A What are the boundary conditions for rotational flow?

[surfwavesfreak]

Hello everyone,
The boundary condition :
P=0, z=ζ
is very common when studying irrotational flows. When cast with the Bernoulli equation, it gives rise to the famous dynamic boundary conditionn, which is much more convenient :
∂_tφ+½(∇φ)²+gζ=0, z=ζ
But what happens if the motion is rotational ? What would be the analog of the dynamic BC ?
This condition is more complex than it seems ...

Thanks a lot !

 

[Simon Bridge]

Have you seen:
http://www.bakker.org/dartmouth06/engs150/06-bound.pdf

 

[surfwavesfreak]

Thanks for your link, but I did not see any equation like
∂tφ+½(∇φ)2+gζ=0, z=ζ
where the pressure is actually removed from the variables.
Any idea ?

 

[Simon Bridge]

What makes you think there should be one?
The point of the suggestion was to hep you understand how to apply boundary conditions for the situation that flow may be rotational.
Once you can understand that, then you can approach your question.

 

[surfwavesfreak]

I was thinking that may be you could cast p=0, z=ζ and the navier stokes equations :
∂_tu_i+u_j∂_xju_i=-∂_xip/ρ+gδ_iz
which I assumed to be valid everywhere, especially at z=ζ
As p=0, ∂xip=0 as well, and you are left with :
∂_tu_i+u_j∂_xju_i=gδ_iz, z=ζ
but I've never seen that anywhere, and I think there may be something wrong somewhere ...

 

[Simon Bridge]

You need to motivate your boundary conditions from the physics you are trying to model.

 

[surfwavesfreak]

Yes you are right. Following what you said, may be a very straightforward boundary condition would be :
∇p×∇ζ=0, z=ζ
as the pressure is constant along the surface, its gradient should always be directed ortohogonally to the surface of the fluid.
Then you get an equation that you can easily cast with the momentum equations (through the pressure gradient).

 

[Simon Bridge]

You don't start with the boundary conditions you want, you start with the physical system and deduce what would count as reasonable boundary conditions.
Start with the specific kind of rotational flow you want to model. How does it arise? What boundary conditions will be consistent? Whatever you get out is the model for that situation under the assumptions you made. That's the best you can do.

Note: you can set BCs, and then ask: what sort of flow has those boundary conditions ... but that is not what you were doing either.