表示湍流场的一种新设想

本文仿照量子场论中描述基本粒子产生湮灭的方法来描述湍流中涡旋的产生和消灭.因为当某一基本粒子存在的时候,我们可以认为它是一个不变实体,而湍流中涡旋则在时间过程中不断变化和耗散,所以在类比应用量子场论方法时首先要解决怎样的湍流涡旋可认为是同一个涡旋.根据线性化理论的特点,我们认为在时间过程中按相似性规律变化时湍流涡旋才算是同一个涡旋,而把不具有相似性的涡旋出现或消失,看成是方程(2.6)中相互作用项φ_i所引起的湮火和产生的结果.然后,我们采用和量子场论相类似的产生算符和消灭算符来描述湍流涡旋系统所处的状态.最后,我们利用原N-S方程中相互作用项来构成涡旋相互作用的“Schr?dinger”方程以描述其状态的变化.这样就得类似于量子场论的湍流涡旋相互作用理论.     

定常Navier-Stokes方程的一种高效稳定有限元方法

提出了二维定常Navier-Stokes(N-S)方程的一种两层稳定有限元方法.该方法基于局部高斯积分技术,通过不满足inf-sup条件的低次等阶有限元对N-S方程进行有限元求解.该方法在粗网格上解定常N-S方程,在细网格上只需解一个Stokes方程.误差分析和数值试验都表明:两层稳定有限元方法与直接在细网格上采用的传统有限元方法得到的解具有同阶的收敛性,但两层稳定有限元方法节省了大量的工作时间.     

钝体分离旋涡流动的区域分解、杂交数值模拟——Ⅰ.理论方法及其应用

为克服涡旋法不能精确预计物体附近小尺度流动结构的理论缺陷,减少高Reynolds数流动N-S方程差分解的困难,本文提出一种区域分解、杂交耦合N-S方程有限差分解及涡旋法的新的数值模型和理论方法.将流场分解为内外两区,在靠近物体表面、范围为O(R)的内区进行N-S方程有限差分解,外区作Lagrange-Euler涡旋法解,建立了分区流动的联结、耦合条件,给出了杂交耦合求解的数值计算方法.用本方法作了Re=10²,10³的圆柱绕流计算,考察了区域交界面位置变化时解的稳定性.与全场N-S方程解及实验结果的比较表明本文方法能精确预计流动分离及近场流动的详细结构,并可有效地计算流动的总体特性,且比全场N-S方程解显著节省机时和计算量.     

三维非定常/定常不可压缩流动N-S方程基于人工压缩性方法的数值模拟

基于人工压缩性方法提出—中心与迎风混合的算法,以数值模拟N-S方程的定常/非定常解．对半离散方程的左端采用中心差分, 方程右端数值流量采用迎风Roe近似算法,其精度可达三阶．湍流模式利用Baldwin-Lomax代数模式．计算例子包括二维平板、机翼剖面、扁椭球、颅动脉瘤等．计算结果表明,压力和摩擦系数与实验符合,在分离涡旋区计算值与实验有差别,这或许是由于湍流模式不够精确的缘故．     

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     

运动壁面槽道流动的直接数值模拟

采用谱方法,在曲线坐标系下对不可压缩Newton流体的N-S方程进行求解,采用定义在物理空间中的流动物理量以避免使用协变、逆变形式的控制方程．在计算空间采用Fourier-Chebyshev谱方法进行空间离散,时间推进采用高精度时间分裂法．为了减小时间分裂带来的误差,采用了高精度的压力边界条件．与其他求解协变、逆变形式控制方程的谱方法相比,该方法在保持谱精度的同时减小了计算量．首先通过静止波形壁面和行波壁面槽道湍流的直接数值模拟,对该数值方法进行了验证;其次,作为初步应用,利用该方法研究了槽道湍流中周期振动凹坑所产生的流动结构．     

确定湍流的涡输运系数的新方法

本文发展了一种从基本运动方程确定湍流的涡粘度、涡扩散率和湍流Prandtl数的理论方法,区别于唯象方法,不需要借助经验参数;应用于均匀湍流,得到与实验结果一致的湍流Prandtl数;并且能计算Boussinesq模型的误差,评估各种方法的优劣.     

