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AIA A JOURNAL Vol. 32, No. 8, August 1994 Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications F. R. Menter* NASA Antes Research Center, Moffett Field, California 94035 Two new two-equation eddy-viscosity turbulence models will be presented. They combine different elements of existing models that are considered superior to their alternatives. The first model, referred to as the baseline (BSL) model, utilizes the original k-u model of Wilcox in the inner region of the boundary layer and switches to the standard A>e model in the outer region and in free shear flows. It has a performance similar to the Wilcox model, but avoids that model's strong freestream sensitivity. The second model results from a modification to the definition of the eddy-viscosity in the BSL model, which accounts for the effect of the transport of the principal turbulent shear stress. The new model is called the shear-stress transport-model and leads to major improvements in the prediction of adverse pressure gradient flows. Introduction (e.g., boundary-layer flows) will not lead to a deterioration for another class of equally important flows (e.g., free shear HIS paper is concerned with two-equation eddy-viscosity flows). The author feels that the slow progress in engineering T turbulence models with emphasis on an engineering per- turbulence modeling, and the confusing picture it often pre- spective. It is based on the experience of the author in testing sents, result to no small extent from an overemphasis of a large number of turbulence models against a wide variety of theoretical concepts and a virtual denial of the empirical na- experimental test cases. The test flows cover a significant ture of the subject. range of flow situations typically encountered in aerodynamic Following an empirical approach, the author has developed computations and are believed to allow some conclusions two new turbulence models based on elements of existing about a model's ability to perform in engineering applications. models which are considered to be superior to their alterna- Two new turbulence models will be presented. They are based tives. A description of these new models follows as well as an on a combination of what the author believes to be the best explanation of the rationale behind the choices that have been elements of existing eddy-viscosity models. made in different areas of the flow and an address to antici- There is a discrepancy between the large number of publica- pated criticism. tions about two-equation models and the slow pace of im- The A:-co model1 is the model of choice in the sublayer of the provement in accuracy that has been achieved since their intro- boundary layer. Unlike any other two-equation model, the duction. The basic problem of two-equation models, namely, A:-co model does not involve damping functions and, as will be their failure to correctly predict the onset and amount of shown, allows simple Dirichlet boundary conditions to be separation in adverse pressure gradient flows, is still unre- specified. Because of its simplicity, the k-co model is superior solved. Furthermore, there is no agreement on the standards to other models, especially with regard to numerical stability. by which to measure the improvement achieved by proposed Furthermore, it is as accurate as any other model in predicting new models, or alterations to existing models. Many times new the mean flow profiles. Wilcox1 has developed modifications models are based on theoretical concepts, which by themselves that allow the treatment of rough walls and surface mass involve severe assumptions about the nature of turbulence, injection which can be used in the new model without change. not even approximately satisfied in aerodynamic flows (homo- One point of criticism is that the &-co model (like many other geneous turbulence, small pressure gradients, low Reynolds models) does not correctly predict the asymptotic behavior of number, flow equilibrium, etc.). It has been the author's the turbulence as it approaches the wall. However, the Taylor experience that small changes (5-10%) in modeling constants series expansion of the Navier-Stokes equations that underlies can lead to a significant improvement (or deterioration) of the analysis is only valid in the immediate wall proximity. So model predictions. None of the available theoretical tools close to the surface the eddy viscosity is much smaller than the (dimensional analysis, asymptotic expansion theory, use of molecular viscosity and the asymptotic behavior of the mean direct numerical simulations (DNS) data, renormalization flow profile is independent of the asymptotic form of the group (RNG) theory, rapid distortion theory, etc.) can provide turbulence. Therefore, even if the turbulence model is not constants to that degree of accuracy. The only way to establish asymptotically consistent, the mean flow profile and the wall the validity of theoretical arguments under those conditions is skin friction are still predicted correctly. A second point of to carefully test the resulting model against a number of criticism is that the k -co model does not accurately represent challenging and well-documented research flows. Unfortu- the k and e distribution in agreement with DNS data. A nately, this is not general practice, and it is often unclear significant number of damping functions have been developed whether the