J and ti,, ', the scbemebecomes.

[ ~ t t r ( J[1 - ~ t t r ~ ]~ y = t R v n , (4i)where

4E-)

We incorporate this approximate faetorizatiollICherne into a aubileration proce5ti in the fashion of

SteinlhorMon :: Shih (22]. .-here the splitting error. t \ t 2 U ( U , , ~ yis law< one subiteration.

[ ..1ttr ] .1v(t) = t R v n- ~ t n ((..1,.- _ . . 1 ~ . ) t _l .[ ..1tU.,J..1y(tl = v i


15/23

The second .tep equation bas been used to rewrite the o.ooaeplittiq error 011 the ri&ht-band side in the first ,tep;~ v -is a coovenieat definition . 0.0021

In practice this 'plittin& perfOl'111& quite well. Usin& 0.11024

a coutant t throupout the field, we have yet to en- 00022COUDter a lituatioa where a oeptive turbulence update .ocean eYeD on the first subitentioo . This iDcludes tests 0.0031wltere t . . . . varied over 1e1r2l"aI orden between 0.001aDd 1000 0.001'. }

From numerical experimentation. we chooee ~ t = 0.001510 aDd Mop IUbiteratin& wbeo the normalized cban&e1f4V/ 1f+ I b is reduced below O.O . Typically. 10 to 0.001420 .biteratioae are aeeded for initialilatioa transients 0 .0012aDd approximately 30 are required wbeo tile traDllitiOlltrip fuaction "waRs up". Within a few time steps only 0.0010one eubit.eration ill Deeded. 0.0001

Tbis procedure d dividin, tbroup by the diap-aaI elements at.o improYei the scheme', reaistance to 0.000 11OUIld-oft"error in the inviKid rqion where v s ,maD .

i

~

~

2

II

IiII

o ~ j1

~ \ I o- Model i~ I

~ II- - - - l

\ if\,

Ii

0 'I

3z

The eoIutioa module wiD still function properly if the Fisu re 7: Ski-friction coefficient in Samuel -Joubert flow ,r r - t r eam Voo or the initial suees for ii is eet to zero. based 011 Uooo z in meters.In c:omparison. round-oft" errors may cause De&ative up-dates, even at cOllver,ence of tbe ,ubiteratiOlll, if either 0 .024(45) or (46) is used . 0 ;

The model shows promisin, conver,ence to steady-8ta&e wbeo coupled to the solver for the velocity field.Cooversence is typically as ,000 or hetter than tbatfor the Baldwin-Lomax aJ&ebraic model. This may bedue to the ah.ence or "blinkin," pbenomenoo in thepresent model . In aJ&ebraic models , conver,ence tolteady-tltate may be adversely affected by dilcontinu l U I behavior. For example, the poIitioo _here the eddyviKosity switches hetween inner and outer formulationsmay _aDder back and fortb between adjacent cells. slowin, cODver,ence or causin, a limit cycle.

Results

Boundary-Lug Calculatjons

Only iDcompreaible boundary layers have been conIidered. With zero prel5ure padient, the model obeysthe accepted ReyuoId . number Kalin 1 0 tbe resultsIhowD at R. = 104 enlUre apeement with the curmat tIaeories. The model &ives satisfactory results inattadaed boundary layers with preeaure padients, typical of tille Stanford lQ68 cues. We only present reault.for the .mk low aDd the Samuel-Joubert low [24] as

tile other CUllS with moderate &r&dients Ihow the sametread. Darmofal's code was used asain.

In the sink flow. with an acceleration parameterK v/U;, dUoo/dz) equal to 1.5 x 10- ' , we obtainC = .00535, H = 1.35, and R. = 60. ~ resultsare weD within the experimental raDse , which is about

14

0.022

0 .020

o o ~ .

6"0 .016

0 .014

0 .012

0 .010

,O.ClOl0.005

0.004

0.002

I

III

I

j . . - -I

....r

2

I

7/

Mode /_ V E ' ~

/, ~ 7 /

Y / iV ,

I

Fisu re 8: Momentum and displacement thicknesses inSamuel-Joubert flow. in meters.

[0 .0050.0.0057] for C [ .35.1 .42J for H. and [700 . S00}for R.. The eddy-viscoaity profile is atypical . Becauseof the lack or entrainment in the sink flo" ., it does notshow a front at the edse of the boundary layer. Ins tead.it extends iDto the freestream re,iOll . This did not d i ~ -turb tbe veloc:ity profile. which is s a t i s f a c t o r ~ bot h illterms or thicknea and shape.