二维剪切流的粘性-无粘湍流干扰理论

对二维不可压缩近壁剪切湍流,本文提出一个粘性-无粘湍流干扰理论.主要内容有:从分子粘性考虑出发确定干扰湍流的流动结构及其物理尺度,导出空间为小尺度的局部流动结构随顺流距离的演变规律,导出支配干扰湍流流动的简化Reyno-lds(SR)方程和扩散抛物化K-ε方程.该SR方程是作者早先提出的简化Navier-Stokes(SNS)方程的湍流形式,它的重要性质是“简化运算”和时间Reynolds平均运算的顺序可以交换.关于最大湍流剪应力、本理论计算值与实验测量值很好相符.经典湍流边界层理论、Clauser平衡湍流边界层以及湍流分离Triple-deck理论均是本文理论的特例.证实了顺流方向长度尺度随干扰增强而显著减小的实验结论.     

二维趋化N-S方程解的唯一性准则

本文研究一类二维趋化N-S方程解的唯一性问题.利用Littlewood-Paley理论和Besov空间理论以及做差法,获得这一类二维趋化N-S方程弱解唯一性的唯一性准则.     

湍流的耗散及弥散相互作用理论

本文推导了表征耗散与弥散相互作用的新的湍流控制方程组,其特点是:用稳定性分析得到湍流动能产生项,再根据广义熵增原理推出并列存在的分别适用于强弱涡量的两个湍流动量方程。运用该理论已成功地计算了一些典型的湍流问题:湍流边界层中的马蹄涡拟序结构、钝体尾涡区的湍流能量逆转、湍流涡团散裂弛豫及各向异性分布,文中还给出了部分算例。     

Uncertainty Quantification for Systems with Random Initial Conditions Using Wiener–Hermite Expansions

A number of engineering problems, including laminar-turbulent transition in convectively unstable flows, require predicting the evolution of a nonlinear dynamical system under uncertain initial conditions. The method of Wiener–Hermite expansion is an attractive alternative to modeling methods, which solve for the joint probability density function of the stochastic amplitudes. These problems include the "curse of dimensionality" and closure problems. In this paper, we apply truncated Wiener–Hermite expansions with both fixed and time-varying bases to a model stochastic system with three degrees of freedom. The model problem represents the combined effects of quadratic nonlinearity and stochastic initial conditions in a generic setting and occurs in related forms in both classical dynamics, turbulence theory, and the nonlinear theory of hydrodynamic stability. In this problem, the truncated Wiener–Hermite expansions give a good account of short-time behavior, but not of the long-time relaxation characteristic of this system. It is concluded that successful application of truncated Wiener–Hermite expansions may require special adaptations for each physical problem.     

The Kolmogorov–Obukhov Statistical Theory of Turbulence

In 1941 Kolmogorov and Obukhov postulated the existence of a statistical theory of turbulence, which allows the computation of statistical quantities that can be simulated and measured in a turbulent system. These are quantities such as the moments, the structure functions and the probability density functions (PDFs) of the turbulent velocity field. In this paper we will outline how to construct this statistical theory from the stochastic Navier–Stokes equation. The additive noise in the stochastic Navier–Stokes equation is generic noise given by the central limit theorem and the large deviation principle. The multiplicative noise consists of jumps multiplying the velocity, modeling jumps in the velocity gradient. We first estimate the structure functions of turbulence and establish the Kolmogorov–Obukhov 1962 scaling hypothesis with the She–Leveque intermittency corrections. Then we compute the invariant measure of turbulence, writing the stochastic Navier–Stokes equation as an infinite-dimensional Ito process, and solving the linear Kolmogorov–Hopf functional differential equation for the invariant measure. Finally we project the invariant measure onto the PDF. The PDFs turn out to be the normalized inverse Gaussian (NIG) distributions of Barndorff-Nilsen, and compare well with PDFs from simulations and experiments.     

