On the Effects of Nonparallelism, Three‐Dimensionality, and Mode Interaction in Nonlinear Boundary‐Layer Stability

Abstract

A self‐consistent theoretical investigation is described for the nonlinear stability, and spatial development, of disturbances in a plane boundary layer subject to a number of three‐dimensional modes, their nonlinear interactions, and the effects of nonparallelism of the basic flow. For the largest weakly nonlinear disturbances considered, nonparallel‐flow effects appear to be negligible at first sight, and primary, secondary, and/or tertiary bifurcations, usually supercritical but not always so, can occur when two fundamental modes are present. As a result the flow downstream then always has three ultimate possibilities: a unique stable disturbed state, two or more possible stable states, or no stable state possible. It is here that the nonparallel‐flow effects exert their crucial influence. For nonparallelism comes into play significantly during the initial growth or decay of a disturbance, and that initial spatial development, from given initial conditions upstream, controls what happens subsequently as the disturbance increases. Thus in the first possibility above, the stable state is achieved through a smooth bifurcation, due to nonparallelism; in the second possibility the nonparallelism decides which stable state is attained (smoothly) from the initial conditions; and in the third possibility the nonparallel flow effects force the disturbance to terminate in a singular fashion. This singularity then leads to a fully nonlinear effect, locally on the boundary‐layer flow. More complicated interactions can arise if more than two three‐dimensional modes are present. The novel effect of the nonparallelism has a connection with related Navier‐Stokes calculations even at near‐critical Reynolds numbers.