## 9.4.2 Theory

Definition of the Periodic Velocity

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

 (9.4-1)

where is the position vector and is the periodic length vector of the domain considered (see Figure  9.4.2).

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:

 (9.4-2)

If one of the density-based solvers is used, 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, , and the gradient of a linearly-varying component, :

 (9.4-3)

where is the periodic pressure and 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 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 occurs in the pressure correction step of the SIMPLE, SIMPLEC, or PISO algorithm where the value of 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 , as described in Section  9.4.3.

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