Mach Number Limit Research Articles

Low Mach number limit for quasi-one-dimensional isentropic Euler flow

This paper is devoted to studying the low Mach number limit for the quasi-one-dimensional isentropic Euler equations in $BV\\cap~L^1$ space. Under assumptions that the total variation of the cross section of the nozzleis sufficiently small and the initial data is away from vacuum, by using the $L^1$-stability estimatesand the method of the standard Riemann semigroup, we rigorouslyprove that the solution can be expandedin the powers of small Mach number and that its coefficients satisfy a linear acoustic equation with the source term.

High order semi-implicit weighted compact nonlinear scheme for the all-Mach isentropic Euler system

The computation of compressible flows at all Mach numbers is a very challenging problem. An efficient numerical method for solving this problem needs to have shock-capturing capability in the high Mach number regime, while it can deal with stiffness and accuracy in the low Mach number regime. This paper designs a high order semi-implicit weighted compact nonlinear scheme (WCNS) for the all-Mach isentropic Euler system of compressible gas dynamics. To avoid severe Courant-Friedrichs-Levy (CFL) restrictions for low Mach flows, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components. A third-order implicit-explicit (IMEX) method is used for the time discretization of the split components and a fifth-order WCNS is used for the spatial discretization of flux derivatives. The high order IMEX method is asymptotic preserving and asymptotically accurate in the zero Mach number limit. One- and two-dimensional numerical examples in both compressible and incompressible regimes are given to demonstrate the advantages of the designed IMEX WCNS.

Lattice BGK model for time-fractional incompressible Navier–Stokes equations

In this paper, a novel D2Q9 lattice Boltzmann model with BGK operator (LBGK) is proposed for the incompressible time-fractional Navier–Stokes equations with Caputo-type fractional derivative. First the fractional derivative is divided into the history part and the local part, in which the former is approximated using the efficient algorithm for the evaluation of the Caputo fractional derivative, while the latter is simply approximated by rectangle formula to keep the time-dependent characteristics of Navier–Stokes equations as the evolution equations. Then, a LBGK model is constructed for Navier–Stokes equations after approximation of the Caputo derivative. Through Chapman–Enskog analysis the macroscopic equations can be recovered from this model in the small Mach number limit. At the end of this paper, a numerical example with analytic solutions is carried out to show that the LBGK model is efficient.

Low Mach number limit for the compressible inertial Qian-Sheng model of liquid crystals: Convergence for classical solutions

<p style='text-indent:20px;'>In this paper we study the incompressible limit of the compressible inertial Qian-Sheng model for liquid crystal flow. We first derive the uniform energy estimates on the Mach number <inline-formula><tex-math id="M1">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> for both the compressible system and its differential system with respect to time under uniformly in <inline-formula><tex-math id="M2">\begin{document}$ \epsilon $\end{document}</tex-math></inline-formula> small initial data. Then, based on these uniform estimates, we pass to the limit in the compressible system as <inline-formula><tex-math id="M3">\begin{document}$ \epsilon \rightarrow 0 $\end{document}</tex-math></inline-formula>, so that we establish the global classical solution of the incompressible system by compactness arguments. We emphasize that, on global in time existence of the incompressible inertial Qian-Sheng model under small size of initial data, the range of our assumptions on the coefficients are significantly enlarged, comparing to the results of De Anna and Zarnescu's work [<xref ref-type="bibr" rid="b6">6</xref>]. Moreover, we also obtain the convergence rates associated with <inline-formula><tex-math id="M4">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm with well-prepared initial data.

Zero Mach Number Limit of the Compressible Primitive Equations: Well-Prepared Initial Data

This work concerns the zero Mach number limit of the compressible primitive equations. The primitive equations with the incompressibility condition are identified as the limiting equations. The convergence with well-prepared initial data (i.e., initial data without acoustic oscillations) is rigorously justified, and the convergence rate is shown to be of order mathcal {O}(varepsilon ) , as varepsilon rightarrow 0^+ , where varepsilon represents the Mach number. As a byproduct, we construct a class of global solutions to the compressible primitive equations, which are close to the incompressible flows.

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.

