Solution Of Compressible Navier-Stokes Equations Research Articles

Vanishing viscosity limit of the compressible Navier-Stokes equations with finite energy and total mass

Assume the initial data of compressible Euler equations has finite energy and total mass. We can construct a sequence of solutions of one-dimensional compressible Navier-Stokes equations (density-dependent viscosity) with stress-free boundary conditions, so that, up to a subsequence, the sequence of solutions of compressible Navier-Stokes equations converges to a finite-energy weak solution of compressible Euler equations. Hence the inviscid limit of the compressible Navier-Stokes is justified. It is worth pointing out that our result covers the interesting case of the Saint-Venant model for shallow water (i.e., α=1,γ=2).

An efficient high-order least square-based finite difference-finite volume method for solution of compressible Navier-Stokes equations on unstructured grids

This paper presents an efficient high-order finite volume method for solution of compressible Navier-Stokes equations on unstructured grids. In this method, a high-order polynomial which is based on Taylor series expansion is applied to approximate the solution function within each control cell. The derivatives in the Taylor series expansion are approximated by the functional values at the cell centers of the considered cell and its neighboring cells using the mesh-free least square-based finite difference (LSFD) scheme. Naturally, this least square-based finite difference-finite volume (LSFD-FV) method inherits appealing characteristics of the LSFD scheme, i.e., the simple algorithm and easy implementation. In addition, since the LSFD scheme is mesh-free in nature, the developed high-order LSFD-FV method is endowed with the flexibility to handle the multi-dimensional problems with complex geometries on arbitrary grids. Different from other high-order methods, this LSFD-FV method applies a novel and effective strategy, i.e., the discrete gas-kinetic flux solver (DGKFS), to compute numerical fluxes at the cell interface. In this way, the inviscid and viscous fluxes are simultaneously and effectively evaluated by local reconstruction of solutions for the Boltzmann equation. The efficient time marching strategy coupled with the high-order LSFD-FV method is adopted to solve the resultant ordinary differential equations with the matching accuracy. A series of numerical examples are presented to validate the accuracy, robustness and flexibility of the developed method on unstructured grids. Numerical results demonstrate the superior performance of the developed high-order method on simulating compressible viscous flows, in comparison with the low-order counterpart and the k-exact method of the same order of accuracy.

압축성 나비어-스톡스 방정식을 풀기 위한 GPU-가속 ADI 기법

A GPU-accelerated ADI method for solutions of compressible Navier-Stokes equations is developed to reduce the computational cost of simulating turbomachinery flow. Despite the advantages of implicit time advancement in stability and convergence speed, implicit schemes tend to rely on linear systems of equations which benefit less from parallel computing than explicit schemes. In particular, the Alternating Direction Implicit (ADI) method applied to compressible Navier-Stokes equations on a multi-block grid requires the solution of block-tridiagonal systems, which are challenging to accelerate on GPUs. The present study suggests an algorithm for GPU-acceleration of the ADI method in the context of compressible flows. A divide-and-conquer-type algorithm is recursively applied to block-tridiagonal matrices to reduce it into a number of smaller sub-matrices well suited for GPU acceleration. Using a single NVIDIA Titan V GPU, a speedup of 26 ⨯ compared to a single-core Intel i7-9800X CPU is achieved on 4.6 million grid points.

Existence and uniqueness of low-energy weak solutions to the compressible 3D magnetohydrodynamics equations

We prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics (MHD) equations. We first obtain the existence of weak solutions with small L2-norm which may display codimension-one discontinuities in density, pressure, magnetic field and velocity gradient. The weak solutions we consider here exhibit just enough regularity and structure which allow us to develop uniqueness and continuous dependence theory for the compressible MHD equations. Our results generalise and extend those for the intermediate weak solutions of compressible Navier-Stokes equations.

Vanishing Viscosity Limit of the Navier–Stokes Equations to the Euler Equations for Compressible Fluid Flow with Vacuum

We establish the vanishing viscosity limit of the Navier-Stokes equations to the Euler equations for three-dimensional compressible isentropic flow in the whole space. It is shown that there exists a unique regular solution of compressible Navier-Stokes equations with density-dependent viscosities, arbitrarily large initial data and vacuum, whose life span is uniformly positive in the vanishing viscosity limit. It is worth paying special attention that, via introducing a "quasi-symmetric hyperbolic"--"degenerate elliptic" coupled structure, we can also give some uniformly bounded estimates of $\displaystyle\Big(\rho^{\frac{\gamma-1}{2}}, u\Big)$ in $H^3$ space and $\rho^{\frac{\delta-1}{2}}$ in $H^2$ space (adiabatic exponent $\gamma>1$ and $1<\delta \leq \min\{3, \gamma\}$), which lead the strong convergence of the regular solution of the viscous flow to that of the inviscid flow in $L^{\infty}([0, T]; H^{s'})$ (for any $s'\in [2, 3)$) space with the rate of $\epsilon^{2(1-s'/3)}$. Further more, we point out that our framework in this paper is applicable to other physical dimensions, say 1 and 2, with some minor modifications. This paper is based on our early preprint in 2015.

