Transient growth in two- and three-dimensional boundary layers

Abstract

The evolution of localized three-dimensional disturbance in two- and three-dimensional laminar boundary layers is examined. The linearized Navier–Stokes equations for three-dimensional disturbances in a three-dimensional parallel shear flow are solved numerically using Fourier transform Chebyshev collocation techniques. Modal analysis shows that substantial short-term energy growth can be obtained even when all instability waves are damped. This transient growth can increase the initial disturbance energy by two or three orders of magnitude, at which stage nonlinear interactions might lead to a breakdown to turbulent flow, bypassing the traditional Tollmien–Schlichting instability mechanism. The dependence of the transient growth on wave number, Reynolds number, sweep angle and Hartree parameter is determined and a method for predicting the maximum transient growth is proposed and found to be reasonably accurate over a wide parameter range. Localized disturbances are also examined and it is found that the bypass growth mechanism can enhance the formation of cross-flow vortices in a three-dimensional flow. Some implications are discussed, particularly with respect to the observed effects of roughness on transition location.