Topics in Geophysical Fluid Dynamics: I. Natural Convection in Shallow Cavities. II. Studies of a Phenomenological Turbulence Model

Citation

Cormack, Donald Edward (1975) Topics in Geophysical Fluid Dynamics: I. Natural Convection in Shallow Cavities. II. Studies of a Phenomenological Turbulence Model. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/132p-k784. https://resolver.caltech.edu/CaltechTHESIS:01212022-000323683

Abstract

Part I

The problem of natural convection in a cavity of small aspect ratio with differentially heated end walls is considered. It is shown by use of matched asymptotic expansions that the flow consists of two distinct regimes: a parallel flow in the core region and a second, non-parallel flow near the ends of the cavity. An analytical solution valid at all orders in the aspect ratio, A, is found for the core region, while the first several terms of the appropriate asymptotic expansion are obtained for the end regions. Parametric limits of validity for the parallel flow structure are discussed. Asymptotic expressions for the Nusselt number and the single free parameter of the parallel flow solution, valid in the limit as A → O, are derived.

Also presented are numerical solutions of the full Navier-Stokes equations, which cover the parameter range Pr = 6.983, 10 ≤ Gr ≤ 2 X 10⁴ and 0.05 ≤ A ≤ 1. A comparison with the asymptotic theory shows excellent agreement between the analytical and numerical solutions provided that A ≾ 0.1 and Gr²Pr²A³ ~ 10⁵. In addition, the numerical solutions demonstrate the transition between the shallow-cavity limit and the boundary-layer limit, A fixed Gr → ∞.

Finally, the effect of upper surface boundary conditions on the flow structure within differentially heated shallow cavities is examined. Matched asymptotic solutions, valid for small cavity aspect ratios are presented for the cases of uniform shear stress with zero heat flux, uniform heat flux with zero shear stress, and a heat flux linearly dependent on surface temperature with zero shear stress. It is shown that these changes in surface boundary conditions have an important influence on temperature and flow structure within the cavity.

Part II

The rational closure technique proposed by Lumley and Khajeh-Nouri (1974), in which each unknown correlation is represented as an expansion about the homogeneous, isotropic state, is applied to the approximate closure of the mean Reynolds stress tensor, and rate of dissipation equations for turbulent flows. The high Reynolds number turbulence model which results is similar in many respects to that presented by Lumley et al. However, a more detailed effort is made to evaluate systematically the numerous parameters. Particular emphasis is placed on the suitability and quality of the experimental data which is used for the estimation of model parameters and on the uniqueness and universality of the resulting parameters.

A quantitative comparison of the present turbulence model to those proposed by Daly and Harlow (1970), Hanjalic and Launder (1972b), Shir (1973) and Wyngaard, Cote and Rao (1973), indicates that the present model gives the best overall prediction of the dynamic response for the homogeneous flows of Uberoi (1956, 1957), Champagne, Harris and Corrsin (1970) and Tucker and Reynolds (1968). A further comparison, which evaluates the ability of these turbulence models to predict profiles of the triple-velocity correlation, the rate of intercomponent transfer and the rate of turbulence energy dissipation for inhomogeneous flows indicates that, of the previous turbulence models, that of Hanjalic and Launder is most consistent with the data examined. However, the present model shows promise to yield an even better approximation to the experimental data.

Thesis Files