Unstructured Viscous Time Step Estimate
If any body could help,
I am looking for an unstructured viscous, or Von Neumann, time step estimate. I have come across a few papers that address this for structured grids based on eignvalue analysis. Swanson, NASA TP 1336 is one such paper. I have done an extensive literature search, and I have found nothing when it comes to unstructured grids. Everybody uses on a inviscid, or CFL based time step estimate. Most of the analyses where done on 1D or 2D systems. It is a lot of work to translate their analysis into unstructured termonolgy. (In fact, I am better of rederiving everything.) If anybody could help, or would like to collaborate, I would appreciate it. Thanks, Tony 
Re: Unstructured Viscous Time Step Estimate
(1). Can you consider the structured mesh as a special case of unstructured mesh? (2). I don't think that the ordering of the mesh will have anything to do with the stability of the method. (3). The stability of the method will require some information about the mesh size and the velocity. That's about all you need. (4). Distortion and the skewness will be a different story, because I don't think you will be able to resolve that within a cell. Normally, the distance is the minimum distance within a cell. (5). My suggestion is: create a structured mesh as if it is a special case of the unstructured mesh, and study the stability of the numerical scheme on it. Then it is just a function of the scheme used.

Re: Unstructured Viscous Time Step Estimate
John,
Thanks for the response. Let me clarify what I am doing. There are two ways, that I have found, to go about implementing a viscous, or Von Neumannm stability limit. 1) I can use the basic 1D VNN stabilty limit from the convection diffusion equation. I believe it is this: deltat = deltax / 2 * nu where nu is the kinetmatic viscosity. Most people use this in structured codes, because it is simple and apparently works. In structured codes, you can get different time step in different directions, because strucutred codes in general handle grid stretching much better. (Because of the solution process.) Now, I really don't care to much about getting an optimal time step in each direction for an unstructured code, because this is nearly impossible for unstructured codes. I think the easiest thing to do is get a representative deltax for and unstructured grid. I could do this by taking the cell volume and dividing by the surface area to get some distance, or I could take the smallest deltax and use this criteria. (Which is probably how I would do it.) 2) The way I would like to do this, is to use the maximum eignvalue of the viscous fluxes, analgous to using the spectral radius, the maximum value of the invicid fluxes. Roy Swanson gives this criteria for structured grids. However, it is not easily transfered to unstructured grids, mainly because everything in the unstrucured sense is based on cross diffusion terms. Also, this was only done in 2D, and I cannot figure out what some constants are representing. So, I would painfully have to rederive this for unstructured grids. I was hoping someone could send me a reference. Since I am dealing with internal fluid mechanics problems, viscous terms are just as important as inviscid terms. Hence, I would get a much better solution if I use this additional stability. Thanks, Tony 
Re: Unstructured Viscous Time Step Estimate
(1). You may want to tell us more about your definition of the "structured grid" and the "unstructured grid".

Re: Unstructured Viscous Time Step Estimate
John,
By structured grid, I mean a grid that is parameterize by 2 coordinates in 2D, and 3 coordinates in 3D. Because of this fact, algorithm are designed to take advantage of grid streching. Unstructured grid, I mean a grid that is parameterized by nodes, which are defined by x,y,z coordinates, which elements (or FVM cells) are defined. This grids cannot be parameterized by simply two or three coordinate systems. Tony 
Re: Unstructured Viscous Time Step Estimate
(1). So, you are talking about the global coordinates. (2). If you study the behavior in a cell or element or grid, then you are always talking about the local coordinates.(3). If you have a foursided quad, is it a structured mesh or unstructured mesh?

Re: Unstructured Viscous Time Step Estimate
Yes, for unstructured meshes, I am talking only about global coordinates. Nothing is done with our code in local coordinate systems. So I am always looking at the cell in global coordinates.
Quads and Hexes are still unstructured, because they are not defined by only two and three coordinate systems. They are computed respect to global coordinates. Tony 
Re: Unstructured Viscous Time Step Estimate
(1). I don't have a solution to this problem. (2). Apparently, stability analysis is based on analytical property which requires the use of coordinates and derivatives. This can be a problem to do when you have random points (or vetices).

Re: Unstructured Viscous Time Step Estimate
Try to get some earlier papers of Mavriplis in ICASE. I remember he discussed this problem in some one.

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