improvements presented for one type of flow in the last years for the k-e model which lead to an improved agreement with DNS data. In Refs. 2 and 3, a number of k-e Presented as Paper 93-2906 at the AIAA 23rd Fluid Dynamics, models with different damping functions have been tested for Plasmadynamics, and Lasers Conference, Orlando, FL, July 6-9, a significant number of flows, with the conclusion that the 1993; received July 26, 1993; revision received Dec. 30, 1993; accepted specific form of the damping functions has little to no effect for publication Jan. 26, 1994. Copyright © 1993 by the American on the predicted velocity profiles and the skin friction of Institute of Aeronautics and Astronautics, Inc. No copyright is as- high-Reynolds-number flows. It should not be forgotten that serted in the United States under Title 17, U.S. Code. The U.S. the main (and often the only) information the mean flow Government has a royalty-free license to exercise all rights under the solver gets from the turbulence model is the eddy viscosity. It copyright claimed herein for Governmental purposes. All other rights is not clear why fitting the DNS data for k and e should lead are reserved by the copyright owner. *Research Scientist, Fluid Dynamics Division, MS 229-1. to an improved eddy-viscosity distribution. In the end, the Downloaded by UNIV OF CALIFORNIA SAN DIEGO on December 14, 2016 | http://arc.aiaa.org | DOI: 10.2514/3.12149 MENTER: EDDY-VISCOSITY TURBULENCE MODELS agreement with DNS data might only be a matter of interpre- lent shear stress. The resulting model will be called the shear- tation. In the sublayer, Wilcox equates the quantity k in his stress transport (SST) model. It is this second step that leads to model as being proportional to the normal component (with a major improvement in performance over both the original respect to the wall) of the turbulent kinetic energy. This inter- &-co and the standard k-e model. pretation leads to a very good agreement with experimental and DNS data. In cases where the agreement with DNS data is Turbulence Model considered important, the damping functions developed by New Baseline Model Wilcox4 can be applied to the present model. The idea behind the BSL model is to retain the robust and The A:-co model is also used in the logarithmic part of the accurate formulation of the Wilcox k-u model in the near wall boundary layer. It has been shown1'5 that the behavior of the region, and to take advantage of the freestream independence A:-co model in the logarithmic region is superior to that of the of the k-e model in the outer part of the boundary layer. To k-e model in equilibrium adverse pressure gradient flows and achieve this, the k-e model is transformed into a &-co formula- in compressible flows. tion. The difference between this formulation and the original In the wake region of the boundary layer, the A:-co model has A:-co model is that an additional cross-diffusion term appears to be abandoned in favor of the k-e model. The reason for this in the co equation and that the modeling constants are different switch is that the A:-co model has a very strong sensitivity to the (A small additional diffusion term is neglected in the transfor- (quiie arbitrary) freestream values co/ specified for co outside mation. It is shown in Ref. 9 that the term has virtually no the boundary layer. It has been shown in Ref. 6 that the eddy effect on the solutions). The original model is then multiplied viscosity in boundary and free shear layers can be changed by by a function F\ and the transformed model by a function more than 100% by simply reducing the value of co/. It has (1 - FI), and both are added together. The function F will be also been shown in Ref. 6 that the k-e model does not suffer designed to be one in the near wall region (activating the from this deficiency. There is no mathematical theory to date original model) and zero away from the surface. The blending which distinguishes between two-equation models that suffer will take place in the wake region of the boundary layer. The from the freestream dependency and those that do not. It is left-hand side of the following equations is the Lagrangian therefore of great importance that the influence of freestream derivative: D/Dt: = d/dt + values on the solutions of newly developed models is tested Original A:-co model: very carefully. The mathematical analysis of the behavior of two-equation DpA: dk models in adverse pressure gradient flows has been largely — - -E (D restricted to the logarithmic region.1'7 Although the behavior of the model in the logarithmic region is of importance, espe- cially in flows with moderate pressure gradients, it is the level 7i £l P) of the eddy viscosity in the wake region that ultimately deter- Dt v mines the ability of an eddy-viscosity model to predict strong adverse pressure gradient flows. This has been clearly demon- Transformed k-e model: strated by the improvement that the Johnson-King model8 DpA: dui achieved over standard algebraic models by reducing the wake £l 0) region eddy viscosity in adverse pressure gradient flows. The Dt dxi limited influence of the logarithmic region on the results for strong adverse pressure gradients is also evident in the failure Dpco of the original A:-co model to accurately predict pressure-in- duced separation (as will be shown later) despite its superior log-region characteristics. The basic idea behind the Johnson- 1 aA: aco