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Application of Two-Equation TurbulenceModels in Aircraft Design.G. Kalitzin�, A.R.B. GouldBritish Aerospace (Operations) Ltd, Sowerby Research Centre,Filton, Bristol, UK.J.J. BentonBritish Aerospace (Regional Aircraft) Ltd, AVRO International Aerospace Division,Woodford, Cheshire, UK.AbstractCFD methods have been used successfully for manyyears in aircraft design. The primary Navier-Stokesmethod in use at British Aerospace is an explicit, struc-tured, Multiblock system. One of the key targets in thedevelopment programme for this system is the implemen-tation of a Second Moment turbulence closure. Beforeproceeding to implement such a model it is essential tohave an e�cient and robust two-equation transport ca-pability. This paper describes how this capability hasbeen achieved. The selection and assessment of two-equation turbulence models for use within this systemis discussed. Two models are considered in detail, onefrom the k-" family of models, and one based on the k-!class. To provide stable and accurate solutions for com-plex three-dimensional ow�elds various modi�cationsare required to both types of model. Several 2D and 3Dcivil aircraft applications are used to validate the system.These also serve to demonstrate the use of transport tur-bulence modelling in the design environment.IntroductionFor many years British Aerospace has been developingCFD methods for use in the design of aerospace vehi-cles. One suite of programs which has emerged from thisdevelopment is based around a 3D, structured, Multi-block mesh generation system. This system is targetedat the modelling of external ows over complex geome-tries and is now being used for design applications re-lated to military aircraft, civil aircraft and high speedweapons. This forms part of a wider initiative to inte-grate CFD with CAD packages which exploits the factthat mesh re-generation for small geometry changes canbe automated once the initial topology and grid have�Permanent address: Otto-von-Guericke-Universit�at, Institutf�ur Str�omungstechnik und Thermodynamik, Universit�atsplatz 2,39106 Magdeburg, Germany.Copyright c 1996 British Aerospace plc. Published by theAmerican Institute of Aeronautics and Astronautics, Inc. withpermission.

been created. This is because the Multiblock topologydescription embodies a generic speci�cation of line-gridstretching so that small changes in geometry require nochange to the topology.The primary ow solver within the Multiblock systemis a 3D, Reynolds-averaged, compressible, cell-centred,central di�erencing method known as RANSMB whichhas been developed by the third author. This makes useof Runge-Kutta multigrid time marching and arti�cialviscosity to produce steady-state solutions. Both Eulerand full Navier-Stokes calculations can be made and sev-eral turbulence closures are available, from the algebraicmodels of Baldwin-Lomax and Johnson-King to one andtwo-equation transport models.Whilst the underlying solution method was �rst de-scribed 10-15 years ago1; 2 it provides an unparalleledcombination of e�ciency and maturity. Development ofthe solver continues in many areas. However, one of thepacing items in its development and wider application isturbulence modelling.The traditional algebraic models, including Baldwin-Lomax3, have been applied with a surprising degree ofsuccess to two and three dimensional ows. Althoughsuch models are cheap and robust their direct depen-dence on empirical mixing length assumptions severelylimit the complexity of the ows which can be adequatelypredicted. Transport models of turbulence would ap-pear to be more appropriate for modelling ows aroundmulti-component geometries in three-dimensions. Sec-ond Moment Closure is an advanced manifestation of thisclass of models. By providing a transport equation foreach Reynolds-stress component, SMC models have thepotential to provide predictions of complex ows whereanisotropic `non-linear' e�ects are important. The penal-ties that accompany such a capability include high cost,due to the large number of additional equations, andpotentially poor numerical properties which may leadto implementation di�culties. Whilst SMC remains thelong term goal for a generalised, ow modelling system,1

