Different Pressure Equation Formulation!

godfatherBond · April 20, 2018, 10:17

Hi OF Community Members,
Greetings!

I need your expertise with the following problem.

I was going through Prof. Jasak's paper "Multi-dimensional simulation of thermal non-equilibrium channel flow!" where a different type of pressure equation was formulated as opposed to what is used in the variable density/compressible solvers of OpenFOAM.
https://www.sciencedirect.com/scienc...0193220900192X

In it the compressible/multiphase continuity is split from
$\frac{\partial \bar{\rho} }{\partial t} + \nabla \rho \tilde{U} = 0$

to

$\nabla \cdot \tilde{U} = -\frac{1}{\bar{\rho}} \frac{D \bar{\rho}}{Dt}$

Now considering compressible flow only $\bar{ \rho} = \bar{\rho} (\bar{p})$ .

Hence material derivative is expanded as
$\nabla \cdot \tilde{U} = -\frac{1}{\bar{\rho}} \frac{D \bar{\rho}}{Dt}=-\frac{1}{\bar{\rho}} \frac{\partial \bar{\rho}}{\partial \bar{p}} \frac{D \bar{p}}{D t}= -\frac{\psi}{\bar{\rho}} \big( \frac{\partial \bar{p}}{\partial t} + \nabla(\tilde{U} \bar{p}) - \bar{p}\nabla \cdot \tilde{U} \big)$ .

It would be really helpful if you could share some thoughts on the following questions:

1.For a variable density flow with face flux as the Mass flux $\rho U$ at face, the above equation will have volumetric flux phi, which is to be used to find mass flux by interpolation of density to the faces.
Where should this in a (say) compressible solver be done( just after phi update or elsewhere)?

2.If we treat the pressure terms on the RHS i.e $\nabla(\tilde{U} \bar{p})$ , temporal term and source term $\bar{p}\nabla \cdot \tilde{U}$ so as to impart numerical stability, then following phi = phi - pEqn.Flux() How will one update the volumetric fluxes as the pEqn now has a new convective term on the RHS following Dr.Jasak Thesis?

3.How is this kind of formulation different(physics wise) from that used in the regular openfoam solvers i.e. $\psi \frac{\partial \bar{p}}{\partial t} + \nabla \cdot (\bar{\rho} \vec{U}) = 0$

Note: The $\tilde {}$ are used for a Favre averaged framework. Also the $\nabla \cdot U$ is treated as done in incompressible solvers in OF.

Correct me if I am wrong anywhere!
Thanks in advance

godfatherBond · April 26, 2018, 06:52

Quote:

Originally Posted by godfatherBond

Hi OF Community Members,

I was going through Prof. Jasak's paper "Multi-dimensional simulation of thermal non-equilibrium channel flow!" where a different type of pressure equation was formulated as opposed to what is used in the variable density/compressible solvers of OpenFOAM.
https://www.sciencedirect.com/scienc...0193220900192X

In it the compressible/multiphase continuity is split from
$\frac{\partial \bar{\rho} }{\partial t} + \nabla \rho \tilde{U} = 0$

to

$\nabla \cdot \tilde{U} = -\frac{1}{\bar{\rho}} \frac{D \bar{\rho}}{Dt}$

Now considering compressible flow only $\bar{ \rho} = \bar{\rho} (\bar{p})$ .

Hence material derivative is expanded as
$\nabla \cdot \tilde{U} = -\frac{1}{\bar{\rho}} \frac{D \bar{\rho}}{Dt}=-\frac{1}{\bar{\rho}} \frac{\partial \bar{\rho}}{\partial \bar{p}} \frac{D \bar{p}}{D t}= -\frac{\psi}{\bar{\rho}} \big( \frac{\partial \bar{p}}{\partial t} + \nabla(\tilde{U} \bar{p}) - \bar{p}\nabla \cdot \tilde{U} \big)$ .

My questions are:

1.For a variable density flow with face flux as the Mass flux $\rho U$ at face, the above equation will have volumetric flux phi, which is to be used to find mass flux by interpolation of density to the faces.
Where should this in a (say) compressible solver be done( just after phi update or elsewhere)?

