Anisotropy and Anomalous Scaling in Steady Homogeneous Turbulence in the Presence of an Average Velocity Gradient

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. The anisotropy caused by the velocity gradient is studied 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 the longitudinal velocity difference 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 ...