Computation of NACA0012 Airfoil Transonic Buffet Phenomenon with Unsteady Navier-Stokes Equations

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1 5th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition 9-2 January 22, Nashville, Tennessee AIAA Computation of NACA2 Airfoil Transonic Buffet Phenomenon with Unsteady Navier-Stokes Equations Juntao Xiong, Feng Liu Shijun Luo, University of California Irvine, Irvine, CA, 92697, USA Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ A computational study of transonic buffet phenomenon over NACA2 airfoil is presented using unsteady Reynolds-averaged Navier-Stokes Equations. Computations are conducted at freestream Mach numbers from.82 to.89 at zero angle of attack for Reynolds numbers of 3. and. million with transition fixed at 5% chord length. Numerical simulation reveals the transonic buffet occurs in a narrow range of freestream Mach numbers of It is close to the range of freestream Mach numbers of in which multiple solutions occur for the steady, inviscid flow. The unsteady flow fields around the airfoil are examined and characterized. The shock motion frequency is found to depend on the distance between time-averaged shock wave position and trailing edge. The shock motion frequency is larger when the time-averaged shock position is closer to the trailing edge. The computational results are verified using present transonic wind tunnel experiment data. Finally the effect of a trailing edge splitter for buffet suppression is studied. Nomenclature a = speed of sound c = chord of airfoil c p = pressure coefficient E = total internal energy F c = inviscid convective flux F d = inviscid diffusive flux f = frequency f = reduced frequency, f = πfc/u k = turbulent kinetic energy l = length of tailing edge splitter M = Mach number p = static pressure t = time u, v, w = velocity components W = conservative variable vector x, y, z = Cartesian coordinates x s = shock wave position x s m = time-averaged shock wave position α = angle of attack, closure coefficient β, β = closure coefficients γ = specific heat ratio µ L = molecular viscosity µ T = turbulent viscosity ω = specific dissipation rate ρ = density σ = closure coefficient Postdoc Researcher, Department of Mechanical Aerospace Engineering, Member AIAA Professor, Department of Mechanical and Aerospace Engineering, Associate Fellow AIAA Researcher, Department of Mechanical Aerospace Engineering of 5 Copyright 22 by the authors. Published by the, Inc., with permission.

2 ν = S-A turbulence model work variables τ = stress tensor I. Introduction Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ Multiple solutions have been found for the steady, inviscid flow past some airfoils under certain conditions of transonic Mach numbers and angles of attack. 9 Jameson 2 reported lifting solutions of the transonic full potential flow equation of a symmetric Joukowski airfoil at zero angle of attack in a narrow band of freestream Mach numbers. Luo et al. 3 showed the multiple solutions of transonic small-disturbance equation for NACA2 airfoil at zero angle of attack at certain range of freestream Mach numbers. Later Jameson 6 showed multiple solutions using Euler equations for some lifting airfoils in a narrow band of freestream Mach number. Hafez and Ivanova research groups 7 9 studied the multiple solutions for a 2% thick symmetric airfoil with flat and wavy surface using potential equations, Euler equations and Navier-Stokes equations. In order to study the mechanism of multiple solutions, Williams et al 5 studied stability of multiple solutions of unsteady transonic small-disturbance equation. Caughey performed unsteady simulations of Euler equations to gain a better understanding of the multiple solutions evolution and stability. Liu et al. presented a linear stability analysis of multiple solutions of transonic small-disturbance potential equation of NACA2 airfoil at zero angle of attack. They found the symmetric solutions are not stable and asymmetric solutions are stable. Crouch et al demonstrated that NACA2 airfoil buffet onset is related to the global stability of the flow field. Based on these researches, it can be conjectured the inviscid instability might be a contributing factor to trigger buffet. The goal of this work is to investigate unsteady viscous flow field behaviors over symmetric NACA2 airfoil under the same conditions which multiple solutions occur for inviscid flow to gain more understanding. The computational code is first validated against available experiments. 2 6 Thereafter the code is used to simulate the unsteady flow field of NACA2 airfoil. The unsteady flow fields around NACA2 airfoil are examined and characterized. Finally the elimination of buffet by adding a trailing edge splitter plate is examined. II. Computational Method The computational fluid dynamics code used here is known as PARCAE 7 and solves the unsteady three-dimensional compressible Navier-Stokes equations on structured multiblock grids using a cell-centered finite-volume method with artificial dissipation as proposed by Jameson et al. 8 Information exchange for flow computation on multiblock grids using multiple CPUs is implemented through the MPI (Message Passing Interface) protocol. The Navier-Stokes equations are solved using the eddy viscosity type turbulence models. All computations presented in this work are performed using Menter SST k-ω model. 9 The main elements of the code are summarized below. The differential governing equations for the unsteady compressible flow can be expressed as follows: W t + (F c F d ) = () The vector W contains the conservative variables (ρ, ρu, ρv, ρw, ρe) T. The fluxes consist of the inviscid convective fluxes F c and the diffusive fluxes F d, defines as F c = ρu ρv ρw ρuu + p ρuv ρuw ρvu ρvv + p ρvw ρwu ρwv ρww + p ρeu + pu ρev + pv ρew + pw (2) 2 of 5