2.If we treat the pressure terms on the RHS i.e $\nabla(\tilde{U} \bar{p})$ , temporal term and source term $\bar{p}\nabla \cdot \tilde{U}$ so as to impart numerical stability, then following phi = phi - pEqn.Flux() How will one update the volumetric fluxes as the pEqn now has a new convective term on the RHS following Dr.Jasak Thesis?

3.How is this kind of formulation different(physics wise) from that used in the regular openfoam solvers i.e. $\psi \frac{\partial \bar{p}}{\partial t} + \nabla \cdot (\bar{\rho} \vec{U}) = 0$

Note: The $\tilde {}$ are used for a Favre averaged framework. Also the $\nabla \cdot U$ is treated as done in incompressible solvers in OF.

Correct me if I am wrong anywhere!
Thanks in advance

godfatherBond · June 13, 2018, 23:05

Hi FOAMers,
Still looking for an explanation.

It would be beneficial if you could spare some time and have a look into it.

Thanks

April 20, 2018, 10:17	Different Pressure Equation Formulation!	#1
godfatherBond Member Maximus Arelius Join Date: Jan 2017 Location: Morocco Posts: 36 Rep Power: 9	Hi OF Community Members, Greetings! I need your expertise with the following problem. I was going through Prof. Jasak's paper "Multi-dimensional simulation of thermal non-equilibrium channel flow!" where a different type of pressure equation was formulated as opposed to what is used in the variable density/compressible solvers of OpenFOAM. https://www.sciencedirect.com/scienc...0193220900192X In it the compressible/multiphase continuity is split from $\frac{\partial \bar{\rho} }{\partial t} + \nabla \rho \tilde{U} = 0$ to $\nabla \cdot \tilde{U} = -\frac{1}{\bar{\rho}} \frac{D \bar{\rho}}{Dt}$ Now considering compressible flow only $\bar{ \rho} = \bar{\rho} (\bar{p})$ . Hence material derivative is expanded as $\nabla \cdot \tilde{U} = -\frac{1}{\bar{\rho}} \frac{D \bar{\rho}}{Dt}=-\frac{1}{\bar{\rho}} \frac{\partial \bar{\rho}}{\partial \bar{p}} \frac{D \bar{p}}{D t}= -\frac{\psi}{\bar{\rho}} \big( \frac{\partial \bar{p}}{\partial t} + \nabla(\tilde{U} \bar{p}) - \bar{p}\nabla \cdot \tilde{U} \big)$ . It would be really helpful if you could share some thoughts on the following questions: 1.For a variable density flow with face flux as the Mass flux $\rho U$ at face, the above equation will have volumetric flux phi, which is to be used to find mass flux by interpolation of density to the faces. Where should this in a (say) compressible solver be done( just after phi update or elsewhere)? 2.If we treat the pressure terms on the RHS i.e $\nabla(\tilde{U} \bar{p})$ , temporal term and source term $\bar{p}\nabla \cdot \tilde{U}$ so as to impart numerical stability, then following phi = phi - pEqn.Flux() How will one update the volumetric fluxes as the pEqn now has a new convective term on the RHS following Dr.Jasak Thesis? 3.How is this kind of formulation different(physics wise) from that used in the regular openfoam solvers i.e. $\psi \frac{\partial \bar{p}}{\partial t} + \nabla \cdot (\bar{\rho} \vec{U}) = 0$ Note: The $\tilde {}$ are used for a Favre averaged framework. Also the $\nabla \cdot U$ is treated as done in incompressible solvers in OF. Correct me if I am wrong anywhere! Thanks in advance __________________ -- 🃏Maximus🃏 Last edited by godfatherBond; May 7, 2018 at 00:36.

June 13, 2018, 23:05		#3
godfatherBond Member Maximus Arelius Join Date: Jan 2017 Location: Morocco Posts: 36 Rep Power: 9	Hi FOAMers, Still looking for an explanation. It would be beneficial if you could spare some time and have a look into it. Thanks __________________ -- 🃏Maximus🃏

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