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t&AMdkA AIAA 2001-2564 The Lag Model, a Turbulence Model for Wall Bounded Flows Including Separation M. E. Olsen T. J. Coakley NASA Ames Research Center Moffett Field, CA 94035 15th AIAA Fluid Dynamics Conference June 1 1 - June 14, 2001 /Anaheim, CA For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1801 Alexander Bell Drive, Suite 500, Reston, VA 20191-4344 AIAA 2001-2564 The Lag Model, a Turbulence Model for Wall Bounded Flows Including Separation M. E. Olsen * T. J. Coakley * NASA Ames Research Center Moffett Field, CA 94035 A new class of turbulence model is described for wall bounded, high Reynolds number flows. A specific turbulence model is demonstrated, with results for favorable and adverse pressure gradient flowflelds. Separation predictions are as good or better than either Spalart Almaras or SST models, do not require specification of wall distance, and have similar or reduced computational effort compared with these models. Introduction One difficulty with current one and two-equation turbulence models is the inability to account di- rectly for non-equilibrium effects such as those encoun- tered in large pressure gradients involving separation and Shockwaves. Current turbulence models such as Spalart 's one-equation model, 5 the classic k - e and Wilcox's k - cu 1 two-equation models have been de- signed and tuned to accurately predict equilibrium flows such as zero-pressure gradient boundary- layers and free shear layers. Application in more complex flows can be problematical at best. Although there have been many attempts to modify or correct basic one- and two-equation models, most of these attempts have been only marginally successful in predicting complex flows. More complex models such as Reynolds stress mod- els have been investigated extensively, primarily for relatively simple flows but also for complex flows. In most cases these models give somewhat better predic- tions than the simpler one and two equation models, but for complex flows they do not perform much bet- ter than the simpler models. One theoretical advan- tage of Reynolds stress models is that they directly account for non-equilibrium effects in the sense that the Reynolds stresses do not respond instantaneously to changes to the strain rate but more realistically lag them in time and/or space. Unfortunately, The Reynolds stress models are usually considerably more complicated and numerically stiff than the one- and two- equation models, and this has prevented their wide application for complex flows. In this paper we introduce a new three equation model designed to account for non-equilibrium effects * Research Scientist, NASA Ames Research Center, Associate Fellow AIAA t Research Scientist, NASA Ames Research Center, Associate Fellow AIAA Thie paper ia a work of the U.S. Government and >• not subject to copy- right protection in the United States. 2001 without invoking the full formalism of the Reynolds . stress models. The basic idea is to take a baseline two-equation model and to couple it with a third (lag) equation to model the non-equilibrium effects for the eddy viscosity. The third equation is designed to predict the equilibrium eddy viscosity in equilibrium flows. We show results obtained with a lag model based on the Wilcox k-cu model. Applications to four flows are given including an essentially incompressible flat plate flows, an essentially incompressible adverse pressure gradient flow with separation, 2 a transonic bump flow 3 with a shock wave and separation, a three dimensional transonic wing flow. 4 Results using the new model are compared with results obtained with Spalart's model 5 and Menter's k-cu SST model. 6 Results obtained with the new model show encouraging improvements over results obtained with the other models. The Lag Model The differential equations defining the lag model are p^ = Pic-eic + Vtm-o-icp-VtVk) (1) Dtu - Pa J -ecu + V(M- + o- a ,pv t Vtu) (2) 5^1 = a(R T )pcuK E -^t) r Dt where: y tE = k/tu Vy. - pvtS 2 e k = 6*pkw S Sij = (3) Rt = pk/Vtu V w = apS 2 £„, = Pptu 2 V^SijSij 2 9xj dxi American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautic* & Astronautics or Published with Permission of Authors) and/or Author(s)' Sponsoring Organization. a(R T ) - _ f(RT + RT.)l ^1(Rt + Rt«o)J with parameters « =■ 5/9 do >= 0.35 P = 0.075 Rt. - 1 P* = 0.09 RToo - •<» a k =* 0.5 c t = 05 (4) The model equations are composed of an underlying model, (k — w) with the lag equation augmenting the system. The k — w model, given by the first two equations is unchanged from the standard model, except that v t is now given by a field PDE, Eq.