research-article

Effective low-Mach number improvement for upwind schemes

Authors:

Shu-sheng Chen,

Xing-hao XiangAuthors Info & Claims

Volume 75, Issue 10

Pages 3737 - 3755

https://doi.org/10.1016/j.camwa.2018.02.028

Published: 15 May 2018 Publication History

Abstract

In this paper, we present an effective low-Mach number improvement for upwind schemes. The artificial viscosity of upwind schemes scales with 1 ∕ M a incurring a loss of accuracy for the Mach number approaching zero. The remedy is based on three steps: (i) the jump of the left and right states is split into the density diffusion part and velocity diffusion part; (ii) the velocity diffusion part is rescaled by multiplying a scaling function; (iii) the scaling function is only related to the local Mach number without the cut-off reference Mach number and meantime restricted by a shock senor. The resulting modification is very easily implemented and applied within Roe, HLL and Rusanov, etc. Then, asymptotic analysis and numerical experiments for a wide Mach number demonstrate that this novel approach is equipped with these attractive properties: (1) free from the cut-off global problem and damping checkerboard modes; (2) satisfying the correct M a 2 scaling of pressure fluctuations, the divergence constraint and a Poisson equation; (3) independent of Mach number in terms of accuracy; (4) better accurate and higher resolution in low Mach number regimes when compared with existing upwind schemes, and applicable for moderate or high Mach numbers. Thus, the proposed modification is expected to provide as an excellent and reliable candidate to simulate turbulent flows at all Mach numbers.

References

[1]

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

[2]

Harten A., Lax P.D., van Leer B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983) 35–61.

Digital Library

[3]

Rusanov V.V., Calculation of interaction of non-steady shock waves with obstacles, J. Comput. Math. Phys. USSR 1 (1961) 267–279.

[4]

K.M. Peery, S.T. Imlay, Blunt body flow simulations, AIAA paper 88-2924, 1988.

[5]

Nishikawa H., Kitamura K., Very simple, carbuncle-free, boundary-layer-resolving, rotated-hybrid Riemann solvers, J. Comput. Phys. 227 (2008) 2560–2581.

[6]

Volpe G., Performance of compressible flow codes at low Mach number, AIAA J. 31 (1993) 49–56.

[7]

Liou M.S., A sequel to AUSM, Part II: AUSM+-up for all speeds, J. Comput. Phys. 214 (2006) 137–170.

Digital Library

[8]

Thornber B., Mosedale A., Drikakis D., Youngs D., Williams R., An improved reconstruction method for compressible flows with low Mach number features, J. Comput. Phys. 227 (2008) 4873–4894.

Digital Library

[9]

Turkel E., Preconditioned methods for solving the incompressible and low speed compressible equations, J. Comput. Phys. 72 (1987) 277–298.

[10]

Turkel E., Vatsa V.N., Local preconditioners for steady and unsteady flow applications, Math. Modelling Numer. Anal. 39 (2005) 515–535.

[11]

Weiss J.M., Smith W.A., Preconditioning applied to variable and constant density flows, AIAA J. 33 (1995) 2050–2057.

[12]

Choi Y.H., Merkle C.L., The application of preconditioning in viscous flows, J. Comput. Phys. 105 (1993) 207–223.

[13]

Turkel E., Preconditioning techniques in computational fluid dynamics, Ann. Rev. Fluid Mech. 31 (1999) 385–416.

[14]

Guillard H., Viozat C., On the behaviour of upwind schemes in the low Mach number limit, Comput. Fluids 28 (1999) 63–86.

[15]

Park S.H., Lee J.E., Kwon J.H., Preconditioned HLLE method for flows at all Mach numbers, AIAA J. 44 (2006) 2645–2653.

[16]

Li X.-S., Gu C.-W., An all-speed Roe-type scheme its asymptotic analysis of low Mach number behaviour, J. Comput. Phys. 227 (2008) 5144–5159.

[17]

Thornber B., Drikakis D., Numerical dissipation of upwind schemes in low Mach flow, Int. J. Numer. Methods Fluids 56 (2008) 1535–1541.

[18]

Fillion P., Chanoine A., Dellacherie S., Kumbaro A., FLICA-OVAP: a new platform for core thermal–hydraulic studies, Nucl. Eng. Des. 241 (2011) 4348–4358.

[19]

Rieper F., A low-Mach number fix for Roe’s approximate Riemann solver, J. Comput. Phys. 230 (2011) 5263–5287.

[20]

Rhie C., Chow W., Numerical study of the turbulent flow past an airfoil with trailing edge separation, AIAA J. 21 (1983) 1525–1532.

[21]

van Leer B., Towards the ultimate conservation difference scheme V: A second-order sequal to Godunov’s method, J. Comput. Phys. 32 (1979) 101–136.

[22]

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

[23]

Yee H.C., Klopfer G.H., Montagne J.L., High-resolution shock-capturing schemes for inviscid and viscous hypersonic flows, J. Comput. Phys. 88 (1990) 31–61.

