Analytical Kink-Type Solutions and Streak Formation in Turbulent Channel Flow

TL;DR

This paper develops an analytical model for turbulent channel flow that accurately predicts velocity profiles and describes the formation of streaks and secondary flows, linking transverse fluctuations to coherent structures.

Contribution

It introduces a novel analytical framework based on Alexeev hydrodynamic equations that captures streak formation and mean velocity profiles in turbulent flows.

Findings

Velocity profiles match experimental data within 3% at high Reynolds numbers.

Analytical expressions describe transverse velocity structures.

Kink-type solutions represent streamwise streaks consistent with observations.

Abstract

An analytical framework for turbulent channel flow is developed based on the Alexeev hydrodynamic equations, focusing on the coupled behavior of streamwise and transverse velocity components. The mean streamwise velocity is represented as a superposition of a laminar (parabolic) component and a nonlinear turbulent contribution, yielding velocity profiles that agree with experimental data from channel and pipe flows over a wide range of Reynolds numbers, $3\times10^3 \le Re \le 3.5\times10^7$, with deviations of approximately $1\%$ at moderate Reynolds numbers and up to $3\%$ at the highest Reynolds numbers. The transverse velocity component is analyzed using a simplified form of the governing equations, leading to analytical expressions that capture its dominant spatial structure. The coupling between transverse velocity and streamwise momentum is then examined, revealing that the streamwise turbulent component admits a family of kink-type solutions. These solutions exhibit localized monotonic transitions separating regions of nearly uniform velocity and are interpreted as analytical representations of streamwise streaks. The model predicts characteristic streak properties, including spacing, thickness, intensity, and streamwise extent, which are shown to be consistent in order of magnitude with experimental observations of near-wall streaks. The results provide a unified analytical description of mean velocity profiles, secondary flows, and streak formation in wall-bounded turbulence, and suggest a mechanism linking transverse velocity fluctuations to the emergence of coherent streamwise structures.