AIAA Paper 2004-1056 42nd AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada 5 - 8 Jan 2004
A NUMERICAL STUDY OF TRANSONIC BUFFET ON A SUPERCRITICAL AIRFOIL
Q. Xiao1 and H. M. Tsai2 Temasek Laboratories National University of Singapore Singapore 119260
F. Liu3 Department of Mechanical and Aerospace Engineering University of California, Irvine CA 92697-3975
Abstract Self-excited oscillation has been investigated both experimentally and computational since the early work 1 The steady/unsteady flow of the BGK No. 1 of McDevitt et al. (e.g., Refs.1-18) Previous supercritical airfoil is investigated by the solution of the experimental and numerical studies of transonic flow Reynolds-Averaged Navier-Stokes equations with a over an 18% thick circular-arc airfoil have indicated two-equation k-ω turbulence model. The dual-time that, the flow over a circular-arc aerofoil exhibit varied stepping method is used to march in time. Two steady behavior depending on the flow conditions. Three distinct regions have been observed for a fixed free- cases ( Ma = 0.71,α =1.396 and Ma = 0.71,α = 9.0 ) stream Reynolds number but varying Mach number. and one unsteady case ( Ma = 0.71,α = 6.97 ), both Below a critical Mach number, the flow is steady and with a far-stream Re = 20x106 , are computed. The characterized by a weak shock wave near the mid-chord results are compared with experimental data obtained with trailing-edge flow separation. For larger Mach by Lee et al. The mechanism of the self-excited numbers, shock induced separation is encountered and oscillation for supercritical airfoils is explored. The the flow becomes unsteady, with unsteady shock oscillation period calculated based on the mechanism motions on the upper and lower surface that are out of proposed by Lee is consistent with the Fourier analysis phase with each other. As the Mach number is further of the computed lift coefficient. increased, a steady shock reappears. It is sufficiently strong to induce flow separation.
Introduction Compared to the non-lifting circular-arc airfoil, the study of other lifting supercritical airfoils is less Transonic buffeting flow appears in many aeronautical matured. The computation and experiment of applications ranging from internal flows such as around NASASC(2)-0714 have been reported in the paper of 7 8 turbo-machinery blades to flows over aircraft. Unsteady Bartels and Bartels et al. . An extensively set of data shock/boundary-layer interaction affects aerodynamic for the buffeting problem studied at the National performance. For flows around an airfoil self-excited Research Council, Canada of the flow over the Bauer- shock oscillations can cause flow separation and Garabedian-Korn (BGK) No. 1 supercritical airfoil at 9 10 increase drag significantly. The capability to accurately transonic speed has been reported by Lee , Lee, et al. , 11,12 13 predict such phenomena is of technological significance Lee , and Lee, et al. . The experiments were and a challenge. The accuracy of computation is conducted over a range of Mach number and angle of influenced by both the accuracy of the numerical attack. A map of the shock induced trailing edge technique and the accuracy of the physically model, separation regions together with steady and unsteady particularly, the turbulence model. pressure measurements were reported for various shock/boundary-layer interactions. The investigation of the supercritical airfoil indicates, different from the circular-arc airfoil, the supercritical airfoil has a much ––––––––––––––––––––––– weaker trailing edge separation in the onset region and 1. Research Scientist, email: [email protected] experience shock oscillation only on one side of the 2. Principal Research Scientist, email: [email protected] Member airfoil. AIAA. 3. Professor, email: [email protected] Associate Fellow AIAA. Copyright © 2004 by the authors. Published by the American The mechanism of the self-excited oscillation is not Institute of Aeronautics and Astronautics, Inc with permission well understood although some explorations have been done by several researchers. (Refs. 11, 13-18). The investigation on the bi-convex airfoil suggested that the