the technical challenges posed by such a model meanthat an interim capability is required.Two-equation models can be seen as an advance onalgebraic models, since they are transport in nature andtherefore can take account of history e�ects. On theother hand, by making use of the Boussinesq approx-imation there is no longer a need for a six-componentReynolds-stress closure and the number of equations isreduced. The present study discusses how a robust ande�cient two-equation solution method has been achievedand demonstrates the capabilities of the system using re-cent BAe AVRO research on a new regional jet.Solution MethodAs its name suggests, a two-equation model requires thesolution of two transport equations in addition to theNavier-Stokes equations. The turbulence equations aredi�erent from the mean ow equations in that they in-clude important source terms which can cause the systemto become numerically very sti�. The approach adoptedby BAe is to solve the turbulence equations explicitlyin the same manner as the mean ow. The system iskept stable through the use of relaxation on the multi-grid, limiters applied to certain quantities, and carefulapplication of the arti�cial viscosity. In RANSMB thedissipation takes the form of blended second and fourthdi�erences. For the mean ow the blending uses a shocksensor to switch on the second di�erence at shocks. Forthe turbulence equations the second di�erencing is in-voked at discontinuities via a non-linear second di�er-ence sensor on the turbulence quantities, and has a co-e�cient large enough to give a �rst order TVD prop-erty. An alternative method of solving such equationsin such an explicit scheme is given by Davidson4, whoproposes a semi-implicitmethod for the turbulence equa-tions, which is coupled in some way to the explicit mean ow solution. This method has proved successful, butdue to its implicit nature becomes complex to imple-ment, especially in a Multiblock environment.Choice of ModelIn order to install a transport turbulence model into anindustrial CFD code the developer faces a number ofissues which may determine whether or not a certainmodel is viable. In addition to the numerical sti�ness,there are further features of the turbulence equationswhich are unique to each new model. When assessing aparticular turbulence model for use in RANSMB thereare certain properties which are desirable in the contextof a structured, Multiblock method. These include:

1. No use of normal-to-wall distance. Most threedimensional calculations contain areas where twoor more boundary layers are interacting, such aswing-body junctions. It is di�cult, therefore, tolocate a suitable wall distance for cells in such aregion. For a Multiblock scheme it is also possiblethat the upper part of a boundary layer will beseparated from the solid surface by a block bound-ary. Locating the relevant surface in a generalisedMultiblock mesh is a complicated and expensiveprocess.2. Simple source terms. The turbulence modelsource terms should be functions of readily avail-able quantities. For example, models which makeuse of second derivatives are undesirable. Sourceterms which become extremely large very close towalls cause sti�ness problems.3. Straightforward boundary conditions. Theturbulence model variables should have boundaryconditions that can be readily applied by setting`halo'-cell values. The use of more complex con-ditions, such as forcing the dependent variables tomatch a function over the near-wall region is un-desirable since this imposes severe constraints onthe grid generation process.The degree to which a transport model can representkey physical phenomena is also very important. For at-tached ows, most traditional models perform very well.However, these models often fail to predict more com-plex ows containing features such as boundary layerseparation, either shock induced or pressure gradientinduced5; 6. Work is underway to analyse and imple-ment some of the numerous corrections and modi�ca-tions which have been proposed to overcome such limi-tations. The present study is concerned with the choiceof the basic model on which to proceed with such work.All two-equation models use the Boussinesq approx-imation to model the turbulent stresses:� �u0iu0j = �t �( @ui@xj + @uj@xi )� 23 @uk@xk �ij�� 23�k�ij ; (1)where the eddy viscosity is given by:�t = C��k2" : (2)The widely accepted transport equation for turbulentkinetic energy, k, is modelled as:@(�uik)@xi = @@xi �(�+ �t�k ) @k@xi �+ Pk � �"; (3)2

where Pk is the production of k, and " is the speci�cdissipation.To achieve closure of the system an expression for thedissipation, ", is required. A large number of closureshave been proposed over the last 20-30 years. Most ofthese give a transport equation for " itself, based on the`standard' form:@(�ui")@xi = @@xi �(� + �t�" ) @"@xi �+C"1 "kPk�C"2�"2k : (4)An alternative closure method is provided byWilcox7. This is written in terms of ! which representsthe ratio between " and k:! = "��k : (5)The transport equation for ! is given as:@(�ui!)@xi = @@xi �(� + �t�! ) @!@xi �+ �!k Pk � ��!2 : (6)Near-wall asymptotic analysis8; 9 shows that thesebasic forms of k-" and k-! models give incorrect pre-dictions of key turbulent quantities approaching a solidsurface. In the case of the k-" equations, it is necessaryto add damping terms or use wall functions to reproduceexperimentally observed behaviour. In the case of the k-! model it has been found that despite the near-wallinconsistencies, good predictions of boundary layer pro-�les and skin friction can be obtained. The suitability ofa large number of both the k-" and k-! type of modelshave been assessed for use within the BAe solver.To test the basic implementations, and to illustratethe following discussions, a at plate boundary layer testcase is used. This is calculated for a ow at M=0.5,Re=7 � 106, using a grid with cell distributions similarto those typically used for aerofoil or wing calculations.The cell nearest the wall is within y+ � 1 and there arearound 30 cells below y+=1000. Transition is applied atx=l = 0:1 and the pro�les are extracted at x=l = 0:9.Assessment of k-"As already mentioned, wall-e�ects are not su�cientlywell modelled by the standard k-" equations, and theseequations are only valid away from a surface, in highturbulent Reynolds number areas of the ow. The lowReynolds number region can be bridged with wall func-tions, which allow relatively coarse meshes to be used.However, no signi�cant increase in capability over thealgebraic models is obtained. An alternative is to use

