Non-Reflecting Boundary Conditions in OpenFOAM


There are two approximate non-reflecting boundary conditions available in OpenFOAM:

They determine the boundary value by solving the following equation

\begin{align}
\frac{D \phi}{D t} = \frac{\partial \phi}{\partial t} + \boldsymbol{U} \cdot \nabla \phi = 0, \tag{1} \label{eq:advection}
\end{align}

where \(D/Dt\) is the material derivative and \(\boldsymbol{U}(\boldsymbol{x}, t)\) is the advection velocity.

We assume that the advection velocity \(\boldsymbol{U}\) is parallel to the boundary (face) normal direction and rewrite the eqn. \eqref{eq:advection} as

\begin{align}
\frac{D \phi}{Dt} \approx \frac{\partial \phi}{\partial t} + U_{n} \cdot \frac{\partial \phi}{\partial \boldsymbol{n}}= 0, \tag{2} \label{eq:advection2}
\end{align}

where \(\boldsymbol{n}\) is the outward-pointing unit normal vector.

These boundary conditions are different in how the advection speed (scalar quantity) \(U_{n}\) is calculated and it is calculated in advectionSpeed() member function.

advective B.C.

The advection speed is the component of the velocity normal to the boundary

\begin{align}
U_n = u_n. \tag{3} \label{eq:advectiveUn}
\end{align}

waveTransmissive B.C.

The advection speed is the sum of the component of the velocity normal to the boundary and the speed of sound \(c\)

\begin{align}
U_n = u_n + c = u_n + \sqrt{\gamma/\psi}, \tag{4} \label{eq:waveTransmissiveUn}
\end{align}

where \(\gamma\) is the ratio of specific heats \(C_p/C_v\) and \(\psi\) is compressibility.

 

Author: fumiya

CFD engineer in Japan

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