82
4. Finite Volume Methods
the domain boundary require special treatment. These boundary fluxes must
either be known, or be expressed as a combination of interior values and
boundary data. Since they do not give additional equations, they should
not introduce additional unknowns. Since there are no nodes outside the
boundary, these approximations must be based on one-sided differences or
extrapolations.
Usually, convective fluxes are prescribed at the inflow boundary. Convective fluxes are zero at impermeable walls and symmetry planes, and are
usually assumed to be independent of the coordinate normal to an outflow
boundary; in this case, upwind approximations can be used. Diffusive fluxes
are sometimes specified at a wall e.g. specified heat flux (including the special case of an adiabatic surface with zero heat flux) or boundary values of
variables are prescribed. In such a case the diffusive fluxes are evaluated using one-sided approximations for normal gradients as outlined in Sect. 3.7.
If the gradient itself is specified, it is used to calculate the flux, and an approximation for the flux in terms of nodal values can be used to calculate
the boundary value of the variable. This will be demonstrated in an example
below.
4.6 The Algebraic Equation System
By summing all the flux approximations and source terms, we produce an
algebraic equation which relates the variable value at the center of the CV
to the values a t several neighbor CVs. The numbers of equations and unknowns are both equal to the number of CVs so the system is well-posed.
The algebraic equation for a particular CV has the form (3.42), and the system of equations for the whole solution domain has the matrix form given
by Eq. (3.43). When the ordering scheme of Sect. 3.8 is used, the matrix A
has the form shown in Fig. 3.5. This is true only for structured grids with
quadrilateral or hexahedral CVs; for other geometries, the matrix structure
will be more complex (see Chap. 8 for details) but it will always be sparse.
The maximum number of elements in any row is equal to the number of near
neighbors for second order approximations. For higher-order approximations,
it depends on the number of neighbors used in the scheme.
4.7 Examples
In order to demonstrate the FV method and to display some of the properties
of the discretization methods presented above, we shall present two examples.
First consider the problem, illustrated in Fig. 4.4, of transport of a scalar
quantity in a known velocity field. The latter is given by u, = x and uy = -y,
which represents the flow near a stagnation point. The streamlines are the
4. Finite Volume Methods
the domain boundary require special treatment. These boundary fluxes must
either be known, or be expressed as a combination of interior values and
boundary data. Since they do not give additional equations, they should
not introduce additional unknowns. Since there are no nodes outside the
boundary, these approximations must be based on one-sided differences or
extrapolations.
Usually, convective fluxes are prescribed at the inflow boundary. Convective fluxes are zero at impermeable walls and symmetry planes, and are
usually assumed to be independent of the coordinate normal to an outflow
boundary; in this case, upwind approximations can be used. Diffusive fluxes
are sometimes specified at a wall e.g. specified heat flux (including the special case of an adiabatic surface with zero heat flux) or boundary values of
variables are prescribed. In such a case the diffusive fluxes are evaluated using one-sided approximations for normal gradients as outlined in Sect. 3.7.
If the gradient itself is specified, it is used to calculate the flux, and an approximation for the flux in terms of nodal values can be used to calculate
the boundary value of the variable. This will be demonstrated in an example
below.
4.6 The Algebraic Equation System
By summing all the flux approximations and source terms, we produce an
algebraic equation which relates the variable value at the center of the CV
to the values a t several neighbor CVs. The numbers of equations and unknowns are both equal to the number of CVs so the system is well-posed.
The algebraic equation for a particular CV has the form (3.42), and the system of equations for the whole solution domain has the matrix form given
by Eq. (3.43). When the ordering scheme of Sect. 3.8 is used, the matrix A
has the form shown in Fig. 3.5. This is true only for structured grids with
quadrilateral or hexahedral CVs; for other geometries, the matrix structure
will be more complex (see Chap. 8 for details) but it will always be sparse.
The maximum number of elements in any row is equal to the number of near
neighbors for second order approximations. For higher-order approximations,
it depends on the number of neighbors used in the scheme.
4.7 Examples
In order to demonstrate the FV method and to display some of the properties
of the discretization methods presented above, we shall present two examples.
First consider the problem, illustrated in Fig. 4.4, of transport of a scalar
quantity in a known velocity field. The latter is given by u, = x and uy = -y,
which represents the flow near a stagnation point. The streamlines are the