Group analysis of the von Kármán–Howarth equation. Part I. Submodels

We study the von Kármán–Howarth (KH) equation by group theoretical methods. This scalar partial differential equation involves two dependent variables (closure problem) and, it has been derived from the Navier–Stokes equations. The equivalence Lie algebra L has been found to be infinite-dimensional and, it is spanned by the four operators. The subalgebra of L is spanned by the three operators. Furthermore, ideal comprises one operator. Optimal systems of one-, two- and three-dimensional subalgebras have been obtained. Normalizers for the one- and two-dimensional subalgebras have been calculated. Finally we have obtained the submodels of the KH equation corresponding to optimal system of one- and two-dimensional subalgebras. This merely suggests alternative solutions to the closure problem of isotropic turbulence.     

Kuramoto-Sivashinsky Equation and Free-Interface Models in Combustion Theory

In combustion theory, a thin flame zone is usually replaced by a free interface. A very challenging problem is the derivation of a self-consistent equation for the flame front which yields a reduction of the dimensionality of the system. A paradigm is the Kuramoto-Sivashinsky (K-S) equation, which models cellular instabilities and turbulence phenomena. In this survey paper, we browse through a series of models in which one reaches a fully nonlinear parabolic equation for the free interface, involving pseudo-differential operators. The K-S equation appears to be asymptotically the lowest order of approximation near the threshold of stability.     

Deterministic representation of chaos with application to turbulence

Chaos and unpredictability in some classical dynamic systems are eliminated by referring the governing equation to a specially selected rapidly oscillating (non-inertial) frame of reference in which the stabilization effect is caused by inertia forces. The resulting motion is found as a sum of smooth and non-smooth (rapidly oscillating) parts. The solution is stable and reproducible in the sense that small changes in initial conditions lead to small changes in both smooth and non-smooth components. In this interpretation, conceptually the closure problem in turbulence is reduced to the problem of finding such a frame of reference where the high Reynolds number instability is eliminated. The usefulness of the approach is illustrated by examples.     

Progress in modelling multilayer salt-fingering structures

Real-world salt fingering structures (consisting of convective layers interlaced with thin salt-finger layers) are modeled by a second-order closure technique. In order to achieve physically realistic behaviour in the limit of vertical motion within the salt fingering layer (hereafter SFL) the convective turbulence model of Zeman and Lumley is modified to behave properly in the limit of one-dimensional “turbulence” motion. With the assumption of perfect correlations within the SFL and the length scale (spacing) of salt fingers the system of model equation is closed. Numerical results indicate that the model predicts the observable features of the two-layer laboratory experiments.     

完整系统三阶Lagrange方程的一种推导与讨论

从牛顿运动方程出发,推导了完整系统关于广义加速度的Lagrange方程.讨论了该方程与传统分析力学中的Lagrange方程的相容性问题.结果显示,三阶Lagrange方程可以通过对Lagrange方程求一阶时间导数得到,表明它们是相容的.因此三阶Lagrange方程提供了一种不同于传统Lagrange方程方法的求解物体运动方程的途径.     

完整系统三阶Lagrange方程的一种推导与讨论

从牛顿运动方程出发，推导了完整系统关于广义加速度的Lagrange方程.讨论了该方程与传统分析力学中的Lagrange方程的相容性问题.结果显示，三阶Lagrange方程可以通过对Lagrange方程求一阶时间导数得到，表明它们是相容的.因此三阶Lagrange方程提供了一种不同于传统Lagrange方程方法的求解物体运动方程的途径.     

The King-Werner method for solving nonsymmetric algebraic Riccati equation

For the nonsymmetric algebraic Riccati equation arising from transport theory, we concern about solving its minimal positive solution. In [1], Lu transferred the equation into a vector form and pointed out that the minimal positive solution of the matrix equation could be obtained via computing that of the vector equation. In this paper, we use the King-Werner method to solve the minimal positive solution of the vector equation and give the convergence and error analysis of the method. Numerical tests show that the King-Werner method is feasible to determine the minimal positive solution of the vector equation.     

On a zero equation model of turbulence

In this paper we study a widely used zero equation model of turbulence. The governing equations are derived by applying to the incompressible Navier-Stokes equations the Reynolds time averaging procedure. We achieve closure by employing the eddy viscosity concept. Using the Implicit Function Theorem we obtain an existence and uniquencess result. We also discuss the existence of nonsingular solutions. Finally, we present an algorithm for solving the modeled equations.