1 American Institute of Aeronautics and Astronautics transonic periodic flows are initiated by an asymmetric Sofialidis and Prinos25 were studied with limited unsteady disturbance. The shock induced separation success. changes the effective geometry of the airfoil, which would cause the forward and rearward movement of One difficulty in most turbulence models is the inability shock depending on whether the streamtube decreases to account directly for non-equilibrium effects found in or increases. The necessary but not sufficient condition separated flows. A new class of model was proposed by for the periodic flow to appear is the shock wave strong Olsen & Coakley26, which is termed the lag model. The enough to cause boundary separation. The Mach basic idea of the lag model is to take a baseline two- number just upstream of the shock should be in the equation model and couple it with a third (lag) equation range of 1.14< M1 <1.24. For the supercritical airfoil to model the non-equilibrium effects for the eddy (Lee11), a model to predict the shock motion was viscosity. Recently, Xiao et al.27 incorporated the lag formulated. The model assumed that the movement of model with the baseline k-ω model and computed both shock leads to the formation of pressure waves which steady and unsteady transonic nozzle flows. Their propagate downstream in the separated flow region. On computations show notable improvements for strong reaching the trailing edge the disturbance generates shock cases, where strong non-equilibrium effect is upstream-moving waves, which interact with the shock present. For separated flows, good turbulence modeling wave and impart energy to maintain the limit cycle. The is necessary to compute the overall flow physics. The period of the oscillation should be the time for a lagged k-ω model also enjoys the advantage of not disturbance to propagate from shock to the trailing edge requiring specification of wall distance, and has similar plus the duration for an upstream moving wave to reach or reduced computational effort compared with the shock from the trailing edge via the region outside Reynolds stress models. the separated flow. In the present work we employ the lag model coupled A number of difficulties are particular to the simulation with the two-equation k-ω model for the study of of unsteady flow. Dual-time stepping method 19 transonic buffeting flow for the BGK No.1 supercritical introduced by Jameson provides an alternative to such airfoil, which is well documented in the paper of Lee9, direct computation of flow involving oscillation. This Lee, et al.10, and Lee11,12. The computational results will method uses an implicit physical time discretization. At be compared with the experiments for the each physical time step, the equations are integrated in steady/unsteady flow. The exploration of mechanism a fictitious pseudo-time to obtain the solution to the for the self-sustained oscillation will be conducted from steady state in pseudo-time. When marching in pseudo- the computational flow field. To the author’s time, this method permits the use of acceleration knowledge, no computational work has been done for techniques such as local time step and multigrid to this supercritical airfoil. speed up the steady flows calculations. The application of dual-time stepping method of self-excited flow 3 5 In the next section, the numerical implementation of the includes Rumsey et al. and Oliver et al. for the 18% dual-time stepping method for the two-dimensional circular-arc airfoil. Navier-stokes equations will be briefly presented. The computational results and discussions will follow. The assessment of turbulence models in unsteady aerodynamic flow featuring buffet and/or dynamic-stall has attracted less attention than the steady flow. In the early work of Levy2, computations were done using the Numerical Method Baldwin-Lomax20 model which showed the unsteadiness in the flow field. It is sufficiently strong to The governing equations for the unsteady compressible induce flow separation. In