High-order localized dissipation weighted compact nonlinear scheme for shock- and interface-capturing in compressible flows

Simulations of single- and multi-species compressible flows with shock waves and discontinuities are conducted using a weighted compact nonlinear scheme (WCNS) with a newly developed sixth order localized dissipative interpolation. In smooth regions, the scheme applies the central nonlinear interpolation with minimum dissipation to resolve fluctuating flow features while in regions containing discontinuities and high wavenumber features, the scheme suppresses spurious numerical oscillations by hybridizing the central interpolation with the more dissipative upwind-biased nonlinear interpolation. In capturing material interfaces between species of different densities, a quasi-conservative five equation model that can conserve mass of each species is used to prevent pressure oscillations across the interfaces. Compared to upwind-biased interpolations with classical nonlinear weights [1,2] and improved weights [3], and the interpolation with adaptive central-upwind weights for scale-separation [4], it is shown that WCNS with the proposed localized dissipative interpolation has better performance to simultaneously capture discontinuities and resolve smooth features.

[1]  Shigeru Obayashi,et al.  Numerical (error) issues on compressible multicomponent flows using a high-order differencing scheme: Weighted compact nonlinear scheme , 2012, J. Comput. Phys..

[2]  J. Jacobs,et al.  PLIF flow visualization and measurements of the Richtmyer–Meshkov instability of an air/SF6 interface , 2002, Journal of Fluid Mechanics.

[3]  Eric Johnsen,et al.  Implementation of WENO schemes in compressible multicomponent flow problems , 2005, J. Comput. Phys..

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

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

[6]  Santhosh K. Shankar,et al.  Numerical Simulation of Multicomponent Shock Accelerated Flows and Mixing using Localized Artificial Diffusivity Method , 2010 .

[7]  Derek M. Causon,et al.  On the Choice of Wavespeeds for the HLLC Riemann Solver , 1997, SIAM J. Sci. Comput..

[8]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[9]  Dong Yan,et al.  Cures for numerical shock instability in HLLC solver , 2011 .

[10]  Tim Colonius,et al.  Finite-volume WENO scheme for viscous compressible multicomponent flows , 2014, J. Comput. Phys..

[11]  Keh-Ming Shyue,et al.  An Efficient Shock-Capturing Algorithm for Compressible Multicomponent Problems , 1998 .

[12]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[13]  Taku Nonomura,et al.  Robust explicit formulation of weighted compact nonlinear scheme , 2013 .

[14]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[15]  Sanjiva K. Lele,et al.  An artificial nonlinear diffusivity method for supersonic reacting flows with shocks , 2005, J. Comput. Phys..

[16]  Xiaogang Deng,et al.  Developing high-order weighted compact nonlinear schemes , 2000 .

[17]  Taku Nonomura,et al.  Increasing Order of Accuracy of Weighted Compact Non-Linear Scheme , 2007 .

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

[19]  Grégoire Allaire,et al.  A five-equation model for the simulation of interfaces between compressible fluids , 2002 .

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

[21]  E. Toro,et al.  Restoration of the contact surface in the HLL-Riemann solver , 1994 .

[22]  P. Roe,et al.  On Godunov-type methods near low densities , 1991 .

[23]  Chi-Wang Shu,et al.  Development of nonlinear weighted compact schemes with increasingly higher order accuracy , 2008, J. Comput. Phys..

[24]  Chi-Wang Shu,et al.  A new class of central compact schemes with spectral-like resolution II: Hybrid weighted nonlinear schemes , 2015, J. Comput. Phys..

[25]  M. Baer,et al.  A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials , 1986 .

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

[27]  M. Pino Martín,et al.  Optimization of nonlinear error for weighted essentially non-oscillatory methods in direct numerical simulations of compressible turbulence , 2007, J. Comput. Phys..

[28]  B. Sturtevant,et al.  Experiments on the Richtmyer–Meshkov instability: single-scale perturbations on a continuous interface , 1994, Journal of Fluid Mechanics.

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

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

[31]  Man Long Wong,et al.  Improved Weighted Compact Nonlinear Scheme for Flows with Shocks and Material Interfaces: Algorithm and Assessment , 2016 .

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

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

[34]  Rémi Abgrall,et al.  Computations of compressible multifluids , 2001 .

[35]  R. Abgrall.  How to Prevent Pressure Oscillations in Multicomponent Flow Calculations , 1996 .

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

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

[38]  W. Cabot,et al.  A high-wavenumber viscosity for high-resolution numerical methods , 2004 .

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

[40]  S. A. Orsag,et al.  Small-scale structure of the Taylor-Green vortex , 1984 .

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

[42]  Andrew W. Cook,et al.  Short Note: Hyperviscosity for shock-turbulence interactions , 2005 .

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

[44]  P. Moin,et al.  Effect of numerical dissipation on the predicted spectra for compressible turbulence , 2022 .