A very-high-order TENO scheme for all-speed gas dynamics and turbulence

Abstract In this paper, we propose a new scale-separation formula and develop a very-high-order targeted ENO scheme, which shows exceptional performance in conventional compressible gas dynamics, high-Mach-number simulations with vacuum or near-vacuum region, incompressible and compressible turbulence prediction. The modified TENO weighting strategy can be extended to arbitrarily very-high-order reconstruction in a straightforward manner. The proposed 10-point TENO10-A scheme is optimized to satisfy an approximate-dispersion relation while maintaining the 8th-order accuracy in smooth regions. For conventional gas dynamics at low to moderate Mach numbers, the TENO10-A scheme is robust, and shows low numerical dissipation in resolving small-scale physical fluctuations while capturing the sharp discontinuities. For high-Mach number simulations, TENO10-A is numerically stable and preserves the ENO property with the assistance of a positivity-preserving flux limiter. In terms of turbulent flows, TENO10-A faithfully predicts energy transfer, and resolves vorticity, entropy and acoustic modal fluctuations. A set of benchmark simulations is considered to assess the performance of proposed scheme.

[1]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[2]  D. Pullin,et al.  Hybrid tuned center-difference-WENO method for large eddy simulations in the presence of strong shocks , 2004 .

[3]  Chi-Wang Shu,et al.  Efficient Implementation of Weighted ENO Schemes , 1995 .

[4]  G. A. Gerolymos,et al.  Very-high-order weno schemes , 2009, J. Comput. Phys..

[5]  H. Huynh,et al.  Accurate Monotonicity-Preserving Schemes with Runge-Kutta Time Stepping , 1997 .

[6]  Lin Fu,et al.  A low-dissipation finite-volume method based on a new TENO shock-capturing scheme , 2019, Comput. Phys. Commun..

[7]  Nikolaus A. Adams,et al.  A new class of adaptive high-order targeted ENO schemes for hyperbolic conservation laws , 2018, J. Comput. Phys..

[8]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[9]  Nikolaus A. Adams,et al.  An adaptive central-upwind weighted essentially non-oscillatory scheme , 2010, J. Comput. Phys..

[10]  Nikolaus A. Adams,et al.  An adaptive local deconvolution method for implicit LES , 2005, J. Comput. Phys..

[11]  S. Orszag,et al.  Small-scale structure of the Taylor–Green vortex , 1983, Journal of Fluid Mechanics.

[12]  Chi-Wang Shu,et al.  Runge-Kutta Discontinuous Galerkin Method Using WENO Limiters , 2005, SIAM J. Sci. Comput..

[13]  Sergio Pirozzoli,et al.  Numerical Methods for High-Speed Flows , 2011 .

[14]  Nikolaus A. Adams,et al.  A physically consistent weakly compressible high-resolution approach to underresolved simulations of incompressible flows , 2013 .

[15]  Chi-Wang Shu,et al.  Anti-diffusive flux corrections for high order finite difference WENO schemes , 2005 .

[16]  Stefan Hickel,et al.  Subgrid-scale modeling for implicit large eddy simulation of compressible flows and shock-turbulence interaction , 2014 .

[17]  Xiangxiong Zhang,et al.  Positivity-preserving high order finite difference WENO schemes for compressible Euler equations , 2012, J. Comput. Phys..

[18]  Xiangxiong Zhang,et al.  On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes , 2010, J. Comput. Phys..

[19]  Soshi Kawai,et al.  Assessment of localized artificial diffusivity scheme for large-eddy simulation of compressible turbulent flows , 2010, J. Comput. Phys..

[20]  Nikolaus A. Adams,et al.  A family of high-order targeted ENO schemes for compressible-fluid simulations , 2016, J. Comput. Phys..

[21]  V. Gregory Weirs,et al.  A bandwidth-optimized WENO scheme for the effective direct numerical simulation of compressible turbulence , 2006, J. Comput. Phys..