terms of accuracy, the turbulent flow are expressed as follows: algebraic turbulence models such as the Baldwin- Mass conservation: ∂ρ ∂ Lomax model do not perform well (Girodroux-Lavigne + (ρu ) = 0 (1) 14 6 ∂ ∂ j and LeBalleur ). Barakos & Drikakis more recently t x j presented an assessment of more advanced turbulence Momentum conservation: closures in transonic buffeting flows over airfoils. ∂ ∂ ∂ ∂τˆ ρ + ρ = − p + ji (2) Various turbulence closures such as algebraic model, ( ui ) ( u jui ) 21 ∂t ∂x ∂x ∂x the one-equation model of Spalart and Allmaras , the j i j Launder-Sharma22 and Nagano-Kim23 linear k- ε models as well as the non-linear eddy-viscosity model of Craft et al. 24 and the non-linear k-ω model of
2 American Institute of Aeronautics and Astronautics Mean energy conservation: ∂ ∂ Where Pr and Pr are the laminar and turbulent (ρE) + (ρu H ) = L T ∂t ∂x j j (3) Prandtl numbers, respectively. The other coefficients are: ∂ ⎡ ∂k ⎤ ⎢uτˆ + (µ +σ *µ ) − q ⎥ a = 0.35, R = 1, R = 0.01, ε = 5/ 9 , ε * = 1, ∂ i ij T ∂ j 0 T 0 T∞ x j ⎣⎢ x j ⎦⎥ β = 0.075, β * = 0.09 , σ = 0.5 , σ * = 0.5 . Turbulent mixing energy:
∂ ∂ ∂u (ρk) + (ρu k) = τ i − β * ρωk It is known that conventional one- and two-equation ∂t ∂x j ij ∂x j j (4) turbulence models generate Reynolds stresses that respond too rapidly to changes in mean flow conditions ∂ ⎡ ∂k ⎤ + ⎢(µ +σ * µ ) ⎥ partially due to the need to accurately reproduce ∂ T ∂ x j ⎣⎢ x j ⎦⎥ equilibrium flows. As a result, these baseline turbulence Specific dissipation rate: models give unsatisfactory results for flows with significant separation under adverse pressure gradients ∂ ∂ ∂u or across shock waves. In the present computation, the ρω + ρ ω = εω τ i − βρω 2 ( ) ( u j ) ( / k) ij standard k-ω turbulence model is coupled with the lag ∂t ∂x ∂x 26 j j (5) model proposed by Olsen and Coakley to calculate the ∂ ⎡ ∂ω ⎤ turbulent eddy viscosity. + µ +σµ ⎢( T ) ⎥ ∂x ⎢ ∂x ⎥ j ⎣ j ⎦ The basic numerical method used to solve the above system of equations in this paper follows that described Turbulent eddy viscosity: in detail by Liu and Ji28. The integral forms of the ∂ ∂ conservation equations are discretized on quadrilateral (ρν ) + (ρu ν ) = a(R )ωρ(ν −ν ) (6) ∂t t ∂x j t T tE t cells using the finite volume approach. A staggered j scheme is used for the coupling of the Navier-Stokes ρ where t is time, xi position vector, is the density, equations and the k-ω and lag equations. A central µ difference scheme is used to discretize the diffusive ui velocity vector, p pressure, dynamic molecular terms and an upwind Roe’s scheme is used for viscosity, ν kinematic equilibrium turbulent eddy tE convective terms in the Navier-Stokes and the k-ω viscosity, k turbulent mixing energy, and ω the equations. specific dissipation rate. The total energy and enthalpy are E = e + k + u u 2 and H = h + k + u u 2 , After being discretized in space, the governing i i i i equations are reduced to a set of ordinary differential = + ρ = []γ − ρ respectively, with h e p / and e p ( 1) . equations with only derivatives in time, which can be The term γ is the ratio of specific heats. Other solved using a multi-stage Runge-Kutta type scheme. quantities are defined in the following equations: To accelerate the convergence, unsteady multigrid 19 µ = ρν (7) method proposed by Jameson and further T t implemented by Liu and Ji28 is applied in the present ν = ε * ω tE k (8) study for all 7 equations. = ρ µ ω RT k T (9)
1 ∂u ∂u S = ( i + j ) (10) ij ∂ ∂ Results and Discussions 2 x j xi 1 ∂u 2 The numerical method presented above is applied to the τ = 2µ (S − k δ ) − ρkδ (11) ij T ij 3 ∂x ij 3 ij Bauer-Garabedian-Korn (BGK) No. 1 supercritical k airfoil which is extensively investigated experimentally 1 ∂u τˆ = 2µ(S − k δ ) +τ (12) at the National Research Council, Canada. The ij ij ∂ ij ij geometry of the airfoil is shown in Fig. 1. The 3 xk experiments have been conducted at a chord Reynolds µ µ ∂h q = −( + T ) (13) 6 j Pr Pr ∂x number of 20x10 with free transition. The Mach L T j numbers in the investigation ranged from 0.5 to 0.792 ⎡(R + R )⎤ with angle of attack α varying from 1.3 to 11.5 degree. a(R ) = a T T 0 (14) T 0 ⎢ + ⎥ It is reported, from the force and pressure power ⎣(RT RT∞ )⎦ spectrum, strong shock oscillation occurs for Mach
