A grid-free numerical scheme is provided for the solution of unsteady flow in the hydraulic turbines. The Lagrangian vortex method is utilized to evaluate the convection and stretch of the vorticity, and the boundary element method is used to solve the Neumann problem to define the potential flow. The no-slip boundary condition is satisfied by generating vortex sticks at the solid surface. A semi-analytical regularization technique is applied to evaluate the singular boundary surface integrals of the potential velocity and its gradients accurately. The fast multipole method (FMM) has been extended to evaluate the velocity and velocity gradients induced by the discretized vortex blobs in the Lagrangian vortex method. The successful simulation for the unsteady flow through a hydraulic turbine's runner has manifested the effectiveness of the proposed method.
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