Thermal Boundary Conditions

Boundary Inputs

For non-isothermal wall boundaries, depending on the thermal specification, you prescribe the following variables:

Inputs	Adiabatic	Convection	Heat Flux	Heat Source	Temperature
Static temperature $T_{\mathrm{spec}}$
Heat flux ${\dot{q}}_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}}$
Heat source $S_Q$
Ambient temperature $T_{\mathrm{\infty }}$
Heat transfer coefficient $h_{\mathrm{ext}}$
Thermal resistance $R_{\mathrm{th}}$

Computed Values

Adiabatic

An adiabatic wall does not permit heat transfer across the boundary. Therefore:

$$\dot{q}=0$$

(1670)

The fluid temperature at the wall is calculated as:

$$T_w=T_s^{\mathrm{ext}}$$

(1671)

where $T_s^{\mathrm{ext}}$ is the static fluid temperature extrapolated from the near-wall cell center at the interior of the domain.

Temperature

The fluid temperature at the wall is set to the user-specified temperature:

$$T_w=T_{\mathrm{spec}}$$

(1672)

When the temperature is specified at the boundary, Simcenter STAR-CCM+ calculates the heat flux (without radiation) from the wall into the domain as:

$$\dot{q}=k\nabla T_w$$

(1673)

where $\nabla T_w$ is the temperature gradient at the wall and $k$ is the conductivity.

This is discretized as described by Eqn. (898):

$D_f={\Gamma }_f\nabla {\varphi }_f\cdot \mathbf{\text{a}}={\Gamma }_f\left[\left({\varphi }_1-{\varphi }_0\right)\stackrel{\rightharpoonup }{\alpha }\cdot \mathbf{\text{a}}+\overline{\nabla \varphi }\cdot \mathbf{\text{a}}-(\overline{\nabla \varphi }\cdot \mathbf{\text{ds}})\stackrel{\rightharpoonup }{\alpha }\cdot \mathbf{\text{a}}\right]$

where:

• ${\Gamma }_f=k$
• $\nabla {\varphi }_f=\nabla T_w$
• $\nabla {\varphi }_0=\nabla T_c$ (the temperature gradient in the near-wall cell)
• ${\varphi }_f=T_w$
• ${\varphi }_0=T_c$

Prior to taking the dot product with area a, this expression is:

$k\nabla T=k[\left(T_w-T_c\right)\stackrel{\rightharpoonup }{\alpha }+\nabla T_c-(\nabla T_c\cdot \mathbf{\text{ds}})\stackrel{\rightharpoonup }{\alpha }]$

Which can be rearranged to:

$k\nabla T=k[\nabla T_c-(\nabla T_c\cdot \mathbf{\text{ds}})\stackrel{\rightharpoonup }{\alpha }]-k\stackrel{\rightharpoonup }{\alpha }{T}_c+k\stackrel{\rightharpoonup }{\alpha }{T}_w+0{T_w}^4$

The wall heat flux $\stackrel{\mathrm{.}}{\mathbf{\text{q}}}$ is linearized as:

$$\stackrel{\mathrm{.}}{\mathbf{\text{q}}}″=A+B{T}_c+C{T}_w+D{\left(T_w\right)}^4$$

(1675)

where $T_c$ and $T_w$ are the cell and wall temperatures, respectively. A, B, C, and D are the net wall heat flux coefficients.

The internal heat flux coefficients are then obtained as:

• $A_{\text{internal}}$ = $k[\nabla T_c-(\nabla T_c\cdot \mathbf{\text{ds}})\stackrel{\rightharpoonup }{\alpha }]$
• $B_{\text{internal}}$ = $-k\stackrel{\rightharpoonup }{\alpha }$
• $C_{\text{internal}}$ = $k\stackrel{\rightharpoonup }{\alpha }$
• $D_{\text{internal}}$ = 0

Heat Flux

For this thermal specification, you specify the heat flux density $\dot{q}$ at a wall. From this heat flux density, Simcenter STAR-CCM+ computes the fluid temperature at the wall according to:

$$\nabla T_w=\frac{{\dot{q}}_{\mathrm{s}\mathrm{p}\mathrm{e}\mathrm{c}}}{k}$$

(1676)

The same linearization applies as described for the temperature specification.

Convection

The convection wall boundary takes into account a convective heat flux from the environment to the external side of the boundary:

$$\dot{q}=h_{\mathrm{e}\mathrm{x}\mathrm{t}}({\mathrm{T}}_{\mathrm{w}}-{\mathrm{T}}_{\mathrm{\infty }})$$

(1677)

where $T_{\infty }$ is the user-specified ambient temperature and $h_{\mathrm{ext}}$ is the user-specified heat transfer coefficient outside the fluid domain.

The conductive heat flux is given by Eqn. (1673). The heat flux $\dot{q}$ and the fluid temperature at the wall $T_{\mathrm{w}}$ are then determined iteratively.

Through a user-specified thermal resistance $R_{\mathrm{th}}$ , you can model a fictitious thickness of the wall boundary:

$$\mathrm{\Delta }=\frac{1}{R_{\mathrm{th}}}$$

(1678)

This fictitious wall thickness allows you to consider heat conduction through the wall as if it were a solid. However, the thermal resistance does not take into account the thermal capacity nor heat conduction in the lateral direction of the wall.

Heat Source

You specify the total heat $\dot{\mathrm{Q}}$ over the entire wall boundary. Simcenter STAR-CCM+ imposes:

$$\dot{\mathrm{q}}=\frac{\dot{\mathrm{Q}}}{\mathrm{A}}$$

(1680)

where $\mathrm{A}$ is the area of the wall boundary.