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Because we have two conditions along each boundary so that the generation of a smooth rVe distribution from second order equation (such as Poisson equation) will not work, we need to have at least a fourth order equation so as to satisfy all the boundary conditions; a natural choice would be to use the biharmonic equation where V (rVe) = R(r,z) (4.1) 2 )2 as a means to control the swirl distribution. As in the original work by Yang, however, R(r,z) is taken to be identically zero here for simplicity. The swirl distribution is controlled by the specification of the value of rVe along the hub and the shroud. In this way, the problem is reduced from specifying the swirl in a two dimensional region to a one dimensional region, i.e., along the boundary only. 4.3 Method of Specifying rVe along the Hub and the Shroud In general, we only need to specify rVe along the leading edge, the hub, the shroud and the trailing edge. As in the original work by Yang, the hub or the shroud region is divided into three sections (see Fig. 4.1): from the leading edge (L.E.) to a point called A, from point A to a second point called B, and from point B to the trailing edge (T.E.). We will specify rVe as polynomials in those three sections and their degrees in each section is determined to ensure certain continuity condition in rVe distribution across the regions. Because the biharmonic and the governing equations require the evaluation of v2rVa (see Eq.(2.26) and Eq.(4.1)), we need to enforce the continuity of the second derivative of rVe along the hub and shroud. This implies that the polynomials will 2 += The forcing function R(r,z) can be chosen appropriately and it can actually be usedPDF Image | Radial Inflow Splitter Blades in Three-Dimensional
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