3 Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ with F d = τ xx τ xy τ xz τ yx τ yy τ yz τ zx τ zy τ zz Θ x Θ y Θ z Θ = u : τ ( µ L Pr L + µ T Pr T ) T (4) The stress tensor τ depends on the viscosity µ = µ L + µ T, where the subscripts L and T represent laminar and turbulent contributions, respectively. Pr L and Pr T are the laminar and turbulent Prandtl numbers, respectively. The closure model used to evaluate the turbulent viscosity µ T is the k ω SST turbulence model, given by the equations ρk t + (ρku µ k k) = ρs k ρω t where µ k = µ L + σ k µ T, µ ω = µ L + σ ω µ T, µ T = + (ρωu µ ω ω) = ρs ω (5) S k = ρ τ : u β ωk S ω = γ τ : u βω 2 + 2( f ) k ω µ τ ω ρa k max(a ω;ωf 2). The source term S k and S ω are In the above equations, f and f 2 are blending functions. The parameters σ k, σ ω, β, β, and γ are closure coefficients. The flow and turbulence equations are discretized in space by a structured hexahedral grid using a cell-centered finite-volume method. Since within the code each block is considered as a single entity, only flow and turbulence quantities at the block boundaries need to be exchanged. The governing equations are solved using dual-time stepping method for time accurate flow. At sub-iteration the five stage Runge-Kutta scheme is used with local-time stepping, residual smoothing, and multigrid for convergence acceleration. The turbulence model equations are solved using stagger-couple method. Further details of the numerical method can be found in Ref. 2,2. III. Results & Discussion The computations are first validated against experimental data. 2 4 Then the code is used to simulate the unsteady flow field of NACA2 airfoil. III.A. Code Validation Grid independence study is performed using NACA2 steady simulation cases. Three C-type computational grids with different spatial resolution coarse, medium, and fine are generated. Table shows the dimensions of the three grids. Figure shows the fine grid. Table. Dimensions of NACA2 computational grids (3) N x N y Coarse M edium Fine Steady computations are run on the three grids at Mach number.75 and.775 of Reynolds number. million with angle of attack.99 and 2.5, respectively. Figure 2 shows the comparisons of pressure 3 of 5

