## Thermal Boundary Conditions in OpenFOAM

Many thermal boundary conditions are available in OpenFOAM. I will upload some basic cases that explain the usage of these boundary conditions.

Source Code
src/TurbulenceModels/compressible/turbulentFluidThermoModels/derivedFvPatchFields/

• convectiveHeatTransfer

It calculates the heat transfer coefficients from the following empirical correlations for forced convection heat transfer:
\begin{eqnarray}
\left\{
\begin{array}{l}
Nu = 0.664 Re^{\frac{1}{2}} Pr^{\frac{1}{3}} \left( Re \lt 5 \times 10^5 \right) \\
Nu = 0.037 Re^{\frac{4}{5}} Pr^{\frac{1}{3}} \left( Re \ge 5 \times 10^5 \right) \tag{1} \label{eq:NuPlate}
\end{array}
\right.
\end{eqnarray}
where $$Nu$$ is the Nusselt number, $$Re$$ is the Reynolds number and $$Pr$$ is the Prandtl number.

• externalCoupledTemperature
• externalWallHeatFluxTemperature

This boundary condition can operate in the following two modes:
Mode#1 Specify the heat flux $$q$$

-k \frac{T_p – T_b}{\vert \boldsymbol{d} \vert} = q + q_r \tag{2} \label{eq:fixedHeatFlux}

* $$k$$: thermal conductivity
* $$q_r$$: radiative heat flux
* $$T_b$$: temperature on the boundary

Mode#2 Specify the heat transfer coefficient $$h$$ and the ambient temperature $$T_a$$ (Fig. 1)

-k \frac{T_p – T_b}{\vert \boldsymbol{d} \vert} = \frac{T_a – T_b}{R_{th}} + q_r \tag{3} \label{eq:fixedHeatTransferCoeff}

* $$R_{th}$$: total thermal resistance of convective and conductive heat transfer

R_{th} = \frac{1}{h} + \sum_{i=1}^{n} \frac{l_i}{k_i} \tag{4} \label{eq:Rth}

• fixedIncidentRadiation
• lumpedMassWallTemperature

There is a dimensionless quantity called the Biot number, which is defined as

Bi = \frac{l/k}{1/h} = \frac{hl}{k}, \tag{5} \label{eq:Biot}

where $$h$$ is the heat transfer coefficient, $$k$$ is the thermal conductivity of a solid and $$l$$ is the characteristic length of the solid. As the definition in Eq. \eqref{eq:Biot} indicates, it represents the ratio of the internal conduction resistance $$l/k$$ and the external convection resistance $$1/h$$. If the Biot number is small ($$Bi \ll 1$$), the solid may be treated as a simple lumped mass system of an uniform temperature. This boundary condition calculates the uniform temperature variation $$\Delta T$$ on the boundary from the following equation:

m c_p \Delta T = Q \Delta t. \tag{6} \label{eq:lumpedmass}

* $$m$$: total mass [kg]
* $$c_p$$: specific heat capacity [J/(kg.K)]
* $$Q$$: net heat flux on the boundary [W]
* $$\Delta t$$: time step [s]

• outletMappedUniformInletHeatAddition
• totalFlowRateAdvectiveDiffusive
• wallHeatTransfer
• compressible::thermalBaffle1D
• compressible::turbulentHeatFluxTemperature
• compressible::turbulentTemperatureCoupledBaffleMixed
• compressible::turbulentTemperatureRadCoupledMixed
• compressible::alphatJayatillekeWallFunction
• compressible::alphatPhaseChangeWallFunction
• compressible::alphatWallFunction

## 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.