AIAA SciTech Forum 9-13 January 017, Grapevine, Texas 55th AIAA Aerospace Sciences Meeting AIAA 017-0495 Wall-modeled large-eddy simulation of transonic buffet over a supercritical airfoil at high Reynolds number Yuma Fukushima 1 and Soshi Kawai Tohoku University, 6-6-01 Aramaki-Aza-Aoba, Aoba-ku, Sendai 980-8579, JAPAN In this study, the wall-modeled large-eddy simulation (LES) of transonic buffet phenomena over the OAT15A supercritical airfoil is conducted. The computational results are compared with zonal detached eddy simulation (DES) and experiment. By using the wall-modeled LES, the buffet onset is successfully predicted. The small separation near the trailing edge is also accurately estimated with buffet phenomena. Velocity properties show good agreement with experiment except the region in which the shock wave moves. From the trace of the shock wave movement, the buffet frequency which nearly equals to experimental value is obtained. The self-sustained oscillation mechanisms are investigated from the computational results. As a results, Lee s acoustic feedback loop model cannot predict the buffet frequency. On the other hand, the possibility of the proposed model is confirmed. Nomenclature ρ = density u i = velocity p = static pressure E = total energy T = temperature γ = ratio of specific heats R = gas constant τ ij = stress tensor q j = heat flux vector μ = dynamic viscosity μ t = turbulent eddy viscosity β = bulk viscosity S ij = strain rate tensor Pr = Prandtl number Pr t = turbulent Prandtl number c s = sound speed μ t,wm = turbulent eddy viscosity in wall model Pr t,wm = turbulent Prandtl number in wall model U = wall-parallel velocity c p = specific heat at constant pressure κ = von Karman constant y + = distance from the wall in wall units u + = wall-parallel velocity in wall units Cp = pressure coefficient h wm = interface height between large-eddy simulation and wall model t=t u /c = nondimensionalized time x s = shock position Re c = Reynolds number based on chord length c = chord length = free-stream Mach number M 1 Postdoctral Fellow, Department of Aerospace Engineering, Member AIAA. Associate Professor, Department of Aerospace Engineering, Senior Member AIAA. 1 Copyright 017 by Yuma Fukushima and Soshi Kawai. Published by the, Inc., with permission.
α St s f s = angle of attack = Strouhal number of shock oscillation = nondimensionalized frequency of shock oscillation I. Introduction Computational fluid dynamics (CFD) has achieved a significant progress in its algorithms and applications with increasing computational resource and became a necessary tool for developing fluid machines such as aircrafts. Due to the progress in the turbulence modelling, reliable results can be obtained from the Reynolds averaged Navier-Stokes (RANS) simulation of the cruise condition of aircrafts. However, it is still difficult to predict the flight performance in off-design conditions including flow separation. 1 The reliability in the off-design condition including flow separation is one of the challenging topics for current CFD. It is important for current CFD to enhance the reliability for unsteady phenomena and expand CFD applicability for a whole flight envelope. Transonic buffet is one of the famous unsteady phenomena related to a flying aircraft. Buffet phenomena involve a vibration related to weak flow separation after a shock wave on an airfoil and affect the comfort and safeness of aircraft operation. Many computational and experimental researches have been conducted to predict the buffet onset and understand a mechanism of the buffet phenomena. -15 For the computational investigation, large eddy simulation (LES) is suitable for such kind of unsteady flow including the flow separation. However, a turbulent boundary layer at high Reynolds number induces an unreachable computational cost in a wall-resolved LES. Chapman estimated the required number of grid points for the wall-resolved LES as N total =Re c 1.8, where Re c is a chord Reynolds number. 16 Choi and Moin revisited this estimation as N total =Re c 13/7. 