Degree Type


Date of Award


Degree Name

Doctor of Philosophy


Mechanical Engineering


This research effort deals with a numerical and analytical study of multicellular flow instability due to natural convection between narrow horizontal isothermal cylindrical annuli;Buoyancy-induced steady or unsteady flow fields between the annuli are determined using the Boussinesq approximated two-dimensional (2-D) Navier-Stokes (N-S) equations and the viscous-dissipation neglected thermal-energy equation. The vorticity-stream function formulation of the N-S equations is adopted;Both thermal and hydrodynamic instabilities are explored. An asymptotic expansion theory is applied to the N-S equations in the double-limit of Rayleigh number approaching infinity and gap width approaching zero. This double-limiting condition reduces the governing equations to a set of Cartesian-like boundary-layer equations. These equations are further simplified by considering the extreme limits of Pr →[infinity] and Pr → 0. The former limit yields an energy equation which retains the nonlinear convective terms, while the vorticity equation reduces to a Stokes-flow equation, signifying the potential for thermal instability. In the latter limit, the nonlinear terms in the vorticity equation remain, while the energy equation collapses to a one-dimensional conduction equation, signifying the potential for hydrodynamic instability;Thermal instability of air near the top portions of narrow annuli is considered for various size small gap widths. For these narrow gaps, the Rayleigh numbers corresponding to the onset of steady multicellular flow are predicted. Numerical solutions of the 2-D N-S equations also yield hysteresis behavior for the two-to-six and two-to-four cellular states, with respect to diameter ratios of 1.100 and 1.200. In contrast, an unsteady hydrodynamic multicellular instability is experienced near the vertical sections of narrow annuli when the Pr → 0 boundary-layer equations are solved numerically;In addition, analytical steady-state perturbative solutions to the boundary-layer equations are obtained. These results compare favorably to related numerical solutions of both N-S and Pr → 0 simplified equations;In all cases, finite-differenced solutions to the governing equations are obtained using a stable second-order, fully-implicit time-accurate Gauss-Seidel iterative procedure.



Digital Repository @ Iowa State University,

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Daniel Bartholemew Fant



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262 pages