Hi Praveen:

I'm working with the reference of Bassi and Rebay. More precisely, I'm studying the sensitivity of DG methods with respect to the solid wall boundary condition.

I have these two papers. In Bassi-Rebay they said that they don't use artificial viscosity (in space) but I'm not sure that their RK scheme is more dissipative than my fourth order RK.

When I say that I have oscillations I'm not refering to the convergence. I know that it is typical. I have a oscillatory solution. I'm using the same meshes of Bassi-Rebay and for example I have no convergence in the case of mesh 1 with isoparametric quadratic elements. If I increase my order of interpolation my oscillations grow (typical without dissipation).

Do you think that my loss of convergence in the same test case is given by the time integrator scheme.

Many thanks for your comments.

Ruben, as you said your oscillation develop behind the cylinder, a question comes to my mind: what's the area ratio of the cylinder cross section and comp. domain? I mean, how large is the wake region. for there are, as of now, in my opinion, two main possible causes: 1. Euler equation (inviscid flow) are not very suitable for a flow of 0.3M (subject to bitter discussion), 2.treatment of boundary condition; once you have simple outlet bc (p=p2) instead of more proper non-reflecting (even its simplest form of mirror image ghost cell works well) you can expect oscillations inbetween boundary and "disturbance"

I hope that speaking on disturbance you mean lo- & hi-pressure spots shedding at cylinder and travelling towards domain border where they actually can bounce back However, I somehow did not remark whether Ruben solves steady flow case or trying to find out when the unsteady flow calms down to steady one...? What are your initial conditions, Ruben?

1. Euler equation (inviscid flow) are not very suitable for a flow of 0.3M (subject to bitter discussion)

Nice statement ... what do you base it on?

And - are they (the Euler equations) appropriate to solve flow with Ma=3.0 or Ma=0.03??

What's wrong with Ma=0.3? As it has been mentioned earlier, it is a really good test-case to check the dissipation in your scheme.

What *physics* do the Euler equations describe?

convection and wave propagation

Good so far... next question...

What fluids under either of these two physical scenarios (convection, or waves) have kinematic viscosity *equal to* zero?

It is a model, just like many other things. Surprisingly often you could achieve good results with an Euler simulation plus added friction from a boundary layer computation (or a simple flat plate, back of the envelope assessment).

There are many applications where inviscid computations can be very helpful:

- wave drag of a ship - wave drag of a rocket - pressure wave propagation - etc.

and the Ma=0.3 cylinder it is a brilliant test case to validate your numerics - any dissipation is wrong!!

Then, as it is only a model, without friction/dissipation/dispersion in the system, it should bounce onto eternity. Thus, its only use would be to estimate how poor your model is at estimating a frictionless flow situation. Is that the point of the OP?

Should his model show wave activity... absolutely... *should* they reach a steady-state? Never on your life - if modeled correctly.

If you truly wanted, at M=0.3, to model a more realistic physical situation, then I would encourage you to go to the N-S. Anything else is playing with numbers, IMHO. :-/

With N-S, at M=0.3, you will have some fun...

diaw...

>Inviscid flow past a cylinder at around M=0.3 is a good test case to study numerical dissipation in the scheme

Why? To see if you have enough artificial dissipation? Most people are aiming for less of it, not more. What exactly do you conclude for your scheme, once you have added enough dissipation to make it work (or found a dissipative enough iteration method)?

>good scheme should give a good approximation to potential solution

Why? Don't you think going from rotational to irrotational flow is a step backwards? Why should my Euler solution (which includes the capability of solving for rotational flow and is prone to flow instability) be forced to step down and reproduce unrealistic potential flow? I don't doubt that you can get pretty close to potential flow if you just repress all instability by adding enough dissipation. It has been done. I just don't see the point of it. To make any sense of flow over a blunt body, we should actually solve for viscous flow (which isn't all that difficult to do). Viscous effects determine steady or unsteady separation and wake behavior. Those are the things that are essential for flow over blunt bodies, unless you talk about Stokes flow, where viscosity is again essential, for different reasons.

It's generally a good idea to verify if a high-fidelity model also produces lower order solutions accurately. I just think in this particular case you shouldn't insist on it too hard. Your grid-based Euler method includes some amount of numerical dissipation (regardless how hard you try to reduce it, you'll need some for stability). Associated with that numerical dissipation is a pseudo-diffusion, pseudo-viscosity or, if you will, pseudo-Reynolds-number. Your Euler code will have some (inaccurate, numerical) viscous effects included, which causes flow separation around corners, or at blunt trailing edges of airfoils. That pseudo-viscosity creates trouble trying to solve steady flow over a blunt body, because as soon as you have viscosity (especially when it is small but non-zero, like at high Reynolds number), you'll have a flow instability.