a one-equation model in the near-wall region, which ismatched to the two-equation model at a certain distancefrom the wall. The most commonly used near-wall modelis that of Wolfshtein10, which solves Eqn.(3) in conjunc-tion with an algebraic expression for ".This approach has been shown to be very e�ective6.A major drawback in the context of the present studyis the need to �nd a suitable location for the transitionfrom the near-wall model to the two-equation model. Ina structured grid system it is usual to select a particularparallel-to-wall grid line which represents a y+ value ofaround 200. For three-dimensional ows with severalintersecting surfaces it becomes di�cult to locate a gridsurface which will be satisfactory.The most appropriate method would appear to be theuse of a single model which is modi�ed through the addi-tion of wall damping terms to allow its use in both highand low Reynolds number regions. A large number ofsuch models exist in the literature. Many of these havebeen tested including Jones-Launder11, Chien12, Lam-Bremhorst13, Lien-Leschziner14 and Speziale9. In prac-tice it has been found that all these models give verysimilar solutions for most ow problems. They do, how-ever, di�er substantially with regard to the ease withwhich they can be made to produce stable solutions.Implementation of k-"One key issue encountered when applying low Reynoldsnumber k-" models is the boundary condition for ".Whilst k tends to zero as the wall is approached, " doesnot vanish. The low Reynolds number models fall intotwo categories, those which apply a speci�c boundarycondition for ", and those which replace " with new vari-ables, such as ~" or � , which are de�ned such that theygo to zero at the surface.One of the most popular of this latter category, andone which has been in use at BAe for many years, is theChien k-~" model. It is worthwhile repeating the de�ni-tion of Chien's model here, to facilitate the descriptionof the implementation.The new dissipation is related to " by:~" = " � 2�ky2 ; (7)where y is the normal-to-wall distance.The new dissipation equation is:@(�ui~")@xi = @@xi �(�+ �t�" ) @~"@xi �+ C"1 ~"kPk� C"2� ~"2k f2 � 2�~"y2 e�C4y+ : (8)3

Here, f2 is a wall damping function,f2 = 1� 0:22e�(Ret=6)2 : (9)One additional damping function is used to factor theeddy viscosity in Eqn.(2),f� = 1� e�C3y+ : (10)Whilst the boundary conditions for ~" are now straight-forward the damping functions contain the wall-distance,y. In addition, the wall coordinate, y+, is used. Thede�nition:y+ = u� y=�, contains the friction velocity, u� ,which is itself a function of velocity gradient at the wall.At separation points the resultant y+ can be zero. Thisis undesirable in the context of the damping functions,and an alternative de�nition is used. This is derivedusing the assumption that Reynolds shear stress is pro-portional to kinetic energy and is written as15:y+ = C 14� �pky� : (11)Comparing this model with the three criteria listedabove it is clear that, although the boundary conditionsappear to be straightforward (k=0, ~"=0), the normal-to-wall distance measure, y, is used. For mesh blocks whichdo not have a wall boundary the damping functions arenot applicable, and the model reverts to the `standard'high Reynolds number " model, Eqn.(4). This is a validassumption, except immediately downstream of a trail-ing edge, where although there is no solid surface, theboundary layer(s) from the wing surface create a ow,which for a short distance, approximates a wall bounded ow. Without a special treatment for wake regions thesudden transition from ~" to " here can lead to conver-gence di�culties.Further, it can be seen that Eqn.(8) contains sourceterms which become very large in the near-wall regionas y ! 0. Small imbalances in these terms can lead tolarge production or destruction of ~", with undesirableconsequences for the convergence of the solution.To overcome the near-wall and wake problems a smalllayer of cells next to the surface is solved using the Wolf-shtein model. This approach resembles the method men-tioned above to bridge the gap between high Reynoldsnumber models and the wall. However, since the Chienmodel is low Reynolds number compatible, the choice ofgrid line for separating the two regions is not critical. InRANSMB four to eight cells next to a wall are used forthe one-equation model.The distance, y, is found by calculating the distancealong a normal grid line from the solid surface to a given

cell. This procedure is used for all blocks adjacent towalls. This requires that the grid be generated in sucha way that blocks adjacent to walls are su�ciently largeto contain the whole low Reynolds number region. Forsituations such as wing-body junctions where more thanone surface is present, y is set to the smaller of the twowall distances.0 1 2 3 4 5

