Mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence

Main mathematical concepts and their physical foundation in the nonstandard analysis theory of turbulence are presented and discussed. The underlying fact is that there does not exist the absolute zero fluid-volume. Therefore, the physical object corresponding to the absolute point is just the uniform fluid-particle. The fluid-particle, in general, corresponds to the monad. The uniform fluid-particle corresponds to the uniform monad, while the nonuniform fluid-particle to the nonuniform monad. There are two kinds of the differentiations, one is based on the absolute point, and the other based on the monad. The former is adopted in the Navier--Stokes equations, and the latter in the fundamental equations presented in this paper for the nonstandard analysis theory of turbulence. The continuity of fluid is elucidated by virtue of the concepts of the fluid-particle and fluid-particle at a lower level. Furthermore, the characters of the continuity in two cases, i.e. in the standard and nonstandard analyses, are presented in this paper. And the difference in discretization between the Navier--Stokes equations and the fundamental equations given herein is also pointed out.     

New Sedov-Type Solution of Isotropic Turbulence

The starting point lies in the results obtained by Sedov （1944） for isotropic turbulence with a self-preserving hypothesis. A careful consideration of the mathematical structure of the Karman-Howarth equation leads to an exact analysis of all cases possible and to all admissible solutions of the problem. I study this interesting problem from a new point of view. New solutions are obtained. Based on these exact solutions, some physical significant consequences of recent advances in the theory of self-preserved homogeneous statistical solution of the Navier-Stokes equations are presented.     

湍流理论的近代发展

这篇论文的内容是关於不可压缩流体的湍流理论近代发展的综合性介绍舆分析。我们首先评述了根据Reynolds的平均运动方程所建立的混合长度理论。其次,分析关於均匀各向同性湍流的主要理论工作。第三,讨论了运用Reynolds的平均运动方程和根据速度涨落方程求得的速度关联函数的动力学方程来处理具有Reynolds剪应力的普通湍流运动问题。同时说明这个方法虽然能够给出比混合长度理论舆实验较为接近的理论结果并能提出速度涨落平方平均值的理论分布,但是由於在求出的速度关联的动力学方程中出现高次元的速度关联,它继续地导致不封闭的微分方程组因而遇到不易克服的困难。因此,从以上湍流理论发展的回顾和最近关於均匀各向同性湍流在后期衰变运动的涡性结构工作,我们在最后提出了对今后湍流理论研究工作的新看法:湍流运动的基本组成部分是流体粘性作用所引起的涡旋运动;这个涡旋运动的动力学根据是用平均的方法后Navier-Stokes方程所导出的Reyonlds的平均运动方程典带度涨落方程。我们并着重说明Reynolds认识到湍流运动可分作平均运动与涨落运动的重要性。今后的理论工作则在於求这两组动力学方程的涡旋运动解,而这种类型的解并须满足像Колмогоров在高Reynolds数运动的局部各向同性湍流理论中所提出的统计条件,方能使解满足惟一性并可舆实验结果相比较。     

Mathematics of dispersive water waves

A commuting hierarchy of dispersive water wave equations makes a three-Hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed. The modified water wave hierarchy is a bi-Hamiltonian system; its modification bifurcates. The water wave hierarchy, and the hierarchies of the Korteweg-de Vries and the modified Korteweg-de Vries equations, as well as the classical Miura map, are given new representations through various specializations of nonstandard systems.     

Single point velocity distributions in homogeneous isotropic turbulence

The starting point for this paper lies in the results obtained by Tatsumi (2004) for isotropic turbulence with the self-preserving hypothesis. A careful consideration of the mathematical structure of the one-point velocity distribution function equation obtained by Tatsumi (2004) leads to an exact analysis of all possible cases and to all admissible solutions of the problem. This paper revisits this interesting problem from a new point of view, and obtains a new complete set of solutions. Based on these exact solutions, some physically significant consequences of recent advances in the theory of homogenous statistical solution of the Navier--Stokes equations are presented. The comparison with former theory was also made. The origin of non--Gaussian character could be deduced from the above exact solutions.     

