Curvilinear coordinate systems cylindrical coordinates

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. 

Figure 3.6-3. Curvilinear coordinate systems (a) cylindrical coordinates, (b) spherical coordinates.

For the example in Figure 2.14 it would be possible to perform the coordinate transformation analytically by introducing cylindrical coordinates. However, in general, geometries are too complex to be described by a simple analytical transformation. There are a variety of methods related to numerical curvilinear coordinate transformations relying on ideas of tensor calculus and differential geometry [94]. The fimdamental idea is to establish a numerical relationship between the physical space coordinates and the computational space curvilinear coordinates The local basis vectors of the curvilinear system are then given as... 

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. 

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... 

One immediate comment before discussing the simplification and solution of these equations is that they appear much more complex than the same equations for a Cartesian coordinate system. To cite just one example, we note that the r component of V2u on the right-hand side of (3-56) not only contains second derivatives with respect tor, 0, and z, but a number of additional terms. These arise because the cylindrical coordinate system is curvilinear, and indeed are often called curvature terms because they arise as a consequence of the curvature of the coordinate lines. The same is true for the u Vu terms in which, for example, (uVu)r/u-V n again because of curvature terms. 

The solution of these equations by means of standard eigenfunction expansions can be carried out for any curvilinear, orthogonal coordinate system for which the Laplacian operator V2 is separable. Of course, the most appropriate coordinate system for a particular application will depend on the boundary geometry. In this section we briefly consider the most common cases for 2D flows of Cartesian and circular cylindrical coordinates. 

Whether the simulation is on a direct discretisation of the equations in cylindrical or transformed coordinates, the discretisation process results in a (usually) linear system of ordinary differential equations, that must be solved. In two dimensions, the number of these will often be large and the equation system is banded. One approach is to ignore the sparse nature of the system and simply to solve it, using lower-upper decomposition (LUD) [212]. The method is very simple to apply and has been used [133,213,214]—it is especially appropriate in curvilinear coordinates and multipoint derivative approximations, where the system is of minimal size [214], and can outperform the more obvious method, using a sparse solver such as MA2 8 (see later). However, many simulators tend to prefer other methods, that avoid using implicit solution in two dimensions simultaneously but still are implicit. Of these, two stand out. 

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

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