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13.4.2 Theory

Streamwise-periodic flow with heat transfer from constant-temperature walls is one of two classes of periodic heat transfer that can be modeled by FLUENT. A periodic fully-developed temperature field can also be obtained when heat flux conditions are specified. In such cases, the temperature change between periodic boundaries becomes constant and can be related to the net heat addition from the boundaries as described in this section.

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Periodic heat transfer can be modeled only if you are using the pressure-based solver.



Definition of the Periodic Temperature for Constant- Temperature Wall Conditions


For the case of constant wall temperature, as the fluid flows through the periodic domain, its temperature approaches that of the wall boundaries. However, the temperature can be scaled in such a way that it behaves in a periodic manner. A suitable scaling of the temperature for periodic flows with constant-temperature walls is [ 278]


 \theta = \frac{T(\vec{r}) - T_{\rm wall}}{T_{\rm bulk,inlet} - T_{\rm wall}} (13.4-1)

The bulk temperature, $T_{\rm bulk,inlet}$, is defined by


 T_{\rm bulk,inlet} = \frac{\int_{A} T \vert\rho \vec{v} \cdo... ...{A}\vert } {\int_{A} \vert\rho \vec{v} \cdot d \vec{A}\vert } (13.4-2)

where the integral is taken over the inlet periodic boundary ( $A$). It is the scaled temperature, $\theta$, which obeys a periodic condition across the domain of length $L$.



Definition of the Periodic Temperature Change $\sigma$ for Specified Heat Flux Conditions


When periodic heat transfer with heat flux conditions is considered, the form of the unscaled temperature field becomes analogous to that of the pressure field in a periodic flow:


 \frac{T(\vec{r} + \vec{L}) - T(\vec{r})}{L} = \frac{T(\vec{r} + 2\vec{L}) - T(\vec{r} + \vec{L})}{L} = \sigma. (13.4-3)

where $\vec{L}$ is the periodic length vector of the domain. This temperature gradient, $\sigma$, can be written in terms of the total heat addition within the domain, $Q$, as


 \sigma=\frac{Q}{\dot{m} c_p L} = \frac{T_{\rm bulk,exit} - T_{\rm bulk,inlet}}{L} (13.4-4)

where $\dot{m}$ is the specified or calculated mass flow rate.


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