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t/Figure 1: Log law comparison (left) and eddy viscos-ity pro�le for the at plate boundary layer using theChien(+Wolfshtein) model.Towards the outer edge of the boundary layer, anew problem is introduced by the ratio of k2=" whichis present in the eddy viscosity, Eqn.(2). As the bound-ary layer edge is approached, both k and " fall to zero.Any slight error in the evaluation of these quantities maycause the ratio to become signi�cant. This e�ect cancause large values of eddy viscosity to be predicted, eventhough the turbulent quantities themselves are small.To control the production of `spurious' eddy viscosityoutside the boundary layer a minimum limiter for " isintroduced16, "min = k 32lmax : (12)This is derived from dimensional considerations, andthe maximum length scale, lmax, is typically set to 0:1�aerofoil chord. This limiter improves stability and alsohelps to reduce the production of spurious eddy viscosity.However, this eddy viscosity is not eliminated altogether.The at plate solution for the Chien-plus-Wolfshteinmodel, together with the corresponding pro�le of eddyviscosity, �t, is shown in Fig. 1. A satisfactory compar-ison with the log-law velocity pro�le is obtained. How-ever, it can be seen that the eddy viscosity remains at alevel which is still much higher than the laminar viscosityoutside the boundary layer.4

Assessment of k-!The k-! model appears to meet two of the three crite-ria immediately. The ! equation, Eqn.(6), can be inte-grated all the way to the wall without the requirementfor damping functions or normal-to-wall distance, andthe source terms appear to be straightforward.The boundary conditions, however, pose a signi�-cant challenge. Since ! / "=k it tends to in�nity asa wall is approached. This is entirely unsatisfactory.Various approximations and alternative boundary con-ditions have been tested. Even when using boundaryconditions which appear entirely inappropriate, such as(@!=@n)wall = 0, good solutions can occasionally be ob-tained. However, such approaches are very dependent onthe near-wall mesh spacing and are not appropriate forgeneral use. The method proposed by Wilcox8 is to �t! to a function which is derived from near-wall analysisof Eqn.(6). Wilcox suggests that this function shouldbe applied over seven to ten cells next to a surface, andthat these cells should be in the region y+ � 2:5. Forall practical applications considered in the present studythis is too restrictive since there are typically only oneor two cells inside this region.Analysis of Eqn.(6) highlights a further shortcomingof this model. In areas of the ow�eld where the turbu-lent energy, k, is negligible, the equation for ! de-couplesfrom the k equation, and a non-zero solution for ! canexist everywhere. This issue manifests itself as a sensi-tivity to freestream, or in�nity, conditions for !. It hasbeen demonstrated by Menter17 that the level of eddyviscosity in a shear layer can be profoundly a�ected bythe speci�ed value of !1.Implementation of k-!To overcome the boundary condition problem, the !equation was inverted15 and expressed in terms of:� = 1��! : (13)The variable � now has units of time, and goes to zeroat a solid surface. The new transport equation for � is:@(�ui� )@xi = @@xi �(� + �t�� ) @�@xi �� ��kPk+ � ��� ���+ �t�� � 2� @�@xi @�@xi : (14)This model would now appear to ful�ll all three of the re-quirements listed above, although the freestream depen-dency is still present. As with the ! equation, a non-zerosolution of the � equation can be obtained even when the

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Figure 2: Log-layer comparison (left) and � pro�lesfor the k-� model with and without the R limiterapplied to the �-equation source terms.local turbulence levels are negligible. The value of � inthe freestream has been found to have a strong in uenceon the turbulent quantities in the boundary layer. InMenter's study of the freestream dependency of the !equation, it is suggested that in uence on the boundarylayer is minimised provided that the freestream value of! is high enough. This implies that a small value of �1is required.The at-plate solution which results from usingEqn.(14) in conjunction with a very small value of �1at the in ow boundary is shown in Fig. 2, together withthe corresponding � pro�le. It can be seen that � con-tinues to grow outside the boundary layer, reaching amaximumwell away from the surface, where k is knownto be negligible. High levels of � are now present in thefreestream. Due to the convective and di�usive termsthese high levels of � are maintained further downstream.This is undesirable in situations where a new shear layeris developing. One relevant example of this would be thecase of a ap deployed behind a wing. For the boundarylayer on the ap, the large levels of � produced by theshear layers on the wing give an e�ective �1 which ismuch too high, and the growth of turbulence on the apis now incorrect.To suppress these high levels of � , the source termsin Eqn.(14) are modi�ed by introducing a limiter basedon local eddy viscosity. For the k-� model, the equationfor �t is: �t = ���k�; (15)and the limiter is de�ned as:R = max(0:01�; ���k� ): (16)5

