190
7. Solution of the Navier-Stokes Equations
mass flux through each CV face is evaluated using the existing velocity field
and is assumed known:
This kind of linearization is essentially the first step of Picard iteration; other
linearizations for implicit schemes were described in Chap. 5. Unless specifically stated otherwise, all variables in the remainder of this section belong to
the mth outer iteration. The mass fluxes (7.83) satisfy the continuity equation
on the 'scalar' CV, see Fig. 7.4. Mass fluxes at the faces of the momentum
CVs must be obtained by interpolation; ideally, these fluxes would provide
mass conservation for the momentum CV but this can be guaranteed only
to the accuracy of the interpolation. Another possibility is to use the mass
fluxes from the scalar CV faces. Since the east and west faces of a u-CV are
halfway between scalar CV faces, the mass fluxes can be calculated as:
The mass fluxes through the north and south faces of the u-CV can be approximated as half the sum of the two scalar CV face mass fluxes:
The superscript u denotes that the indices refer to the u-CV, see Fig. 7.4.
The sum of the four mass fluxes for the u-CV is thus half the sum of mass
fluxes into the two adjacent scalar CVs. They therefore satisfy the continuity
equation for the double scalar CV so the mass fluxes through the U-CV
faces also conserve mass. This result also holds for v-momentum CVs. It is
necessary to ensure that the mass fluxes through the momentum CVs satisfy
the continuity equation; otherwise, momentum will not be conserved.
The convective flux of ui-momentum through the 'el-face of a u-CV is
then (see Sect. 4.2 and Eq. (7.83)):
The CV face value of ui used in this expression need not be the one used to
calculate the mass flux, although an approximation of the same accuracy is
desirable. Linear interpolation is the simplest second order approximation.
We call this a central difference scheme (CDS), although no differencing is
involved. This is because, on uniform grids, it results in the same algebraic
equations as the CDS finite difference method.
Some iterative solvers fail to converge when applied to the algebraic equation systems derived from central difference approximations of convective
7. Solution of the Navier-Stokes Equations
mass flux through each CV face is evaluated using the existing velocity field
and is assumed known:
This kind of linearization is essentially the first step of Picard iteration; other
linearizations for implicit schemes were described in Chap. 5. Unless specifically stated otherwise, all variables in the remainder of this section belong to
the mth outer iteration. The mass fluxes (7.83) satisfy the continuity equation
on the 'scalar' CV, see Fig. 7.4. Mass fluxes at the faces of the momentum
CVs must be obtained by interpolation; ideally, these fluxes would provide
mass conservation for the momentum CV but this can be guaranteed only
to the accuracy of the interpolation. Another possibility is to use the mass
fluxes from the scalar CV faces. Since the east and west faces of a u-CV are
halfway between scalar CV faces, the mass fluxes can be calculated as:
The mass fluxes through the north and south faces of the u-CV can be approximated as half the sum of the two scalar CV face mass fluxes:
The superscript u denotes that the indices refer to the u-CV, see Fig. 7.4.
The sum of the four mass fluxes for the u-CV is thus half the sum of mass
fluxes into the two adjacent scalar CVs. They therefore satisfy the continuity
equation for the double scalar CV so the mass fluxes through the U-CV
faces also conserve mass. This result also holds for v-momentum CVs. It is
necessary to ensure that the mass fluxes through the momentum CVs satisfy
the continuity equation; otherwise, momentum will not be conserved.
The convective flux of ui-momentum through the 'el-face of a u-CV is
then (see Sect. 4.2 and Eq. (7.83)):
The CV face value of ui used in this expression need not be the one used to
calculate the mass flux, although an approximation of the same accuracy is
desirable. Linear interpolation is the simplest second order approximation.
We call this a central difference scheme (CDS), although no differencing is
involved. This is because, on uniform grids, it results in the same algebraic
equations as the CDS finite difference method.
Some iterative solvers fail to converge when applied to the algebraic equation systems derived from central difference approximations of convective