Direct numerical simulation of two dimensional turbulent thermal convection of an incompressible fluid (Prandtl number = 0.71) in a laterally heated rigid box has been carried out for a range of Rayleigh numbers. The nonlinear dynamics of the turbulent convection is studied using tools of Dynamical Systems Theory. The Karhunen-Loeve dimension of the turbulent flow is found to vary from 49 at Rayleigh number $Ra = 4\times10\sp8$ to 127 at $Ra = 10\sp9$. The characterization of turbulence as spatio-temporal chaos has been investigated in detail by studying the spatial variation of local dynamics whose properties vary throughout the domain. Spatial maps of the dimensional complexity and Lyapunov exponent of the local dynamics are constructed. Analysis of the spatial variation of these quantities revealed significant heterogeneity and sharp gradients. Numerical support for the relationship between the distribution of local Lyapunov exponents and that of the energy dissipation rate as conjectured earlier by Ruelle (1991) is presented. The implications of the factorization of the global dynamics due to the spatial localization of the local chaotic dynamics are discussed. It is shown that hierarchical partitioning of the global system into spatially coherent subsystems can in principle lead to a lower dimensional description of turbulence. A quantity called coherence parameter is introduced for detecting the spatio-temporal coherence of a turbulent flow. / Source: Dissertation Abstracts International, Volume: 53-11, Section: B, page: 5753. / Major Professor: J. F. Magnan. / Thesis (Ph.D.)--The Florida State University, 1992.

The complex yet important unsteady flow problem of dynamic stall was studied using the random-walk vortex blob method. A fast summation technique was used, which reduced the computational operations for n vortices from $O(n\sp2)$ to O(n log n). Viscous flows about a NACA 0012 airfoil and about a parabolic leading edge were investigated. The Reynolds number based on the chord length was 5000. The purposes of this study were to enhance understanding of the physical process of dynamic stall, and to investigate efficient ways to prevent the stall. The numerical results showed that the development of unsteady separation at the leading edge caused the development of a dynamic stall vortex. In the further development, the dynamic-stall vortex dominated the flow field, and its motion affected the lift greatly. Flow control methods to improve the lift coefficient were also investigated. These included a moving surface and suction through the surface. It was found that a careful implementation of these two methods could be effective for dynamic stall prevention. The computational results also demonstrated that early application was necessary to control dynamic stall. / Source: Dissertation Abstracts International, Volume: 56-12, Section: B, page: 6885. / Major Professor: Leon L. Van Dommelen. / Thesis (Ph.D.)--The Florida State University, 1995.

The Formal Variational Calculus is applied to the equations of fluid dynamics. In particular, it is shown that the equations for one-dimensional isentropic compressible flow have infinitely many higher order symmetries and a first order conservation law. The notion of a generalized fluid equation is introduced and is shown that the incompressible fluid equations inherit their Hamiltonian structure from the compressible ones. Finally, the behavior of certain Hamiltonian structures under a change in dependent variables is examined / acase@tulane.edu

In an effort to understand better the flow in the core of the Earth, we investigate the steady rise of an infinitely long vertical rigid cylinder parallel to its axis in a rotating electrically conducting fluid in the presence of uniform prescribed transverse magnetic field. The rotation and magnetic-field vectors have arbitrary orientation. We suppose the circular cylinder is forced to rise with a constant speed and investigate the structure of the flow and calculate the drag on the cylinder. The flow structure is found by solving a two-dimensional (independent of the axial coordinate) mixed boundary value problem. Approximate analytic solutions for velocity field and perturbed magnetic field are obtained. The buoyancy driven rise speed of the cylinder is calculated. The results are consistent with the those derived from Moore and Saffman (1969) and given by Hasimoto (1960) as limiting cases. The numerical value of dimensional rise speed obtained is in good agreement with the typically quoted rise speed in geophysics. / Source: Dissertation Abstracts International, Volume: 56-04, Section: B, page: 2067. / Major Professor: David Loper. / Thesis (Ph.D.)--The Florida State University, 1995.

