Turbulent Channel Flow: Analytical Solutions of Navier-Stokes and Alexeev Hydrodynamic Equations with Experimental Validation

Abstract

Abstract

Understanding turbulent channel flow remains a central problem in fluid dynamics, particularly in relation to near-wall structures and scaling behavior. In this work, we investigate analytical solutions of turbulent channel flow using both the classical Navier-Stokes (NS) equations and the Alexeev hydrodynamic equations (AHE), derived from the Generalized Boltzmann Equation, which accounts for non-local effects and finite collision-time dynamics. Analytical solutions are obtained under the assumption that stationary states correspond to mean flow quantities. The AHE solutions are validated against numerical results and compared with NS predictions and experimental data from multiple studies spanning Reynolds numbers from 3×103 to 3.5×107. The AHE-based solutions show consistently improved agreement with experimental velocity profiles across this wide range, while the NS-based solutions exhibit systematic deviations. The analysis reveals a new similarity parameter associated with the boundary-layer thickness, providing additional structure to turbulent scaling. A reduced formulation for the transverse velocity component yields analytical solutions that are consistent with experimental observations of secondary flows. Furthermore, coupled numerical solutions for the streamwise and transverse velocities exhibit a family of kink-type structures, which are interpreted as representations of streamwise streaks. This provides a mechanistic link between transverse velocity dynamics and the formation of coherent near-wall structures. Overall, the results offer a unified framework for mean velocity profiles, secondary flows, and streak formation in turbulent channel flow, and suggest that non-local hydrodynamic effects play an important role in wall-bounded turbulence.