Modeling flows over gravel beds by a drag force method and a modified S–A turbulence closure

A double-averaged Navier–Stokes equations (DANS) model has been developed for depth-limited open channel flows over gravels. Three test cases are used to validate the model: an open-channel flow over a densely packed gravel bed with small-scale uniform roughness (D/d50 ∼ 13, d50 = median diameter of roughness elements, D = water depth), open-channel flows over large-scale sparsely distributed roughness elements (D/Δ ∼ 2.3–8.7, Δ = roughness height) and steep slope gravel-bed river flows with D/d50 ∼ 7–25. Various methods of treatment of the gravel-induced resistance effect have been investigated. The results show that the wall function approach (WFA) is successful in simulating flows over small gravels but is not appropriate for large gravels since the vertical profile of the longitudinal velocity does not follow the logarithmic–linear relationship. The drag force method (DFM) performs better but the non-logarithmic velocity distribution generated by sparsely distributed gravels cannot be simulated accurately. Noting that the turbulence length scale within the gravel layer is governed by the gravel size, the DANS model incorporating the DFM and a modified Spalart–Allmaras (S–A) turbulence closure is proposed. The turbulence length scale parameter in the S–A model is modified to address the change in the turbulence structure within the gravel layer. The computed velocity profiles agree well with the corresponding measured profiles in all cases. Particularly, the model reproduces the S-shape velocity profile for sparsely distributed large size roughness elements. The modeling methodology is robust and can be easily integrated into the existing numerical models.