King model is to enforce Bradshaw's observation that the - — — (4) CO OXj OXj principal turbulent shear stress is proportional to the turbulent kinetic energy in the wake region of the boundary layer. Now, Eq. (1) and Eq. (2) are multiplied by F and Eq. (3) and Enforcing this proportionality introduces a lag effect into the Eq. (4) are multiplied by (1 - FI) and the corresponding equa- equations that accounts for the transport of the principal tions of each set are added together to give the new model: turbulent shear stress. It will be shown that the classical for- mulation of the eddy viscosity in two-equation models violates DpA: dk Bradshaw's relation and thereby misses this important effect. + — 0* + °kp ) T- (5) In the new model the eddy-viscosity formulation will be mod- ified to take the transport effects into account. Dpco 7 d |~ dcol Finally, in free shear layers away from surfaces, the stan- x —— fa + 0^) —— -=- - dard k-e model will be utilized. There does not seem to be a dXj I dXj] Dt v dXj model that accurately predicts all free shear flows (wake, jet, 1 8k mixing layer) and the k-e seems to be a fair compromise. 2p(l - F!)(7 - — — (6) w2 To achieve the desired features in the different regions, the standard high-Reynolds-number version of the k-e model will Let 0! represent any constant in the original model (CT , . . .), be transformed to a A:-co formulation. It will then be multiplied </> any constant in the transformed k-e model (a , . . .) and </> by a blending function (1 — FI) and added to the original A:-co 2 k2 the corresponding constant of the new model (o . . .), then the model times FI . The blending function F\ will be designed to relation between them is: be one in the sublayer and logarithmic region of the boundary layer and to gradually switch to zero in the wake region. This means that the new model will be based on a A:-co formulation, (7) with the original Wilcox model activated in the near wall All constants, as well as the function F are given in the region and the standard k-e model activated in the outer wake l5 Appendix. region and in free shear layers. This first step leads to a new model that will be termed the baseline (BSL) model. The BSL Shear-Stress Transport Model model has a performance very similar to that of the original &-co model, but without the undesirable freestream dependency. One of the major differences between eddy-viscosity and full Reynolds-stress models, with respect to aerodynamic applica- In a second step, the definition of the eddy viscosity will be tions, is that the latter accounts for the important effect of the modified to account for the transport of the principal turbu- Downloaded by UNIV OF CALIFORNIA SAN DIEGO on December 14, 2016 | http://arc.aiaa.org | DOI: 10.2514/3.12149 1600 MENTER: EDDY-VISCOSITY TURBULENCE MODELS transport of the principal turbulent shear stress r - : -pu'v' Model Versatility and Generality (obvious notation) by the inclusion of the term The price for avoiding the freestream dependency and achieving the improved performance due to the modification DT dr in the eddy viscosity lies mainly in the necessary computation (8) D? = :Tt + l dx of the blending functions FI, F and the additional cross-diffu- sion term. The blending functions involve the distance from The importance of this term has clearly been demonstrated by the surface which, however, has to be computed only once (as the success of the Johnson-King (JK) model.8 Note that the long as there is no grid deformation). Note that the distance main difference between the JK model and the Cebeci-Smith from the surface is uniquely defined as being the shortest model lies in the inclusion of this term in the former, leading distance between the present point and all no-slip boundaries to significantly improved results for adverse pressure gradient (distance does not have to be measured normal to a surface— flows. The JK model features a transport equation for the e.g., backward-facing step). In most application codes, the turbulent shear stress T that is based on Bradshaw's assump- boundary points have a marker and the computation of the tion that the shear stress in a boundary layer is proportional to distance function can therefore be automated. The increase in the turbulent kinetic energy k: complexity from the Wilcox model to the present model is mainly in terms of coding. The overall computing time, as well T = (9) as the stability of the code are not affected. Once the model is implemented, it offers a wide variety of with ai being a constant. On the other hand, in two-equation options. An example is a two-layer k-e model12 with the origi- models, the shear stress is computed from: nal k-u model in the sublayer and the k-e model in the high- Reynolds-number region. This can be achieved by changing (10) T=/i, 0 the argument of F\ for the BSL model from Eq. (A9) (see Appendix) to: with Q = (du/dy). For conventional two-equation models, Eq. (10) can be rewritten to give: (13) /Product Production i ^ - = p i ~. —— —— (11) N Dissipat Dissipation i ^ The modification ensures that FI is zero for y+ > 70. This two-layer k-e model utilizes the superior sublayer characteris- as noted in Ref. 10. In adverse pressure gradient flows the tics of the £-co model in much the same way that the model in ratio of production to dissipation can be significantly larger Ref. 12 introduces an algebraic expression into the e equation. than one, as found from