Depth‐integrated nonhydrostatic free‐surface flow modeling using weighted‐averaged equations

Abstract

Summary In this study, a depth‐integrated nonhydrostatic flow model is developed using the method of weighted residuals. Using a unit weighting function, depth‐integrated Reynolds‐averaged Navier‐Stokes equations are obtained. Prescribing polynomial variations for the field variables in the vertical direction, a set of perturbation parameters remains undetermined. The model is closed generating a set of weighted‐averaged equations using a suitable weighting function. The resulting depth‐integrated nonhydrostatic model is solved with a semi‐implicit finite‐volume finite‐difference scheme. The explicit part of the model is a Godunov‐type finite‐volume scheme that uses the Harten‐Lax‐van Leer‐contact wave approximate Riemann solver to determine the nonhydrostatic depth‐averaged velocity field. The implicit part of the model is solved using a Newton‐Raphson algorithm to incorporate the effects of the pressure field in the solution. The model is applied with good results to a set of problems of coastal and river engineering, including steady flow over fixed bedforms, solitary wave propagation, solitary wave run‐up, linear frequency dispersion, propagation of sinusoidal waves over a submerged bar, and dam‐break flood waves.

Vertically Averaged and Moment Equations for Dam-Break Wave Modeling: Shallow Water Hypotheses

“…Many efforts are conducted, therefore, to increase the range of validity of the SWE by relaxing one or some of the shallow water hypotheses. In the river environment, these studies are rather recent [8][9][10]. However, there is a long tradition in maritime hydraulics working with higher order models accounting for a non-hydrostatic pressure distribution ( [11][12][13][14][15][16][17][18][19][20][21][22], among others).…”

“…A third family of vertically averaged models to simulate non-hydrostatic flows is based on the application of the weighted residual method to derive vertically averaged and moment equations from the RANS equations [43][44][45]. This third family of non-hydrostatic models is known as vertically averaged and moment (VAM) equations model [10]. The VAM model is a physical system of equations that is not yet really familiar to the scientific community.…”

Mathematical modeling of shallow-water flows on steep slopes

“…Ancey et al (2008), Cao et al (2015) and Fernandez-Feria (2006) applied the modified version of the Saint-Venant equations in local coordinates to model shallow water flows on uniform slopes. As establishing models in global coordinates can simplify the mathematical expressions of governing equations and the process of data handling, Denlinger and O'Connell (2008) followed the nonhydrostatic model proposed by Denlinger and Iverson (2004), which was then further developed by Cantero-Chinchilla et al (2017) and Castro-Orgaz and Hager (2017). On the other hand, Juez et al (2017), Van Emelen (2014) and Van Emelen et al (2014) built hydrodynamic models in global coordinates following Juez et al (2013).…”

Discussion of “Vertical 2D Nonhydrostatic Model Using Mode Splitting for Dam-Break Flows” by Yonghui Zhu and Dechao Hu

“…A detailed comparison with laboratory observations and 2D solutions of dam-break waves is accomplished to illustrate the previous points. Other models, like the vertically averaged and moment (VAM) equations could be alternatively used (Steffler and Jin 1993;Khan and Steffler 1996;Cantero-Chinchilla et al 2018).…”