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Large-eddy simulation of transonic buﬀet over a supercritical airfoil E. Garnier1 and S. Deck2 1 ONERA, Applied Aerodynamics Department, 8 rue des Vertugadins, 92190 Meudon, France. eric.garnier@onera.fr 2 ONERA, Applied Aerodynamics Department, 8 rue des Vertugadins, 92190 Meudon, France. sebastien.deck@onera.fr 1 Introduction The transonic buﬀet is an aerodynamic phenomenon that results in a large- scale self-sustained periodic motion of the shock over the surface of the airfoil. The time scale associated to this motion is much slower than the one of the wall bounded turbulence. It is then an appropriate case for URANS approaches and ﬁrst attempts with these methods have been reasonably successful in reproducing the mean features of such ﬂows. Nevertheless, as shown by Thiery and Coustols [2] results are very sensitive to the turbulence model. Moreover, with some models, it is necessary to increase the angle of attack with respect to the experiment to obtain an unsteady ﬂow. Furthermore, they have evidenced a signiﬁcant sensitivity of the results to the conﬁnement due to the wind tunnel walls. The ﬁrst hybrid RANS/LES computation on this conﬁguration was performed by Deck [1] who has demonstrated that zonal DES (ZDES) generally improves the results with respect to URANS computations carried out with the Spalart-Allmaras model. In particular, the spectral content of the pressure ﬂuctuations in the separated zone is much more closer to the experimental data with ZDES than with URANS. In this latter computation, the shock/boundary layer interaction was treated in RANS mode and one of the purpose of the present study is to assess the improvement that can be obtained from a fully turbulent treatment of the boundary layer on the suction side of the airfoil by means of LES. More generally, the main objective of this study is to assess the capabil- ities of LES to capture the buﬀet phenomenon. The large amount of data provided by these simulations could then support the progress in the physical understanding of such ﬂows. The validation of the computation is performed against the very comprehensive experiment performed at ONERA by Jacquin et al. [3] which was also used by Thiery and Coustols [2] and Deck [1]. 2 E. Garnier and S. Deck 2 Description of the computation The supercritical OAT15A airfoil was computed in the same ﬂow conditions than in the experiment by Jacquin et al.[3]. This airfoil has a chord of 230 mm and a relative thickness of 12.3 %. Its angle of attack is equal 3.5 degrees. The free-stream Mach number was set to 0.73 and the Reynolds number based on the chord length is equal to 3 106 . The ﬂow solver is the structured multiblock code FLU3M developed at ONERA. It is second-order accurate in space and time. The numerical scheme dedicated to the computation of the convective ﬂuxes is based on a Roe scheme which was modiﬁed to adapt locally its dissipation using the Ducros et al. sen- sor [4]. The Selective Mixed Scales Model has been chosen for this study [5]. The time step has been imposed to 3.10−7 s in order to ensure the convergence of the subiterative process of the temporal implicit scheme using 5 subitera- tions. In order to limit the required computational eﬀort, the ﬂow is computed in RANS mode on the pressure side of the airfoil and in LES mode on the suction side and in the wake. Moreover, RANS zones are treated in 2D. The grid reﬁnement criteria commonly used in LES of attached ﬂows are satisﬁed (∆x+ ≈ 50 in the longitudinal direction, ∆z + ≈ 20 in the spanwise direction + and ∆ymin ≈ 1 in the wall-normal direction). Despite the zonal treatment of the ﬂow, 20.8 millions of cells are necessary to compute a domain width of only 3.65 % of chord in the grid A (Nz=140). The span and consequently the number of points were doubled to construct the grid B (Nz=280). This may be insuﬃcient but the grid size results from a compromise with the long integration time required to capture few buﬀeting periods. 