316
10. Compressible Flow
These boundary conditions can be implemented by extrapolating the pressure
from the interior of the solution domain to the boundary and then calculating
the velocity there with the aid of Eqs. (10.16) and (10.17). These velocities
can be treated as known within an outer iteration. The temperature can be
prescribed or it can be calculated from the total temperature:
This treatment leads to slow convergence of the iterative method as there
are many combinations of pressure and velocity that satisfy Eq. (10.16). One
must implicitly take into consideration the influence of the pressure on the
velocity at the inflow. One way of doing this is described below.
Fig. 10.1. A control volume next to an
inlet boundary with a prescribed flow direction
At the beginning of an outer iteration the velocities a t the inflow boundary (side 'w' in Fig. 10.1) must be computed from Eqs. (10.16) and (10.17)
and the prevailing values of the pressure; they will then be treated as fixed
during the outer iteration of the momentum equation. The mass fluxes a t
the inflow are taken from the preceding outer iteration; they should satisfy
the continuity equation. From the solution of the momentum equation, (uF*,
u F f ) , a new mass flux rizm' is computed. The 'prescribed' velocities on the
inflow boundary are used to compute the mass flux there. In the following
correction step, the mass flux (including its value at the inflow boundary) is
corrected and mass conservation is enforced. The difference between the mass
flux correction on the boundary and that a t interior control volume faces is
that, a t the boundary, only the velocity and not the density is corrected. The
velocity correction is expressed in terms of the pressure correction and not
its gradient:
~ k , , = u:,, tan P .
The coefficient Cu is determined with the aid of Eq. (10.16):
10. Compressible Flow
These boundary conditions can be implemented by extrapolating the pressure
from the interior of the solution domain to the boundary and then calculating
the velocity there with the aid of Eqs. (10.16) and (10.17). These velocities
can be treated as known within an outer iteration. The temperature can be
prescribed or it can be calculated from the total temperature:
This treatment leads to slow convergence of the iterative method as there
are many combinations of pressure and velocity that satisfy Eq. (10.16). One
must implicitly take into consideration the influence of the pressure on the
velocity at the inflow. One way of doing this is described below.
Fig. 10.1. A control volume next to an
inlet boundary with a prescribed flow direction
At the beginning of an outer iteration the velocities a t the inflow boundary (side 'w' in Fig. 10.1) must be computed from Eqs. (10.16) and (10.17)
and the prevailing values of the pressure; they will then be treated as fixed
during the outer iteration of the momentum equation. The mass fluxes a t
the inflow are taken from the preceding outer iteration; they should satisfy
the continuity equation. From the solution of the momentum equation, (uF*,
u F f ) , a new mass flux rizm' is computed. The 'prescribed' velocities on the
inflow boundary are used to compute the mass flux there. In the following
correction step, the mass flux (including its value at the inflow boundary) is
corrected and mass conservation is enforced. The difference between the mass
flux correction on the boundary and that a t interior control volume faces is
that, a t the boundary, only the velocity and not the density is corrected. The
velocity correction is expressed in terms of the pressure correction and not
its gradient:
~ k , , = u:,, tan P .
The coefficient Cu is determined with the aid of Eq. (10.16):