ABSTRACT
Since numerical solutions require a defi ned coordinate system of uniquely identifi ed grid points, the geometry issue will be addressed in this chapter. Pseudo-one-dimensional fl ows utilize control volumes in which convection along one direction is to be evaluated. Rectangular Cartesian coordinates and cylindrical coordinates constitute the remaining practical control volumes which may be defi ned with algebraic coordinate systems. More general geometries require curvilinear orthogonal or nonorthogonal coordinates. Vector and tensor analysis is the mathematical language
invented to describe curvilinear geometry. Tensor analysis describes both coordinate systems and components of vector and tensor quantities. It also serves as a type of shorthand, but this is a trivial consequence. Such formalism looks concise on paper, but it introduces considerable unnecessary complexity. Th e defi nition of coordinate lines is a necessity, but base vectors which change direction within the computational domain cannot be conveniently integrated throughout the fl owfi eld. As an alternative, one may defi ne curvilinear coordinates as a mathematical transformation of independent variables into what is called function space or vector space. Th e scalar-dependent variables from a coordinate system which utilizes base vectors which are invariant in direction are used without change. Th is is a straightforward, although complex, process, which is described in this chapter. Be advised! Mathematicians like to generalize, engineers do not. Th is causes some confusing literature which, hopefully, will be elucidated in this chapter. Th e essentials of these issues are presented in this chapter in order to arrive at the discretized equations which are to be solved by the CTP code.
