scholarly journals Construction of a low Mach finite volume scheme for the isentropic Euler system with porosity

2021 ◽  
Author(s):  
Pascal Omnes ◽  
Stéphane Dellacherie ◽  
Jonathan Jung
Keyword(s):  
Mach Number ◽  
Finite Volume ◽  
Euler System ◽  
Finite Volume Scheme ◽  
Linear Wave ◽  
Finite Volume Schemes ◽  
Linear Wave Equation ◽  
Continuous Space ◽  
Low Mach Number ◽  
Non Linear System

Classical finite volume schemes for the Euler system  are not accurate at low Mach number and some fixes have to be used and were developed in a vast literature over the last two decades. The question we are interested in in this article is: What about if the porosity is no longer uniform? We first show that this problem may be understood on the linear wave equation taking into account porosity. We explain the influence of the cell geometry on the accuracy property at low Mach number. In the triangular case, the stationary space of the Godunov scheme approaches well enough the continuous space of constant pressure and divergence-free velocity, while this is not the case in the Cartesian case. On Cartesian meshes, a fix is proposed and accuracy at low Mach number is proved to be recovered. Based on the linear study, a numerical scheme and a low Mach fix for the non-linear system, with a non-conservative source term due to the porosity variations, is proposed and tested.

scholarly journals Construction of modified Godunov-type schemes accurate at any Mach number for the compressible Euler system

2016 ◽  
Vol 26 (13) ◽  
pp. 2525-2615 ◽  
Author(s):  
S. Dellacherie ◽  
J. Jung ◽  
P. Omnes ◽  
P.-A. Raviart
Keyword(s):  
Wave Equation ◽  
Mach Number ◽  
Stability Result ◽  
Linear Analysis ◽  
Euler System ◽  
Linear Wave ◽  
Godunov Scheme ◽  
Linear Wave Equation ◽  
Number Problem ◽  
Theoretical Results

This paper is composed of three self-consistent sections that can be read independently of each other. In Sec. 1, we define and analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov-type schemes applied to the linear wave equation to make this scheme accurate at any Mach number. This allows to define an all Mach correction and to propose a linear all Mach Godunov scheme for the linear wave equation. In Sec. 2, we apply the all Mach correction proposed in Sec. 1 to the case of the nonlinear barotropic Euler system when the Godunov-type scheme is a Roe scheme. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in this nonlinear case by showing how this construction is related with the linear analysis of Sec. 1. At last, we apply in Sec. 3 the all Mach correction proposed in Sec. 1 in the case of the full Euler compressible system. Numerous numerical results proposed in Secs. 1–3 justify the theoretical results and show that the obtained all Mach Godunov-type schemes are both accurate and stable for all Mach numbers. We also underline that the proposed approach can be applied to other schemes and allows to justify other existing all Mach schemes.

scholarly journals On Low Mach Number Preconditioning of Finite Volume Schemes

PAMM ◽  
2005 ◽  
Vol 5 (1) ◽  
pp. 759-760 ◽  
Author(s):  
Philipp Birken ◽  
Andreas Meister
Keyword(s):  
Mach Number ◽  
Finite Volume ◽  
Finite Volume Schemes ◽  
Low Mach Number

scholarly journals Low-Mach number treatment for Finite-Volume schemes on unstructured meshes

2018 ◽  
Vol 336 ◽  
pp. 368-393 ◽  
Author(s):  
Nicholas Simmonds ◽  
Panagiotis Tsoutsanis ◽  
Antonis F. Antoniadis ◽  
Karl W. Jenkins ◽  
Adrian Gaylard
Keyword(s):  
Mach Number ◽  
Finite Volume ◽  
Unstructured Meshes ◽  
Finite Volume Schemes ◽  
Low Mach Number

scholarly journals The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

Mathematics ◽  
2018 ◽  
Vol 6 (10) ◽  
pp. 211 ◽  
Author(s):  
Haoyu Dong ◽  
Changna Lu ◽  
Hongwei Yang
Keyword(s):  
Finite Volume ◽  
Time Discretization ◽  
Euler System ◽  
Weno Scheme ◽  
Local Characteristic ◽  
Finite Volume Scheme ◽  
Mesh Structure ◽  
Hyperbolic Conservation Law ◽  
Higher Derivative ◽  
Accuracy Order

