Two-dimensional vertically averaged and moment equations for shallow free-surface flows

The classical depth averaged de St. Venant equations, which are used for most of the computational models in open channels, are based on the fundamental assumptions of uniform velocity and hydrostatic pressure distributions. The depth averaging process used to derive de St. Venant equations neglects flow details over the vertical dimension to reduce computational effort. Thus, they are limited in their applicability to cases where vertical details are not of importance. Alternative two-dimensional vertically averaged and moment equations are developed to account for problems where more vertical details are significant and essential. These problems include flow cases with non-uniform velocity and non-hydrostatic pressure distributions.;The vertically averaged equations are derived by vertically integrating the fundamental three dimensional Reynolds equations, whereas the new moment equations required to solve for the extra degrees of freedom are derived by a moment weighted residual method from the same Reynolds equations. The equations are derived in a general way that can suit different shapes of velocity as well as pressure distributions. The derivation of the two-dimensional vertically-averaged and moment equations is presented in detail.;The implicit Petrov-Galerkin finite element scheme is applied in this study. Triangular elements with linear basis functions are used for all variables.;The vertically averaged and moment equations model is used to analyze a wide variety of hydraulic problems involved in open channel flow. These problems include flow in channel transitions with rapid contraction and/or expansion and flow in curved channels with different degrees of curvature. Linear and quadratic distribution shapes are proposed for the horizontal velocity components. In addition, quadratic vertical velocity and pressure distribution shapes are considered in these simulations.;The numerical model developed in this study is applied to two sets of experimental data. Computed values for water surface profile, depth averaged longitudinal and transverse velocities across the channel width and vertical profiles of longitudinal and transverse velocities are compared to the observed experimental data. For the rapidly varied flow simulations, the proposed model simulates the observed water surface profile slightly better than the conventional de St. Venant model. For the curved open channel simulations, this study recommends the replacement of the standard conventional de St. Venant model by the proposed model in terms of depth or vertically averaged modeling. This should be true for large-scale models where the generated numerical meshes are not very fine. This study also recommends that very fine finite element meshes, in which the numerical discretizations are of the order of the flow depth, be used when a high degree of accuracy of the predicted secondary flows near the channel walls is sought.;The present vertically averaged and moment model is found to be efficient, robust and converges to the correct solutions in a 2-D setting.