Kinematic Laplacian Equation Method: A Velocity-Vorticity Formulation for the Navier-Stokes Equations

In this work, a novel procedure to solve the Navier-Stokes equations in the vorticity-velocity formulation is presented. The vorticity transport equation is solved as an ordinary differential equation (ODE) problem on each node of the spatial discretization. Evaluation of the right-hand side of the ODE system is computed from the spatial solution for the velocity field provided by a new partial differential equation 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, and details of the implementation of the KLE by isoparametric finite element discretization are given. Validation results of the KLE method applied to the study of the classical case of a circular cylinder in impulsive-started pure-translational steady motion are presented. The problem is solved at several Reynolds numbers in the range $5<Re<180$ comparing numerical results with experimental measurements and flow visualization plates. Finally, a recent result from a study on periodic vortex-array structures produced in the wake of forced-oscillating cylinders is included.