In the Samuel-Joubert flo" the .sre ement i5 r a t h~ Soad for the skiD friction, fi g . 7, and th e t h i c l m e s ~~ .


16/23

1.65

1.10 I 0iI

I1 515 I

1.80~ /

Exp ' -4: : ~ ~ Ic OJ1.45

1.400 c V

i - - - . /1.35

2 3z

Figure 9 : Shape factor in Samuel-Joubert flow .

Fig. 8. The model produces slightly higher skin friction, but lower thicknesses . The shape factors (Fig. 9)are in disa&reement even before the pressure gradient isapplied, and are the root of the disagreement in thicknesses. Wit h an adverse gradient, t he skin-Criction termloses its authority in the momentum equation. The experimental values for z between 1 and 2m, ~ 1.39,are surprisingly high considering the weak pressure gradient and the Reynolds number , R ~ 6500 . Interestingly, the calculated shape factor is catching up withthe experimental one Cor z > 3m .

Figure 10 shows the velocity at z = 3.4m . The position o the boundary-layer edge is good , but the computed profile is Cuner than the experimental profile . Theshear-stress profiles in Fig. 11 show good agreemen t forthe outer values, but the near-wan agreement may bepoor enough to partly explain the differences in the ve-locity profiles. Stress disagreements can of course becompounded by the convection and pressure terms. Curiously Dr. F. Menter, who was kind enough to test themodel in his Navier-Stokes code [25], obtained similara&reement Cor the stress profiles, but better agreementfor the velocity.

The Samuel-Joubert results suggest a mild but genuine tendency to underpredict the shape factor and

0 .1 2 ' ~ : - - - , - - - , - - , - - - , . - - - , - - - -0.1' J . . . -- -----;--+--+---.1--+---,..--

I i O . ' O r - + - - + - - 1 r - - - t - - + - - + - i- ~ ~0 .

08

i--+--+--+-+----if---+--.;... ~i It:0 .08 + - - - f - - - + - - I f - - + - - - ; - - - + - - i - - J - :.f1O 0 7 + - - + _ + _ - + - - t - ~ - - I - - + J S o L

V :0 . 0 6 + - _ + _ + _ - + - - t - ~ - _ 1 - - 4 -

Exp .... / ~O0 5 + - _ + _ + _ - + _ - t __ _ - - : ~ ~ ~ L _i ~ o d e l

O 0 4 + - - + - - + - - - 1 : . . - - - t - - + ' < - - - - o , . q : .

0.03+- _ + - - + _ + - - + . - . < - . . j , 4 7 _ :/ I0 2 + - _ + _ + _ - + - - ; ; . . q - r ~ - - 1 - - + - -

, , /0.01 t - - - t - - r - - : : l I F ; . , - & - - + - - r - - + - - f - - - - '~0.00 ~ _ . . . . _ ~ : . . . . J L - . J - - . . L - _ _ 1 _ - - . J - - -

0.2 0.. 0 .6 0 .11 ' .0U

Figure 10 : Velocity profile at z = 304m in SamuelJoubert flow. U normalized with edge \ elocity. y inmeters .

0 .'2

0."

0. '0

0.08

0.011

0.07

Y0060.05

0.04

0.03

0 02

0.01

II

"

0.000.0000

I

i

~".............

0.0004

iI

I

k-.........

-...... jl odeI

~,,

Exp . ~ ~ i7 )

~ .. ~ I- l - - t--- P i ,0.0008 0.00 '2 0.0016 0.0020

r

Figure 11: Shear-stress profile at r = 3.39m in SamuelJoubert flow . normalized with edge velocity. y inmeters.

thickneMeI in adverse pressure sradients . This may this type of 80w. Neverthl less, the behavior of the

make the model a litt le more resilient to separation than newm o d ~ l

is not disappointing and vindicates the no-it would ideally be . The tendency is not as strong as tion that a model calibrated or mixing layers , wakes.with the Cebeci-Smith, Baldwin-Lomax, and IN mod- and zero-gradient boundary layers has a gooa chance inels, but our comparison with experiment is not quit e adverse-gradent boundary layers, which are an interas Sood as that obtained by Menter \II-ith the Johnson- mediate sit'Jation . In addition , an improvement of theKing and Ir-w models [25J. Both models have been dt'


17/23

~ a v i e r - S t o k e sCalculations

We present three cues or the RAE 2822 airfoil; thefirst two have a sharp trailing edge but shock interactiooa or different strengths. and the tbird has a blunttrailing edge.

