Sponge functions

Sponge functions#

The sponge functions introduce additional source terms into the governing equations to stabilize numerical calculations. They are mainly used as an additional boundary treatment for outflow or far-field boundaries. Including a sponge ensures that perturbations decay in the outflow or far field, thus mimicking open boundary conditions and improving spectral accuracy. In FELiCS, if a sponge coefficient is given, sponge functions are incorporated by iterating over all equations and adding the corresponding damping terms to each. The sponge coefficient can be specified as a spatial field in the mean flow file.

Assumptions:

  • fluctuations can be smoothly damped with a specifically tailored sink term

References:

Nomenclature:

  • \(\sigma\): sponge coefficient, defined by user. Should be zero in the main parts of the domain where a physical solution is desired, and be greater than zero in the respective boundary region.

  • \(\phi_{i}\): state variables for which the equations are solved (e.g.: fluctuations/modes)

  • \(\phi_{i, target}\): target value for the state variables in the boundary region; assumed to be zero for linear problems; needs to be defined by the user if a non-linear problem is solved.

  • \(X_{\phi_{i}}\): FEM test function for the corresponding equation

Nonlinear equations#

When the sponge is activated, the following term is added to the nonlinear equations for all state variables \(\phi_{i}\):

\[ -\sigma (\phi_{i} -\phi_{i, target}) \]

Weak form#

The weak form of the nonlinear sponge term, as implemented in FELiCS is

\[ \int_\Omega -\mathrm{j} \sigma (\overline{\phi_{i}} - \phi_{i,target})\cdot X_{\phi_{i}} dx \]

Linear equations#

The linear form of the sponge term is

\[ -\sigma \phi_{i}' \]

Weak form#

The weak form of the linearized sponge term, as implemented in FELiCS, is

\[ \int_\Omega -\mathrm{j}\sigma \phi_{i} ' \cdot X_{\phi_{i}} dx \]

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