Unable to reduce numerical dissipation to absolute zero (you could try increasing grid resolution...), your best chance is to go the other way --> increase numerical dissipation to suppress the instability, hoping that it won't affect the steady solution too much (you should check if your cylinder flow is symmetric, front to back, like potential flow). And I suspect increasing dissipation is exactly what people have done (knowingly or not) to obtain an inviscid incompressible steady-state solution for flow over a cylinder.

It is no secret to experimentalists that you can get a pretty 'potential' looking flow by actually using very viscous fluids at low velocity (Stokes flow). Seems like a paradox to approximate inviscid potential flow by extremely viscous (or numerically dissipative) flow, but that's what you have to do to supress all separation and instability.

"An Euler Code that can compute Potential Flow" M. Rad, P. Roe, University of Michigan, Ann Arbor, USA Finite Volumes for Complex Applications - Problems and Perspectives July, 19-22, 1999 / Uni Duisburg, Germany

can't be so stupid atfer all ... ;-)

Nobody said 'stupid'. But also nobody has explained the value of this endeavor, so far. Would you care quoting from this paper? What did the authors attempt to prove, and why do you think your code needs to be able to replicate their results? Maybe we can shine some light on this.

Potential flow is a model; it does not correspond to reality and nobody has made that claim in this thread. Maybe the questioner is not interested in reality. Potential flow equations have some characteristics which we want the numerical scheme to reproduce.

I dont know what reality is; reality is not a quality you can test with litmus paper. - Hawking

Are you sure that Bassi-Rebay have not used some upwind numerical flux function ? I dont have the paper now but will check it on Monday. I doubt that it will work without some upwind flux or some added dissipation.

These schemes make use of an elliptic-hyperbolic decomposition of the Euler equations. Some very interesting developments are taking place in this area. Apart from the cylinder test case they also show that their scheme can obtain potential flow past a large aspect ratio ellipse, which is known to be a very tough case for upwind euler schemes or schemes with added dissipation, see Pulliam. It is definitely worth reading.

References

T. Pulliam, A computational challenge - Euler solution for ellipses

Thanks for all the comments.

I would to say that I'm not interested in real flows (at this moment). I'm testing the code and the inviscid flow past a cylinder is one of the most useful tests for the numerics.

I found another classical test that I'm just implementing: the same problem but changing the geometry. I'm using a channel with a circular bump.

Which is the best way to add some dissipation in my scheme?

Hi Praveen:

I have no access to the reference: T. Pulliam, A computational challenge - Euler solution for ellipses

Can you send me the paper?

>Potential flow equations have some characteristics which we want the numerical scheme to reproduce

Still vague. What I am trying to get are some concrete answers (to satisfy my own curiosity). Are people doing this for the study of elliptic vs. hyperbolic flow (other post)? Is that what Ruben is studying? What part of potential cylinder flow is hyperbolic, anyway?? Do I have to read that paper to find out?

There are people in this forum who are trying to make this case work for their code, or have been trying in the past. These people must have some concrete idea of 'why' they are doing it, other than 'I think it might be a good idea'.

My point is: To accurately approximate flow with a CFD method, you must try to keep dissipation at a minimum. In view of that, is it good advice to tell Ruben to calibrate his Euler code against potential flow over a cylinder?? Think of the consequences. Now all he can think of is: How do I increase dissipation... ?

Ruben, unless you have a very good reason to do so, e.g. you do research in the same area as the authors of the referenced papers, you don't need to make you Euler code work for this flow. My advice is to try more reasonable cases (you mentioned a bump), or implement viscous fluxes and predict vortex shedding on the cylinder. Inviscid flow over airfoils should be tried as well, and if you're computing compressible flow, there are many more opportunities to validate your Euler method, without ever having to force it to do something it's not meant to do, sacrificing accuracy for robustness.

The simplest numerical flux function is the Lax-Friedrichs flux, which seems to be commonly used in DG methods. It is however highly dissipative. A local lax-friedrichs flux based on a characteristic decomposition is much better. For this see

B Cockburn, San-Yih Lin and Chi-Wang Shu, TVD Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws III: One Dimensional Systems, JCP, 84, 90-113 (1989)

See paper no. 39 here

http://people.nas.nasa.gov/~pulliam/...lications.html