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A numerical scheme has been obtained rigorously from the initial-boundary value problem for the Navier-Stokes (NS) equations in the context of the vortex method. The technique is based on a transformation of the NS equation into a parabolic equation which has an exact solution. The numerical scheme is derived by expanding the exact solution in Taylor series in powers of a small time interval. Numerical implementation is developed with use of vortex particles to represent the vortex flow domain. The method is used to solve practical engineering problems. The technique can also incorporate turbulence modelling.