test cases for 1D inviscid incompressible flow
 

Hello, guys
I have always been focusing on compressible flow. However, I would like to solve some incompressible problems. I prefer somebody could send me 1D test cases governing by Euler equations. The test case had better be a benchmark problem with exact solution. I would be grateful if anyone could share his wisdom and experiences.

Best Regards,

sorry but your question is not clear...for incompressible flows you mean divergence-free flow, rhight? But in 1D that reduce to have du/dx=0 so that u is constant along x. You cannot define a PDE for the governing of u motion.

Thank you for reply.
Maybe I did not express myself clear. I want to solve incompressible flow with artificial compressibility. So a pseudo time will be added into the mass conservation law. I just want to ignore viscous terms first. Maybe there remains some problems.

In 1D the general mass equation is:

d rho/dt + d(rho*u)/dx=0

how do you enforce the "incompressible" constraint?

With the constraint rho=constant. And to make original equations into a time-marching problems, a pseudo term will be added into mass equations.
Say d rho*/d t*, and t* is a pseudo time, rho* is a pseudo time. p= beta×rho*, p is pressure and beta is a parameter to be determined. We can apply time-marching strategy into incompressible flow problems with AC(artificial compressibility).

See in:
Chorin, Alexandre Joel. "A numerical method for solving incompressible viscous flow problems." Journal of computational physics 2.1 (1967): 12-26.

The first paper for AC. Hope I express myself clear this time.

but at a physical steady state with rho=constant you simply get du/dx= 0...
therefore your solution is a constant state for the velocity...It can only vary with time according to du/dt + dp/dx=0. But dp/dx is rho dependent..

The paper of Chorin is for a 2D flow...

--Yeah, the velocity should be constant at steady state. It seems the case is too simple, i should take viscous term account at least.

--I intend to combine AC with high order method for time-marching problem. I don't know whether this way can work, so I want to begin the simple case: 1D.

What is your suggestion?

even with viscous term in 1D you get u= constant ... the simplest 1D model is the Burgers equation in which du/dx does not vanish (therefore is a compressible model)... otherwise you must solve a 2D problem enforcing Div v =0

Ok. Thank your very much. I will try 2D.

And still, my original purpose is to ask for benchmark cases.

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