Why isn't front tracking mass conserved

Pmaroul · December 4, 2018, 12:09

Hello every body.

I study on the multi-phase & phase change flow simulation by the Front tracking model.

It's noted in the papers that the front tracking is not mass conserved. In the other word, the mass imbalance in this solver, is natural. My questions are presented as follows:

1- Why doesn't this model solve multi-phase flows as mass conserved?

2- What is the remedy for this deficiency (weakness) ?

Thanks in advance for any attention & advice.

P.Maroul

LuckyTran · December 5, 2018, 11:53

The motion of the phase front is exactly that. It has nothing to do with mass. Front tracking methods are usually level-set methods, they are ballistic. Ballistic equations don't describe anything except the trajectory.

Mass conservation come from a separate equation (the continuity equation). First you need an equation that guarantees mass conservation (i.e. the continuity equation). Then your choice of solving the discretized equations must also support this. FEM for example breaks conservation in the discretized equations, FVM preserves the conservation.

Too often someone says something about oranges and somehow apples are involved.

sbaffini · December 6, 2018, 05:13

Not an expert on this but, on a general level, conservation of stuff (mass, momentum, etc.) is not guaranteed whenever it depends on interpolation.

As for the LuckyTran example, FE (and FD) methods do not typically discretize conservation equations, while FV methods do (and conservation is discretely fullfilled at any grid resolution).

In the multiphase world, VOF methods actually discretize mass conservation equations while, in contrast, Level Set ones (and front tracking ones by extension) discretize, somehow, the domain of the phases and interpolations are involved on the boundaries between them.

Pmaroul · December 7, 2018, 03:03

Dear Sirs.

Therefore-Based on your explanations-if I use FVM, the solution will be mass conserved. won't it?

Is it enough?

sbaffini · December 7, 2018, 04:48

Not in my understanding. If you use Level-set or front tracking with FVM it will still be mass leaking. You need VOF in FVM to have mass conservation.

AliE · December 7, 2018, 04:52

Hi, sbaffini is right! Level set is not mass conserving even in fvm. You need a special procedure called re-initialization to improve mass conservation. VoF is conservative but badley evaluate curvature giving raise to spurious currents (which you can correct with special force balanced methods). Thus as usally happens, the blanket is always short ��

Pmaroul · December 8, 2018, 09:53

Quote:

Originally Posted by LuckyTran

The motion of the phase front is exactly that. It has nothing to do with mass. Front tracking methods are usually level-set methods, they are ballistic. Ballistic equations don't describe anything except the trajectory.

Mass conservation come from a separate equation (the continuity equation). First you need an equation that guarantees mass conservation (i.e. the continuity equation). Then your choice of solving the discretized equations must also support this. FEM for example breaks conservation in the discretized equations, FVM preserves the conservation.

Too often someone says something about oranges and somehow apples are involved.

Therefore, according to you , some modifications on the front tracking method can guarantee mass conservation. Can't they?

Pmaroul · December 8, 2018, 09:59

Quote:

Originally Posted by AliE

Hi, sbaffini is right! Level set is not mass conserving even in fvm. You meed a special procedure called re-initialization to improve mass conservation. VoF is conservative but badley evalaute curvature giving raise to spurious currents (which you can correct with special force balanced methods). Thus as usally happens, the blanket is always short ��

Therefore, according to you, the combination of the front tracking & re- initialization can prevent the mass leakage. Can't it?

AliE · December 8, 2018, 10:04

If you are using level set method, yes or it can at least mitigate the problem. There are some papers in jcp about this procedure. For a good reference manual and more bibio, you can look also at the book on multiphase flows by tryggvason & prosperetti. Hightly reccomended for a nice overview!

Pmaroul · December 8, 2018, 11:31

Dear ALiE

I use the front tracking not Level Set. I imagine they are slightly different. Aren't they?

What's the term jcp stand for?

Would you plz indicate some papers only by paper address/number?

Thanks anyway.

sbaffini · December 8, 2018, 11:38

Journal of Computational Physics

Pmaroul · December 8, 2018, 12:29

Quote:

Originally Posted by sbaffini

Not an expert on this but, on a general level, conservation of stuff (mass, momentum, etc.) is not guaranteed whenever it depends on interpolation.

