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andy June 15, 2006 06:24

Limited Gradients for Viscous fluxes and Sources
Dear all,

I've had good results with going from first to second order using linear reconstruction and then a limiter for scalar equations and the euler equations with step profile convection and shock tube simulations, respectively. This is on a cell-vertex based mesh using a Roe type scheme.

However, when I take a well converged first order solution, and then restart it with linear reconstruction with a limiter for a viscous flow, I get strange results. I am using the limited gradients for the viscous terms and the source terms for the turbulence model. I thought this would be correct and consistent, but when the simulation is run, the turbulence dies away.

When I change the source terms and the viscous fluxes to use the old unlimited grads, the solution appears to least as mentioned.

Is it correct to use the limited grads throughout, and should I expect to get a normal answer, hence I have a bug?? Or should I be using the limited grads just for the convective terms?

Does anybody have any experience with viscous flow behaving strangely when using limited grads?

Many thanks,


AJS June 16, 2006 03:05

Re: Limited Gradients for Viscous fluxes and Sourc
Andy, The justification for limiting gradients while reconstructing is the TVD criterion. WHat is the justification for limiting gradients in the diffusive fluxes? In fact, for the source terms you are not using the gradient to project values anywhere. So why limit it? You should be using unlimited gradients for these terms. Also, which turbulence model are you using?


andy June 16, 2006 03:42

Re: Limited Gradients for Viscous fluxes and Sourc

I'm well aware of that fact that in the viscous and source terms no project is required. However, to solve the stress tensor at an interface, and to compute the k-e product term, as well as I dilatation term for a chemistry model, I require gradients. Since the limiter I have implemented is a gradient limiter, I first compute the gradients, then overwrite them with a limiter gradient - the details of which are un-important. Now I compute the convective fluxes using an upwind type method. In the turbulent case I then have to compute the stress tensor at a cell interface, and the previously mentioned source terms also require gradients.

The question I was asking was, to be consistent, should I use the same limited nodal gradients within the viscous and source terms, all-be-it for a different reason, or should I bother to additionally store the unlimited gradients, and use these for the viscous and source terms.

I'm well aware of the fact that TVD does not admit real viscosity. But from a consistency point of view, it would appear cleaner to use the same grads throughout. There is a wide field of work on the oscillatory nature of gradients for the pure diffusion problem, no convection, for which people remove the oscillations through limiters - hence I don't think my question is that stupid.

None-the-less I can't ignore the strange results I get when I use these limited grads.


Praveen. C June 16, 2006 04:17

Re: Limited Gradients for Viscous fluxes and Sourc
Your observations are definitely interesting. In my own work on vertex-based FVM, I have used unlimited gradients for the viscous terms. I use a MUSCL-type scheme where I store unlimited gradients. During the loop over the cell faces, I use a limited reconstruction to obtain the two states on either side of the cell face (see the CFD book by Lohner). The viscous terms are actually treated using a Galerkin approximation which has good stability properties for elliptic operators.

andy June 16, 2006 04:56

Re: Limited Gradients for Viscous fluxes and Sourc
You can see my point though, right? I mean based on consistancy. It's interesting that you've found the same thing. Could I ask what limiter? I'd also be interested in a paper reference/short explanation on you treatment of ellipic terms. Certainly, I use a method to remove any odd-even decoupling at an interface for the viscous terms, but I'd like to hear what you do.



Praveen. C June 16, 2006 06:09

Re: Limited Gradients for Viscous fluxes and Sourc
Sorry if I gave you the wrong impression. I use limiter only for the convective flux, while viscous terms are treated using a Galerkin approximation. It is not a fully finite volume method. Some people refer to it as Galerkin-Finite-Volume method. This method is also used in NSC2KE. You can check the report on NSC2KE

andy June 16, 2006 06:16

Re: Limited Gradients for Viscous fluxes and Sourc
No you didn't give the wrong impression.



Angen June 16, 2006 11:38

Re: Limited Gradients for Viscous fluxes and Sourc

Could you give us references to works on the osillatory nature of gradients for the pure diffusion problem (no convection) and the type of limiters that were used to remove them.

I am not sure if it is helpful in your case but I was solving convetion-diffusion-reaction equations with correction designed based on Flux Corrected Transport methodology. The best results were obtained when the amount of artificial diffusion that warranty non-oscillatory solutions for pure convection was diminished by the amount of already excisting natural diffusion. I used uncorrected values of variables to calculate source terms.


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