the experimental data of Driver,11 However, in the present approach the blending between the and therefore Eq. (11) leads to an overprediction of r. To two regions is performed automatically and without user input. satisfy Eq. (9) within the framework of an eddy-viscosity The versatility of the model makes it possible to give the model, the eddy viscosity is redefined in the following way: user a number of options, without making it necessary to program various models. a\k (12) Numerical Method The mean flow equations are solved by the INS3D code of where F is a function that is one for boundary-layer flows and Rogers and Kwak13 which is based on a pseudocompressibility zero for free shear layers. In an adverse pressure gradient method. Important details about the discretization of the tur- boundary layer, production of k is larger than its dissipation bulence model are given in Ref. 9. All computations have been (or Q>tf!co) and Eq. (12) therefore guarantees that Eq. (9) is performed on different grids to ensure that the presented satisfied whereas the original formulation v - k/u is used for solutions are grid independent. The airfoil computations were the rest of the flow. performed on a standard grid kindly provided by Rogers.14 To recover the original formulation of the eddy viscosity for The standard k-e model is coded as given in Ref. 15. free shear layers [where Bradshaw's assumption, expressed in Eq. (9) does not necessarily hold] the modification to the Results shear-stress transport (SST) model is limited to wall bounded Flat Plate Boundary Layer flows. This is achieved in the same way as it is for the BSL model by applying a blending function F (also defined in the To demonstrate the freestream dependency of the original appendix). For general flows Q is taken to be the absolute k-u model, flat plate zero pressure gradient boundary-layer value of the vorticity. computations with different freestream values for co have 2.0 .2.0 ———— k-w BSL (high ——— k-w org. (high «f) - - k-o) org. (low Wf) - - - k-w BSL (low 1.5 1.5 1.0 £ 1.0 0.5 0.5 0.0 0.0 0 12.34 5 0 1 234 5 ix /u.6'x100 Fig. 1 Freestream dependency of the eddy viscosity for the original and the BSL A:-co model. Downloaded by UNIV OF CALIFORNIA SAN DIEGO on December 14, 2016 | http://arc.aiaa.org | DOI: 10.2514/3.12149 MENTER: EDDY-VISCOSITY TURBULENCE MODELS the models to assess their potentials for these types of flows. The author has reached a similar conclusion in Ref. 10. It is therefore important to test models under more demanding conditions, with stronger adverse pressure gradients and possi- bly separation. The following flowfield, reported by Driver,11 k-co SST - has proven to be a highly self-consistent and demanding test -w BSL - case. k-w org. In Driver's flow, a turbulent boundary layer develops in.the k-e JL I Experiment - axial direction of a circular cylinder. An adverse pressure gradient is imposed by diverging wind tunnel walls and suction -0.2 applied at these walls. The pressure gradient is strong enough to cause the flowfield to separate. The inflow Reynolds num- ber is 2.8 • 105 based on the diameter D of the cylinder (140 mm). A 60 x 3 x 60 grid10 was used for the present computa- Fig. 2 Wall pressure distribution for Driver's adverse pressure-gradi- tions. A computation on a 100 x 3 x 100 grid gave almost ent flow. identical results. Figure 2 shows the wall pressure distribution for Driver's flow as computed by the different models. The SST model been performed. For the first set of computations, the correct gives superior results to the other models due to its ability to freestream values as given in Ref. 6 have been specified at the account for the transport of the principal turbulent shear inflow boundary freestream for both the original and the BSL stress. As expected, the JL k-e model produces the least accu- k-u model. Then, the preceding value was reduced by four rate results, with the BSL and the original £-co model being orders of magnitude and the computations were repeated with close to each other in the middle. both models. Note that the freestream value of k was also Figure 3, depicting the wall shear-stress distribution for reduced to keep the freestream value of the eddy-viscosity Driver's flow, shows that the SST model predicts the largest constant (the freestream value of the eddy viscosity has no amount of separation, whereas the JL model stays firmly influence, as long as it is small compared to its values inside attached. Again, the BSL and the original A:-co model produce the boundary layer). Figure 1 shows eddy-viscosity profiles for very similar results. the original and the BSL A:-co model. The eddy viscosity of the The differences between the models can be seen in Fig. 4, original model changes by almost 100% due to the changes in which shows the velocity profiles. The SST model clearly co/, whereas the BSL model gives the same results for both produces the best agreement with the experiments. The larger cases. The strong sensitivity of the original model to co/ is displacement effect predicted by this model is reflected in the clearly unacceptable and can lead to a severe deterioration of flattening of the c distribution as was observed in Fig. 2. The the results for complex flows, as will be shown later. Results p original k -co model predicts slightly better results than the BSL of the SST model are also independent of co/. A more detailed model, and the JL k-e model