3 American Institute of Aeronautics and Astronautics numbers between 0.69 and 0.733. For the present conditions develops into unsteady with shock-induced numerical simulation, three test cases are conducted oscillation. The average lift coefficient is 1.03 which is with a fixed chord Reynolds number of 20x106 and close to the experimental data of 1.016. Fourier Mach number of 0.71 at three different angles of attack, analysis result shown in Fig. 6 reveals a predominant = π case 1: α =1.396 ; case 2: α = 6.97 ; and case 3: reduced frequency ( k fc U ∞ ) equal to 0.16, which α = 9.0 . Based on Lee’s experimental results, the flow is about 0.4 of the bi-convex circular airfoil. This value is steady for the flow conditions of case 1 and case 3. is about 60% lower than the experimental value of 10 For case 2, however the flow is unsteady with shock Lee . induced oscillation. It should be noted that the Mach number and angle of attack used in the computation are The Mach number contours and skin friction coefficient the same as the experimental ones without any wind Cf plots at different time instants during one oscillation tunnel correction. period are presented in Fig. 7. The instantaneous shock positions on the upper surface of the airfoil are shown The computational grid has a C-topology. Most of the in Fig. 8. The t* equal to zero corresponds to the time computations are performed on a 640x64 mesh with instant at which the lift coefficient Cl is the minimum. 512 points on the surface of the airfoil, 128 nodes in the Both Fig. 7 and Fig. 8 reveal that, different from the wake. The computational domain extends non-lifting bi-convex circular-arc airfoil, the shock approximately 20 chords up and down stream. The oscillation here occurs on the upper side of the airfoil average minimum normal spacing at the wall is only. The shock motion is mainly of the type A 29 approximately 10-6 chord, which corresponds to reported by Tijdeman . Detailed flow features can be + obtained from the Mach number contours and the skin a y value of about 0.6 for an unsteady calculation at friction distributions in Fig. 7. At t*=0, a weak shock Ma = 0.71 with α = 6.97 . The grid independence test occurs at a location of approximately one quarter of the has been conducted and the results will be presented in airfoil, and there is a small separation region after the detail later. shock. During the first half cycle (0 4 American Institute of Aeronautics and Astronautics computations show an increase in amplitude with converged steady result. The time-mean lift coefficient increasing x/c for x/c<0.4, subsequently a decrease with and oscillation frequency are found to be the same for increasing x/c within the range of 0.4 5 American Institute of Aeronautics and Astronautics of Tijdeman29 formula is accessed by Lee et al13. The and a p is the velocity of the downstream pressure wave propagation of pressure disturbance is analyzed by the in the separated flow region. The can be calculated a p nonlinear transonic small disturbance equation. The from the pressure wave phase angle versus x/c relation. results for the NACA 64A006 airfoil compare For the time taken by a disturbance to travel from the favorably with the numerical computations when the integration of the Eq. (20) is carried out on a line trailing edge to the shock (Tp2 ), there are several starting at the trailing edge at 170 deg angle. In the methods available. The first uses the empirical formula paper of Lee11, the method is used to estimate the given by Erickson and Stephenson30 as follows: oscillation frequency for the BGK No.1 supercritical a(1− M ) f = (17) airfoil, their calculated frequency is found to be fair − 4(c xs ) with the measured shock oscillation frequency from the where c is the chord length, a the far-stream sound balance force spectra. That provides evidence for the speed, M the far-stream Mach number and f the possible mechanism of self-sustained shock