论文标题
湍流及其应用中的自相似性
Self-similarity in turbulence and its applications
论文作者
论文摘要
首先，我们讨论了Navier-Stokes方程的非高斯类型的自相似解决方案。我们重新审视了Canonne-Planchon（1996）研究的一类自相似解决方案。为了阐明它，我们详细研究了1D汉堡方程的自相似解决方案，完成了可能拥有的最一般形式的相似性概况。特别是，在众所周知的源型解决方案之上，我们确定了串联型解决方案。它由汇合的超几何函数之一表示，即。 Kummer的功能$M。$ 对于2D Navier-Stokes方程，除了著名的汉堡涡流外，我们还为相关的Fokker-Planck方程提供了另一个解决方案。就像上面的扭结型解决方案一样，这可以被视为汉堡涡流的“共轭”。已经确定了这种解决方案的某些渐近特性。建议对3D Navier-Stokes方程的含义。 其次，我们讨论了自相似解决方案的应用，以探索更通用的解决方案。特别是，基于3D Navier-Stokes方程的源类型的自相似解决方案，我们考虑了我们可以说出更多通用解决方案的内容。
First, we discuss the non-Gaussian type of self-similar solutions to the Navier-Stokes equations. We revisit a class of self-similar solutions which was studied in Canonne-Planchon (1996). In order to shed some light on it, we study self-similar solutions to the 1D Burgers equation in detail, completing the most general form of similarity profiles that it can possibly possess. In particular, on top of the well-known source-type solution we identify a kink-type solution. It is represented by one of the confluent hypergeometric functions, viz. Kummer's function $M.$ For the 2D Navier-Stokes equations, on top of the celebrated Burgers vortex we derive yet another solution to the associated Fokker-Planck equation. This can be regarded as a 'conjugate' to the Burgers vortex, just like the kink-type solution above. Some asymptotic properties of this kind of solution have been worked out. Implications for the 3D Navier-Stokes equations are suggested. Second, we address an application of self-similar solutions to explore more general kind of solutions. In particular, based on the source-type self-similar solution to the 3D Navier-Stokes equations, we consider what we could tell about more general solutions.
