Non-reflective, buffer zone-type boundary conditions

Sponge layer BC (Israeli & Orszag, 1981)

Additional damping term

An additional damping term is added to the differential equation, which neutralises any deviation from the designated value. For the arbitrary variable q it can be formulated as follows:

$\frac{\partial \underline {\hat q}}{\partial t} = -\underline F_{phys.}(\un... ...e {\hat q})\textcolor{red}{- R_d(x, r) (\underline {\hat q} - \underline q_0)}$

The F_phys. denotes the physically obtained flux, and the red term represents the change in the model that reduces the quantity with propagation into the buffer zone.

Application at the sound source

This method is directly applicable at the sound source if enough data is available, e. g. from CFD, MS or analytical solution. We simulate an overlapping zone as sound source forcing the value to the given one for this case. This condition is valid if no wave is reflected back from the sound source.

A Perfectly Matched Layer (PML) BC (Hu, 2001)

No change of the group velocity (dispersion relation) please!

The condition, which delivers a Perfectly Matched Layer (PML) boundary condition, keeps the dispersion relation and the related (real part) of the wavenumber constant:

$\det(-\omega \underline E + k_x \underline A + \mu_{mn} [\underline B + m \underline C]) = 0$

If the wave number remains constant, no reflection is possible. As the damping is applied only in one direction, the effect on the other direction will be zero. This is observed e. g. with a plane wave propagating along the PML.

Historically: split PML

The first layer of this kind is the split PML formulated by Berenger (1994) for electrodynamics. Hu (1996) followed this route at first, however unlike with electrodynamics the mean flow must be carefully monitored, as shall be seen later. For the derivation we assume that the damping part of the PDE will only effect one direction.

$\frac{\partial \underline {\hat q}_x}{\partial t} = - \underline A \cdot\... ...underline q} }{\partial x} \textcolor{red}{- \sigma_x \underline {\hat q}_x}$

The damping coefficient σ _x, which is similar to the damping given above in the PML is in theory arbitrarily distributed. As shown below, the PML is a corrected sponge layer, with propagation allowed along the PML. The directional dependency of the PML is obtained by splitting all variables, and therefore this approach leads to unphysically split variables in the whole domain. A storage overhead by the factor of space dimensions is observed.

Instability in flow conditions

The main disadvantage is the instability in the presence of flow, which has been observed and discussed by many authors following publication of the PML formulation. This formulation is the starting point for the un-split PML proposed by Hu in 2001. This condition is unconditionally stable with a 1D mean flow and any kind of propagating perturbation mode (acoustic, entropy and vorticity).

A stable un-split PML

We start with a formulation of the split PML in Fourier space employing the angular frequency. The multiplication of the remaining undamped equation by e. g.:

$1+\frac{\mathrmit{i} \sigma_x}{\omega}$

allows the combination of the different direction parts of any acoustic field variable back to the un-split variable. The auxiliary variable:

$\underline q_1:=\int \underline {\hat q}\; \mathit{d} t$

must be obtained by integration, although this is only necessary in the buffer layer. The resulting equation is of the same kind as that shown above for the sponge layer. The main difference is the correction of the damping to the parallel component along the buffer layer.