4 distribution around NACA2 airfoil between computational and experimental 2 data. Fine grid results show excellent agreement with the experimental data and predicts accurate shock wave position. So unsteady flow field simulations are only performed using fine grid. The 8% thick arcfoil is used to validate the unsteady code, because there exist unsteady data 3 6 for this airfoil at zero angle of attack and supercritical Mach numbers. The grid with finest dimensions is used to simulate unsteady flow field. Figure 3 shows the lift coefficient evolution history and Fast Fourier Transform (FFT) analysis. The FFT amplitude is normalized by largest peak. The primary frequency corresponding to the largest peak is f =.46. It agrees with the experimental data, f = ,5,6 Figure 4 shows the comparison of time-averaged pressure distribution around arcfoil between computational and experimental 4 data. The computational result agrees well with experimental data. The code can predict accurate time-averaged shock wave position and shock motion primary frequency. The validation of the code to study shock boundary layer interaction problem is confirmed by the good comparison of the fine grid results on steady and unsteady cases. Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ III.B. Steady Flow computations The steady computations had been used to predict the onset of buffet. 22 Furthermore, a good convergence with steady methods probably means good numerical and flow stability. Figure 5 shows continuity equation residuals history at freestream Mach numbers from.82 to.89 at zero angle of attack for Reynolds number million. At Mach number.86 and.87 the residuals do not converge. The numerical unsteadiness might be caused by the buffet phenomenon. It might indicate the buffet happends in Mach number range of.86 and.87. In the next section the unsteady simulations are performed to study the problem. III.C. Unsteady Flow Simulations In this section the unsteady flow field simulations for NACA2 airfoil at freestream Mach numbers from.82 to.89 at zero angle of attack for Reynolds numbers. million with transition fixed at 5% chord length are performed to study unsteady viscous flow behaviors. The computational results reveal that the flow is steady when the Mach number is between.82 and.84. When the Mach number gets to.85, the flow suddenly changes into an unsteady oscillatory mode. The shock waves on the upper and lower surfaces of the airfoil begin to move back and forth in a periodic motion. The unsteadiness is caused by the shock wave interaction with the boundary layer over the airfoil surface. It can be categorized to type A shock wave motion. The unsteady flow pattern persists as the Mach number is further increased until it gets to.88 where the flow becomes steady again. The buffet occurs in a narrow band of freestream Mach numbers of It is close to the range of freestream Mach numbers of in which multiple solutions occur for the steady, inviscid flow. It might indicate the inviscid instability is another contributing factor to trigger buffet. Figure 6 shows the lift coefficients evolution history for the three Mach numbers of.85,.86, and.87. The c l is largest when the Mach number is.86. It means the unsteadiness is strongest at that Mach number. This can be also observed in Fig. 7. The FFT amplitudes are normalized by largest peak between the three Mach numbers. When the Mach number is.86, FFT has largest amplitude. The amplitude drops very quickly as the Mach number increases to.87. The lift coefficient oscillation frequency increases as the Mach number increases. Figure 8 shows the time-averaged isentropic Mach number distribution on the airfoil surface at Mach numbers from.82 to.89. The time-averaged shock wave positions are determined with the positions where isentropic Mach number is.. It is clear to show the time-averaged shock wave position moves to trailing edge as the Mach number increases. Figure 9 shows time dependent shock wave positions against Mach number. In the figure the squares are time-averaged shock wave postion. It clearly shows the flow is steady when Mach number is lower than.85 or higher than.87. Figure shows the shock oscillation frequency against the distance between the time-averaged shock wave position and the trailing edge at different Mach numbers. The shock oscillation frequency depends on the shock wave mean position. The distance between the time-averaged shock wave position and the trailing edge is an important length scale to determine shock wave oscillation frequency. This length scale represents the distance which disturbance waves travel through. Figures -2 show the Mach number contours and streamlines at 4 sequential time instants during one period of the shock oscillation at Mach number.86. The figures show the shock waves on the upper and lower surface of NACA2 airfoil move back and forth in a periodic motion. As the upper surface shock 4 of 5

