The background to the SIMPLE, PISO and PIMPLE pressure-velocity algorithms can be demonstrated using the incompressible, inviscid flow equations, comprising the momentum equation:
\[ \ddt{\u} + \div (\u \otimes \u) = - \grad p, \]
\[ \div \u = 0. \]
Disctretising 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, \( \vec{S}_f \):
\[ \phi = \u_f \dprod \vec{S}_f = \left( \frac{\vec{H}}{A} \right)_f \dprod \vec{S}_f - \left( \frac{1}{A} \right)_f \vec{S}_f \dprod \snGrad p \]
Discretistation of the continuity equation yields the constraint:
\[ \div \phi = 0. \]
Substituting the flux equation leads to the pressure equation:
\[ \div \left[ \left( \frac{1}{A} \right)_f \grad p \right] = \div \left( \frac{\vec{H}}{A} \right)_f. \]
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