17 To avoid the large computational cost, methods of modelling an inner-layer turbulence are proposed instead of resolving the turbulence in the inner part of the boundary layer. Especially for buffet phenomena, zonal detached eddy simulation (DES) has been used by Deck 4 and applied to the transonic buffet over the OAT15A supercritical airfoil. The results showed a good agreement with experiment by changing the thickness of a RANS region manually. 15 Subsequently, the new version of DES was proposed as delayed DES and applied to the buffet problem. 10,18 Another category of modelling the inner-layer turbulence is a wall-stress-model. 19 This kind of model is mainly based on the momentum conservation in a parallel shear flow. In the wall-modeled LES, only the inner layer of the boundary layer is modeled, and the outer layer is computed by LES. Kawai and Larsson pointed out the numerical and subgrid modeling errors derived from the LES computation near the wall and proposed the simple solution to bypass this errors in the wall-modeled LES computation. 0 This model was extended to the flow without equilibrium assumption and applied to a shock/boundary-layer interaction problem 1 and a separated flow over an airfoil near stall condition at high Reynolds number. From results, it is considered that the wall-models have potential to predict the buffet phenomena involving shock wave and turbulent boundary layer interactions. It is also noted that the wallmodels can predict the turbulent boundary layer of high Reynolds number without any experimental parameters. In this study, the proposed wall-models are applied to the transonic buffet simulation around the OAT15A to investigate the capability of the wall-modeld LES. Buffet flow conditions are Re c=3.0 10 6, α=3.5deg, and M =0.73. Computational results are compared with the experimental data 3 and the zonal DES. 4 From the computational results, the self-subtained oscillation mechanisms are investigated. First, the buffet frequency is predicted by the acoustic feedback loop model. Next, we propose a new self-sustained oscillation model. The possibility of the proposed model is investigated. II. Numerical Methods A. Outer-layer LES equations The compressible spatially-filtered Navier-Stokes equations are solved in the LES mesh that is designed to resolve the outer-layer turbulence with modeled shear stresses and heat fluxes at the wall u j 0 (1) t x j p ij ui uiu j t x x x j i j E q E pu j ijui t x x x j j j j () (3)
p 1 E u 1 p RT k where the quantities are spatially-filtered and ρ is the density, u is the velocity, p is the static pressure, E is the total energy, T is the temperature, γ is the ratio of specific heats, and R is the gas constant. The stress tensor τ ij and heat flux vector q j are ij t S ij t Skk 3 (6) S ij 1 u u i j xj x i 1 t cs q j (8) 1Pr Prt x j where μ is the dynamic viscosity that is computed by Sutherland s law, β is the bulk viscosity, Pr is the Prandtl number, and c s is the sound speed. A selective mixed-scale model 3 is used to calculate the sub-grid scale turbulent eddy viscosity μ t and turbulent Prandtl number Pr t=0.9. B. Inner-layer equilibrium wall model The wall model with equilibrium assumption 0 solves the equilibrium boundary layer equations in an overlapping layer between the wall location y=0 and the interface height y=h wm d du t,wm 0 (9) dy dy d du t,wm dt t,wm U cp 0 (10) dy dy Pr Pr t,wm dy where U and T are the wall-parallel velocity and the temperature. According to the equations, solving the equilibrium wall model only needs the wall-normal grid connectivity. These equations are derived from the conservation equations for streamwise momentum and total energy with use of the standard approximations in equilibrium-boundary-layer flow. The pressure is assumed to be wall-normal independent and equal to the outer-layer LES solution at the interface height y=h wm. The mixing-length eddy-viscosity model with near-wall damping is used to determine the μ t,wm in the inner-layer wall model as (11) w t,wm y D D 1 exp y / A where τ w /ρ is the velocity scale in a boundary layer with the local instantaneous density and wall stress, y + is the distance from the wall in wall units and A + =17. The von Karman constant κ=0.41 and turbulent Prandtl number Pr t,wm=0.9. In this study, μ t,wm is computed from x/c=0.07 on both sides of the airfoil to model the forced laminarturbulent transition according to the experiment. (4) (5) (7) (1) C. Numerical schemes The spatial derivatives for interior nodes LES equations are evaluated by the sixth-order compact differencing scheme. 