Large amplitude ion-acoustic solitary waves in a warm negative ion plasma with superthermal electrons: The fast mode revisited

Large amplitude ion-acoustic fast mode solitary waves in a negative ion plasma with kappa electrons are revisited, using the Sagdeev pseudopotential approach. As is well known, this plasma supports the propagation of both compressive and rarefactive solitons, and there exist a range of parameter values in which the two types of structures coexist. This is confirmed by the present study, which is based on well-established soliton existence domains. After investigating the existence of solitons in terms of the lower and upper Mach number limits for broader regions in the parameter space, we have found that as a result of the ion thermal effects, the range of the allowed Mach numbers is reduced and only small amplitude rarefactive solitons propagate in this plasma, an effect that is enhanced by the superthermal behavior of the electrons. Rearranging our analytical work so as to get a two-positive ion plasma, our results show the presence of stopbands in the soliton existence domains, as reported by Nsengiyumva et al. [Phys. Plasmas 21, 102301 (2014)], despite the use of different normalization and different parameter space. This suggests that the observed stopbands are a real phenomenon, which needs consideration when studying plasma waves.

Zero Mach number limit of the compressible Euler\u2013Korteweg equations

In this paper, we investigate the zero Mach number limit for the three-dimensional compressible Euler–Korteweg equations in the regime of smooth solutions. Based on the local existence theory of the compressible Euler–Korteweg equations, we establish a convergence-stability principle. Then we show that when the Mach number is sufficiently small, the initial-value problem of the compressible Euler–Korteweg equations has a unique smooth solution in the time interval where the corresponding incompressible Euler equations have a smooth solution. It is important to remark that when the incompressible Euler equations have a global smooth solution, the existence time of the solution for the compressible Euler–Korteweg equations tends to infinity as the Mach number goes to zero. Moreover, we obtain the convergence of smooth solutions for the compressible Euler–Korteweg equations towards those for the incompressible Euler equations with a convergence rate.

Analysis of artificial pressure equations in numerical simulations of a turbulent channel flow

Recently, several methods have been proposed to simulate incompressible fluid flows using an artificial pressure evolution equation, avoiding the resolution of a Poisson equation. These methods can be seen as various levels of approximation of the compressible Navier–Stokes equation in the low Mach number limit. We study the simulation of incompressible wall-bounded flows using several artificial pressure equations in order to determine the most relevant approximations. The simulations are stable using a finite difference method in a staggered grid system, even without diffusive term, and converge to the incompressible solution, both in direct numerical simulations and for coarser meshes, to be used in large-eddy simulations. A pressure equation with a convective and a diffusive term produces a more accurate solution than a compressible solver or methods involving more approximations. This suggests that it is near to an optimal level of approximation. The presence of a convective term in the pressure evolution equation is in particular crucial for the accuracy of the method. The rate of convergence of the solution in terms of artificial Mach number is studied numerically and validates the theoretical quadratic convergence rate. We demonstrate that this property can be used to accelerate the rate of convergence using an extrapolation in terms of artificial Mach number. Since the approach is based on an explicit and local system of equations, the numerical procedure is massively parallelisable and has low memory requirements.

Low Mach number limit of multidimensional steady flows on the airfoil problem

In this paper, we justify the low Mach number limit of the steady irrotational Euler flows for the airfoil problem, which is the first result for the low Mach number limit of the steady Euler flows in an exterior domain. The uniform estimates on the compressibility parameter $\varepsilon$, which is singular for the flows, are established via a variational approach based on the compressible-incompressible difference functions. The limit is on the H\"{o}lder space and is unique. Moreover, the convergence rate is of order $\varepsilon^2$. It is noticeable that, due to the feature of the airfoil problem, the extra force dominates the asymptotic decay rate of the compressible flow to the infinity. And the effect of extra force vanishes in the limiting process from compressible flows to the incompressible ones, as the Mach number goes to zero.

Low Mach number limit of steady flows through infinite multidimensional largely-open nozzles

In this paper, we justify of the low Mach number limit of the steady irrotational Euler flow through infinite multidimensional largely-open nozzles. Firstly, the existence and uniqueness of the incompressible flow are proved. Then, the uniform estimates on the compressibility parameter ε, which is singular for the flows, are established via a variational approach based on the compressible-incompressible difference functions. The limit is in the Hölder space and is unique. Moreover, the convergence rate is of order ε2 including the pressure. And, for the compressible case, the extra force influences the asymptotic behaver to the open end, while the effect vanishes in the limiting process from compressible flows to the incompressible ones. Also, for the compressible flows with general conservative forces, the well-posedness of subsonic solution are given, which leads to the existence of subsonic-sonic limit solutions.