The source terms in Eqn.(14) are multiplied by the ratio�t=R and the transport equation for � is now written as:@(�ui� )@xi = @@xi �(�+ �t�� ) @�@xi �� �����2R Pk+� ��� �tR � �� + �t�� � 2���kR @�@xi @�@xi : (17)The limiter, R, is chosen such that in regions of the owwhere the eddy viscosity is larger than 1% of the laminarviscosity the source terms are una�ected. A useful side-e�ect of such a limiter is that the primary turbulencevariables are now only used in the denominator whenthe local turbulence level is above the 1% cut-o�. Thisprecludes the possibility of division by zero.The solution for this modi�ed model on the at plateis also given in Fig. 2. Once the edge of the boundarylayer is reached, the eddy viscosity decreases rapidly, andthe limiter is activated, causing � to be reduced smoothlyto a very low level in the freestream. Meanwhile, thevelocity pro�le is seen to remain largely una�ected.Care is required in selecting the appropriate valuefor the limiter. Very close the wall the eddy viscosityis again small. All the calculations presented here usemeshes where the minimum wall cell height has a y+value between 0.5 and 2. For these meshes the limit of0:01� is found to work well. For �ner meshes, a lowerlimit would be required.-1 0 1 2 3 4 5

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Figure 3: Flat plate velocity pro�les for the k-�model, Eqn.(18) and the k-g model, Eqn.(19).Concerning the velocity pro�le predicted by the k-� model, it is apparent in Fig. 2 that the model givesan incorrect prediction of U+ implying that there is anover-prediction of skin friction. This can been tracedto an error in the calculation of the gradient of � veryclose to the solid surface. It is known that as the wall is

approached, � approaches zero as a function of y2. Sincethe method used to calculated local gradients uses �rstorder averages (as in most CFD methods), the quadraticnature of � is not properly accounted for. The last termin Eqn.(17) contains a product of the gradients of � , andit is the incorrect evaluation of this source term whichleads to the poor modelling of the ow very close to thewall.To eliminate this discretisation error a higher ordergradient calculation method could be employed near thesurface. A simpler alternative is to modify the model toeliminate the quadratic behaviour. A further transfor-mation is carried out by the �rst two authors to re-castthe � equation in terms of a new variable, g, which rep-resents p� . Using the same source term limiter, R, thenew model can be written as:@(�uik)@xi = @@xi �(� + �t�k ) @k@xi �+ Pk � ���2k2R ; (18)@(�uig)@xi = @@xi �(� + �t�g ) @g@xi �� ����g32R Pk+��2kg2R � �� + �t�g� 3���kgR @g@xi @g@xi : (19)The eddy viscosity is:�t = ���kg2; (20)and the constants are:�g � �� � �! = 2:0; �� � C� = 0:09;� = 5=9; � = 0:075The new solution is shown in Fig. 3 and it can be seenthat the correct velocity pro�le is now obtained.A similar transformation has been carried out byGibson and Dafa'Alla18 to convert a k-" model to q-�, where q represents pk. However, this transformationadds new source terms to the k equation. Since the kdi�usion term is the only term involving gradients of k inthe k-g model the discretisation errors at the wall havea much reduced e�ect. Hence, a transformation of the kequation does not appear to be necessary in the contextof the present study.2D ValidationBoth the k-~" and k-g implementations described abovehave been validated on a large number of external owsfor both 2D and 3D geometries. To demonstrate thecapabilities of both models a smaller number of test caseswill be described. First, a transonic aerofoil, and a low6

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Figure 4: Pressure (left) and friction coe�cients forRAE2822 Case 9.Mach number, high-lift, multi-element aerofoil are pre-sented and discussed. Following this, some examples ofthree-dimensional applications related to current BAeAVRO research work are given.A well known, and geometrically simple test case isthe RAE2822 aerofoil. Test ow `Case 9' is comparedwith experimental data19 for surface pressure and fric-tion in Fig. 4. The corrected conditions are M = 0:734,� = 2:54o and Re = 6:5� 106. It can be seen that bothmodels give very similar predictions, which correspondwell with the experimental data. In particular, the fric-tion coe�cient plot demonstrates that the k-g model isperforming well in the near-wall region.Figure 5: Streamlines over the L1T2 three-elementaerofoil at 21.69o predicted by the k-g model.