(3), instead of tytu. The boundary conditions are those of a converted scalar: specified on inflow boundaries and extrapo- lated on outflow boundaries. At inflow boundaries, k, to and -v t are set to constants (hereafter referred to as JCo, Woo, and -v teo = Wca*,). In this paper, Wt& was chosen to be 0.0001, corresponding to a turbulent intensity of 0.8%. The value of k external to the wall bounded flows is, of course, substantially lower than this, since k decays in the absence of mean strains. tOoo is chosen to yield a low value of v tl (here chosen as one tenth the molecular v) , and -Vt is set to v t| . The model's predictions ore insensitive to the values chosen for these constants as is shown in the results. At the solid walls a "no lag" boundary condition is enforced: the eddy viscosity is set to its equilibrium value: zero for hydraulicly smooth walls, finite for rough walls. The cases discussed in this paper all use the smooth wall conditions. 7 Rough wall boundary conditions are also possible in the manner of the k — to model, 2 but are not discussed in this paper. The turbulent eddy viscosity is governed by Eq.(3). This equation simply says that the eddy viscosity goes to its equilibrium value ("v tc ) along a streamline like a first order dynamical system with a time constant given by l/(euo). The stability of the turbulence model as a dynamical system is ensured by the form of this source term, given a stable underlying model. There is no diffusion term in this equation, and evo- lution of the eddy viscosity is dependent only on its upstream history and the underlying equilibrium eddy viscosity at that point. The a term of Eq.3 of the source for the -vt Eq.(4) governs the amount of lag present in the model. This term is made up of three factors. The leading con- stant, ao, controls the amount of lag in the model. The higher the value of ao, the less lag(shorter time constant) in the system, and the closer will follow the underlying turbulence model. The second factor in Eq.4 goes monotonically from 100 to one as Rt goes from to infinity(Fig. 1). This in effect stiffens the 100 SO 20 10 - 2 * i ii"~ • • • •*" i "* J — " (R t +1)/(R t +.01) 10" 100 Fig. 1 Lag Source Rt Factor model at low values of Rt so that It will have less lag, and will act more like the unmodified model in these conditions. In effect, this term "turns on" the lag only in turbulent regions (Rt » 1). The model requires the storage of one additional field variable over the SST model, but does not require the wall distance information used in both the SST and SA models, thus freeing up the storage required for that variable. As the model is computationally simple, not requiring the Vvt • V>t or Vk • Vcv terms present in the other models, it actually requires less CPU time per iteration than either SST or SA models, and similar or fewer iterations for convergence. Niim^rirpl Method The solutions obtained in this paper were done with a modified version of the OVERFLOW 8 '* code. For the mean flow equations, the existing 3rd order upwind scheme or the central/matrix dissipation 10 method was used. The full Navier Stokes equations were solved, as opposed to utilizing the thin layer approxi- mation. Converged solutions were indistinguishable in terms of skin friction and velocity profiles. Matrix dis- sipation was used with 2nd and 4th order smoothing coefficients of 2 and 0.1, respectively. The eigenvalue limiters were set to zero, and Roe averaging was used to form the matrix. Multigrid was employed, both as grid sequencing for startup, and during the relaxation process. 3 levels of multigrid were used for all solutions presented in this paper. The turbulence model equations were spatially dis- cretized using a 2nd order upwind method with a min-mod limiter. This is a departure from the 2 equa- tion models implemented in OVERFLOW, which use 1st order upwind. 1st order upwind was the initial American Institute of Aeronautics and Astronautics presented in this paper. The turbulence model equations was spatially dis- cretized using a 2nd order upwind method with a min-mod limiter. This is a departure from the 2 equa- tion models implemented in OVERFLOW, which use 1st order upwind. 