## 5.10Pressure-velocity coupling

The previous section combined equations for and , 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

 (5.16)
where:
• is a linear term in ;
• is a function of 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

 (5.17)
The equation provides an update to , based on current values of and substituted on the r.h.s. In other words, and are calculated explicitly.

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

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

### Flux corrector

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

 (5.18)

### Pressure equation

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

 (5.19)
Notes on CFD: General Principles - 5.10 Pressure-velocity coupling