We develop a Lax–Wendroff scheme on time discretization procedure for finite volume weighted essentially non-oscillatory schemes, which is used to simulate hyperbolic conservation law. We put more focus on the implementation of one-dimensional and two-dimensional nonlinear systems of Euler functions. The scheme can keep avoiding the local characteristic decompositions for higher derivative terms in Taylor expansion, even omit partly procedure of the nonlinear weights. Extensive simulations are performed, which show that the fifth order finite volume WENO (Weighted Essentially Non-oscillatory) schemes based on Lax–Wendroff-type time discretization provide a higher accuracy order, non-oscillatory properties and more cost efficiency than WENO scheme based on Runge–Kutta time discretization for certain problems. Those conclusions almost agree with that of finite difference WENO schemes based on Lax–Wendroff time discretization for Euler system, while finite volume scheme has more flexible mesh structure, especially for unstructured meshes.

scholarly journals Exact solution and the multidimensional Godunov scheme for the acoustic equations

2022 ◽  
Author(s):  
Wasilij Barsukow ◽  
Christian Klingenberg
Keyword(s):  
Exact Solution ◽  
Mach Number ◽  
Euler Equations ◽  
Two Dimensions ◽  
Finite Volume Scheme ◽  
Godunov Scheme ◽  
Low Mach Number ◽  
Low Mach Number Limit ◽  
Acoustic Equations ◽  
Spatial Dimensions

The acoustic equations derived as a linearization of the Euler equations are a valuable system for studies of multi-dimensional solutions. Additionally they possess a low Mach number limit analogous to that of the Euler equations. Aiming at understanding the behaviour of the multi-dimensional Godunov scheme in this limit, first the exact solution of the corresponding Cauchy problem in three spatial dimensions is derived. The appearance of logarithmic singularities in the exact solution of the 4-quadrant Riemann Problem in two dimensions is discussed. The solution formulae are then used to obtain the multidimensional Godunov finite volume scheme in two dimensions. It is shown to be superior to the dimensionally split upwind/Roe scheme concerning its domain of stability and ability to resolve multi-dimensional Riemann problems. It is shown experimentally and theoretically that despite taking into account multi-dimensional information it is, however, not able to resolve the low Mach number limit.

scholarly journals Investigation on spatial distribution of acoustic resonance in annular cavity: Frequency and intensity

2020 ◽  
Vol 19 (1-2) ◽  
pp. 73-94
Author(s):  
Zhifei Guo ◽  
Peiqing Liu ◽  
Hao Guo
Keyword(s):  
Spatial Distribution ◽  
Wave Equation ◽  
Mach Number ◽  
Analytical Model ◽  
Pressure Fluctuation ◽  
Standing Waves ◽  
Broadband Noise ◽  
Linear Wave ◽  
Freestream Velocity ◽  
Low Mach Number

This paper studies the acoustic behavior inside the deep annular and cylindrical cavity at low Mach number. The turbulent shear layer above the cavity acts as a broadband noise source and drives resonant standing waves inside the cavity for various modes. According to previous investigation, those resonant standing waves inside the cavity play an important role in the aeroacoustic resonance of cavity noise, which gives perfect prediction of tonal frequency from the solution of wave equation. From the perspective of engineering application, it is more important to predict the spatial distribution of tonal intensity. It is needed to point out that the solution of the linear wave equation also provides the relative spatial distribution tonal intensity and the absolute value of tonal intensity can be determined from the acoustic experiments that is measured only at some locations. Based on this idea, a scheme is setup and validated to predict the amplitude spatial distribution of tonal intensity of aeroacoustic resonance. For example, an analytical model is established to provide the relative mode shape of aeroacoustic resonance in a simple geometry of cavity, which is realized by solving the wave equation with boundary conditions in a semi-closed space. This model considers the freestream velocity scaling and the depth correction factor varying with the Helmholtz number. The experimental aeroacoustic result is acquired by measuring the pressure fluctuation at some locations of cavity internal wall with the use of surface microphones. The experimental results are used to supplement and validate this analytical model. The amplitude spatial distribution at any freestream velocity (low Mach number) can be acquired by measuring the pressure fluctuation once at the leading edge or trailing edge of cavity bottom at an arbitrary Mach number, as the amplitude of most modes reaches its maximum here.

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