PLIC-VOF with a Conservative formulation at high density ratio

abhishek13 · February 9, 2018, 04:54

Greetings!
I am trying to develop a Two Phase VOF solver with a conservative formulation. I have successfully implemented the following:

PLIC-VOF in 2D (for coupling with a Naviers Stokes Solver)
Non-conservative formulation for a dam-break problem with density ratio of 1000
Conservative formulation for a dam-break problem with denstiy ratio of 10

At higher density ratios, while using the conservative formulation, there is an unphysical increase in the velocity near the interface. I know that this is caused by a sudden jump in density which causes an ill-posed matrix while solving for pressure. I would like to know how to rectify the issue to make the conservative solver work at higher density ratios.

beer · February 9, 2018, 05:36

Basically what you have to do is not to use the average density for the interface density in the mass flux. Rather use a mass flux, which is calculated from the mass fluxes of the two phases:
if the face densities for the two phases are $\rho_{f1}$ and $\rho_{f2}$ , the face normal velocity is

, the face are is

, the overall volume flux on the face is

$\dot{V}_{Sum}=v_F*A$ .

Now, the volume flux for the second phase $\dot{V}_2$ is known from the PLIC reconstruction, because that is what you need PLIC for. This means you can calculate the volume flux for the first phase from these numbers:

$\dot{V}_1=\dot{V}_{Sum} - \dot{V}_2$

Now just use these two volume fluxes to calculate the mass flux using the density for the two phases on the face seperately:

$\dot{m}_f=\dot{V}_{f1}*\rho_1 + \dot{V}_2*\rho_{f2}$

This procedure worked very well for me. I can't give you a reference though, because I kind of came up with it myself. However, I am fairly sure someone has done and published it before.

abhishek13 · February 9, 2018, 08:46

Hello! Thanks for the reply. Are you suggesting that the density at the faces should be calculated using the amount of both fluids present in the volume (face velocity*dt)*A?

beer · February 9, 2018, 10:28

No, sorry I should have stated that. You should calculate the face density for each phase seperately. I edited my answer above.

abhishek13 · February 11, 2018, 22:44

Sorry but i still dont get what you are trying to say. Where do i need to use the mass flux?

nkalkote · February 12, 2018, 03:38

Dear Abhishek,

This is caused due to disparity in Gamma (specific heat ratio calculation) Please refer the following paper,

https://www.sciencedirect.com/scienc...21999115005021

titled "Preventing numerical oscillations in the flux-split based finite difference method for compressible flows with discontinuities".

As this paper is highly condensed you might have to cross refer the references for more details.

Regards,

beer · February 15, 2018, 04:07

Quote:

Originally Posted by abhishek13

Sorry but i still dont get what you are trying to say. Where do i need to use the mass flux?

For the discretization of the convection term and for the source term in the pressure correction equation.

Saideep · February 15, 2018, 10:23

Might be of use. A simple way is to filter the volume fraction several times so that you dont see a sharp jump in the fluid properties but a gradual jump. This is something that the software "Gerris Flow Solver" using PLIC-VOF does.

However, I did find that for some density ratios (air water) keeping the viscosity ratio as 1, the solver really struggles.

February 9, 2018, 04:54	PLIC-VOF with a Conservative formulation at high density ratio	#1
abhishek13 New Member Abhishek Join Date: Feb 2018 Posts: 4 Rep Power: 8	Greetings! I am trying to develop a Two Phase VOF solver with a conservative formulation. I have successfully implemented the following: PLIC-VOF in 2D (for coupling with a Naviers Stokes Solver) Non-conservative formulation for a dam-break problem with density ratio of 1000 Conservative formulation for a dam-break problem with denstiy ratio of 10 At higher density ratios, while using the conservative formulation, there is an unphysical increase in the velocity near the interface. I know that this is caused by a sudden jump in density which causes an ill-posed matrix while solving for pressure. I would like to know how to rectify the issue to make the conservative solver work at higher density ratios.

February 9, 2018, 05:36		#2
beer Member Join Date: Dec 2012 Posts: 92 Rep Power: 13	Basically what you have to do is not to use the average density for the interface density in the mass flux. Rather use a mass flux, which is calculated from the mass fluxes of the two phases: if the face densities for the two phases are $\rho_{f1}$ and $\rho_{f2}$ , the face normal velocity is , the face are is , the overall volume flux on the face is $\dot{V}_{Sum}=v_FA$ . Now, the volume flux for the second phase $\dot{V}_2$ is known from the PLIC reconstruction, because that is what you need PLIC for. This means you can calculate the volume flux for the first phase from these numbers: $\dot{V}_1=\dot{V}_{Sum} - \dot{V}_2$ Now just use these two volume fluxes to calculate the mass flux using the density for the two phases on the face seperately: $\dot{m}_f=\dot{V}_{f1}\rho_1 + \dot{V}_2\rho_{f2}$ This procedure worked very well for me. I can't give you a reference though, because I kind of came up with it myself. However, I am fairly sure someone has done and published it before. Last edited by beer; February 9, 2018 at 10:27.*

February 11, 2018, 22:44		#5
abhishek13 New Member Abhishek Join Date: Feb 2018 Posts: 4 Rep Power: 8	Sorry but i still dont get what you are trying to say. Where do i need to use the mass flux? Last edited by abhishek13; February 13, 2018 at 02:46.

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February 9, 2018, 08:46		#3
abhishek13 New Member Abhishek Join Date: Feb 2018 Posts: 4 Rep Power: 8	Hello! Thanks for the reply. Are you suggesting that the density at the faces should be calculated using the amount of both fluids present in the volume (face velocitydt)A?

February 9, 2018, 10:28		#4
beer Member Join Date: Dec 2012 Posts: 92 Rep Power: 13	No, sorry I should have stated that. You should calculate the face density for each phase seperately. I edited my answer above.

February 12, 2018, 03:38		#6
nkalkote New Member Nikhil Join Date: Mar 2016 Location: India Posts: 15 Rep Power: 10	Dear Abhishek, This is caused due to disparity in Gamma (specific heat ratio calculation) Please refer the following paper, https://www.sciencedirect.com/scienc...21999115005021 titled "Preventing numerical oscillations in the flux-split based finite difference method for compressible flows with discontinuities". As this paper is highly condensed you might have to cross refer the references for more details. Regards,

February 15, 2018, 10:23		#8
Saideep Senior Member Saideep Join Date: Apr 2015 Location: INDIA Posts: 203 Rep Power: 12	Might be of use. A simple way is to filter the volume fraction several times so that you dont see a sharp jump in the fluid properties but a gradual jump. This is something that the software "Gerris Flow Solver" using PLIC-VOF does. However, I did find that for some density ratios (air water) keeping the viscosity ratio as 1, the solver really struggles.