Abstract
We will propose in this paper an improved HSMAC (Highly Simplified MAC) scheme. The proposed scheme is constructed based on the Helmholtz-Hodge theorem; i.e. U = gradφ+curlψ {ψ : divψ=0}, and numerically verified with basic and comparable problem; i.e. velocity profiles on the symmetric section of 3D driven cavity with proportin lx×ly×lz=1×1×6 in comparison with 2D cavity result by Ghia et al. Velocity profiles by the conventional HSMAC method for this proportion with Re-number1000 do not perfectly agree with 2D result, although 2D result perfectly agree. This study began from this recognition point. The substantial difference of the Navier-Stokes equation between 3D and 2D is that the vorticity stretching term exists or not. The influence of the stretching term to the symmetric section appears only with the term ω3∂U3/∂x3 while the symmetry of the whole system is kept. The above 3D numerical result kept sill symmetry. The objective of this paper is to improve the HSMAC scheme using Helmholtz's expression; i.e. the velocity expression in the HSMAC scheme is further modified by scalar potential elements φ: curlψ=U-gradφ{φ: div(gradφ)≠0}.
