On the non-linear mechanics of hydrodynamic stability

TLDR

Traditional stability theory of laminar flow considers only infinitesimal disturbances and predicts only initial exponential growth, but the nonlinear Reynolds stress that arises at finite amplitudes can modify the mean flow and potentially lead to an equilibrium where energy transfer balances viscous dissipation. The paper aims to extend stability theory to finite‑amplitude disturbances and analyze how nonlinear hydrodynamic effects govern disturbance growth. A theoretical framework based on energy‑flow assumptions is developed to describe how mean‑flow distortion alters energy transfer to disturbances, predicting both growth and equilibrium states for Poiseuille flow and rotating‑cylinder configurations. The model predicts the equilibrium distorted mean flow and the torque needed to drive rotating cylinders, matching well with Taylor’s experimental torque data for the inner‑cylinder‑rotating case.

Abstract

In most work on the theory of stability of laminar flow, infinitesimal disturbances only have been considered, so that only the initial growth of the disturbance has been determined. It is the object of the present paper to extend the theory to larger amplitudes and to study the mechanics of disturbance growth with the inherent non-linearity of the hydrodynamical system taken into account. The Reynolds stress (where averages are taken with respect to some suitable space coordinate) is the fundamental consequence of the non-linearity, and its effects can be anticipated as follows. Initially a disturbance grows exponentially with time according to the linear theory, but eventually it reaches such a size that the transport of momentum by the finite fluctuations is appreciable and the associated mean stress (the Reynolds stress) then has an appreciable effect on the mean flow. This distortion of the mean flow modifies the rate of transfer of energy from the mean flow to the disturbance and, since this energy transfer is the cause of the growth of the disturbance, there is a modification of the rate of growth of the latter. It is suggested that, in many cases, an equilibrium state may be possible in which the rate of transfer of energy from the (distorted) mean flow to the disturbance balances precisely the rate of viscous dissipation of the energy of disturbance. A theory based on certain assumptions about the energy flow is given to describe both the growth of the disturbance and the final equilibrium state, and application is made to the cases of Poiseuille flow between parallel planes and flow between rotating cylinders. The distorted mean flow in the equilibrium state can be calculated and from this, in the latter case, the torque required to maintain the cylinders in motion. Good agreement is obtained with G. I. Taylor's measurements of the torque for the case when the inner cylinder rotates and the outer cylinder is at rest.