Energy Averaging for Rotating Fluids with Stationary Wall Boundaries

For rotating machinery simulations, Simcenter STAR-CCM+ provides the option to model the rotating fluid by using a moving (rotating) reference frame. Typically, the fluid rotates with the reference frame at a specified rotational speed, and the stationary shroud is modeled as a wall boundary residing on another reference frame, which is stationary. Due to the rotation of the fluid (and associated blades), the heat transfer profile at the stationary wall boundary needs to be averaged. The circumferential averaging takes place on radial bins. To calculate the averaged heat flux on the internal side of a rotating fluid region and the averaged thermal effect on the surrounding stationary wall boundary, consider the observer placed on the stationary boundary and looking inside the rotating internal medium.

Depending on the wall boundary thermal specification, the energy averaging is calculated differently.

Heat Flux

The general equation for heat flux at the wall $q_w$ is:

$$\overline{h_c{T}_c}-\overline{h_c}{T}_w+\overline{q_{absp,w}^r}-\sigma {ϵ}_w^{\text{total}}{T}_w^4=-q_w$$

(1821)

where:

• $\overline{h_c}$ is the area-averaged heat transfer coefficient of the fluid, on the internal side, between the cell center and boundary face.
• $\overline{T_c}$ is the area-averaged cell temperature.
• $T_w$ is the wall temperature.
• $\overline{q_{\text{absp},w}^r}$ is the absorption of radiation heat flux at the boundary.
• ${ϵ}_w^{\text{total}}$ is the total wall emissivity.

In terms of boundary heat flux coefficients, the equation for $q_w$ is:

$$\overline{a}+\overline{b{T}_c}+\overline{c}{T}_w+d{T}_w^4=-q_w$$

(1822)

To facilitate energy linearization at the boundary, Eqn. (1822) is written as:

$$\overline{a}+\overline{b_{\text{eff}}}{T}_c+\overline{c}{T}_w+d{T}_w^4=-q_w$$

(1823)

where:

• $a=\overline{q_{\text{absp},w}^r}$
• $b=\overline{h_c}$
• $c=-\overline{h_c}$
• $d=-\sigma {ϵ}_w^{\text{total}}$
• $\overline{b_{\text{eff}}}=\frac{\overline{b{T}_c}}{T_c}$

With external radiation active at the boundary, the above equations Eqn. (1821), Eqn. (1822), and Eqn. (1823) also include radiative emission and absorption terms $(\sigma {ϵ}_{w,ext}^{\text{total}}{T}_w^4-\overline{q_{\text{absp},w,\text{ext}}^r})$ for the external surface on the right-hand side of the equations.

Adiabatic

This is a special case of heat flux condition where $q_w=0$ . Averaging is as for the heat flux condition.

Heat Source

This is similar to heat flux condition, except heat source term is first converted into heat flux applied per boundary face.

Temperature

Eqn. (1823) is used here, except the boundary temperature $T_w$ is known and heat flux $q_w$ is calculated.

Convection

In the absence of thermal resistance

The averaging is same as for Eqn. (1821) except the right-hand side has the convection heat flux instead of a specified heat flux:

$$\overline{h_c{T}_c}-\overline{h_c}{T}_w+\overline{q_{\text{absp},w}^r}-\sigma {ϵ}_w^{\text{total}}{T}_w^4=h_{\infty }(T_w-T_{\infty })$$

(1824)

where $T_{\infty }$ and $h_{\infty }$ are the temperature and heat transfer coefficient of the surroundings.

Using boundary heat flux coefficients, this equation can be written as:

$$\overline{a}+\overline{b_{\text{eff}}}{T}_c+\overline{c}{T}_w+d{T}_w^4=h_{\infty }(T_w-T_{\infty })$$

When external radiation is active, the above equation and Eqn. (1824) also include the radiative emission and absorption terms for the external surface on the right-hand side.

In the presence of thermal resistance

The boundary condition takes into account the effects of the ambient environment and a thermal resistance.

The equations to be solved are:

• For the internal side:

  $$\overline{h_c{T}_c}-\overline{h_c}{T}_{w0}+\overline{q_{absp,0}^r}-\sigma {ϵ}_0^{\text{total}}{T}_{w0}^4=\frac{T_{w0}-T_{w1}}{R_w}$$

  (1825)
• For the external side:

  $$h_{\infty }(T_{\infty }-T_{w1})+\overline{q_{absp,1}^r}-\sigma {ϵ}_1^{\text{total}}{T}_{w1}^4=\frac{T_{w1}-T_{w0}}{R_w}$$

  (1826)

where:

• $T_{w0}$ is the temperature of the wall on the inside surface.
• $T_{w1}$ is the temperature of the wall on the outside surface.
• $R_w$ is the thermal resistance of the wall.

In terms of boundary heat flux coefficients, these equations become:

• For the internal side:

  $$\overline{a_0}+\overline{b_{0\text{eff}}}{T}_{c0}+\overline{c_0}{T}_{w0}+d_0{T}_{w0}^4=\frac{\mathrm{\Delta }{T}_w}{R_w}$$

  (1827)

  where $\mathrm{\Delta }{T}_w=T_{w0}-T_{w1}$ .

• For the external side:

  $$a_1+h_{\infty }{T}_{\infty }-h_{\infty }{T}_{w1}+d_1{T}_{w1}^4=\frac{-\mathrm{\Delta }{T}_w}{R_w}$$

  (1828)