Navier-Stoke8 calculations for- Cues 6 and 10 OIl theRAE 2822 airfoil [26] were performed using the presentmodel, & I well u the Baldwin-Lomax and Johnson-Kingturbulence models. T he John80n-King results were obtained Uling the Navier-Stokes code or Swanson [27].Results for- each model were computed on 384 x 80 and768 )( 160 grids to minimize numerieal errors. Thesegrids were generated by the elliptic method of Wigton r23]. The medium and fine grids contain 257 and513 points on the airfoil. respectively.

AU calculations for Case 6 were performed at tbe sameconditions: M = 0.725. He = 6.5 X 106 a prescribedlift coefficient of C l = 0.743. and transition trips at3% chord. Results on the 768 x 160 grid are shown inFigs. 12 and 13. AU models converged solidly on bothgrids.

Figure 12 sbows a comparison of surface pretllluresfor- Cue 6 obtained with the three turbulence modelsand experiment [26]. The shock for- tbe present modelis about 1.5% chord farther forward than in the experiment or- as predicted with the Baldwin-Lomax andJohnson-King models. but well within the a catter of various models as reported at the Viscous Transonic AirfoilWor-kshop (VTAW) [6]. The new model is farther fromthe experiment near tbe leading edge on the upper surface. but cloeer to the experiment near the trailing edge

on the lower surface. All the calculations reveal smallpressure glitches near the trailing edge. With the newmodel on the fine grid. the angle of attack was 2.37.the drag coefficient 0.0121. and the moment coefficient-0.091, compared with 2 . 9 ~ .0.0127. and -0.095 in theexperiment, re8pectively.

'1.4

1.2I x p e r ~p t e s e n t ~ ~ I -L..-

- - - . a.tdwi".lomu, - . . .IcIhNon-KJngI

I

i.0

-0.1

\\;\,

~ iI

l.r' ~ I

-0.6-CI

-0.4

V ~ N ;~ I,-0.2

0.0

0.2

0.40.0

/

0.2

~

0.4 0.6

z/c

N .'" ~ .:J,I

0.8 1.0

Figure 12: Pressure distribution for Case 6, RAE 2822airfoil.

0.007 . . . . - . . . . --- .-- . .- . . . . -- . . . . ---- r-------_

pteNnt model- - - lIaIdwin-L . . . . ._ . - . ..Ic>hrcon-K"'I

0 .005 , . : ~ _ I : : ~ _ _ _ - ' - -

T I ~ j ~ ~ _ + - - - ~ ~ ~ ~ - - - ~ ~ ~ - ~ - ~ ~ ~ - - + _ ~ ~ ~ ~0.004 I

I003 + - - - - < ~ _ + - - + - - + - - - + * _ I - - _ + - - + - - ~ _C,

Figure 13 shows the upper-surface skin-friction coefficient for the same case. Between the three models thereis a difference or up to 10% upstream of the shock. andsimilar but reversed differences downstream of it . Thenew model d0e8 not predict wall-shear reversal. Theother two predict reversal at the foot of the shock, butonly on the finest grid (768 x 160). Reversal was neverpredieted at the VTAW, where the finest grid used was369 x 65 . It appears that with current codes the simple

qUe8tion of whether reversal occurs is not firmly answered even at grid resolutions that are considered veryfine. At the shock. the new model produces a largerstep up for the boundary-layer thicknesses than withBaldwin-Lomax, resulting in a mor-e f o r ~ a r dposition.Intere8tingly, the size or the displacement effect is notcorrelated with the occurrence of reversal at the wall.

- O . O O , - - ~ - 4 - - L - - - - L - - - < ~ ~ - - ~ - -

16

0.0 0.2 0.4 0 .6 0.8 ' .0z/c

Figure 13: Skin-friction coefficient on upper surface forCase 6, RAE 2822 airfoil. C, based on ;at:.

Flow conditions for the RAE 2822 Case 10 are M =0.75, e = 6.5 X 106 , a prescribed lift coefficient ofCl = 0.743, and trips at 3% chord . The new moddobtained a 8teady solution on the 384 x 80 grid. butproduced a limit cycle on the finer 768 x 160 grid . Allthe cyclic variation was in the separation bubble. Thenew model also produced a limit cycle on the 384 x 8(1grid when the artificial dissipation in the ~ a \ i e r - S t o k e ssolver ~ a s cut in half . Baldwin-Lomax and Johnson -

.


