For the closure of the algebraic equation systems that arise from the discretization, the boundary conditions have to be implemented. The boundary values are specified at the boundary cell faces which are related to only one cell (see figure 3.3). Therefore, a special attention has to be paid to implementation of boundary conditions, which depend on their type. In what follows, the boundary conditions used in this study and their implementation into the discretization procedure are described. B n ξ 0 B B
Near boundary cell
s Boundary cell−face
d P
P
Figure 3.3: Near-boundary cell with definitions and notation.
3.4.1
Inlet boundaries
At inlet boundaries the values of all variables are usually known. This implies the Dirichlet boundary conditions and simplifies implementation. All convective fluxes can be calculated using given boundary values and diffusion fluxes can be approximated using given boundary values and one-sided finite differences in approximations of diffusive fluxes.
3.4. Implementation of boundary conditions 27
3.4.2
Outlet boundaries
Unlike at inlet, at outlet boundaries we do not know exact values of the dependent variables but need to approximate them. Therefore, outlet boundary should be placed as far downstream from the region of interest as possible. Furthermore, it should be placed at a location where the flow is everywhere directed outwards in order to avoid propagation of any error introduced by estimations of the outlet conditions. The variable values at the outlet boundary may be extrap- olated from the flow domain. The simplest approximation is that of zero gradient extrapolation which, for simple backward approximation, reduces toφPb =φP0. Velocity components require
a special treatment since it has to be ensured that the overall mass conservation is satisfied. The velocity components are first estimated by extrapolation from interior. These values are used to calculate outlet mass fluxes. The velocity components and mass fluxes are then corrected by mul- tiplying them with the ratiom˙I/m˙O, wherem˙I is the total mass inflow andm˙O is the total mass
outflow. This ensures that the global mass conservation is fulfilled in each outer iteration, which is important for the solution of the pressure-correction equation [11] (see also section 5.3.3).
3.4.3
Symmetry boundaries
If the flow is to be symmetrical with respect to a line or a plane, the first condition which has to be fulfilled is that there is no flow across the boundary, i.e. the normal velocity at a symmetry boundary must be zero (and so are all convective fluxes). Using this condition, the velocity vector at the boundary can be obtained from the known velocity vector at the CV-center next to the boundary (see figure 3.4)
vB =vP0 −(vP0 ·n)n. (3.35)
This velocity can now be used to calculate diffusion fluxes in the usual way, but this approach can lead to poor convergence. Symmetry boundary conditions can be implemented in the mo-
n P P B v v v P Symmetry P n 0 B dB
mentum equations using the fact that only the gradient of the normal velocity component in the direction normal to the boundary is non-zero, i.e. that only the normal stress τnn = 2µ
∂vn
∂n
makes, contribution to the force coming from the viscous part of the stress tensor [26]:
fsym = Z SB τnnndS = Z SB 2µ ∂vn ∂n ! ndS ≈ " 2µ ∂vn ∂n ! Sn # B (3.36)
wherevn =vP0 ·n is the normal velocity component and the normal derivative is approximated using a one-side difference (see figure 3.4):
∂vn ∂n ≈ vn δn = vP0·n db ·n . (3.37)
Contributions of the force calculated by equation (3.36) are distributed to corresponding momen- tum equations for each velocity component.
For all scalar quantities, normal derivatives at the symmetry boundary must be zero – a boundary condition of Neumann type is applied (zero diffusion fluxes).
3.4.4
Wall boundaries
Solid walls are considered as impermeable and a no-slip boundary condition is applied, i.e. the fluid velocity is equal to the wall velocity. Hence, the Dirichlet boundary condition is directly applicable for the momentum equation. Implementation is similar as for the inlet, except that, due to the wall impermeability, the convective fluxes are zero. For scalar quantities (e.g. temperature) either the Dirichlet or the Neumann boundary condition can be applied, depending on whether the variable value or its gradient is prescribed at the boundary. In the former case diffusion fluxes are usually approximated using one-sided differences and in the latter case, the fluxes can be computed directly (usually they are already given) and inserted into the conservation equation for the near-wall CV.
3.4.5
Boundary conditions for the pressure-correction equation
Since the pressure-correction equation is different from the other basic equations, the implemen- tation of boundary conditions for this equation deserves some attention. When the mass flux through a boundary is prescribed, which is the case for all types of boundary conditions men- tioned above, its correction is zero at the boundary. It implies, according to equation (3.27), that the normal derivative of pressure-correction is zero at the boundary, i.e. the Neumann boundary condition is applied.
Another possibility is that the pressure at the boundary is given. In that case the value of the pressure-correction at the boundary is zero, implying the Dirichlet boundary condition for pressure-correction. If the pressure is prescribed at the boundary, velocity cannot be specified there but has to be extrapolated from the interior. This is done in the same way as for inner cell