Mini-course: Fluid Flows at a High Reynolds number

Time:09:00:05/07/2016 to  11:00:08/07/2016

Venue/Location: B4-705 VIASM

Speaker: Asst. Prof. Nguyen Trong Toan – Pennsylvania State University, USA.

Content:

Course Description: :

The transition from laminar or well-organized fluid motion to turbulent flows, governed by the so-called Reynolds number, is striking, but not fully understood, with the formation of complicated patterns. In addition to being of great theoretical interest, laminar-to-turbulence transition is of significant engineering importance due to its role in heat transfer, its influence on momentum mixing, and its effect on drag.
In describing the early stage of the transition, physicists traditionally rely on the emergence of so-called Tollmien-Schlichting instability waves that arise within a thin layer near the boundary (viscous boundary layers). The instability is highly counter-intuitive, precisely due to the presence of small viscosity, and was pointed out by Heisenberg (1924), and C. C. Lin and Tollmien in the 40s.

In these lectures, I will present the recent complete mathematical proof of the mentioned Heisenberg’s viscous destabilization phenomenon (cf., Grenier-Guo-Nguyen, Advances in Math. and Duke Math J., both to appear), as well as the recent advances on the Prandtl’s asymptotic expansions in the inviscid limit or infinite Reynolds number limit (cf.,http://blog.toannguyen.org/ for course lecture notes). The lectures are aimed at the level of graduate students.

A tentative plan is as follows:

+ Description of fluid motion. Reynolds number.

+ The classical Orr-Sommerfeld equations. Spectral instability.

+ The invalidity of Prandtl’s asymptotic expansions in the inviscid limit.

+ Grenier’s nonlinear iterative scheme.