1st order upwind was the initial implementation, but this proved too dissipative, and led to excessive grid density requirements for grid in- dependent solutions. The reason for this can be seen in the 3rd equa- tion, which implements the history effects (lag) of the model. Using a 1st order upwind on this equation is analogous to using a 1st order time integrator to in- tegrate an ODE, with grid spacing analogous to the ODE time step size. When the 1st order upwinding was replaced with a 2nd upwinding, the gridding re- quirements dropped back to what would normally be required for grid independent solutions with other tur- bulence models. The additional work required to solve the third equation is offset by the relative simplicity of the underlying model. Convergence is rapid and robust, as implemented in OVERFLOW. Results Dissipation of Isotropic Turbulence Isotropic turbulence has no mean strain, so that the decay of k and u> follow those of the underlying model, here the Wilcox k- tu model. The -v t equation uncou- ples from the other two equations, and the equation governing the evolution of the eddy viscosity becomes: d-v t IT = Qo(k- arvt) This decoupling is aesthetically beautiful, from the viewpoint of "v t as a ratio of turbulent stess to mean strain. When the mean strain vanishes the turbulent stresses are not effected by the value of the eddy vis- cosity. Similarly, the lag equation does not affect the decay of turbulent kinetic energy built into the under- lying model under conditions of zero mean strain. Equilibrium Channel Flows For fully developed channel shear flows, such as Cou- ette, or fully developed pipe flows, the model again decouples and reproduces the results of the underly- ing model. The differential equation becomes: u^— =Q (k-urv t ) ox and as ^ = for fully developed Couette or chan- nel flows, this simply enforces "v t = k/cu. In the same manner as in the decay of isotropic turbulence, if the underlying model does a good job under these con- ditions (which k - w does) then the Lag model will also. Subsonic Flat Plate The fine grid for this case is 101(streamwise)x 101 (wall normal). Nearly iden- tical results were obtained on a 51x51 grid. The wall normal grid stretching for these cases was 1.1 and 1.2 respectively, with initial y+ spacings of 0.1 and 0.2. The initial 4 wall normal points were equispaced. 5x10" 4x10 - 3x10" 2x10" M=0.2 Hot Plate ° Karman-Schoener Lag - - k — cj 500 1000 2000 5000 10* R »2x10 4 Fig. 2 Subsonic Flat Plate Skin Friction 30 25 20 15 10 *■ Law of the Wan ';; -M'-ic D P". , 95 L " Re ,= 20.35 10 100 1000 10' Fig. 3 Law of the Wall Velocity Profiles For flat plates, the model roughly reproduces the original model performance, since the flow is slowly varying in the streamwise direction, and the model forces -v t to its equilibrium values. Comparison for low speed flows (actually Moo = 0.2, Fig. ) shows a good comparison with Karman-Schoener correlation. This was expected from the underlying k — u> model's predictions for smooth flat plates. The law of the wall is also well reproduced (Fig. 3). American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Authors) and/or Author^)' Sponsoring Organization. 5x1 o~ 3 4x10 -3 3x10" J - 2x10 -3 • • ' R,-10" M.-0.2. k„«10"V. "~ ~" Kormon-Schoenherr ■. •. V — •• p - in * ' ' ' ' I ' — 500 1000 2000 ■ i i i i i 5000 10* 2x10 4 Fig. 4 u>oo Dependence Fig. 5 Driver CSO Flowfleld Upstream boundary conditions were constant total pressure and temperature, with static pressure allowed to vary and velocity direction aligned with the cylin- der axis. The outer streamline was treated as an invisdd wall. The viscous wall(the surface of a cylin- der) is a no-slip, adiabatic wall. The downstream static pressure was adjusted to match the experimen- tal static pressure (Fig. 6) upstream of the interaction region(x « -.438m). Upstream length of the cylin- drical body was adjusted so that the computed the boundary layer thickness at x = -0.438m matched the experiment. This "entry length" was the same for all three models. The standard OVERFLOW low Mach number pre- conditioning wc£ employed. The grid used in these cal- culations was 200(axial) by 160(radial), an extremely fine grid. The calculations were repeated with a wall distance information, and are under consideration for use as future baseline models. The y + require- **10 ments were also investigated, and similar behaviour to the underlying k - to model were found. T Improve- ments and extensions of the which Include roughness are available for the k— w model, 1 ' and are also