As for the LuckyTran example, FE (and FD) methods do not typically discretize conservation equations, while FV methods do (and conservation is discretely fullfilled at any grid resolution).

In the multiphase world, VOF methods actually discretize mass conservation equations while, in contrast, Level Set ones (and front tracking ones by extension) discretize, somehow, the domain of the phases and interpolations are involved on the boundaries between them.

Dear Sirs

I didn't find out the reason.

I present mass continuity& mass transfer equations involved in the front tracking procedure and are probably the cause for the mass leak as follows:

$\frac{{d{\mathbf x}}_f}{dt}=V_n{\mathbf n}{\mathbf \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left({\mathbf 1}\right)$
$\nabla .\left(\rho {\mathbf u}\right){\mathbf =}\int_{{\mathbf A}}{({\rho }_l-{\rho }_g)V_n\delta ({\mathbf x}{\mathbf -}{{\mathbf x}}_{{\mathbf f}})ds}{\mathbf \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (}{\mathbf 2}{\mathbf )}\$
$V_n=\frac{1}{2}\left(u_l+u_g\right)-\frac{\dot{m}}{2}\left(\frac{1}{{\rho }_l}+\frac{1}{{\rho }_g}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\$

which above expression (equation) is responsible for the mass imbalance?

Thanks for any answer.

Pmaroul · December 8, 2018, 12:35

Quote:

Originally Posted by sbaffini

Journal of Computational Physics

Thanks Sir

sbaffini · December 8, 2018, 13:53

Quote:

Originally Posted by Pmaroul

Dear Sirs

I didn't find out the reason.

I present mass continuity& mass transfer equations involved in the front tracking procedure and are probably the cause for the mass leak as follows:

$\frac{{d{\mathbf x}}_f}{dt}=V_n{\mathbf n}{\mathbf \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left({\mathbf 1}\right)$
$\nabla .\left(\rho {\mathbf u}\right){\mathbf =}\int_{{\mathbf A}}{({\rho }_l-{\rho }_g)V_n\delta ({\mathbf x}{\mathbf -}{{\mathbf x}}_{{\mathbf f}})ds}{\mathbf \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (}{\mathbf 2}{\mathbf )}\$
$V_n=\frac{1}{2}\left(u_l+u_g\right)-\frac{\dot{m}}{2}\left(\frac{1}{{\rho }_l}+\frac{1}{{\rho }_g}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\$

which above expression (equation) is responsible for the mass imbalance?

Thanks for any answer.

As I said, I'm not an expert on this matter. My understanding is that in going from one time step to the next, even if you can transport exactly the front, it is only a discrete representation of it, and you are going to need interpolation to actually reconstruct it. As soon as interpolation is involved (versus a discrete conservation principle) you get conservation issues.

You should follow the references backward starting from the source where you first read about the mass conservation issue and you will certainly find some that will go into the details.

FMDenaro · December 8, 2018, 19:11

Quote:

Originally Posted by Pmaroul

Dear Sirs

I didn't find out the reason.

I present mass continuity& mass transfer equations involved in the front tracking procedure and are probably the cause for the mass leak as follows:

$\frac{{d{\mathbf x}}_f}{dt}=V_n{\mathbf n}{\mathbf \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\left({\mathbf 1}\right)$
$\nabla .\left(\rho {\mathbf u}\right){\mathbf =}\int_{{\mathbf A}}{({\rho }_l-{\rho }_g)V_n\delta ({\mathbf x}{\mathbf -}{{\mathbf x}}_{{\mathbf f}})ds}{\mathbf \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (}{\mathbf 2}{\mathbf )}\$
$V_n=\frac{1}{2}\left(u_l+u_g\right)-\frac{\dot{m}}{2}\left(\frac{1}{{\rho }_l}+\frac{1}{{\rho }_g}\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)\$

which above expression (equation) is responsible for the mass imbalance?

Thanks for any answer.

