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11.3 Boundary Conditions 329

momentum equations (P2.2.2) to obtain the velocity field for a known pressure,
and then solve Eq. (11.2.1) to obtain an updated (or corrected) pressure field.
The MAC and SIMPLE algorithms both use the Poisson equation to couple the
pressure and velocity fields.
Another way is to modify the continuity equation so that it becomes hy-
perbolic in time so that the methods developed for compressible flows can be
used with only a slight modification accounting for this transformation. The
hyperbolic character can be obtained by introducing a time derivative d/dt to
the continuity equation, which would vanish upon reaching steady state. The
method can therefore be applicable only to steady flows. The choice of the
scalar quantity is obtained by elimination. The density cannot be chosen since
it is constant, and the velocity already appears in the time derivative of the
momentum equations. The pressure remains the only choice yielding
1 dp dgui
(11.2.2)
/3 dt dxi
where f3 is the artificial compressibility parameter introduced by Chorin [5]. This
parameter can be obtained from dimensional analysis and has the dimensions
2
of velocity . Since this is not a non-dimensional constant, it is not universal and
its effect on the iterative procedure depends on the problem being considered.
Its effect is discussed in subsection 11.5.3.

11.3 Boundary Conditions

The incompressible Navier-Stokes equations require boundary conditions. The
system of equations, which contains four variables in three-dimensions (velocity
field and temperature) needs the specification of three variables along the in-
flow and outflow boundaries (Table 11.1). In practice, if the far-field boundaries
are placed far enough from the immerse body, then inviscid incompressible flow
boundary conditions can be used. Specification of all variables at inflow bound-
aries is required together with one boundary condition at outflow boundaries.
This leads to under specification of the mathematical formulation that could
result in non-unique solutions [6]. Note that, in the case of inviscid flow, the

Table 11.1. Physical and numerical boundary conditions for the 3D incompressible flow
equations.
Navier-Stokes Euler

Physical conditions Inflow 3 Inflow 3
Outflow 3 Outflow 1
Numerical conditions Inflow 1 Inflow 1
Outflow 1 Outflow 3