Finite element solution of the Navier–Stokes equations by a velocity–vorticity method

Abstract

A velocity–vorticity formulation of the Navier–Stokes equations is presented as an alternative to the primitive variables approach. The velocity components and the vorticity are solved for in a fully coupled manner using a Newton method. No artificial viscosity is required in this formulation. The pressure is updated by a method allowing natural imposition of boundary conditions. Incompressible and subsonic results are presented for two‐dimensional laminar internal flows up to high Reynolds numbers.

GDQ Method for Natural Convection in a Cubic Cavity Using Velocity–Vorticity Formulation

“…In the classical numerical solution algorithms [5,10], an iterative solution procedure is adopted to resolve the coupling between Eqs. (12)-(14).…”

“…Wong and Baker [5] used a second-orderaccurate Taylor's series expansion to compute the vorticity values at the boundaries. Davis and Carpenter [4] used an integral approach for vorticity definition at the boundary, as followed by Guevremont et al [10]. Such low-order schemes have been commonly employed with the finite-element methods [4,5,8,9] and the finite-difference methods [2,3] in the solution of velocity-vorticity equations.…”

An advanced meshless LBIE/RBF method for solving two-dimensional incompressible fluid flows

“…(12) should be employed. However, due to the singularity that appears in the diagonal elements [26], the tangent (13) or the normal (14) forms of this equation should be used.…”

“…However, in spite of their success, the FDM and FEM suffer of mesh problems, solution instabilities when the stability inf-sup conditions are not satisfied and difficulties associated with the treatment of the incompressibility condition [18]. Although many of these problems can be circumvented with the use of a velocity-vorticity approach, instead of the velocity-pressure formulation of Navier-Stokes equations, the problem of requiring good quality meshes remains ( [6,13,14]). Many of the aforementioned problems with FEM have already been successfully solved with the development of advanced multiscale and related schemes.…”

A Penalty Method for the Vorticity–Velocity Formulation

“…The vorticity-velocity formulation of the Navier-Stokes equations has emerged as an attractive alternative to the velocity-pressure formulation in simulating incompressible flows [8,16,32,40]. Several general advantages of this formulation are often cited in the literature: (1) it deals with the physically relevant variables of vortex dominated flows; (2) it works in both two-and three-dimensions; (3) it eliminates the pressure term, which leads to a simple diffusion operator rather than the Stokes operator; (4) boundary conditions can be easier to implement in external flows where the vorticity at infinity is easier to set than the pressure boundary condition; and (5) no additional computational work is required to evaluate noninertial terms since all noninertial effects arising from rotation and translation of the reference frame enter into solution through the initial and boundary conditions [45].…”