Asymptotic Preserving Low Mach Number Accurate IMEX Finite Volume Schemes for the Isentropic Euler Equations

In this paper, the design and analysis of a class of second order accurate IMEX finite volume schemes for the compressible Euler equations in the zero Mach number limit is presented. In order to account for the fast and slow waves, the nonlinear fluxes in the Euler equations are split into stiff and non-stiff components, respectively. The time discretisation is performed by an IMEX Runge–Kutta method, therein the stiff terms are treated implicitly and the non-stiff terms explicitly. In the space discretisation, a Rusanov-type central flux is used for the non-stiff part, and simple central differencing for the stiff part. Both the time semi-discrete and space-time fully-discrete schemes are shown to be asymptotic preserving. The numerical experiments confirm that the schemes achieve uniform second order convergence with respect to the Mach number. A notion of accuracy at low Mach numbers, termed as the asymptotic accuracy, is introduced in terms of the invariance of a well-prepared space of constant densities and divergence-free velocities. The asymptotic accuracy is concerned with the closeness of the compressible solution with that of its incompressible counterpart in a low Mach number regime. It is shown theoretically as well as numerically that the proposed schemes are asymptotically accurate.

Effects of reflection of electrons and negative ions on magnetized electronegative and collisional plasma sheath

The effects of the reflection of electrons and negative ions in magnetized electronegative and collisional plasma sheath on the Bohm criterion and the sheath structure are numerically investigated. The Bohm criterion expression of the sheath with considering the reflection of electrons and negative ions is derived theoretically. The lower limit of ion Mach number versus parameters and the distribution curve of charged particle density in sheath are obtained by numerical simulation when Boltzmannian model and reflection model are applied to electrons and negative ions. The results show that the upper limit of ion Mach number is identical to that of Boltzmannian model, but their lower limit expressions are different. The lower limit of ion Mach number in the reflection model is also related to the wall potential, and with the increase of the wall potential, ion Mach number first increases and then remains unchanged after reaching the same value as that from Boltzmannian model, and the speeds of their reaching the maximum values are different due to the difference in sheath edge negative ion concentration and temperature. In both Boltzmannian and the reflection model, the lower limit of the ion Mach number decreases with the concentration of the negative ion at the sheath edge increasing and the negative ion temperature decreasing, but the maximum value is smaller in the reflection model. The lower limit of ion Mach number for each of the two models increases with sheath edge electric field increasing, but increases faster and the final value is larger in Boltzmannian model. The lower limit of ion Mach number for each of the two models decreases with the increase of collision parameter or magnetic field angle, but decreases faster in Boltzmannian model with the increase of collision parameter or magnetic field angle. The lower limits of ion Mach number in the two models tend to be the same with the increase of magnetic field angle. When the wall potential is small, the reflection of electrons and negative ions has a great influence on the sheath structure. When the wall potential is large, the reflection of electrons and negative ions have little effect on the density distribution of charged particles in the sheath.

On the Low Mach Number Limit for Quantum Navier--Stokes Equations

We investigate the low Mach number limit for the 3-D quantum Navier-Stokes system. For general ill-prepared initial data, we prove strong convergence of finite energy weak solutions to weak solutions of the incompressible Navier-Stokes equations. Our approach relies on a quite accurate dispersive analysis for the acoustic part, governed by the well-known Bogoliubov dispersion relation for the elementary excitations of the weakly-interacting Bose gas. Once we have a control of the acoustic dispersion, the a priori bounds provided by the energy and Bresch-Desjardins entropy type estimates lead to the strong convergence. Moreover, for well-prepared data we show that the limit is a Leray weak solution, namely it satisfies the energy inequality. Solutions under consideration in this paper are not smooth enough to allow for the use of relative entropy techniques.

