Steady Homogeneous Turbulence in the Presence of an Average Velocity Gradient

TL;DR

This paper investigates homogeneous turbulence with a constant average velocity gradient, revealing a different energy spectrum and anomalous scaling laws, using a novel finite-scale Lyapunov analysis and extending classical turbulence equations.

Contribution

It introduces a new analysis of turbulence with velocity gradients, deriving an evolution equation valid in rotating frames and showing deviations from classical Kolmogorov scaling.

Findings

Energy spectrum varies as κ^{-2} instead of κ^{-5/3}

Structure functions exhibit anomalous scaling ζ_n ≈ n/2

Correlation scales are smaller than in isotropic turbulence

Abstract

We study the homogeneous turbulence in the presence of a constant average velocity gradient in an infinite fluid domain, with a novel finite-scale Lyapunov analysis, presented in a previous work dealing with the homogeneous isotropic turbulence. Here, the energy spectrum is studied introducing the spherical averaged pair correlation function, whereas the anisotropy caused by the velocity gradient is analyzed using the equation of the two points velocity distribution function which is determined through the Liouville theorem. As a result, we obtain the evolution equation of this velocity correlation function which is shown to be valid also when the fluid motion is referred with respect to a rotating reference frame. This equation tends to the classical von K\'arm\'an-Howarth equation when the average velocity gradient vanishes. We show that, the steady energy spectrum, instead of following the Kolmogorov law $\kappa^{-5/3}$, varies as $\kappa^{-2}$. Accordingly, the structure function of the longitudinal velocity difference $<\Delta u_r^n> \approx r^{\zeta_n}$ exhibits the anomalous scaling $\zeta_n \approx n/2$, and the integral scales of the correlation function are much smaller than those of the isotropic turbulence.