Laminar and Turbulent Dissipation in Shear Flow with Suction

The rate of viscous energy dissipation in a shear layer of incompressible Newtonian fluid with injection and suction is studied by means of exact solutions, nonlinear and linearized stability theory, and rigorous upper bounds. For large enough values of the injection angle a steady laminar flow is nonlinearly stable for all Reynolds numbers, while for small but nonzero angles the laminar flow is linearly unstable at high Reynolds numbers. The upper bound on the energy dissipation rate—valid even for turbulent solutions of the Navier-Stokes equations—scales precisely the same as that in the steady laminar solution with regard to the viscosity in the vanishing viscosity limit. Both the laminar dissipation and the upper bound on turbulent dissipation display scaling in which the energy dissipation rate becomes independent of the viscosity for high Reynolds numbers. Hence the laminar energy dissipation rate and the largest possible turbulent energy dissipation rate for flows in this geometry differ by just a prefactor that depends only on injection angle. This result establishes the sharpness of the upper bound's scaling in the vanishing viscosity limit for these boundary conditions, and this system provides an analytic illustration of the delicacy of corrections to scaling (e.g., logarithmic terms as appearing in the "law of the wall") to perturbations in the boundary conditions.