Turbulent Transition in Plane Couette Flows

Abstract

The energy gradient theory has been proposed with the aim of better understanding the mechanism of flow transition from laminar flow to turbulent flow. In this theory, it is demonstrated that the transition to turbulence depends on the relative magnitudes of the transversal energy gradient which amplifies the disturbance and the energy loss from friction which damps the disturbance. For a given flow geometry and fluid properties, when the maximum of the function $K$ (the ratio of the energy gradient in the transverse direction to the energy loss in the streamwise direction) in the flow field is larger than a certain critical value, it is expected that instability would occur for some initial disturbances. In this paper, using the energy analysis, the equation for calculating the function $K$ for plane Couette flow is derived. It is demonstrated that the critical value of $K$ at turbulent transition, which is observed from experiments, is about 370 for plane Couette flow. This value is about the same as for plane Poiseuille flow and pipe Poiseuille flow (385–389). This is in agreement with the theoretical analysis of the disturbance energy evolution. Therefore, it is concluded that the critical value of $K$ at turbulent transition is about 370–389 for wall-bounded parallel shear flows which include both pressure and shear driven flows.