On the spectrum for the artificial compressible system

Stability of stationary solutions of the incompressible Navier–Stokes system and the corresponding artificial compressible system is considered. Both systems have the same sets of stationary solutions and the incompressible system is obtained from the artificial compressible one in the zero limit of the artificial Mach number ? which is a singular limit. It is proved that if a stationary solution of the incompressible system is asymptotically stable and the velocity field of the stationary solution satisfies an energy-type stability criterion by variational method with admissible functions being only potential flow parts of velocity fields, then it is also stable as a solution of the artificial compressible one for sufficiently small ?. The result is applied to the Taylor problem.     

On singular limit for the compressible Navier–Stokes system with non-monotone pressure law

We consider a singular limit for the compressible Navier–Stokes system with general non-monotone pressure law in the asymptotic regime of low Mach number and large Reynolds numbers. We show that any dissipative weak solution approaches the solution of incompressible Euler equation both for well-prepared initial data and ill-prepared initial data.     

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.     

Low Mach number limit for a model of radiative flow

We consider a simplified model of compressible Navier–Stokes–Fourier coupled to the radiative transfer equation introduced by Seaïd, Teleaga and al., and we study its low Mach number limit. We prove the convergence toward the incompressible Navier–Stokes system coupled to a system of two stationary transport equations.     

Well-posedness of a low Mach number system

This Note deals with a short-time existence result for a system of nonlinear partial differential equations modelling a diphasic flow. The so-called Dlmn system is derived from the compressible Navier–Stokes equations under the assumption that the Mach number is small. A classical solution is obtained by means of a Picard iteration process. The proof of convergence relies on estimates associated to hyperbolic and parabolic equations. This procedure results in conditions on the time of existence of the solution.     

Rotating compressible fluids under strong stratification

We consider the Navier–Stokes system written in the rotational frame describing the motion of a compressible viscous fluid under strong stratification. The asymptotic limit for low Mach and Rossby numbers and large Reynolds number is studied on condition that the Froude number characterizing the degree of stratification is proportional to the Mach number. We show that, at least for the well prepared data, the limit system is the same as for the problem without stratification—a variant of the incompressible planar Euler system.     

Low Mach Number Limit for the Navier–Stokes System on Unbounded Domains Under Strong Stratification

We consider the low Mach number singular limit problem for the Navier–Stokes system describing the motion of a compressible fluid under strong stratification in an exterior domain. It is shown that the limit problem is represented by the so-called anelastic approximation. In particular, strong (pointwise) convergence of the acoustic components of the velocity is established by means of an abstract result of Tosio Kato.     

Suitable weak solutions: from compressible viscous to incompressible inviscid fluid flows

We establish the asymptotic limit of the compressible Navier–Stokes system in the regime of low Mach and high Reynolds number on unbounded spatial domains with slip boundary condition. The result holds in the class of suitable weak solutions satisfying a relative entropy inequality.     

Vanishing viscosity and Debye-length limit to rarefaction wave with vacuum for the 1D bipolar Navier–Stokes–Poisson equation

In this paper, we consider the one-dimensional (1D) compressible bipolar Navier–Stokes–Poisson equations. We know that when the viscosity coefficient and Debye length are zero in the compressible bipolar Navier–Stokes–Poisson equations, we have the compressible Euler equations. Under the case that the compressible Euler equations have a rarefaction wave with one-side vacuum state, we can construct a sequence of the approximation solution to the one-dimensional bipolar Navier–Stokes–Poisson equations with well-prepared initial data, which converges to the above rarefaction wave with vacuum as the viscosity and the Debye length tend to zero. Moreover, we also obtain the uniform convergence rate. The results are proved by a scaling argument and elaborate energy estimate.     