18/23

(

,

-'.4

f '1.2-,.0

+- j.

- : ; - --aa -aa

-C.4 4

-o.z

0.0

G.2

J

.rJV

~,J

I.4

0.0 0.2

.--1' ~~ ~ ; .

- - - - 8 e l d w i ~ L o m u

~ \ - ' - ~ K i n g. \\

,I\

I~ '

\1

\ ~~

V ~ .~ .\ ~

, 'I

~. .

0.4 0.6 0.1 ' .0rIc

F i p r ~14: Pressure distribution for Case 10, RAE 2822airfoiL

0.007

I p r w e n t ~- - - ' 8 e I d w i ~ .~

- - ~ K i n g

f'. , ~ ~ I,I .'.,: - -, I1 ---

1 I' - . .- \ :

\ I

O.CIIII

0.005

0.004

. \~ ,\ ,. I

\\. .- \\

I ,. ,

-

~ \

0.002

0.001

0.000

' .'-0.00'0.0 0.2 0 .4 0.6 0.1 ' .0

rIc

Fipre 15: Skin-friction coeflkient on upper surface forCu e 10. RAE 2822 airfoil. C, based on U .

K i n ~produced steady solutions on both gids. Resultsfor Cue 10 a r ~preaented in F ip . 14 and 15 for aU threet u r b ~ n c emodels on the 384 x 80 grid .

As uaual , Case 10 produces larger differences thanCue 6. With our policy of a t c h i n ~the Mach numberand lift coefficient the new model ~ i v e sa better answerthan Baldwin-Lomax, and slightly bett er than JohnsonKing (see Fig 14). This is in terms oCahock locationand pressures near the leading edge . upper surface, and

I i

trailing edge . lower surface . All the mod els fail to agre ewith experiment for r ic> 0.7. upper surface . The newmodel predicts a flattening pressure for r ic> 0 .9. upper surface, in qualitative disagreement .....ith the experiment. The Johnson-King model has shown the &ametrend, but not as strongly and only in Coakley's implementation (6). The Baldwin-Barth model also producesthe flattening (8).

This behavior of the C. is c o r r ~ l a t e dwith that ofthe akin-friction coefficient, Fig . 15. Downstream ofthe shock, the experiment and our Johnson-King reIUlta ahow the akin friction returning to .trong positive values. The flow reat taches firmly. The BaldwinLomax results show weakly positive skin friction overa short stretch. With the new model t h ~skin frictiongazes zero before again taking moderate negati\'e \'alues. There is no reattachment, and the flattening preslUre dist ribut ion reflects it. Note that a similar behavior has been observed with algebraic models: if th eskin friction approaches zero smoothly enough the VanDriest damping can, erroneously, shut dO 'n t h ~eddyviscosity across the whole layer. This is not what ishappening here .

The limit cycle behavior of the new model on the finegrid shows an oscillation between slightly negative andslightly positive sk in friction near ric =0.8.

The apparent failure to reattach is not a favorableresult, and this behavior 'as also observed in anotherseparated case (B. Paul , personal communication) . TheIOlutions show a very 10' eddy \'iscosity near the wall.A run with a more diffusive model (i.e lower u thatcould have helped the eddy viscosity diffuse towards thewall failed t o show much difference. However we do notbelieve there is a structural reason for the failure to reattach, baaed on two tests . The first is a cakulation ofreattachment on our boundary -layer code , modified toa time-developing code; a layer of reversed ftov. near thewall was eliminated as could be expected . The secondis an unpublished calculation by Menter of a. separationbubble; again the skin friction returned to positive lues without hesitation. Numerical errors in our currentNavier-StolEes/turbulence-model solver may be playinga role.

For Case 10. with the ne ,' model on the 384 x 80gid, the angle of attack was 2.52. the drag coefficient0.0238, and the moment coefficient 0 .104 . comparedwith 3.19 , 0.0242, and -0.106 in the experiment , respectively.

