I am looking for any book/article reference about the mathematical description of zero normal flux boundary condition for shallow water equations. My concern is that for a near-shore case how it is obvious to have zero normal flux. Physically, it does make sense that we have a near-shore case and on the boundary, there is no flow in the normal direction. How to mathematical explain it using the continuity equation in the case when there is a steady flow? The continuity equation suggests that $\partial h / \partial t + u. \partial h/ partial x = 0$. If we take steady flow then it is clear to me to get zero normal flux condition. But what if the first term is not zero? or do we say that at the boundary the flow is always steady?
Hi, I am not sure to understand your question. Continuity equation gives $\partial h/ \partial t + \partial (hu)/ \partial x =0$ , and zero normal flux corresponds to $u=0$ on the boundary, then we have $\partial h/ \partial t + h \partial u/ \partial x =0$. And, of course, flow is not always steady at the boundary (I assume that we considere only the two dimensional (x,z) direction, else, of course, tangencial component has more degree of freedom).
Bernard Di Martino Specifying my question, how do we get zero normal flux from the continuity equation? I read in the mentioned below article that if we take steady flow, then \partial h / \ partial t becomes zero, which gives the zero flux condition. But my concern is that what if the flow is non-steady, how do we derive zero normal flux condition from the continuity equation in that case? For reference to the article please see section 4.2.3 (the kinematic boundary condition) https://personalpages.manchester.ac.uk/staff/matthias.heil/Lectures/Fluids/Material_2019/Chapter4.pdf
Shallow Water Hydrodynamics: Mathematical Theory and Numerical Solution for a Two-dimensional System of Shallow Water Equations
ElsevierTan Weiyan (Eds.)Year:1992
Lattice Boltzmann Methods for Shallow Water Flows
Springer-Verlag Berlin HeidelbergDr. Jian Guo Zhou (auth.)Year:2004
Numerical Methods for Shallow-Water Flow
Springer NetherlandsC. B. Vreugdenhil (auth.)Year:1994
Okay, I better understand your question that concerns the conditions on the surface of the fluid. The approach is normally the opposite. A kinematic boundary conditions are imposed on surface that reflect the impermeability of the surface (equation 4.5 in the paper) and this condition appear naturaly we when we take the integral between the bottom to the surface of the incompressibility equation div(U) (where U=(u,v,w) is the 3d velocity). Note that we need to use the Leibtiz formulae to take the integrale since the boundary is a function of x.
Bernard Di Martino Thank you very much for your explanation. I wrote explicitly in mathematical form in the attached file, what I got from your explanation. Could you please have a look at it? It will be helpful to have a second opinion.
Maged Gumaan Bin-Saad Thank you very much for your suggestion. This book is indeed one of the best I have studied so far on shallow water equations and boundary conditions.
I think you're confusing a lot of things. In the paper you sent, it's about the impermeability of the surface on the non integrated model. This condition is used to obtain the shallow water mode. See the file SW_model.jpg. You equation 158 is not the usual continuity equation since. The notation 2 D model is not clear. We can considère a 3d model (u,v,w), that gives the 2d model (\bar u, \bar v), or the 2d model (u,w) that gives the 1d model (\bar u). But if the non integrated model has an incompressible hypothesis, the integrated model (for me, the shallow water model) need to verify equation (2) and not your equation (158). From (2), if the lateral boundaries are wall, the integral on the domain gives the conservation of the total mass of water (since \int_\omega (\d (hu)\dx) dx = 0, the \d (\int_\omega h dx) \dt =0).
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