The Zero Mach Number Limit of the Three-Dimensional Compressible Viscous Magnetohydrodynamic Equations

This paper is concerned with the zero Mach number limit of the three-dimension- al compressible viscous magnetohydrodynamic equations. More precisely, based on the local existence of the three-dimensional compressible viscous magnetohydrodynamic equations, first the convergence-stability principle is established. Then it is shown that, when the Mach number is sufficiently small, the periodic initial value problems of the equations have a unique smooth solution in the time interval, where the incompressible viscous magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number goes to zero. Moreover, the authors prove the convergence of smooth solutions of the equations towards those of the incompressible viscous magnetohydrodynamic equations with a sharp convergence rate.     

A note on the zero Mach number limit of compressible Euler equations

This note presents a short and elementary justification of the classical zero Mach number limit for isentropic compressible Euler equations with prepared initial data. We also show the existence of smooth compressible flows, with the Mach number sufficiently small, on the (finite) time interval where the incompressible Euler equations have smooth solutions.

     

The combined quasineutral and low Mach number limit of Euler–Poisson equations coupled to a magnetic field

In this paper, we consider the combined quasineutral and low Mach number limit of compressible Euler–Poisson system coupled to a magnetic field. We prove that, as the Debye length and the Mach number tend to zero simultaneously in some way, the solution of compressible Euler–Poisson system coupled to a magnetic field will converge to that of ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.     

Zero Mach number limit to compressible quantum magnetohydrodynamic equations with vanishing viscosity coefficients

The connection between the compressible viscous quantum magnetohydrodynamic model with low Mach number and the ideal incompressible magnetohydrodynamic equations is studied in a periodic domain. More precisely, for well‐prepared initial data, we prove the convergence of classical solutions of the compressible viscous quantum magnetohydrodynamic model to the classical solutions of the incompressible ideal magnetohydrodynamic equations with a convergence rate when the Mach number, viscosity coefficient, and magnetic diffusion coefficient simultaneously tend to zero.     

On the stability of the solutions to the compressible Navier-Stokes equations when the Mach number goes to zero

In this paper an asymptotic stability result is estabilished for the compressible navier-Stokes equations. Since the Mach number tends to zero, the incompressible limit solution of compressible Navier-Stokes equations is proved to be stable exponentially. Some results of Stokes' problem are used.     

一类可压缩流的马赫数极限

该文证明了在二或三维情形下, 当马赫数趋于零时, 一类完全可压缩Navier-Stokes方程的解收敛到相应的完全不可压缩Navier-Stokes方程的解.     

A note on incompressible limit for compressible Euler equations

This paper presents a simple justification of the classical low Mach number limit in critical Besov spaces for compressible Euler equations with prepared initial data. As the first step of this justification, we formulate a continuation principle for general hyperbolic singular limit problems in the framework of critical Besov spaces. With this principle, it is also shown that, for the Mach number sufficiently small, the smooth compressible flows exist on the (finite) time interval where the incompressible Euler equations have smooth solutions, and the definite convergence orders are obtained. Copyright © 2011 John Wiley & Sons, Ltd.     

Life span of 2‐D irrotational compressible fluids in the halfplane

We consider the Euler equations of barotropic inviscid compressible fluids in the half plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In dimension two such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial velocity. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small. We consider smooth irrotational solutions. First, we study the life span, i.e. the largest time interval T(ε) of existence of classical solutions, when the initial data are a small perturbation of size εfrom a constant state. For the proof of this result we use a combination of energy and decay estimates. Then, the estimate of the life span allows to show, by a suitable scaling of variables, the existence of irrotational solutions on any arbitrary time interval, for any small enough Mach number. Copyright © 2002 John Wiley & Sons, Ltd.     

The low Mach number limit in Besov spaces for compressible magnetohydrodynamics equations

In this paper, we develop a continuation principle for general hyperbolic singular limit problems in more general Besov spaces, which covers the cases of usual Sobolev spaces with higher regularity in and the critical Besov space. As an application, we give a simple justification for the low Mach number limit of compressible magnetohydrodynamics equations. More precisely, for the Mach number sufficiently small, the smooth compressible flows exist on the (finite) time interval where the incompressible magnetohydrodynamics equations have smooth solutions, and the definite convergence orders are also obtained. Copyright © 2014 John Wiley & Sons, Ltd.     

Incompressible limits of the Navier-Stokes equations for all time

We study the asymptotic behaviors of the regular solutions to the compressible Navier-Stokes equations for “well-prepared” initial data for all time as the Mach number tends to zero, by deriving a differential inequality with certain decay property. The estimates obtained in this paper are uniform both in time and Mach number.     

Low Mach number limit for the non-isentropic Navier-Stokes equations

We study the low Mach number limit of the local in time solutions to the compressible Navier-Stokes equations with zero heat conductivity coefficient as the Mach number tends to zero. A uniform existence result for the one-dimensional initial-boundary value problem is proved provided that the initial data are “well-prepared” in the sense that the temporal derivatives up to order two are bounded initially.     

