Hi!

I'm writting a code for solving the Euler equations of compressible flow.

I'm interested in state state solutions and I have problems with the oscillations. Even in the subsonic case I have spurious oscillations and the convergence is too slow. I think that I need a time integration scheme with damping. What scheme can I use?

Many thanks!

Which scheme are you using (in time and space)? On what type of grid are you solving?

I'm using Discontinuous Galerkin in space and I have implemented two integration schemes: fourth order explicit pade and fourth order explicit runge-kutta.

I'm solving in different grids but the main problems are in an structured grid around a circle.

Does your particular problem *have* a steady-state solution?

diaw...

of course

I have no experience with the Discontinuous Galerkin method.

Gut feeling would suggest you don't have enough spatial dissipation. With a standard FV scheme you can easily converge a steady solution using Runge-Kutta. You get waves running back and fors for quite a while, though.

Can you trace where the oscillation develop? Around circle, at comp. domain boundary....?

Around circle

diaw wrote:

Does your particular problem *have* a steady-state solution?

Ruben replies:

of course

diaw writes:

How can you be so sure? If a Steady-state solution does indeed exist, how long does the flow take to reach this 'steady' condition? Is it achievable in your lifetime, in a system with no dispersion? Food for thought.

is the galerkin method "the most classical one", i.e. something between fvm and fem? or have you modified it somehow? as you wrote you solve euler equations for compressible flow, I'd guess you only account for friction between the flow and the surface. you neglect viscosity at all. I also guess you use 2-D planr approach. then I guess your oscillation develop on trailing edge/half/portion. then, in my opinion, they arise due to area enlargement and clearly need a slope/flux limiter / recovery technique

Yes, the method is the classical Discontinuous Galerkin. I'm neglecting the viscosity and I'm working in the 2D case. Do you think that a time integration scheme with damping is not sufficient?

You are computing an inviscid cylinder - correct?

What is your boundary condition on the surface?

I hope you are NOT considering friction between fluid and surface, as suggested by "faber"!

You said the oscillation develop on the surface. Where are your oscillations? At the stagnation points, or in the region of the highest Mach number?

Some authors say that the steady state can be reached before 100.000 runge-kutta time steps. In these numerical experiments no artificial viscosity is added but I think that they use a runge-kutta method with damping.

You are computing an inviscid cylinder - correct? Yes

What is your boundary condition on the surface? Solid wall

I hope you are NOT considering friction between fluid and surface, as suggested by "faber"! Correct

You said the oscillation develop on the surface. Where are your oscillations? At the stagnation points, or in the region of the highest Mach number? The oscillations apear behind the cylinder

How did you implement the Euler wall? No convective flux? Or did you prescribe (somehow) a parallel flow direction? What is the free-stream Mach number, btw. ?

How did you implement the Euler wall? No convective flux? Or did you prescribe (somehow) a parallel flow direction? Parallel flow direction

What is the free-stream Mach number, btw. ? 0.3

...

the only idea left is that the numerical dissipation might become very low in the rear stagnation area (Ma -> 0). I really don't know how your scheme would behave there. Some FV methods (e.g. classical AUSM) might produce pressure oscillations in such a region.

Ruben wrote:

Some authors say that the steady state can be reached before 100.000 runge-kutta time steps. In these numerical experiments no artificial viscosity is added but I think that they use a runge-kutta method with damping.

diaw's reply:

If you are modeling the pure Euler equations, then you have no inherent dispersion (damping) in the governing equation & would have to bounce until eternity unless you work in some 'numeric' or 'artificial' dissipation of some sort.

diaw...

There are so many possible sources for oscillations, it's very hard to judge without knowing your code. The scheme is important, but so are the boundary conditions (solid wall, far field, reflecting versus non-reflecting...)

I can't really comment... but I am curious: Inviscid flow over a cylinder at Mach = 0.3??? How does that relate to any real flow? Are you trying to use your Euler solver to get a potential flow solution? The fact that other people have been successful in obtaining a steady state solution (where there is none in reality) for these conditions, may simply mean that their schemes are extremely dissipative. That your code won't give you an answer is not necessarily something bad.

Maybe you should try viscous flow.

Inviscid flow past a cylinder at around M=0.3 is a good test case to study numerical dissipation in the scheme. A good scheme should give a good approximation to potential solution, with left-right and top-bottom symmetry.

What is the order of basis functions used in your simulations ? What numerical flux function do you use ? What do you mean by oscillations ? Oscillations in the solutions or that convergence is highly oscillatory ? Can you post some pictures of your results ?

The DG method is sensitive to resolution of boundaries. If the boundary is only approximated by piece-wise linear curves, then the solutions can be grossly innacurate, see [1,2]. How fine is your grid ? Are you using isoparametric boundary elements ?

References

1. L. Krivodonova and M. Berger. High-order Accurate Implementation of Solid Wall Boundary Conditions in Curved Geometries. J. of Comp. Physics, Vol. 211:492-512, 2006.
   
2. F. Bassi and S. Rebay, High-order accurate discontinuous finite element solution of the 2D Euler equations, JCP, Volume 138 , Issue 2 (December 1997), Pages: 251 - 285