Ph.D. Thesis Defense by Charlotte K. C. Song
Monday, March 24, 2003

(Drs. Cyrus Aidun and Marc Smith, co-advisors)

"Hydrodynamic Stability of Confined Shear-Driven Flows"


Driven cavity flow has received much attention over the past few decades. Interest in this problem has been both practical and academic. Many coating and lubrication processes have flow regimes similar to driven cavity hydrodynamics. Furthermore, the simple domain and boundary conditions of this enclosed shear-driven flow render an ideal benchmark system for validation of numerical algorithms. While the velocity, pressure, and vorticity fields in the driven cavity have been computed using several numerical approaches, the system's stability to disturbances has yet to be thoroughly explored.

The present study employed a global spectral technique within a suitable Hilbert space to approximate the two-dimensional driven cavity basic state and to solve the generalized eigenvalue problem resulting from the linearized disturbance equations. Homogeneous boundary conditions were required and achieved by eliminating the nonhomogeneous velocity boundary condition on the lid. The velocity field was reduced using a divergence-free, separable function of the spatial coordinates; this function was the product of a regularized boundary condition on the lid with the exact solution of the asymptotic shallow cavity case. The results of this study assisted in relieving the ambiguity between previous experimental results and numerical primary instability computations.

The basic state was initially assumed to be completely two-dimensional and linear stability analyses were performed on states having Reynolds numbers greater than the experimentally observed critical value. The present three digit accurate analysis concluded that a stationary normal mode challenged the base state at a Reynolds number of 945 having a spatial wavelength of 35.5% of the unit cavity height. In a physical cavity, an Eckman-type boundary layer develops on the cavity endwalls. Its effect on the core flow within the cavity is modeled by the introduction of constant-gradient, pressure-driven cross-flow. The linear stability of this modified basic state flow was evaluated and compared to that of the fully two-dimensional basic state.