[24]

Einfeldt B., On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal. 25 (2) (1988) 294–318.

Digital Library

[25]

Godunov S.K., Finite difference methods for the computation of discontinuous solutions of the equations of fluid dynamics, Math. USSR Sbornik 47 (1959) 271–306.

[26]

Shima E., Kitamura K., Parameter-free simple low-dissipation AUSM-family scheme for all speeds, AIAA J. 49 (2011) 1693–1709.

[27]

Oßwald K., Siegmund A., Birken P., Hannemann V., Meister A., L2Roe: A low-dissipation version of Roe’s approximate Riemann solver for low Mach numbers, Int. J. Numer. Methods Fluids 81 (2016) 71–86.

[28]

Y. Wada, M.S. Liou, A flux splitting scheme with high-resolution and robustness for discontinuities, AIAA-94-0083, NASA Technical MemorandUm 106452, 1994.

[29]

Klein R., Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics. I: one-dimensional flow, J. Comput. Phys. 121 (1995) 213–237.

[30]

Sod G.A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. 27 (1978) 1–31.

[31]

Lax P.D., Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7 (1954) 159–193.

[32]

Yoon S., Jameson A., Lower-upper Symmetric-Gauss–Seidel method for the Euler and Navier–Stokes equations, AIAA J. 9 (1988) 1025–1026.

[33]

Bassi F., De Bartolo C., Hartmann R., Nigro A., A discontinuous Galerkin method for inviscid low Mach number flows, J. Comput. Phys. 228 (2009) 3996–4011.

[34]

Feistauer M., Kucera V., On a robust discontinuous Galerkin technique for the solution of compressible flow, J. Comput. Phys. 224 (2007) 208–221.

[35]

Kim K.H., Kim C., Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows Part II: Multi-dimensional limiting process, J. Comput. Phys. 208 (2005) 570–615.

[36]

Gresho P.M., Chan S.T., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. II: Implementation, Int. J. Numer. Methods Fluids 11 (1990) 621–659.

[37]

Coles D., Wadcock A.J., Flying-hot-wire study of flow past an NACA 4412 airfoil at maximum lift, AIAA J. 17 (1979) 163–171.

[38]

Garcia-Uceda Juarez A., Raimo A., Shapiro E., Thornber B B., Steady turbulent flow computations using a low mach fully compressible scheme, AIAA J. 52 (2014) 2559–2575.

[39]

Turbulence Modeling Resource Website, http://turbmodels.larc.nasa.gov/index.html.

[40]

Taylor G.I., Green A.E., Mechanism of the production of small eddies from large ones, Proc. R. Soc. London A 158 (1937) 499–521.

[41]

J. Debonis, Solutions of the Taylor–Green Vortex problem using high-resolution explicit finite difference methods, AIAA Paper 2013-0382, 2013.

[42]

Castro M., Costa B., Don W.S., High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws, J. Comput. Phys. 230 (2011) 1766–1792.

Digital Library

[43]

Boris J.P., Grinstein F.F., Oran E.S., Kolbe R.L., New insights into Large Eddy simulation, Fluid Dyn. Res. 10 (1992) 199–228.

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Published In

cover image Computers & Mathematics with Applications

Computers & Mathematics with Applications Volume 75, Issue 10

May 2018

374 pages

ISSN:0898-1221

Issue’s Table of Contents

Elsevier Ltd.

Publisher

Pergamon Press, Inc.

United States

Publication History

Published: 15 May 2018

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https://dl.acm.org/doi/10.1016/j.jcp.2024.113653
Carlino MBoscheri W(2024)Arbitrary-Lagrangian-Eulerian finite volume IMEX schemes for the incompressible Navier-Stokes equations on evolving Chimera meshesJournal of Computational Physics10.1016/j.jcp.2024.112764501:COnline publication date: 15-Mar-2024
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Bourgeois RTremblin PKokh SPadioleau T(2024)Recasting an operator splitting solver into a standard finite volume flux-based algorithm. The case of a Lagrange-projection-type method for gas dynamicsJournal of Computational Physics10.1016/j.jcp.2023.112594496:COnline publication date: 27-Feb-2024
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Kemm F(2024)New options for explicit all Mach number schemes by suitable choice of time integration methodsJournal of Computational Physics10.1016/j.jcp.2023.112583496:COnline publication date: 27-Feb-2024
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Seraj SYildirim AAnibal JMartins J(2023)Dissipation and time step scaling strategies for low and high Mach number flowsJournal of Computational Physics10.1016/j.jcp.2023.112358491:COnline publication date: 15-Oct-2023
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Kemm F(2023)Numerical investigation of Mach number consistent Roe solvers for the Euler equations of gas dynamicsJournal of Computational Physics10.1016/j.jcp.2023.111947477:COnline publication date: 15-Mar-2023
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Jung JPerrier V(2022)Steady low Mach number flowsJournal of Computational Physics10.1016/j.jcp.2022.111462468:COnline publication date: 1-Nov-2022
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