5.10 Pressure-velocity coupling

The previous section combined equations for eqn and eqn, governing momentum and mass conservation, in a sequential solution.

The algorithms used to couple these equations, in a manner which is convergent, uses the following notation to describe terms in the momentum equation, e.g. Eq. (2.67 ), excluding eqn

Au H(u) @u-+ r (uu) r ( ru) -(T;p)g; @t 0 \relax \special {t4ht=
  • eqn is a linear term in eqn;
  • eqn is a function of eqn and other sources.

Momentum corrector

A momentum corrector equation is formed by expressing the momentum equation, e.g. Eq. (2.67 ) in terms of the notation of Eq. (5.16 ), and rearranging to give

 H(u)- -1 u ℬ A A rp: \relax \special {t4ht=
The equation provides an update to eqn, based on current values of eqn and eqn substituted on the r.h.s. In other words, eqn and eqn are calculated explicitly.

In the algorithms described in the following sections, eqn is chosen to be the extracted diagonal coefficients eqn eqn of the matrix eqn corresponding to the momentum terms in Eq. (5.16 ).

PICT\relax \special {t4ht=

The split between eqn and eqn is shown above. As a matrix with diagonal components only, eqn has one value per cell so can be represented as a scalar field. Setting eqn to be the diagonal coefficients is a natural choice for implicit treatment of eqn within the coupling algorithm.

Flux corrector

A flux corrector equation follows from Eq. (5.17) by interpolating eqn to cell faces and evaluating eqn according to

 H(u)- jSfj f ℬ Sf A f A frn pf: \relax \special {t4ht=

Pressure equation

A pressure equation is then created by substituting fluxes eqn from Eq. (5.18 ) into the mass conservation Eq. (2.46 ) in discrete form eqn. The resulting expression is equivalent to a discretised pressure equation, with coefficients containing eqn and eqn,

 1- H(u)- r A rp = r A : \relax \special {t4ht=
Notes on CFD: General Principles - 5.10 Pressure-velocity coupling