##### What we want to achieve

When we simulate fluid flow, we have to cut a finite computational domain out of an entire flow region. For accurate simulation, we need to let fluid and sound wave flow smoothly out of the domain through the boundary.

The reflection at the boundary has a larger effect on the solution especially when we perform a compressible flow simulation as can be seen in the following movie (Upper: With reflection, Lower: Without reflection of sound wave).

In OpenFOAM, we can use two approximate non-reflecting boundary conditions:

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}

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template<class Type> Foam::tmp<Foam::scalarField> Foam::advectiveFvPatchField<Type>::advectionSpeed() const { const surfaceScalarField& phi = this->db().objectRegistry::template lookupObject<surfaceScalarField> (phiName_); fvsPatchField<scalar> phip = this->patch().template lookupPatchField<surfaceScalarField, scalar> ( phiName_ ); if (phi.dimensions() == dimDensity*dimVelocity*dimArea) { const fvPatchScalarField& rhop = this->patch().template lookupPatchField<volScalarField, scalar> ( rhoName_ ); return phip/(rhop*this->patch().magSf()); } else { return phip/this->patch().magSf(); } } |

##### 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.

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template<class Type> Foam::tmp<Foam::scalarField> Foam::waveTransmissiveFvPatchField<Type>::advectionSpeed() const { // Lookup the velocity and compressibility of the patch const fvPatchField<scalar>& psip = this->patch().template lookupPatchField<volScalarField, scalar>(psiName_); const surfaceScalarField& phi = this->db().template lookupObject<surfaceScalarField>(this->phiName_); fvsPatchField<scalar> phip = this->patch().template lookupPatchField<surfaceScalarField, scalar>(this->phiName_); if (phi.dimensions() == dimDensity*dimVelocity*dimArea) { const fvPatchScalarField& rhop = this->patch().template lookupPatchField<volScalarField, scalar>(this->rhoName_); phip /= rhop; } // Calculate the speed of the field wave w // by summing the component of the velocity normal to the boundary // and the speed of sound (sqrt(gamma_/psi)). return phip/this->patch().magSf() + sqrt(gamma_/psip); } |

##### What do lInf and fieldInf mean?

Coming soon.