Physics of Flow Instability and Turbulent Transition in Shear Flows

TL;DR

This paper introduces the energy gradient method, a physics-based model explaining flow instability and turbulent transition in shear flows, validated by experimental data and analytical results.

Contribution

It proposes a novel energy gradient model derived from physics that explains flow instability, turbulence onset, and vortex formation in shear flows.

Findings

The energy gradient parameter Kmax is constant (~370-389) at transition.

The disturbance amplitude threshold scales with Reynolds number as Re^{-1}.

The model aligns well with experimental observations of flow instability locations.

Abstract

In this paper, the physics of flow instability and turbulent transition in shear flows is studied by analyzing the energy variation of fluid particles under the interaction of base flow with a disturbance. For the first time, a model derived strictly from physics is proposed to show that the flow instability under finite amplitude disturbance leads to turbulent transition. The proposed model is named as "energy gradient method." It is demonstrated that it is the transverse energy gradient that leads to the disturbance amplification while the disturbance is damped by the energy loss due to viscosity along the streamline. It is also shown that the threshold of disturbance amplitude obtained is scaled with the Reynolds number by an exponent of -1, which exactly explains the recent modern experimental results by Hof et al. for pipe flow. The mechanism for velocity inflection and hairpin vortex formation are explained with reference to analytical results. Following from this analysis, it can be demonstrated that the critical value of the so called energy gradient parameter Kmax is constant for turbulent transition in wall bounded parallel flows, and this is confirmed by experiments and is about 370-389. The location of instability initiation in the flow field accords well with the experiments for both pipe Poiseuille flow (r/R=0.58) and plane Poiseuille flow (y/h=0.58). It is also inferred from the proposed method that the transverse energy gradient can serve as the power for the self-sustaining process of wall bounded turbulence. Finally, the relation of "energy gradient method" to the classical "energy method" based on Rayleigh-Orr equation is discussed.