of the Navier-Stokes Equations on Cell-Centered Colocated Grids

Ecole Centrale de Nantes

LMF UMR CNRS C6598, CFD Group,

1, rue de la Noe, B.P. 92101

F-44321 Nantes cedex 3, FRANCE

Abstract

These last twenty years, the search of robust and efficient strategies for the
numerical resolution of the steady or unsteady incompressible Navier-Stokes
equations has been a crucial task giving rise to numerous methods. The major
obstacle of these equations lies in the absence of pressure terms in the
incompressibility constraint. Several ways have been suggested to overcome
this difficulty. The first trend is
represented by pressure correction methods like
SIMPLE or PISO procedures. The main drawback of
such techniques lies in the slowing down of convergence when the number of grid
points or the clustering ratios over curvilinear grids increase. A possible
cure
consists in using non-linear multigrid with sequential
pressure correction methods as basis solvers (or smoothers). The second trend
is to solve the incompressible Navier-Stokes equations in a
locally or fully coupled manner, where momentum and continuity equations are
solved simultaneously. Coupled strategies for the resolution of the
Navier-Stokes equations in
primitive
variables allow to develop robust solvers and subsequently to
understand the limitations of segregated methods.

A fully coupled method for the resolution of the incompressible
Navier-Stokes equations is presented. This method previously used by
Deng et al. (1991) employs a cell-centered colocated grid, standard
linearization of convection terms, central difference discretization for both
convective and diffusive terms and a pressure Poisson equation approach,
leading
to deduce from the incompressibility constraint an equation for the pressure
variable. The originality of this present work is to introduce auxiliary
variables --the so-called pseudo-velocities-- to make easier the flux
reconstruction step. The resulting structure of the nodal unknowns matrix
consists in seven or nineteen bands of sparse blocks. Direct solvers have been
used to solve
coupled systems but their use for
three-dimensional applications
is penalized by strong storage limitations. In order to improve the global
efficiency of the algorithm by seeking grid independent convergence rates, a
non-linear multigrid approach is chosen by implementing a FMG-FAS (Full
Multigrid-Full Approximation Scheme) procedure for
the resolution of the coupled system. Numerical treatments and implementation
aspects of this non-linear procedure are detailed.

Steady laminar lid-driven
cavity flows calculations have been performed on three-dimensional
geometries to discuss the performances of this approach. The retained
test-problems were the three-dimensional versions of the ones investigated by
Demirdzic et al. (1992) and Oosterlee et al. (1993) : regular, skewed and
L-shaped lid-driven cavities. The
non-linear multigrid approach (MGC) is compared
with the
single grid coupled method (SGC) and decoupled methods
based
on the sequential PISO algorithm (DC, DC-MG) with different pressure
linear
solvers respectively Krylov subspace solver and a linear
multigrid solver. Computations were performed on cartesian, orthogonal and
stretched grids for the regular cubic cavity, cartesian and
non-orthogonal grids for the skewed lid-driven cavity
(skewness angle : 30 degrees)
and finally curvilinear grids for the L-shaped cavity. The chosen Reynolds
numbers (Re) are equal to 100 or 400. From
the whole numerical results, the main conclusion is that the non-linear
multigrid approach seems quite powerful, at least more efficient than decoupled
or single grid coupled methods.