under consideration for future model improvement. Driver CSO Separated Cage This is a low speed separated case. 3 The aodsym- metric geometry (Fig. 5) is defined by an external streamline determined from experimental data, and wall pressures are available in addition to velocity pro- files and skin friction. ■ ■ » i i ■ i i i i i . -0.3 0.5 x(m) Fig. 9 Driver CSO Surface Pressures 3x10 2x10 -0.3 A «(m) Fig. 7 Driver CSO Skin Friction 100 x 80 grid and demonstrated grid independence in the same manner as Bardinaet. al. T Surface pressures, skin friction and velocity profiles agreed with the fine grid results. This case was one which demonstrated the need for handling the turbulence model convection operator with 2nd order upwind (minmod limiter), as the solutions on the two grids are effectively identical when the' 2nd order operator is employed, but show slight differences when computed with a first order up- wind operator for the turbulence model. The pressure variation predicted by the Lag model is closer to the experimental data than either the Spalart-Almaras or SST models, although both give reasonable agreement. This improvement in pressure variation prediction could be important for internal American Institute of Aeronautics and Astronautics t 1 1 I ■ ' ■ I ' ' ' — 1_J I I I I ■ ■ ' ■ ■ ■ ■ I ■ -, f = -0.463m 0.02 0.01 0.06 0.04 0.02 ■\' *' ' 1 1 1 1 1 1 1 1 1 1 1 ' i ' ' 0O20.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 0.2 0.4 0.6 Fig. 8 Driver CSO Velocity Profiles 0.02 - 0.01 -2x10 , _i_. L. 1 .... 1 . 0.1 : 1 = -0.0127 y • 0.08 : A 0.06 : 0.04 -_ 0.02 - f = +0.058 f = +0.101 -2x10" { = + 0.152 -2x10 -2x10 uV/U. Fig. 9 Driver CSO Stress Profiles American Institute of Aeronautics and Astronautics (c)200 i American Institute of Aeronautics * Astronautics or Published with Permission of Authors) and/or Authors)' Sponsoring Organization. low speed flows with small separations. The model's ability to more accurately predict the pressure varia- tions will be repeated in later test eases. The skin friction prediction of all four models is shown in Fig. 7. All four predict the skin friction rea- sonably well, with the SA model predicting a reattach- ment point slightly downstream of the experimental data and the k — tv model failing to predict separa- tion. All models predict Cf too low Just upstream of separation, from x = — .1 to x = 0. The velocity profiles (Fig. 8) also show good agree- ment. The differences between the underlying k — w model and the lag model are relatively small upstream of separation, but the lag model predicts the sepa- rated profiles more accurately in the separated region (x>0). The shear stress profiles are similarly well predicted (Fig. 9), for both the evolution of the maximum shear stresss and its location. The lag of the model is evident most clearly in this figure. Upstream of the separa- tion, the shear stress predicted by the lag model is true to it's name, and lags the underlying k— u> model especially evidently at the £. = —0.076 station. By the t = 0.101 station, all of the models are predicting roughly the same shear stress, even though the velocity profile predictions (Fig. 8) show the greatest scatter at this location. One of the model's slight imperfections can be seen in Fig.8 and Fig.9, as the outer edge of the boundary layer has a kink not seen in the other models. Bachalo-Johnaon Bump This test case features the transonic Interaction of a fully developed turbulent boundary layer with the pressure field created by a circular arc bump on the surface of a cylinder. The surface pressure distribu- tions for various freestream Mach numbers are avail- able, and velocity and Reynolds stress profiles are available for the Moo = 0.875 case. Upstream boundary conditions were constant total pressure and temperature, with static pressure allowed to vary and velocity direction aligned with the cylin- der axis. The outer edge of the flowfield was treated by extending the grid 8 bump chords away from the wall, and utilizing characteristic(no reflection) bound- ary conditions. The viscous wall(the surface of a cylinder) is a no-slip, adiabatic wall. The downstream static pressure was held at p TO Upstream length of the cylinder was adjusted to match computed boundary layer thickness at x = —0.25m. to experiment as done in the Driver case, and again this "entry length" was the same for all three models. The grid for this case is 181(8treamwise)x78(wall normal). Fine grid solutions with a 358 x 161 grid were indistinguishable to plotting accuracy, in terms of both surface pressures and velocity profiles at both Moo = 0.875 and Moo = 0.925 cases. The wall normal grid spacing for both normal and fine grids had a y + less than 0.17 upstream of the shock. 