A three-dimensional, time-accurate, incompressible, Navier-Stokes, finite element CFD algorithm

Author

Paul Thomas Williams

Date of Award

5-1993

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Engineering Science

Major Professor

A. J. Baker

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 dissertation 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 semi-discretization of a Galeikin weak statement, equal-order interpolation for all state-variables, a &theta-implicit time-integration scheme, and a quasi-Newton iterative procedure extended by a Taylor Weak Statement (TWS) formulation for dispersion error control. Original contributions to algorithmic theory include: (a) formulation of the unsteady evolution of the divergence error, (b) investigation of the role of non-smoothness in the discretized continuity-constraint function, (c) development of a uniformly H^{I Galerkin weak statement for the Reynolds-averaged Navier-Stokes pressure Poisson equation, (d) derivation of physically and numerically well-posed boundary conditions, and (e) investigation of sparse data structures and iterative methods for solving the matrix algebra statements generated by the algorithm. 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 non-smooth continuity-constraint function to control inherent dispersive-error mechanisms. Dominated by this dispersive-error mode, the non-smoothness of the discrete continuity-constraint function is proven to play a critical role in its ability to remove the divergence error in the discrete velocity distribution. 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. A broad range of 3-dimensional verification, benchmark, and validation test problems, as computed by the code CFDL.PHI3D, completes this dissertation.}