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Figure 6: Lift/Incidence and Lift/Drag polars for theL1T2 at M = 0:197, Re = 3:52� 106.To test the models for ows with high pressure gra-dients, and greater geometrical complexity, a detailledstudy has been carried out for a three-element high-liftcon�guration. The geometry was de�ned as part of theNational High-Lift Programme (NHLP) and is known as`L1T2'. Experiments carried out on this geometry aredescribed by Moir20. Calculations have been performedwith both k-~" and k-g over the complete incidence rangeat M = 0:197, Re = 3:52 � 106. The geometry andstreamlines for a representative ow are shown in Fig. 5.Of particular interest are the interactions between thewake from the slat and the boundary layer on the uppersurface of the wing, and the wing wake and ap bound-ary layer. There are also complex re-circulation regionsin the `cove' areas which exist on the lower surface of theslat, and in the ap cut-out on the lower surface of thewing.Polar plots of lift against drag and lift against inci-dence are given in Fig. 6. The predictions of lift fromthe k-g model correspond well with experiment, whilethose of the k-~" implementation are slightly lower. Thelift/drag curve shows similar di�erences. The k-g solu-tion is closer to experiment while the Chien model over-predicts drag at every point. A closer examination ofthe solutions at 20:18o shows that there is little di�er-ence in the predictions of surface pressure coe�cient,shown in Fig. 7. The lower lift predicted by k-~" resultsfrom a slightly lower suction over the upper surface ofthe wing and slat. It is well documented8 that k-! tends7

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Figure 7: Surface pressure coe�cient for L1T2 atM = 0:197, � = 20:18o, Re = 3:52� 106.to give better predictions for adverse pressure gradient ows. It would appear that this behaviour is preservedin the present k-g formulation. Pro�les of total pressureat the same ow conditions are also available and aregiven in Fig. 8. These are presented for the ow abovethe wing at 35% chord, and at the leading edge, midchord and trailing edge of the ap. Here, larger di�er-ences between the models can be seen. The k-~" solutionappears to give incorrect spreading of the wakes, with arather more di�use total pressure distribution. The k-gmodel is consistently closer to experiment.For an explanation of these di�erences, it is usefulto examine the way in which the turbulent ow is han-dled by these two models. Fig. 9 shows contours of nor-malised eddy viscosity for the 21:18o case. Immediatelyevident is the large number of block boundaries which arepresent, this being a consequence of the mesh generationprocedure which demands a single boundary conditiontype on a block face. Some of the block boundaries arehowever utilised for grid control by means of user speci-�ed line-grid stretching functions. The block boundaries,when they are positioned close to shear layers, can causeproblems for models which require wall distance mea-sures. For many test cases it is possible to ensure thatall low Reynolds number regions are contained withinthe block nearest the wall. However, for moderatelycomplex geometries, such as the L1T2, this may not befeasible. As previously stated, the k-~" model implemen-tation reverts to the high Reynolds number form oncethe opposite face of a wall block is reached. It is likelythat, at such high incidences, the model is switching tooearly. This a�ects the region above the ap in particular,where the wake and boundary layer mixing regions arestill at low turbulent Reynolds number conditions. Thecontours of normalised eddy viscosity highlight another

shortcoming of the Chien implementation. A certainamount of `spurious' eddy viscosity outside the boundarylayers can be seen, which a�ect a large area of the owabove the geometry. As observed in the at plate testcase, this value of eddy viscosity is controlled, but noteliminated, by the use of limiters. The presence of sucheddy viscosity can trigger undesirable di�usion or evenproduction of k which in turn can a�ect the levels of tur-bulence at the boundary layer edge further downstream.Meanwhile, the k-g implementation behaves well at theboundary layer edge, and the turbulence is seen to returnto negligible levels smoothly.-0.5 0.0 0.5 1.0

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sur

face

Figure 8: Pro�les of total pressure coe�cient forL1T2 at M = 0:197, � = 20:18o, Re = 3:52� 106.8

Figure 9: Contours of eddy viscosity (�t=�) for theChien model (top) and the k-g model. Values rangefrom 0 (Red) to 800 (Violet).3D ApplicationsTwo applications are described from current BAe AVROresearch studies on regional jet con�gurations. Atpresent, the primary requirement for this ow solver istransonic cruise conditions with very accurate drag pre-dictions for a wide spread of lift coe�cients around thedesign point extending well into the wave drag rise. Thewing designer would routinely compute CL/CD polarsover a CL range of 0.1 to 0.7. Initial accuracy of �1%of total aircraft drag is sought together with drag incre-ments to substantially less than one count. These in-crements may be due to variation of shape or Reynolds

