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Hi，everyone

There is a question that confused me. The eauqtion solved in telemac2d is non-conservative Saint-Venant equation in the depth-velocity form:
(dT/dt)+(u.grad(T))-(1/h.div(h.vt.grad(T)))=0
So the diffusion term is 1/h.div(h.vt.grad(T))

But when OPTION FOR THE DIFFUSION OF TRACERS=1,the diffusion term is div(vt*grad(T))
but when OPTION FOR THE DIFFUSION OF TRACERS=2,the diffusion term is 1/h*div(h*vt*grad(T))

I do not understand about it.

zqzuoan

Hello,

Yes, only option 1 is correct in theory. Option 2 is a simplification that removes the problem of dry zones. However actually the diffusion in Shallow water equations also includes the dispersion due to the heterogeneity of velocities on the vertical, so it is in fact also impossible to state that option 1 is fully correct (sometimes the dispersion is even treated as an extra advection). It is a clear example showing that simplifying equations (by depth-averaging) brought unexpected extra problems and we do not have this problem in 3D.

With best regards,

Jean-Michel Hervouet

Thank you, sir

You mean that, If I chose OPTION FOR THE DIFFUSION OF TRACERS=1, the eauqtion solved in telemac2d is conservative Saint-Venant equation. And if OPTION FOR THE DIFFUSION OF TRACERS=2, the eauqtion solved in telemac2d is non-conservative Saint-Venant equation in the depth-velocity form. Is it right?

zqzuoan

Hello,

In fact option 2 is not even the non conservative form, it is just an approximation without justification, that is however true if the depth varies slowly in space.

With best regards,

Jean-Michel Hervouet

Thank you, sir

So, I had better chose option 1 if I consider the tidal flats? And the equation solved in option 1 is :
(dhT/dt)+div(hTu)-div(h.vt.grad(T)))=0

That is right?

zqzuoan

Hello,

Actually not, in fact option 1 or 2 only changes the diffusion term, we always solve the equation in the form (dT/dt)+... or, to explain more in detail, we discretise the equation (d(hT)/dt)+... and transform it into (dT/dt)+... after discretisation, so that it is mathematically equivalent, and this gives us a mass conservation proof (except for the diffusion term...).

Regards,

JMH

OK, the equation solved in telemac2d is non-conservative Saint-Venant equation in the depth-velocity form, right? In option 1, the diffusion term is div(h*vt*grad(T)), so the equation is h*(dT/dt)+h(u.grad(T))-div(h*vt*grad(T))=0. In option 1,the diffusion term is 1/h*div(h*vt*grad(T)), the equation is divided by h， that is right?

zqzuoan

Hello,

Actually :

option 1 : dT/dt+u.grad(T)-div(vt*grad(T))=0

option 2 : dT/dt+u.grad(T)-(1/h)div(h*vt*grad(T))=0

Regards,

JMH

Soryy, I have not understand yet. Where is the h in option 1...the equation solved in telemac2d is non-conservative Saint-Venant equation in the depth-velocity form, shouldn't it be: h*dT/dt+h*u*grad(T)-div(h*vt*grad(T))=0 ?

zqzuoan

Hello,

and the equation h*dT/dt+h*u*grad(T)-div(h*vt*grad(T))=0 is divided by h to give: dT/dt+u*grad(T)-(1/h)*div(h*vt*grad(T))=0.

JMH