shows very little sensitivity to the study of the freestream dependency can be found in Refs. 6 pressure gradient, as was already reflected in Figs. 2 and 3. and 9. The reasons for the different behavior of the models can be In each of the following comparisons between the different seen in the following two pictures. Figure 5 compares turbu- models, co/was always chosen according to the formula given lent shear-stress profiles at different stations. The JL model in Ref. 6. Zero pressure gradient flat plate boundary layer computa- tions are given in Ref. 9. All models give good agreement with the experimental correlations for u + versus y+ and c/. The 0.4 k-« SST A:-co models can be run with the first gridpoint as far out as k-w BSL k—u orq. y+ = 3 without a deterioration of the results. k-e JL 0.2 Experiment Free Shear Layers x_ For free shear layers the SST and the BSL models reduce to u~ the same model (F\ = 0; F = 0), and are virtually identical to 0.0 the standard k-e model. Because the behavior of the k-e model for free shear layers is well known, and because of space limitations, results are not shown here, but can be found in -0.2 Ref. 9. Reference 9 also shows the ambiguity of the results of -4 -2 0 the original A:-co model1 with respect to the freestream values. x/D Adverse Pressure Gradient Flows Fig. 3 Wall shear-stress distribution for Driver's adverse pressure- gradient flow. One of the most important aspects of a turbulence model for aerodynamic applications is its ability to accurately predict adverse pressure gradient boundary-layer flows. It is especially important that a model be able to predict the location of flow separation and the displacement effect associated with it. The test case most widely used to measure the performance of turbulence models under adverse pressure gradient condi- tions is the flow reported by Samuel and Joubert.16 Results for this flow are shown in Ref. 9 and are not reproduced here due to space limitations. It was found in Ref. 9 that all three A:-co models reproduce the experimental data very well, whereas the JL k-e model gives values that are too high for c/. The small differences between the solutions reported in Ref. 9, especially between the different A:-co models, do not allow final conclusions about the abilities of the models to predict adverse pressure gradient flows. It appears that the Samuel- Fig. 4 Velocity profiles for Driver's adverse pressure-gradient flow at x/D = -0.091, 0.363, 1.088, 1.633, and 2.177. Joubert flow does not pose a sufficiently strong challenge to Downloaded by UNIV OF CALIFORNIA SAN DIEGO on December 14, 2016 | http://arc.aiaa.org | DOI: 10.2514/3.12149 1602 MENTER: EDDY-VISCOSITY TURBULENCE MODELS obviously predicts significantly higher shear-stress levels than the other models in the region where separation is approached. 0.4 This in turn leads to the firmly attached velocity profiles of Fig. 4. The differences between the models can also be seen from the eddy-viscosity distributions. Figure 6 shows the max- 0.2 imum value of the kinematic eddy-viscosity profiles for all streamwise (x) stations, nondimensionalized by u d*. The SST k-o SST model predicts the reduction of this quantity due to the ad- - - - - k-w BSL 0.0 verse pressure gradient in very good agreement with the exper- ------- k—« org. iments. The BSL and the original A:-co model are very close to _..._... _ k € JL each other up to separation (around x/D = 0), whereas the ^ Experiment original model is closer to the experiments in the recovery -0.2 region. Both models give consistently too high values for the 10 20 30 maximum eddy viscosity in the adverse pressure gradient re- x/H gion. The k-e model falls only barely below the value of Fig. 7 Wall shear-stress distribution for backward-facing step flow. 0.0168 recommended by Clauser for equilibrium boundary layers (and used in the Cebeci-Smith model) and thereby fails to account for the nonequilibrium effects altogether. Backward-Facing Step Flow Results for the flow over a backward-facing step as reported by Driver and Seegmiller17 will be discussed next. This flow- field was a test case in the 1981 Stanford conference for the evaluation of turbulence models. However, most of the com- putations at the time were performed on comparatively coarse grids and there is substantial evidence that significantly finer grids have to be used to obtain grid-independent results.18 The present computations have been performed on a 120 x 120 grid, with substantial grid refinement near the step. As with Fig. 8 Velocity profiles for backward-facing step flow at the stream- the other flowfields, a grid refinement study was made. The wise locations: x/H = 2.0, 4.0, 6.5, 8.0, 14.0, and 32.0. present results are virtually identical to those performed on a 90 X 90 and on a 240 x 240 grid. Figure 7 shows a comparison of computed and experimental the four models are 6.5 (SST), 5.9 (BSL), 6.4 (original A:-co), skin friction distributions. The k-u models all perform signif- and 5.5 (JL k-e) compared to a value of about 6.4 in the icantly better than the k-e model. The reattachment length of experiments. The reattachment length predicted by the k-e model is better than previously reported, certainly as a result of the fine grid employed in the present computations (see also 0.6 ————— k-w SST - - - - k-w BSL Ref. 18). However, the model predicts variations of c/which ------- k-a org. are significantly too large in the recirculation and the reattach- — --- — - - k-e JL O Exp. ment region. 