The blunt-trailing edge airfoil is RAE 2822, truncatedat 94% ofthe original chord (hue height 1 1 4 ~chord) .Our objective is to explore the behavior of the modeland ofthe numerics at corner-induced separation. withhigh-lift applications in mind . Accordingly we c h o s~Case 1 (29) which is 8ubcritical : of = 0.6i6, Re =5.4 x 106 , and lift coefficient CI = .451 . Calculat iom


19/23

:IlUminIU dg

0.6

0.5

I ,,

V :.. ..V

)0.4

0.3

/.2 0.1 II.O I L - _ - - - - - - - - - - - - - - - -

1 i.0i0Y

IiI

. . .. . - - - - . . .. :- - - - - - : , 02::---xIC--:-' -.,:-----:--.., . . -D.1

Figure 16: Velocity contours near the trailing edge forCase 1. blunt RAE 2822 airfoil. Levels: -0.007 (small

1.00 1.02 1.0


20/23

O l f = = = : ~ ~ = = ~ ~ ~ g : : ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~eddy visc06ity blends its boundary-layer behavior (as: in fig. 4) into its wake behavior (a bell-shaped distri-bution). There is little basis to judge its level in therecirculation region.

. 1L--------:=====

Finall)' Fig. 20 shows the pressure distribution . Theagreement is good, especially considering the 0.415' difference in angle ofattack, except that the calculated C pis shifted down. This was also observed by Drela for hisnumerical method (31). We also ran Case 4 to measurethe Reynold&-number effect. In the experiment, Case 4had the same angle of attack as Case 1, but a Reynoldsnumber of only 1.9 x 106 , resulting in a 1066 of 0.036 inC/ We ran Case 4 with imposed C = 0.415, and theangle of attack settled at 1.94 0 The lift-curve slope isabout 0.155 per degree . This implies that the C/ I06Sdue to the lower Reynolds number is about 0.022. Thisis only about 60% of the lift I06S observed in the exper-iment

....

~ : ~ ~ = = = = ~ : :1 .00 1.0 1.at XI l.OJ 1.04 Summary of the results

Figure 19: Eddy-viscosity contours near the trailingedge for Case 1, blunt RAE 2822 airfoil.

-1.0

,-O.s -0.11

o . ~

p

-0.2

0.0

0.2 Ij

0.40.0

ElOP

- . - o.t. .:

Y

1I

0.2

I,

: If\ j I

x I

lY \

", ~ x l . J0 .4 0.11 0.' 1.0

ri

Figure 20: Pressure distribution for Case 1, blunt RAE2822 airfoil.

saddle, isolating the two nodes . There is no reason forsuch a pattern to occur in the absence of symmetry, and

it is not stable. The saddle and the lower node could alsoeliminate each other under other conditions . Streamlinepatterns can overstate the importance of regions with10\\' velocity and little dynamical significance.

Figure 19 shows eddy-viscosity contours . Continuity between the grid blocks, "'hich is hard to achievewith algebraic models, is of course observed. The

19

We have exercised the model outside its domain ofcalibration and with the full Na"ier-Stokes equations .instead of just the boundary-layer equations. Its compatibility with unstructured grids has yet to be used.but there is no need to demonstrate it. In a few caseswith shock-induced separation the new model yieldeda limit cycle , with a pulsation of the bubble . when thealgebraic models yielded steady solutions. Since timeaccurate solutions Ilfe very expensive, steady solutionsmay be greeted as successes whether they are ~ -cally correct or not . Outright divergence ofthe iterativeprocess has never occurred, and the model seems to bereasonably "friendly" to the relaxation process, without

any attention being paid to the initial condition . Thus.the model appears robust enough to be implemented byindependent users, in a variety of codes and physicalsituations, and it should be particularly attractive tounstructured-grid \lIers.

The accuracy 1 0 rar is consistent with our expectations . The model's response to grad ual or steep pressuregradients, and to the removal of the blocking effect ofthe wall, is encouraging . The post-shnck reattachmentin an adverse pressure gradient has proven to be difficultfor the model. Menter also reported somewhat disappointing results over a backward-facing step. traced toan excessively-rapid build-up of the shear stress (perIOnal communication) . This weakness also afHicts thek- model, and may respond to a modification of the

/", function. The quality of our results with the blunttrailing edge indicate that this problem cannot be verysevere. It appears that the calibration cases are indeed representative enough of the flows of interest toensure decent performance in non-trivial situations, andto warrant extensive applications and tests ofthe modelin its present form .