4 At boundary points 1 and and correspondingly at imax and imax-1, second-order one-sided and secondorder central formulas are employed. Aliasing errors are contained by applying an eight-order low-pass filter to the conserved variables with a filtering coefficient of 0.48. 5 Since the LES mesh only resolves the outer-layer turbulence and does not resolve the viscous sublayer, the first grid point off the wall in the LES mesh is in the log-layer. Due to the large mesh size, the time-step size can be increased by one- or two-order of magnitude depending on Reynolds number compared to the traditional wall-resolved LES. The third-order total variation diminishing Runge-Kutta scheme 6 is used for time integration. The localized artificial diffusivity method is used to robustly capture the shock waves while using the sixth-order compact differencing scheme. 7 3
The boundary condition at y=0 for the wall model equations is the adiabatic non-slip condition. At y=h wm, the wall parallel velocity U and the temperature T are imposed from the instantaneous LES solution. After solving the system of two coupled ordinary differential equations, the shear stress τ w and the heat flux q w at the wall can be directly computed and fed back to the LES as a boundary condition at the wall. D. Computational conditions In the wall-modeled LES computation, at least 0 mesh points shold resolve the boundary layer. Because the required grid resolution depends on the local boundary layer thickness, the small mesh size should be designed near the transition region to resolve the thin boundary layer and smoothly stretched in the streamwise direction along with the boundary layer development. A C-type structured mesh is generated as shown in Fig. 1. Outer boundary is located 80c away from the airfoil in η direction. The spanwise length is set to 0.065c which is the maximam separation width at x/c=0.75. Mesh points are N ξ N η N ζ=4603 169 565. The mesh resolution is Δξ + =Δη + =Δζ + =9.6 based on the wall unit at x/c=0. and h wm + =60. The nondimensionalized time step is 1.5 10-5 which results in the maximum Courant-Friedrichs-Levy number of 0.4. ξ Airfoil η y x (a) overall view y y x x z (b) two-dimensional grid, every tenth grid point (c) three-dimensional grid, every tenth grid point Fig. 1 Wall-modeled LES mesh III. Results and Discussions Figure shows the instantaneous Mach number distributions when the shock wave is at the most upstream and downstream position. During the shock oscillation, the shock wave moves from about 47%c to 59%c. At the trip position, expansion wave appears regardless of shock wave position because of the forced transition (Fig. (c)(d)). Behind the shock wave, the boundary layer is separated, and separated region is larger when the shock wave is in the upstream position (Fig. (e)(f)). Near the trailing edge, the size of separation largely varies depending on the shock wave position. (Fig. (g)(h)). Figure 3(a) shows the averaged Cp distribution. To investigate the capability of the buffet onset prediction, the result of RANS computation with Baldwin-Lomax turbulence model 8 at buffet condition and the wall-modeled LES 4
