[aerosjc]

Hi,

We need some methods to handle the complex geometry in the industrial applications. The natural way is the unstructured grid along with the Finite Volume method or the Discontinuous Galerkin method or the Flux Reconstruction method.
Alternatives include the Level Set method and Immersed Boundary method on the Cartesian grid.
I know the Level Set method is an alternative of the Volume of Fluid method for the multi-phase flow. Let us limit the discussion to the single phase flow.
The question is: what is the advantages/disadvantages of the Levet Set method or Immersed Boundary method over the Unstructured Grid when used to handle the complex geometry in the industrial applications?
For example, when the channel has a rough wall with many small elements, the Immerser Boundary method would be a suitable choice.

[sbaffini]

Level Set, as you note, is originally dedicated to multi-phase flows. But, as solids are, indeed, a different phase, there are several VOF or level set like methods that are indeed used as an immersed boundary approach.

In fact, Immersed Boundary embraces a lot of different techniques. Some of them have limitations on their flexibility (for example, VOF like methods might have problems for thin surfaces or with wall functions), others have leaks in the fluxes on the boundaries because of interpolations.

In my opinion, if body fitted meshing is feasible, structured or not, you should do it. Otherwise, immersed boundary is today mature enough to give reasonable answers to most problems, but you need the right software, not just a random one

[aerosjc]

The Cartesian grid is simple. The unstructured grid is complex and coding of it is relatively harder. So if I could use the Level Set method to handle the complex geometry on the Cartesian grid well, why would I use the unstructured grid? The Immersed Boundary method cannot strictly maintain the conservation of flux. That is a key disadvantage. How about the Levet Set method?

[sbaffini]

Originally Posted by aerosjc

The Cartesian grid is simple. The unstructured grid is complex and coding of it is relatively harder. So if I could use the Level Set method to handle the complex geometry on the Cartesian grid well, why would I use the unstructured grid? The Immersed Boundary method cannot strictly maintain the conservation of flux. That is a key disadvantage. How about the Levet Set method?

Let me start by highlighting that cartesian is not necessarily easier. In fact, for an actual professional code (that is, a sufficiently complex code), the cartesian part easily becomes the largest pain to handle and maintain, besides bloating the code with a lot of unmaintainable indices and triple loops which get copy pasted around a lot.

But, I see what you mean. For a single man, short-term project (or something on a similar scale), cartesian is the easiest way to go, and if you can do some sort of immersed boundary you are also ready to go.

However, as a matter of fact, unstructured grid methods are easier to write and maintain. You need to handle the connectivity, which has a computational cost, but that is a far easier to maintain book keeping part with respect to the cartesian method ones.

Also, IB makes just the same sense on unstructured grids as structured ones. Actually, my very point is to use IB only if an unstructured grid gets close to impossible. The right code would actually mix the two approaches seamlessly. Moreover, cartesian methods have a point in going very high order, but the immersed boundary treatment is at most second order if you want to keep it simpler than actual meshing, thus with no actual requirement for a cartesian grid at all.

Second, while I don't know much of several IB approaches, let me clarify that IB methods can be roughly distinguished in continuous forcing methods (either the original Peskin forcing or VOF like approaches, which need a full grid also inside the body), discrete forcing methods (which, either notionally or practically truncate the grid somewhere and apply interpolated boundary conditions somewhere near the truncation) or cut cell methods (which again, notionally or practically cut the grid in a way to more closely follow the actual geometry, yet not exactly as an unstructured grid generator would).

It is the discrete forcing approach as a whole that has flux leaks at boundaries. It is so because boundary fluxes are a result of interpolation (body fitted methods, instead, explicitly set 0 convective fluxes on walls) and then go into the boundary getting lost. The first category doesn't have this problem because the fluxes naturally rebalance themselves inside the body (not necessarily with accuracy with respect to the body fitted solution). However, the discrete forcing approach has advantages in treating complex boundary conditions and thin surfaces.

I know very little of Level Set IB approahes, but I'm under the impression that they might fall in either the first or second category (maybe even the third), depending from the implementation, with the resulting pros and cons.

[aerosjc]

Thank you for the explanations. Now what I learn is that although IB is able to handle the complex geometry, it has multiple defects. The question I asked is too simple and too general. Only when faced with a specific problem, we can talk about which one is suitable.

[sbaffini]

Exactly, but let me say that they are limitations more than defects. If your starting point is that you want to solve a problem with approximate boundary conditions (because that's what IB is really about), you must be willing to abandon something for the additional flexibility. There is simply no free lunch.

Sometimes it legitimately makes sense. For example, a very complex geometry that must be analyzed only for some global quantity. Or when you need to run a lot of cases, maybe automatically in an optimization loop, and you certainly don't want to make tens or hundreds of meshes. Or, more importantly, for moving geometries, as originally framed.