The key of the problem is in Eq.(2) where the Dirac function is approximated on the discrete grid of finite size

Pmaroul · December 10, 2018, 18:35

A fairly complete explanation of this topic is given in the paper "A Front-Tracking Method for the Computations
of Multiphase Flow by G. Tryggvason and et al" as follows

"
While the momentum equations are usually solved in the conservative from, the advection of the front is not conservative. Unlike the VOF method, for example, errors are likely to result in changes in the total mass. Accurate advection of the front points minimizes this error and we have done numerous simulations of bubbles, for example, where the change in mass remains within 1–2% during a time when the bubbles move about 100 diameters. In some cases, particularly for very long runs with many bubbles or drops where the resolution of each particle is relatively low, we have encountered changes in mass that are unacceptably high. In these cases, we correct the size of the particles every few time steps. Since the correction is very small at each time, the effect on the result is negligible. The inaccuracy in the advection of the front is due to errors coming from the interpolation of the velocities and the integration scheme. Increasing the accuracy of the front advection by using a higher order time stepping method is straightforward. The error due to the interpolation comes from the fact that although the discrete velocity field may be divergence-free (for incompressible flows), the interpolated velocity field is not necessarily divergence-free. An interpolation scheme that produces a divergence-free velocity at the front points has been developed by Peskin and Printz [89]. The result is, however, a more complex pressure equation, and we have not implemented this technique. Interpolation errors appear primarily to be due to poor resolution and should therefore generally be small. A test of the accuracy of the time integration has been done by Juric [59] who advected an initially circular blob of fluid by a prescribed velocity field that deformed the blob into a long ligament. Mass was conserved very well during the simulation and when the velocity was reversed, the circle was recovered nearly perfectly. This test has been used for several other methods that either track or capture interfaces, and it is generally found that tracking produces superior results. Adding and deleting front points and elements can also lead to changes in area and volume. However, by using a relatively large number of points and inserting points using a second-order curve fit, this effect is minimized."

I have specified possible reasons for the mass leak as red colour.
However, the real reason for mass leakage is still ambiguous for me.

Thanks anyone for the more detailed description.

FMDenaro · December 11, 2018, 04:15

Yes, I strongly suggest to read the series of papers of Peskin when the issue is analysed. The source of the error is the "interpolation" induced by the use of the discrete Dirac function

Alex C. · December 11, 2018, 09:19

Touré, Fahsi and Soulaïmani (2016) proposed a level-set correction that extends the mass conservation of the method by constraining the level-set field to remain a signed distance function.

December 4, 2018, 12:09	Why isn't front tracking mass conserved	#1
Pmaroul Member Peter Maroul Join Date: May 2018 Posts: 52 Rep Power: 8	Hello every body. I study on the multi-phase & phase change flow simulation by the Front tracking model. It's noted in the papers that the front tracking is not mass conserved. In the other word, the mass imbalance in this solver, is natural. My questions are presented as follows: 1- Why doesn't this model solve multi-phase flows as mass conserved? 2- What is the remedy for this deficiency (weakness) ? Thanks in advance for any attention & advice. P.Maroul

December 7, 2018, 04:52		#6
AliE Senior Member Join Date: Dec 2017 Posts: 153 Rep Power: 8	Hi, sbaffini is right! Level set is not mass conserving even in fvm. You need a special procedure called re-initialization to improve mass conservation. VoF is conservative but badley evaluate curvature giving raise to spurious currents (which you can correct with special force balanced methods). Thus as usally happens, the blanket is always short �� sbaffini likes this. Last edited by AliE; December 8, 2018 at 09:59.

December 8, 2018, 10:04		#9
AliE Senior Member Join Date: Dec 2017 Posts: 153 Rep Power: 8	If you are using level set method, yes or it can at least mitigate the problem. There are some papers in jcp about this procedure. For a good reference manual and more bibio, you can look also at the book on multiphase flows by tryggvason & prosperetti. Hightly reccomended for a nice overview! Pmaroul likes this.

