Numerical methods for discontinuous grid interfaces?
 

In multi-block methods and self-adaptive methods, one needs to find algorithms for grid interfaces. People may use grid interfaces where grid spacing change gradually. However, there is great demand to deal with grid interfaces where grid spacings are discontinuous or change abruptly at the interfaces.

There are some works for the problem, such as that by Rai, Berger, Shyy and etc. But in these works grid spacings do not change much, often 1:2-3.

Is there any work about interface treatments with large ratio of grid spacing? such as 1:5?

I suppose this is an open question. I would like to discuss with you.

Thank you much.

Re: Numerical methods for discontinuous grid interfaces?
 

Assuming that on the left-hand side of the interface you have a fine mesh because you need to resolve a shear layer, this shear layer information would disappear on the right-hand side of the interface where you have a coarse mesh which is much larger than the thickness of the shear layer. On the other hand, if the fine mesh solution is linear, the loss of information would be negligible on the coarse mesh side. So, it all depends on the complexity of the problem you are trying to solve.( that is the variation of the flow field variable.) If a flow field is constant, it is probably not sensitive to the mesh system used at all. Otherwise, you need to have adequate mesh density to define the solution profiles.

The problem is far from solved
 

Yes, you are right; the mesh system is related to the problem itself. However, in general a flow distribution is not known before calculations. Near a wall the distribution is neither constant nor linear in practical calculations. What we at most may assume is that a flow is smooth.

In my mind, a good grid interface treatment should provide reasonable accuracy ( both locally and gloabely, i.e., near the interface and away from there ), stability, and exhibits no oscillation near the interface.

I do not know which algorithm has above properties. I guess such algorithm is yet to be developed.

Actually, I have not found a paper giving a good and comprenhensive survey about the exisiting schemes for grid interfaces.

Re: The problem is far from solved
 

You are right about it. The problem is far from solved, because it has not been defined yet. For a conventional structured , non-uniform mesh, the mesh stretching ratio should be kept below 1:2. In practical applications, the stretching ratio should be kept between 1.2 to 1.5 in order to have accurate results. A ratio of 1:2 will likely give you flow oscillation ( numerically induced oscillation through finite-difference schemes). For un-structured mesh with discontinuous grid, you need to come up with a method of interpolation such that solution from the coarse mesh side can be transfered to the fine mesh side.( discontinuity in grid simply means that the fine mesh reaches the dead-end road at the interface. there is no corresponding node on the coarse mesh side.) You maybe able to use only a limited number of mesh points in the calculation, if you use higher-order parametric curves to represent your solution for your numerical scheme. This is another direction you can take, that is ,use a coarse mesh with a much-higher-order numerical scheme. You can minimize the numerical solution errors by using the exact solution in your numerical scheme. That's why it is very important to know your problem and your solution. The Law of the Wall treatment near a wall is an extreme case for this approach, the wall shear stress is the only parameter used to characterize the complex profiles near the wall. If you can do the same thing, 9 points would be enough to solve a 2-D problem. ( similar to the control volume approach of boundary layer equations.) It's also straightforward to find out whether your scheme will run into trouble with 1:5 grid spacing ratio. To do this :1) between two end points,write down an analytical profile of order 4 ( 4th order polynomial or higher),2) place a node point in between two end points such that the grid spacing ratio is 1:5,3) derive your numerical representation of the 1-D equation, 4) check the numerical solution against the exact profile you specified.( that is , use the exact solution to develop the numerical scheme, and/or non-uniform grid systems.) Grid generation alone really has no meaning at all, unless you also relate it to the solution, the governing equation and the numerical scheme. Ideally, you would like to generate the mesh after the solution is known. In this way, you can distribute mesh points at the right locations to reduce errors.

Re: The problem is far from solved
 

Gentlemen:

Our CFD-ACE flow solver uses multi-block "many-to-one" local mesh refinement and is used routinely for complex problems with large meshes.

Check out CFD-ACE

There are some example problems on our website at: Combustion Applications

R. Sukumar

Can the CFD-ACE deal with the problem?
 

Could you tell us some details about how the CFD-ACE handle the problem?

It sounds that your 'many to one' is something like the cases Rai and Shyy deal with. In their cases mesh ratio is not large in the direction normal to grid interfaces.

Hansong Tang

let's get to the bottom of the problem
 

Your idea is absolutely good, and it sounds fesiable for some problems.

But, what we are interested in is a general case; no given flow profile, no specific interior scheme. So the problem may be formulated as:

given the following:

1) a smooth flowfield with a wall,

2) two patched grids a with large spacing ratio in the direction both normal and tangental to the grid interface ( the fine mesh may be in or cover the inertial layer ),

3) two interior node schemes of same accuracy, each for one grid,

find an interface treatment which is stable, leads the whole solution to converges well, and maintain the accuracy of the interior schemes.