3 Mean ﬁeld analysis After a transient of 2 periods, the ﬂow has been averaged over only one pe- riod of the buﬀet phenomenon for the case A. The span was then doubled to generate the grid B and, after a transient of one period, the statistics were collected over one another period. Figure 1 presents an isovalue of the Q cri- terion colored by the longitudinal velocity. The separation occurs after the location of the shock identiﬁed by one isovalue of the pressure (in purple). On this snapshot which corresponds to a situation where the shock moves down- stream, the ﬂow is separated under the lambda shock and near the trailing edge. Figure 2 (left) shows the averaged pressure distribution on the airfoil. The buﬀet zone is shifted downstream by 6 % of chord with respect to the experi- ment. This shift appears more clearly on the pressure ﬂuctuation distributions presented in ﬁgure 2 (right). The use of a doubled span (grid B) signiﬁcantly reduces the ﬂuctuations near the trailing edge. The analysis of the instanta- neous ﬁelds obtained on grid A has evidenced that this overestimation was Large-eddy simulation of transonic buﬀet over a supercritical airfoil 3 Fig. 1. Q criteria near the wing wall and one isovalue of the pressure to mark the shock location due to the presence of intense bidimensional coherent structures developing when the ﬂow separates from the shock up to the trailing edge. The span of the grid B allows the tridimensionalisation of these structures which limits their intensity and subsequently the wall pressure ﬂuctuations. -2 0.5 -1.5 LES A LES A LES B 0.4 LES B Exp. Exp. -1 0.3 Prms/Qo -0.5 Kp 0 0.2 0.5 0.1 1 1.5 0 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 x/c x/c Fig. 2. Averaged pressure coeﬃcient distribution (left) and rms pressure distribu- tion (right) The proﬁles of averaged and ﬂuctuating longitudinal velocity at x/c=0.35 are plotted in ﬁgure 3. These data evidence that, upstream from the interac- tion, the velocity ﬁeld is well estimated by the LES. This result was far from being trivial since the ﬂow undergoes a numerically forced transition at the same station than in the experiment (x/c=0.07). Downstream from the interaction (at x/c=0.75), one can observe in ﬁgure 4 that the agreement of the LES with both the averaged and the ﬂuctuating longitudinal velocity proﬁles is more than satisfactory. It is however worth- while to notice that between x/c=0.4 and x/c=0.6, both experimental and numerical velocity proﬁles diﬀer signiﬁcantly since the shock is not located at the correct mean position. 4 E. Garnier and S. Deck 0.014 0.014 0.013 0.013 y (m) y (m) 0.012 0.012 0.011 0.011 0.01 0.01 0 50 100 150 200 250 300 350 400 0 20 40 60 80 U(m/s) Urms (m/s) Fig. 3. Mean longitudinal velocity (left) and longitudinal velocity ﬂuctuations (right) proﬁles at x/c=0.35 (Grid B). 0.03 0.03 0.02 0.02 y (m) y (m) 0.01 0.01 0 0 0 50 100 150 200 250 300 0 50 100 150 U (m/s) Urms (m/s) Fig. 4. Mean longitudinal velocity (left) and longitudinal velocity ﬂuctuations (right) proﬁles at x/c=0.75 (Grid B). Nevertheless, it is believed that the results quality is suﬃcient to initiate a physical analysis of the ﬂow. 4 spectral analysis Due to the short duration of the LES simulations, an auto-regressive (AR) model method has been used to compute the Power Spectral Density of the pressure. Indeed, this method is well adapted to study short data that are known to consist of sinusoids in white noise[8]. The AR parameters are ob- tained with Burg’s method[9]. The pressure spectrum for x/c = 0.9 is com- pared to experiment in ﬁgure 5. The occurrence of strong harmonic peaks highlights the periodic nature of the motion. On the experimental side, the main peak at 69 Hz represents the frequency of the self-sustained motion of the shock over the airfoil. A higher frequency near 76 Hz is found in the computation. This 10 % error can be considered here as acceptable. Large-eddy simulation of transonic buﬀet over a supercritical airfoil 5 180 LES exp. S3Ch 160 SPL 140 120 100 1 2 3 4 10 10 10 10 frequency (Hz) Fig. 5. PSD of pressure ﬂuctuations. 