206
7. Solution of the Navier-Stokes Equations
CV, where the pressure is calculated. When colocated arrangement is used, all
CVs extend to the boundary and we need the boundary pressure to calculate
the pressure forces in the momentum equations. We have to use extrapolation
to obtain pressure at the boundaries. In most cases, linear extrapolation is
sufficiently accurate for a second order method. However, there are cases in
which a large pressure gradient near a wall is needed in the equation for the
normal velocity component to balance a body force (buoyancy, centrifugal
force etc.). If the pressure extrapolation is not accurate, this condition may
not be satisfied and large normal velocities near the boundary may result.
This can be avoided by calculating the normal velocity component for the first
CV from the continuity equation, by adjusting the pressure extrapolation, or
by local grid refinement.
The boundary conditions for the pressure-correction equation also deserve
some attention. When the mass flux through a boundary is prescribed, the
mass flux correction in the pressure correction equation is also zero there.
This condition should be directly implemented in the continuity equation
when deriving the pressure-correction equation. It is equivalent to specifying
a Neumann boundary condition (zero gradient) for the pressure correction.
At the outlet, if the inlet mass fluxes are given, extrapolation of the velocity to the boundary (zero gradient, e.g. UE = up) can usually be used for
steady flows when the outflow boundary is far from the region of interest and
the Reynolds number is large. The extrapolated velocity is then corrected to
give exactly the same total mass flux as at inlet (this cannot be guaranteed
by any extrapolation). The corrected velocities are then considered as prescribed for the following outer iteration and the mass flux correction at the
outflow boundary is set to zero in the continuity equation. This leads to the
pressure-correction equation having Neumann conditions on all boundaries
and makes it singular. To make the solution unique, one usually takes the
pressure at one point to be fixed, so the pressure correction calculated at that
point is subtracted from all the corrected pressures. Another choice is to set
the mean pressure to some value, say zero.
Another case is obtained when the pressure difference between the inlet
and outlet boundaries is specified. Then the velocities at these boundaries
cannot be specified - they have to be computed so that the pressure loss is
the specified value. This can be implemented in several ways. In any case the
boundary velocity has to be extrapolated from the inner nodes (in a manner
similar to the interpolation for cell faces in a colocated arrangement) and
then corrected. An example of how specified static pressure can be handled
is given in Chap. 10.
7.8 Examples
In this section we shall demonstrate the accuracy and efficiency of the implicit
SIMPLE method for steady incompressible flows on staggered and colocated
7. Solution of the Navier-Stokes Equations
CV, where the pressure is calculated. When colocated arrangement is used, all
CVs extend to the boundary and we need the boundary pressure to calculate
the pressure forces in the momentum equations. We have to use extrapolation
to obtain pressure at the boundaries. In most cases, linear extrapolation is
sufficiently accurate for a second order method. However, there are cases in
which a large pressure gradient near a wall is needed in the equation for the
normal velocity component to balance a body force (buoyancy, centrifugal
force etc.). If the pressure extrapolation is not accurate, this condition may
not be satisfied and large normal velocities near the boundary may result.
This can be avoided by calculating the normal velocity component for the first
CV from the continuity equation, by adjusting the pressure extrapolation, or
by local grid refinement.
The boundary conditions for the pressure-correction equation also deserve
some attention. When the mass flux through a boundary is prescribed, the
mass flux correction in the pressure correction equation is also zero there.
This condition should be directly implemented in the continuity equation
when deriving the pressure-correction equation. It is equivalent to specifying
a Neumann boundary condition (zero gradient) for the pressure correction.
At the outlet, if the inlet mass fluxes are given, extrapolation of the velocity to the boundary (zero gradient, e.g. UE = up) can usually be used for
steady flows when the outflow boundary is far from the region of interest and
the Reynolds number is large. The extrapolated velocity is then corrected to
give exactly the same total mass flux as at inlet (this cannot be guaranteed
by any extrapolation). The corrected velocities are then considered as prescribed for the following outer iteration and the mass flux correction at the
outflow boundary is set to zero in the continuity equation. This leads to the
pressure-correction equation having Neumann conditions on all boundaries
and makes it singular. To make the solution unique, one usually takes the
pressure at one point to be fixed, so the pressure correction calculated at that
point is subtracted from all the corrected pressures. Another choice is to set
the mean pressure to some value, say zero.
Another case is obtained when the pressure difference between the inlet
and outlet boundaries is specified. Then the velocities at these boundaries
cannot be specified - they have to be computed so that the pressure loss is
the specified value. This can be implemented in several ways. In any case the
boundary velocity has to be extrapolated from the inner nodes (in a manner
similar to the interpolation for cell faces in a colocated arrangement) and
then corrected. An example of how specified static pressure can be handled
is given in Chap. 10.
7.8 Examples
In this section we shall demonstrate the accuracy and efficiency of the implicit
SIMPLE method for steady incompressible flows on staggered and colocated