The KLE Method: A Velocity-Vorticity Formulation for the Navier-Stokes Equations

Abstract

In this work, 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 dis-
cretization 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 vari-
ables in a vorticity-in-time/velocity-in-space split algorithm reduces the number of un-
knowns 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 im-
plementation 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 visual-
ization plates; and finally, a recent result from a study on periodic vortex-array structures
produced in the wake of forced-oscillating cylinders.