Coupling of compressible and incompressible flow regions using the multiple pressure variables approach

In many cases, multiphase flows are simulated on the basis of the incompressible Navier–Stokes equations. This assumption is valid as long as the density changes in the gas phase can be neglected. Yet, for certain technical applications such as fuel injection, this is no longer the case, and at least the gaseous phase has to be treated as a compressible fluid. In this paper, we consider the coupling of a compressible flow region to an incompressible one based on a splitting of the pressure into a thermodynamic and a hydrodynamic part. The compressible Euler equations are then connected to the Mach number zero limit equations in the other region. These limit equations can be solved analytically in one space dimension that allows to couple them to the solution of a half‐Riemann problem on the compressible side with the help of velocity and pressure jump conditions across the interface. At the interface location, the flux terms for the compressible flow solver are provided by the coupling algorithms. The coupling is demonstrated in a one‐dimensional framework by use of a discontinuous Galerkin scheme for compressible two‐phase flow with a sharp interface tracking via a ghost‐fluid type method. The coupling schemes are applied to two generic test cases. The computational results are compared with those obtained with the fully compressible two‐phase flow solver, where the Mach number zero limit is approached by a weakly compressible fluid. For all cases, we obtain a very good agreement between the coupling approaches and the fully compressible solver. Copyright © 2014 John Wiley & Sons, Ltd.     

Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations

In this paper, we study the incompressible limit of the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the convergence-stability principle, we show that, when the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient are sufficiently small, the initial-value problem of the model has a unique smooth solution in the time interval where the ideal incompressible magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient go to zero. Moreover, we obtain the convergence of smooth solutions for the model forwards those for the ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.     

Asymptotic Expansions and Numerical Methods for Compressible and Low Mach Number Fluid Flow

The results of a formal asymptotic low Mach number analysis [5, 6] of the Euler equations of gas dynamics are used to extend the validity of a numerical method from the simulation of compressible inviscid flow fields to the low Mach number regime. Although, different strategies are applicable [7, 8, 5, 9] in this context we focus our view to a preconditioning technique recently proposed by Guillard and Viozat [16]. We present a finite volume approximation of the governing equations using a Lax‐Friedrichs scheme whereby a preconditioning of the incorporated numerical dissipation is employed. A discrete asymptotic analysis proves the validity of the scheme in the low Mach number regime.     

The Incompressible Limits of Compressible Navier-Stokes Equations in the Whole Space with General Initial Data

It is showed that, as the Mach number goes to zero, the weak solution of the compressible Navier-Stokes equations in the whole space with general initial data converges to the strong solution of the incompressible Navier-Stokes equations as long as the later exists. The proof of the result relies on the new modulated energy functional and the Strichartz＇s estimate of linear wave equation.     

基于Roe格式的可压与不可压流的统一计算方法

摘要：以Navier-Stokes方程为基础，基于有限体积的时间推进的预处理技术．提出了一个可以用来求解可压与不可压流场的统一的计算方法，原始变量选用压力、速度与温度，通过矩阵变换与重构，使得对流项系数矩阵在可压与小可压条件下都不会奇异．将可压与不可压流场的计算方法统一起来。采用Roe格式计算对流通量，采用中心差分格式计算扩散通量．算例表明，该方法可以进行高Mach数、中等Mach数、低Mach数及不可压流场的计算。由于采用了Roe格式，该方法还可以捕获不连续流场的间断面。     

Low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data

The low Mach number limit for the full compressible magnetohydrodynamic equations with general initial data is rigorously justified in the whole space R³${\mathbb{R}}^3$. First, the uniform-in-Mach-number estimates of the solutions in a Sobolev space are established on a finite time interval independent of the Mach number. Then the low Mach number limit is proved by combining these uniform estimate with a theorem due to Métivier and Schochet (2001) [45] for the Euler equations that gives the local energy decay of the acoustic wave equations.     

Incompressible limit and stability of all-time solutions to 3-D full Navier-Stokes equations for perfect gases

This paper studies the incompressible limit and stability of global strong solutions to the threedimensional full compressible Navier-Stokes equations, where the initial data satisfy the "well-prepared" conditions and the velocity field and temperature enjoy the slip boundary condition and convective boundary condition, respectively. The uniform estimates with respect to both the Mach number ∈(0, ∈] and time t ∈ [0, ∞) are established by deriving a differential inequality with decay property, where ∈∈(0, 1] is a constant.As the Mach number vanishes, the global solution to full compressible Navier-Stokes equations converges to the one of isentropic incompressible Navier-Stokes equations in t ∈ [0, +∞). Moreover, we prove the exponentially asymptotic stability for the global solutions of both the compressible system and its limiting incompressible system.     

Incompressible limit of all-time solutions to 3-D full Navier–Stokes equations for perfect gas with well-prepared initial condition

In this paper, we prove the incompressible limit of all-time strong solutions to the three-dimensional full compressible Navier–Stokes equations. Here the velocity field and temperature satisfy the Dirichlet boundary condition and convective boundary condition, respectively. The uniform estimates in both the Mach number \({\epsilon\in(0,\overline{\epsilon}]}\) and time \({t\in[0,\infty)}\) are established by deriving a differential inequality with decay property, where \({\overline{\epsilon}\in(0,1]}\) is a constant. Based on these uniform estimates, the global solution of full compressible Navier–Stokes equations with “well-prepared” initial conditions converges to the one of isentropic incompressible Navier–Stokes equations as the Mach number goes to zero.