December 10, 2018, 18:35		#16
Pmaroul Member Peter Maroul Join Date: May 2018 Posts: 52 Rep Power: 8	A fairly complete explanation of this topic is given in the paper "A Front-Tracking Method for the Computations of Multiphase Flow by G. Tryggvason and et al" as follows " While the momentum equations are usually solved in the conservative from, the advection of the front is not conservative. Unlike the VOF method, for example, errors are likely to result in changes in the total mass. Accurate advection of the front points minimizes this error and we have done numerous simulations of bubbles, for example, where the change in mass remains within 1–2% during a time when the bubbles move about 100 diameters. In some cases, particularly for very long runs with many bubbles or drops where the resolution of each particle is relatively low, we have encountered changes in mass that are unacceptably high. In these cases, we correct the size of the particles every few time steps. Since the correction is very small at each time, the effect on the result is negligible. The inaccuracy in the advection of the front is due to errors coming from the interpolation of the velocities and the integration scheme. Increasing the accuracy of the front advection by using a higher order time stepping method is straightforward. The error due to the interpolation comes from the fact that although the discrete velocity field may be divergence-free (for incompressible flows), the interpolated velocity field is not necessarily divergence-free. An interpolation scheme that produces a divergence-free velocity at the front points has been developed by Peskin and Printz [89]. The result is, however, a more complex pressure equation, and we have not implemented this technique. Interpolation errors appear primarily to be due to poor resolution and should therefore generally be small. A test of the accuracy of the time integration has been done by Juric [59] who advected an initially circular blob of fluid by a prescribed velocity field that deformed the blob into a long ligament. Mass was conserved very well during the simulation and when the velocity was reversed, the circle was recovered nearly perfectly. This test has been used for several other methods that either track or capture interfaces, and it is generally found that tracking produces superior results. Adding and deleting front points and elements can also lead to changes in area and volume. However, by using a relatively large number of points and inserting points using a second-order curve fit, this effect is minimized." I have specified possible reasons for the mass leak as red colour. However, the real reason for mass leakage is still ambiguous for me. Thanks anyone for the more detailed description. AliE likes this.

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December 5, 2018, 11:53		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,754 Rep Power: 66	The motion of the phase front is exactly that. It has nothing to do with mass. Front tracking methods are usually level-set methods, they are ballistic. Ballistic equations don't describe anything except the trajectory. Mass conservation come from a separate equation (the continuity equation). First you need an equation that guarantees mass conservation (i.e. the continuity equation). Then your choice of solving the discretized equations must also support this. FEM for example breaks conservation in the discretized equations, FVM preserves the conservation. Too often someone says something about oranges and somehow apples are involved.

December 6, 2018, 05:13		#3
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Not an expert on this but, on a general level, conservation of stuff (mass, momentum, etc.) is not guaranteed whenever it depends on interpolation. As for the LuckyTran example, FE (and FD) methods do not typically discretize conservation equations, while FV methods do (and conservation is discretely fullfilled at any grid resolution). In the multiphase world, VOF methods actually discretize mass conservation equations while, in contrast, Level Set ones (and front tracking ones by extension) discretize, somehow, the domain of the phases and interpolations are involved on the boundaries between them.

December 7, 2018, 03:03		#4
Pmaroul Member Peter Maroul Join Date: May 2018 Posts: 52 Rep Power: 8	Dear Sirs. Therefore-Based on your explanations-if I use FVM, the solution will be mass conserved. won't it? Is it enough?

December 7, 2018, 04:48		#5
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Not in my understanding. If you use Level-set or front tracking with FVM it will still be mass leaking. You need VOF in FVM to have mass conservation.

December 8, 2018, 11:31		#10
Pmaroul Member Peter Maroul Join Date: May 2018 Posts: 52 Rep Power: 8	Dear ALiE I use the front tracking not Level Set. I imagine they are slightly different. Aren't they? What's the term jcp stand for? Would you plz indicate some papers only by paper address/number? Thanks anyway.

December 8, 2018, 11:38		#11
sbaffini Senior Member Paolo Lampitella Join Date: Mar 2009 Location: Italy Posts: 2,192 Blog Entries: 29 Rep Power: 39	Journal of Computational Physics

December 11, 2018, 04:15		#17
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,882 Rep Power: 73	Yes, I strongly suggest to read the series of papers of Peskin when the issue is analysed. The source of the error is the "interpolation" induced by the use of the discrete Dirac function

December 11, 2018, 09:19		#18
Alex C. Member Join Date: Jul 2013 Posts: 56 Rep Power: 13	Touré, Fahsi and Soulaïmani (2016) proposed a level-set correction that extends the mass conservation of the method by constraining the level-set field to remain a signed distance function.