Conservative hybrid compact-WENO schemes for shock-turbulence interaction. (English) Zbl 1045.76029

Pirozzoli, Sergio

J. Comput. Phys. 178, No. 1, 81-117 (2002).

From the summary: We propose an efficient hybrid compact-WENO scheme to obtain high resolution in shock-turbulence interaction problems. The algorithm is based on a fifth-order compact upwind algorithm in conservation form to solve for the smooth part of the flow field, which is coupled with a high-resolution weighted essentially nonoscillatory (WENO) scheme to capture the discontinuities. The derivation of the compact scheme is discussed in detail, and a stability study of the full discretization is included. The performance of the numerical algorithm has been assessed by performing preliminary simulations on benchmark problems, such as the interaction of a shock wave with entropy and vortical disturbances.

MSC:

76M20	Finite difference methods applied to problems in fluid mechanics
76L05	Shock waves and blast waves in fluid mechanics
76F99	Turbulence

Keywords:

shock-capturing scheme; fifth-order compact upwind algorithm

References:

[1]	Adams, N. A.; Shariff, K., A high-resolution hybrid compact-ENO scheme for shock-turbulence interaction problems, J. Comput. Phys., 127, 27 (1996) · Zbl 0859.76041
[2]	Adams, N. A., Direct numerical simulation of turbulent compression corner flow, Theor. Comp. Fluid Dyn., 12, 109 (1998) · Zbl 0931.76033
[3]	Arora, M.; Roe, P. L., On postshock oscillations due to shock capturing schemes in unsteady flows, J. Comput. Phys., 130, 25 (1997) · Zbl 0869.76050
[4]	Balsara, D.; Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160, 405 (2000) · Zbl 0961.65078
[5]	M. Carpenter, D. Gottlieb, and, C.-W. Shu, On the conservation and convergence to weak solutions of global schemes, J. Sci. Comput, in press.; M. Carpenter, D. Gottlieb, and, C.-W. Shu, On the conservation and convergence to weak solutions of global schemes, J. Sci. Comput, in press. · Zbl 1030.65097
[6]	Chang, C. T., Interaction of a plane shock and oblique plane disturbances with special reference to entropy waves, J. Aero. Sci., 24, 675 (1957) · Zbl 0080.19402
[7]	Chu, B. T.; Kovásznay, L. S.G., Non-linear interactions in a viscous heat-conducting compressible gas, J. Fluid Mech., 3, 494 (1957)
[8]	Cockburn, B.; Shu, C.-W., Nonlinearly stable compact schemes for shock calculation, SIAM J. Numer. Anal., 31, 607 (1994) · Zbl 0805.65085
[9]	Collatz, L., The Numerical Treatment of Differential Equations (1966), Springer-Verlag: Springer-Verlag New York · Zbl 0221.65088
[10]	Colonius, T.; Lele, S.; Moin, P., Boundary conditions for direct computation of aerodynamic sound, AIAA J., 31, 1574 (1993) · Zbl 0785.76069
[11]	Deng, X.; Maekawa, H., Compact high-order accurate nonlinear schemes, J. Comput. Phys., 130, 77 (1997) · Zbl 0870.65075
[12]	Deng, X.; Zhang, H., Developing high-order weighted compact nonlinear schemes, J. Comput. Phys., 165, 22 (2000) · Zbl 0988.76060
[13]	Freund, J. B., Proposed inflow/outflow boundary condition for direct computation of aerodynamic sound, AIAA J., 35, 740 (1997) · Zbl 0903.76081
[14]	Gaitonde, D.; Shang, J. S., Optimized compact-difference-based finite-volume schemes for linear wave phenomena, J. Comput. Phys., 138, 617 (1997) · Zbl 0898.65055
[15]	Garnier, E.; Mossi, M.; Sagaut, P.; Comte, P.; Deville, M., On the use of shock-capturing for large-eddy simulation, J. Comput. Phys., 153, 273 (1999) · Zbl 0949.76042
[16]	Garnier, E.; Sagaut, P.; Deville, M., A class of explicit ENO filters with application to unsteady flows, J. Comput. Phys., 170, 184 (2001) · Zbl 1011.76056
[17]	Grasso, F.; Pirozzoli, S., Simulations and analysis of the coupling process of compressible vortex pairs: free evolution and shock induced coupling, Phys. Fluids, 13, 1343 (2001) · Zbl 1184.76194
[18]	Gustafsson, B., The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comput., 29, 396 (1975) · Zbl 0313.65085
[19]	Hannapel, R.; Hauser, T.; Friedrich, R., A comparison of ENO and TVD schemes for the computation of shock-turbulence interaction, J. Comput. Phys., 121, 176 (1995) · Zbl 0834.76058
[20]	Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non oscillatory schemes, III, J. Comput. Phys., 71, 213 (1987)
[21]	Hirsch, C., Numerical Computation of Internal and External Flows (1988), Wiley: Wiley New York · Zbl 0662.76001
[22]	Jiang, G. S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202 (1996) · Zbl 0877.65065