I guess maybe a combination, with some development, of several exisiting techniques can tackle the problem. Two of them are Kujii ( J. Comput. Phys. 118, 1995 ) and Thomposon ( Appl. Nmer. Math., 9, 1992 ).

Do you think so?

Hansong Tang

Re: Can the CFD-ACE deal with the problem?
 

Hansong:

Obviously smaller mesh ratios will be more accurate. Overall accuracy is a function of how well you interpolate at the "many-to-one" interface as well as how accurately you interpolate values from the cell center to the cell face (CFD-ACE is a colocated grid code). How we code this capability is not something I can discuss in the open forum, since it is a competitive advantage over other codes.

It is hard to make generalizations since different classes of problems involve different physics. If you'd like to email me details of your problem, I'd be glad to discuss it with you.

R. Sukumar

Re: let's get to the bottom of the problem
 

I don't have the time to check out the two papers you mentioned. But I think, any idea would be better than no idea. The only way to succeed is to fail first. Not just once, but many times. My feeling is that, when you are short of computer memory, you have no choice but to use coarse mesh and reserve the fine mesh for the area where the flow gradient is high. This is especially true for adaptive, un-structured mesh approach,where the mesh refinement is done after the initial coarse mesh. The user-interactive mesh refinement approach can create very un-even non-uniform mesh size distribution. It is very hard to control.( it maybe flexible but very hard to control the size and error distribution when compared to the structured mesh approach ). What I am trying to say is: un-structured mesh is great for complex geometry ,especially when you don't want to spend a lot of time generating a mesh. But the difficulty with the un-structured mesh is: the smoothness of the mesh size distribution is very hard to control after a few level of refinement. Sometimes, the convergence process will fail because of a couple of ill-sized, or ill-shaped cell. For a 2-D problem, you may be able to locate these cells and do something about it. For 3-D problems, it's not easy to tackle the problem. How to get the mesh-independent solution from adaptive,un-structured mesh approach is itself a difficult problem.

Re: Numerical methods for discontinuous grid interfaces?
 

Dear Hansong,

There is an algorithmus named as CHIMERA grid approach to treat the discontinuity of mesh interfaces. I have used it several years for moving grids and find it satisfactory. I recommand you some papers:

Terry L. Holst: Numerical Solution of the Full Potential Equation Using a Chimera Grid Approach, AIAA Journal, Vol. 35, No. 9, September 1997

Kai-Hsiung Kao, Meng-Sing Liou: Grid Adaption Using Chimera Composite Overlapping Meshes, AIAA Journal, Vol. 32, No. 5, May 1994

If you want to know further about my experience of using this algorithmus, send me an e-mail. I am very glad to give you information.

Beat regards

X. Ye

Re: Numerical methods for discontinuous grid interfaces?
 

Actually, the probelm is to deal with grid interfaces where grid size changes a lot, such as 1:5. Do you have this kind of experience?

I think that overset grids proposed by Steger etc also face difficulties in such cases. THis technique may be ok for ordinary overset grids ( e. g., Hansong Tang and Tie Zhou, why nonconservative algorithms may be applicable: analysis for Chimera grids, Proc. 7th Int Sypm. CFD, p336, 1997 ).

Hansong Tang

Re: Numerical methods for discontinuous grid interfaces?
 

Yes, I have the experience of using Chimera approach to treat this kind of grid interface (1:5). But such kind of interfaces were in uncritical regions. There is hence no such problems as you mentioned.

By the way, would you please explain me your thesis "why nonconservative algorithms may be applicable: analysis for Chimera grids" with a few words. I am very intereted in your opinion. As I see, the 2D or 3D interpolation approach of Chimera is very critical for a supersonic region where the wave propagation can go only in constrained directions.

Xiangyang Ye

Re: Numerical methods for discontinuous grid interfaces?
 

Dear Ye,

It is nice that you have experience dealing with grid interfaces with big change in grid size. Can I find your results about this somewhere?

One of the major conclusions in the paper I mentioned deals with the convergent solution as mesh size approaches zero:

1) if the solution is almsot continuous across grid interfaces, or,

2) if it is discontinuous there but a R-H like condition is satisfied,

then, the soulution is a weak solution and thus its conservation error is zero.

1) and 2) correspond to a situation when shocks pass through the interfaces and stop right there, respectively. Although 1) and 2) do not tell you how to construct a noconservative interface algorithm that leads to a weak solution, they can tell you, when a nonconservative interface teatment is used, if a limited solution is a weak solution or not.

Tang