Surface flow formula (Water Overlay)

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Imbalances in water levels across the grid drive the flow of water until a state of equilibrium is reached in terms of h (the height of the water column) and flux. Behavior of the flow is described by a second-order semi-discrete central-upwind scheme produced by Kurganov and Petrova (2007)[1], which is based on the 2-D Saint-Venant equations (a.k.a. shallow water equations):

{\displaystyle {\begin{aligned}{\frac {\partial h}{\partial t}}&+{\frac {\partial (hu)}{\partial x}}+{\frac {\partial (hv)}{\partial y}}=0,\\[3pt]{\frac {\partial (hu)}{\partial t}}&+{\frac {\partial }{\partial x}}\left(hu^{2}+{\frac {1}{2}}gh^{2}\right)+{\frac {\partial (huv)}{\partial y}}=-gh{\frac {\partial B}{\partial x}}-ghn^{2}u{\sqrt {u^{2}+v^{2}}}h^{-{\frac {4}{3}}},\\[3pt]{\frac {\partial (hv)}{\partial t}}&+{\frac {\partial (huv)}{\partial x}}+{\frac {\partial }{\partial y}}\left(hv^{2}+{\frac {1}{2}}gh^{2}\right)=-gh{\frac {\partial B}{\partial y}}-ghn^{2}v{\sqrt {u^{2}+v^{2}}}h^{-{\frac {4}{3}}},\end{aligned}}}

where

 u is the velocity in the x-direction v is the velocity in the y-direction h is the water depth B is the bottom elevation g is the acceleration due to gravity n is the Gauckler–Manning coefficient

References

1. Kurganov A, Petrova G (2007) ∙ A Second-Order Well-Balanced Positivity Preserving Central-Upwind Scheme for the Saint-Venant System ∙ found at: http://www.math.tamu.edu/~gpetrova/KPSV.pdf (last visited 2019-04-11)