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July 10, 2003, 06:59 
aspect ratio

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I'm trying to solve the oil flow in a bearing. In the gap (which is very narrow, of course) I must introduce elements with big aspect ratio (A.R. = 500) to avoid an excesive number of elements. It is usually prescribed that the aspect ratio of cell elements should be limited (to 100?), at least in FVM calculations. 1 I wonder what is the numerical issue beneath this restriction. 2 Can a proper coordinate transformation handle this restriction?? Can anyone help me, please? Thanks a lot, Gorka 

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July 10, 2003, 11:01 
Re: aspect ratio

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the general reason is if you have a large aspect ratio then the disturbances will propagate faster in one direction then the other direction. This is not desired when you have high gradients in flow field. there are many mathematical issues too. try to read some book on grid generation for details
J 

July 10, 2003, 20:23 
Re: aspect ratio

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Centaur's remarks are well taken. Let me speculate and you can think about it.
The scales are quite different here. For steady flow you have the strong boundary effects, shear across the channel, but maybe little happening along the channel. Very long thin elements would seem appropriate (if the transformations are handled correctly). If there is a possibility of instabilities developing due to the strong shear, then these instabilities might be of similar scale (in each direction), and long thin elements would probably not be appropriate (insert Centaur's remarks here). 

July 11, 2003, 04:27 
Re: aspect ratio

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First, let me give you the rap on the "optimal cell shape": if you've got a rapidly varying solution in one direction and a slowly varying solution in the other, you should adjust the shape of the CV such that the rate of change across the CV is approximately the same in both directions (this comes in a number of variations, including "isotropic under apporpriate measure" and similar).
The numerical issue is the noise in the gradient: as the aspect ratio grows, the error in the gradient evaluation in the two (three) directions also grows. You know you've overdone it when the noise from the "slow" direction polutes the other directions and the thing goes boom! With a decent code you should be able to go over 100 though... ad 2: not to my knowledge. The most reliable 2nd order gradient calculation for the FVM is least squares, which inverts a geometrical matrix  hence the pollution. On Gauss gradients things are worse. If you solve the problem please tell us how you did it Hrv 

July 11, 2003, 06:24 
Re: aspect ratio

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Hi guys,
can someone give me a reference that talks aobut the issues raised above? Thanks. 

July 11, 2003, 15:47 
Re: aspect ratio

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Just a thought ...
If the aspect ratio is so large  based on the assumption that the variation along the shorter side is faster, then what is the logic against using a locally 1D solution. Again, the variation along the long side is supposed to be very small, and for all practical purposes it seems to me that the solution can be decoupled in each direction. I'd like to learn from the experts why this is a bad idea! Adrin Gharakhani 

July 11, 2003, 18:27 
Re: aspect ratio

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solutions cant be decoupled. lets take a simple example. say u r solving a 2D case for example famous driven cavity problem. when u discritize the Xmomentum eqn over the doamin, you would have velocities of Y directions and the cell length in Y direction in calculating coefficients for xmomentum eqn same way Ymomentum eqn will have velocities in xdirection as well as cell width in that direction. so they are coupled system and there is no way you can decouple it. If you can decouple the solution in each direction then that would be real great research.
J 

July 12, 2003, 15:58 
Re: aspect ratio

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When the variation in one direction is very small compared to the other, then for all practical purposes the former can be assumed constant, and as such you can "decouple". Parabolized NS equations come to mind, though for lubrication type flows even more simplifications can be made (unless, of course, there is massive flow separation)
Adrin Gharakhani 

July 14, 2003, 04:30 
Re: aspect ratio

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Adrin you're effectively right. If the aspect ratio is large you can derive an asymptotic expansion for the soultion of the NavierStokes equation  This is exactly what is done in Shallow water theory, Lubrication theory in a small gap and classical boundary layer theory (to name but a few cases).
Actually the problem of bondary layer stability is a good example of the conjecture of Jonas in an earlier post. Here the wavelengths of the instability waves are much shorter than longitudal development scale of the boundary layer. Tom. 

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