number, or the addition of extra components such aswinglets and ap track fairings. Also required is theability to reliably detect the bu�et boundary.Primary areas of interest in Navier-Stokes results willbe associated with shock locations and degradation ofthe boundary layer towards the trailing edge including3D e�ects, separations, and trailing edge pressure rise.Related design areas where CFD has a particular contri-bution to make are in the details of ow in wing-fuselageand pylon-wing junctions, these being very sensitive toReynolds number dependent separations, often in thepresence of quite strong shocks in the wing-nacelle gap.Thus validation of the code for representative cases isvital.Figs. 10 and 11 show results for a BAe AVRO re-gional jet wing-body research con�guration for whichtunnel results are available. Flow conditions are tran-sonic cruise at M = 0:79, � = 1:0o, transition at 15%chord, CL � 0:42, but at a tunnel Reynolds numberbased on c of 2.7 million. The mesh has 1.2 millioncells in 230 blocks with a �rst cell depth of y+ � 2 andaround 25 cells in the boundary layer. Pressure con-tours in Fig. 10 are obtained by use of the k-g turbulencemodel. Fig. 11 shows surface streamlines using both k-gand k-~" models. These streamlines can be compared towind tunnel oil- ow visualisations. The k-~" results showa separation at the wing root trailing edge which is notpresent in the wind tunnel. This separation is not shownby the k-g results, for which the streamlines match thebehaviour seen in the wind-tunnel. Note that the wingis attached to a belly fairing which presents a verticalwall at the wing root thus avoiding acute angles at thewing-fuselage junction, and that the leading edge rootfairing further suppresses any tendency to separation.Elsewhere the two sets of streamlines show very similarproperties, notably the curvature and incipient separa-tion near the trailing edge crank which is also suggestedin the tunnel oil ow. Although not clear in these �g-ures, the surface ow pattern around the leading edgefairing is well predicted by both models.It is signi�cant that trailing edge root separationssimilar to that seen with k-~" are also produced for thiscase when Baldwin-Lomax and Johnson-King turbulencemodels are used in RANSMB, the former model beingnoted for its resistance to separation. The one commonfeature in these three models that is not shared with k-gis the need for wall distance, which may be one sourceof error.Fig. 12 shows pressure contours for a di�erentwing/body plus pylon/nacelle research con�guration us-ing the k-g model. Conditions are M = 0:8, � = 1:0o,9

Figure 10: Contours of pressure coe�cient on a typical regional jet transport wing/body predicted by thek-g model. M = 0:79, � = 1:0o, Rec = 2:7� 106.

Figure 11: Skin friction lines on the same wing/body for k-~" (left), and k-g.10

Figure 12: Pressure coe�cient contours for a wing/body/pylon/nacelle research con�guration calculatedusing the k-g model. M = 0:8, � = 1:0o, Re = 6:0� 106.11

transition again at 15% chord, Rec = 6 � 106, andCL � 0:41. The mesh has 2 million cells in 900 blockswith a �rst cell depth of y+ = 2. This is part ofan ongoing study into nacelle-pylon installation and isshown here with a very closely coupled nacelle and over-wing pylon. Flow disturbance can be seen at the wing-fuselage junction. A small horseshoe vortex appears atthe leading edge root. No belly or wing root fairings arepresent in this simulation and the wing-fuselage junc-tion presents an acute angle at the trailing edge. Thiscould be expected to lead to incipient separation in thewing-fuselage junction. Tunnel tests are planned.Computational CostThe wing-body solutions required 23 Mwords and 10hours on a Cray YMP machine at 60 M op/s as partof a CL/CD polar calculation. The wing-body-pylon-nacelle solution required 41 Mwords and 75 hours on aCray J90 machine at 32 M op/s from a free-stream start.Although wing-body CL/CD polars are routinely com-puted on the YMP, work is underway to implement the ow solver on the Cray T3D massively parallel proces-sor to facilitate this and extend the capability to nacellecase polars. ConclusionsThe way in which particular two-equation transportmodels are chosen for implementation in an industrialdesign system has been described. This process is gov-erned almost entirely by numerical considerations. Onek-" model and one k-! model have been selected andthe properties of these two models are measured againstthree criteria: no normal-to-wall distance, simple sourceterms and straightforward boundary conditions. Bothmodels require manipulation before a stable implemen-tation is achieved. The Chien k-~" model has simpleboundary conditions, but makes use of wall distance andalso requires damping functions which become quite sti�.These issues are overcome using a small one-equationsub-layer next to wall surfaces, and by making use ofa limiter for ~". Despite these precautions the solutionsobtained are often adversely a�ected by the numericaldi�culties encountered. The Wilcox k-! model has nodamping functions and simpler source terms, but theboundary condition for ! makes it di�cult to use. Thismodel is rewritten to use a new variable, g, and allthree criteria can now be satis�ed. The dependency onfreestream conditions is controlled through the carefuluse of a source term limiter which has no e�ect in bound-ary layers and wakes. Results of two-dimensional tran-sonic and low Mach number test cases demonstrate that,

