The kinematic Laplacian equation method

A novel procedure for the Navier–Stokes equations in the vorticity–velocity formulation is presented. The time evolution of the vorticity is solved as an ODE problem on each node of the spatial discretization, using at each step of the time discretization the spatial solution for the velocity field provided by a new PDE expression called the kinematic Laplacian equation (KLE). This complete decoupling of the two variables in a vorticity-in-time/velocity-in-space split algorithm reduces the number of unknowns to solve in the time-integration process and also favors the use of advanced ODE algorithms enhancing the efficiency and robustness of time integration. The issue of the imposition of vorticity boundary conditions is addressed, as well as the details of the implementation of the KLE by isoparametric finite element discretization. We shall see some validation results of the KLE method applied to the classical case of a circular cylinder in impulsive-started pure-translational steady motion at several Reynolds numbers in the range 5 < Re < 180, comparing them with experimental measurements and flow visualization plates; and finally, a recent result from a study on periodic vortex-array structures produced in the wake of forced-oscillating cylinders.