Abstract
This paper presents an optimized low-dissipation monotonicity-preserving (MP-LD) scheme for numerical simulations of high-speed turbulent flows with shock waves. By using the bandwidth dissipation optimization method (BDOM), the linear dissipation of the original MP scheme of Suresh and Huynh (J. Comput. Phys. 136, 83–99, 1997) is significantly reduced in the newly developed MP-LD scheme. Meanwhile, to reduce the nonlinear dissipation and errors, the shock sensor of Ducros et al. (J. Comput. Phys. 152, 517–549, 1999) is adopted to avoid the activation of the MP limiter in regions away from shock waves. Simulations of turbulent flows with and without shock waves indicate that, in comparison with the original MP scheme, the MP-LD scheme has the same capability in capturing shock waves but a better performance in resolving small-scale turbulence fluctuations without introducing excessive numerical dissipation, which implies the MP-LD scheme is a valuable tool for the direct numerical simulation and large eddy simulation of high-speed turbulent flows with shock waves.
References
Balsara, D., Shu, C.-W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160, 405–452 (2000)
Barre, S., Alem, D., Bonnet, J.P.: Experimental study of a normal shock/homogeneous turbulence interaction. AIAA J. 34, 968–974 (1996)
Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3191–3211 (2008)
Capdeville, G.: A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes. J. Comput. Phys. 227, 2977–3014 (2008)
Daru, V., Tenaud, C.: High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations. J. Comput. Phys. 193, 563–594 (2004)
Ducros, F., Ferrand, V., Nicoud, F., Weber, C., Darracq, D., Gacherieu, C., Poinsot, T.: Large-eddy simulation of the shock/turbulence interaction. J. Comput. Phys. 152, 517–549 (1999)
Gaitonde, D.V., Visbal, M.R.: Padé-type higher-order boundary filters for the Navier-Stokes equations. AIAA J. 38, 2103–2112 (2000)
Garnier, E., Mossi, M., Sagaut, P., Comte, P., Deville, M.: On the use of shock-capturing schemes for large-eddy simulation. J. Comput. Phys. 153, 273–311 (1999)
Garnier, E., Sagaut, P., Deville, M.: Large eddy simulation of shock/homogeneous turbulence interaction. Comput. Fluids 31, 245–268 (2002)
Gloerfelt, X., Lafon, P.: Direct Computation of the Noise induced by a turbulent flow through a diaphragm in a duct at low Mach number. Comput. Fluids 37, 388–401 (2008)
Gottlieb, J.J., Groth, C.P.T.: Assessment of Riemann solvers for unsteady one-dimensional inviscid flows of perfect gases. J. Comput. Phys. 78, 437–458 (1988)
Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge-Kutta schemes. Math. Comput. 67, 73–85 (1998)
Grube, N.E., Taylor, E.M., Martín, M.P.: Direct numerical simulation of shock-wave/isotropic turbulence interaction. Paper 2009-4165, American Institute of Aeronautics and Astronautics (2009)
Hannappel, R., Friedrich, R.: Direct numerical simulation of a Mach 2 shock interacting with isotropic turbulence. Appl. Sci. Res. 54, 205–221 (1995)
Hill, D.J., Pullin, D.I.: Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks. J. Comput. Phys. 194, 435–450 (2004)
Jammalamadaka, A., Li, Z., Jaberi, F.A.: Large-eddy simulation of turbulent boundary layer interaction with an oblique shock wave. Paper 2010-110, American Institute of Aeronautics and Astronautics (2010)
Jamme, S., Cazalbou, J.-B., Torres, F., Chassaing, P.: Direct numerical simulation of the interaction between a shock wave and various types of isotropic turbulence. Flow Turbul. Combust. 68, 227–268 (2002)
Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995)
Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)
Johnsen, E., Larsson, J., Bhagatwala, A.V., Cabot, W.H., Moin, P., Olson, B.J., Rawat, P.S., Shankar, S.K., Sjögreen, B., Yee, H.C., Zhong, X., Lele, S.K.: Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. J. Comput. Phys. 229, 1213–1237 (2010)
Larsson, J., Lele, S.K.: Direct numerical simulation of canonical shock/turbulence interaction. Phys. Fluids 21, 12610 (2009)
Larsson, J., Lele, S.K., Moin, P.: Effect of numerical dissipation on the predicted spectra for compressible turbulence. In: Annual Research Briefs, pp. 45–57. Center for Turbulence Research, Stanford (2007)
Lee, S., Lele, S.K., Moin, P.: Direct numerical simulation of isotropic turbulence interacting with a weak shock wave. J. Fluid Mech. 12, 533–562 (1993)
Lee, S., Lele, S.K., Moin, P.: Interaction of isotropic turbulence with shock waves: effect of shock strength. J. Fluid Mech. 340, 225–247 (1997)
Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–43 (1992)
Lele, S.K., Larsson, J.: Shock-turbulence interaction: what we know and what we can learn and what we can learn from peta-scale simulations. J. Phys. Conf. Ser. 180, 1–10 (2009)
Li, Z., Jaberi, F.A.: A high-order finite difference method for numerical simulations of supersonic turbulent flows. Int. J. Numer. Methods Fluids 68, 740–766 (2011)
Lo, S.C., Blaisdell, G.A., Lyrintzis, A.S.: High-order shock capturing schemes for turbulence calculations. Int. J. Numer. Methods Fluids 62, 473–498 (2009)