0.4 Figure 8 shows a comparison of the velocity profiles. All models fail to capture the relaxation downstream of reattach- ment correctly. The results of Ref. 19 show that this is also true for a more complex model which accounts for anisotropy 0.2 effects. NACA 4412 Airfoil Flow The following set of computations is for the flow around a NACA 4412 airfoil at 13.87 deg angle of attack. The Reynolds 0.000 0.005 0.010 0.015 -uV/LL2 number with respect to the chord length is Re = 1.52-106. Experimental data for this flow have been reported by Coles Fig. 5 Turbulent shear-stress profiles for Driver's adverse pressure- and Wadcock.20 The grid for the computations consists of gradient flow at x/D = -0.091, 0.363, 1.088, 1.633, and 2.177. 241 x 61 points and was made available by Rogers.14 It is similar to the one used in Ref. 21. Figure 9 shows a comparison of the computed and the experimental velocity profiles at different streamwise stations. k-w SST The results are similar to those for the separated case of k-« BSL k-w org. Driver, Fig. 4. Again, the SST model predicts the displace- k-6 JL o 3 ment effect in very good agreement with the experiments. The Clauser (Cebeci-Smith) Experiment BSL model is showing some response to the pressure gradient, and produces results similar to those reported in Ref. 21 for the Baldwin-Barth model. Another interesting result of this computation is that the original A:-co model predicts velocity profiles even further away from the experiments than does the Jones-Launder k-e model. The reason for the poor perfor- mance of the original k-w model lies in its freestream depen- dency (for details see Ref. 9). To prove this point, Fig. 9 also shows computations for the SST model and the original k-u -2 model with different freestream values for co. In the curves x/D labeled with high co/, the value of co was prevented from decaying between the inflow boundary and the leading edge of Fig. 6 p;(max)/i/<,5* distribution for Driver's adverse pressure-gra- dient flow. the airfoil, so that the freestream value co/ was about fifty Downloaded by UNIV OF CALIFORNIA SAN DIEGO on December 14, 2016 | http://arc.aiaa.org | DOI: 10.2514/3.12149 MENTER: EDDY-VISCOSITY TURBULENCE MODELS 1603 layer flows, the BSL model is very similar to the original &-co 0.10 k-w SST k—o j SST hiqh o; f model, but it avoids the strong freestream sensitivity of that k-w BSL f 0.08 - k— w org. model. k—0 3 orq. hiqh u* k-e JL In the second step, a modification to the eddy viscosity has 0.06 Experimi been introduced. It is based on the philosophy underlying the 0.04 Johnson-King model, which holds that the transport of the principal turbulent shear stress is of vital importance in the 0.02 prediction of severe adverse pressure gradient flows. The re- 0.00 sulting model is termed the shear-stress transport (SST) model. Both models have been carefully fine tuned and tested for a Fig. 9 Velocity profiles on the upper surface of a NACA 4412 airfoil large number of challenging research flows. The original £-co, at 13.87 deg angle of attack; streamwise stations x/c = 0.675, 0.731, as well as the standard k-e model are included in the compar- 0.786, 0.842, 0.897, and 0.953. ison. As expected, the BSL model gives results very close to the original A:-co model of Wilcox but avoids its freestream dependency. The SST model leads to a significant improve- M = 0.925 1.0 ment for all flows involving adverse pressure gradients and k-o SST should be the model of choice for aerodynamic applications. k-w BSL It is the only available two-equation model that has demon- k-cj orq. k-e std. strated the ability to accurately predict pressure-induced sepa- Experiment 0.5 ration and the resulting viscous-inviscid interaction. The new models require an increased amount of program- ming effort compared to the original A:-co model. However, once programmed, the new models consume only insignifi- cantly more computing time and more importantly, they have proven to be very stable even in complex applications.23 The concept underlying the new models is very flexible and lends itself to a multitude of different combinations. An example -0.5 given in the text is a two-layer k-e model. 0.4 0.6 0.8 1.2 1.6 x/c It is the author's conviction that a turbulence model has to be tested rigorously for a large number of flows, to establish Fig. 10 Comparison of surface pressure distributions for transonic the boundaries of its usefulness. Because of the limitations of bump flow at M = 0.925. the available theoretical tools and the severe assumptions in- volved, this is also true for models based on more theoretical arguments. The new models are presently tested for transonic times larger than it was in the previous computations. The flows with very encouraging results. An early version of the change in co/had very little impact on the computation with the SST model has been tested for complex three-dimensional SST model (small changes might be due to a slight influence flows in Ref. 24. The results compare very favorably with the on the transition behavior), whereas the original A:-co model results of a full Reynolds-stress model, but significantly more predicts significantly different results. The results of the origi- testing in three-dimensional flows will be necessary. nal model for the high co/ are very close to those of the BSL model, as had to be expected