Abstract:
The objective of this paper is to seek the general solution of the
complete Navier-Stokes equations governing heterogeneous, time-dependent, incompressible, viscous, laminar fluid dynamics through
digital simulation. In particular, two-layered stratified flows are
treated by using a numerical algorithm called the Marker-and-Cell
extended method (MACE), which is an extension of the MAC code
originated by the Los Alamos Scientific Laboratory.
To provide the necessary background, the paper begins with a
mathematical description of stratified flows. The controlling equations
are the Navier-Stokes equations, the continuity relation, and the
incompressible condition. These equations and various alternative
formulations are presented in axiomatic form; care has been taken in
this exposition so as to exhibit the hypotheses involved in analytical hydrodynamics. Necessary boundary conditions characterizing the
physical states of the stratified fluids are also stated. The finite difference
scheme of MACE algorithm is discussed in detail, and
specific boundary conditions are derived for the MACE code allowing
inflow and outflow capabilities, no-slip and free-slip boundary-layer
considerations, and free-surface formulation.
In order to place the MACE method on a solid mathematical
foundation, a discussion of numerous aspects of stability and error
are derived for the Navier-Stokes equations. The stability of the
MACE numerical scheme and the convergence of the numerical solution
to the theoretical solution are proven.
An example involving two-layered stratified fluid flow with inflow
and outflow is presented to show how the previously non-realizable
solutions of the general Navier-Stokes equations can be computed
systematically through this algorithm. The example was simulated
digitally to obtain movie-frames illustrating various stages of the
transient state of stratified flow patterns. A complete set of flow diagrams
demonstrating the logical sequence of calculations by the MACE
method is provided; a FORTRAN IV program listing is appended.
Natural extensions of the MACE method are proposed to handle
general heterogeneous fluids of both the continuum and the discrete layered
stratification. The MACE method has great potential, for
continuations of this method may lead to discover whether or not viscous,
laminar fluid flow can be fully described by the fundamental
Navier-Stokes equations.
