Nonlinear Stability of Kolmogorov Flow with Bottom-Friction Using the Energy Method

Abstract

Stability of Kolmogorov flow: U=-sin y to any finite disturbances is treated by using the energy method. The linear damping term -λ u due to the bottom friction is taken into account. The Euler-Lagrange equation is solved numerically and analytically to determine the critical Reynolds number, R_Ec, below which subcritical instability cannot occur. It is shown numerically that R_Ec and the linear critical Reynolds number, R_Lc are of the same order in 0<λ <200. The critical wavenumber, (α_Ec, , β_Ec) is always (0, , 0) when λ <49.1; otherwise α _Ec≠ 0. By using a small wavenumber expansion, it is obtained that R_E= [8λ (λ +1)]^1/2 at α=0. In the limit λ → ∞, numerical results suggest that R_Ec→ 2λ and α_Ec→ ∞. In this limit for general parallel flow U(y) the relation: (2/M)λ EcLc=(1/σ₀)λ is obtained analytically where M=max_y, | ∂ U/∂ y|, and σ₀ is the inviscid maximum growth rate.