Second-order hydrodynamic equations for flows over mobile boundaries and their applications for solving environmental problems

Most environmental flows often exhibit high levels of spatial and temporal heterogeneity in hydrodynamic fields due to the effects of multi-scale roughness elements and their mobility. These effects are especially profound in the near-bed region and at low-submergence conditions typical for such flows. The conventional Reynolds-Averaged Navier-Stokes (RANS) equations are not suitable for these conditions and the double-averaged (in time and space) hydrodynamic equations have been increasingly used in recent years. Although these equations have become nearly a standard tool in many studies of fixed rough-bed flows their applications for mobile-bed conditions require theoretical clarification of bed mobility effects as well as their experimental assessment, particularly for higher-order equations. The goal of this paper is therefore twofold: (1) to review and refine the double-averaged hydrodynamic equations, particularly focusing on the second-order equations for mobile-bed conditions; and (2) to briefly discuss potential applications of these equations in treating mobile-bed flows such as gravel-bed rivers during active bed-load or flows with flexible in-stream vegetation. The key outcomes of this study include: (1) RANS for mobile-bed flows; (2) balance equations for spatially-averaged Reynolds stresses and form-induced stresses; and (3) preliminary suggestions for closure models. It is anticipated that the proposed equations will expand the hydraulics toolbox to better address multiple problems associated with flows over mobile boundaries. Examples include flow interactions with vegetated beds, moving gravel, fish schools, wind and tidal turbine farms.