Calculation of low Mach number acoustics : a comparison of MPV, EIF and linearized Euler equations
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 3, p. 561-576

The calculation of sound generation and propagation in low Mach number flows requires serious reflections on the characteristics of the underlying equations. Although the compressible Euler/Navier-Stokes equations cover all effects, an approximation via standard compressible solvers does not have the ability to represent acoustic waves correctly. Therefore, different methods have been developed to deal with the problem. In this paper, three of them are considered and compared to each other. They are the Multiple Pressure Variables Approach (MPV), the Expansion about Incompressible Flow (EIF) and a coupling method via heterogeneous domain decomposition. In the latter approach, the non-linear Euler equations are used in a domain as small as possible to cover the sound generation, and the locally linearized Euler equations approximated with a high-order scheme are used in a second domain to deal with the sound propagation. Comparisons will be given in construction principles as well as implementational effort and computational costs on actual numerical examples.

DOI : https://doi.org/10.1051/m2an:2005016
Classification:  41A60,  65M55,  76Q05
Keywords: aero-acoustics, low Mach number flows, asymptotic expansion, heterogeneous domain decomposition
@article{M2AN_2005__39_3_561_0,
author = {Roller, Sabine and Schwartzkopff, Thomas and Fortenbach, Roland and Dumbser, Michael and Munz, Claus-Dieter},
title = {Calculation of low Mach number acoustics : a comparison of MPV, EIF and linearized Euler equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {39},
number = {3},
year = {2005},
pages = {561-576},
doi = {10.1051/m2an:2005016},
zbl = {1130.76072},
mrnumber = {2157150},
language = {en},
url = {http://www.numdam.org/item/M2AN_2005__39_3_561_0}
}

Roller, Sabine; Schwartzkopff, Thomas; Fortenbach, Roland; Dumbser, Michael; Munz, Claus-Dieter. Calculation of low Mach number acoustics : a comparison of MPV, EIF and linearized Euler equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 39 (2005) no. 3, pp. 561-576. doi : 10.1051/m2an:2005016. http://www.numdam.org/item/M2AN_2005__39_3_561_0/

[1] M. Dumbser and C.-D. Munz, Arbitrary High Order Discontinuous Galerkin Schemes. IRMA series in mathematics and theoretical physics. | MR 2186376

[2] R. Fortenbach and C.-D. Munz, Multiple Scale considerations for sound generation in low Mach number flow, in Proc. The GAMM Jahrestagung, Augsburg, Germany, March 25-28 (2002). | Zbl pre05020049

[3] K. Geratz, Erweiterung eines Godunov-Typ-Verfahrens für zwei-dimensionale kompressible Strömungen auf die Fälle kleiner und verschwindender Machzahl. Ph.D. Thesis, RWTH Aachen (1997).

[4] J. Hardin and D. Pope, An acoustic/viscous splitting technique for computational aeroacoustics. Theoret. Comput. Fluid Dynamics 6 (1994) 323-340. | Zbl 0822.76057

[5] S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible Limit of Compressible Fluids. Comm. Pure Appl. Math. 34 (1981) 481-524. | Zbl 0476.76068

[6] R. Klein, 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. | Zbl 0842.76053

[7] R. Klein, N. Botta, L. Hofmann, A. Meister, C.-D. Munz, S. Roller and T. Sonar, Asymptotic adaptive methods for multiscale problems in fluid mechanics. J. Engrg. Math. 39 (2001) 261-343. | Zbl 1015.76071

[8] A. Meister, Asymptotic single and multiple scale expansions in the mow Mach number limit. SIAM J. Appl. Math. 60 (1999) 256-271. | Zbl 0941.35052

[9] B.E. Mitchell, S.K. Lele and P. Moin, Direct computation of the sound from a compressible co-rotating vortex pair. J. Fluid Mech. 285 (1995) 181-202. | Zbl 0848.76085

[10] C.-D. Munz, S. Roller, R. Klein and K.J. Geratz, The extension of incompressible flow solvers to the weakly compressible regime. Comput. Fluids 32 (2003) 173-196. | Zbl 1042.76045

[11] S. Roller, Ein numerisches verfahren zur simulation schwach kompressibler Strömungen. Ph.D. Thesis, University of Stuttgart (2004).

[12] S. Roller and C.-D. Munz, A low Mach number scheme based on multi-scale asymptotics. Comput. Visual. Sci. 3 (2000) 85-91. | Zbl 1060.76630

[13] T. Schneider, N. Botta, K. Geratz and R. Klein, Extension of finite volume compressible flow solvers to multi-dimensional, variable density zero Mach number flow. J. Comput. Phys. 155 (1999) 248-286. | Zbl 0968.76054

[14] T. Schwartzkopff, Finite-Volumen Verfahren hoher Ordnung und heterogene Gebietszerlegung für die numerische Aeroakustik. Ph.D. Thesis, University of Stuttgart (2005).

[15] T. Schwartzkopff and C.-D. Munz, Direct simulation of aeroacoustics, in Proc. Applied Mathematics and Mechanics (GAMM 2002) 2 (2002). | Zbl pre02151300

[16] T. Schwartzkopff and C.-D. Munz, Direct simulation of aeroacoustics, in Analysis and Simulation of Multifield Problems, W. Wendland and M. Efendiev, Eds., Springer. Lect. Notes Appl. Comput. Mech. 12 (2003) 337-342. | Zbl pre02151300

[17] T. Schwartzkopff, M. Dumbser and C.-D. Munz, CAA using domain decomposition and high order methods on structured and unstructured meshes, in 10th AIAA/CEAS Aeroacoustic Conference, Manchester, GB (2004).

[18] T. Schwartzkopff, M. Dumbser and C.-D. Munz, Fast high order ADER schemes for linear hyperbolic equations. J. Comput. Phys. 197 (2004) 532-539. | Zbl 1052.65078

[19] T. Schwartzkopff, C.-D. Munz, E. Toro and R. Millington, ADER-2d: A very high-order approach for linear hyperbolic systems, in Proceedings of ECCOMAS CFD Conference 2001 (September 2001). | Zbl 1022.76034

[20] E. Toro and R. Millington, ADER: High-order non-oscillatory advection schemes, in Proceedings of the 8th International Conference on Nonlinear Hyperbolic Problems, preprint (February 2000).