Separation of the two-dimensional unsteady Prandtl boundary layers under an adverse pressure gradient

TL;DR

This paper analyzes the conditions under which back flow and boundary layer separation occur in two-dimensional unsteady Prandtl boundary layers with adverse pressure gradients, providing criteria and examples for back flow formation.

Contribution

It establishes that the first critical point of the tangential velocity must be on the boundary under adverse pressure, and provides growth rate conditions for back flow occurrence.

Findings

Back flow occurs at the boundary when the first critical point exists.

A growth rate condition on initial velocity predicts back flow.

Examples show back flow depends on flow distance and initial velocity growth.

Abstract

In this paper, we study the back flow of the two-dimensional unsteady Prandtl boundary layer under an adverse pressure gradient. The occurrence of back flow is an important physical event in the evolution of boundary layer, which eventually leads to separation. For the two-dimensional unsteady Prandtl boundary layer equations, when the initial tangential velocity is strictly monotonic with respect to the normal variable, and the pressure gradient of the outer flow is adverse, we obtain that the first critical point of the tangential velocity profile with respect to the normal variable, if exists when the boundary layer evolves in time, must appear on the boundary. Moreover, we give a condition on the growth rate of the initial tangential velocity such that there is a back flow point of the Prandtl boundary layer under the adverse pressure gradient. In the appendix, we introduce two examples showing that back flow occurs either when the flow distance is long in the streamwise direction for a given initial monotonic tangential velocity field, or when the initial tangential velocity grows slowly in a large neighborhood of the boundary for a fixed flow distance in the streamwise direction.