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Jared Barber (jared_barber)
Member
Username: jared_barber

Post Number: 43
Registered: 01-2007
Posted on Tuesday, June 23, 2009 - 10:26 pm:   

Hello,

In the file "viscous", a way is presented to maintain incompressibility in the steady Stokes flow equations by using the equation:

div(grad(p)) = penalty*(dx(u)+dy(v))

The problem in the file "viscous" supposedly models an object/square blocking channel flow.

Introduction of the previously mentioned equation presents a problem. Normal Navier-Stokes flow only requires stipulation of 2 boundary conditions on the domain edge plus an additional global constraint to determine the pressure's unknown "integration constant". Now, however, we have to supply three boundary conditions along the domain edge. The question is, how do we best do that and why? In particular, I am interested in the specification of the pressure boundary conditions along the domain edge including the boundary conditions on the obstructing object/square.

Why, for instance, was load(p) = natural(p) = normal(grad(p)) = 0 used on the object? Hopefully I've applied the natural boundary conditions correctly here. If this boundary condition makes normal(grad(p)) = approximately 0, then this implies that normal(div(grad(u)),div(grad(v))) = approximately zero which I'm not sure should be true. (I set density to zero for ease.) If it is, it'd be great to know why. If it's not, it would be good to know why this boundary condition approximation is ok and how good of an approximation it is. I would also be interested in the justification for the other pressure boundary conditions in viscous.

If anyone has ideas about this, it'd be great to hear. Thanks,

Jared
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Jared Barber (jared_barber)
Member
Username: jared_barber

Post Number: 44
Registered: 01-2007
Posted on Wednesday, June 24, 2009 - 01:02 am:   

Also, I can't find any references that talk about this particular method. Any suggestions. Gresho and Sani don't seem to have it. Did I miss it? If anyone knows of any textbooks or papers that discuss the use of this method, that'd be great to have them, thanks.

Jared
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Robert G. Nelson (rgnelson)
Moderator
Username: rgnelson

Post Number: 1272
Registered: 06-2003
Posted on Tuesday, June 30, 2009 - 01:47 pm:   

1.
Prof. Backstrom in his book "Simple Fields of Physics by FEA" p267 (available from our bookstore link) follows Gresho and Sani (Intl J for Numerical Methods in Fluids, Vol 7, 1111-1145(1987) ) in defining
natural(P) = normal(BodyForce + visc*del2(V) - dens*(V.grad)V)
This method seems to be more ill-conditioned that natural(P)=0, but this depends on parameter relationships.

2.
Simple reorganization of the penalty equation gives
div(grad(P) - penalty*V) = 0
If you write the pressure equation this way, then
Natural(P) = penalty*normal(V)
Note that writing
div(grad(P)) = penalty*(dx(u)+dy(v))
does not cause FlexPDE to integrate the penalty term by parts, and therefore the penalty term should not be included in the BC.

3.
You can simply DEFINE P = -penalty*div(V), which embeds it in the momentum equations, and the boundary condition issue disappears. Sometimes this system is ill-conditioned and becomes harder to solve, but this depends on parameter relationships and I don't know a rule for determining how it will act. Try it and see.

I have not investigated all these issues, so I am not able to give any guidance on the matter.

In the cases I have tried, Natural(P)=0 produced results indistinguishable from method 1, but perhaps I didn't try a problem that was sensitive to the BC.


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Jared Barber (jared_barber)
Member
Username: jared_barber

Post Number: 45
Registered: 01-2007
Posted on Wednesday, July 15, 2009 - 01:50 pm:   

Thanks for the info, I will look into this.

Jared

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