Zero-viscosity-capillarity limit to the planar rarefaction wave for the 2D compressible Navier–Stokes–Korteweg equations

We shall consider the two-dimensional (2D) isentropic Navier–Stokes–Korteweg equations which are used to model compressible fluids with internal capillarity. Formally, the 2D isentropic Navier–Stokes–Korteweg equations converge, as the viscosity and the capillarity vanish, to the corresponding 2D inviscid Euler equations, and we do justify this for the case that the corresponding 2D inviscid Euler equations admit a planar rarefaction wave solution. More precisely, it is proved that there exists a family of smooth solutions for the 2D isentropic compressible Navier–Stokes–Korteweg equations converging to the planar rarefaction wave solution with arbitrary strength for the 2D Euler equations. A uniform convergence rate is obtained in terms of the viscosity coefficient and the capillarity away from the initial time. The key ingredients of our proof are the re-scaling technique and energy estimate, in which we also introduce the hyperbolic wave to recover the physical viscosities and capillarity of the inviscid rarefaction wave profile.     

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.     

Anelastic Approximation as a Singular Limit of the Compressible Navier–Stokes System

Considering compressible Navier–Stokes system in a slab geometry in the regime when both Mach and Froude numbers vanish at the same rate, we study the behavior of corresponding weak solutions, that are known to exist globally-in-time (for large data). We establish their convergence to a solution of the so-called anelastic approximation when the limit flow is stratified, i.e., the limit density depends effectively on the vertical coordinate.     

Implementation of a Preconditioning–Technique in the Hybrid Flow–Solver TAU+

The present paper describes the implementation of a preconditioning method in the hybrid DLR–TAU⁺–code and its application to nearly incompressible flows. The method is designed in order to get an efficient and accurate solution even for very low Mach numbers using a time stepping scheme for the solution of the compressible Navier–Stokes equations. The algorithm is based on the work of Choi and Merkle. The numerical results obtained for inviscid and viscous flows indicate, that for Mach numbers lower than 0.1 the accuracy as well as the convergence properties are almost independent of the fluid speed, like for incompressible codes.     

On the low Mach number limit of compressible flows in exterior moving domains

We study the incompressible limit of solutions to the compressible barotropic Navier–Stokes system in the exterior of a bounded domain undergoing a simple translation. The problem is reformulated using a change of coordinates to fixed exterior domain. Using the spectral analysis of the wave propagator, the dispersion of acoustic waves is proved by means of the RAGE theorem. The solution to the incompressible Navier–Stokes equations is identified as a limit.     

On compactness of the velocity field in the incompressible limit of the full Navier–Stokes–Fourier system on large domains

The incompressible limit for the full Navier–Stokes–Fourier system is studied on a family of domains containing balls of the radius growing with a speed that dominates the inverse of the Mach number. It is shown that the velocity field converges strongly to its limit locally in space, in particular, the effect of the sound waves is eliminated by means of the local decay estimates for the acoustic wave equation. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

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.     

Low Mach Number Limit of Viscous Polytropic Flows: Formal Asymptotics in the Periodic Case

This article is devoted to the low Mach number limit of weak solutions to the compressible Navier–Stokes equations for polytropic fluids with periodic boundary conditions and ill‐prepared data. We derive formally the equation satisfied by the mean value of the velocity and the equations governing the dynamics of the nonlinear acoustic waves in dimension d= 2 or 3.     

Verification and validation of the foredrag coefficient for supersonic and hypersonic flow of air over a cone of fineness ratio 3

The foredrag coefficient resulting from the supersonic and hypersonic flow of air over a cone was calculated numerically using a finite volume approach based on the compressible Euler and Navier-Stokes equations with constant and variable thermophysical properties. No turbulence model was considered. Simulations were carried out for a cone of fineness ratio 3 under the free-stream Mach numbers 2.73, 3.50, 4.00, 5.05 and 6.28 (the Reynolds number, based on cone length, is within 0.45 and 2.85 million). Up to six grids were employed for numerical calculations, with 60 × 60 to 1920 × 1920 volumes. The numerical error was estimated to be less than 0.01% of the numerical solution for all models. Comparisons of the numerical foredrag coefficients of the three models with the experimental data showed that the Navier–Stokes model with variable thermophysical properties agreed better with the experimental foredrag for the entire Mach number interval studied, taking into account the validation standard uncertainty.