[NotOverUnderated]

Hello,

For typical static meshes, the Navier stokes equations are written in an Eulerian formulation.

But I am not sure about meshes that have some parts that are rotating which are usually modeled using the sliding mesh technique.

Example:

Below is an example of a domain used to simulate the flow around a rotating rectangle object.



1) What is the forumlation name for such simulations? It is certainly not an Eulerian formulation because the inner region is rotating. On the other hand it is not like a Lagrangian formulation either.

Any help is very much appreciated.

Kind regards
ONESP-RO

[FMDenaro]

Have a look to "Chimera methods"

[NotOverUnderated]

Originally Posted by FMDenaro

Have a look to "Chimera methods"

Thank you for your reply.
I know a little bit about overset meshes but I am interested in formulation name for sliding meshes. Is it the same as overset meshes? I have checked some resources but they are heavy on maths beyond what I could understand.

[LuckyTran]

We just call it moving mesh.


Actually it is still Eulerian because sliding interface is what enables the grid to move in-sync with the motion of the boundaries and it is the same Eulerian equations just with a non-constant mesh. The sliding mesh is the only "new" technique at the mesh interface between stationary and moving parts and it effectively only a BC.


The same is true when you have deforming boundaries as long as the grid morphs with the boundary. For these cases, we like to call them dynamic meshes because it requires a re-gridding every time-step whereas the sliding mesh does not require re-gridding. Both are Eulerian.

[NotOverUnderated]

Originally Posted by LuckyTran

We just call it moving mesh.


Actually it is still Eulerian because sliding interface is what enables the grid to move in-sync with the motion of the boundaries and it is the same Eulerian equations just with a non-constant mesh. The sliding mesh is the only "new" technique at the mesh interface between stationary and moving parts and it effectively only a BC.


The same is true when you have deforming boundaries as long as the grid morphs with the boundary. For these cases, we like to call them dynamic meshes because it requires a re-gridding every time-step whereas the sliding mesh does not require re-gridding. Both are Eulerian.

Many thanks for your reply.

Eulerian meshes,by definition, are fixed meshes and have no moving vertices. After delving a little bit in literature I found that the formulation used for sliding meshes is known as Arbitrary-Lagrangian-Eulerian (ALE).

Kind regards

[LuckyTran]

Eulerian reference frames existed long before CFD and meshes. That clearly cannot be the definition! I think it goes all the way back to this guy named Euler.

But I guess it is now 2024 and we let codes can have whatever pronouns they want.

[FMDenaro]

Originally Posted by NotOverUnderated

Many thanks for your reply.

Eulerian meshes,by definition, are fixed meshes and have no moving vertices. After delving a little bit in literature I found that the formulation used for sliding meshes is known as Arbitrary-Lagrangian-Eulerian (ALE).

Kind regards

No, Eulerian moving control volume exist, too.

[NotOverUnderated]

Originally Posted by FMDenaro

No, Eulerian moving control volume exist, too.

Do you mind sharing any references? I cannot find a single reference that states that Eulerian control volumes can move.

Kind regards

[FMDenaro]

Originally Posted by NotOverUnderated

Do you mind sharing any references? I cannot find a single reference that states that Eulerian control volumes can move.

Kind regards

Forget for a moment the CFD and go to the fundamental fluid mechanics textbooks, you will read that a control volume moving at a prescribed arbitrary velocity is still an Eulerian framework. Only when the volume moves by means of the mapping of the flow velocity that is called Lagrangian.

[NotOverUnderated]

Originally Posted by FMDenaro

Forget for a moment the CFD and go to the fundamental fluid mechanics textbooks, you will read that a control volume moving at a prescribed arbitrary velocity is still an Eulerian framework. Only when the volume moves by means of the mapping of the flow velocity that is called Lagrangian.

I believe there is a misunderstanding. Yes we can consider a moving control (with constant velocity or accelerating) as in control volume around an accelerating rocket and consider it as Eulerian Control volume but only because the frame is attached to the control volume itself (side note: in CFD the equivalent of this approach for modeling of rotating objects is to use SRF (single reference frame) or MRF (multiple reference frame)) because the equations are written in a relative reference frame.