5 wave moving upstream, it becomes weaker and the separation bubble becomes smaller. At the same time the lower surface shock moves downstream and becomes stronger. Stronger shock induces flow separation. As the separation region on the lower surface becomes large enough, the lower surface shock wave is pushed upstream and the upper surface shock wave moves downstream. The consequence of this behavior is an unsteady shock oscillation over the airfoil. Figures 3-5 show pressure oscillation FFT analysis at x/c =.7,x/c =.9, and x/c =., respectively. The FFT amplitudes are normalized by the largest peak at x/c =.7. The figures show there are two disturbance waves which are caused by shock wave interaction with boundary layer moving to downstream. The phases of movements of waves differ by exactly 8. The two waves interact at the trailing edge. It offers the possibility of elimination of buffet by blocking the path of the disturbance waves interact. Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ III.D. Reynolds Number effect In the previous section C the unsteady flow simulations for NACA2 airfoil are performed at Reynolds number million. In order to study the effect of Reynolds number unsteady flow simulations are run at 3. million. Figure 6 shows the lift coefficients evolution history for the three Mach numbers of.85,.86, and.87. c l for the three Mach numbers at Reynolds number 3. million are smaller compared with million results. It means the unsteadiness is weaker at lower Reynolds number. Figure 7 shows FFT analysis of lift coefficients. The FFT amplitudes are normalized by largest peak between the three Mach numbers. Figure 8 shows the time-averaged isentropic Mach number distribution on the airfoil surface at Mach numbers from.85 to.87. It is clear to show the time-averaged shock wave position moves to upstream as the Reynolds number decreased to 3. million. Figure 9 shows the shock oscillation frequency against the distance between the time-averaged shock wave position and the trailing edge at different Mach number.85,.86, and.87. Since the time-averaged shock wave positions move to upstream at Reynolds number 3. million, the shock oscillation frequencies become smaller. In order to verify the computational results, transonic wind tunnel unsteady pressure measurements are performed for NACA2 airfoil at Mach numbers from.8 to.89 at zero angle of attack for Reynolds number 3. million. The experiment results without correction show the buffet onset Mach number.88. In general, the wind tunnel sidewall interference would decrease the Mach number in the core flow. So the experiment predicts higher buffet onset point. Figure 2 shows the comparison of time-averaged pressure distribution on the NACA2 surface between computational and experimental 23 data. In the figure the computational Mach number is.85 and the experimental Mach number is.88. The time-averaged shock wave positions are very close. Figure 2 shows the comparison of pressure oscillation FFT analysis at x/c =.75. It shows the primary frequencies of pressure oscillation are very close. Again it shows it is reasonable to use time-averaged shock wave position as shock wave motion length scale. III.E. Elimination of Buffet by Trailing Edge Splitter Plate In the previous section C the unsteady flow field behaviors of buffet are studied. During the shock oscillation period the upper and lower surface shock waves move back and forth, and the disturbance information of upper and lower side surface are exchanged through the trailing edge of the airfoil. If we can somehow block the path of the information exchange, the buffet might be alleviated or eliminated. In this study the adding trailing edge splitter plate method is studied. 3 Two different length plates are used which are shown in Fig. 22. The corresponding pressure oscillation contours on the upper surface of NACA2 airfoil are shown in Fig. 23. When the short plate with 6.% of chord length is used the flow is still unsteady. But the flow becomes steady when the long plate with 25.% of chord length is used. The adding trailing edge splitter plate method can be successfully used to eliminate buffet. IV. Conclusion A computational study is conducted on the flow field of NACA2 airfoil at freestream Mach numbers from.82 to.89 at zero angle of attack for Reynolds numbers 3. and. million with transition fixed at 5% chord length. Computational results show the buffet occurs in a narrow band of freestream Mach numbers from.85 to.87. It is very close to the range of freestream Mach numbers in which multiple solutions occur for the steady, inviscid flow. It shows very interesting relation between buffet in viscous flow 5 of 5

