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9.4.2 Theory

Definition of the Periodic Velocity

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

$\displaystyle u({\vec{r}}) = u({\vec{r}} + {\vec{L}}) = u({\vec{r}} + 2{\vec{L}}) = \cdots$      
$\displaystyle v({\vec{r}}) = v({\vec{r}} + {\vec{L}}) = v({\vec{r}} + 2{\vec{L}}) = \cdots$     (9.4-1)
$\displaystyle w({\vec{r}}) = w({\vec{r}} + {\vec{L}}) = w({\vec{r}} + 2{\vec{L}}) = \cdots$      

where $\vec{r}$ is the position vector and $\vec{L}$ is the periodic length vector of the domain considered (see Figure  9.4.2).

Figure 9.4.2: Example of a Periodic Geometry

Definition of the Streamwise-Periodic Pressure

For viscous flows, the pressure is not periodic in the sense of Equation  9.4-1. 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 (9.4-2)

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}}) (9.4-3)

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 $\beta$, as described in Section  9.4.3.

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