[23]	Jiang, L.; Shan, H.; Liu, C., Weighted compact schemes for shock capturing, Int. J. Comput. Fluid D., 15, 147-155 (2001) · Zbl 1044.76046
[24]	Kobayashi, M. H., On a class of Padé finite volume methods, J. Comput. Phys., 156, 137 (1999) · Zbl 0940.65092
[25]	Kopal, Z., Numerical Analysis (1961), Wiley: Wiley New York · Zbl 0101.33701
[26]	Lee, S.; Lele, S. K.; Moin, P., Direct numerical simulation of isotropic turbulence interacting with a weak shock wave, J. Fluid Mech., 251, 533 (1993)
[27]	Lee, S.; Lele, S. K.; Moin, P., Interaction of isotropic turbulence with shock waves: effect of shock strength, J. Fluid Mech., 340, 255 (1997) · Zbl 0899.76194
[28]	Lele, S. K., Compact finite difference schemes with spectral-like resolution, J. Comput. Phys., 103, 16 (1992) · Zbl 0759.65006
[29]	LeVecque, R., Numerical Methods for Conservation Laws (1990), Birkhäuser-Verlag: Birkhäuser-Verlag Basel · Zbl 0723.65067
[30]	Liu, X. D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115, 200 (1994) · Zbl 0811.65076
[31]	Mao, D.-K., A treatment of discontinuities in shock-capturing finite difference methods, J. Comput. Phys., 92, 422 (1991) · Zbl 0716.65086
[32]	McKenzie, J. F.; Westphal, L. O., Interaction of linear waves with oblique shock waves, Phys. Fluids, 11, 2350 (1968) · Zbl 0172.53203
[33]	Pirozzoli, S.; Grasso, F.; D’Andrea, A., Interaction of a shock wave with two counter-rotating vortices: shock dynamics and sound production, Phys. Fluids, 13, 3460 (2001) · Zbl 1184.76423
[34]	Poinsot, T. J.; Lele, S. K., Boundary conditions for direct simulations of compressible viscous reacting flows, J. Comput. Phys., 101, 104 (1992) · Zbl 0766.76084
[35]	H. S. Ribner, Convection of a Pattern of Vorticity through a Shock Wave; H. S. Ribner, Convection of a Pattern of Vorticity through a Shock Wave
[36]	Rogerson, A.; Meiburg, E., A numerical study of the convergence properties of ENO schemes, J. Sci. Comput., 5, 151 (1990) · Zbl 0732.65086
[37]	Samtaney, R.; Pullin, D. I., On initial-value and self-similar solutions of the compressible Euler equations, Phys. Fluids, 8, 2650 (1996) · Zbl 1027.76642
[38]	Shu, C.-W., Numerical experiments on the accuracy of ENO and modified ENO schemes, J. Sci. Comput., 5, 127 (1990) · Zbl 0732.65085
[39]	C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics edited by, B. Cockburn, C. Johnson, C.-W. Shu, E. Tadmor, and A. Quarteroni, Springer-Verlag, Berlin, 1998, Vol, 1697, p, 425.; C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics edited by, B. Cockburn, C. Johnson, C.-W. Shu, E. Tadmor, and A. Quarteroni, Springer-Verlag, Berlin, 1998, Vol, 1697, p, 425.
[40]	Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439 (1988) · Zbl 0653.65072
[41]	Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes II, J. Comput. Phys., 83, 32 (1989) · Zbl 0674.65061
[42]	Tam, C. K.W.; Webb, J. C., Dispersion-relation-preserving finite difference schemes for computational acoustics, J. Comput. Phys., 107, 262 (1993) · Zbl 0790.76057
[43]	Thompson, K. W., Time dependent boundary conditions for hyperbolic systems, J. Comput. Phys., 68, 1 (1987) · Zbl 0619.76089
[44]	Tolstykh, A. I.; Lipavskii, M. V., On performance of methods with third- and fifth-order compact upwind differencing, J. Comput. Phys., 140, 205 (1998) · Zbl 0936.76059
[45]	Vichnevetsky, R.; Bowles, J. B., Fourier Analysis of Numerical Approximations of Hyperbolic Equations (1982), SIAM: SIAM Philadelphia · Zbl 0495.65041
[46]	Z. J. Wang, and, R. F. Chen, Optimized Weighted Essentially Non-Oscillatory Schemes for Computational Aeroacoustics; Z. J. Wang, and, R. F. Chen, Optimized Weighted Essentially Non-Oscillatory Schemes for Computational Aeroacoustics
[47]	V. G. Weirs, and, G. V. Candler, Optimization of Weighed ENO Schemes for DNS of Compressible Turbulence; V. G. Weirs, and, G. V. Candler, Optimization of Weighed ENO Schemes for DNS of Compressible Turbulence
[48]	Zang, T. A.; Hussaini, M. Y.; Bushnell, D. M., Numerical computations of turbulence amplification in shock-wave interactions, AIAA J., 22, 13 (1984) · Zbl 0534.76062
[49]	Zhuang, M.; Cheng, R. F., Optimized upwind dispersion-relation-preserving finite difference schemes for computational aeroacoustics, AIAA J., 36, 2146 (1998)
[50]	Yee, H. C.; Sandham, N. D.; Djomehri, M. J., Low-dissipative high-order shock-capturing methods using characteristic-based filters, J. Comput. Phys., 150, 199 (1999) · Zbl 0936.76060