Nonstandard Fock spaces

A theory of nonstandard inner product spaces is developed using methods of nonstandard analysis. Various results concerning nonstandard operators and their spectra are proved. The theory is applied to construct nonstandard Fock spaces which extend the standard Fock spaces. Moreover, a rigorous framework for the field operators of quantum field theory is presented. The results are illustrated for the case of Klein-Gordon fields.     

Model of Non-Laminar Low-Temperature Plasma Flow with Intensive Self-Mixing

A new model of turbulence is presented. It consists in treating the turbulence as an essentially self-mixing process. The mass, momentum and energy transport is assumed to proceed along interlacing curvilinear cones. A relevant mixing field is introduced. It consists in relating to each point of space an infinite set of instantaneous velocities momentums and energies. Into each transport equation a self-mixing term of the proper form is introduced. The closure procedures for modyfied transport equations are also discussed.     

Toward a rigorous quantum field theory

This paper outlines a framework that may provide a mathematically rigorous quantum field theory. The framework relies upon the methods of nonstandard analysis. A theory of nonstandard inner product spaces and operators on these spaces is first developed. This theory is then applied to construct nonstandard Fock spaces which extend the standard Fock spaces. Then a rigorous framework for the field operators of quantum field theory is presented. The results are illustrated for the case of Klein-Gordon fields.     

The numerical solution of problems in calculus of variation using Chebyshev finite difference method

The Chebyshev finite difference method is used for finding the solution of the ordinary differential equations which arise from problems of calculus of variations. Our approach consists of reducing the problem to a set of algebraic equations. This method can be regarded as a non-uniform finite difference scheme. Some numerical results are also given to demonstrate the validity and applicability of the presented technique. The method is easy to implement and yields very accurate results.     

Application of the equivalent surface source method to the acoustics of duct systems with non-uniform wall impedance

The paper outlines the application of the equivalent surface source method to the analysis of the acoustic field in a partially lined duct with arbitrarily non-uniform wall impedance. Lined sections of the duct wall are represented by unsteady mass source singularities, the strengths of which are determined by solving integral equations. The method is applicable to lined walls of impedance which is non-uniform in the streamwise and/or circumferential direction. Numerical examples are given to show the effects of various design parameters on sound attenuation. Some advantageous features of circumferentially non-uniform wall impedance are demonstrated.     

重油裂解反应器内多相流动分析

重油裂解反应器内多相流动分析吴邦贤（中国科学院工程热物理研究所北京１０００８０）关键词重油裂解，多相流，直升反应器通过重油裂解反应生产轻油已成为现代炼油业中的关键转换过程。在重油裂解反应器内同时存在众多的物理，化学，流动过程，其中包括：重油液滴雾化，...     

非均匀等离子体湍流的重正化准线性理论(Ⅰ)——Misguich-Balescu理论的推广

本文将Misguich和Balescu的理论推广到非均匀等离子体的情形,导出了包括非均匀性效应的扩散方程与扩散张量的表式以及色散关系。指出Pelletier等人对等离子体湍流可作马尔科夫近似处理的论证完全不适用于非均匀等离子体湍流的问题。 关键词：     

TOKAM-3D: A 3D fluid code for transport and turbulence in the edge plasma of Tokamaks

We present a new code aiming at giving a global and coherent approach for transport and turbulence issues in the edge plasma of Tokamaks. The TOKAM-3D code solves 3D fluid drift equations in full-torus geometry including both closed field lines and SOL physics. No scale separation is assumed so that interactions between large scale flows and turbulence are coherently treated. Moreover, the code can be run in transport regimes ranging from purely anomalous diffusion to fully established turbulence. Specific numerical schemes have been developed which can solve the model equations whether the presence of a limiter in the plasma is taken into account or not. Example cases giving an overview of the field of application of the code as well as verification results are also presented.     

