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Pressure-velocity algorithms

note

This page is under construction.

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

∂∂t(u)+∇ ∙ (u⊗u)=−∇p, \ddt{\u} + \div (\u \otimes \u) = - \grad p,

and continuity equation

∇ ∙ u=0. \div \u = 0.

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

M[u]=−∇p M[\u] = - \grad p

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

M[u]=Au−H M[\u] = A\u - \vec{H}

The discretised momentum equation therefore becomes:

Au−H=−∇p A\u - \vec{H} = - \grad p

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

u=HA−1A∇p. \u = \frac{\vec{H}}{A} - \frac{1}{A} \grad p.

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

ϕ=uf ∙ Sf=(HA)f ∙ Sf−(1A)fSf ∙ ∇f⊥p \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:

∇ ∙ ϕ=0. \div \phi = 0.

Substituting the flux equation leads to the pressure equation:

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