Restricted Euler dynamics in free-surface turbulence

The small-scale velocity gradient on the free surface is related to the large-scale properties of free-surface flows, such as the upwelling/downwelling motions from and into the fluid underneath and the transport along the free surface itself. The Lagrangian dynamics of the velocity gradient can be simplified and represented by a dynamic system using the restricted Euler model that neglects the viscous and nonlocal pressure Hessian terms in the Navier-Stokes equation. In this work, we derive the restricted Euler model for free-surface turbulence in the absence of surface deformation, and discuss the associated stable/unstable manifolds. The model is compared with the data collected on the free surface of a turbulent tank with negligible surface waves. We show that the joint probability density function of the velocity gradient invariants exhibits a distinct pattern, which differs from the one displayed by generic two-dimensional sections of three-dimensional turbulence and agrees with the predictions of the restricted Euler model. The latter, therefore, may be a powerful tool to examine the dynamics of free-surface turbulence.