6 and multiple solution in inviscid flow. The unsteady shock wave motion frequency is significantly correlated to the time-averaged shock wave position. The effect of Reynolds number on unsteady flow field is studied. The time-averaged shock wave position moves to upstream as Reynolds number decreases. The shock motion frequency decreases as well. The computational results are further validated using present transonic wind tunnel tests. Flow buffet can be eliminated by adding tailing edge splitter plate with sufficient length. References Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ Liu, Y., Liu, F., and Luo, S., Abstract: MX.4 : Linear Stability Analysis on Multiple Solutions of Steady Transonic Small Disturbance Equation, APS, Nov Steinhoff, J. and Jameson, A., Multiple Solutions of Transonic Potential Flow Equation, AIAA Journal, Vol. 2, No., 982, pp Luo, S., Shen, H., and Liu, P., Transonic Small Transverse Perturbation Equation and its Computation, Proc. Symposium on Computing the Future II: Computational Fluid Dynamics Advances and Applications, ed. D. A. Caughey and M. M. Hafez, Salas, M. D., Jameson, A., and Melnik, R. E., A Comparative Study of the Nonuniqueness problem of the Potential Equation, AIAA , July Williams, M. H., Bland, S. R., and Edwards, J. W., Flow Instablities in Transonic Small-Disturbance Theory, AIAA Journal, Vol. 23, No., 985, pp Jameson, A., Airfoils admitting non-unique solutions of Euler Equations, AIAA , June Ivanova, A. V. and Kuz min, A. G., Nonuniqueness of the Transonic Flow past an Airfoil, Fluid Dynamics, Vol. 39, No. 4, 24, pp Hafez, M. M. and Guo, W. H., Nonuniqueness of Transonic Flows, Acta Mechanica, Vol. 38, 999, pp Hafez, M. M., Non-uniqueness Problems in Transonic Flows, Computational Fluid Dynamics, Vol., No., 22, pp Caughey, D. A., Stability of Unsteady Flow past Airfoils Exhibiting Transonic Non-uniqueness, Computational Fluid Dynamics, Vol. 3, No. 3, 24, pp Crouch, J. D., Garbaruk, A., Magidov, D., and Travin, A., Origin of Transonic Buffet on Aerofoil, J. Fluid Mech., Vol. 628, 29, pp Mcdevitt, J. B. and Okuno, A. F., Static and Dynamic Pressure Measurements on a NACA 2 Airfoil in the Ames High Reynolds Number Facility, NASA TP 2485, Mcdevitt, J. B., Supercitical Flow About a Thick Circular-Arc Airofil, NASA TM 78549, Mcdevitt, J. B., Levy, J. J., and Deiwert, G. S., Transonic flow about a thick circular-arc airfoil, AIAA Journal, Vol. 4, No. 5, 976, pp Levy, J. J., Experimental and computational steady and unsteady transonic flows about a thick airfoil, AIAA Journal, Vol. 6, No. 6, 978, pp Marvin, J. G. and Levy, J. J., Turbulence modelling for unsteady transonic flows, AIAA Journal, Vol. 8, No. 5, 98, pp Xiong, J., Nielsen, P., Liu, F., and Papamoschou, D., Computation of High-Speed Coaxial Jets with Fan Flow Deflection, AIAA Journal, Vol. 48, No., 2, pp Jameson, A., Schmift, W., and Turkel, E., Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time Stepping Schemes, AIAA , Janurary Menter, F., Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications, AIAA Journal, Vol. 32, No. 8, 994, pp Liu, F. and Zheng, X., A Strongly Coupled Time-Marching Method for Solving the Navier-Stokes and k-ω Turbulence Model Equations with Multigrid, Journal of Computational Physics, Vol. 28, No., 996, pp Xiao, Q., Tsai, H. M., and Liu, F., Numerical Study of Transonic Buffet on a Supercritical Airfoil, AIAA Journal, Vol. 44, No. 3, 26, pp Chung, I., Lee, D., and Reu, T., Prediction of Transonic Buffet Onset for An Airfoil with Shock Induced Separation Bubble Using Steady Navier-Stokes Solver, AIAA , June Zhao, Z., Gao, C., Ren, X., Xiong, J.,, Liu, F., and Luo, S., Pressure Distributions over NACA 2 Airfoil in a Transonic Wind Tunnel, Submitted to 42th AIAA Fluid Dynamics Conference. 6 of 5

7 Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ c p x/c Figure. NACA2 airfoil computational fine mesh. Fine Medium Coarse Experiment c p x/c Fine Medium Coarse Experiment Figure 2. Comparisons of pressure distribution around NACA2 airfoil. M =.75, α =.99, Re = 6 ; M =.775, α = 2.5, Re = 6. 7 of 5

8 .5.8 Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ c l t (s) Normalized magnitude Reduced frequency Figure 3. Evolution of lift coefficient and FFT analysis for 8% thick arcfoil, M =.76, α =., Re = 6. Lift coefficient; FFT analysis. c p -2 - Experiment Computation x/c Figure 4. Comparison of time-averaged pressure distribution around 8% thick arcfoil, M =.76, α =., Re = 6. 8 of 5

