Orthogonal curvilinear coordinate systems

It is not unusual to encounter a problem that is not conveniently posed in one of the common coordinate systems (i.e., cartesian, cylindrical, or spherical). As an illustration consider the flow behavior for the system shown in Fig. 5.20. The analysis seeks to understand the details of the flow field and pressure drop in the narrow conical gap between the movable flow obstruction and the conical tube wall. Intuitively one can anticipate that the flow may have a relatively simple behavior, with the flow parallel to the gap. However, such simplicity can only be realized when the flow is described in a coordinate system that aligns with the gap. An orthogonal curvilinear coordinate system can be developed to model this problem. 

Figure 5.24 illustrates an elbow section in a cylindrical channel where the radius of curvature of the section R is comparable to the channel radius r,-. Analysis of the flow field in this section may be facilitated by the development of a specialized orthogonal curvilinear coordinate system, (r, 6, a). The unit vectors are illustrated in the figure. Referenced to the cartesian system, the angle 6 is measured from the x axis in the x-y plane. The angle a is measured from and is normal to the x-y plane. The distance r is measured radially outward from the center of the toroidal channel. 

The differential forms of the conservation equations derived in the appendixes for reacting mixtures of ideal gases are summarized in Section 1.1. From the macroscopic viewpoint (Appendix C), the governing equations (excluding the equation of state and the caloric equation of state) are not restricted to ideal gases. Most of the topics considered in this book involve the solutions of these equations for special flows. The forms that the equations assume for (steady-state and unsteady) one-dimensional flows in orthogonal, curvilinear coordinate systems are derived in Section 1.2, where specializations accurate for a number of combustion problems are developed. Simplified forms of the conservation equations applicable to steady-state problems in three dimensions are discussed in Section 1.3. The specialized equations given in this chapter describe the flow for most of the combustion processes that have been analyzed satisfactorily. 

The conservation equations in orthogonal, curvilinear coordinate-systems may easily be derived from equations (l)-(4). Since the complete form is complicated and will not be required in subsequent problems, the... 

By utilizing the expression for the divergence of a diagonal tensor in orthogonal, curvilinear coordinate systems, one can show that equation (2) reduces to... 

The transport equations can be written in many different forms, depending on the coordinate system used. Generally, we may select the orthogonal curvilinear Cartesian-, cylindrical-, and spherical coordinate systems, or the non-orthogonal curvilinear coordinate systems, which may be fixed or moving. In reactor engineering we frequently apply the simple curvilinear... 

C.2.4 Orthogonal Curvilinear Coordinate Systems and Differential Operators... 

In this section the relevant differential operators are defined for generalized orthogonal curvilinear coordinate systems. 

The resulting expression for the nabla operator (C.86) are then employed to deduce the transformation formulas for the gradient, divergence, and curl operators in any orthogonal curvilinear coordinate system [11] ... 

The metrics (or scale factors) for a large number of orthogonal curvilinear coordinate systems can be found in the appendix by Happel and Brenner (1973) J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Noordhoff International, Leyden, The Netherlands, 1973). 

An analytic solution for such a problem can thus be sought as a superposition of separable solutions of this equation in any orthogonal curvilinear coordinate system. The most convenient coordinate system for a particular problem is dictated by the geometry of the boundaries. As a general rule, at least one of the flow boundaries should coincide with a coordinate surface. Thus, if we consider an axisymmetric coordinate system (f, rj, ), then either f = const or r] = const should correspond to one of the boundaries of the flow domain. 

Substituting this expression into Helmholtz equation and by separating variables we obtain two normal differential equations of the second order. It is important to emphasize here that the method of separation of variables is applicable only for some orthogonal curvilinear coordinate systems. 

Suppose that we find a periodic orbit, it defines a line in configuration space. We will use this line to define the coordinate system a periodic orbit defines a line of constant u. We further suppose that the location of the periodic orbit is a continuous function of an external parameter - the total energy of the system. Then we find a continuous set of lines of constant u and in this manner define an orthogonal curvilinear coordinate system.For each value of u, there exists a value of the energy g(u) for which we find at u a periodic orbit with energy E=g(u). 

In this section the relevant differential operators are defined for generalized orthogonal curvilinear coordinate systems. Let qi,q2,qi) be curvilinear orthogonal coordinates connected with the Cartesian coordinates (jc,y,z) by the vector relation... 

For instance, it can be easily shown that the cylindrical coordinates and spherical coordinates are orthogonal curvilinear coordinate systems. 

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