A time-accurate finite element solution of the Navier-Stokes equations

A finite element formulation of the Reynolds-averaged Navier-Stokes equations (RANS) for three-dimensional unsteady, viscous, compressible, turbulent flow, written in conservation form, is presented. The equations are linearized using the Newton method. They are discretized in space using a Galerkin finite element approach and are integrated in time using the Gear scheme: a multi-level, implicit, unconditionally stable method, with an order of accuracy that can be controlled by using a number of preceding time levels at each time step. The resulting set of algebraic equations for continuity and momentum, at each time step, are solved in a fully-coupled manner by a preconditioned iterative solver. To reduce memory requirements, the ;The solution of the unsteady viscous Burgers equation, a model equation for the Navier-Stokes system, is presented by both the Gear method and the more popular Crank-Nicolson method, and the results used as a justification for the selection of the Gear method.;The outcome of this Thesis has been embedded into a Concordia-Pratt & Whitney Canada code, NS3D. The unsteady capabilities of the code have been validated against two incompressible test cases: the laminar flow over a circular cylinder at Re = 100 and the turbulent flow around a triangular flame holder Re = 45,000. The von Karman vortex street shedding, observed experimentally, is captured in both cases and is computed shedding frequency is shown to be within 5% of the measured ones.