If the equations are written in a absolute fixed frame, one can see that the analysis above cannot be used to model a true rotating object (as in my first post above) where the angle of the rectangle is changing over time (See this animation for illustration: https://www.youtube.com/watch?v=b0Gf8TRdWGo).

My conclusion is that an Eulerian description for trully moving meshes in absolute frames does not make much sense.

I appreciate any suggestions/corrections

[LuckyTran]

The approach you show in your own post is an example of a case where the mesh is attached to the reference frame. Even if you provided no picture, it would be implied anyway if you are use a sliding mesh. Because otherwise, there would be no reason to be using a sliding mesh.

Not all moving meshes are Eulerian but that was not your question. Still, even if you were modeling flow in aortic valves mounted on a rocket engine orbiting a star, that would still (most likely) be an Eulerian formulation as long as you are using simple definitions of derivatives. You can have arbitrary mesh motion and it remains Eulerian. It only becomes Lagrangian when you start using the material derivative. Lagrangian formulations exist in specialized codes but that is extremely far fetched from what your question is. It doesn't become Lagrangian just because it moves, Eulerian grids can move anywhere they like.

So, answered already. There is no general special name for what you show in your picture because it's not special.

[JBeilke]

http://sokocalo.engr.ucdavis.edu/~je...EPO/CM2410.pdf

[NotOverUnderated]

Originally Posted by LuckyTran

The approach you show in your own post is an example of a case where the mesh is attached to the reference frame. Even if you provided no picture, it would be implied anyway if you are use a sliding mesh. Because otherwise, there would be no reason to be using a sliding mesh.

Not all moving meshes are Eulerian but that was not your question. Still, even if you were modeling flow in aortic valves mounted on a rocket engine orbiting a star, that would still (most likely) be an Eulerian formulation as long as you are using simple definitions of derivatives. You can have arbitrary mesh motion and it remains Eulerian. It only becomes Lagrangian when you start using the material derivative. Lagrangian formulations exist in specialized codes but that is extremely far fetched from what your question is. It doesn't become Lagrangian just because it moves, Eulerian grids can move anywhere they like.

So, answered already. There is no general special name for what you show in your picture because it's not special.

I do not think so. The moving mesh in a sliding mesh is not Eulerian. For reference, take a look at section 6.2 in this classical reference [1], quoted below:

The ALE kinematical description avoids excessive mesh distortion (see Figure 7). For this problem, a computationally efficient rezoning strategy is obtained by dividing the mesh into three zones: (1) the mesh inside the inner circle is prescribed to move rigidly attached to the rectangle (no mesh distortion and simple treatment of interface conditions); (2) the mesh outside the outer circle is Eulerian (no mesh distortion and no need to select mesh velocity); (3) a smooth transition is prescribed in the ring between the circles (mesh distortion under control).




Figure legend: Details of finite element mesh around the rectangle. The ring allows a smooth transition
between the rigidly moving mesh around the rectangle and the Eulerian mesh far from it

Reference:

[1] DONEA, Jean, HUERTA, Antonio, PONTHOT, J.‐Ph, et al. Arbitrary Lagrangian–Eulerian Methods. Encyclopedia of computational mechanics, 2004.

[FMDenaro]

Originally Posted by NotOverUnderated

I do not think so. The moving mesh in a sliding mesh is not Eulerian. For reference, take a look at section 6.2 in this classical reference [1], quoted below:





Reference:

[1] DONEA, Jean, HUERTA, Antonio, PONTHOT, J.‐Ph, et al. Arbitrary Lagrangian–Eulerian Methods. Encyclopedia of computational mechanics, 2004.

In my opinion, a Lagrangian grid is defined only when it moves according to the flow. Any other type of arbitrary moving mesh remains Eulerian.

[LuckyTran]

The whole point of ALE is to use mesh coordinates, allowing a purely Eulerian framework to solve the problem of moving boundaries.


Every unstructured FVM solver I've ever seen uses mesh coordinates since you must always compute the face fluxes based on adjacent neighbor pairs. This would make the vast majority of FVM (and CFD) codes ALE codes. Even if I assume that the politically correct name for such a formulation is ALE and not almost purely Eulerian (APE), that would make virtually all CFD codes ALE and not APEs. Again, not special. ALE's are the CFD version of the Apache Helicopter meme.