Internal waves are generated in a fluid if the density increases continuously with depth. The variation in density with depth, or stratification, defines the natural frequency of the fluid N. Two common examples of stratified fluids are the ocean and atmosphere; internal waves are generated continuously in both mediums. Although there are many internal wave generation mechanisms, one common and frequently studied method is tidal flow over oceanic bathymetry. If the local natural frequency of the water near the topography is greater than the tidal frequencyω, internal waves will be generated by the tidal flow over the topography. If N=ω, only evanescent waves will be formed. Unlike internal waves, evanescent waves decay rapidly as they move vertically away from their generation site. As evanescent waves pass from an evanescent region (N=ω),through a turning depth (N=ω) and into a propagating region (N=ω), they become propagating internal waves. Because internal waves can propagate energy across large distances, they play an important role in oceanic mixing and the overall energy budget of the ocean. Knowing where these waves are formed from evanescent waves and their corresponding energy improves understanding of the impact on their surrounding area.Kinetic energy density of evanescent and internal waves formed from oscillatory flow over topography in evanescent regions is first estimated using synthetic schlieren experiments and a novel linear theory model. Experiments were performed with two Gaussian topographies in an exponential density profile. The linear theory model, which uses a set of equations that links the evanescent and propagating regions with the Airy function to overcome the discontinuity inherent with a turning depth, was compared to the experiments. Both methods showed that increasing Fr1,the strength of the evanescent region relative to the excitation frequency, causes the propagating kinetic energy to decrease. In addition, kinetic energy decreased with increasing distance between the topography and the turning depth. Because the model does not account for non-linearities such as turbulence generation, it regularly overestimates propagating kinetic energy relative to the experiments. After comparing the model with synthetic schlieren experiments, it was used to estimate that 25% of the evanescent wave energy generated by an oceanic topography located at 15◦N, 130◦E can become propagating wave energy.The influence of topography shape and fluid density profile on kinetic energy density was also explored through a combination of experiments, a linear theory model, and numerical simulations. From numerical simulations, kinetic energy can be directly calculated with the velocity pro-file and indirectly with the density perturbation field, in the same manner as the synthetic schlieren experiments. Average propagating internal wave kinetic energy (KE∗ 2) as a function of Fr1D/H,which combines Fr1 with the relative distance between the topography and the turning depth D/H,was compared for all methods. KE∗ 2 decreases with increasing Fr1D/H for all methods. Also, far from the turning depth, the direct and indirect simulations indicate similar kinetic energy when in the propagating region, where a distance from the turning depth can be quantified based on N and ω. This work was expanded to include a medium Gaussian, steep Gaussian, sinusoidal, and complex topography with two layer linear, parabolic, cubic, and exponential density profiles to investigate the validity of assuming an average natural frequency in the evanescent region and the impact of the topographic slope on KE∗ 2. A comparison of the density profiles indicated that using a two layer linear density profile has similar results compared to the other density profiles for estimating KE∗ 2 as a function of Fr1D/H. Also, KE∗ 2 is non-negligible for Fr1D/H<4. Increasing the maximum slope of a topography shape decreases the kinetic energy of the generated internal waves, though it was found that the energy is dependent upon the actual shape of the topography as well.Particle image velocimetry (PIV) experiments were performed and compared to synthetic schlieren (SS). While SS experiments generally resulted in an overestimate of kinetic energy relative to the PIV results, the trends from each experimental method matched well. It is recommended that SS be used in regions away from turning depths, but that either are valid in the evanescent and propagating regions. PIV methods should be used when results near the turning depth or the topography are desired.

<p> An experimental study of stratified fluid flow phenomena for two equal depth, different density stratified liquids in a rectangular channel is presented. Two two fluid combinations were used, a sugar water and fresh water, and fresh water and varsol. The critical value of the determined densimetric Froude number at which the upper fluid began to participate in the flow was obtained and found to be 0.28 as against Huber's (1) predicted value of 2.76. It was concluded that the interfacial mixing and viscous effects are largely responsible for this difference. </p> <p> An attempt to extend Harleman's (7) work was made. The results obtained in present work were in good agreement with Harleman's (7) experimental work. </p> / Thesis / Master of Engineering (ME)