oscillation proposed by Lee11. frequency of the upstream wave. Tp2 is then determined by T = 1/ f . Mabey31 computed the In the present study, the above mechanism has been p2 explored using the computational results. Two-point π reduced frequency ( 2 fc/U ∞ ) using Eq. (17), the result pressure cross correlation are used to determine the is approximately 1.43 compared to about 1 from propagation direction of the pressure fluctuations. For experiments. the separated flow region, the cross correlation analysis is conducted for the upper surface pressure at the points The second method to determine Tp2 is based on the from E to T, (shown in Fig.1). The point nearest the assumption of Tijdeman29. Optical studies of self- trailing edge (T) is used as the reference in each cross sustained oscillation show that the wake oscillates in a correlation. For the outside separation region, pressure synchronous manner with the shock oscillations and cross correlation are done for points A to D using A as behaves like a flap. Tijdeman29 has pointed out that the the reference (shown in Fig. 25). The results are shown flap can be treated as an acoustic source placed at the in Figs. 26 and 27, respectively. Inside the separated hinge line, and the time required for a disturbance to region, the cross correlation in Fig. 26 shows that the travel from the trailing edge to the shock is given by the surface pressure disturbances propagate downstream following expression: from shock to the trailing edge. However, negative time delay is observed in Fig. 27 for the pressure outside the = xc dx (18) Tp ∫ separation region, which indicates that the disturbances 2 c − (1 M loc )aloc are moving upstream. This appears to be consistent with the hypothesis of Lee11. where M loc and aloc are the local Mach number and The shock oscillation period is calculated by using Eq. speed of sound, respectively. M is approximated by loc (15) and the time of downstream pressure wave the following empirical formula: propagation (Tp1 ) is obtained by Eq. (16). The value M = R[M − − M ∞ ] + M ∞ (19) loc loc S of a is computed from the pressure wave phase angle where M and M are the Mach number at the p loc−S ∞ versus x/c relation shown in Fig. 12. The almost linear surface and far stream, respectively, and R is 0.7. relation can be seen for x/c>0.25 and the average velocity of the pressure wave for this particular figure is Lee11 expressed Eq.(18) in the following form: about 0.12U ∞ . = xc Tp ∫ 1/ au dx (20) 2 c where a is the velocity of the upstream wave outside The time for upstream wave propagation from the u trailing edge to the shock is calculated by two methods. the separated region. As this value is not easily The first (Method 1) uses Eq. (17), which is measured experimentally, Lee et al. 13 estimated it using 29 independent of the local flow field, but is related to the the empirical formula given by Tijdeman as: far stream Mach number and sound speed only. The = − au (1 M loc )aloc (21) time-mean shock location xs is about 0.15 chord for Mabey et al.32 related Eqs. (18-19) with reduced = and α = . The second method (Method frequency and found the values were 3.6 and 2.3 for Ma 0.71 6.97 2) uses Eq (20). Different from Lee11’s method, the M = 0.81 and M = 0.88 which are higher than the upstream wave velocity ( ) here is calculated from experimental data of about 1.0. The accuracy of the use au 6 American Institute of Aeronautics and Astronautics the above pressure correlations of the computed results It is thought that the shock induced oscillation on the instead of the empirical formula of Eqns. (19) and (21). airfoil is initiated by an unsteady asymmetric disturbance and sustained by the communication across The average propagation speed au is obtained in a the trailing edge. From the above results and cross correlation by using the spatial separation discussion, it is best described as the case where the between the points and the time delay corresponding to the correlation peak. Results for the calculated reduced disturbance is caused by the formation of the separation bubble which is inherently unsteady