Energy Gradient Theory of Hydrodynamic Instability

A new universal theory for flow instability and turbulent transition is proposed in this study. Flow instability and turbulence transition have been challenging subjects for fluid dynamics for a century. The critical condition of turbulent transition from theory and experiments differs largely from each other for Poiseuille flows. This enigma has not been clarified so far owing to the difficulty of the problem. In this paper, a new mechanism of flow instability and turbulence transition is presented for parallel shear flows and the energy gradient theory of hydrodynamic instability is proposed. It is stated that the total energy gradient in the transverse direction and that in the streamwise direction of the main flow dominate the disturbance amplification or decay. Thus, they determine the critical condition of instability initiation and flow transition under given initial disturbance. A new dimensionless parameter K for characterizing flow instability is proposed for wall bounded shear flows, which is expressed as the ratio of the energy gradients in the two directions. It is thought that flow instability should first occur at the position of Kmax which may be the most dangerous position. This speculation is confirmed by Nishioka et al’s experimental data. Comparison with experimental data for plane Poiseuille flow and pipe Poiseuille flow indicates that the proposed idea is really valid. It is found that the turbulence transition takes place at a critical value of Kmax of about 385 for both plane Poiseuille flow and pipe Poiseuille flow, below which no turbulence will occur regardless the disturbance. More studies show that the theory is also valid for plane Couette flows and Taylor-Couette flows between concentric rotating cylinders. It is concluded that the energy gradient theory is a universal theory for the flow instability and turbulent transition which is valid for both pressure and shear driven flows in both parallel flow and rotating flow configurations.