It is suggested in the literature to use Hybrid scheme for -2<Pe<2. I was wondering if Upwind scheme fails to provide acceptable results, or it's just matter of better accuracy.

I would appreciate your comments on this.

You must provide more information. Questions of the type "which is better" must always be posed in the particular context of your interest since there is no scheme which is universally better than all the rest. Where available please quote reference papers so that people know what you are talking about.

I am running a code for two phase flow, molding, with composition variation and I have linked the momentum, energy and mass balance equations. I have used the upwind but I have oscilations in my composition profile in the two-phase region where the Pe number is small. I was wondering if it is related to the Upwind scheme or the problem might be from somewhere else. Thanks for your comments.

Since you have oscillations in your solution then I assume that your upwind scheme is of an order of accuracy above one. From Godunov the computation of convective fluxes using a scheme with an order of accuracy above one will yield oscillations, sometimes called wiggles in the literature. It is related to the fact that implicitly or explicitly you are required to calculate a gradient to advance your order of accuracy from 1 to 2 or above. In two cells the gradients will be mismatched and hence this translates into an oscillation in your solution. Please note that this is not related explicitly to upwind or hybrid or even a bog-standard central difference, apart from that the schemes are trying to reconstruct the convective terms using an order of accuracy above one, since first order accurate solutions are far to dissipative, they will however not give undershoots and overshoots-oscillations. The only way to overcome this is to incorporate a limiter into your reconstruction procedure. This limiter should be differentiable, i.e. not a min max switch, and should not hinder convergence. A limiter works by calculating weights to multiply the derivatives by in order that at an interface between two cells they match. If chosen correctly you do not need a priori knowledge of the bounds of your variable, the limiter will correctly limit to these bounds. In other words if you use a min max switch you require a number to test whether or not the variable is oscillating, this is not always a priori knowledge, and if your limiter is not differentiable then you reduce an order of accuracy in the region of discontinuities like shocks.

Hope this helps, by far one of the most advance limiters that is derived in the context of TVD and edge based scheme is the following, although their methodology can be adapted to other solution procedures.

A high-resolution procedure for Euler and Navier-stokes computations on unstructured grids, in JCP 164, 165-203. P.Jawahar et al.

Cheers

Andy

Thanks a lot Andy, however, I am using a fixed grid control volume method with first order upwind scheme. The oscilation happens in the two phase region when in one grid two phases may exist at the same time while at the adjacent grid one phase exists. I have used central difference for approximation of diffusive terms and first order upwind scheme for the convective terms, there is no problem in the velocity or temperature profiles, the problem is in the composition. Initially there is no oscilation and as the saturation of the other phase increases the compoisition variation in the two phase region starts oscilating. This eventually affect the convergence of energy equation. I hope this clarifies my problem. I have tried the "Deferred-correction" method which uses the upwind scheme implicity and combines it with thecentral difference and upwind scheme calculated at the previous time step, but had no luck.

I am guessing here and could be wrong but do you model the composition with a passive scalar, or even a scalar with a source? Is it possible to model the flow as inviscid? Don't care if it is non-physical that is not of concern, but it will tie down whether or not the convective terms are responsible for the overshoots and undershoots, as they should be the terms causing the problems. I would go back to my original theory and say that I still do not understand how a first order scheme can give overshoots!! It could be something to do with your time-integration, I take it from your wording that you are writing the source? I would not imagine there can be anything else that is causing it, since all other terms are dissipative and would kill the overshoots. How exactly have you modelled the convective terms, I am guessing it is first order Roe? Not sure what else to say, although I would try the inviscid test...

Cheers

Andy

I don't know how work two phases flow computations. Do you have source terms ? In this case, it is possible that they give you oscillations and conservativity losses because schemes you used are not adaptated to non homogenous equations system. In the other case, please ignore this

There are lots of things that could cause your wiggles. You have to go back to simpler testcases to figure out what's wrong. For example: are you sure that your code works for a problem where you have only one phase? If no, test it. If yes, something is wrong with the multiphase implementation. Maybe the wrong equation of state (just a wild guess ;-)

Yours,

Philipp

Thanks a lot for your comments. I am using the fixed gird approach. The single phase works very well. Even in the two phase flow the Momentum and energy equations are smooth. The sourse term appears in the momentum and energy equations. I haver tried fixed time step but still have the same problem with composition variation. In the two phase region I have a sharp variation of composition that can be regarded as a shock. The Upwind first order scheme is able to handle it in the initial times, however, for longer times it starts to fail. I tried to remove the diffusive parts from the mass balance expression and the code worked fine, but the trend of the results were different. As for the diffusive terms in the compositional part I am using central difference which is widely used in the literature.

Try to have a look a this Lecture Series from VKI :

LS 1999-03, "Rankine-Hugoniot-Riemann Solver and Convergence acceleration for Low Speed Combustion" by B. Muller and P. Jenny.

In the first part, they explain why source terms can cause loss of conservativity and how to remedy to this.

Dear kevin,

Can you tell me where can I get this lecture notes?

Many thanks for your help.

The web site of the von Karman Institute is http://www.vki.ac.be/.

To order old Lecture Series, I think you should contact Mr Van Haelen ( vanhaelen@vki.ac.be ).

Good luck