不可压缩Navier-Stokes方程数值模拟

建立不可压缩Navier-Stokes方程的Crank-Nicolson有限差分方法,数值模拟了在初始正弦波下的二维水槽内流体受到倾斜激励时流场和涡的演变机制.数值结果表明,初始波为正弦波时,流场内出现一个单涡,单涡下沉变成了两个小涡,两个小涡消失后流场内部出现三条规则的流场带,最后这三条流场带演变成一个尖涡,尖涡在周围流体的作用下演变成一个单涡,最后单涡在自由面消失,当耗散系数和Reynolds数增大时,流场和涡演化的周期变小.     

不可压缩Navier-Stokes方程解的非线性连续性

研究了不可压缩的Navier-Stokes方程弱解的粘性的连续依赖性.首先利用Moser迭代得到当T> 0时,在Ω×(0,T)上速度■的L^∞范数估计.其次讨论了对粘度μ的连续依赖性,并给出了精确的估计.     

Navier-Stokes方程的脉动速度方程的最优动力系统建模和动力学分析

研究了基于Navier-Stokes方程的脉动速度方程的最优低维动力系统建模理论.最优目标泛函为脉动速度基函数的不可压缩性和正交性.数值计算了充分发展的并排双方柱绕流问题,并基于双尺度全局最优化方法,建立了它的脉动速度的最优动力系统模型.对其相空间轨道、Poincaré截面、分岔特性、功率谱和Lyapunov指数集等动力学特性进行了分析.随着Reynolds数的增加,双方柱绕流的脉动速度方程最优动力系统具有复杂的类倍周期分岔行为.     

含压力基Navier-Stokes方程最优动力系统建模和分析

研究了采用压力基函数和速度基函数的Navier-Stokes方程的最优截断低维动力系统建模理论.在黏性不可压缩流体中模拟了并排三方柱绕流流场,对此流场进行了含压力基函数和速度基函数的Navier-Stokes方程的最优动力系统建模,并以此为工具分析了三方柱绕流最优动力系统的动力学特性.该研究得到了如下结论:三方柱绕流的...     

四维不可压缩Navier-Stokes方程的能量守恒

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$L^qleft([0, T] ; L^pleft(mathbb{R}^4right)right)$条件,保证该方程能量守恒.     

三维不可压缩Navier-Stokes 方程的整体适定性

本文证明了, 在临界Besov 空间中, 速度的竖直方向具有大的初始值的三维不可压缩Navier-Stokes 方程的整体解是唯一存在的. 首先, 引进合适的权函数, 用以控制方程中的非线性项; 其次, 充分利用流体的不可压缩性质, 分别估计速度的水平分量和竖直分量以及压力的水平方向梯度和竖直方向梯度; 最后, 通过适当选取权函数的系数, 得到封闭的能量估计, 从而得到方程的整体适定性.     

二维非齐次不可压缩Navier-Stokes/Vlasov-Fokker-Planck方程组的渐近分析

研究了二维空间中非齐次不可压缩Navier-Stokes/Vlasov-Fokker-Planck方程组的渐近分析,此模型用于塑造流体-粒子的相互作用.运用紧性方法得到ρ~ε,u~ε的强收敛,最终得到由关于粒子宏观密度的对流-扩散方程及不可压缩Navier-Stokes方程组成的极限方程组.本文将相关文献的结果推广到非齐次不可压缩的情形.     

不可压Navier-Stokes方程迎风格式及后验误差估计

1引言不可压Navier-Stokes方程作为流体力学的基本方程,其数值计算一直是科学与工程计算关心的问题．本文考虑定常问题: -ε△u (u·▽)u ▽p = f x∈Ω,▽·u=0 x∈Ω, (1) u =0 x∈(?)Ω．这里ε=1／Re是Reynolds数的倒数,u=(u1,u2,…,ud)为待求流速场,p是待求压力场,f=(f1,f2,…,fd)是给定的体力．Ωv(?) Rd(d=2,3)是有界区域,且具有分片Lipschitz连续边界(?)Ω．     

解不可压缩Navier-Stokes方程的非精确块因子分解预处理子

针对相关于不可压缩Navier-Stokes方程数值求解的一类3×3块结构的线性方程组,基于线性方程组的等价形式,构造了一个非精确的块因子分解预处理子,在新的特征值等价矩阵形式的基础上,得到了预处理矩阵特征值实部和虚部的上下界估计.数值实验表明,与已有的预处理子相比,所构造的预处理子可以使得GMRES迭代方法对网格尺寸,网格形式以及粘度系数的依赖性都比较弱,且在迭代步数和CPU时间上都占优.     

非定常不可压Navier-Stokes方程的高效和稳健的差分格式Ⅱ

A second order accurate implicit finite difference scheme CNMT2 is proposed in this paper for the unsteady incompressible Navier-Stokes equations. It is proved that the scheme is unconditionally nonlinearly stable on smoothly nonuniform halfstaggered meshes; this stability also holds for this scheme with the pressure correction projection method. However, it is found that the pressure correction projection method may lead to deviation problems in practical simulation of high Re flow;the reason and the cure is given in this paper in terms of differential-algebraic equations.     

