An accurate numerical solution algorithm for 3D velocity–vorticity Navier–Stokes equations by the DQ method

Abstract

SUMMARYA numerical scheme based on a di erential quadrature (DQ) method is discussed for the solution of three-dimensional velocity-vorticity Navier-Stokes equations. Numerical solutions of the six ÿeld variables, three velocities and three vorticities are obtained by adopting a coupled numerical solution procedure. Hence, it is required to specify only the velocity boundary conditions, whereas the vorticity values at the boundary are computed implicitly, thus without seeking boundary vorticity values by an iterative procedure followed in other explicit numerical schemes. Since the DQ method approximates a given di erential operator with higher-order polynomials, the vorticity deÿnition is accurately enforced at the boundary. This ensures a divergence-free solution for the ow ÿeld. Flow simulations obtained for a lid-driven cubic cavity for Stokes ow and Re = 100, 400 and 1000 show excellent agreements with other numerical schemes. A grid independence study demonstrates that the present numerical scheme could predict benchmark solutions with coarser meshes than other numerical schemes.

Assessment of a vorticity based solver for the Navier–Stokes equations

“…It solves directly for the vorticity, and it has been argued that methods that do so are more physically accurate, particularly near boundaries [8]. Using vorticity equations for fluid dynamics solvers has a long history and has been a subject of intensive studies, see, e.g., [16,18,24,25,34,35] for a sample of results. Furthermore, it was pointed out recently in [30], see also the discussion in [14], that the discrete vorticity w h from the finite element vorticity equation is a more natural quantity than r  u h for the discrete balance laws for vorticity, enstrophy and helicity when the forcing terms are conservative.…”

On Error Analysis for the 3D Navier–Stokes Equations in Velocity-Vorticity-Helicity Form

“…Initial testing of VVH has shown great promise, and there are several novel properties of the formulation that warrant further development and testing. First, VVH is a velocity-vorticity system, which can provide more accurate solutions than velocity-pressure systems [8,10,11,18,19,24,26,28,27]. Typically such an improvement in accuracy comes at a cost, but in [22] an efficient iterative scheme for VVH is devised that performed very well on initial tests.…”

A fluid solver based on vorticity–helical density equations with application to a natural convection in a cubic cavity

“…Incompressible viscous flows of a Newtonian fluid are modeled by the system of the Navier-Stokes equations typically written in 'primitive' (velocity-pressure-density) variables. The first application of vorticity equations in CFD may be traced back to the late 1970s [1]; the review paper in [2] summarizes many aspects of the approach (see also [3][4][5][6] for more recent applications of the velocity-vorticity formulation to both isothermal and buoyancy-driven flows). A popular numerical approach for such models utilizes the velocity-vorticity form of the Navier-Stokes equations.…”

“…The vorticity equations are typically complemented with the vector Poisson equation linking velocity and vorticity u D r w (2) and possibly the convection-diffusion temperature equation. The first application of vorticity equations in CFD may be traced back to the late 1970s [1]; the review paper in [2] summarizes many aspects of the approach (see also [3][4][5][6] for more recent applications of the velocity-vorticity formulation to both isothermal and buoyancy-driven flows). The advantages of using the vorticity 984 M. A. OLSHANSKII equation (1) for numerical simulations include the following: it allows access of the physically relevant variables of vortex dominated flows, simpler elliptic operators arise rather than the saddlepoint problems because the pressure term is eliminated, and boundary conditions can be easier to implement in external flows where the vorticity at infinity is easier to set than the pressure boundary condition.…”