Development of new solution methods in fluid dynamics based on first integrals of Navier-Stokes equations

计算流体力学(CFD)已成为几乎所有技术领域的重要研究工具。尽管有成熟的数学理论和可用的商业软件代码，但运动控制方程的解的计算仍然具有挑战性，特别是由于所涉及的非线性。与第一个项目相比，第一个项目主要涉及一般理论的发展和通过将求解过程应用于原型特征的流动进行检验，该项目的第二部分更侧重于当前科学意义上的选定应用，特别是在这样的问题上，其中使用运动方程的第一积分而不是原始的N-S方程似乎是有益的。这些问题特别是：(I)各种具有自由表面的二维流动问题，因为这里提供了场方程的复杂公式和动态边界条件的Dirichlet/Neumann公式，以及(Ii)三维涂层在圆锥和半球体上的流动，使用润滑近似来降低问题的复杂性。(Iii)在第一积分框架内的湍流流动的公式是一个需要深入研究的一般性课题，因为湍流在一般情况下具有特殊的重要性。

Computational fluid dynamics (CFD) has emerged as an essential investigative tool in nearly every field of technology. Despite a well-developed mathematical theory and available commercial software codes the computation of solutions of the governing equations of motion is still challenging, especially due to the nonlinearity involved. In contrast to the first project which was mainly related to the development of the general theory and its testing by applying the solving procedure to flows of prototypic character, this second part of the project is more focused on selected applications of current scientific relevance, especially on such problems where the use of the first integral of the equations of motion instead of the original Navier-Stokes equations appears to be beneficial. These are in particular: (i) various 2D flow problems with free surface, since a complex formulation of the field equations and a Dirichlet/Neumann formulation of the dynamic boundary condition are available here and (ii) 3D coating flows over cones and half-spheres, using lubrication approximation for reduction of complexity of the problem.Apart from the above, (iii) the formulation of turbulent flows within the framework of the first integral is a general topic requiring a thorough investigation, motivated by the particular importance of turbulent flows in general.