I'd like to simulate fluid flow over an ellipse

[pavelkomarov]

Is there a good way to simulate fluid flow over an ellipse with pyclaw? I think it's a 2d Euler equations problem, but no existing example is close enough for me to understand how I'd do this using this library.

[mandli]

I think there are two approaches you could take:

1. You could take an annulus (periodic domain in one dimension) and map it to the domain you are interested in, running a 2D Euler example with the boundary conditions along the ellipse that you were interested in (e.g. wall boundary conditions). There is not a specific example to do this but a few that use periodic boundary conditions and map to a torus such as this example.
2. I would suggest taking a look at the 2D Variable Coefficient example. It shows how to map a quadrilateral grid to a shape and assigns different acoustic parameters to 2 circular inclusions. You could modify this example to have an ellipse inclusion and provide different coefficients to the Euler equations for inside and out side the ellipse.

Hope this gives you a start. If you get stuck on something let us know.

[pavelkomarov]

I don't really want acoustics, do I? I'm trying to do a fluid flow. The ellipse is a solid shape, which the flow can not penetrate. I want to simulate a whole rectangle, where the ellipse lives in the middle, not just the border of the shape. I'm basically trying to do primitive CFD.

The example I keep looking at is the bubble bursting, but that's not really what I want either, and the $\gamma$ and all that seems only tenuously related to the Euler equations given in comments at the top of the file.

[mandli]

The Euler equations in the context of hyperbolic PDEs are usually compressible so support acoustic waves. I think what you are perhaps suggesting is the incompressible version, which PyClaw does not do by itself. There are a number of ways to do this but there are no examples of it. The setup for the hyperbolic portion of the equations though can be used with an appropriate operator splitting. In that case I would suggest starting from option (1) above.

As an aside, $\gamma$ is the ideal gas constant for the compressible version of the Euler equations that is needed to close the system.

[pavelkomarov]

Well I would like compressible flow, but I want the whole fluid to flow, not just to simulate a single acoustic wave. Forgive my ignorance, but PyClaw is a hyperbolic PDE solver, so it can't at all do elliptic or parabolic PDEs? Which class do the compressible Euler equations fall in to?

[mandli]

The full Euler equations do more than just acoustics, the linear acoustics equations are linearizations of the Euler equations. The compressible Euler equations are also hyperbolic usually. PyClaw and the rest of Clawpack are well suited to the hyperbolic parts of PDEs but an operator splitting approach can be used to add an additional solver step to handle parabolic terms. For instance, you could take an advection-diffusion equation such as

$$u_t + u_x = u_{xx}$$

and split it into two-steps

$$u_t + u_x = 0$$

$$u_t = u_{xx}$$

incurring a splitting error. @rjleveque's book on finite difference methods gives a good introduction to how this works and also how to do this in Clawpack in general.

[ketch]

@pavelkomarov Welcome to the Clawpack community! It's great to have a new user who is also willing to help with development.

I believe the nearest example is clawpack/pyclaw/examples/euler_2d/shock_forward_step.py. But you'll need to make some significant changes. As @mandli explained, you will want to use a mapped grid, to fit a non-cartesian obstacle.