Thompson, Warsi, Mastin - Numerical Grid Generation, страница 11

If completecontinuity is not required, then the surrounding layer is not required and the interfaces wouldbe treated in the same manner as are physical boundaries. However, the surrounding layerand the point correspondence thereon discussed above might still be needed for thenumerical solution to be done on the grid.The extension of all of the above concepts and structures to three dimensions is direct,the illustrations having been given in two dimensions only for economy of presentation.Exercises1. Sketch the grid when the physical region shown below is transformed to an emptyrectangle as indicated.2. Locate the cuts on the grid in the physical region for the grids shown on pp.

35-37.3. Sketch the grid for a O-grid, a C-grid, and a slit configuration for a cascade arrangement(a periodic stack of bodies, e.g., turbine blades.)4. For the configuration shown below, let body II transfrom form to a slit in a C-type systemabout I. Sketch the grid lines and show the transformed region configuration.5. Sketch the surface grid on a sphere with eight "corners" on the sphere. This is analogousto the 2D grid for the circle with four "corners" on p. 23.

This configuration would be moreappropriate for embedding a spherical region in a composite structure than would the usualpolar system.6. Diagram the transformed region configuration for a polar coordinate system between twoconcentric spheres. Here the polar axis will map to an entire side of the transformed region.7. Sketch the surface grid on a circular cylinder with a hemispherical cap for two cases: (1)with a cylindrical-coordinate type grid on the hemisphere and (2) with four "corners" on theintersection of the hemisphere with the cylinder.

The latter case would be more appropriatefor a composite system.8. For the two-slit configuration on p. 33, diagram a composite system composed of emptyblocks. Show all cuts and the correspondence between pairs thereof9. Sketch the grid and diagram the blocks for a composite two-block system for a circularpipe T-section. It is necessary to use a cross-sectional grid of the type shown on p. 23,having four "corners" on the circle, since cylindrical grids would not join with linecontinuity.10. Diagram the block structure and grid for a six-block composite system for the regionbetween two concentric spheres, based on the surface grid of Exercise 10. Note that no polaraxis occurs with this configuration.11.

Consider a region between two boundaries, both of which are formed of cylinders withhemi-spherical caps, these being coaxial with one inside the other. Sketch the grid anddiagram the blocks for a three-block system, with one block corresponding to the annularregion between the caps, for the following two configurations: (1) with the polar axisconnecting the caps and (2) with no polar axis. In the latter case each of four sides of oneblock will correspond to one of four portions of one side of the other block.12. Diagram the point correspondence across all the cuts in the two-body 0-grid on p.34.Also give the relation between the indices of corresponding points on the cuts.

Finally givethe relation between the indices of points on a surrounding layer of points and points insidethe field inside the cuts.13. For one block of the system on p. 52, give the correspondence between indices of pointson the surrounding layers and points inside adjacent blocks for the cuts.14. Sketch the grid for a 2-D composite system having two circular regions embedded in agrid which is generally rectangular. Let one of the circular regions have acylindrical-coordinate type of grid and the other have a grid of the type with four "corners"on the circle as on p. 23.15.

Show that is is not possible to handle the point correspondence across the cuts in theembedded slab type system shown on p. 46 (a 2-block system) by using an extra layer ofpoints just inside the slab in the outer system. Also show that it is possible to represent thecorrespondence across the cuts using surrounding layers if a 4-block composite system isused.III. TRANSFORMATION RELATIONSThe transformation relations from cartesian coordinates to a general curvilinear systemare developed here using certain concepts from differential geometry and tensor analysis,which are introduced only as needed.

Warsi [15] has given an extensive collection ofconcepts from tensor analysis and differential geometry applicable to the generation ofcurvilinear coordinate systems. Another discussion is given in Eiseman [16], where theseconcepts are developed as part of a general survey on the generation and use of curvilinearcoordinate systems.

Eiseman includes a discussion on differential forms, which is afundamental part of modern differential geometry, but primarily restricts his development toEuclidean space. In contrast, Warsi has given a classical development that includes curvedspace, but not differential forms.Partial derivatives with respect to cartesian coordinates are related to partialderivatives with respect to curvilinear coordinates by the chain rule which may be written ineither of two ways. If A is a scalar-valued function, then(1)or, equivalently,(2)Either formulation may be used to relate the cartesian and curvilinear derivatives of thefunction A.

However, there is a difference in the transformation derivatives which must beinserted in these relations. In the first ease one must be able to evaluate (or approximate) thevectorswhereas, the second case requiresThus all the transformation relations may be based on either of these two sets of vectors.Various properties of, and relationships between, these vectors are developed and applied inthis chapter to provide the necessary transformation relations.1. Base VectorsThe curvilinear coordinate lines of a three-dimensional system are space curvesformed by the intersection of surfaces on which one coordinate is constant.

One coordinatevaries along a coordinate line, of course, while the other two are constant thereon. Thetangents to the coordinate line and the normals to the coordinate surface are the base vectorsof the coordinate system.A. CovariantConsider first a coordinate line along which only the coordinatevaries:Clearly a tangent vector to the coordinate line is given by(Coordinates appearing as subscripts will always indicate partial differentiation.) Thesetangent vectors to the three coordinate lines are the three covariant base vectors of thecurvilinear coordinate system, designated(3)where the three curvilinear coordinates are represented by i (i=1,2,3), and the subscript iindicates the base vector corresponding to the i coordinate, i.e., the tangent to thecoordinate line along which only i varies.B.

ContravariantbyA normal vector to a coordinate surface on which the coordinate:is constant is givenThese normal vectors to the three coordinate surfaces are the three contravariant base vectorsof the curvilinear coordinate system, designated(4)Here the coordinate index i appears as a superscript on the base vector to differentiate thesecontravariant base vectors from the covariant base vectors. The two types of base vectors areillustrated in the following figure, showing an element of volume with six sides, each ofwhich lies on some coordinate surface.C. OrthogonalityOnly in an orthogonal coordinate system are the two types of base vectors parallel,since for a non-orthogonal system, the normal to a coordinate surface does not necessarilycoincide with the tangent to a coordinate line crossing that surface:Also for an orthogonal system the three base vectors of each type are obviously mutuallyperpendicular.2.

Differential ElementsThe differential increments of arc length, surface, and volume, which are needed forthe formulation of the respective integrals, can be generated directly from the co-variant basevectors. The general arc length increment leads also to the definition of a fundamental metrictensor.A.

Covariant metric tensorThe general differential increment (not necessarily along a coordinate line) of aposition vector is given byAn increment of arc length along a general space curve then is given byThe general arc length increment thus depends on the nine dot products,(i=1,2,3)and (j= 1,2,3), which form a symmetric tensor. These quantities are the covariant metrictensor components:(5)Thus the general arc length increment can be written as(6)B. Arc length elementAn increment of arc length on a coordinate line along whichi variesis given by(7)C. Surface area elementAlso an increment of area on a coordinate surface of constanti isgiven by(8)Using the vector identity(9)we have(10)so that the increment of surface area can be written as(11)D.