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Arbitrary Choosing of the Solution Domain  Navier Stokes and Manufactured Solutions 

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January 11, 2015, 10:33 
Arbitrary Choosing of the Solution Domain  Navier Stokes and Manufactured Solutions

#1 
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Fabian Gabel
Join Date: Oct 2014
Location: Darmstadt
Posts: 13
Rep Power: 12 
I want to verify a finitevolume solver (SIMPLEAlgorithm) for the incompressible NavierStokes equations by using a manufactured solution. I use Dirichlet boundary conditions for the velocity at all boundaries. The manufactured solution for the velocities I use, is constructed as the curl of a vector field and thereby fulfills the continuity equation at any point.
Depending on how I choose the domain on which to solve the equations I run into convergence problems: The residual of the pressure correction equation stagnates after an initial decrease. I am approximating the mass fluxes through boundary faces using the midpoint rule and apparently this leads to a net loss/gain of mass considering the entire problem domain. Is this the normal behavior? Are there any remedies other than choosing a manufactured solution that identically vanishes at the boundaries? Would it be an option to provide the exact mass flux instead of the approximate (not sure if this is allowed, since the Dirichlet boundary condition only fixes the velocity)? 

January 11, 2015, 15:53 

#2 
Senior Member

The principle underlying the manufactured solution approach to the verification is that, for a given solver, you should be able to normally setup the problem according to the known solution (thus, the appropriate set of Dirichlet/Neumann b.c. and source terms, when encessary).
If your solver does not produce the expected results than there is something wrong in it. Is this an inhouse code or some commercial package? 

January 11, 2015, 16:54 

#3 
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Fabian Gabel
Join Date: Oct 2014
Location: Darmstadt
Posts: 13
Rep Power: 12 
The Solver I use is neither commercial nor inhouse  it is a modification of caffa I implemented for my masters thesis.
Apparently the global mass balance is not fulfilled when the boundary mass fluxes aren't calculated precise enough or the manufactured solution doesn't vanish on the boundaries. On the other side I was able to create solutions that are symmetric in the sense, that the mass fluxes on different sides of the problem domain complement each other, leading to a zero net mass flux over the domain boundaries. I think a higher order of integration for mass fluxes at the boundaries might help. Note that I already verified the solver on problem domains that have the afore mentioned beneficent properties. Unfortunately I don't have the ability and the time to confirm my suspicion by using another cfd tool. I was hoping someone of the community already had stumbled upon this exact problem and could share some insight. 

January 11, 2015, 19:38 

#4 
Senior Member

I am not very confident on the Peric codes but, is it possible that this is a staggered grid issue? I can't see why a 0 boundary value should be different from any other numerical value as long as the overall boundary integration still results in mass conservation.


January 12, 2015, 04:20 

#5  
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Fabian Gabel
Join Date: Oct 2014
Location: Darmstadt
Posts: 13
Rep Power: 12 
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This effect can be weakened by using a finer grid (since my discretization is consistent) or by using a higher order of integration at the boundaries. 

January 12, 2015, 08:35 

#6 
Senior Member

I edited the post because i finally got your issue.
If i got it correctly, your manufactured solution is such that it is locally divergence free but it preserves global continuity only for certain domain configurations. Obviously what you are doing cannot be done and there certainly is a problem with either your manufactured solution or the domain over which you are solving for it. The domain you can use must be such that the manufactured solution preserves the global continuity. This, at least, for incompressible solvers. It is a mathematical constraint for the well posedness of the pressure equation. 

Tags 
boundary conditions, mass conservation, navierstokes, simple algorithm 
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