\underConstruction

The background to the SIMPLE, PISO and PIMPLE pressure-velocity algorithms can be demonstrated using the incompressible, inviscid flow equations, comprising the momentum equation:

$\ddt{\u} + \div (\u \otimes \u) = - \grad p,$

and continuity equation

$\div \u = 0.$

Discretising the momentum equation leads to a set of algebraic equations of the form:

$M[\u] = - \grad p$

where the matrix $$M[\u]$$ comprises the diagonal and off-diagonal contributions using the decomposition:

$M[\u] = A\u - \vec{H}$

The discretised momentum equation therefore becomes:

$A\u - \vec{H} = - \grad p$

which on re-arranging leads to the velocity correction equation:

$\u = \frac{\vec{H}}{A} - \frac{1}{A} \grad p.$

The volumetric flux corrector equation is then derived by interpolating $$\u$$ to the faces and dotting the result with the face area vectors, $$\vec{S}_f$$:

$\phi = \u_f \dprod \vec{S}_f = \left( \frac{\vec{H}}{A} \right)_f \dprod \vec{S}_f - \left( \frac{1}{A} \right)_f \vec{S}_f \dprod \snGrad p$

Discretisation of the continuity equation yields the constraint:

$\div \phi = 0.$

Substituting the flux equation leads to the pressure equation:

$\div \left[ \left( \frac{1}{A} \right)_f \grad p \right] = \div \left( \frac{\vec{H}}{A} \right)_f.$