I have questions on the loss of mass. In the literature, some researchers insist that "this method causes the loss of mass while our method is mass-conservative method". In fact, I do not understand the physical meaning of the loss of mass. If mass is lost in the computation, what happened? What's wrong with the mass loss? And how can we figure out whether the scheme is mass-loss or conservative? Do you know how to measure the loss of mass?

Thanks.

Conservation of mass underlies the entire structure of a continuum hypothesis. The mass conservation equation is referred to as the continuity equation because it derives precisely from the assumption of a continuous field. Loss of conservation of any of the conserved quantities is a bad thing. Typically if you are losing mass in a computation then it means you are solving an equation that contains pseudo-mass sources or sinks, and eventually your solution will be garbage. The simplest approach to evaluate your scheme for mass conservation is to turn off the boundary conditions and see if your scheme can maintain a uniform flow field. If it can't then you will probably have serious issues when you run it for a real flow.

In some practices, I experienced unphysical wave reflections from boundaries. Is it also due to the loss of mass?

Dear Jinwon,

That is due to the boundary conditions not being non-reflecting. You need to use absorbing BCs and the non-physical reflections will be gone.

Hope this helps

Regards,

Ganesh

What is mainly meaned by *mass loss* in literature is when we explicitely deal with mass conservation (advection of materials interface multiphase flow etc). For example it is well known that Level set method is not mass conservative (without special treatment) while conservation of VoF method is very supperior.

Dealing with incompressible flow, mass loss is sticked to divergence of velosity field, non-zero divergence is equivalent with avirtual mass source/sink. In sharp interface multiphase flow, also mass loss could be happen in exactly divergence free velosity field, due to numerical diffusion in advection step.

Hope this helps ...

I am solving compressible two-fluid flows by the RKDG coupled with the level set method. Many papers said that the LSM is not mass conservative. What does it mean? Nevertheless, LSM is one of most commonly used method in CFD community as far as I know. Could you tell me how this mass loss play in compressible two-fluid flows? If LSM is not mass conservative, it is not good for CFD simulations. Or is it sufficient for some situations?

In "Level set method is not mass conservative (without special treatment)", what is special treatment? Is it the reinitialization of LS function?

first about level set it has the most public method dealing with interfaces (CFD, image proceccing, etc, just google on it to see, if you refre to Sethian, inventor of LS, web, you see more about aplication), -------------

but why in use in spite of some (probably) drawback: due to several reasons (first note the mass loss is the main drawback but it is related to numerical treatment and so enrichment of numerical method anihilate this):

- in some application mass loss in not important, or is tollerable

- its benefits, mainly second order normal curvature computation while in VoF due least square fit and exess effort is required.

- ease of implementation, it is completely algebraic and easy to implement in any type of grid in any dimension, while VoF is more geometric and is indeed difficult to implement (it is the same for front tracking too).

- better presentation of method and exploring more application by its inventors (if you check publication of S. Osher and J. Sethian you see that they try to explor new application and enter LS into several application).

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mass loss in compressible flow, i am not expert in this field, but i think that as you track two distinct phases by a level set so mass loss could leads to shrink of one phase more than what should actually happen, in the other word, what is happen in numerical solution is that level set solver (virtually) uses a velosity field that is not equal to your computed velosity field and used velosity field, and so after advection position of interface is not what should be if level set be mass conservative.

To figure out mass conservation of level set, the simplest test is grid resolution study, with increase grid resolution numerical diffusion is decreased and so mass loss, to decouple this from flow solver, you could use a fine grid just for level set, but be careful when interpolate result of coarse grid to fine one.

any treatment that decrease numerical diffusion:

- increase grid resolution at vicinity of interface

- adding hybrid particles to improve reinit step (Fedkiw & Enrich)

- high order methods, WENO ...

- semi lagrangian methods showed lower diffusion too.

- semi lagrangian countoring (Jon Strain) showed promizing results too (but this is no longer actual level set and has geometric nature and is difficult to implement)

- variational level set, adding some additional terms, it not only improve mass loss but also does not ussually needs reinitiallization.

- ...

I added the reinitialization process after each LS evolution. Compared to the analytic solution, the locations of shock front and interface were in good agreement.

> the locations of shock front and interface were in good agreement

so it seems that (your) LS is sufficient for your application, note that based on flow regime,behaviour could be different, e.g., when you have vortical flow that size of vortexes are not very larger than grid size (small vortex), serious conservation problem could happen (simply eny time that numerical diffusion of advection be seriuos LS encounter problem)

Yes. I think so. In incompressible flows, it often suffered from the loss of mass.

Your original PDE's are derived as conservation laws for mass, momenta (1, 2, or 3 eqns), and conservation of energy.

One hopes to approximate those PDE's with discrete equations (FEM, FDM, FVM, etc) that produce solutions that satisfy the underlying conservation laws in some fashion.

To go beyond this, testing 'convervation' requires some attention to definitions. Otherwise you're just arm waving. This applies to conservation of momenta and energy as well as mass

Is conservation an important property of an algorithm? Complicated question. C. W. Hirt has shown that a particular FDM is, while not conserving exactly (to machine accuracy), is formally one order of magnitude more accurate than the corresponding FVM on an indentical mesh.

You can read Hirt's explanation here:

http://www.flow3d.com/cfd101/cfd101_cons_not.html

Please not that I have no connection with Flow Science Inc. or its staff. I used their product for a while in the 1990's. I would claim a friendship with Tony Hirt.

I can give a simple example, I was simulating a rectangular bubble to study surface tension effect, using Phase field model(Allen Cahn not cahn hilliard). Physically the rectangular bubble deform to circular one but start shrinking and after some time disappear. then i added lagrange multiplier to the phase field model and finally the masss was conserve. means a rectangular bubble deform to cirular one due to surface tension effect with no mass loss.

by using Cahn hilliard, there will be no need for lagrange multiplier. I suggest you replace level set by phsae field model, this will need you little more work.

Good luck