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-   -   Hi Resolution Scheme (http://www.cfd-online.com/Forums/cfx/20753-hi-resolution-scheme.html)

Craig Johansen October 26, 2004 13:23

Hi Resolution Scheme
 
2 Questions:

1) I know that the Hi-Resolution scheme keeps Beta (blend factor) as close to 1 as possible. This keeps the advection scheme as close to second order accurate as possible. How is Beta Calculated? Is it based upon a CFL criteria?

2) I couldn't figure out how to plot the distribution of Beta throughout my solution in CFX post. Is this possible?

Thanks

Glenn Horrocks October 26, 2004 18:18

Re: Hi Resolution Scheme
 
Hi Craig,

1) beta is calculated on boundedness considerations, not CFL. Have a look at the documentation, page 235, Numerical considerations.

2) You can plot beta in Post. The beta variables will appear as "[variable].beta" under the additional variable list.

Glenn Horrocks

Craig Johansen October 26, 2004 20:21

Re: Hi Resolution Scheme
 
Thanks Glenn,

I read the documentation on page 235 regarding beta before I posted the message. It says Beta is calculated to be as close to 1 as possible without VIOLATING boundeness principles. It is not calculated on boundedness considerations, it is just assuring that Beta does not exceed a certain value(3^0.5).

It says the 'recipe' for Beta is given by Barth and Jesperson (1989). I went and got that paper but I still do not know how Beta is calculated.

I am under the impression that Beta is a function of local stability. If the local position in the computational grid is unstable, CFX will decrease the value to Beta applied to that area. If this is true, what stability criteria does is follow to calculate Beta. If it is not based on a cfl number, is it based upon a local cell Reynolds number?

Thanks for the information about Post.

Glenn Horrocks October 27, 2004 18:57

Re: Hi Resolution Scheme
 
Hi Craig,

I think you have misunderstood the documentation. Boundedness is different to numerical stability criteria such as CFL. Boundedness refers to keeping the solution within the possible physical limits, such as the temperature of any part of the fluid between a hot and cold plate must be between the hot and cold plate temperatures. It is not physically possible to have a temperature colder than the cold plate, but in a numerical simulation with poor boundedness it can occur.

I have not seen the B&J paper, but if that does explain things adequately, try the textbook "Computational Fluid Mechanics" by Roache. It has an excellent description of the various issues for accurate CFD simulations such as numerical stability and boundedness.

Glenn Horrocks

Neale October 30, 2004 21:06

Re: Hi Resolution Scheme
 
Beta is used for the high resolution advection scheme. In that scheme it needs an "upwinded" value of "phi" at integration points:

phi_ip = phi_upwind + beta * r dot grad(phi)

-phi_upwind is determined based on the sign of the mass flow rate at the integration point.

-r is the vector pointing from the upwind node to the integration point.

-grad(phi) is the control volume gradient of phi at the upwind node

For first order upwind, beta =0, for second order beta = 1.

The high resolution scheme calculates beta such that the value of "phi" at the integration point will not create an overshoot or undershoot with respect to the local nodal values.

There must be an equation in the Barth and Jesperson paper that has something like max(..., min(...), etc...) This equation is called a "limiter". The calculation of beta is based on that.

Neale



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