1.4 Momentum Conservation
7
where (see Eqs. (1.9) and (1.10)):
Here bi stands for the ith component of the body force, superscript means
transpose and ii is the Cartesian unit vector in the direction of the coordinate
xi. In Cartesian coordinates one can write the above expression as:
A vector field can be represented in a number of different ways. The basis
vectors in terms of which the vector is defined may be local or global. In curvilinear coordinate systems, which are often required when the boundaries are
complex (see Chap. 8) one may choose either a covariant or a contravariant
basis, see Fig. 1.1. The former expresses a vector in terms of its components
along the local coordinates; the latter uses the projections normal to coordinate surfaces. In a Cartesian system, the two become identical. Also, the basis
vectors may be dimensionless or dimensional. Including all of these options,
over 70 different forms of the momentum equations are possible. Mathematically, all are equivalent; from the numerical point of view, some are more
difficult to deal with than others.
Fig. 1.1. Representation of a vector through different components: u i - W t e s i a n
components; vi - contravariant components; v i - covariant components [ V A = V B ,
( % ) A = ( u ~ ) B , ( V i ) A # ( V i ) B , ( v i ) ~
# ( v i ) ~ ]
The momentum equations are said to be in "strong conservation form" if
all terms have the form of the divergence of a vector or tensor. This is possi-
7
where (see Eqs. (1.9) and (1.10)):
Here bi stands for the ith component of the body force, superscript means
transpose and ii is the Cartesian unit vector in the direction of the coordinate
xi. In Cartesian coordinates one can write the above expression as:
A vector field can be represented in a number of different ways. The basis
vectors in terms of which the vector is defined may be local or global. In curvilinear coordinate systems, which are often required when the boundaries are
complex (see Chap. 8) one may choose either a covariant or a contravariant
basis, see Fig. 1.1. The former expresses a vector in terms of its components
along the local coordinates; the latter uses the projections normal to coordinate surfaces. In a Cartesian system, the two become identical. Also, the basis
vectors may be dimensionless or dimensional. Including all of these options,
over 70 different forms of the momentum equations are possible. Mathematically, all are equivalent; from the numerical point of view, some are more
difficult to deal with than others.
Fig. 1.1. Representation of a vector through different components: u i - W t e s i a n
components; vi - contravariant components; v i - covariant components [ V A = V B ,
( % ) A = ( u ~ ) B , ( V i ) A # ( V i ) B , ( v i ) ~
# ( v i ) ~ ]
The momentum equations are said to be in "strong conservation form" if
all terms have the form of the divergence of a vector or tensor. This is possi-