Thermal Boundary Conditions

In non-isothermal viscous flow, a thermal boundary condition must be specified on each boundary. The boundary condition is represented by the last term on the lefthand side of the Eqn. (1031). On the velocity inlet, mass flow inlet, and stagnation inlet, a Dirichlet condition is imposed, that is, the static temperature at the boundary is specified. On the pressure outlet, a Neumann condition is imposed, that is, the viscous flow solver assumes zero flux at the boundary and the boundary term in Eqn. (1031) is dropped.

Wall and Free Stream

For the case of free stream or wall, you specify one of the following options:

Temperature

A Dirichlet condition is set for the temperature at the wall or free stream boundary.

Figure 1. EQUATUION_DISPLAY

T = \frac{T_{spec}}{T_{c}}

(1063)

where $T$ is the non-dimensional temperature, $T_{c}$ is the characteristic temperature, and $T_{spec}$ is the user-specified static temperature.

Adiabatic

The adiabatic boundary condition describes an insulated boundary with no heat flux.

Figure 2. EQUATION_DISPLAY

- \nabla T \cdot e_{n} = 0

(1064)

where $e_{n}$ is the dimensionless unit normal vector.

Heat Flux

At the heat flux boundary, a constant heat flux is imposed:

Figure 3. EQUATION_DISPLAY

- \nabla T \cdot e_{n} = q

(1065)

where $q$ is the non-dimensional heat flux at the boundary, given by:

q = q_{spec} l_{c} / k

where

q_{spec}

is the user-specified heat flux,

l_{c}

is the characteristic length scale, and

k

is the fluid thermal conductivity.

Convection

In the convection boundary condition, the conduction flux is calculated from Newton's cooling law:

Figure 4. EQUATION_DISPLAY

- \nabla T \cdot e_{n} = Nu (T - T_{\infty})

(1066)

where $Nu$ is the Nusselt number, given by:

Nu = h l_{c} / k

where $h$ is the heat transfer coefficient and $T_{\infty}$ is the ambient temperature.

Radiation

In the radiation boundary condition, the radiation flux is calculated from the Stefan-Boltzmann law:

Figure 5. EQUATION_DISPLAY

- \nabla T \cdot e_{n} = α (T^{4} - T_{\infty}^{4})

(1067)

where $α = σ ϵ T_{c}^{3} l_{c} / k$ and:

$σ$ is the Stefan-Boltzmann constant.
$ϵ$ is the emissivity ( $ϵ = 1$ for a gray body).
$T_{c}$ is the characteristic temperature.

Convection and Radiation

The convection and radiation boundary condition combines the components of the previous two conditions:

Figure 6. EQUATION_DISPLAY

- \nabla T \cdot e_{n} = Nu (T - T_{\infty}) + α (T^{4} - T_{\infty}^{4})

(1068)