as it interacts with frequency ( k = 2πfc U ) by the two methods are given ∞ the shock wave. The unstable separation bubble in Table one as well as the Fourier analysis result of lift changes the effective geometry of an airfoil similar to coefficient Cl. It should be pointed out that, in the the effect of a trailing edge flap oscillating upwards in a experiment the oscillation frequency is derived from the periodic manner. The separated flow being a low normal force spectra. However, this value is obtained momentum flow region reduces the effective chamber from the Fast Fourier Transformation of Cl in the CFD. and hence the resulting reduction of lift. The separated It is treated as the basis for the comparison with the flow also creates an upstream wave, which provides the computation from Method 1 and Method 2. feedback from the rear of the airfoil to perturb the shock upstream. This mechanism can be self sustaining Table 1. Comparison of reduced frequency obtained by as we have seen in this case of the BGK supercritical different method. airfoil at Ma=0.71 and angle of attack at 6.97 degrees. Method 1 Method 2 FFT result Conclusions 0.41 0.39 0.32 The computation of self-excited oscillation on BGK Both computations with Method 1 and 2 are close to the No. 1 supercritical airfoil is conducted using the lagged Fourier analysis result for Cl. The good agreement two-equation k-ω turbulence model. This work appears indicates the validity of Eq. (15), i.e.: the oscillation to be the first for modeling the BGK. No. 1 period is equal to the sum of the time for a disturbance supercritical airfoil. The results are presented and traveling from shock to the trailing edge and the time of compared with experimental data. The mechanism of the disturbance propagating from the trailing edge to the shock oscillation is also investigated. The main the shock. This provides strong evidence to the conclusions can be drawn as follows: mechanism of self-sustained oscillation proposed by Lee. 1) The unsteady results are generally in good agreement with experiment except the oscillation To gain further insight into the unsteady flow features, frequency is lower than that of experiment. the streamline plots near the trailing edge at different time instants during one oscillation cycle are shown in 2) The calculations of oscillation frequency based on Fig. 28. The separated flow region at t*=0 extends from the suggestion of Lee12 are close to the FFT result x/c=0.65 on the airfoil to x/c=1.35 in the wake region. of lift coefficient Cl. This provides solid evidence At subsequent time instants, the separation location to the mechanism of self-sustained oscillation by moves downstream, and the separated flow convects Lee11. downstream and eventually disappears on the top region of the airfoil about the time from t*=0.1 to 0.2. 3) For the fully separated unsteady flow of At a later time separated flow reappears and a bulge can Ma = 0.71 and α = 6.97 , notable improvement be observed around x/c=0.55 at t*=0.3 which then can be observed by using the lag model on the forms a new vortex. At the time instants of t*=0.4 and surface pressure distribution and lift coefficient. 0.5, the bulge becomes larger and a new separation For the oscillation frequency, the impact is small. region appears at t*=0.5. This also corresponds to the instants of maximum lift. In the subsequent time instants (t*=0.6, 0.7 and 0.8), this vortex increases its size in both upstream and downstream directions with a corresponding loss of lift. A secondary vortex is also formed in the vicinity of the trailing edge at t*=0.7 and 0.8. The vortices coalesce and form one large separation flow region which decreases in size in the successive time instant t*=0.9. The cycle of separated flow and shock movement then repeats itself. 