High Mach number limit of one-dimensional piston problem for non-isentropic compressible Euler equations: Polytropic gas

We study the high Mach number limit of the one dimensional piston problem for the full compressible Euler equations of polytropic gas, for both cases that the piston rushes into or recedes from the uniform still gas, at a constant speed. There are two different situations, and one needs to consider measure solutions of the Euler equations to deal with the concentration of mass on the piston or formation of vacuum. We formulate the piston problem in the framework of Radon measure solutions and show its consistency by proving that the integral weak solutions of the piston problems converge weakly in the sense of measures to (singular) measure solutions of the limiting problems, as the Mach number of the piston increases to infinity.

Vectorial Kinetic Relaxation Model with Central Velocity. Application to Implicit Relaxations Schemes

We apply flux vector splitting (FVS) strategy to the implicit kinetic schemes for hyperbolic systems. It enables to increase the accuracy of the method compared to classical kinetic schemes while still using large time steps compared to the characteristic speeds of the problem. The method also allows to tackle multi-scale problems, such asthelow Machnumber limit, for which wavespeeds with larger atioare involved. We present several possible kinetic relaxation schemes based on FVS and compare them on one-dimensional test-cases. We discuss stability issues for this kind of method

Low Mach number limit of compressible two-fluid model

In this paper, we consider the low Mach number limit of the weak solutions to the viscous compressible two-fluid model with one velocity and a pressure of two components in three spatial dimensions. For well-prepared initial data, we prove rigorously that the weak solutions of the compressible two-fluid model converge to the strong solution of the incompressible Navier–Stokes equations as the Mach number tends to zero. Here, we allow that the two densities converge to different limits. Furthermore, the rates of convergence are also obtained.

Drag force on spherical particle moving near a plane wall in highly rarefied gas

The drag force on a sphere in tangential and normal motion to a plane wall is evaluated in the limit of large Knudsen number and small Mach (and Strouhal) number assuming isothermal conditions and diffuse reflection of gas molecules on walls. In the limit of free molecular flow, the molecular distribution function of the gas is evaluated using a set of coupled Fredholm integral equations. The results are compared with direct simulation Monte Carlo calculations and extended for finite Knudsen numbers. In all cases stronger dependence of the force on the width of the gap is found for normal compared to tangential motion. When the flow within the gap can be considered as essentially collisionless in nature, a similar dependence of the force on the gap width is observed at finite Knudsen numbers as in the free molecular case.

Low Mach number limit of the three-dimensional full compressible Navier–Stokes–Korteweg equations

In this paper, we justify the low Mach number limit for the three-dimensional full compressible Navier–Stokes–Korteweg equations rigorously within the framework of smooth solution. Under the assumptions of small density and temperature perturbation, we show that for sufficiently small Mach number, the initial-value problem of the three-dimensional full compressible Navier–Stokes–Korteweg equations admits a unique smooth solution on the time interval where the smooth solution of the corresponding incompressible Navier–Stokes equations exists. Moreover, we obtain the convergence of smooth solutions for the full compressible Navier–Stokes–Korteweg equations toward those for the incompressible Navier–Stokes equations with a convergence rate.

A low dissipation method to cure the grid-aligned shock instability

The grid-aligned shock instability prevents an accurate computation of high Mach number flows using low-dissipation shock-capturing methods. In particular one manifestation, the so-called carbuncle phenomenon, has been investigated by various different groups over the past decades. Nevertheless, the mechanism of this instability is still not fully understood and commonly is suppressed by the introduction of additional numerical dissipation. However, present approaches may either significantly deteriorate the resolution of complex flow evolutions or involve additional procedures to limit stabilization measures to the shock region.Instead of increasing the numerical dissipation, in this paper, we present an alternative approach that relates the problem to the low Mach number in transverse direction of the shock front. We show that the inadequate scaling of the acoustic dissipation in the low Mach number limit is the prime reason for the instability. Our approach is to increase the “numerical” Mach number locally whenever the advection dissipation is small compared to the acoustic dissipation. A very simple modification of the eigenvalue calculation in the Roe approximation leads to a scheme with less numerical dissipation than the original Roe flux which prevents the grid-aligned shock instability. The simplicity of the modification allows for a detailed investigation of multidimensional effects. By showing that modifications in flow direction affect the shock stability in the transverse directions we confirm the multidimensional nature of the instability. The efficiency and robustness of the modified scheme is demonstrated for a wide range of test cases that are known to be particularly prone to the shock instability. Moreover, the modified flux also is successfully applied to multi-phase flows.