Problem at the poles ...
 

Hi Folk,

I have used a hybrid Fourier-Chebyshev spectral method of collocation to simulate the outer layer of the atmosphere of a planet or a star. Basically written and solved in spherical coordinates, but assuming axisymmetry around the axis of rotation, therefore I ended up solving a two dimensional problem in r and theta (also assuming symmetry at the equator, so that theta goes from 0 to 90 degres only). Mainly the problem is that of a compressible flow (though shocks are not expected) with artificial viscosity, rotation and gravity. There is a density equation and 3 velocity (or otherwise momentum) equations and option for an energy equation (though isothermal or polytropic was usually assumed). Now this "object" is accreting matter, that is to say that matter is added (stream of gas) from a region near the equator and this matter eventually ends up on the surface of the (celestial) obejct. Then as the matter loses its angular momentum it migrates towards the poles. I managed to take care of the outer boundary conditions by imposing them on the characteristics of the flow (open boundary, execept where there is an inflow), and the inner radial boundary is the rigid (rotating) surface of the star/planet. Now the simulations look fine untill the matter that is streaming inwards eventually reaches the poles. Then strong oscillations appear there and the code just 'explodes' (as spectral methods sometimes can be explosively unstable). I was told that the problem at the pole is a common one and that there is not much that can be done to handle the sudden accumulation of gas at the pole: Increase resolution? Increase viscosity? Not so possible in my case for CPU reasons.

Has anyone deals with that kind of flow (I guess people from Atmospheric reasearch might have?)?

Thanks, Patrick

Re: Problem at the poles ...
 

I'm not sure that your problem at the poles is the same as the one that occurs in atmospheric flows (although it could be related). In atmospheric flows the problem arises because of the convergence of the grid (this shouldn't be a problem for your axisymmetric flow). This convergence makes it almost impossible to enforce the CFL condition on the difference equations (up until fairly recently all atmospheric codes were explicit in time).

Have you looked at the possibility of Taylor expanding about the pole to get similarity boundary conditions on the flow?

Re: Problem at the poles ...
 

I am unfamiliar with spectral and astrofluid. But my experience with closed cavity is that the total mass must be conserved. Since in your case there is external fluid added in near the equator region, there must be BCs where fluid exsts. If mass accumulates, computation will diverge.

Problem NEAR the poles ...
 

Hi,

Thanks ALL for your comments.

The flow is 'free' to escape radially outwards, in addition due to the gravity it is compressed and can accumulate. I am not looking for a realy steady state, since it is impossible in that case, but just for state that evolves and does not diverge. The problem is of course time dependent, and solved using a 4th order Runge-Kutta scheme which works best for Chebyshev Methods.

At the pole itselft I actually skip the pole, since I am taking the grid (collocation) points shifted by half a grid spacing, such that the first point is just off the pole and the last one is just off the equator, but the same amount. The symmetry conditions are then applied using fourier expansion, where some of the variables have a symmetry of 2pi and other of only pi, and also some have antisymmetry relation at pi or 2pi, etc... Then all the variables are extended to either pi or 2pi and the spatial derivatives are taken using the Fourier's rule. In that manner I avoid the divergence at the pole in the euqation and do not need to expand in Taylor series there.

Patrick

Re: Problem NEAR the poles ...
 

The advantage of Taylor expanding about the pole is that, even if you don't use it explicitly numerically, it will tell you something about the nature of the solution near the pole. If I understand your problem correctly you have mass inflow at the equator and mass outflow at the pole (the fact that there is a build up at the pole suggests that the outflow condition near the pole is possiblby incorrect). The Taylor expansion near to the pole should give you a clue to what's wrong with your ouflow condition and will also tell you what the correct boundary condition should be.