256
8. Complex Geometries
8.10.3 Impermeable Walls
At an impermeable wall, the following condition applies:
Ui = ui,waU .
(8.72)
This condition follows from the fact that viscous fluids sticks to solid boundary (no-slip condition).
Since there is no flow through the wall, convective fluxes of all quantities
are zero. Diffusive fluxes require some attention. For scalar quantities, such
as thermal energy, they may be zero (adiabatic walls), they may be specified
(prescribed heat flux), or the value of the scalar may be prescribed (isothermal
walls). If the flux is known, it can be inserted into the conservation equation
for the near-wall CVs, e.g. at the south boundary:
where f is the prescribed flux per unit area. If the value of 4 is specified
at wall, we need to approximate the normal gradient of 4 using one-sided
differences. From such an approximation we can also calculate the value of
4 at the wall when the flux is prescribed. Many possibilities exist; one is to
calculate the value of 4 at an auxiliary point P' located on the normal n , see
Fig. 8.15, and use the approximation:
where 6n = ( r s - r p , ) - n is the distance between points P' and S. If the nonorthogonality is not severe, one can use dp instead of dp,. Shape functions
or extrapolated gradients from cell centers can also be used.
Y
lib
*i n x
mentation of boundary conI i ,
ditions at a wall
Diffusive fluxes in the momentum equations require special attention. If
we were solving for the velocity components v,, vt and v,, we could use the
approach described in Sect. 7.7. The viscous stresses at a wall are:
8. Complex Geometries
8.10.3 Impermeable Walls
At an impermeable wall, the following condition applies:
Ui = ui,waU .
(8.72)
This condition follows from the fact that viscous fluids sticks to solid boundary (no-slip condition).
Since there is no flow through the wall, convective fluxes of all quantities
are zero. Diffusive fluxes require some attention. For scalar quantities, such
as thermal energy, they may be zero (adiabatic walls), they may be specified
(prescribed heat flux), or the value of the scalar may be prescribed (isothermal
walls). If the flux is known, it can be inserted into the conservation equation
for the near-wall CVs, e.g. at the south boundary:
where f is the prescribed flux per unit area. If the value of 4 is specified
at wall, we need to approximate the normal gradient of 4 using one-sided
differences. From such an approximation we can also calculate the value of
4 at the wall when the flux is prescribed. Many possibilities exist; one is to
calculate the value of 4 at an auxiliary point P' located on the normal n , see
Fig. 8.15, and use the approximation:
where 6n = ( r s - r p , ) - n is the distance between points P' and S. If the nonorthogonality is not severe, one can use dp instead of dp,. Shape functions
or extrapolated gradients from cell centers can also be used.
Y
lib
*i n x
mentation of boundary conI i ,
ditions at a wall
Diffusive fluxes in the momentum equations require special attention. If
we were solving for the velocity components v,, vt and v,, we could use the
approach described in Sect. 7.7. The viscous stresses at a wall are: