The Limit of High Reynolds Number for Shear Layers

首席研究员将使用涡旋法、谱法和有限差分法三种不同的方法研究不可压缩流体中剪切层的高雷诺数极限。经典的极限被认为是涡旋片，但最近的研究表明涡旋片在有限时间内发展出曲率奇点，并且在奇点形成时，涡旋片从欧拉方程的弱解变为测量值解。此外，不同的正则化可能导致不同的结果，使得消失粘度极限的性质成为一个中心问题。考虑了不同的数值方法，以便对结果的准确性进行直接检查，对这些方法在剪切流中的性能进行直接比较，并确定每种方法中需要改进的地方。数值计算的结果将用于研究粘性效应如何改变涡旋片发展曲率奇点的趋势。数值结果也将用于帮助回答开放的数学问题：当正则化消失时，欧拉方程的不同正则化是否收敛到不同的极限？

8903484 Baker The principal investigator will study the limit at high Reynolds number of shear layers in incompressible fluid using three different methods--vortex methods, spectral methods, and finite difference methods. Classically, the limit is thought to be a vortex sheet, but recent studies suggest that vortex sheets develop curvature singularities in finite time, and that at the time of singularity formation, the vortex sheet changes from a weak solution to a measure-valued solution of Euler equations. Moreover, different regularizations may lead to different results, making the nature of the limit of vanishing viscosity a central concern. The different numerical methods are considered in order to obtain a direct check on the accuracy of the results, to make a direct comparison of the performance of these methods on shear flows and to identify areas for improvement in each method. Results of the numerical calculations will be used to study how viscous effects modify the tendency for vortex sheets to develop curvature singularities. Numerical results will also be used to help answer the open mathematical question: do different regularizations of the Euler equations converge to different limits as the regularization vanishes?