[22]  Sergio Pirozzoli,et al.  On the spectral properties of shock-capturing schemes , 2006, J. Comput. Phys..

[23]  Nikolaus A. Adams,et al.  Positivity-preserving method for high-order conservative schemes solving compressible Euler equations , 2013, J. Comput. Phys..

[24]  Chi-Wang Shu,et al.  On positivity preserving finite volume schemes for Euler equations , 1996 .

[25]  Yuan Liu,et al.  A Robust Reconstruction for Unstructured WENO Schemes , 2013, J. Sci. Comput..

[26]  Björn Sjögreen,et al.  Numerical dissipation control in high order shock-capturing schemes for LES of low speed flows , 2016, J. Comput. Phys..

[27]  J. P. Boris,et al.  New insights into large eddy simulation , 1992 .

[28]  Lin Fu,et al.  A Targeted ENO Scheme as Implicit Model for Turbulent and Genuine Subgrid Scales , 2019, Communications in Computational Physics.

[29]  Raphaël Loubère,et al.  A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods , 2005 .

[30]  Wai-Sun Don,et al.  An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws , 2008, J. Comput. Phys..

[31]  Eleuterio F. Toro,et al.  Finite-volume WENO schemes for three-dimensional conservation laws , 2004 .

[32]  Klaus A. Hoffmann,et al.  Minimizing errors from linear and nonlinear weights of WENO scheme for broadband applications with shock waves , 2013, J. Comput. Phys..

[33]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[34]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[35]  Dimitris Drikakis,et al.  On the implicit large eddy simulations of homogeneous decaying turbulence , 2007, J. Comput. Phys..

[36]  Lei Luo,et al.  A sixth order hybrid finite difference scheme based on the minimized dispersion and controllable dissipation technique , 2014, J. Comput. Phys..

[37]  Soshi Kawai,et al.  Localized artificial diffusivity scheme for discontinuity capturing on curvilinear meshes , 2008, J. Comput. Phys..

[38]  Yuxin Ren,et al.  A characteristic-wise hybrid compact-WENO scheme for solving hyperbolic conservation laws , 2003 .

[39]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[40]  Ivan Fedioun,et al.  Comparison of improved finite-difference WENO schemes for the implicit large eddy simulation of turbulent non-reacting and reacting high-speed shear flows , 2014 .

[41]  S. Osher,et al.  Weighted essentially non-oscillatory schemes , 1994 .

[42]  Nikolaus A. Adams,et al.  Targeted ENO schemes with tailored resolution property for hyperbolic conservation laws , 2017, J. Comput. Phys..

[43]  Nikolaus A. Adams,et al.  Scale separation for implicit large eddy simulation , 2011, J. Comput. Phys..

[44]  Yong-Tao Zhang,et al.  Resolution of high order WENO schemes for complicated flow structures , 2003 .

[45]  Sergio Pirozzoli,et al.  Conservative Hybrid Compact-WENO Schemes for Shock-Turbulence Interaction , 2002 .

[46]  Wai-Sun Don,et al.  High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws , 2011, J. Comput. Phys..

[47]  Andrew W. Cook,et al.  Artificial Fluid Properties for Large-Eddy Simulation of Compressible Turbulent Mixing , 2007 .

[48]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[49]  Nail K. Yamaleev,et al.  A systematic methodology for constructing high-order energy stable WENO schemes , 2009, J. Comput. Phys..

[50]  F. Nicoud,et al.  Large-Eddy Simulation of the Shock/Turbulence Interaction , 1999 .

[51]  Chi-Wang Shu,et al.  Monotonicity Preserving Weighted Essentially Non-oscillatory Schemes with Increasingly High Order of Accuracy , 2000 .

[52]  Parviz Moin,et al.  Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves , 2010, J. Comput. Phys..

[53]  Bruno Costa,et al.  An improved WENO-Z scheme , 2016, J. Comput. Phys..

[54]  J. M. Powers,et al.  Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points , 2005 .