RADINS equations for aerated shallow water flows over rough beds

Depth-averaged equations describing turbulent free-surface flows are usually derived assuming that the free surface is constant over a statistical ensemble of realizations. This work presents the derivation of the Reynolds-Averaged-Depth-Integrated Navier-Stokes (RADINS) equations obtained if this assumption is removed. These equations are applied to transient turbulent open-channel flow over a rough bed with an aerated free surface containing air bubbles. The RADINS equations contain terms which explicitly include the contribution of turbulent free-surface fluctuations to the large-scale volume and momentum transfer. These terms are evaluated in a laboratory experiment which involves a turbulent bore climbing up a rough slope. The new RADINS equations are compared with the conventional depth-integrated-Reynolds-averaged Navier-Stokes (DIRANS) equations. The RADINS and DIRANS equations are identical only for flows with negligible depth fluctuations. If this condition is not satisfied, the equations are different and the terms in the RADINS equations have a clearer physical meaning.