7 American Institute of Aeronautics and Astronautics REFERENCES 14. Girodroux-Lavigne, P., LeBalleur J.C., “Time consistent computation of transonic buffet 1. McDevitt, J.B. Levy, L.L. and Deiwert, G.S. over airfoils” 1988, ONERA TP No. 1988-97. “Transonic flow about a thick circular-arc 15. Mabey, D.G., “Physical phenomena associated airfoil,” AIAA Journal, Vol.14, No.5, 1976, with unsteady transonic flow,” Chapter 1 in pp.606-613. Unsteady transonic aerodynamics, ed. by 2. Levy L.L. “Experimental and computational D.Nixon, Vol.120 in AIAA Progress in steady and unsteady transonic flows about a Aeronautics and Astronautics, 1989. thick airfoil,” AIAA Journal, Vol.16, No.6, 16. 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Xiao, Q., Tsai, H.M. and Liu, F., “Computation of transonic diffuser flows by a 8 American Institute of Aeronautics and Astronautics lagged k-ω turbulence model,” Journal of National Advisory Committee for Aeronautics, Propulsion and Power, vol. 19, no. 3, May- December 1947. June, 2003.pp. 473-483. 31. Mabey, D.G., “Oscillatory flows from shock- 28. Liu, F. and Ji, S., “Unsteady flow calculations induced separations on biconvex airfoils of with a multigrid Navier-Stokes Method,” AIAA varying thickness in ventilated wind tunnels.” Journal, Vol. 34, No. 10, 1996, pp. 2047- AGARD CP-296, Boundary layer effects on 2053. unsteady airfoils, Aix-en-Provence, France, 29. Tijdeman, H., “Investigation of the transonic 14-19 Sept. 1980, pp.11.1-14. flow around oscillation airfoils” NLR TR 32. Mabey, D.G., Welsh, B. L. and Cripps, B. E. 77090 U, National Aerospace Laboratory, The “Periodic flows on a rigid 14% thick biconvex Netherlands, 1977. wing at transonic speeds.” TR81059, Royal 30. Erickson, A.L., Stephenson, J.D., “A Aircraft Establishment, May 1981. suggested method of analyzing transonic flutter of control surfaces based on available experimental evidence.” NACA RM A7F30, 9 American Institute of Aeronautics and Astronautics 1 0.9 0.8 0.7 0.6 0.5 Cl 0.4 0.3 0.2 0.1 0 0 10 20 30 40 50 60 70 80 Realtime Fig.1 Geometry of the BGK No.1 supercritical airfoil. = (From Ref. 9) Fig.2 Evolution of lift coefficient Cl of Ma 0.71 and α =1.396 . 1.5 Ma 1.1822 1.02457 0.866947 1 0.70932 0.551694 0.394067 0.23644 0.0788134 0.5 -Cp 0 -0.5 -1 0.25 0.5 0.75 1 x/c Fig. 3 Mach contour of Ma = 0.71 and α =1.396 . Fig. 4 Pressure distribution along the upper surface of airfoil. Ma = 0.71 and α =1.396 . 1.6 1 0.9 1.4 0.8 1.2 0.7 0.6 1 Cl 0.5 0.8 0.4 0.3 0.6 Normalized magnitude 0.2 0.4 0.1 0 0 50 100 150 200 250 300 0 0.5 1 1.5 2 Realtime Reduced frequency Fig. 5 Evolution of lift coefficient Cl of Ma = 0.71 and Fig. 6 Fourier analysis of reduced frequency for α = 6.97 . Ma = 0.71 and α = 6.97 . 10 American Institute of Aeronautics and Astronautics 1.4 1.3 t*=0.4 t*=0.6 1.2 1.1 Cl 1 t*=0.2 0.9 t*=0.8 0.8 t*=1.0 t*=0 0.7 0 5 10 15 20 25 Realtime in the final cycle Ma Ma Ma 1.4281 1.4281 1.4281 t*=0 1.3329 t*=0.2 1.3329 t*=0.4 1.3329 1.23769 1.23769 1.23769 1.14248 1.14248 1.14248 1.04728 1.04728 1.04728 0.952069 0.952069 0.952069 0.856862 0.856862 0.856862 0.761655 0.761655 0.761655 0.666448 0.666448 0.666448 0.571241 0.571241 0.571241 0.476034 0.476034 0.476034 0.380827 0.380827 0.380827 0.285621 0.285621 0.285621 0.190414 0.190414 0.190414 0.0952069 0.0952069 0.0952069 Ma Ma Ma 1.4281 t*=0.8 1.4281 1.4281 t*=0.6 1.3329 1.3329 t*=1.0 1.3329 1.23769 1.23769 1.23769 1.14248 1.14248 1.14248 1.04728 1.04728 1.04728 0.952069 0.952069 0.952069 0.856862 0.856862 0.856862 0.761655 0.761655 0.761655 0.666448 0.666448 0.666448 0.571241 0.571241 0.571241 0.476034 0.476034 0.476034 0.380827 0.380827 0.380827 0.285621 0.285621 0.285621 0.190414 0.190414 0.190414 0.0952069 0.0952069 0.0952069 0.012 0.012 0.012 0.01 t*=0 0.01 t*=0.2 0.01 t*=0.4 0.008 0.008 0.008 0.006 0.006 0.006 0.004 0.004 0.004 f Cf Cf C 0.002 0.002 0.002 0 0 0 -0.002 -0.002 -0.002 -0.004 -0.004 -0.004 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 x/c x/c x/c 0.012 0.012 0.012 0.01 t*=0.6 0.01 t*=0.8 0.01 t*=1.0 0.008 0.008 0.008 0.006 0.006 0.006 0.004 0.004 0.004 f f f C C C 0.002 0.002 0.002 0 0 0 -0.002 -0.002 -0.002 -0.004 -0.004 -0.004 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 x/c x/c x/c Fig. 7 Mach number contour and skin coefficient at different instant time in one period. Ma = 0.71 and α = 6.97 . 