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# CHAPTER ■ Computational Coordinates and Conservation

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CHAPTER ■ Computational Coordinates and Conservation book

# CHAPTER ■ Computational Coordinates and Conservation

DOI link for CHAPTER ■ Computational Coordinates and Conservation

CHAPTER ■ Computational Coordinates and Conservation book

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## 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.