In this work we address the development of the velocity decomposition algorithm, a numerical flow solution method that incorporates both velocity potential and Navier-Stokes-based solution procedures. The motivation for this is so that the field discretization required by the Navier-Stokes solver can be reduced to the region of the flow domain in which the flow is vortical. Specific advantages are that the computational cost is reduced, it is easier to discretize the flow domain, and difficult problems such as the simulation of ships maneuvering in a seaway are closer to being within reach. The target applications are broad, ranging from vortex shedding on bluff objects such as risers, to the wave induced loads on a platform in a current and irregular seas. In previous work, the algorithm has been successfully applied to address steady flows of 3-D non-lifting bodies without water waves, or 2-D bodies that can have lift and be near a water surface. In this paper, the velocity decomposition approach is extended to numerically solve for the unsteady flow of single-phase viscous flows. The velocity vector is decomposed into irrotational and vortical components. A boundary element method is used to solve for the irrotational component (designated as the viscous potential) by applying a viscous boundary condition to the body boundary. A field method is used to solve for the total velocity on a reduced domain where the flow is vortical. The new algorithm investigates two approaches to solve the unsteady problem based on different types of time-dependence exhibited by the solution. The unsteady velocity decomposition method is demonstrated on two cases, and the solutions are compared to those generated by a conventional viscous flow solver. The results by the new algorithm agree well with the benchmark solutions and exhibit a reduction in time.

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