CAA broadband noise prediction for aeroacoustic design

The current status of a computational aeroacoustics (CAA) approach to simulate broadband noise is reviewed. The method rests on the use of steady Reynolds averaged Navier-Stokes (RANS) simulation to describe the time-averaged motion of turbulent flow. By means of synthetic turbulence the steady one-point statistics (e.g. turbulence kinetic energy) and turbulent length- and time-scales of RANS are translated into fluctuations having statistics that very accurately reproduce the initial RANS target-setting. The synthetic fluctuations are used to prescribe sound sources which drive linear perturbation equations. The whole approach represents a methodology to solve statistical noise theory with state-of-the-art CAA tools in the time-domain. A brief overview of the synthetic turbulence model and its numerical discretization in terms of the random particle-mesh (RPM) and fast random particle-mesh (FRPM) method is given. Results are presented for trailing-edge noise, slat noise, and jet noise. Some problems related to the formulation of vortex sound sources are discussed.     

Classical integrability of two-dimensional non-linear sigma models

The conditions under which a general two-dimensional non-linear sigma model is classically integrable are given. These requirements are found by demanding that the equations of motion of the theory are expressible as a zero curvature relation. Some new integrable two-dimensional sigma models are then presented.     

Deindividuation phenomenon: Links between mesodynamics and macroscopic phenomenology of turbulence

The paper develops the earlier declared original approach to elaboration of a turbulence theory based on mathematical modeling of deindividuation of particles moving in a turbulent medium. It is shown that modeling of the deindividuation inevitably leads to gauge structures. It is found that the existing statistical concepts of turbulence are inapplicable to this type of problems. In the approach, new links between mesodynamics and macroscopic phenomenology of turbulence are established making possible new solutions based on synthesis of relevant equations.     

Numerical solutions of the isotropic 3-wave kinetic equation

We show that the isotropic 3-wave kinetic equation is equivalent to the mean-field rate equations for an aggregation–fragmentation problem with an unusual fragmentation mechanism. This analogy is used to write the theory of 3-wave turbulence almost entirely in terms of a single scaling parameter. A new numerical method for solving the kinetic equation over a large range of frequencies is developed by extending Lee’s method for solving aggregation equations. The new algorithm is validated against some analytical calculations of the Kolmogorov–Zakharov (KZ) constant for some families of model interaction coefficients. The algorithm is then applied to study some wave turbulence problems in which the finiteness of the dissipation scale is an essential feature. Firstly, it is shown that for finite capacity cascades, the dissipation of energy becomes independent of the cut-off frequency as this cut-off is taken to infinity. This is an explicit indication of the presence of a dissipative anomaly. Secondly, a preliminary numerical study is presented of the so-called bottleneck effect in a wave turbulence context. It is found that the structure of the bottleneck depends non-trivially on the interaction coefficient. Finally, some results are presented on the complementary phenomenon of thermalisation in closed wave systems which demonstrates explicitly for the first time the existence of so-called mixed solutions of the kinetic equation which exhibit aspects of both KZ and equilibrium equipartition spectra.     

非均匀采样的功率谱反演大气湍流相位屏的快速模拟

对大气湍流功率谱非均匀采样可以有效改善传统功率谱反演法低频采样严重不足的缺陷, 实现高精度的大气湍流相位屏的模拟. 但采用的直接求和运算计算复杂度高, 相位屏的模拟速度极慢. 将非均匀快速傅里叶变换(NUFFT)引入到大气湍流相位屏的模拟, 可以实现相位屏的快速模拟. 从随机过程的谱分解出发, 将大气湍流相位随机过程表示为有限谐波分量叠加和的均方极限. 通过一个高斯核函数的卷积, 将非均匀分布的谐波复振幅映射到均匀网格空间, 进而利用快速傅里叶变换, 降低计算复杂度, 加快大气湍流相位屏的模拟速度. 以大气湍流的Kolmogorov 谱为例, 利用NUFFT仿真得到大气湍流相位屏, 并对相位屏的模拟精度、模拟速度和误差进行统计分析. 结果表明, NUFFT的引入可以实现快速、高精度的大气湍流相位屏的模拟.     

Entropy and gravitation

Einstein's gravitational theory is analyzed from a thermodynamic point of view. A thermodynamic potential characterizing the sources of gravitational fields is presented. By means of this potential the entropy production density is derived. Einstein's equations with dissipative terms appear as linear phenomenological laws in the sense of irreversible thermodynamics. Some thermodynamic influences on gravitational phenomena are discussed.