Investigation of Turbulent Transition in Plane Couette Flows Using Energy Gradient Method (Advances in Applied Mathematics and Mechanics, 2018)
The energy gradient method has been proposed with the aim of betterunderstanding the mechanism of flow transition from laminar flow toturbulent flow. In this method, it is demonstrated that the transitionto turbulence depends on the relative magnitudes of the transverse gradientof the total mechanical energy which amplifies the disturbance and the energyloss from viscous friction which damps the disturbance, for given imposeddisturbance. For a given flow geometry and fluid properties, when the maximumof the function $K$ (a function standing for the ratio of the gradient of totalmechanical energy in the transverse direction to the rate of energy loss due toviscous friction in the streamwise direction) in the flow field is larger than acertain critical value, it is expected that instability would occur for someinitial disturbances. In this paper, using the energy gradient analysis, theequation for calculating the energy gradient function $K$ for plane Couette flowis derived. The result indicates that $K$ reaches the maximum at the moving walls.Thus, the fluid layer near the moving wall is the most dangerous position to generateinitial oscillation at sufficient high $\operatorname{Re}$ for given same level ofnormalized perturbation in the domain. The critical value of $K$ at turbulent transition,which is observed from experiments, is about 370 for plane Couette flow when two wallsmove in opposite directions (anti-symmetry). This value is about the same as that forplane Poiseuille flow and pipe Poiseuille flow (385-389). Therefore, it is concludedthat the critical value of $K$ at turbulent transition is about 370-389 for wall-boundedparallel shear flows which include both pressure (symmetrical case) and shear drivenflows (anti-symmetrical case).