Non-Linear Stability of a Two-Dimensional Stagnation Flow

Abstract

A slightly non-linear case of stability is treated when the non-dimensional amplitude A₀ of the disturbances in a two-dimensional stagnation region of a cylindrical blunt body is not so large that the derivative of A₀² with respect to a non-dimensional time τ can be expressed as dA₀²/d_τ = 2A₀² (α₀ + α_lA₀²) where, α₀ and α_l denote the respective growth factors of the first and second degrees. The mutual combination between the signs of α₀ and α_l has an important significance for the problem of non-linear stability. In this paper, the results of α_l < 0 is obtained by determining the value of α_l such that the integration of the kinetic energy involved in the disturbances becomes minimum up to the higher order of magnitude. This result suggests that the flow in the stagnation region reaches the state of super-critical equilibrium when the Reynolds number exceeds the critical value.