11 American Institute of Aeronautics and Astronautics 0.7 2 0.6 1.5 0.5 0.4 1 p C Xs/C 0.3 - 0.5 0.2 0 0.1 0 0 5 10 15 20 25 30 -0.5 Realtime 0 0.25 0.5 0.75 1 x/c Fig. 8 Shock position in one oscillation cycle. Fig. 9 Time-mean pressure comparison with experimental results. Closed circle: experiment; Line: computation. 2 1 1.8 1.6 E 0.8 1.4 G 1.2 0.6 H p 1 C I - J 0.8 K Amplitude 0.4 L 0.6 N O P Q 0.2 0.4 R T 0.2 0 0 0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 x/c Realtime Fig. 10 Pressure distribution vs time at the different location of Fig. 11 Amplitude of unsteady pressure on airfoil. upper surface due to shock oscillation. Closed circle: experiment; open circle: computation. 180 1.6 135 1.4 90 1.2 45 0 1 Cl Phase-45 angle 0.8 -90 0.6 -135 -180 0.4 0 0.2 0.4 0.6 0.8 1 0 50 100 150 200 x/c Realtime Fig. 12 Phase angle of unsteady pressure on airfoil. Closed Fig. 13 Lift coefficient Cl evolution of Ma = 0.71 and circle: experiment; open circle: computation α = 9.0 . 12 American Institute of Aeronautics and Astronautics 2 Ma 1.46678 1.369 1.27121 1.5 1.17343 1.07564 0.977856 0.880071 0.782285 1 0.684499 0.586714 0.488928 -Cp 0.391142 0.293357 0.5 0.195571 0.0977856 0 -0.5 0 0.25 0.5 0.75 1 x/c Fig.14 Mach contour of Ma = 0.71 and α = 9.0 . Fig. 15 Pressure distribution along the upper surface of airfoil. Ma = 0.71 and α = 9.0 . 1.4 1 0.9 0.8 1.2 0.7 0.6 1 Cl 0.5 0.4 0.3 0.8 Normalized frequency 0.2 0.1 0.6 0 0 50 100 150 200 250 300 0 0.5 1 1.5 2 Realtime Reduced frequency Fig. 16 Effect of mesh on lift coefficient Cl evolution for Fig. 17 Effect of mesh on the oscillation frequency. Ma = 0.71 and α = 6.97 . Solid line: 640x64 grid; Dotted line: Solid line: 640x64 grid; Dotted line: 552x48 grid; 552x48 grid; Dashed line: 320x32 grid. Dashed line: 320x32 grid. 2 1.5 1.4 1.5 1.3 1.2 1 1.1 Cl -Cp 1 0.5 0.9 0.8 0 0.7 0.6 -0.5 0 50 100 150 200 250 0 0.25 0.5 0.75 1 Realtime x/c Fig. 18 Effect of mesh on the time-mean pressure distribution Fig. 19 Effect of initial condition on the lift coefficient along the upper surface of airfoil. Solid line: 640x64 grid; Cl evolution. Solid line: start from uniform field; Dotted Dotted line: 552x48 grid; Dashed line: 320x32 grid. line: start from the converged steady solution. 13 American Institute of Aeronautics and Astronautics 2 1.5 1.4 1.5 1.3 1.2 1 1.1 1 Cl -Cp 0.5 0.9 0.8 0 0.7 0.6 -0.5 0.5 0 0.25 0.5 0.75 1 0 100 200 300 400 x/c Realtime Fig. 20 Effect of initial condition on the lift coefficient Cl Fig. 21 Effect of lag model on lift coefficient Cl evolution. Solid line: start from uniform field; Dotted line: start evolution for Ma = 0.71 and α = 6.97 . Solid line: with from the converged steady solution. the lag model; Dotted line: without the lag model. 2 1 1.5 0.8 1 0.6 -Cp 0.5 0.4 Normalized magnitude 0 0.2 -0.5 0 0.2 0.4 0.6 0.8 1 0 0 0.5 1 1.5 2 x/c Reduced frequency Fig. 23 Effect of lag model on the time mean pressure Fig. 22 Effect of lag model on the oscillation frequency. Solid distribution along the upper surface of airfoil. Solid line: line: with the lag model; Dotted line: without the lag model. with the lag model; Dotted line: without the lag model; Circle: experiment. D C B A A B C D x/c 1.0 0.795 0.591 0.5 y/c 0 0.0873 0.118 0.125 Fig. 25 Locations of the analysis for outside separation Fig. 24 Model of self-sustained oscillation. (From Ref. 12) region cross-correlation. 14 American Institute of Aeronautics and Astronautics R(A,A,τ) R(T,T,τ) 1 1 R(T,P,τ) R(A,B,τ) 0.8 R(T,N,τ) 0.8 τ R(T,K,τ) R(A,C, ) 0.6 τ 0.6 R(T,I, ) τ τ R(A,D, ) 0.4 R(T,E, ) 0.4 0.2 0.2 0 p 0 -Cp -0.2 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 -20 -10 0 10 20 -20 -10 0 10 20 Realtime Realtime Fig. 26 Cross-correlation of computed unsteady Fig. 27 Cross-correlation of computed unsteady pressure on airfoil, indicating downstream pressure pressure outside boundary-layer, indicating upstream wave inside the separation region. propagating wave. 15 American Institute of Aeronautics and Astronautics t*=0 t*=0.1 t*=0.2 t*=0.3 t*=0.4 t*=0.5 t*=0.6 t*=0.7 t*=0.8 t*=0.9 t*=1.0 Fig. 28 Streamline plots at different instant time in one period. Ma = 0.71 and α = 6.97 . 16 American Institute of Aeronautics and Astronautics