21/23

Outlook

The development of the model will nevertheless continue. It will praerve the b sic favorable features of

the model: ainpe trauport equation, local formulation,moderate resolution requirements, good numericalltability, iD8eDlitivity to the freestream value, ready control of trauition. The invariance principles wiII allO beupheld. Additional tests will be made, notably in threedimelllioal. No additional difficulties are anticipated.Tests in more Itronpy stimulated Oows such a massiveIeparatioD, wakes in pressure gradients, or free vortices,are likely to reveal w e a k _ in the model. They willa o aue the current Navier-Stokea codes, in two respects. The first .. the detai1accuracy, for instance thecooaervatioo of momentum in boundary layers, whichis far from perfect in our experience. This problem haabeen ot.tructed by the difficulty in computing the thicknellel6 and IJ in Navier-Stokes codes, but it must nowbe

addreased,i f

only on a flat plate at low Mach number.The other aspect is the question of steady solutions

in Oows with medium- or large-scale separation. Anclassic example is the flow past a circular cylinder; apressing industrial example is the Oow past a staDedairfoil, near its maximum lift coefficient C mazo. Exceptat very low Reynolds number the Oow is unsteady andthree-dimensional. On the other hand, provided the ge-ometry is time-independent, it is legitimate to define itstime-average and to hope for a code that would compute it as a ateady solution. t is Iso legitimate to re-quest the low-frequency component (i.e., with Strouhalnumber of order 1) of the solution, particularly the Oue-tuating loads for Itructural purposes . The ideal turbulence would include a Iwitch between these two options.

We do not know, in general, whether the exact solution of the Navier-Stokes/turbulence-model system inits pretent form is steady. In the design of some models,decisions have been made solely on the basis of a preference for the candidate that yielded s teady solutions. Weare far from having the capability of routinely performing time-accurate calculations to explore the issue. Allthe routine calculations aim at Iteady solutions. Failureto converge, nd the generation of a limit cycle, m. bea lign that the exact solution is unsteady. t may allobe just a numerical problem.

Enviaioo a ateady solution, say past a cylinder. AnaJebraic modelluch as, for instance, Baldwin-Lomaxwould be UIed out of ts range, but this does not stop eager UIe t I . In the recirculating re&ion it would yield eddy

viscosities on the order of U R with U the freestreamvelocity and R the cylinder radius. We presume herethat the Baldwin-Lomax m n condition would occurin the wake, and not near the wall . A balance between production, on one side, and diffusion and destruction, on the other, suggests that the new modelalso could produce a IOlution with an eddy vilCOIiity

20

of order Uoo in the recirculation region. A Reynoldsnumber based on tbat viscosity would be finite, whichshows that there is no immediate lCaJing argument thatcan predict whether a steady solution can be stable.

This question must await extensive tests. The pressureto produce time-accurate Navier-Stokes codes will onlyincrease, even i f he steady-st ate codes have not reachedperfection.

Even when unsteady solutions of the modeled equations are obtained, their meaning will need scrutiny.Many of the properties of Reynolds-averaging hold approximately i f here is a separation of scales (i.e., a spectral gap) hetween the resolved motions and those thatare left to the turbulence model. We have little evidencein that domain. An example of legitimate decomposition would be a mixing layer that wanders on a timescale much longer that its internal time scale which, after modeling, is 6 AU (with 6 the thickness, and A ,the velocity difference). Note that what we envision

here is different from Large-Eddy Simulation . As thegrid is refined the model does not change, the way itdoes in LES (through a narrowing of the filter, so thatthe limit is direct simulation) . The difficulty of LES isthe filtering without spectr al gap. Instead, v e convergeto a smooth solution of the modeled equations .

Some of the near-future directions have been hintedat above. There is the choice of S between the vorticity, the strain rate, or another scalar norm of the deformation tensor. There is the use of an approximationof the turbulent kinetic energy Ii to give the Reynoldsstress tensor a plausible trace (or set of eigen\lllues).There is a modification of the / , function in the regionr 1, that would alter the results only on adverse preslure gradients. The thi rd modification could be assessed

in the Samuel-Joubert or airfoil flows, but obvious testcases for the first two are not at hand.

The current model has no c o m p r e s s i i l i t ~ terms. Empirical te rms baaed on the turbulent Mach number, suchas the one in the Secundov model, or on the quantityVp.V;; are available. The former may be calibratedin supersonic mixing layers. The la tte r may assist t hemodel in shock/boundary-layer interactions if a consistent trend is found that shows an erroneous shock lo-cation, and we can extricate that trend from numer i-cal concerns (e.g., artificial dissipation) and from theendIeaa corrections of tranlOnic airfoil testing. In thatrange of density variations it may be enough to writethe transport equation in terms of p instead of ii. taking advantage of the flexibility in placing p inside or

outside various derivatives. This is consistent ,ith thespirit in which the model was devised. To tJ .:s date.efforts to devise a curvature term with the required invariance pr(\perties and no d.dependence have failed.Note that curvature effects have been obsef\ed in freeshear flows. Three-dimensional effects in boundary layers (Le., pressure-gradient vector at an angle to the ve


22/23

locity vector) are also delicate to introduce even empirically without violating the i n ~ - a r i a n c eprinciples . Noplana have been made to depart (rom a acalar eddy visc05ity.

