Transformations and grids

Abstract

If all CFD applications dealt with physical problems where a uniform, rectangulargrid could be used in the physical plane, there would be no reason to alter thegoverning equations derived in Chap. 2. We would simply apply these equations inrectangular (x, y, z, t) space, finite-difference these equations according to the differencequotients derived in Chap. 5, and calculate away, using uniform values ofδx, δy, δz and δt. However, few real problems are ever so accommodating. For example,assume we wish to calculate the flow over an airfoil, as sketched in Fig. 6.1,where we have placed the airfoil in a rectangular grid. Note the problems with thisrectangular grid:(1) Some grid points fall inside the airfoil, where they are completely out of theflow. What values of the flow properties do we ascribe to these points?(2) There are few, if any, grid points that fall on the surface of the airfoil. Thisis not good, because the airfoil surface is a vital boundary condition for thedetermination of the flow, and hence the airfoil surface must be clearly andstrongly seen by the numerical solution.As a result, we can conclude that the rectangular grid in Fig. 6.1 is not appropriatefor the solution of the flow field. In contrast, a grid that is appropriate is sketched inFig. 6.2(a). Here we see a non-uniform, curvilinear grid which is literally wrappedaround the airfoil. New coordinate lines, ξ and n, are defined such that the airfoilsurface becomes a coordinate line, n = constant. This is called a boundary-fitted coordinatesystem, and will be discussed in detail later in this chapter. The importantpoint is that grid points naturally fall on the airfoil surface, as shown in Fig. 6.2(a).What is equally important is that, in the physical space shown in Fig. 6.2(a), thegrid is not rectangular, and is not uniformly spaced. As a consequence, the conventionaldifference quotients are difficult to use. What must be done is to transformthe curvilinear grid mesh in physical space to a rectangularmesh in terms ofξ and n. This is shown in Fig. 6.2(b), which illustrates a rectangular grid in termsof ξ and n. The rectangular mesh shown in Fig. 6.2(b) is called the computationalplane. There is a one-to-one correspondence between this mesh, and the curvilinearmesh in Fig. 6.2(a), called the physical plane. For example, points a, b and c inthe physical plane (Fig. 6.2a) correspond to points a, b and c in the computationalplane, which involves uniform δξ and uniform δn. The computed information isthen transferred back to the physical plane. Moreover, when the governing equationsare solved in the computational space, they must be expressed in terms of thevariables ξ and n rather than x and y; i.e., the governing equations must be transformedfrom (x, y) to (ξ, n) as the new independent variables.The purpose of this chapter is to first describe the general transformation of thegoverning flow equations between the physical plane and the computational plane.Following this, various specific grids will be discussed. This material is an exampleof a very active area of CFD research called grid generation. © Springer-Verlag Berlin Heidelberg 2009.