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

TL;DR

The paper tackles the insufficiency of Navier–Stokes descriptions for turbulent boundary layers by adopting the Alexeev Hydrodynamic Equations (AHE), derived from the Generalized Boltzmann Equation (GBE) which includes finite-particle-size effects via a second material derivative with timescale $\tau$. It develops both full AHE and simplified stationary models for 2D channel flow, obtaining an approximate analytical turbulent solution and validating it against numerical solutions and high-quality experiments across $Re$ from $3\times 10^3$ to $3.5\times 10^7$. The turbulent mean velocity is expressed as a superposition of laminar and turbulent components, $U(y)=U_{0}[\gamma(1-e^{1-e^{y/\delta}})+(1-\gamma)4y(L-y)/L^{2}]$, with $\gamma\approx 0.6$--$0.7$ and a boundary-layer scale $\delta$ that matches the Kolmogorov microscale, typically near $0.6$ mm for common fluids. Across channel and pipe experiments (Wei & Willmarth, Doorne, Zagarola et al.), the AHE predictions align with measurements far better than NS predictions, suggesting a robust framework for turbulence generation and potential control via transverse pressure and wall suction/injection mechanisms. This work thus links kinetic-scale physics to macroscopic turbulent behavior and offers practical insight into turbulence control strategies with small wall-normal actuation.

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.

Figures (6)

• Figure 1: (a) Two numerical solutions for the transverse velocity $V^T(y)$ in the channel at Re = 14914, computed using equation (\ref{['eq:2d_VT2a']}). The initial approximations were $sin(n\pi y)$, for $n = 2, 4$. The analytical solution (\ref{['eq:VTsol']}) is also shown (red line). (b) The solution for the streamwise turbulent velocity $U^T(y)$ from equation (\ref{['eq:2d_mom_UT2']}) (blue line), corresponding to the $V^T_1(y)$ solution (the only non-zero numerical solution), and the analytical solution $U^T(y)$ (\ref{['eq:UTsol']}) (red line) are shown.
• Figure 2: Comparison with Wei (1989) experiments at Re = 2970: the mean streamwise velocity (circles), the Navier-Stokes laminar solution $U^L$ for the streamwise velocity (parabolic function, green line), the turbulent $U^T$ solution the streamwise velocity (super-exponential function, blue line), and their superposition $U$, the solution of the Alexeev hydrodynamic equations. The solution $U$ fits the experimental data well. The vertical black line shows the value of $\delta$.
• Figure 3: Comparison of streamwise mean velocity from Wei (1989) experiments in (Y+, U+) coordinates at Re = 2970 (circles) with the Navier-Stokes laminar solution $U^L$ (parabolic solution, green line), THE turbulent solution $U^T$ (super-exponential solution, blue line), and their superposition $U$ (solution of the Alexeev hydrodynamic equations). The vertical dashed black line indicates $\delta$.
• Figure 4: Velocity profiles for turbulent flow in channel: the experimental data (dots) by Wei (1989) Wei_1989, analytical solution of the Navier-Stokes laminar solution (parabolic function, green line) $U^L$, turbulent $U^T$ solution (super-exponential function, blue line), and Alexeev hydrodynamic equations solution $U$ (red line), the superposition of $U^L$ and $U^T$, for Reynolds number Re=22776, that fits well the experimental data (red dots) for Re=22776. The log law by von Karman, $U^+ = 1/k~log (y^+) + B$ (log law) is shown as olive-green line.
• Figure 5: Comparison of experimental data for streamwise mean velocity $U_{exp}=u/U_0$ versus radius $r/D$ ($D$ is the diameter) (blue circles) for the van Doorne experiment Doorne_2007 at Re=7200 with the laminar (parabolic) solution $U^L$ of the Navier-Stokes equations (green line), the turbulent (super-exponential) solution $U^T$ (blue line), and the solution $U$ of the Alexeev hydrodynamic equations (red line). The analytical solution $U$ fits the experimental velocity profile well, while other solutions deviate significantly from the data.