When a multi-component liquid is cooled and solidified, commonly, the solid phase advances from the cold boundary into the liquid as a branching forest of dendritic crystals. This creates a region of mixed solid and liquid phases, referred to as a mushy zone, in which the solid forms a rigidly connected framework with the liquid occurring in the intercrystalline gaps. When the fluid seeps through the dendrites, further freezing occurs which fills in pores of the matrix and reduces its permeability to the liquid flow. In particular, if a binary alloy (for example, NH$\sb4$Cl-H$\sb2$O solution) is cooled at bottom and a dense component (for example, NH$\sb4$Cl) is solidified, buoyant material released during freezing in the pores returns to the melt only through thin, vertical, but widely separated, 'chimneys', the flow through the matrix between them being organized to supply these chimneys. / We presented photos of a mush-chimney system obtained from the ammonium chloride experiment, and we studied how convection with horizontal divergence affects the structure and flow of the mush-chimney system. We use a simple ODE system in the mush derived by assuming that the temperature depends on vertical coordinate only. We find that the mass fraction of solid increases and the depth of a mush decreases when the strength of convection increases. / We present an axisymmetric model containing only one chimney to analyze the structure of the mush-chimney system. We find solutions of the temperature, the solid fraction, and the pressure in the chimney wall. In particular, the pressure expression shows that the fluid flow needs a huge pressure in order to pass through the chimney wall if its permeability is very small. / We assume that a ratio of composition is large, which allows us to neglect the pressure contribution of the chimney wall. We use the knowledge of the variables in the mush, evaluated on the chimney wall, to find the fluid flow in the chimney and the radius of chimney. Our procedure employs the von Karman-Pohlhausen technique for determining chimney flow (Roberts & Loper, 1983) and makes use of the fact that the radius of the chimney is much less than the thickness of the mush. We find a relation between a parameter measuring the ratio of viscous and buoyancy forces in the chimney and the vertical velocity component on the top of the mush, and estimate numerically the value of this velocity measuring the strength of convection. The results obtained show reasonably good agreement with theoretical and experimental works (Roberts & Loper (1983), Chen & Chen (1991), Tait & Jaupart (1992), Hellawell etc. (1993), Worster (1991)). / Source: Dissertation Abstracts International, Volume: 56-07, Section: B, page: 3802. / Major Professor: David Loper. / Thesis (Ph.D.)--The Florida State University, 1995.

An experimental study of a high-temperature, supersonic jet operating at ideal and off-design conditions is conducted using both standard pressure and temperature instrumentation, and particle image velocimetry (PIV). The purpose is to determine the effects of shock-cell structure and of temperature on the growth and development of the jet plume. The study uses a two stage approach to introduce shock and expansion waves into the flow. First a single axisymmetric, convergent-divergent nozzle having a design exit Mach number of 2.0, is operated at three varying pressure ratios, corresponding to isentropic flow Mach numbers of 1.8, 2.0, and 2.15. In the second stage, three axisymmetric, convergent-divergent nozzles with matched throat areas, and varying area ratios are utilized. These nozzles consist of the Mach number 2.0 nozzle previously used, as well as nozzles having design exit Mach numbers of 1.8 and 2.15. These nozzles are operated at a constant pressure ratio corresponding to an isentropic Mach number of 2.0. Results indicate that shock cells in the near field do not significantly affect the growth of the shear layers. Furthermore, flow in the fully developed region of the jet appear unaffected by the presence of the shocks. Increasing temperature ratio results in an increase of the jet spreading rates and centerline velocity decay. Normalized shear layer growth rates in the initial jet regions are consistent with that of a planer shear layer. The PIV data is found to be consistent with that measured using probe instrumentation. / Source: Dissertation Abstracts International, Volume: 56-03, Section: B, page: 1673. / Major Professor: Anjaneyulu Krothapalli. / Thesis (Ph.D.)--The Florida State University, 1995.

