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On numerical solutions of the general Navier-Stokes equations for two-layered stratified flows

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  • 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.
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