Lockhard, D.P., Brentner, K.S., Atkins, H.L.: High-accuracy algorithms for computational aeroacoustics. AIAA J. 32, 246–251 (1995)
Mahesh, K., Lele, S.K., Moin, P.: The influence of entropy fluctuations on the interaction of turbulence with a shock wave. J. Fluid Mech. 334, 353–379 (1997)
Martín, M.P.: Direct numerical simulation of hypersonic turbulent boundary layers part 1. Initialization and comparison with experiments. J. Fluid Mech. 570, 347–364 (2007)
Martín, M.P., Taylor, E.M., Wu, M., Weirs, V.G.: A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence. J. Comput. Phys. 220, 270–289 (2006)
Mittal, R., Moin, P.: Suitability of upwind-biased finite difference schemes for large-eddy simulation of turbulent flows. AIAA J. 35, 1415–1417 (1997)
Oliveira, M., Lu, P., Liu, X., Liu, C.: A new shock/discontinuity detector. Int. J. Comput. Math. 87, 3063–3078 (2010)
Pirozzoli, S.: Numerical methods for high-speed flows. Annu. Rev. Fluid Mech. 43, 163–194 (2011)
Poinsot, T.J., Lele, S.K.: Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys. 101, 104–129 (1992)
Ponziani, D., Pirozzoli, S., Grasso, F.: Development of optimized weighted-ENO schemes for multiscale compressible flows. Int. J. Numer. Methods Fluids 42, 953–977 (2003)
Priebe, S., Wu, M., Martín, M.P.: Direct numerical simulation of a reflected-shock-wave /turbulent-boundary-layer interaction. AIAA J. 47, 1173–1185 (2009)
Qiu, J., Shu, C.-W.: On the construction, comparison, and local characteristic decomposition for high-order central WENO schemes. J. Comput. Phys. 183, 187–209 (2002)
Rider, W.J., Margolin, L.G.: Simple modifications of monotonicity-preserving limiter. J. Comput. Phys. 174, 473–488 (2001)
Rizzetta, D.P., Visbal, M.R.: Application of large-eddy simulation to supersonic compression ramps. AIAA J. 40, 1574–1581 (2002)
Roe, P.L.: Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43, 357–372 (1981)
Rogallo, R.S.: Numerical experiments in homogeneous turbulence. In: NSA Technical Memorandum, NASA TM-813155 (1981)
Shen, Y., Zha, G.C.: Improvement of weighted essentially non-oscillatory schemes near discontinuities. Paper 2009-3655, American Institute of Aeronautics and Astronautics (2009)
Shi, J., Zhang, Y.T., Shu, C.-W.: Resolution of high order WENO schemes for complicated flow structures. J. Comput. Phys. 186, 690–696 (2003)
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys. 83, 32–78 (1989)
Sjögreen, B., Yee, H.C.: Multiresolution wavelet based adaptive numerical dissipation control for high order methods. J. Sci. Comput. 20, 211–255 (2004)
Steger, J.L., Warming, R.F.: Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods. J. Comput. Phys. 40, 263–293 (1981)
Suresh, A., Huynh, H.T.: Accurate monotonicity–preserving schemes with Runge–Kutta time stepping. J. Comput. Phys. 136, 83–99 (1997)
Tam, C.K.W., Webb, J.C.: Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys. 107, 262–281 (1993)
Taylor, E.M., Wu, M., Martín, M.P.: Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence. J. Comput. Phys. 223, 384–397 (2007)
Touber, E., Sandham, N.D.: Large-eddy simulation of low-frequency unsteadiness in a turbulent shock-induced separation bubble. Theor. Comput. Fluid Dyn. 23, 79–107 (2009)
Tu, G.H., Yuan, X.J.: A characteristic-based shock-capturing scheme for hyperbolic problems. J. Comput. Phys. 225, 2083–2097 (2007)
Visbal, M.R., Gaitonde, D.V.: Shock capturing using compact-differencing-based methods. Paper 2005-1265, American Institute of Aeronautics and Astronautics (2005)
Wang, Z.J., Chen, R.F.: Optimized weighted essentially nonoscillatory schemes for linear waves with discontinuity. J. Comput. Phys. 174, 381–404 (2001)
Weirs, V.G., Candler, G.V.: Optimization of weighted ENO schemes for DNS of compressible turbulence. Paper 97-1940, American Institute of Aeronautics and Astronautics (1997)
Wu, M., Martín, M.P.: Direct numerical simulation of supersonic turbulent boundary layer over a compression ramp. AIAA J. 45, 879–889 (2007)
Yee, H.C., Sjögreen, B.: Adaptive filtering and limiting in compact high other methods for multiscale gas dynamics and MHD systems. Comput. Fluids 37, 593–619 (2008)
Acknowledgements
We are grateful for the valuable comments and suggestions made by the reviewers, which are significant contributions in improving the quality of this paper. L. Lu was partially supported by the National Natural Science Foundation of China (51136003, 50976010, 51006006), the National Basic Research Program of China (2012CB720205), the Aeronautical Science Foundation of China (2010ZB51025), the 111 Project (B08009) and the Astronautical Technology Innovation Foundation of China.
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Fang, J., Li, Z. & Lu, L. An Optimized Low-Dissipation Monotonicity-Preserving Scheme for Numerical Simulations of High-Speed Turbulent Flows. J Sci Comput 56, 67–95 (2013). https://doi.org/10.1007/s10915-012-9663-y
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DOI: https://doi.org/10.1007/s10915-012-9663-y