computation at non-buffet condition (Re c=3.0 10 6, α=3.5deg, and M =0.715) are also shown. In the RANS computation, the shock wave does not moves, and typical Cp distribution is obtained. Zonal DES can predict the shock oscillation by tuning the size of RANS region. 15 However, the region in which the shock wave moves is estimated more upstream than experiment. Furthermore, the separation near the trailing edge is predicted larger. At the nonbuffet condition (M =0.715) of wall-modeled LES computation, the shock wave does not moves as well as RANS computation. Wall-modeled LES at the buffet condition (M =0.73) can predict the shock oscillation and Cp slope is also obtained. In addition, the small separation near the trailing edge is precisely predicted. The region in which the shock wave moves has some difference between the experiment and the wall-modeled LES. Previous studies pointed out the difference between setting and actual flow conditions in the experiment of OAT15A airfoil. 9,30 With confinement effect of the wind tunnel wall, the shock wave position moves more upstream than that in the infinitely far field condition. Figure 3(b) shows the surface pressure fluctuation. Zonal DES computation estimates the larger pressure fluctuation than experiment near the shock wave and the trailing edge. Wall-modeled LES provides small fluctuation at non-buffet condition and large fluctuation at buffet condition. At the buffet condition, the shock wave moves at more downstream region in the wall-modeled LES. Therefore, the flow is accelerated and the shock wave become stronger than that in the experiment. As a results, the pressure fluctuation becomes larger. Figure 4(a) shows the wall-normal velocity compared with log-law (ln(y + )/0.41+5.) at x/c=0.3 from h wm + =60. The velocity distribution satisfies the log-law. Figure 4(b) also shows the Reynolds stress distribution compared with RANS results at x/c=0.3 from h wm+ =60. Turbulent boundary layer is fully developed upstream of the shock wave. Figure 5 shows the velocity distribution. In the zonal DES computation by Deck, the computational region is divided in some zones, and the zones upstream of the shock are computed by D RANS simulation. Therefore, the velocity fluctuation of the zonal DES is much smaller than the others at x/c=0.8. Wall-modeled LES can predict the velocity fluctuation as shown in Fig. 5(b). At the region in which the shock wave moves, the averaged velocity and the velocity fluctuation depends on the shock wave position. In the zonal DES computation, the shock wave travels up to x/c=0.3. Therefore, the shock wave and the low speed region behind the shock wave reach up to x/c=0.35. As a results, the averaged velocity is estimated smaller and the velocity fluctuation is overestimated than experiment. Wallmodeled LES can predict the velocity distributions relatively well. On the other hand, this trend is opposite at x/c=0.55. The shock wave moves down to x/c=0.59 in the wall-modeled LES computation. Therefore, the shock wave and the high speed region ahead of the shock wave reaches down to x/c=0.55. As a results, the averaged velocity and the velocity fluctuation are overestimated in the wall-modeled LES. Velocity distributions downstream the shock wave are comparable with each other. Figure 6 shows the trace of the shock wave position in the wall-modeled LES computation. The shock wave moves from x/c=0.48 to x/c=0.6 at St s=f sc/m =0.065. This Strouhal number is 70.1Hz at c=30mm which nearly equals to the buffet frequency obtained from experiment. 5
(a) Near the airfoil, most upstream (b) Near the airfoil, most downstream (c) Trip position, most upstream (d) Trip position, most downstream (e) Near the shock, most upstream (f) Near the shock, most downstream (g) Near the TE, most upstream (h) Near the TE, most downstream Fig. Instantaneous Mach number distribution 6
(a) Averaged Cp (b) Pressure fluctuation Fig. 3 Pressure distribution (a) wall-normal velocity (b) Reynolds stress Fig. 4 Turbulent boundary layer properties at x/c=0.3 7
(a) Averaged velocity, x/c=0.8 (b) Velocity fluctuation, x/c=0.8 (c) Averaged velocity, x/c=0.35 (e) Averaged velocity, x/c=0.55 (d) Velocity fluctuation, x/c=0.35 (f) Velocity fluctuation, x/c=0.55 (g) Averaged velocity, x/c=0.75 (h) Velocity fluctuation, x/c=0.75 Fig. 5 Velocity distribution 8