while both models give robust convergence on moder-ately complex geometries, the solutions given by the k-gmodel are at least as good, and often better than thoseof the k-~" model. When applied to three-dimensionaldesign tests, stable solutions can be obtained with bothtransport models. The k-g model, with its lack of de-pendency on wall distance, appears particularly usefulfor complex aircraft con�gurations. Initial results indi-cate that accurate and reliable solutions can be obtained.The increased con�dence in the solutions which are nowbeing obtained is helping to increase the use of CFD bythe design teams.AcknowledgementThe �rst author's stay at BAe Sowerby was sponsored bythe European Commission under speci�c training agree-ment No. AER2-CT93-5006H.References[1] Jameson A., Schmidt W. & Turkel E. Numericalsolutions of the Euler equations by �nite volumemethods using Runge-Kutta time-stepping schemes.AIAA Paper 81-1259, 1981.[2] Jameson A. & Baker T. Multi-grid solutions of theEuler equations for aircraft con�gurations. AIAAPaper 84-0093, 1984.[3] Baldwin B.S. & Lomax H. Thin Layer Approxima-tion and Algebraic Model for Separated TurbulentFlows. AIAA Paper 78-257, 1978.[4] Davidson L. Implementation of a Semi-Implicitk-� Turbulence Model Into an Explicit Runge-Kutta Navier-Stokes Code. CERFACS Report,TR/RF/90/25, 1990.[5] Gould A.R.B. Validation of Turbulence Models forTwo- and Three-dimensional Flows. In: ECARP(Validation Area) Report, to be published in Noteson Numerical Fluid Mechanics Series, by Vieweg,1996.[6] Lien F.S. & Leschziner M.A. Modelling 2D Sep-aration from High-Lift Aerofoil with Non-LinearEddy-Viscosity Models and Second-Moment Clo-sure. UMIST Report, TFD/94/05, 1994.[7] Wilcox D.C. Reassessment of the scale determin-ing equation for advanced turbulence models. AIAAJournal, 26 pp1311-1320, 1988.[8] Wilcox D.C. Turbulence modelling for CFD. DCWIndustries Inc., 1993.12

[9] Speziale C.G., Abid, R. & Anderson E.C. Criti-cal Evaluation of two-equation models for near-wallturbulence. AIAA Journal, 30 pp324-331, 1992.[10] Wolfshtein M.W. The Velocity and TemperatureDistribution in One-Dimensional Flow with Turbu-lence Augmentation and Pressure Gradient. Int.Journal of Heat and Mass Transfer, 12 pp301-312,1969.[11] Jones W.P. & Launder B.E. The prediction of lami-narisation with a two-equation model of turbulence.Int. Journal of Heat and Mass Transfer, 15 pp301-314, 1972.[12] Chien K-Y. Predictions of Channel and Boundary-Layer Flows with a Low-Reynolds-Number Turbu-lence Model. AIAA Journal, 20 pp33-38, 1982.[13] Lam C.K.G. & Bremhost K.A. Modi�ed Form ofk-� Model for Predicting Wall Turbulence. ASMEJournal of Fluids Engineering, 103 456-460, 1981.[14] Lien F.S. & Leschziner M.A. Computational Mod-elling of 3D Turbulent Flow in S-Di�user and Tran-sition Ducts. Engineering Turbulence Modellingand Measurements, 2 Elsevier, 1993.[15] Kalitzin G. Validation and Development of Two-Equation Turbulence Models. In: ECARP (Vali-dation Area) Report, to be published in Notes onNumerical Fluid Mechanics Series, by Vieweg, 1996.[16] Benton J.J. Description of Methods Used by BritishAerospace. In: `EUROVAL, A European Initiativeon Validation of CFD Codes.' Haase, Brandsma,Elsholz, Leschziner, Schwamborn (eds.) Notes onNumerical Fluid Mechanics Vol 42, Vieweg, 1993.[17] Menter F.R. In uence of Freestream Values on k-!Turbulence Model Predictions. AIAA Journal, 30pp1657-1659, 1992.[18] Gibson M.M. and Dafa'Alla A.A. A Two-EquationModel for Turbulent Wall Flow. AIAA Journal, 33pp1514-1518, 1995.[19] Cook P.H., MacDonald M.A. & Firmin M.C.P.Aerofoil 2822 - Pressure Distributions, BoundaryLayer and Wake Measurements. Appendix A6,AGARD AR-138 , 1979.[20] Moir I.R.M. Measurements on a Two-DimensionalAerofoil With High Lift Devices. Appendix A2,AGARD AR-303 , 1994. 13