OpenFOAM guide/The SIMPLE algorithm in OpenFOAM

1 The SIMPLE algorithm

The Navier-Stokes equations for a single-phase flow with a constant density and viscosity are the following:

$\nabla \cdot \left( \rho \vec{U} \right) = 0$

$\frac{\partial U}{\partial t} + \nabla \cdot \left( \vec{v} \vec{v} \right) - \nabla \cdot \left( \nu \nabla \vec{v} \right) = - \nabla p$

The solution of this couple of equations is not straightforward because an explicit equation for the pressure is not available. One of the most common approaches is to derive an equation for the pressure by taking the divergence of the momentum equation and by substituting it in the continuity equation.

The momentum equation can be re-written in a semidiscretized form as follows:

$a_p \vec{U_p} = H(\vec{U}) - \nabla p \Longleftrightarrow \vec{U_p} = \frac{H(\vec{U})}{a_p} - \frac{\nabla p}{a_p}$