224
8. Complex Geometries
a fixed basis leads to a fully conservative form of the momentum equations.
To ensure momentum conservation, it is desirable to use such a basis and the
simplest one is the Cartesian basis. When the flow is three-dimensional, there
are no advantages to using any other basis (e.g. grid-oriented, covariant, or
contravariant). Only when the choice of another vector basis leads to the
simplification of the problem is it worth abandoning the use of Cartesian
components. An example of such a case is the flow in a pipe or other axisymmetric geometries. In the absence of swirl, the velocity vector has only
two non-zero components in the polar-cylindrical basis, but three non-zero
Cartesian components. The problem therefore has three dependent variables
in terms of the Cartesian components, but only two if the polar-cylindrical
components are used, a substantial simplification. If the swirl is present, the
problem is three-dimensional in any case, so there is no advantage in using
polar-cylindrical components rather then Cartesian ones.
8.3.1 Grid-Oriented Velocity Components
If grid-oriented velocity components are used, non-conservative source terms
appear in the momentum equations. These account for the redistribution of
the momentum between the components. For example, if polar-cylindrical
components are used, the divergence of the convection tensor pvv leads to
two such source terms:
In the momentum equation for the r-component, there is a term pvi/r,
which represents the apparent centrifugal force. This is not the centrifugal
force found in rotating flows (e.g. pump or turbine passages) - it is solely
due to the transformation from Cartesian to polar-cylindrical coordinates.
This term describes the transfer of 0-momentum into r-momentum due to
the change of direction of ve.
In the momentum equation for the 0-component, there is a term -pv,ve/rl
which represents the apparent Coriolis force. This term is source or sink
of 0-momentum, depending on the signs of the velocity components.
In general curvilinear coordinates, there are more such source terms (see
books by Sedov, 1971; Truesdell, 1977, etc). They involve Christoffel symbols
(curvature terms, higher-order coordinate derivatives) and are often a source
of numerical errors. The grid is required to be smooth - the change of grid
direction from point to point must be small. Especially on unstructured grids,
in which grid lines are not associated with coordinate directions, this basis is
difficult to use.
8.3.2 Cartesian Velocity Components
In this book we shall use Cartesian vector and tensor components exclusively. The discretization and solution techniques remain the same if other
8. Complex Geometries
a fixed basis leads to a fully conservative form of the momentum equations.
To ensure momentum conservation, it is desirable to use such a basis and the
simplest one is the Cartesian basis. When the flow is three-dimensional, there
are no advantages to using any other basis (e.g. grid-oriented, covariant, or
contravariant). Only when the choice of another vector basis leads to the
simplification of the problem is it worth abandoning the use of Cartesian
components. An example of such a case is the flow in a pipe or other axisymmetric geometries. In the absence of swirl, the velocity vector has only
two non-zero components in the polar-cylindrical basis, but three non-zero
Cartesian components. The problem therefore has three dependent variables
in terms of the Cartesian components, but only two if the polar-cylindrical
components are used, a substantial simplification. If the swirl is present, the
problem is three-dimensional in any case, so there is no advantage in using
polar-cylindrical components rather then Cartesian ones.
8.3.1 Grid-Oriented Velocity Components
If grid-oriented velocity components are used, non-conservative source terms
appear in the momentum equations. These account for the redistribution of
the momentum between the components. For example, if polar-cylindrical
components are used, the divergence of the convection tensor pvv leads to
two such source terms:
In the momentum equation for the r-component, there is a term pvi/r,
which represents the apparent centrifugal force. This is not the centrifugal
force found in rotating flows (e.g. pump or turbine passages) - it is solely
due to the transformation from Cartesian to polar-cylindrical coordinates.
This term describes the transfer of 0-momentum into r-momentum due to
the change of direction of ve.
In the momentum equation for the 0-component, there is a term -pv,ve/rl
which represents the apparent Coriolis force. This term is source or sink
of 0-momentum, depending on the signs of the velocity components.
In general curvilinear coordinates, there are more such source terms (see
books by Sedov, 1971; Truesdell, 1977, etc). They involve Christoffel symbols
(curvature terms, higher-order coordinate derivatives) and are often a source
of numerical errors. The grid is required to be smooth - the change of grid
direction from point to point must be small. Especially on unstructured grids,
in which grid lines are not associated with coordinate directions, this basis is
difficult to use.
8.3.2 Cartesian Velocity Components
In this book we shall use Cartesian vector and tensor components exclusively. The discretization and solution techniques remain the same if other