Instability of Taylor-Couette Flow between Concentric Rotating Cylinders

TL;DR

This paper applies the energy gradient theory to analyze the instability and turbulent transition in Taylor-Couette flow, successfully predicting critical conditions and explaining turbulence mechanisms, thus suggesting the theory's broad applicability.

Contribution

The study extends the energy gradient theory to Taylor-Couette flow, deriving critical instability conditions that align with experiments and explaining turbulence generation mechanisms.

Findings

Critical energy gradient parameter matches experimental data.

Spiral turbulence can be explained by energy gradient theory.

Theory applicable to both pressure-driven and shear-driven flows.

Abstract

The energy gradient theory is used to study the instability of Taylor-Couette flow between concentric rotating cylinders. This theory has been proposed in our previous works. In our previous studies, the energy gradient theory was demonstrated to be applicable for wall-bounded parallel flows. It was found that the critical value of the energy gradient parameter Kmax at turbulent transition is about 370-389 for wall-bounded parallel flows (which include plane Poiseuille flow, pipe Poiseuille flow and plane Couette flow) below which no turbulence occurs. In this paper, the detailed derivation for the calculation of the energy gradient parameter in the flow between concentric rotating cylinders is provided. The calculated results for the critical condition of primary instability (with semi-empirical treatment) are found to be in very good agreement with the experiments in the literature. A possible mechanism of spiral turbulence generation observed for counter-rotation of two cylinders can also be explained using the energy gradient theory. The energy gradient theory can serve to relate the condition of transition in Taylor-Couette flow to that in plane Couette flow. The latter reasonably becomes the limiting case of the former when the radii of cylinders tend to infinity. It is our contention that the energy gradient theory is possibly fairly universal for analysis of flow instability and turbulent transition, and is found valid for both pressure and shear driven flows in parallel and rotating flow configurations.