An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations

We prove that bounded weak solutions of the compressible Euler equations will conserve thermodynamic entropy unless the solution fields have sufficiently low space-time Besov regularity. A quantity measuring kinetic energy cascade will also vanish for such Euler solutions, unless the same singularity conditions are satisfied. It is shown furthermore that strong limits of solutions of compressible Navier-Stokes equations that are bounded and exhibit anomalous dissipation are weak Euler solutions. These inviscid limit solutions have non-negative anomalous entropy production and kinetic energy dissipation, with both vanishing when solutions are above the critical degree of Besov regularity. Stationary, planar shocks in Euclidean space with an ideal-gas equation of state provide simple examples that satisfy the conditions of our theorems and which demonstrate sharpness of our $L^3$-based conditions. These conditions involve space-time Besov regularity, but we show that they are satisfied by Euler solutions that possess similar space regularity uniformly in time.

On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum

In this paper we present a sufficient condition for the global well-posedness of classical solutions to an initial value problem of compressible isentropic Navier-Stokes equations in the whole space $\mathbb{R}^3$. As an immediate result, the main theorem obtained implies that the Cauchy problem of compressible Navier-Stokes equations with vacuum has a global unique classical solution, provided the initial energy is sufficiently small, or the shear viscosity coefficient is sufficiently large, or the upper bound of the initial density is suitably small and the adiabatic exponent $\gamma\in (1,3/2)$. These results particularly extend the recent ones due to Huang-Li-Xin [7], where the global well-posedness of classical solutions with small initial energy was established.

High regularity of solutions of compressible Navier-Stokes equations

We study the barotropic compressible Navier-Stokes equations in a bounded or an unbounded domain $\Omega $ of $ \mathbf{R}^3$. The initial density may vanish in an open subset of $\Omega$ or be positive but vanish at space infinity. We first prove the local existence of solutions $(\rho^{(j)}, u^{(j)})$ in $C([0,T_* ]; H^{2(k-j)+3} \times D_0^1 \cap D^{2(k-j)+3} (\Omega ) )$, $0 \le j \le k, k \ge 1$ under the assumptions that the data satisfy compatibility conditions and the initial density is sufficiently small. To control the non-negativity or decay at infinity of density, we need to establish a boundary-value problem of a $(k+1)$-coupled elliptic system which may not be, in general, solvable. The smallness condition of the initial density is necessary for the solvability of the elliptic system; this is not necessary when the initial density has positive lower bound. Secondly, we prove the global existence of smooth radially symmetric solutions of isentropic compressible Navier-Stokes equations by controlling every regularity with $|\rho|_{L^\infty}$.

Behaviour of weak solutions of compressible Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor

The aim of this paper is to study the behaviour of a weak solution to Navier-Stokes equations for isothermal fluids with a nonlinear stress tensor for time going to infinity. In an analogous way as in [18], we construct a suitable function which approximates the density for time going to infinity. Using properties of this function, we can prove the strong convergence of the density to its limit state. The behaviour of the velocity field and kinetic energy is mentioned as well.

Une approche locale de la limite incompressible

We prove in this Note a local weak convergence result of solutions of compressible Navier-Stokes equations, as the Mach number goes to 0, to a Leray solution of incompressible Navier-Stokes equations.

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.

Boundary conditions for the solution of compressible navier-Stokes equations by an implicit factored method

A method is presented for formulating the boundary conditions in implicit finite-difference form needed for obtaining solutions to the compressible Navier-Stokes equations by the Beam and Warming implicit factored method. The usefulness of the method was demonstrated (a) by establishing the boundary conditions applicable to the analysis of the flow inside an axisymmetric piston-cylinder configuration and (b) by calculating velocities and mass fractions inside the cylinder for different geometries and different operating conditions. Stability, selection of time step and grid sizes, and computer time requirements are discussed in reference to the piston-cylinder problem analyzed.