2.5x10" I*-/". 2 R .. o o 10" s 10 • M) ■ :0 o 10"« 10 o • 10 ,: •«•■" ' » i t a t t I o o s o -#-4-H -t I 1 H */C 0.5 1.5 Fig. 10 Moo - 0.875 Skin Friction Insenaitivity to k«o and Rtoo values The insenaitivity of the solution to freestream choices of R teo and koe is illustrated in Fig. 10. Here, the predicted x component of skin friction is shown for a range of choices of these parameters. The koo range corresponds to an initial freestream turbulence intensities from 0.08% to 2.5%. The R too range corre- sponds to Initial eddy viscosities of from 0.1 to 10~ 3 molecular viscosity. The surface pressure prediction variations are just as insensitive to these variations in freestream turbulence values. The Lag model reproduces the experimental pres- sure distributions (Fig. 11) as well as either the Spalart-Allmaras or Menter SST models, a distinct improvement over the underlying k — w model, which consistently misses the shock location and underpre- dicts the extent of the flow separation. The velocity profiles show the progression of the flowfield through the separation(Fig. 12). The Lag model predicts a separation point intermediate be- tween the predictions of the SST and SA models, and has a flow recovery better than the SST, though it still does not recover as rapidly as experiment. In these ve- locity profiles, there is no obvious kink at the edge of the boundary layer in contrast to the CS0 flowfield. The shear stress profiles(Fig. 13) show the Lag model's increased prediction of T max downstream of the separation point, though it is not as large as mea- sured in the experiment. Note that this plot has the wall distance logarithmic, expanding the inner region of the boundary layer. There is a consistent under- prediction of the stresses in the inner layer by all of the models in the separated region, and all the models American Institute of Aeronautics and Astronautics 0.6 - ■ ■ ■ ■ i M =0.925 x/c i — | — i — i — i — i — | — i — i — i — ■ — i i r 0.5 1 1-5 -i — i — | — i — i — i — i — | — i — i — i i i ' r 0.5 1 1.5 0.5 Fig. 11 Moo = 0.875,0.900,0.925 Bachalo- Johnson Bump Pressure Distributions ■ i . . . i , 0.5 1 Fig. 12 Bump Velocity Profile Comparisons, Moo = 0.875 American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautic* & Astronautics or Published with Permission of Authors) and/or Authors)' Sponsoring Organization. -0.01 Fig. IS Moo > 0.875 Bump Shear Stress Profiles American Institute of Aeronautics and Astronautics tween the predictions of the SST and SA models, and has a flow recovery better than the SST, though it still does not recover as rapidly as experiment. In these ve- locity profiles, there is no obvious kink at the edge of the boundary layer in contrast to the CSO flowfield. The shear stress profiles(Fig. 13) show the Lag model's increased prediction of x max downstream of the separation point, though it is not as large as mea- sured in the experiment. Note that this plot has the wall distance logarithmic, expanding the inner region of the boundary layer. There is a consistent under- prediction of the stresses in the inner layer by all of the models in the separated region, and all the models seriously underpredict the measured maximum shear stress. ONERA M6 Wing The ONERA M6 4 is a venerable 3D test case. In the conditions from a = 3° to a = 5° range from a nearly attached flowfield to a relatively extensive separation, as shown in Fig. 14. The grid used in these computations was the one shipped as a test case with OVERFLOW, which has grid dimensions of 269(streamwise) x 35(spanwise) x 67( wall normal) , with 201 points streamwise along the wing surface. The y + of the first point off the surface was below 1.25 over the entire wing surface. No grid resolution study was performed. The C p predictions of the model are compared with experiment in Fig. 15. The Lag model produces a good prediction over this entire range of conditions. The largest discrepancies in this range are at the wing root, and are more likely due to the tunnel wall interference, as the flow is attached in this region. As can be seen in the "oil flow" pictures of Fig. 14, all these cases have some separation, from a "incipient separation" at 3° to the rather