5.9 Systems of equations
Most CFD calculations involve solving a system of equations that represent the physics of the problem. For example, laminar flow by natural convection can be represented by the equations introduced in Sec. 2.20 , reproduced below.
The system provides 3 equations (1 vector, 2
scalar) which can be solved for 3 unknowns, 
, 
and 
.
As discussed in Sec. 3.4
, the finite volume method
solves an individual matrix equation for each variable,
e.g. 
for 
. The vector
equation for 
is decoupled into 3 matrix equations for individual
components, i.e.
,
and 
.
Each individual matrix equation for one solution
variable, e.g. 
, incorporates
terms from other variables, e.g. 
, into the source
vector 
. The contribution to 
is calculated using
current values of the respective variables. Systems of equations
are thereby solved by successive
substitution of solved variables into the source vectors of
subsequent equations.
An iterative solution for a single equation, like
the one in Sec. 5.7
, can be extended to a
system of equations. Time 
is incremented by 
and equations are
solved in sequence, before returning to start the next time step
with the increment of 
.
The substitutions in the momentum and pressure
equations are particularly important, culminating in corrections to 
and the advective
flux 
, discussed in
Sec. 5.10
.
At the start of any time step the current
becomes 
for the discretisation of the 
term in the momentum
equation. The advection term 
is discretised by
Eq. (3.8
), treating one
as
flux 
and the other as the advected quantity.
The equation is solved for 
. The new solution for
is
substituted into the 
equation which is solved for 
. The new solution
for 
is then used to correct 
in order to help
enforce the mass conservation constraint 
(
).
Before the current solution step is completed,
is
also corrected to reduce the error in the discretisation of
when it then becomes 
in the following solution step. The correction
also provides a better “initial guess” 
for the matrix
solution of the next momentum equation, which reduces the solution
time.
