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An Optimized Low-Dissipation Monotonicity-Preserving Scheme for Numerical Simulations of High-Speed Turbulent Flows

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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.

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References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barre, S., Alem, D., Bonnet, J.P.: Experimental study of a normal shock/homogeneous turbulence interaction. AIAA J. 34, 968–974 (1996)

    Article  Google Scholar 

  3. 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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Capdeville, G.: A central WENO scheme for solving hyperbolic conservation laws on non-uniform meshes. J. Comput. Phys. 227, 2977–3014 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Daru, V., Tenaud, C.: High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations. J. Comput. Phys. 193, 563–594 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. 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)

    Article  MATH  Google Scholar 

  7. Gaitonde, D.V., Visbal, M.R.: Padé-type higher-order boundary filters for the Navier-Stokes equations. AIAA J. 38, 2103–2112 (2000)

    Article  Google Scholar 

  8. 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)

    Article  MATH  Google Scholar 

  9. Garnier, E., Sagaut, P., Deville, M.: Large eddy simulation of shock/homogeneous turbulence interaction. Comput. Fluids 31, 245–268 (2002)

    Article  Google Scholar 

  10. 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)

    Article  MATH  Google Scholar 

  11. 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)

    Article  MATH  Google Scholar 

  12. Gottlieb, S., Shu, C.-W.: Total variation diminishing Runge-Kutta schemes. Math. Comput. 67, 73–85 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

  14. Hannappel, R., Friedrich, R.: Direct numerical simulation of a Mach 2 shock interacting with isotropic turbulence. Appl. Sci. Res. 54, 205–221 (1995)

    Article  MATH  Google Scholar 

  15. 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)

    Article  MATH  Google Scholar 

  16. 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)

  17. 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)

    Article  MATH  Google Scholar 

  18. Jeong, J., Hussain, F.: On the identification of a vortex. J. Fluid Mech. 285, 69–94 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Larsson, J., Lele, S.K.: Direct numerical simulation of canonical shock/turbulence interaction. Phys. Fluids 21, 12610 (2009)

    Article  Google Scholar 

  22. 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)

    Google Scholar 

  23. 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)

    Article  Google Scholar 

  24. 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)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–43 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. 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)

    Article  Google Scholar 

  27. 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)

    Article  MathSciNet  Google Scholar 

  28. 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)

    MathSciNet  Google Scholar 

  29. Lockhard, D.P., Brentner, K.S., Atkins, H.L.: High-accuracy algorithms for computational aeroacoustics. AIAA J. 32, 246–251 (1995)

    Article  Google Scholar 

  30. 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)

    Article  MathSciNet  MATH  Google Scholar 

  31. 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)

    Article  MATH  Google Scholar 

  32. 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)

    Article  MATH  Google Scholar 

  33. Mittal, R., Moin, P.: Suitability of upwind-biased finite difference schemes for large-eddy simulation of turbulent flows. AIAA J. 35, 1415–1417 (1997)

    Article  MATH  Google Scholar 

  34. Oliveira, M., Lu, P., Liu, X., Liu, C.: A new shock/discontinuity detector. Int. J. Comput. Math. 87, 3063–3078 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pirozzoli, S.: Numerical methods for high-speed flows. Annu. Rev. Fluid Mech. 43, 163–194 (2011)

    Article  MathSciNet  Google Scholar 

  36. Poinsot, T.J., Lele, S.K.: Boundary conditions for direct simulations of compressible viscous flows. J. Comput. Phys. 101, 104–129 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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)

    Article  MathSciNet  MATH  Google Scholar 

  38. 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)

    Article  Google Scholar 

  39. 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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rider, W.J., Margolin, L.G.: Simple modifications of monotonicity-preserving limiter. J. Comput. Phys. 174, 473–488 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  41. Rizzetta, D.P., Visbal, M.R.: Application of large-eddy simulation to supersonic compression ramps. AIAA J. 40, 1574–1581 (2002)

    Article  Google Scholar 

  42. Roe, P.L.: Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43, 357–372 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  43. Rogallo, R.S.: Numerical experiments in homogeneous turbulence. In: NSA Technical Memorandum, NASA TM-813155 (1981)

  44. Shen, Y., Zha, G.C.: Improvement of weighted essentially non-oscillatory schemes near discontinuities. Paper 2009-3655, American Institute of Aeronautics and Astronautics (2009)

  45. 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)

    Article  MathSciNet  MATH  Google Scholar 

  46. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  47. Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys. 83, 32–78 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  48. Sjögreen, B., Yee, H.C.: Multiresolution wavelet based adaptive numerical dissipation control for high order methods. J. Sci. Comput. 20, 211–255 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  49. 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)

    Article  MathSciNet  MATH  Google Scholar 

  50. Suresh, A., Huynh, H.T.: Accurate monotonicity–preserving schemes with Runge–Kutta time stepping. J. Comput. Phys. 136, 83–99 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  51. Tam, C.K.W., Webb, J.C.: Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys. 107, 262–281 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  52. 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)

    Article  MATH  Google Scholar 

  53. 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)

    Article  MATH  Google Scholar 

  54. Tu, G.H., Yuan, X.J.: A characteristic-based shock-capturing scheme for hyperbolic problems. J. Comput. Phys. 225, 2083–2097 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  55. Visbal, M.R., Gaitonde, D.V.: Shock capturing using compact-differencing-based methods. Paper 2005-1265, American Institute of Aeronautics and Astronautics (2005)

  56. Wang, Z.J., Chen, R.F.: Optimized weighted essentially nonoscillatory schemes for linear waves with discontinuity. J. Comput. Phys. 174, 381–404 (2001)

    Article  MATH  Google Scholar 

  57. 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)

  58. Wu, M., Martín, M.P.: Direct numerical simulation of supersonic turbulent boundary layer over a compression ramp. AIAA J. 45, 879–889 (2007)

    Article  Google Scholar 

  59. 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)

    Article  MathSciNet  Google Scholar 

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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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  1. National Key Laboratory of Science and Technology on Aero-Engine Aero-Thermodynamics, School of Energy and Power Engineering, Beihang University, 37 Xueyuan Road, Beijing, 100191, China

    Jian Fang & Lipeng Lu

  2. Department of Mechanical Engineering, Michigan State University, East Lansing, MI, 48824-1226, USA

    Zhaorui Li

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  1. Jian Fang
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Correspondence to Lipeng Lu.

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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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