from the derivation of the Appendix: Baseline and Shear-Stress Transport Models models. This example clearly shows the dangers of using the Baseline Model original A:-co model for industrial applications. DpA: dUj (Al) Transonic Bump Flow The final test case is the axisymmetric transonic shock- Dpco i . wave/turbulent boundary-layer experiment of Bachalo and - ~ Tfj —L - ftoco2 + — 0* Johnson.22 In this experiment, an axisymmetric boundary layer interacts with a shock wave created by a circular arc. It is beyond the scope of this paper to present a detailed study of (A2) + 2(1 - transonic flows and only the highest Mach number case (M = 0.925) will be shown. The number of gridpoints used The constants </> of the new model are calculated from the was 150 x 3 x 80. Grid independence was established by using constants, <t>\, fa , as follows: different grids (129 x 3 x 60 and 180 x 3 x 100). Figure 10 shows the wall pressure distribution computed by the different (A3) models, compared with the experiment. The SST model pre- dicts significantly better results than the other models, due to The constants of set 1 (</>i) are (Wilcox): its improved transport features. Detailed comparisons for transonic flows will be presented in the future. 0*1 = 0.5, o i = 0.5, fa = 0.0750 (A4) /3* = 0.09, K = 0.41, = fa/0* - a K Conclusions Tl wl Two new turbulence models have been developed on a The constants of set 2 (</> ) are (standard k-e): strictly empirical basis. They are combinations of what the 2 author considers to be the best available elements of existing a*2=1.0, o«2 = 0.856, 02 = 0.0828 eddy-viscosity models to date. Both models are based on a A:-co (A5) formulation which is superior to other formulations with re- 0* = 0.09, K = 0.41, = T2 gard to numerical stability. In a first step, a new baseline (BSL) model has been derived. It utilizes the original A:-co With the following definitions: model in the sub- and log-layer and gradually switches to the standard k-e model in the wake region of the boundary layer. v = — (A6) The k-e model is also used in free shear layers. For boundary- CO Downloaded by UNIV OF CALIFORNIA SAN DIEGO on December 14, 2016 | http://arc.aiaa.org | DOI: 10.2514/3.12149 1604 MENTER: EDDY-VISCOSITY TURBULENCE MODELS . 500A arg = max ( 2 • (A16) (A7) Important detail! In applying this model, it is important that the reader be ) = tanh (arg{) (A8) aware of the following ambiguity in the formulation of the production term of co for the SST model. The definition of the production term of co is sometimes written as: V*50 0 arg, = (A9) co du; P* = 7 T r —- (All) where y is the distance to the next surface and CD is the ku which introduces the nondimensional group v (co//:) in front positive portion of the cross-diffusion term of Eq. (A2): of the strain rate tensor. In the original and in the BSL model this group is equal to one and the two formulations for P are therefore identical. This is not the case for the SST model because of Eq. (A14). The SST model has been calibrated with (2pa - — -^» 1Q-20) (A10) w2 \ a) oXj oXj J respect to Eq. (A2) and Eq. (A17) should therefore not be used. Acknowledgments The term arg! obviously goes to zero far enough away from The author wants to thank S. E. Rogers and J. Bardina for solid surfaces because of the l/y or l/y2 dependency in all providing the mean flow solvers for the incompressible and three terms in Eq. (A9). Inside a boundary layer the three the transonic computations, respectively, as well as their assis- arguments in Eq. (A9) have the following purpose: the first tance in implementing the turbulence models. argument is the turbulent length scale divided by y. It is equal to 2.5 in the log layer and goes to zero towards the boundary- layer edge. The second argument ensures that FI is equal to References one in the sublayer [note that co goes like l/y2 in the near wall !Wilcox, D. C., "Reassessment of the Scale-Determining Equation region and is proportional to l/y in the log region, so that for Advanced Turbulence Models," AIAA Journal, Vol. 26, No. 11, l/(co.y2) is constant near the surface and goes to zero in the log 1988, pp. 1299-1310. region]. The third argument is an additional safeguard against 2Michelassi, V., and Shih, T.-H., "Elliptic Flow Computation by the freestream-dependent solution. It can be shown that the Low Reynolds Number Two-Equation Turbulence Models," NASA last argument ensures that argj goes to zero near the bound- TM-105376, CMOTT-91-11, Dec. 1991. 3Wilcox, D. C., "Comparison of Two-Equation Turbulence Mod- ary-layer edge in case the "degenerate" solution given in Ref. els for Boundary Layers with Pressure Gradient," AIAA Journal, 6 is approached. As arg! goes to zero near the boundary-layer Vol. 31, No. 8, 1993, pp. 1414-1421. edge, so does FI so that the standard k-e is used in that region. 4Wilcox, D. C., "A Half Century Historical Review of the k-w The following choice of freestream values is recommended: Model," AIAA Paper 91-0615, Jan. 1991. 5Huang, P. G., Bradshaw, P., and Coakley, T. J., "Assessment of Closure Coefficients for Compressible-Flow Turbulence Models," (All) NASA TM-103882, March 1992. 6Menter, F. R., "Influence of Freestream Values on k-u Turbu- lence Model Predictions," AIAA Journal, Vol. 30, No. 6, 1992, pp. 1651-1659. where L is the approximate length of the computational domain. 