9 4 3 2 Residual - Ma =.82 Ma =.84 Ma =.85 Ma =.86 Ma =.87 Ma =.88 Ma = Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ Iterations Figure 5. Continuity equation residuals for NACA2 airfoil steady computations, α =., Re = 6. c l t (s) Ma =.85 Ma =.86 Ma =.87 Figure 6. Evolution of lift coefficients for NACA2 airfoil, α =., Re = 6. Normalized magnitude Ma =.85 Ma =.86 Ma = Reduced Frequency Figure 7. FFT analysis of lift coefficients for NACA2 airfoil, α =., Re = 6. 9 of 5

10 Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ Figure 8. Time-averaged isentropic Mach number distributions around NACA2 airfoil, α =., Re = 6. Figure 9. Shock wave position against freestream Mach number for NACA2 airfoil, α =., Re = 6. (Blue square denotes time-averaged shock wave position) Reduced Frequency /(-x sm /c) Figure. Shock wave oscillation frequency position against the distance between time-averaged shock wave position and trailing edge for NACA2 airfoil, M =.85.87, α =., Re = 6. of 5

11 Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ (c) Figure. Mach number contours at different instant time in one period around the NACA2 airfoil, M =.86, α =., Re = 6. T = t ; T = t 2; (c)t = t 3; (d)t = t 4; (c) Figure 2. Streamlines at different instant time in one period around the NACA2 airfoil, M =.86, α =., Re = 6. T = t ; T = t 2; (c)t = t 3; (d)t = t 4; (d) (d) of 5

12 .8 x/c=.7, upper x/c=.7, lower x/c=.7, upper x/c=.7,lower Normalized magnitude.6.4 Phase (Rad) Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ Pressure fluctuation reduced frequence Pressure fluctuation reduced frequency Figure 3. FFT analysis of pressure oscillation at x/c =.7 for NACA2 airfoil, M =.86, α =., Re = 6. FFT amplitude; FFT phase. Normalized magnitude x/c=.9, upper x/c=.9, lower Pressure fluctuation reduced frequence Phase (Rad) x/c=.9, upper x/c=.9,lower Pressure fluctuation reduced frequency Figure 4. FFT analysis of pressure oscillation at x/c =.9 for NACA2 airfoil, M =.86, α =., Re = 6. FFT amplitude; FFT phase. Normalized magnitude x/c=. Phase (Rad) 4.5 x/c= Pressure fluctuation reduced frequence Pressure fluctuation reduced frequency Figure 5. FFT analysis of pressure oscillation at x/c =. for NACA2 airfoil, M =.86, α =., Re = 6. FFT amplitude; FFT phase. 2 of 5

13 .3.2 Ma =.85 Ma =.86 Ma =.87. c l Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ t (s) Figure 6. Evolution of lift coefficients for NACA2 airfoil, α =., Re = Normalized magnitude Ma =.85 Ma =.86 Ma =.87 2 Reduced Frequency Figure 7. FFT analysis of lift coefficients for NACA2 airfoil, α =., Re = M is Ma=.85, Re=3.x 6 Ma=.85, Re=.x 6 Ma=.86, Re=3.x 6 Ma=.86, Re=.x 6 Ma=.87, Re=3.x 6 Ma=.87, Re=.x x/c Figure 8. Time-averaged isentropic Mach number distributions around NACA2 airfoil, α =.. 3 of 5

14 .6 Re=3.x 6 Re=.x 6 Reduced Frequency.4.2 Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ /(-x sm /c) Figure 9. Shock wave oscillation frequency position against the distance between time-averaged shock wave position and trailing edge for NACA2 airfoil, M =.85.87, α =.. c p -2 - Experiment (lower_surface) Experiment (upper_surface) Computation x/c Figure 2. Comparison of time-averaged pressure distribution around NACA2 airfoil, α =., Re = Normalized magnitude Experiment Computation Pressure oscillation frequency (Hz) Figure 2. FFT analysis of pressure oscillation at x/c =.75 for NACA2 airfoil, α =., Re = of 5

15 Downloaded by UNIVERSITY OF CALIFORNIA IRVINE on January 29, 25 DOI:.254/ Figure 22. Configurations of trailing edge splitter plates. l = 6.%c; l = 25.%c Figure 23. Computed pressure contours on NACA2 airfoil upper surface, M =.86, α =., Re = 6. l = 6.%c; l = 25.%c 5 of 5

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