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Why we subtract continuity Eqn into momentum Eqn instead of adding it? 

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April 27, 2015, 03:14 
Why we subtract continuity Eqn into momentum Eqn instead of adding it?

#1 
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Dongyue Li
Join Date: Jun 2012
Location: Torino, Italy
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Hey guys,
This piece of code is quite common in OpenFOAM； Code:
fvm::ddt(alpha1, U1) //mom eqn fvm::Sp(fvc::ddt(alpha1),U1) //continuity eqn + fvm::div(alpha1, U1) //mom eqn  fvm::Sp(fvc::div(alpha1), U1) ////continuity eqn Code:
fvm::ddt(alpha1, U1) //mom eqn +fvm::Sp(fvc::ddt(alpha1),U1) //continuity eqn + fvm::div(alpha1, U1) //mom eqn +fvm::Sp(fvc::div(alpha1), U1) ////continuity eqn 

April 27, 2015, 05:38 

#2 
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ali alkebsi
Join Date: Jan 2012
Location: Strasbourg, France
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I dont really know
normally we seek diagnal dominance, so i would say so but thats the contrary of what is done here ?????? 

April 27, 2015, 11:31 

#3 
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Dongyue Li
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Location: Torino, Italy
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Yep...Wish someone who can explain this. lol


April 27, 2015, 12:32 

#4 
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Cyprien
Join Date: Feb 2010
Location: Stanford University
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Hi,
Actually, we subtract a momentum transfer term because the real equation (left hand side of NS) we want to solve is which is written in a nonconservative form. In FVM, however, we prefer conservative form, These two equations are not the same if the fluid is not incompressible or if there is phase change. You can notice that you can switch from nonconservative to conservative form with, So adding the continuity equation do not have any sense. Cheers, 

April 28, 2015, 02:21 

#5 
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Dongyue Li
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Quote:
Yeah, The problem is why we want to solve the first Eqn in your posts. The reason why I ask this kind of problem is that: "openfoam22x is solving nonconservative equations but 23x is solving conservative eqns."for twofluid model. So I think the thing deeper is : 1. Why we solve Eqn 2 instead of Eqn 1. I saw some posts that Hrv and Henry explain this, They just said the conservative eqn is stable and good. I dont know the reason. 2. Eqn 2 and Eqn 1 should predicts the same results rite? But sometimes it does not. Best, 

April 28, 2015, 11:39 

#6 
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Cyprien
Join Date: Feb 2010
Location: Stanford University
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Hi Dongyue,
Eq 1 and Eq 2 are equivalent only if the continuity equation (equal to zero) is satisfy. I guess that since we deal with an operatorsplitting scheme (first we solve the momentum, then the pressure, then the correction, then we iterate), the continuity equation is not fully respected, and that why we add to add this term, to avoid spurious momentum transfer due to an illevaluation of the continuity. Cheers, 

June 3, 2015, 12:00 

#7  
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Bruno Blais
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Location: Canada
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Quote:
Equation 1 and 2 are not equivalent numerically because they do not allow for the same Riemann invariant. If you take the example of a shockwave or any other type of discontinuity, as a CauchyRiemann classical problem, you find that the wave propagation / shock propagation/ discontinuity propagation will be at the wrong velocity if you do not formulate your problem using the conserved variables. I am not familiar with these issues in the context of two phases flows, but for compressible flows this is an extremely critical issue and you always need to formulate the problem in conservative variables (rho, rho * u, rho * e). I believe that since these multiphase flows contain large amount of discontinuity in the phases (like air on top of water or whatever), a conservative formulation is a lot more appropriate if you want to obtain the right propagation of the interface. There is a very very nice book by Euleterio Toro on hyperbolic systems that discusses these issues if I remember. 

June 4, 2015, 05:20 

#8 
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Daniel Witte
Join Date: Nov 2011
Posts: 139
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Hi,
I am confused with these statements. Sharonyue refers to the alpha1Eq, not the momentum equation. In interDymFoam, I have the following code: Code:
fvScalarMatrix alpha1Eqn ( #ifdef LTSSOLVE fv::localEulerDdtScheme<scalar>(mesh, rDeltaT.name()).fvmDdt(alpha1) #else fv::EulerDdtScheme<scalar>(mesh).fvmDdt(alpha1) #endif + fv::gaussConvectionScheme<scalar> ( mesh, phiCN, upwind<scalar>(mesh, phiCN) ).fvmDiv(phiCN, alpha1) ); I changed to: Code:
fvScalarMatrix alpha1Eqn ( #ifdef LTSSOLVE fv::localEulerDdtScheme<scalar>(mesh, rDeltaT.name()).fvmDdt(alpha1) #else fv::EulerDdtScheme<scalar>(mesh).fvmDdt(alpha1) #endif + fv::gaussConvectionScheme<scalar> ( mesh, phiCN, upwind<scalar>(mesh, phiCN) ).fvmDiv(phiCN, alpha1)  fvc::div(phi)*fvm::Sp(1,alpha1) ); By the way, for compressible solvers, this code is wrong (div phi is not zero, but div rho phi. Regards, Daniel 

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