Browse Theses

A new numerical tool for the direct numerical simulation (DNS) of instability and transition to turbulence is presented. The Navier-Stokes equations for incompressible flow are solved in generalized curvilinear coordinates so that channel flows may be investigated in which the walls of the channel are both curved and wavy. The channel geometry and the flow solution are assumed to be periodic in the streamwise and spanwise directions. A Fourier/Chebyshev spectral collocation method is employed. Both fully- and semi-implicit second-order integration methods are used, with the velocity and pressure fully-coupled. The multi-level iteration scheme used to solve this system consists of Newton's method on the outer level, and the GMRES scheme for sub-iterations. The large linear algebra system obtained from the linearization of the spatial discretization and coupled velocity and pressure is preconditioned through an approximate factorization of the linearized Navier-Stokes operator which decouples the solutions of the velocity and pressure updates in the iterative algorithm. The pressure system is preconditioned by left and right Fourier transform operators followed by a block Jacobi approximation.

This numerical technique is applied to problems of instability and transition in curved channel flows with and without wall waviness. Detailed validations are performed by repeating the results of Finlay, Keller and Ferziger (JFM, vol. 194, 1988) and Ligrani et al. (Phys. Of Fluids A, vol. 4, no. 4, 1992) for two- and three-dimensional Dean vortex flows in a curved channel. New results are obtained for curved channel flows with two-dimensional small amplitude wall waviness. The waviness significantly alters the evolution of both Dean vortex and Tollmien-Schlichting wave instabilities. In particular, the waviness modifies the traveling wave twisting Dean vortex solution of Ligrani et al., for Reynolds number 409, and results in a highly oscillatory state. Waviness also modifies the secondary instability of Tollmien-Schlichting waves at Reynolds number 5000 by forcing asymmetry in the three-dimensional lambda-vortex structures near the upper and lower walls of the channel.

This work is supported, in part, by the Department of Defense NDSEG Fellowship program, NSF grant no. CTS95-12450, Ohio Supercomputer Center grant no. PES070-5, and AFOSR grant no. F49620-93-1-0393.