Acknowled,ements

The initial work was performed by the first author while an employee of NASA Ames Reeearch Cente r . Dr. L. Wilton reviewed the manuscript and ranl O m e 0 the cues. Mr. W. F. NoU8S reviewed themaDUKript. Ms. W. Wilkinson provided some of therue.. We have benefited from comments by Drs. Baldwin. Barth, Birch, Bradabaw, Jou, Menter. Speziale,and Vandromme . Theae comments do not constituteendonemeata.

Appendix; Summary of the Model. Version IWe solve the Reynolds-averaged Navier-Stokes equa

tiODS and a tr ansport equat ion (or the turbulence model.The ReyncJds atreues are given by -1ijl i j = I Sijwhere Sij lJUi/lJZ j + JUj/8z i )/2. The eddy visc05ity v. is pven by

X3 ii"1 = i 1,,1. ~ = 3 3 X: : - . (AI)

X +C.,1 If

v is the molecular visc05ity. ii is the working variableand obeys the transport equation

Dj ; _ 1 [ 2]Di = n [ I - I . ] S 11+; V. If + i)Vii) + cn Vii)

The trip function III is as follows: d, is the distan cefrom the field point to the trip . which is on a wall. ,,:, isthe wall vorticity at the trip, and ~ is the differencebetween the velocity at the field point and that at thetrip. Then" m i n ( O . I . ~ U / I o I I ~ Z )where ~ . r is th.,grid spaein, along the wall at the trip. and

III = n " exp ( -C ,2 :J [d2 + ~ d ; ]. A6)The constants are Cu = 0.1355, tf =2/3, cn = 0.622,

It = 0.41. C",l = C t t l t 2 + (1 + cn)/tf. c .2 = 0.3,Ctl/3 =2, Ct.l = 7.1, Cn = , C , = . C'l = 1.1, Cf4 = 2.Turbulent heat tranlfer obeys .a turbulent Prandtl numher (not to be confused witb tf) equal to 0.9.

References

[1] Baldwin, B . S,' : Lomax, H ., "Thin layer approximation and algebraic model for separated turbulentflows". AIAA-7S-257.

[2] Cebeci, T ., : Smitb, A. M. O ... A finite-differencemethod for calculating compressible laminarand turbulent boundary l a y e r s ~ ,Journal of Bas icEngineering, Vol. 92, No.3, pp. 523-535.

[3] Jobnson, D . A., : King. L. S., A mathematically simple turbulence closure model for attached and separated turbulent boundary l a y e r 5 ~ ,AIAA JoUrnal. Vol. 23, No. 11. 1985, pp. 1684-1692.

[4] Mavriplis, D. J "Algebraic Turbulence Modeling- [C.,tltI/ - : : 112] [ ~ r+ In ~ U 2 A2) for Unstructured and Adaptive Meshes", AIAA-90-

1653.Here S is the magnitude of the vorticity,

[5] Launder. B. E : Tselepidakis , D. P., "Direction- iiS ==S + 242 1 2, A3) in eecond-moment modelling of near-wall turbu

lence" AIAA-91-0219 .and d is the distance to the closest wall.

The function I., is

[1+ I ] 1/ ' -I.,=.? )3 , ,=,,+c w, ( , ,6_,,) . ,,:..... --.+ tl/3 S,,2d2

A4)For wse " I . reaches a cooatant, so lar,e values of "can be truncated to 10 or so .

The wall boundary conditionis

if =o.

In thefreestream 0 is beat. provided numerical errors do notpush ;; to negative values near the edge of the boundary layer (the exact solution can't go negative) . Valuesbelow ,,/10 will be acceptable. The Ame applies to theinitial condition . The /12 function is

1,2 =C 3 exp(-Cr4 :.\'). A5)

21

[6] Holst, T. L., "Viscous transonic airfoil workshop .Compendium of results". AIAA-S7-1460 .

