Equilibration and optimal excitation in viscous shear flow

Initial conditions which result in the evolution of a viscous shear flow into a quasisteady state or into a state susceptible to breakdown leading to transition to turbulence are of special interest. Numerical experiments are carried out to investigate small finite-amplitude two-dimensional initial disturbances which grow rapidly into elliptical vortices, the heuristic of selective decay of enstrophy in choosing a quasisteady state of the flow, and the linear three-dimensional perturbations which gain the most energy from the mean flow.;The theory of secondary instability, which describes a possible path to turbulence for a viscous shear flow, requires the presence of a finite amplitude two-dimensional elliptical disturbance for growth of three-dimensional instabilities to occur. In Poiseuille flow, two-dimensional disturbances which are optimally configured to gain the most energy from the Poiseuille mean flow profile and contain sufficient initial energies are shown to develop nonlinearly on a rapid timescale of a few advection time units into quasiequilibria capable of supporting secondary instabilities. Optimal disturbances in Couette flow are observed to decay rapidly after initial growth.;The conservation of both energy and enstrophy for a two-dimensional inviscid flow results in a cascade of energy to larger scales and of enstrophy to smaller scales. It has been argued that the selective decay of enstrophy should result in initial evolution of a slightly viscous flow into a state which has the smallest enstrophy consistent with the initial values of energy, momentum, and circulation. Time evolution of a variety of flows demonstrates that the trajectory in energy-enstrophy space is more complicated. Energy decays slowly and monotonically as expected. However, enstrophy may increase rapidly before rapid dissipation sets in, and the intermediate state, from which both energy and enstrophy decay on a viscous timescale, may not lie on the minimum enstrophy curve.;Variational methods are applied to the three-dimensional linearized flow equations to find the three-dimensional perturbations that gain the most energy in a given time period. The optimal perturbations found for Couette, Poiseuille, and Blasius mean flow profiles resemble streamwise vortices, which divert the mean flow into streaks of streamwise velocity and enable the energy of the perturbation to grow enormously despite the gradual decay of the streamwise vortex itself. Energy growth rate bounds are plotted against Reynolds number, and regions of absolute stability are plotted in wavenumber space. The energy growth of the three-dimensional optimal perturbations dwarfs that of optimals restricted to the two-dimensional plane of the shear, suggesting that excitation of this larger class of perturbations may play a more important role in transition to turbulence in natural, or undriven, flows than does secondary instability of elliptical vortices.