Evolution of Wavelike Disturbances in Shear Flows: A Class of Exact Solutions of the Navier-Stokes Equations
New classes of exact solutions of the incompressible Navier-Stokes equations are presented. The method of solution has its origins in that first used by Kelvin (Phil. Mag. 24 (5), 188-196 (1887)) to solve the linearized equations governing small disturbances in unbounded plane Couette flow. The new solutions found describe arbitrarily large, spatially periodic disturbances within certain two- and three-dimensional 'basic' shear flows of unbounded extent. The admissible classes of basic flow possess spatially uniform strain rates; they include two- and three-dimensional stagnation point flows and two-dimensional flows with uniform vorticity. The disturbances, though spatially periodic, have time-dependent wavenumber and velocity components. It is found that solutions for the disturbance do not always decay to zero; but in some instances grow continuously in spite of viscous dissipation. This behaviour is explained in terms of vorticity dynamics.
Proceedings of the Royal Society of London Series A
- Pub Date:
- July 1986