[7] Viegas. J. R., Rubesin, M. W., : Horstman, C.C ., "On tbe use of wall functions as boundary conditions for t w ~ d i m e n s i o n lseparated compressibleROWIS", AIAA-85-OISO.

[S] Baldwin, B . S., : Bartb. T. J A one-equationturbulence traoaport model for high Reynolds numher wall-bounded flows". AJAA-9I-061O.

[9] Bradshaw. P Ferriss . D. H., : Atwell . N. P ."Calculation of boundary-layer de\'e1opment using the turbulent energy equation", J . fluid ~ t c h Vol. 2S , 3, 1967, pp . 593-616.


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[10] Nee, V. W., : Kovuznay, L. S. G., Simple phenomenological theory of turbulent ahear flows ,Physics of Fluids, Vol, 12, No.3 1969, pp. 473-484.

[11] Secundov, Smirnova, Koclov, : Gulyaev, Oneequation eddy vileosity model (modified L, S. J.Kovumay model) . Short aummary of the equatioaa. Personal communication, 1990.

[12] MitcJaeltree, R. A., Salas, M. D., : Husan H. A.,One-equation turbulence model for transonic air-

foil lows , AlAA Journal, Vol. 28. No.9 1990, pp.1625-1632.

[13J Bradshaw, P., Launder, B. E., : Lumley, J.L., Collaborative teating of turbulence models ,AlAA-91-0215.

[14] Wilcox, D. C., Reassessment of the scaledetermining equation for advanced turbulencemodels , AIAA Journal, Vol. 26, No. 11, 1988, pp.1299-1310.

[15] Spataft, P. R., Direct simulation of a turbulentboundary layer up to R = 410 , J. Fluid Mech .Vol. 187, 1988, pp. 61-98.

[24] Samuel, A. E., : Joubert, P. N. A b o u n d a r ~ l a y e rdeveloping in an increasingly ad\'erse pressure gradient , J. Fluid Mech., Vol. 66, 3, 19i4, pp. 4$1-505.

[25] Menter, F. R., Performance of popular turbulencemodels for attached and separated adverse pressuregradient flows , AIAA-91-1784.

[26] Cook, P. H., McDonald, M. A., : Firmin, M.C. P., Aerofoil RAE 2822 - Pressure distributions, and boundary layer and wake measurements , AGARD-AR-138, 1979.

[27] Swanson, R. C., : Turkel. E., Artificial Dissipation and Central Difference Schemes for the Euler and Navier-Stokes Equations , AIAA-Si-llOi,1987.

[28] Wigton, L. B., High Quality Grid Generation Us-ing Laplacian Sweeps , Fourth International Sym

posium on Computational Fluid Dynamics, C.Da\'is.Sep. 1991, pp. 1222-1228.

[29] Cook, P. H., : McDonald, M. A., Wind tun-nel measurements in the boundary layer and wakeof 6.n airfoil with a blunt base at high subsonicspeeds , RAE Tech. Rept. 84002. 1984.

[16] Baldwin, B. S., : Barth T. J. A one-equation [30J Henne, P. A., : Gregg, R. D., A new airfoil designturbulence transport model for high Reynolds num- concept , AIAA-89-2201-CPber waIl-bounded flows , NASA TM 102847, 1990.

[171 Townsend, A. A. The structure of turbulent shearflow. Cambridge University Press, New York, 1976.

[18] Schlichting, H. Boundary-layer theory. McGrawHill, New York, 1979.

[19] Hunt, J. C. R. Turbulence structur e in thermal convection and shear-free boundary layers .J. Fluid Mech., Vol. 138, 1984, pp. 161-184.

[2OJ Mellor, G. L., : Herring, H. J., Two methodsof calculating turbulent boundary layer behaviorhue on numerical solution of tbe equations ofmotion . Proc. Conf. Turb. Boundary Layer Pred.,Stanford, 1968.

[21] Martinelli, L., : Jameson, A.. Validation ofa Multigrid Method for the Reynolds AveragedEquations . AIAA-88-0414.

[22J Steintboruon E., : Shih, T. I.-P., A Methodof Reducing Approximate Factorization Errors inTwo- and Three-Factored Schemes , submitted toSIAM J. of Statistical and Scientific Computing.

[23] Shih, T. I.-P., : Chyu, W. J. Approximate Factorisa tion with Source Terms , AIAA Journal , Vol.29, No. 10, October 1991, pp. 1759-1760.

22

[31] Orela, M., Integral boundary layer formulation forblunt trailing edges , AIAA-89-2200.

..j

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