Hagen-Poiseuille Flow Linear Stability Paradox Resolving and Viscous Dissipative Mechanism of the Turbulence Emergence in the Boundary Layer

TL;DR

This paper resolves the Hagen-Poiseuille flow linear stability paradox by introducing a modified Galerkin method that accounts for longitudinal variability, revealing conditions for linear instability and turbulence onset consistent with experimental data.

Contribution

It proposes a new linear stability analysis approach for Hagen-Poiseuille flow that abandons traditional disturbance separation, explaining turbulence emergence.

Findings

Linear instability occurs at Re_th=448, matching experimental thresholds.

Longitudinal disturbance periods critically influence flow stability.

Phase velocities align with observed turbulent puff fronts.

Abstract

In the linear theory of hydrodynamic stability up to now there exist examples of flows for which there is full quantitative distinction, as for cylindrical Hagen-Poiseuille (HP) flow in a pipe with round section, between theory conclusions and experimental data on the threshold Reynolds number Reth. In the present work, we show that to get a conclusion of linear instability of the HP flow for finite Reynolds numbers Re, it is necessary to abandon the use of traditional 'normal' form of disturbances which assumes an opportunity of separation of variables describing disturbances variability depending on radial and longitudinal (along the pipe axis) coordinates. In the result of the absence of such variables separation, in the suggested linear theory, it is proposed to use Bubnov-Galerkin's approximation method modification that gives an opportunity to account longitudinal variability periods distinctions for different radial modes defined a priori in the result of standard Galerkin-Cantorovich's method to the equation of evolution of extremely small axially symmetric velocity field tangential component disturbances. We found that when considering even two linearly interacting radial modes for the HP flow, linear instability is possible only when there exists mentioned above conditionally periodic longitudinal along the pipe axis disturbance variability when Re_th(p) very sensitively depends on the ratio p of two longitudinal periods each of which describes longitudinal variability for its own radial mode only. Obtained for the HP flow linear instability realization minimal value Re_th=448 (when p=1.527) quantitatively agrees with the Tolmin-Shlihting waves in the boundary layer emergence, where also Re_th=420. We get quantitative agreeing of the phase velocity values of the considered disturbances with experimental data on the fronts of the turbulent 'puffs' spreading in the pipe.