ENERGY GROWTH IN VISCOUS CHANNEL FLOWS

In recent work it has been shown that there can be substantial transie nt growth in the energy of small perturbations to plane Poiseuille and Couette flows if the Reynolds number is below the critical value pred icted by linear stability analysis. This growth, which may be as large as O(1000), occurs in the absence of nonlinear effects and can be exp lained by the non-normality of the governing linear operator - that is , the nonorthogonality of the associated eigenfunctions. In this paper we study various aspects of this energy growth for two- and three-dim ensional Poiseuille and Couette flows using energy methods, linear sta bility analysis, and a direct numerical procedure for computing the tr ansient growth. We examine conditions for no energy growth, the depend ence of the growth on the streamwise and spanwise wavenumbers, the tim e dependence of the growth, and the effects of degenerate eigenvalues. We show that the maximum transient growth behaves like O(R2), where R is the Reynolds number. We derive conditions for no energy growth by applying the Hille-Yosida theorem to the governing linear operator and show that these conditions yield the same results as those derived by energy methods, which can be applied to perturbations of arbitrary am plitude. These results emphasize the fact that subcritical transition can occur for Poiseuille and Couette flows because the governing linea r operator is non-normal.