ANSYS FLUENT 12.0 Theory Guide - 1.4.3 Physics of Periodic Flows

The assumption of periodicity implies that the velocity components repeat themselves in space as follows:

$\displaystyle u({\vec{r}}) = u({\vec{r}} + {\vec{L}}) = u({\vec{r}} + 2{\vec{L}}) = \cdots$
$\displaystyle v({\vec{r}}) = v({\vec{r}} + {\vec{L}}) = v({\vec{r}} + 2{\vec{L}}) = \cdots$			(1.4-1)
$\displaystyle w({\vec{r}}) = w({\vec{r}} + {\vec{L}}) = w({\vec{r}} + 2{\vec{L}}) = \cdots$

where $\vec{r}$ is the position vector and $\vec{L}$ is the periodic length vector of the domain considered (see Figure 1.4.2).

For viscous flows, the pressure is not periodic in the sense of Equation 1.4-1. Instead, the pressure drop between modules is periodic:

$\Delta p = p({\vec{r}}) - p({\vec{r}} + {\vec{L}}) = p({\vec{r}} + {\vec{L}}) - p({\vec{r}} + 2{\vec{L}}) = \cdots$

(1.4-2)

If one of the density-based solvers is used, $\Delta p$ is specified as a constant value. For the pressure-based solver, the local pressure gradient can be decomposed into two parts: the gradient of a periodic component, $\nabla \tilde{p}({\vec{r}})$ , and the gradient of a linearly-varying component, $\beta \frac{\vec{L}}{\vert\vec{L}\vert}$ :

${\nabla} p({\vec{r}}) = \beta \frac{\vec{L}}{\vert\vec{L}\vert} + {\nabla} \tilde{p}({\vec{r}})$

(1.4-3)

where $\tilde{p}({\vec{r}})$ is the periodic pressure and $\beta \vert{\vec{r}}\vert$ is the linearly-varying component of the pressure. The periodic pressure is the pressure left over after subtracting out the linearly-varying pressure. The linearly-varying component of the pressure results in a force acting on the fluid in the momentum equations. Because the value of $\beta$ is not known a priori, it must be iterated on until the mass flow rate that you have defined is achieved in the computational model. This correction of $\beta$ occurs in the pressure correction step of the SIMPLE, SIMPLEC, or PISO algorithm where the value of $\beta$ is updated based on the difference between the desired mass flow rate and the actual one. You have some control over the number of sub-iterations used to update $\beta$ . For more information about setting up parameters for $\beta$ in ANSYS FLUENT, see this section in the separate User's Guide.