Energy Gradient Theory of Hydrodynamic Instability

TL;DR

This paper introduces the energy gradient theory as a universal framework for understanding flow instability and turbulence transition in shear flows, proposing a new parameter K that predicts transition points accurately.

Contribution

It presents a novel energy gradient theory and a dimensionless parameter K that effectively predicts flow instability and turbulence transition across various shear flow configurations.

Findings

The critical value of Kmax for turbulence transition is about 385.

Flow instability occurs at the position of maximum K, Kmax.

The theory is validated with experimental data for multiple flow types.

Abstract

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. 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. 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.