In order to understand storm track dynamics, we construct a primitive equations numerical model to simulate stratified fluid motion in a differentially heated annulus with bottom topography. The numerical experiments include different rotation rates, topographic amplitudes and local heating. Throughout these cases we focus on the relationships between the time mean flow and transient wave activity. The results show that transient wave activity is weak when the zonally distributed baroclinicity is weak (e.g. at high rotation rate). The wave activity is also weak when the baroclinicity is too strong. The barotropic wind has a steering effect on the transient wave activity; the higher the zonal wind speed, the further downstream are the storm tracks. / Larger amplitude bottom topography generates a higher degree of zonal variation of baroclinicity. The extent of the localization of the wave activity is controlled by the zonal variability of the baroclinicity. The greater the baroclinicity contrast, the more concentrated are the storm tracks. / It is also found that eddies tend to destroy the baroclinic component and enhance the barotropic component of the zonal current. The ageostrophic geopotential flux redistributes eddy kinetic energy from where it is generated to where it is dissipated. The barotropic energy generation rates are small compared with the baroclinic conversion terms for all cases. / Linear theory accounts well for the eddy structures and wave propagation characteristics observed in the nonlinear simulations. A test of the sensitivity of the storm track to the magnitude of the frictional coefficient using the linear model shows that as the friction coefficient is reduced the scale of the eddies become smaller and the location of the storm track is farther upstream. / Source: Dissertation Abstracts International, Volume: 56-07, Section: B, page: 3810. / Major Professor: Richard L. Pfeffer. / Thesis (Ph.D.)--The Florida State University, 1995.

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The principal motivation for this dissertation is to extend the study of small amplitude high frequency wave propagation in solutions for hyperbolic conservation laws begun by A. Majda and R. Rosales in 1984. It was then that Majda and Rosales obtained equations governing the leading order wave amplitudes of resonantly interacting weakly nonlinear high frequency wave trains in the compressible Euler equations. The equations were obtained through systematic application of multiple scales and result in a pair of nonlinear acoustic wave equations coupled through a convolution operator. The extended solutions satisfy a pair of inviscid Burgers' equations coupled via a spatial convolution operator. Since then, many mathematicians have used this technique to extend the time validity of solutions to systems of equations other than the Euler equations and have arrived at similar nonlinear non-local systems. This work attempts to look at some of the basic features of the linear and nonlinear coupled and decoupled non-local equations, offering some analytic solutions and numerical insight into the phenomena associated with these equations. We do so by examining a single non-local linear equation, and then a single equation coupling a Burgers' nonlinearity with a linear convolution operator. The linear case is completely solvable. Analytic solutions are provided along with numerical results showing the fundamental properties of the linear non-local equations. In the nonlinear case some analytic solutions, including steady state profiles and traveling wave solutions, are provided along with a battery of numerical simulations. Evidence indicates the existence of attractors for solutions of the single equation with a single mode kernel. Provided resonant interaction takes place, the profile of the attractor is uniquely dependent on the kernel alone. Hamiltonian equations are obtained for both the linear and nonlinear equations with the condition that the resonant kernel must be an odd function with respect to the spacial variable. This work also offers some insight into the general understanding of nonlinear non-local systems of equations. It develops working insight for the action of the resonant mechanism between the solution and a known kernel.

This is a collection of essays on weakly and strongly nonlinear systems and possible ways of solving/interpreting them. Firstly, we study sand patterns which are often observed on sea (river) beds. One of the most common features looks like straight rolls perpendicular to the water motion. In many cases, the straight rolls are superimposed on a much longer wave so that two vastly different length scales coexist. In general, there are at least two mechanisms responsible for the growth of periodic sand waves. One is linear instability, and the other is nonlinear coupling between long waves and short waves. One novel feature of this work is to suggest that the latter can be much more important than the former one for the generation of long waves. A weakly nonlinear analysis of the corresponding physical system suggests that the nonlinear coupling leads to the growth of the longer features if the amplitude of the shorter waves has a non-zero curvature. For the case of a straight channel and a tidal shallow sea, we derive nonlinear amplitude equations governing the dynamics of the main features. Estimates based on these equations are consistent with measurements. Secondly, we consider strongly nonlinear systems with randomness. The phenomenon of self-induced transparency (SIT) is reinterpreted in the context of competition between randomness, nonlinearity and dispersion. The problem is then shown to be isomorphic to a problem of the nonlinear Schroedinger (NLS) type with a random (in space) potential. It is proven that the SIT result continues to hold when the uniform medium of inhomogeneously broadened two-level atoms is replaced by a series of intervals in each of which the frequency mismatch is randomly chosen from some distribution. The exact solution of this problem suggests that nonlinearity can improve the transparency of the medium. Also, the small amplitude, almost monochromatic limit of SIT is taken and results in an envelope equation which is an exactly integrable combination of NLS and a modified SIT equation. Some generalizations are made to describe a broad class of integrable systems which combine randomness, nonlinearity and dispersion.