General Curvilinear Coordinates

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. 

A coordinate system that is natural for the conical channel can be established as illustrated in right-hand panel of Fig. 5.20. The origin of the new coordinate system begins on the tube wall at the entrance of the conical section. The x coordinate aligns with the surface of the tube wall and the y coordinate measures the distance across the channel and is normal to the tube wall. The 4 coordinate measures the circumferential angle around the conical 

The transformation task begins by representing a general vector V in the channel as 

The approach to finding the transformation metric factors can be found in most books that discuss vector-tensor analysis (an excellent reference is Malvern [257]). For orthogonal coordinate transformations, metric factors are given generally as 

It should be noted that the metric factors represent diagonal elements of a transformation matrix. It is therefore prudent to check the off-diagonal components to ensure that the new coordinate system is indeed orthogonal. In general, the elements of the metric tensor are given as [257] 

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Figure 5.21 Thin layer between curved surfaces and the general curvilinear coordinate system...

Let qi,q2,q3) be curvilinear orthogonal coordinates connected with the Cartesian coordinates x,y,z) by the vector relation r = r qi, q2, qa), where r is the radius vector of the point P considered. The Cartesian coordinates are then related to the generalized curvilinear coordinates by ... 

N. V. Kantartzis, T. I. Kosmanis, and T. D. Tsiboukis, Fully nonorthogonal higher-order accurate FDTD schemes for the systematic development of 3-D reflectionless PMLs in general curvilinear coordinate systems, IEEE Trans. Magn., vol. 36, no 4, pp. 912—916, July 2000.doi 10.1109/20.877591... 

Consider a 3-D domain that can be adequately described by the generalized curvilinear coordinate system (u, v, w) and that its mappings are adequately smooth to allow consistent definitions. Then, any vector F can be decomposed into three components with respect to the contravariant a , a , a or the covariant a , a, a,a, linearly independent basis system as... 

N. V. Kantartzis, J. S. Juntunen, and T. D. Tsiboukis, An enhanced higher-order FDTD technique for the construction of efficient reflectionless PMLs in 3-D generalized curvilinear coordinates, in Proc. IEEE Antennas Propag. Soc. Int. Symp., Orlando, FL, July... 

E. A. Navarro, C. Wu, R Y. Chung, and J. Litva, Some considerations about the finite difference time domain method in general curvilinear coordinates, IEEE Microw. Guided Wave Lett., vol. 6, pp. 193-195, June 1996.doi 10.1109/75.491502... 

We will need several results from tensor analysis and rather than using general results (see [Ij, [18], [44]) for general curvilinear coordinate systems, we begin by specifying our system and working with the coordinates relevant to liquid jets - we will allow asymmetry, i.e. d dependence, however. The appropriate general curvilinear coordinates here are v = z, V = 6, and the surface can be defined parametrically by the position vector... 

Scribano Y, Lauvergnat DM, Benoit DM (2010) Fast vibrational configuratirai interaction using generalized curvilinear coordinates and self-consistent basis. J Chem Phys 133 094103... 

A general curvilinear coordinate system is introduced. The profile coordinate s defines the material line, and a second coordinate t is perpendicular to the tangential length of the reference axis s . 

In general, curvilinear coordinates are not mutually orthogonal that is,... 

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... 

Another curvilinear coordinate system of importance in two-centre problems, such as the diatomic molecule, derives from the more general system of confo-cal elliptical coordinates. The general discussion as represented, for instance by Margenau and Murphy [5], will not be repeated here. Of special interest is the case of prolate spheroidal coordinates. In this system each point lies at the intersection of an ellipsoid, a hyperboloid and and a cylinder, such that... 

There is finally the possibility of decay-like leakages between the particle-antiparticle spaces, and further that there could be an overall escape out of the presently defined spaces. If so associated Jordan blocks naturally appearing would decelerate this decay via the polynomial delay mechanism described earlier [7-10] with implications to subject matters like problems related to size of the cosmological constant. Also, the account given here should consider a more general decomposition of pz into curvilinear coordinates (cf. Eq. (30)) in order to yield a more appropriate analysis (see e.g. [22] and references therein). 

Adding the analogous contributions from the q2- and 3-directions and dividing by the volume dr, we obtain the general result for the divergence in curvilinear coordinates ... 

Let us consider surfaces in a Cartesian frame, whence these results can be generalized to any set of three coordinates x in an arbitrary coordinate system fixed in space. A surface in 3D space can generally be defined in several different ways. Explicitly, z = F x,y), implicitly, f x,y,z) = 0 or parametrically by defining a set of parametric equations of the form x = x C, rf), y = y C, v), z = z (, p) which contain two independent parameters Q, p called surface coordinates or curvilinear coordinates of a point on the surface. In this coordinate system a curve on the surface is defined by a relation f Q, p) = Q between the curvilinear coordinates. By eliminating the parameters Q, p one can derive... 

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... 

Considering a generalized orthogonal coordinate system, the orthogonal curvilinear coordinates are defined as qa- In this O-system the base vectors Gq, are defined as unit vectors along the coordinates. The position of the point P is given by the coordinates, or by the position vector r = r qa,t). 

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

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. 

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