1. The use of higher-order accurate methods alleviates the numerical dissipation problem; however, there will always be a lower limit on the number of grid points required per core radius of the given vortex. Hence, as the core radius of the vortex gets smaller, the number of grid points that are required for the calculation becomes greater. In situations where the trajectory of the vortex is approximately known (to within a chord or two) special high density grids uith the required nuniber of grid points can be patched into coarser grids. The technology to transfer information from grid to grid in an accurate manner has been developed in Refs. 7-9.
2. In this study ire present a fifth-order 1cc 11- rate upwind scheme that is set in an iterative implicit framework. The scheme is second-order accurate in time and can be made to solve the nonlinear, fully implicit, finite-difference equations corresponding to the Navier-Stokes equations at each time-step. The scheme is dcrived with the Osher flux-differencing approach. but it car 1 bc used in conjunction with other types of flux differencing such as Roe's scheme.ll The new scheme is then tested by calculating the motion of a vortex in a free str'eam and monitoring a measlire of the rate of decay of the vortex. The supcriority of this method as compared to conventional second-order methods iS demonstrated in this case. The new method is then applied to the blade-vortex interaction problem. The bladevortex code is first validated for weak interactions using the experimental data of Ref. 3 and the numerical data of Ref. 4. The code is then used to simulate strong blade-vortex interactions such as "head-on" collisions. The number of grid points required for the calculation is minimized by patching a fine grid into the region of travel of the vortex. Information transfer between patches is performed using the methods developed in Refs. 7-9.
3. 1-1/2d ar + 65 where the i:+1,2, and i: +,2 are numerical fluxes Consistent dith the'dransformed fluxes and P, respectively. The difference scheme [Eq. (4)l is explicit when m = n and is fully implicit when m i n + 1.