INCOMPRESSIBLE COMPUTATIONAL FLUID DYNAMICS AND THE CONTINUITY CONSTRAINT METHOD FOR THE THREE-DIMENSIONAL NAVIER-STOKES EQUATIONS

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Abstract
As the field of computational fluid dynamics (CFD;) continues to mature, algorithms are required to exploit the most recent advances in approximation theory, numerical mathematics, computing architectures, and hardware. Meeting this requirement is particularly challenging in incompressible fluid mechanics, where primitive-variable CFD formulations that are robust, while also accurate and efficient in three dimensions, remain an elusive goal. This monograph asserts that one key to accomplishing this goal is recognition of the dual role assumed by the pressure, i.e., a mechanism for instantaneously enforcing conservation of mass and a force in the mechanical balance law for conservation of momentum. Proving this assertion has motivated the development of a new, primitive-variable, incompressible, CFD algorithm called the continuity constraint method (CCM;). The theoretical basis for the CCM consists of a finite-element spatial semidiscretization of a Galerkin weak statement, equal-order interpolation for all state variables, a 6-implicit time-integration scheme, and a quasi-Newton iterative procedure extended by a Taylor weak statement;(TWS) formulation for dispersion error control. This monograph presents: (I) the formulation of the unsteady evolution of the divergence error, (2) an investigation of the role of nonsmoothness in the discretized continuity-constraint function, (3;) the development of a uniformly H’ Galerkin weak statement for the Reynolds-averaged Navier-Stokes pressure Poisson equation, and(4;) a derivation of physically and numerically well-posed boundary conditions. In contrast to the general family of ‘pressure-relaxation’ incompressible CFD algorithms, the CCM does not use the pressure as merely a mathematical device to constrain the velocity distribution to conserve mass. Rather, the mathematically smooth and physically motivated genuine pressure is an underlying replacement for the nonsmooth continuity-constraint function to control inherent dispersive-error mechanisms. The genuine pressure is calculated by the diagnostic pressure Poisson equation, evaluated using the verified solenoidal velocity field. This new separation of tasks also produces a genuinely clear view of the totally distinct boundary conditions required for the continuity constraint function and genuine pressure.

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