A regularity criterion for the solutions of 3D Navier-Stokes equations

In terms of two partial derivatives of any two components of velocity fields, we give a new criterion for the regularity of solutions of the Navier-Stokes equation in R³. More precisely, let u=(u¹,u²,u³) be a weak solution in (0,T)×R³. Then u becomes a classical solution if any two functions of _∂1u¹, _∂2u² and _∂3u³ belong to Lθ(0,T;Lr(R³)) provided with , .     

Hopf bifurcations to quasi-periodic solutions for the two-dimensional plane Poiseuille flow

This paper studies various Hopf bifurcations in the two-dimensional plane Poiseuille problem. For several values of the wavenumber α, we obtain the branch of periodic flows which are born at the Hopf bifurcation of the laminar flow. It is known that, taking α ≈ 1, the branch of periodic solutions has several Hopf bifurcations to quasi-periodic orbits. For the first bifurcation, calculations from other authors seem to indicate that the bifurcating quasi-periodic flows are stable and subcritical with respect to the Reynolds number, Re. By improving the precision of previous works we find that the bifurcating flows are unstable and supercritical with respect to Re. We have also analysed the second Hopf bifurcation of periodic orbits for several α, to find again quasi-periodic solutions with increasing Re. In this case the bifurcated solutions are stable to superharmonic disturbances for Re up to another new Hopf bifurcation to a family of stable 3-tori. The proposed numerical scheme is based on a full numerical integration of the Navier-Stokes equations, together with a division by 3 of their total dimension, and the use of a pseudo-Newton method on suitable Poincaré sections. The most intensive part of the computations has been performed in parallel. We believe that this methodology can also be applied to similar problems.     

Qualitative properties of stationary measures for three-dimensional Navier-Stokes equations

The paper is devoted to studying the distribution of stationary solutions for 3D Navier-Stokes equations perturbed by a random force. Under a non-degeneracy assumption, we show that the support of such a distribution coincides with the entire phase space, and its finite-dimensional projections are minorised by a measure possessing an almost surely positive smooth density with respect to the Lebesgue measure. Similar assertions are true for weak solutions of the Cauchy problem with a regular initial function. The results of this paper were announced in the short note [A. Shirikyan, Controllability of three-dimensional Navier-Stokes equations and applications, in: Sémin. Équ. Dériv. Partielles, 2005-2006, École Polytech., Palaiseau, 2006].     

推导刚体动力学方程的张量方法

本文提出了基于连续介质力学概念推导刚体动力学方程的张量方法,运用具有零共旋率的惯性张量的时间导数公式,证明了Lagrange方程、Nielsen方程、Gibbs-Appell方程、Kane方程和广义动量式Kane方程等五种方法的等价性,给出了角速度、角加速度之间的一些微分关系式.     

带附加噪声的随机广义2DGinzburg-Landau方程的渐进行为

考虑带附加噪声的随机广义2D Ginzburg-Landau方程．通过先验估计的方法,随机动力系统的紧性得到证明,进一步验证了该随机动力系统在H10存在随机整体吸引子．     

REMARK ON UNIQUE CONTINUATION OF SOLUTIONS TO THE STOKES AND THE NAVIER-STOKES EQUATIONS

1IntroductionUnique continuation of solutions to the linear partial di?erential equations with analyticcoe?cients is well known.There are more general results in elliptic,parabolic and hyperbolicequations(cf.[8-10,12-13]and references therein).The continu…     

Geometric depletion of vortex stretch in 3D viscous incompressible flow

New geometric constraints on vorticity are obtained which suppress possible development of finite-time singularity from the nonlinear vortex stretching mechanism. We find a new condition on the smoothness of the direction of vorticity in the vortical region which yields regularity. We also detect a regularity condition of isotropy type on vorticity in the intensive vorticity region via a new cancellation principle. This is in contrast with the one of isotropy type on the curl of vorticity obtained recently by A. Ruzmaikina and Z. Gruji? [A. Ruzmaikina, Z. Gruji?, On depletion of the vortex-stretching term in the 3D Navier-Stokes equations, Comm. Math. Phys. 247 (2004) 601-611]. We improve as well all of their results by eliminating their assumption that the initial vorticity ω₀ is required to be in L¹.     

Continuous dependence on the time and spatial geometry for the navier-stokes equations

In this paper we derive explicit a priori inequalities which imply stability of the solution of the initial-boundary value problem for the Navier-Stokes equations under perturbations of the initial time geometry and of the spatial geometry. These inequalities bound the solution perturbation In L₂ in terms of some well defined measure of the perturbation in geometry. We establish continuous dependence on spatial geometry in both two and three dimensions and continuous dependence on initial geometry in two dimensions. In the latter problem the three dimensional case will be somewhat more complicated due to the different form of the Sobolev inequality.