50 40 p,m 30 Fig. 6 Shock wave location at 0.c away from the wall IV. Self-sustained Oscillation Mechanisms A. Acoustic feedback loop model It was confirmed that the buffet phenomena was accurately reproduced by WMLES from the previous section. In this section, the self-sustained oscillation mechanisms are investigated. There is an acoustic feedback loop model proposed by Lee as a typical model of the self-sustained oscillation mechanism. This feedback loop model is described as follows. Due to the movement of the shock wave, pressure waves are generated which propagate downstream in the separated flow region at a velocity a p. On reaching the trailing edge, the pressure waves generate upstream propagating acoustic waves at a velocity a u from the trailing edge. Acoustic waves interact with the shock wave and impart energy to maintain the shock oscillation. These two pressure waves are responsible for this feedback loop model, and it is considered that buffet frequency can be estimated by investigating the propagation velocities of these two pressure waves using equations as below. 1 0 0.45 0.55 t 0.65 c x s, ave c xs, ave Sts (13) ap au where St s is the buffet frequency, x s,ave is the mean shock position, a p and a u are velocities of pressure waves. We investigate the propagation velocities a p and a u of these two pressure waves from obtained results. Pressure variation from the averaged value is shown in Fig.7. It can be seen that acoustic waves are generated from the trailing edge and propagate upstream. The propagation velocity is estimated from the time history of the pressure variation. First, we investigate the acoustic waves to upstream. Figure 8(a) shows a time history of pressure variation outside the separation region. In this figure, the propagation velocity of acoustic waves can be estimated from the slope of pressure propagation. However, because Fig. 8(a) includes pressure fluctuations with low frequency due to shock wave oscillation and acoustic waves with high frequency from the turbulent boundary layer, we performs the filtering operation. Figure 8(b) is a low-pass filtered pressure variation with Strouhal number less than 1. Figure 8(c) is a bandpass filtered pressure variation with Strouhal number between 1.0 and 4.6 to remove the pressure variation associated with the shock wave oscillation and the acoustic waves generated from the turbulent boundary layer. From Fig. 8(b), large pressure variation associated with the shock wave oscillation can be confirmed. On the other hand, from the Fig. 8(c), several pressure waves propagating from the downstream to the upstream can be confirmed and their propagation velocity is estimated to be about -0.38U. 9
Next, the velocity of pressure waves propagating in the separated flow is estimated. Figure 9(a) shows the time history of pressure variation on surface. Large pressure variation associated with the shock wave oscillation is confirmed as shown in Fig. 8(a). Periodic fluctuations with a relatively high frequency are also confirmed. Looking at the cross sectional surface(fig. 9(b)), it can be seen that the periodic pressure variations are due to the vortical structures caused by the shock induced separation. The advection velocity of the vortical structures is estimated as 0.64U. Now, buffet frequency is predicted as about 0.5 using Eq. 13 with x s,ave=0.5. This frequency is much different from the buffet frequency estimated from the shock wave movement history. Therefore, we cannot confirm the possibility of Lee's model from the results of WMLES. B. Proposed model We propose a new self-sustained oscillation model in this section. In the proposed model, the shock wave oscillation depends on the pressure ratio forward and backward of the shock wave. Across the shock wave, the Rankine-Hugoniot equation is described in Eqs. 14 and 15. p 1 M1 1 (14) p 1 1 M M1 1 1 / M 1 / 1 where p 1 and p are pressures forward and backward of the shock wave. M 1 and M are the Mach number forward and backward of the shock wave. These equations mean that the effective Mach number of the shock wave is defined by the pressre ratio. When the pressure ratio chenges, the shock wave becomes weak or strong and moves forward or backward to balance the equations. Variation in pressure ratio is caused by the shock induced separation of the boundary layer. Figure 10 express the proposed model. When the shock wave is at the most downstream, relatively large