不可压缩流动的并行数值方法

不可压缩流动的数值模拟是计算流体力学的重要组成部分. 基于有限元离散方法, 本文设计了不可压缩Navier-Stokes (N-S)方程支配流的若干并行数值算法. 这些并行算法可归为两大类: 一类是基于两重网格离散方法, 首先在粗网格上求解非线性的N-S方程, 然后在细网格的子区域上并行求解线性化的残差方程, 以校正粗网格的解; 另一类是基于新型完全重叠型区域分解技巧, 每台处理器用一局部加密的全局多尺度网格计算所负责子区域的局部有限元解. 这些并行算法实现简单, 通信需求少, 具有良好的并行性能, 能获得与标准有限元方法相同收敛阶的有限元解. 理论分析和数值试验验证了并行算法的高效性     

一维粘性守恒律初边值问题粘性激波解的渐近稳定性

本文考虑一维可压缩Navier-Stokes方程有关初边值问题粘性激波解的渐近稳定性,通过L~2-能量估计,证明了在小扰动情况下,粘性激波是稳定的。     

Ladyzhenskaya模型的非协调有限元逼近

§1.引言 Navier-Stokes方程是描述粘性不可压缩流体运动的偏微分方程,它是研究这类非线性问题的很好的数学模型。但是,当速度梯度较大时,方程的整体解是否唯一可解,这个问题尚未得到解决。为此,对于不可压缩粘性流体的定常情形,Ladyzhenskaya提出利用下面的模型来代替Navier-Stokes方程。 设Ω是R~n(n=2或3)中的有界区域,边界?ΩLipshitz连续,u是流体速度,p是     

一维粘性守恒律初边值问题粘性激波解的渐近稳定性

本文考虑一维可压缩Navier-Stokes方程有关初边值问题粘性激波解的渐近稳定性,通过L2-能量估计,证明了在小扰动情况下,粘性激波是稳定的.     

非特征边界的MHD方程的边界层

考查了小粘性时非特征边界情况下MHD方程在边界附近的性质,说明速度在边界上不为零.源于之前非特征边界条件下不可压缩Navier-Stokes方程边界层的工作,证明了边界层的存在性,并得到了当粘性收敛于零时,MHD方程的解收敛于理想MHD方程的解.     

奇数维可压缩液晶流方程解的逐点估计

首先, 本文利用标准的能量估计方法得到高维(3 维及以上) 的液晶流方程组小初值经典解的整体存在性. 然后, 本文运用Green 函数方法, 得到奇数维情形(3 维及以上) 该解的逐点估计. 该结果表明, 密度ρ和动量m同Navier-Stokes 方程组一样满足一般Huygens 原理, 而单位向量场d则没有这种现象, 其有着与热方程的解类似的时空估计.     

随机Navier-Stokes方程数值解的收敛性

Navier-Stokes方程在流体力学中有广泛的应用.通常情况下,大多数Navier-Stokes方程没有精确解,数值方法显得尤为重要.本文根据BDM法,利用It公式,Burkholder-Davis-Gundy不等式,Doob不等式和Gronwall引理对随机Navier-Stokes方程数值解的收敛性进行了讨论,得出数值解均方意义下收敛到解析解.     

不可压流体的边界层问题

研究三维有界区域在边界上有流动的不可压流体的边界层问题,导出了Navier-Stokes方程区域内部的近似方程(Euler方程和线性化的Euler方程)和边界附近近似的方程(零阶边界层方程与一阶边界层方程),证明了这种近似的合理性.     

粘流—无粘干扰流动理论广义解的存在性和唯一性条件

本文证明了粘流-无粘干扰流动理论基本控制方程—简化Navier-Stokes方程原始变量变分形式广义解的存在性,给出了广义解的唯一性条件和Re数上限估计,得到与完全N-S方程的已知结果相应的一系列结论。一、预备知识和基本假设     

四维不可压缩Navier-Stokes方程的能量守恒北大核心CSCD

研究了四维不可压缩Navier-Stokes方程的能量守恒,当该方程的Leray-Hopf弱解(适当弱解)存在维数小于4的奇异集时,基于Wu在文章中关于四维不可压缩Navier-Stokes方程的部分正则性结果,得到了四维空间中Lq([0,T];Lp(R4))条件,保证该方程能量守恒.     

Blowup Criteria for Full Compressible Navier-Stokes Equations with Vacuum State

This paper deals with the global strong solution to the three-dimensional(3D) full compressible Navier-Stokes systems with vacuum.The authors provide a sufficient condition which requires that the Sobolev norm of the temperature and some norm of the divergence of the velocity are bounded,for the global regularity of strong solution to the 3D compressible Navier-Stokes equations.This result indicates that the divergence of velocity fields plays a dominant role in the blowup mechanism for the full compressible Navier-Stokes equations in three dimensions.     

Global well-posedness and time-decay estimates for compressible Navier-Stokes equations with reaction diffusion

Science China Mathematics - We consider the full compressible Navier-Stokes equations with reaction diffusion. A global existence and uniqueness result of the strong solution is established for the...     

