可分离非定常非平行流稳定性问题
Separable unsteady nonparallel flow stability problems
摘要：
所谓的“直接”的方法来分离变量的线性偏微分方程的水动力稳定性问题。在柱坐标系下对完全线性稳定性方程进行了计算。本文定义了几类描述空间发展的非定常流动的纳维-斯托克斯方程的精确解，它们的线性稳定性问题可以严格地归结为常微分方程的特征值问题。这些精确可解的非平行和非定常流动稳定性问题可用于检验近似方法和基于（线性化）纳维-斯托克斯方程直接数值模拟的方法。粘性不可压缩纳维-斯托克斯方程的精确解被确定为基本状态，线性稳定性问题是精确可分的，从理论和工程观点来看，它们本身可能是令人感兴趣的。
英文摘要：
The so-called 'direct' approach to separation of variables in linear PDEs is applied to the hydrodynamic stability problem. Calculations are made for the complete linear stability equations in cylindrical coordinates. Several classes of the exact solutions of the Navier-Stokes equations describing spatially developing and unsteady flows, for which the linear stability problems can be rigorously reduced to eigenvalue problems of ordinary differential equations, are defined. Those exactly solvable nonparallel and unsteady flow stability problems can be used for testing approximate approaches and the methods based on direct numerical simulations of the (linearized) Navier-Stokes equations. The exact solutions of the viscous incompressible Navier-Stokes equations determined as the basic states, for which the linear stability problem is exactly separable, may be themselves of interest from theoretical and engineering points of view.
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