DPMFoam pressure poisson and momentum corrector

bjnieuwboer · March 4, 2019, 10:41

Dear Foamers,

I am writing out the equations for solving the fluid motion as used in DPMFoam and I came to what I thought to be an inconsistency. Can someone help me by pointing out the error I am making in the derivation of the pressure Poisson equation and momentum corrector for DPMFoam? In the derivation I am leaving out the forces of the particles on the fluid for simplicity. I am deriving the pressure poisson and the momentum corrector from two starting points. But I cannot get them matched up with the code.

Method A:
Start with the update formulation for the fluid:

$\alpha_f \boldsymbol{u_c} - \alpha_f \boldsymbol{H} \boldsymbol{A}^{-1} = -\boldsymbol{A}^{-1} \nabla P$
The continuity equation is used to derive the pressure poisson equation

$\frac{\partial \alpha_c }{\partial t} + \nabla \cdot \left( \alpha_c \boldsymbol{u_c} \right)$
These two equations are combined and lead to the wrong pressure poisson equation:
$\nabla \cdot \left( \boldsymbol{A}^{-1} \nabla P \right) =\frac{\partial \alpha_c}{\partial t} + \nabla \cdot \left( \alpha_c \, \boldsymbol{H} \boldsymbol{A}^{-1} \right)$
The momentum predictor is right:
$\boldsymbol{u_c} = \boldsymbol{H} \boldsymbol{A}^{-1} - \frac{1}{\alpha_f} \boldsymbol{A}^{-1} \nabla P$

Method B:
Start with the update formulation without the concentration:
$\boldsymbol{u_c} - \boldsymbol{H} \boldsymbol{A}^{-1} = -\boldsymbol{A}^{-1} \nabla P$
The same continuity equation is used to derive the pressure poisson equation

The two equations are combined and lead to the right pressure poisson equation:
$\nabla \cdot \left( \alpha_c \, \boldsymbol{A}^{-1} \nabla P \right) =\frac{\partial \alpha_c}{\partial t} + \nabla \cdot \left( \alpha_c \, \boldsymbol{H} \boldsymbol{A}^{-1} \right)$
However, the momentum predictor is wrong:
$\boldsymbol{u_c} = \boldsymbol{H} \boldsymbol{A}^{-1} - \boldsymbol{A}^{-1} \nabla P$

At method A the pressure poisson equation is wrong compared with the code and in method B the momentum corrector is wrong. I cannot find literature on the derivation of the pressure Poisson for a fluid with a void fraction. Also in the really nice document of Jeroen Hofman, this is not addressed. Can someone help me address this issue?

March 4, 2019, 10:41	DPMFoam pressure poisson and momentum corrector	#1
bjnieuwboer Member Bas Nieuwboer Join Date: Mar 2013 Posts: 32 Rep Power: 13	Dear Foamers, I am writing out the equations for solving the fluid motion as used in DPMFoam and I came to what I thought to be an inconsistency. Can someone help me by pointing out the error I am making in the derivation of the pressure Poisson equation and momentum corrector for DPMFoam? In the derivation I am leaving out the forces of the particles on the fluid for simplicity. I am deriving the pressure poisson and the momentum corrector from two starting points. But I cannot get them matched up with the code. Method A: Start with the update formulation for the fluid: $\alpha_f \boldsymbol{u_c} - \alpha_f \boldsymbol{H} \boldsymbol{A}^{-1} = -\boldsymbol{A}^{-1} \nabla P$ The continuity equation is used to derive the pressure poisson equation $\frac{\partial \alpha_c }{\partial t} + \nabla \cdot \left( \alpha_c \boldsymbol{u_c} \right)$ These two equations are combined and lead to the wrong pressure poisson equation: $\nabla \cdot \left( \boldsymbol{A}^{-1} \nabla P \right) =\frac{\partial \alpha_c}{\partial t} + \nabla \cdot \left( \alpha_c \, \boldsymbol{H} \boldsymbol{A}^{-1} \right)$ The momentum predictor is right: $\boldsymbol{u_c} = \boldsymbol{H} \boldsymbol{A}^{-1} - \frac{1}{\alpha_f} \boldsymbol{A}^{-1} \nabla P$ Method B: Start with the update formulation without the concentration: $\boldsymbol{u_c} - \boldsymbol{H} \boldsymbol{A}^{-1} = -\boldsymbol{A}^{-1} \nabla P$ The same continuity equation is used to derive the pressure poisson equation The two equations are combined and lead to the right pressure poisson equation: $\nabla \cdot \left( \alpha_c \, \boldsymbol{A}^{-1} \nabla P \right) =\frac{\partial \alpha_c}{\partial t} + \nabla \cdot \left( \alpha_c \, \boldsymbol{H} \boldsymbol{A}^{-1} \right)$ However, the momentum predictor is wrong: $\boldsymbol{u_c} = \boldsymbol{H} \boldsymbol{A}^{-1} - \boldsymbol{A}^{-1} \nabla P$ At method A the pressure poisson equation is wrong compared with the code and in method B the momentum corrector is wrong. I cannot find literature on the derivation of the pressure Poisson for a fluid with a void fraction. Also in the really nice document of Jeroen Hofman, this is not addressed. Can someone help me address this issue?

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