Bounds on Surface Stress-Driven Shear Flow

摘要

The background method is adapted to derive rigorous limits on surface speeds and bulk energy dissipation for shear stress-driven flow in two- and threedimensional channels. By-products of the analysis are nonlinear energy stability results for plane Couette flow with a shear stress boundary condition: when the applied stress is gauged by a dimensionless Grashoff number Gr, the critical Gr for energy stability is 139.5 in two dimensions, and 51.73 in three dimensions. We derive upper bounds on the friction (a.k.a. dissipation) coefficient C_f = τ/u/(1/2)~2, where τ is the applied shear stress and u is the mean velocity of the fluid at the surface, for flows at higher Gr including developed turbulence: C_f ≤ 1/32 in two dimensions and C_f ≤ 1/8 in three dimensions. This analysis rigorously justifies previously computed numerical estimates.