论文标题
实现不可压缩的Navier-Stokes流量作为Clebsch电位的运输过程的叠加
Realization of Incompressible Navier-Stokes Flow as Superposition of Transport Processes for Clebsch Potentials
论文作者
论文摘要
在理想的流体中，Clebsch电位作为与Hamiltonian对Euler方程的描述相关的配对规范变量的出现。本文通过一组完整的配对Clebsch电位表达速度场时探讨了不可压缩的Navier-Stokes方程的属性。首先，结果表明，不可压缩的Navier-Stokes方程可以作为传输系统（对流扩散）方程式施放，其中每个Clebsch电位都扮演着广义分布函数的作用。与每个Clebsch电势相关的扩散操作员因术语而异，这取决于相应的Clebsch对的托架。进一步表明，CLEBSCH电位可用于定义香农型熵度量，即与能量和endrophy不同的功能，其生长速率是非负的。结果，流动必须在平衡下消失。该功能可以解释为衡量速度场拓扑复杂性的量度。此外，CLEBSCH参数化可以鉴定一类流量，该类别大于二维流的类别，具有涡旋拉伸项相同消失的特性，并且entophy的生长速率是非阳性的。
In ideal fluids, Clebsch potentials occur as paired canonical variables associated with the Hamiltonian description of the Euler equations. This paper explores the properties of the incompressible Navier-Stokes equations when the velocity field is expressed through a complete set of paired Clebsch potentials. First, it is shown that the incompressible Navier-Stokes equations can be cast as a system of transport (convection-diffusion) equations where each Clebsch potential plays the role of a generalized distribution function. The diffusion operator associated with each Clebsch potential departs from the standard Laplacian due to a term depending on the Lie-bracket of the corresponding Clebsch pair. It is further shown that the Clebsch potentials can be used to define a Shannon-type entropy measure, i.e. a functional, different from energy and enstrophy, whose growth rate is non-negative. As a consequence, the flow must vanish at equilibrium. This functional can be interpreted as a measure of the topological complexity of the velocity field. In addition, the Clebsch parametrization enables the identification of a class of flows, larger than the class of two dimensional flows, possessing the property that the vortex stretching term identically vanishes and the growth rate of entrophy is non-positive.