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.     

Pressure Boundary Conditions for Blood Flows

Simulations of blood flows in arteries require numerical solutions of fluid-structure interactions involving Navier-Stokes equations coupled with large displacement visco-elasticity for the vessels. Among the various simplifications which have been proposed, the surface pressure model leads to a hierarchy of simpler models including one that involves only the pressure. The model exhibits fundamental frequencies which can be computed and compared with the pulse. Yet unconditionally stable time discretizations can be constructed by combining implicit time schemes with Galerkin-characteristic discretization of the convection terms in the Navier-Stokes equations. Such problems with prescribed pressure on the walls will be shown to be efficient and accurate as an approximation of the full fluid structure interaction problem.     

Boundary layers for compressible Navier-Stokes equations with outflow boundary condition

In this paper, we study the asymptotic relation between the solutions to the initial boundary value problem of the one-dimensional compressible full Navier-Stokes equations with outflow boundary condition and the associated Euler equations. We assume all the three characteristics to the corresponding Euler equations are all negative up to some small time, then we prove the existence and the stability of the boundary layers as long as the strength of the boundary layers is suitably small.     

On the stationary solutions of the full compressible Navier-Stokes equations and its stability with respect to initial disturbance

This paper is concerned with the existence, uniqueness and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier-Stokes equations effected by external force of general form in R³.     

Convergence to rarefaction waves for the nonlinear Boltzmann equation and compressible Navier-Stokes equations

The main purpose of this paper is to study the asymptotic equivalence of the Boltzmann equation for the hard-sphere collision model to its corresponding Euler equations of compressible gas dynamics in the limit of small mean free path. When the fluid flow is a smooth rarefaction (or centered rarefaction) wave with finite strength, the corresponding Boltzmann solution exists globally in time, and the solution converges to the rarefaction wave uniformly for all time (or away from t=0) as ?→0. A decomposition of a Boltzmann solution into its macroscopic (fluid) part and microscopic (kinetic) part is adopted to rewrite the Boltzmann equation in a form of compressible Navier-Stokes equations with source terms. In this setting, the same asymptotic equivalence of the full compressible Navier-Stokes equations to its corresponding Euler equations in the limit of small viscosity and heat conductivity (depending on the viscosity) is also obtained.     

Nonlinear projection methods for multi-entropies Navier-Stokes systems

This paper is devoted to the numerical approximation of the compressible Navier-Stokes equations with several independent entropies. Various models for complex compressible materials typically enter the proposed framework. The striking novelty over the usual Navier-Stokes equations stems from the generic impossibility of recasting equivalently the present system in full conservation form. Classical finite volume methods are shown to grossly fail in the capture of viscous shock solutions that are of primary interest in the present work. To enforce for validity a set of generalized jump conditions that we introduce, we propose a systematic and effective correction procedure, the so-called nonlinear projection method, and prove that it preserves all the stability properties satisfied by suitable Godunov-type methods. Numerical experiments assess the relevance of the method when exhibiting approximate solutions in close agreement with exact solutions.

     

ZERO DISSIPATION LIMIT TO A RIEMANN SOLUTION FOR THE COMPRESSIBLE NAVIER-STOKES SYSTEM OF GENERAL GAS

For the general gas including ideal polytropic gas, we study the zero dissipation limit problem of the full 1-D compressible Navier-Stokes equations toward the superposition of contact discontinuity and two rarefaction waves. In the case of both smooth and Riemann initial data, we show that if the solutions to the corresponding Euler system consist of the composite wave of two rarefaction wave and contact discontinuity, then there exist solutions to Navier-Stokes equations which converge to the Riemman solutions away from the initial layer with a decay rate in any fixed time interval as the viscosity and the heat-conductivity coefficients tend to zero. The proof is based on scaling arguments, the construction of the approximate profiles and delicate energy estimates. Notice that we have no need to restrict the strengths of the contact discontinuity and rarefaction waves to be small.     

Blowup mechanism for viscous compressible heat-conductive magnetohydrodynamic flows in three dimensions

We investigate initial-boundary-value problem for three-dimensional magnetohydrodynamic(MHD)system of compressible viscous heat-conductive flows and the three-dimensional full compressible Navier-Stokes equations. We establish a blowup criterion only in terms of the derivative of velocity field, similar to the Beale-Kato-Majda type criterion for compressible viscous barotropic flows by Huang et al.(2011). The results indicate that the nature of the blowup for compressible MHD models of viscous media is similar to the barotropic compressible Navier-Stokes equations and does not depend on further sophistication of the MHD model, in particular, it is independent of the temperature and magnetic field. It also reveals that the deformation tensor of the velocity field plays a more dominant role than the electromagnetic field and the temperature in regularity theory. Especially, the similar results also hold for compressible viscous heat-conductive Navier-Stokes flows,which extend the results established by Fan et al.(2010), and Huang and Li(2009). In addition, the viscous coefficients are only restricted by the physical conditions in this paper.