Analytical model of flow velocity in gravel-bed streams under the effect of gravel array with different densities

Abstract

The standard exponential and logarithmic laws for flow velocity evaluation cannot be applied for gravel-bed streams due to the effect of large-scale roughness. An analytical model based on a two-layer subdivision approach was developed to predict flow velocity over a gravel array with different densities. The velocity distribution in the gravel layer was derived by solving the Darcy-Forchheimer-Brinkman (DFB) differential equation based on the concept of porous medium flows. The velocity in the surface layer was modeled by the mixing length theory with the length scale modified by taking into account the porous effect of the gravel (ball) arrays. To assess the model, flow velocities over gravel beds simulated by Ping-Pong ball arrays with different densities (or porosities) were measured in a laboratory flume. The results show that the predicted velocities by this subdivision model agree well with experimental data, reflecting that the porosity of the gravel array essentially impacts the flows both in the gravel layer and surface layer. In comparison with the existing models not considering the porous effect of the gravel bed, this model can effectively predict vertical velocity distributions and thereby flow resistances in gravel streams. The permeable interface velocity atop the gravel layer, as the boundary condition for the two-layer model, is sensitive to the bed context effect rather than the hydraulic effect, suggesting that the inflection point of the full velocity profile might be vital in determining the accuracy of the model. Moreover, with other data sources over dense gravel-bed, our model is suitable for data pre-dealt with the spatially averaging, where the spatial non-uniformity effect is great.