Euler equations using FEniCS

Question

Hello!

I am looking for ways to solve the incompressible Euler equations using FEniCS. Standard methods are unstable due to the lack of a viscosity term.

Of particular interest is also how the "free-slip" boundary condition, u \cdot n = 0, is implemented.

Best regards,

Cardilin

Comments

Free-slip in the context of weak formulation usually means that surface is impermeable
$$ \mathbf{u}\cdot\mathbf{n} = 0 $$
and no tagential force is exhibited on the surface
$$ \mathbf{Tn}\cdot\tau = 0 \quad\forall\tau\textrm{ unit tangential vector to the surface}.$$

The latter condition is natural as one can ignore the term
$$\int_{\partial\Omega}\mathbf{Tn}\cdot\mathbf{v}\;\mathrm{d}S = 0 \quad\forall\mathbf{v}\textrm{ velocity test functions}$$
as underlying functions space is constrained by impermeability condition.

Implementation of $\mathbf{u}\cdot\mathbf{n}=0$ in FEniCS is easy when surface is aligned with coordinates surface. Just define

bc_free_slip = DirichletBC(V.sub(0), 0.0, surface)

for V underlying velocity space and surface given by a condition x[0] = const. For complex geometry one must abandon this approach and can stick to Lagrange multipliers, penalty method, etc. Check also http://scicomp.stackexchange.com/questions/7123/fenics-separate-boundary-conditions-in-normal-and-tangential-direction-of-mesh.

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Jan experience points +1