7Rodi, W., and Scheurer, G., "Scrutinizing the k-e Model Under The boundary condition for co at a solid surface is: Adverse Pressure Gradient Conditions," Journal of Fluids Engineer- ing, Vol. 108, June 1986, pp. 174-179. 8Johnson, D. A., and King, L. S., "Mathematically Simple Turbu- at y = 0 (A12) co =10 lence Closure Model for Attached and Separated Turbulent Boundary Layers," AIAA Journal, Vol. 23, No. 11, 1985, pp. 1684-1692. 9Menter, F. R., "Zonal Two Equation k-w Turbulence Models for Aerodynamic Flows," AIAA Paper 93-2906, July 1993. where Ayi is the distance to the next point away from the wall. 10Menter, F. R., "Performance of Popular Turbulence Models for Equation (A 12) simulates the smooth wall boundary condition Attached and Separated Adverse Pressure Gradient Flows," AIAA of Ref. 1 as long as 4y,+ <3. Journal, Vol. 30, No. 8, 1992, pp. 2066-2072. nDriver, D. M., "Reynolds Shear Stress Measurements in a Sepa- rated Boundary Layer," AIAA Paper 91-1787, June 1991. Shear-Stress Transport Model 12Rodi, W., "Experience with a Two-Layer Model Combining the The SST model is identical to the preceding formulation, k-e Model with a One-Equation Model Near the Wall," AIAA Paper except that the constants, 0i, have to be changed to: 91-0216, Reno, NV, Jan. 1991. Set 1 (SST inner): 13Rogers, S. E., and Kwak, D., "An Upwind Differencing Scheme for the Time-Accurate Incompressible Navier-Stokes Equations," AIAA Paper 88-2583, June 1988. (7*! = 0.85, cr = 0.5, 0! = 0.0750, ^ = 0.31 wl 14Rogers, S. E., private communication. NASA Ames, 1993. (A13) 15Launder, B. E., and Sharma, B. I., "Application of the Energy- 18* = 0.09, /c = 0.41, 7i Dissipation Model of Turbulence to the Calculation of Flow near a Spinning Disk," Letters in Heat and Mass Transfer, Vol. 1, No. 1, 1974, pp. 131-138. and the eddy viscosity is defined as: 16Samuel, A. E., and Joubert, P. N., "A Boundary Layer Develop- ing in an Increasingly Adverse Pressure Gradient," Journal of Fluid Mechanics, Vol. 66, Pt. 3, 1974, pp. 481-505. (A14) 17Driver, D. M., and Seegmiller, H. L., "Features of a Reattaching Turbulent Shear Layer in Divergent Channel Flow," AIAA Journal, Vol. 23, No. 2, 1985, pp. 163-172. 18Thangam, S., and Speciale, C. G., "Turbulent Separated Flow where 0 is the absolute value of the vorticity. F is given by: Past a Backward-Facing Step: A Critical Evaluation of Two-Equation Turbulence Models," Institute for Computer Applications in Science F = tanh (arg2) (A15 and Engineering) , ICASE Rept. 91-23, 1991. Downloaded by UNIV OF CALIFORNIA SAN DIEGO on December 14, 2016 | http://arc.aiaa.org | DOI: 10.2514/3.12149 MENTER: EDDY-VISCOSITY TURBULENCE MODELS 19Abid, R., Speciale, C. G., and Thangam, S., "Application of a sonic Turbulent Boundary Layer Separation Generated on an Axisym- New k-r Model to Near Wall Turbulent Flows," AIAA Paper 91- metric Flow Model," AIAA Paper 79-1479, June 1979. 0614, Jan. 1991. 23Rogers, S. E., Menter, F. R., Mansour, N. N., and Durbin, P. A., 20Coles, D., and Wadock, A. J., 'Tlying-Hot-Wire Study of Flow "A Comparison of Turbulence Models in Computing Multi-Element Past an NACA 4412 Airfoil at Maximum Lift," AIAA Journal, Vol. Airfoil Flows," AIAA Paper 94-0291, Jan. 1994. 17, No. 4, 1979, pp. 321-328. 24Menter, F. R., "Assessment of Higher Order Turbulence Models 21Rogers, S. E., Wiltberg, N. L., and Kwak, D., "Efficient Simula- for Complex 2D and 3D Flow Fields," Proceedings of the Second tion of Incompressible Viscous Flow Over Single and Multi-Element International Symposium on Engineering Turbulence Modeling and Airfoils," AIAA Paper 92-0405, Jan. 1992. Measurements (Florence, Italy), Elsevier, Amsterdam, 1993, pp. 22Bachalo, W. D., and Johnson, D. A., "An Investigation of Tran- 229-239. dEr rom writing clerical procedures to nuclear power plant procedures.... this book provides step-by-step help! Procedure Writing Principles and Practices Douglas Wieringa, Christopher Moore, and Valeric Barnes from booting up a per- parts of these documents principles govern the way Procedures are instruc- tions, and this book ex- sonal computer to oper- may be considered pro- it should be written. The cedures if they present authors drawon their more plains how to write in- ating a nuclear power plant during an emer- instructions. No matter than ten years of experi- structions so that others gency. Plans, mission how simple or complex ence and present their prin- can understand them. statements, proposals, the procedure is, certain ciples in this book. Procedures can range from simple to complex; and technical articles are 1993, 211 pages, Paperback they describe anything not procedures, although ISBN 0-935470-68-9, $29.95, Order #: PPP-1(945) Place your order today! Call 1 -800/682-AIAA Sales Tax: CA residents, 8.25%; DC, 6%. For shipping and handling add $4.75 for 1-4 books (call &AIAA for rates for higher quantities). Orders under $100.00 must be prepaid. Foreign orders must be American Institute of Aeronautics and Astronautics prepaid and include a $20.00 postal surcharge. Please allow 4 weeks for delivery. Prices are Publications Customer Service, 9 jay Could Ct., P.O. Box 753, Waldorf, MD 20604 subject to change without notice. Returns will be accepted within 30 days. Non-U.S. residents FAX 301/843-0159 Phone 1-800/682-2422 8 a.m. - 5 p.m. Eastern are responsible for payment of any taxes required by their government. | Downloaded by UNIV OF CALIFORNIA SAN DIEGO on December 14, 2016 | http://arc.aiaa.org | DOI: 10.2514/3.12149
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