separation occurs and the flow path is narrowed. Therefore, the pressure behind the shock wave decreases. As a result, pressure ratio decreases and the shock wave weakens. The weak shock wave moves upstream to balance the Eqs. 14 and 15. On the other hand, when the shock wave is at the most upstream, the separation disappears and the flow path is expanded. The pressure behind the shock wave increases. As a result, pressure ratio increases, and the shock wave becomes strong and moves downstream. Figure 11 shows the time history of the shock wave position, span averaged pressure and local Mach number near the trailing edge and the separation size. The shock wave position is defined by the position with maximum pressure ratio at 0.c away from the wall. Pressure and local Mach number are sampled at the point of x/c=0.9 and z/c=0.19 from the leading edge. The separation size is defined by the thickness from the wall to the position where Mach number is 0.5. In Fig. 11, due to the movement of the shock wave, other values fluctuate largely. When the shock wave is at the most downstream, the separation size begins to increase. As a results, the pressure decreases and the Mach number increases. When the shock wave is at the most upstream, the separation size begins to decrease. As a results, the pressure increases and the Mach number decreases. From the results, it can be seen that there is a time lag between the shock wave position and other values. Figure 1 show the span averaged Mach number at each time position. When the shock wave moves upstream, the size of the separation increases and the flow behind the shock wave is slightly accelerated. On the other hand, when the shock wave moves downstream, relatively low Mach number region behind the shock wave is confirmed. From the results, the possibility of the proposed model is confirmed. (15) 10
t t t t t t u c u c u c 15.7 9.0 0.6 1.0 x/c p Fig. 7 Pressure variation from the averaged value 15.7 9.0 0.6 1.0 x/c (a) Without filter (b) Low-pass filter with St 1.0 (c) Band-pass filter with 1.0 St 4.6 Fig. 8 Pressure variation out of the separation p 15.7 To upstream au=-0.38u 9.0 0.6 1.0 x/c p 11
t t u c 15.7 p To downstream ap=0.64u 9.0 0.6 x/c 1.0 (a) Pressure variation on surface (b) Pressure variation at cross section Fig. 9 Pressure waves to downstream 3Shock moves upstream (a) Shock wave most downstream 3Shock moves downstream (b) Shock wave most upstream Fig. 10 Proposed model Pressure decreases 1Large separation Pressure increases 1Small separation 1
0.81 0.65 p, M local p,m 0.77 0.73 0.69 x s p 0.65 0.45 0 30 40 50 t u t t c Fig. 11 Physical quantities fluctuation during shock wave oscillation (a) Shock wave most upstream M local Separation 0.6 0.55 0.5 X s (b) Shock wave to downstream Separation size (c) Shock wave most downstream (d) Shock wave to upstream Fig. 1 Span-averaged flow field V. Conclusions Transonic airfoil buffet at high Reynolds number is computed by the wall-modeled LES. By using the wallmodeled LES, the buffet onset is successfully predicted. Pressure distributions show the wall-modeled LES can simulate the small separation near the trailing edge with buffet phenomena. The region in which the shock wave moves is estimated more downstream than experiment. The upstream turbulent boundary layer properties show the turbulent boundary layer is fully developed upstream of the shock wave. Velocity distributions upstream and downstream of 13
the shock wave agree well with the experiment. Near the shock wave, velocity distributions depend on the range of the shock wave oscillation. According to the trace of the shock wave position, the shock wave moves from x/c=0.48 to x/c=0.6 at St s=0.065. This frequency nearly equals to the buffet frequency of eximernt. We investigated the self-sustained oscillation mechanisms of buffet phenomena from the obtained result. As a result, Lee's acoustic feedback loop model was unable to predict the buffet frequency. On the other hand, we proposed the new self-sustained oscillation model. In the proposed model, the shock wave oscillation depends on the pressure ratio forward and backward