252
8. Complex Geometries
On structured grids, one can transform the normal pressure derivative
at cell faces into a combination of derivatives along grid line directions and
obtain a pressure-correction equation which involves mixed derivatives; see
Sect. 8.5. If the cross-derivatives are treated implicitly, the computational
molecule of the pressure-correction equation contains at least nine nodes in
2D and nineteen nodes in 3D. The above two-step procedure results in similar
convergence properties as the use of implicitly discretized cross-derivatives
(see PeriC, 1990), but is computationally more efficient, especially in 3D.
8.9 Axi-Symmetric Problems
Axi-symmetric flows are three-dimensional with respect to Cartesian coordinates i.e. the velocity components are functions of all three coordinates, but
they are only two-dimensional in a cylindrical coordinate system (all derivatives with respect to the circumferential direction are zero, and all three
velocity components are functions of only the axial and radial coordinates,
z and T). In cases without swirl, the circumferential velocity component is
zero everywhere. As it is much easier to work with two independent variables
than three, for axi-symmetric flows, it makes sense to work in a cylindrical
coordinate system rather than a Cartesian one.
In differential form, the 2D conservation equations for mass and momentum, written in a cylindrical coordinate system, read (see e.g. Bird et al.,
1962):
8. Complex Geometries
On structured grids, one can transform the normal pressure derivative
at cell faces into a combination of derivatives along grid line directions and
obtain a pressure-correction equation which involves mixed derivatives; see
Sect. 8.5. If the cross-derivatives are treated implicitly, the computational
molecule of the pressure-correction equation contains at least nine nodes in
2D and nineteen nodes in 3D. The above two-step procedure results in similar
convergence properties as the use of implicitly discretized cross-derivatives
(see PeriC, 1990), but is computationally more efficient, especially in 3D.
8.9 Axi-Symmetric Problems
Axi-symmetric flows are three-dimensional with respect to Cartesian coordinates i.e. the velocity components are functions of all three coordinates, but
they are only two-dimensional in a cylindrical coordinate system (all derivatives with respect to the circumferential direction are zero, and all three
velocity components are functions of only the axial and radial coordinates,
z and T). In cases without swirl, the circumferential velocity component is
zero everywhere. As it is much easier to work with two independent variables
than three, for axi-symmetric flows, it makes sense to work in a cylindrical
coordinate system rather than a Cartesian one.
In differential form, the 2D conservation equations for mass and momentum, written in a cylindrical coordinate system, read (see e.g. Bird et al.,
1962):