8.10 Implementation of Boundary Conditions
255
We shall often refer to a local coordinate system (n, t, s), which is a rotated
Cartesian frame with n the outward normal to the boundary and t and s
tangential to the boundary.
8.10.1 Inlet
Usually, at an inlet boundary, all quantities have to be prescribed. If the
conditions at inlet are not well known, it is useful to move the boundary as far
from the region of interest as possible. Since the velocity and other variables
are given, all the convective fluxes can be calculated. The diffusive fluxes are
usually not known, but they can be approximated using known boundary
values of the variables and one-sided finite difference approximations for the
gradients.
8.10.2 Outlet
At the outlet we usually know little about the flow. For this reason, these
boundaries should be as far downstream of the region of interest as possible.
Otherwise, errors may propagate upstream. The flow should be directed out
of the domain over the entire outlet cross-section, and if possible, be parallel.
In high Reynolds number flows, upstream propagation of errors - a t least in
steady flows - is weak so it is easy to find approximations for the boundary
conditions. Usually one extrapolates along grid lines from the interior to the
boundary (or, better, along streamlines). The simplest approximation is that
of zero gradient along grid lines. For the convective flux this means that a
first order upwind approximation is used. The condition of zero gradient on a
grid line can be implemented implicitly. For example, a t the east face the first
order backward approximation gives cj !q = 4 p . When we insert this expression
into the discretized equation for the CV next to boundary, we have:
so the boundary value q5E does not appear in the equation. This does not
mean that the diffusive flux is zero at the outlet boundary, except when the
grid is orthogonal to the boundary.
If higher accuracy is required, one has to use higher-order and one-sided
finite difference approximations of the derivatives at outlet boundary. Both
convective and diffusive fluxes have to be expressed in terms of the variable
values a t inner nodes.
When the flow is unsteady, especially when turbulence is directly simulated, care is needed to avoid reflection of errors a t the outlet boundary.
These issues are discussed in Sect. 9.2.
255
We shall often refer to a local coordinate system (n, t, s), which is a rotated
Cartesian frame with n the outward normal to the boundary and t and s
tangential to the boundary.
8.10.1 Inlet
Usually, at an inlet boundary, all quantities have to be prescribed. If the
conditions at inlet are not well known, it is useful to move the boundary as far
from the region of interest as possible. Since the velocity and other variables
are given, all the convective fluxes can be calculated. The diffusive fluxes are
usually not known, but they can be approximated using known boundary
values of the variables and one-sided finite difference approximations for the
gradients.
8.10.2 Outlet
At the outlet we usually know little about the flow. For this reason, these
boundaries should be as far downstream of the region of interest as possible.
Otherwise, errors may propagate upstream. The flow should be directed out
of the domain over the entire outlet cross-section, and if possible, be parallel.
In high Reynolds number flows, upstream propagation of errors - a t least in
steady flows - is weak so it is easy to find approximations for the boundary
conditions. Usually one extrapolates along grid lines from the interior to the
boundary (or, better, along streamlines). The simplest approximation is that
of zero gradient along grid lines. For the convective flux this means that a
first order upwind approximation is used. The condition of zero gradient on a
grid line can be implemented implicitly. For example, a t the east face the first
order backward approximation gives cj !q = 4 p . When we insert this expression
into the discretized equation for the CV next to boundary, we have:
so the boundary value q5E does not appear in the equation. This does not
mean that the diffusive flux is zero at the outlet boundary, except when the
grid is orthogonal to the boundary.
If higher accuracy is required, one has to use higher-order and one-sided
finite difference approximations of the derivatives at outlet boundary. Both
convective and diffusive fluxes have to be expressed in terms of the variable
values a t inner nodes.
When the flow is unsteady, especially when turbulence is directly simulated, care is needed to avoid reflection of errors a t the outlet boundary.
These issues are discussed in Sect. 9.2.