of the shock wave, and the pressure variation is caused by the shock induced separation of the boundary layer. The possibility of the proposed model was confirmed from the obtained result. The remaining problem is determining the buffet frequency in the proposed model. Acknowledgments This work was supported in part by MEXT as a social and scientific priority issue (Development of Innovative Design and Production Processes that Lead the Way for the Manufacturing Industry in the Near Future) to be tackled by using post-k computer. A part of this research used computational resources of the K computer provided by the RIKEN Advanced Institute for Computational Science (Project ID:hp15054) References 1 E. N. Tinoco, D. R. Bogue, T-J. Kao, N. J. Yu, P. Li and D. N. Ball, Progress Toward CFD for Full Flight Envelop, The Aeronautical Journal, Vol. 109, No. 1100, pp. 451-460, 005. B. H. K. Lee, Self-sustained Shock Oscillations on Airfoils at Transonic Speeds, Progress in Aerospace Sciences, Vol. 37, Issue, pp. 147-196, 001. 3 L. Jacquin, P. Molton, S. Deck, B. Maury and D. Soulenvant, An Experimental Study of Shock Oscillation over a Transonic Supercritical Profile, AIAA Paper 005-490, 005. 4 S. Deck, Numerical Simulation of Transonic Buffet over a Supercritical Airfoil, AIAA Journal, Vol. 43, No. 7, 005. 5 J. D. Crouch, A. Garbaruk, D. Magidov, Predicting the Onset of Flow Unsteadiness based on Global Instability, Journal of Computational Physics, Vol. 4, Issue, pp. 94-940, 007. 6 J. D. Crouch, A. Garbaruk, D. Magidov and A. Travin, Origin of Transonic Buffet on Aerofoils, Journal of Fluid Mechanics, Vol. 68, pp. 357-369, 009. 7 J. P. Thomas and E. H. Dowell, Airfoil Transonic Buffet Calculations Using the OVERFLOW Flow Solver, AIAA Paper 011-077, 011. 8 S. Koga, M. Kohzai, M. Ueno, K. Nakakita, N. Sudani, Analysis of NASA Common Research Model Dynamic Data in JAXA Wind Tunnel Tests, AIAA Paper 013-0495, 013. 9 A. Hartmann, A. Feldhusen and W. Schroder, On the Interaction of Shock Waves and Sound Waves in Transonic Buffet Flow, Physics of Fluids, Vol. 5, No., 013. 10 F. Grossi, M. Braza and Y. Hoarau, Prediction of Transonic Buffet by Delayed Detached-Eddy Simulation, AIAA Journal, Vol. 5, No. 10, pp. 300-31, 014. 11 J. Dandois, Experimental Study of Transonic Buffet Phenomenon on a 3D Swept Wing, Physics of Fluids, Vol. 8, Issue 1, 016. 1 T. Izumi, Y. Ogino, K. Sawada, An Attempt to Improve Prediction Capability of Transonic Buffet Using URANS, AIAA Paper 015-059, 015. 13 K. Kumada and K. Sawada, Improvement in Prediction Capability of Transonic Buffet on NASA-CRM Using URANS, AIAA Paper 016-1346, 016. 14 K. Ishiko, A. Hashimoto, T. Aoyama and K. Takekawa, Detached-Eddy Simulation of NASA-CRM Transonic Buffet, Proceedings of the 6th European Conference on Computational Fluid Dynamics, 014. 15 T. Ishida, K. Ishiko, A. Hashimoto, T. Aoyama and K. Takekawa, Transonic Buffet Simulation over Supercritical Airfoil by Unsteady-FaSTAR Code, AIAA Paper 016-1310, 016. 16 D. R. Chapman, Computational Aerodynamics Development and Outlook, AIAA Journal, Vol. 17, No. 1, 1979. 17 H. Choi and P. Moin, Grid-point requirements for large eddy simulation: Chapman s estimates revisited, Physics of Fluids, Vol. 4, Issue 1, pp. 01170-1 01170-5, 01. 18 P. R. Spalart, S. Deck, M. L. Shur, K. D. Squires, M. Kh. Strelets, A. Travin, A New Version of Detached-Eddy Simulation, Resistant to Ambiguous Grid Densities, Theoretical and Computational Fluid Dynamics, Vol. 0, No. 3, pp. 181-195, 006. 19 J. Larsson, S. Kawai, J. Bodart and I. Bermejo-moreno, Large Eddy Simulation with modeled Wall-stress: Recent Progress and Future Directions, Mechanical Engineering Reviews, Vol. 3, No. 1, p. 15-00418, 016. 0 S. Kawai and J. Larsson, Wall-modeling in Large Eddy Simulation: Length Scales, Grid Resolution and Accuracy, Physics of Fluids, Vol. 4, Issue 1, pp. 015105-1 015105-10, 01. 1 S. Kawai and J. Larsson, Dynamic Non-equilibrium Wall-modeling for Large Eddy Simulation at High Reynolds Numbers, Physics of Fluids, Vol. 5, Issue 1, pp. 015105-1 015105-1, 013. 14
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