I am a graduate student who studies hydrodynamics. I want to solve hyperbolic 2-D shallow water equations (1 continuity eq, 2 momentum eqs) by Finite Volume Method using limiter with unstructued grid. For 1-D Saint-Venant equation, when flow is subcritical we specify 1 boundary condition (discharge or flow velocity) at upstream end and another one(water depth) is calculated from characteristic equation at upstream end. I got to know it becomes complicated when the problem is 2D. The characeristic line is no longer expressed in 2D plane. In 2D shallow water equation the unknowns are u, v, h (u,v=x,y velocity; h=water depth). For subcritical flow two variables(u, v) are known and one variable(h) us unknown at inflow boundary. How should I determine h ?

You might find the answer by looking at a few demonstration cases describing the main details of how PHOENICS , finite-volume code, has been taught to deal with Shallow Water Equations for the sub- and super-critical flows to simulate:

• <a href="http://www.simuserve.com/cfd-shop/uslibr/freesurf/fs4.htm">
  
  Water elevation in an open U-bend</a>
  
• <a href="http://www.simuserve.com/cfd-shop/uslibr/freesurf/fs5.htm">
  
  Whirlpool in a pond</a> and
  
• <a href="http://www.simuserve.com/cfd-shop/uslibr/freesurf/fs6.htm">
  
  Hydraulic jump</a>

ROSA, one of the PHOENICS Special Purpose Programes , makes use of this technique to simulate the dispersion of oil spills and other chemicals in natural rivers.
Regards
Sergei Zhubrin

Thanks for your reply. I visited PHOENICS sites that you said. But the demonstration cases also used fixed flux at inlet boundary,ie. all (3) boundary values are imposed at inlet boundary by Dirichlet type. Anyway, I greatly appreciate your answer.

Regards

Kang, S. K.

If one introduces the terms representing the elevation of bed above arbitrary horizontal datum as a momentum sources, the depths at the inlet and outlet planes can be calculated as a part of the solution without imposing the fixed inlet fluxes.
The corresponding changes of BC are straightforward to derive and implement.
Regards
Sergei Zhubrin

Then, can you please send me the derivations of the terms for super- and sub-critical case privately.

Thanks for your reply.

Regards

Kang, S. K.