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where Ni,j denoting N- and the free index i represents the sum from 1 to 4. If the known values fi and f4 are moved to the RHS, Eq.(3.12) can be rewritten as k] f q (3.13) The above equations are solved for the unknowns f2 and fs on each element. Be- tween the two adjacent elements, an average value is used for f. This procedure is done to solve all the values of f along the span; then it marches to the next stream- wise location using the previous solutions as the new initial conditions, either forward or backward, depending on the location of the stacking axis. The artificial density is solved in the same way. The additional boundary condition on the blade shape, i.e., = 0 at the hub and shroud (see Eq.(2.34)), is imposed explicitly. The blade camber is determined, and then the value of f along the hub and shroud is modified to satisfy this condition. This is done by setting q= 0 and by replacing W, and W, with - cosv and sinV,where v is the angle normal to the boundary; f2 and fs are then solved to satisfy the condition that U = 0. 3.4 The Biharmonic Solver The distribution of the mean swirl schedule is generated through the solution of the biharmonic equation V7 (rVe) = R(r,z) (3.14) ; this will be discussed further in the next chapter. Equation (3.14) can in principle be solved by the usual finite element method as in structural mechanics. However, this method was found to be rather impractical (Ref.1, pg.58). Thus the same method as in the original work by Yang is used; this technique is based on the idea of reformu- lating the above equation as two coupled Poisson equations (Ref.15, 16, 17, 18) in thePDF Image | Radial Inflow Splitter Blades in Three-Dimensional
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