流体动力学的基本方面与不可压缩纳维尔-斯托克斯流体的统计模型的构建有关。后者可以被认为是\textit{deterministic}或\textit{stochastic,}分别为\textit{regular}或\textit{turbulent flows.}在这项工作中，我们声称，这种类型的一个可能的统计公式可以通过1点（本地）的速度空间概率密度函数（PDF，$f_{1}$）来确定在所谓的逆动力学理论（IKT）的框架。这个理论有几个重要的结论。其中特别包括[统计模型${f_{1},Γ} ]$的]初始PDF的表征。这被发现是一般的非高斯PDF，即使在流动的情况下，这是经常在初始时间。此外，对于规则和湍流，它的时间演化提供了一个刘维尔方程，而相应的刘维尔算子被发现只依赖于有限数量的速度矩相同的PDF。因此，它的时间演化（功能上）仅取决于相同的PDF。此外，本文建立的统计模型唯一地确定了$f_{1}.$的初始条件和时间演化。作为一个基本含义，该理论允许\textit{exact construction of the corresponding statistical equation for the stochastic-averaged PDF}和\textit{unique representation of the multi-point PDF}仅用1点PDF表示。作为一个例子，减少2点PDF的情况下，通常采用的NS湍流的统计描述，被认为是。
英文摘要：
Fundamental aspects of fluid dynamics are related to construction of statistical models for incompressible Navier-Stokes fluids. The latter can be considered either \textit{deterministic} or \textit{stochastic,} respectively for \textit{regular} or \textit{turbulent flows.} In this work we claim that a possible statistical formulation of this type can be achieved by means of the 1-point (local) velocity-space probability density function (PDF, $f_{1}$) to be determined in the framework of the so-called inverse kinetic theory (IKT). There are several important consequences of the theory. These include, in particular, the characterization of the initial PDF [for the statistical model ${f_{1},Γ} ]$ . This is found to be generally non-Gaussian PDF, even in the case of flows which are regular at the initial time. Moreover, both for regular and turbulent flows, its time evolution is provided by a Liouville equation, while the corresponding Liouville operator is found to depend only on a finite number of velocity moments of the same PDF. Hence, its time evolution depends (functionally) solely on the same PDF. In addition, the statistical model here developed determines uniquely both the initial condition and the time evolution of $f_{1}.$ As a basic implication, the theory allows the \textit{exact construction of the corresponding statistical equation for the stochastic-averaged PDF}and the \textit{unique representation of the multi-point PDF}'s in solely in terms of the 1-point PDF. As an example, the case of the reduced 2-point PDF's, usually adopted for the statistical description of NS turbulence, is considered.
The Variable Density Model for the Rayleigh-Taylor Instability and its transformation to the diffusive, inhomogeneous, incompressible Navier-Stokes equations
Theory of multi-point probability densities for incompressible Navier-Stokes fluids
在湍流的统计描述中出现的一个公开问题与表征Navier-Stokes流体的所谓多点速度概率密度函数（PDF）的\textit{theoretical prediction based on first principles}有关。在本文中，它将被证明-基于一个合适的公理方法-这个问题的解决方案实际上可以实现基于所谓的逆动力学理论（IKT），最近开发的不可压缩流体。更准确地说，我们打算表明，基于要求\textit{the Boltzmann-Shannon entropy for the s-point velocity PDF ($f_s$) is independent of the order $s$ and is also maximal at all times}，所有多点PDF是\textit{necessarily factorized in terms of the corresponding 1-point velocity PDF}（$f_1$）。因此，通常被认为是湍流的唯象描述的多点PDF可以在理论上预测\textit {基于知识的$% f_1$通过IKT获得。
A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows
Solution to two-dimensional Incompressible Navier-Stokes Equations with SIMPLE, SIMPLER and Vorticity-Stream Function Approaches. Driven-Lid Cavity Problem: Solution and Visualization
