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jojo81 October 15, 2012 12:05

Analytic solution for 2D steady Euler equations

I'm trying to verifiy my CFD code for low Mach number flows with some analytical or manufactured solutions.

The convection and/or diffusion equation for a scalar is fine and now I focus on the Navier Stokes equations:

\nabla \cdot \underline{u} = 0
\frac{\partial \underline{u}}{\partial t} + \left( \underline{u} \cdot \nabla \right) \underline{u} - \nu \Delta \underline{u} = - \frac{1}{\rho_0} \nabla p

First, I want to check the non linear convection term, so I suppress the viscosity considering a perfect fluid and starting from the Green Taylor vorticies solution for Navier Stokes:

u(x,y,t) = - \cos (2 \pi x) \sin (2 \pi y) e^{-8 \pi^2 \nu t}
v(x,y,t) = + \sin (2 \pi x) \cos (2 \pi y) e^{-8 \pi^2 \nu t}
p(x,y,t) = -\frac{1}{4}\left[\cos(4 \pi x) + \cos (4 \pi y)\right] e^{- 16 \pi^2 \nu t}

I get this solution:

u(x,y,t) = - \cos (2 \pi x) \sin (2 \pi y)
v(x,y,t) = + \sin (2 \pi x) \cos (2 \pi y)
p(x,y,t) = -\frac{1}{4}\left[\cos(4 \pi x) + \cos (4 \pi y)\right]

which satisfies the steady Euler equations:

\nabla \cdot \underline{u} = 0
\left( \underline{u} \cdot \nabla \right) \underline{u} = - \frac{1}{\rho_0} \nabla p

I tried this solution with my CFD code setting boundary and initial conditions with the solution, the pressure is also imposed, not computed, with the solution.

At each iteration, I solve the velocity components and then update the convective flux, solve velocity, update the convective flux, etc. I use finite volume, upwind or centered convective scheme.

My problem: after some oscillations, the flow diverges and I don't understand why. Is the solution unstable ? Can't a solver for Navier Stokes solve an Euler problem ?

Did or is anybody try this kind of problem ?


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