Analytical Solutions for Turbulent Channel Flow Using Alexeev and Navier-Stokes Hydrodynamic Equations: Comparison with Experiments

TL;DR

This paper derives and analytically solves the Alexeev hydrodynamic equations for turbulent channel flow, demonstrating superior agreement with experimental data over Navier-Stokes solutions and revealing a new similarity parameter related to turbulence scales.

Contribution

It introduces analytical solutions based on the Alexeev hydrodynamic equations for turbulent flow, showing improved accuracy and identifying a new turbulence similarity parameter.

Findings

Alexeev hydrodynamic equations outperform Navier-Stokes in matching experimental data

A new similarity parameter for turbulence boundary layer thickness is identified

Analytical solutions are validated against numerical and experimental results

Abstract

Understanding turbulent boundary layer flows is important for many application areas. Enhanced theoretical models may provide deeper insights into the fundamental mechanisms of turbulence that elude current models; therefore, the search for improved kinetic equations and their respective hydrodynamic equations continues. In this work, we consider the Generalized Boltzmann Equation (GBE), proposed by Alexeev (1994). The GBE accounts for finite particle size and the variation of the distribution function over timescales of the order of the collision time. The Alexeev hydrodynamic equations are derived from the GBE. In this work, the Alexeev hydrodynamic equations (AHE) and Navier-Stokes (NS) equations are solved analytically for turbulent channel flow under the assumption that stationary solutions yield the mean flow velocity. The analytical solutions of the AHE are validated by numerical solutions and compared with the NS solutions and experimental data for turbulent channel flow from multiple sources, spanning Reynolds numbers from 3,000 to 35,000,000. Solutions of the AHE demonstrate significantly better agreement with experimental data than those obtained from the NS equations. The analytical solution revealed a new similarity parameter: the boundary layer thickness scale, which coincides with the Kolmogorov microscale observed in experiments. The mechanisms for turbulence generation and control are discussed.