extensively separated 5° case. All four models give good predictions at a = 3° and a = 4° conditions, but at a = 5° the separation there is an appreciable difference in the predictions provided by the various models. The wing tip separation pro- gression in particular appears to be well captured by both the Spalart-Almaras and the Lag model. The SST model has more extensive separation than exper- iment, and the separation predicted by the k-cu model is less extensive than experiment. Discussion The lag equation could be coupled to virtually any model. We have chosen to couple it to the k — iv model for this example. Another possible implementa- tion would be to lag the Reynold's stresses, as opposed to the eddy viscosity, via an equation of the form Dt = q'(Rt) "> (2nt E Sij -Tij) to account for anisotropic effects seen in 3D flows. The main feature of this new class of models is to introduce a lag into the response of the eddy viscosity to rapid changes in the mean flowfields so as to em- ulate the responses seen experimentally. Virtually all turbulence models generate Reynolds stresses that re- spond too rapidly to changes in mean flow conditions. Even the Reynolds stress models predict overly rapid response of the Reynolds stresses to changes in mean flow conditions. This is in large part due to the mod- els' need to accurately reproduce equilibrium flows. The lag equation gives the existing models an addi- tional degree of freedom, without tampering with their typically good ability to predict equilibrium flows. Summary The Lag model gives good results for mild to mod- erate 2D separations, and agrees well with 3D cases tested. It works well for skin friction prediction at in- compressible and supersonic Mach numbers, and pre- . diets separation well for incompressible and transonic test cases. The model does not require wall distance, in contrast to both SST and SA models, and its sim- plicity is such that the computational effort is roughly equivalent to the simpler 1 and 2 equation models. Future work will include efforts to remove freestream effects, and will look at free shear flows, along with other experimental test cases. References [l]Wilcox, David C. . "Turbulence Modeling for CFD". DCW Industries, Inc, 1993. ' [2]D.M. Driver. "Reynolds Shear Stress Measurements in a Separated Boundary Layer Flow ". AIAA Paper 91-1787, 1991. [3]Bachalo, W.D. and D.A. Johnson. "Transonic Turbu- lent Boundary Layer Separation Generated on an Axisymmetric Flow Model". AIAA Journal, 24:437-443, 1986. [4]Schmitt V. and Charpin F. "Pressure Distributions on the ONERA-M6 Wing at Transonic Mach Numbers". In AGARD Report, number 138 in AR, pages Bl-1 - Bl-44, 1979. [5]Spalart, P.R. and S.R. Allmaras. "A one equation Tur- bulence Model for Aerodynamic Flows". La Reckerckc Aerospa- tiale, 1:5-21, 1994. [6]Menter, F.R. "Two Equation Eddy Viscosity Model for Engineering Applications". AIAA Journal, 32:1299-1310, 1994. [7]Bardina, Jorge E., Huang, Peter G., and Thomas J. Coakley. "Turbulence Modeling Validation, Testing, and De- velopment". NASATM 110446, April 1997. [8]Buning, Pieter. G et al. Overflow user's manual. Version 1.8, NASA Ames Research Center, February 1998. [9]Jespersen, D, Pulliam, T. H., and P.G. Buning. Recent enhancements to overflow. AIAA Paper 97-0664, January 1997. [10]Swanson, R. C. and Eli Turkel. "On Central-Difference and Upwind Schemes". Journal of Computational Physics, 101:292-306, 1992. [lljF.R. Menter. "Influence of Freestream Values on k - a) Turbulence Model Predictions". AIAA Journal, 30(6): 1657- 1659, 1992. [12]Thomas J. Coakley. "Development of Turbulence Models for Aerodynamic Applications". AIAA Paper 97-2009, 1997. [13]Hellsten, Antti and Seppo Laine. "Extension of the k-cu- SST Turbulence Model for Flows Over Rough Surfaces" . AIAA Paper 97-3577-CP, AIAA AFM Conference, August 11-13 1997, New Orleans, LA, 1997. American Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautics * Astronautics or Published with Permission of Author(s) and/or Authors)' Sponsortrtg Organization. a) a -3° b)«-4* c) «t-5« Fig. 14 Predicted Surface Flow(Lag Model), M « 0.84, Re = 18 x 10* 77=0.2 • 1.5 I "" 1 """" tj-0.65 10 0.5 10 0.5 10 0.5 1 Fig. 15 ONERA M6 C„ Comparisons, a = 3,4 and 5° 0.5 10 American Institute of Aeronautics and Astronautics [14]Menter, Florian R. and Rumsey L. C. "Assessment of Two Equation Turbulence Models for Transonic Flows". AIAA Paper 94-2343, 1994. 11 American Institute of Aeronautics and Astronautics