A direct, implicit, numerical solution algorithm, with second-order accuracy in space and time, is constructed for the three-dimensional unsteady incompressible Navier-Stokes equations formulated in terms of velocity and vorticity, using generalized orthogonal coordinates to achieve the accurate solution of complex viscous flow configurations. A numerically stable, efficient, direct inversion procedure is developed for the computationally intensive divergence-curl elliptic velocity problem. This overdetermined partial differential operator is first formulated as a uniquely determined, nonsingular matrix-vector problem; this aspect of the procedure is a unique feature of the present analysis. The three-dimensional vorticity-transport equation is solved by a modified factorization technique which completely eliminates the need for any block-matrix inversions and only scalar tridiagonal matrices need to be inverted. The method is applied to the test problem of the three-dimensional flow within a shear-driven cubical box. Coherent streamwise vortex structures are observed within the steady-state flow at Re = 100.
[1]
Thomas B. Gatski,et al.
A numerical study of the two-dimensional Navier-Stokes equations in vorticity-velocity variables
,
1982
.
[2]
Derek B. Ingham,et al.
Finite-difference methods for calculating steady incompressible flows in three dimensions
,
1979
.
[3]
Hermann F. Fasel,et al.
Investigation of the stability of boundary layers by a finite-difference model of the Navier—Stokes equations
,
1976,
Journal of Fluid Mechanics.
[4]
S. M. Richardson,et al.
Solution of three-dimensional incompressible flow problems
,
1977,
Journal of Fluid Mechanics.
[5]
J. Wu,et al.
Solutions of the compressible Navier-Stokes equations using the integral method
,
1981
.