5 Space and time scales Once, the main statistical and spectral features of the ﬂow have been found, it is worthwhile to study the kinematics of these pressure waves. To this end, let us consider the ﬂuctuating pressure at diﬀerent stations. The two- ′ (x1 ,t)P ′ (x2 +ξ,t−τ ) point two-time correlation coeﬃcient: Rx1 ,x2 (∆ξ, τ ) = √P √ (P ′ 2 (x1 )) (P ′ 2 (x2 +ξ)) establishes the correlation between two signals located at abscissa x1 et x2 + ξ and separated by a time delay τ . The convection velocity can be obtained as the slope of the linear ﬁtting of the ξ versus τmax (τmax represents the delay where the correlation coeﬃcient reached its maximum), as illustrated in ﬁgure 6. 1 ∞ ∞ 1U 4 U -0 .2 7 -0 .3 6 0.8 0.6 ∆ξ/c 0.4 6 .0 81 0 -2 7. U 18 ∞ 0.2 10 -2 U ∞ 0 0 5 τmaxU ∞/c Fig. 6. Propagation velocities obtained by a least square ﬁtting of the linear relation between the separation distance ∆ξ and time delay τ (ﬁlled symbol: exp, solid line: upper side of the airfoil). On the upper-side of the airfoil, a downstream propagation velocity equal 6.08 10−3U∞ is clearly identiﬁed for the LES and appears to be slightly lower than in the experiment. On the lower side of the airfoil, a forward motion at velocity 0.364 10−3U∞ is evidenced. The latter velocity corresponds exactly to the upstream travelling acoustic waves on the lower side of the airfoil. 6 E. Garnier and S. Deck 6 Discussion To assess the frequency of the motion, Lee[6] proposed that the period of the shock oscillation should agree with the time it takes for a disturbance to propagate from the shock to the trailing edge added to the time needed for an upstream moving wave to reach the shock from the trailing edge. A simpliﬁed model has been used in reference [1] to assess the complete duration c−xs c−xs to complete such a loop: T = vdownstream + |vupstream | where c is the chord and xs is the mean location of the shock wave. xS can be obtained by noting the ﬁrst abscissa where the skewness of pressure ﬂuctuations is zero. One gets (xs /c)LES = 0.52 while (xs /c)exp = 0.45. The velocity of upstream- travelling acoustic waves is vupstream = a(M − 1) where a is the local speed of the sound in the ﬁeld outside the separated area. With M = 0.8 and a = 330 m/s, the Lee’s equation gives f = 1/T ≈ 110 Hz which is higher than the frequency FLES ≈ 80 Hz. More recently, Crouch et al.[7] advocated that transonic buﬀet results from global instability where the unsteadiness is characterized by phase-locked oscillations of the shock and the separated shear layer. Within this scenario, the region downstream from the shock is not the only region contributing to the feedback loop. Indeed, an upstream travelling acoustic motion has been highlighted on the lower surface of the airfoil (see ﬁgure 6). A deeper investigation of these phenomena will follow the present work. 7 Acknowledgments This work has been partly sponsored by the French National Research Agency (project ANR-07-CIS7-009-04). References 1. Deck, S. (2005) AIAA J. 43:1556-1566 2. Thiery, M. and Coustols, E. (2004) Flow, Turbulence and Combustion 74:331- 354 3. Jacquin, L., Molton, P., Deck, S., Maury, B., Soulevant, D. (2005) AIAA paper 2005-4902 4. Ducros, F., Ferrand, V., Nicoud, F., Weber, C., Darracq, D., Gacherieu, C., and Poinsot, T. (1999) J.Comput. Phys. 152:517-549 5. Lenormand, E., Sagaut, P., Ta Phuoc, L., and Comte, P. (2000) AIAA J. 38:1340-1350 6. Lee, B.H.K. (1990) Aeronautical Journal, 143-152 7. Crouch, J.D. and Garbaruk, A. and Magidov, D. and Jacquin, L. (2008), Pro- ceedings of IUTAM conference, Corfou, GREECE 8. Trapier, P. and Duveau, P. and Deck, S. (2006) AIAA J. 44:2354-2365 9. Burg, J.P. (1978) In Modern Spectrum Analysis, Ed. D.G. Childers, 34-41, IEEE Press, New-York