<p> Flows in channels or estuaries may exhibit variations in density arising· from differences in temperature, salinity or suspended solids. In the absence of significant vertical mixing, stable, discrete layers may form with distinct density interfaces. </p> <p> This thesis presents a computational approach for the analysis of two-layer, vertically stratified, one-dimensional horizontal flows in open channels. A variety of such problems are identified and a critical survey of the existing literature is presented. A framework is defined against which these problems are classified and decomposed into analytical problems of the simplest possible scope. Based on the conditions that lead to changes in flow characteristics, four research areas are examined. These are energy balance, interfacial hydraulic jump, lock exchange flows, and long transitions. Although restricted to essentially one-dimensional flows, the analytical study of these four areas is extended to allow for non-uniform velocity distribution the introduction of boundary-layer displacement thicknesses and correction factors for kinetic energy and momentum. Also, a significant feature of the study is the ability to handle channels of arbitrary cross-sectional geometry. </p> <p> The basic philosophy of the approach followed in this study is to develop a relatively simple and computationally econaoical procedure which is applicable to a wide variety of problems involving channels systems of arbitrary geometry and boundary conditions. A library of computer subroutines provides a convenient means of developing an open-ended system of computational techniques for the solution of a wide range of problems. Such a library of computational algorithms may also promote. cooperation and collaboration among researchers and engineers concerned with stratified flow hydraulics. Such algorithms should provide solutions for frequently recurring problems, should be mutually compatible and allow the construction of relatively complex analytical models in a modular fashion. A comprehensive library of routines is developed which consists of fourty-four subroutines and functions. This evolves as a well-defined hierarchy of algorithms in which the most basic algorithms are nested within the more sophisticated ones to the sixth or seventh level. </p> <p> The computational algorithms are tested for theoretical and computational performance. Numerical predictions are compared with available experimental and field data. Moreover, an experimental program is described which is designed and carried ·out to verify the numerical predictions obtained for the first of the above-mentioned four topics. </p> <p> An important aspect of the study is the illustration of the application of the routines in the solution of typical practical problems such as selective withdrawal from stratified water bodies and recirculation of cooling water from power plants. In addition, to facilitate utilization of the programs by others, complete documentation and listings are provided. </p> / Thesis / Doctor of Philosophy (PhD)

Vita. / Digitized by Kansas Correctional Industries

The Rayleigh-Taylor instability is a buoyancy driven instability that takes place in a stratified fluid system with a constant acceleration directed from the heavy fluid into the light fluid. In this study, both experimental data and numerical simulations are presented. Experiments are performed primarily using a lithium-tungstate aqueous solution as the heavy liquid, but sometimes a calcium nitrate aqueous solution is used for comparison purposes. Experimental data is obtained for both miscible and immiscible fluid combinations. For the miscible experiments the light liquid is either ethanol or isopropanol, and for the immiscible experiments either silicone oil or trans-anethole is used. The resulting Atwood number is either 0.5 when the lithium-tungstate solution is used or 0.2 when the calcium nitrate solution is used. These fluid combinations are either forced or left unforced. The forced experiments have an initial perturbation imposed by vertically oscillating the liquid containing tank to produce Faraday waves at the interface. The unforced experiments rely on random interfacial fluctuations, due to background noise, to seed the instability. The liquid combination is partially enclosed in a test section that is accelerated downward along a vertical rail system causing the Rayleigh-Taylor instability. Accelerations of approximately 1g (with a weight and pulley system) or 10g (with a linear induction motor system) are experienced by the liquids. The tank is backlit and digitally recorded with high speed video cameras. These experiments are then simulated with the incompressible, Navier-Stokes code Miranda. The main focus of this study is the growth parameter (ɑ) of the mixing region produced by the instability after it has become apparently self-similar and turbulent. The measured growth parameters are compared to determine the effects of miscibility and initial perturbations (of the small wavelength, finite bandwidth type used here). It is found that while initial perturbations do not affect the instability growth, miscibility does.

Stably stratified flows are common in the environment such as in the atmospheric· boundary layer, the oceans, lakes and estuaries. Understanding mixing and dispersion in these flows is of fundamental importance in applications such as the prediction of pollution dispersion and for weather and climate prediction/models. Mixing efficiency in stratified flows is a measure of the proportion of the turbulent kinetic energy that goes into increasing the potential energy of the fluid by irreversible mixing. This can be important for parameterizing the effects of mixing in stratified flows. In this research, fully resolved direct numerical simulations (DNS) of the Navier-Stokes equations are used to study transient turbulent mixing events. The breaking of internal waves in the atmosphere could be a source of such episodic events in the environment. The simulations have been used to investigate the mixing efficiency (integrated over the duration of the event) as a function of the initial turbulence Richardson number Ri = N2L2/U2, where N is the buoyancy frequency, L is the turbulence length scale, and u is the turbulence velocity scale. Molecular effects on the mixing efficiency have been investigated by varying the Prandtl number Pr = V/K, where v is the viscosity and K is the scalar diffusivity. Comparison of the DNS results with grid turbulence experiments has been carried out. There is broad qualitative agreement between the experimental and DNS results.· However the experiments suggest a maximum mixing efficiency of 6% while our DNS gives values about five times higher. Reasons for this discrepancy are investigated. The mixing efficiency has also been determined using linear theory. It is found that the results obtained for the very stable cases converge on those obtained from DNS suggesting that strongly stratified flows exhibit linear behaviour. Lagrangian analysis of mixing is fundamental in understanding turbulent diffusion and mixing. Dispersion models such as that of Pearson, Puttock & Hunt (1983) are based on a Lagrangian approach. A particle-tracking algorithm (using a cubic spline interpolation scheme following Yeung &Pope, 1988) was developed and incorporated into the DNS code to enable an investigation into the fundamental aspects of mixing and diffusion from a Lagrangian perspective following fluid elements. From the simulations, the ensemble averaged rate of mixing as a function of time indicates clearly that nearly all the mixing in these flows occurs within times of order 3 Vu. The mean square vertical displacement statistics show how the stable stratification severely inhibits the vertical displacement of fluid elements but has no effect on displacements in the transverse direction. This is consistent with the Pearson, Puttock & Hunt model. The important link that asymptotic value of the mean square vertical displacement is a measure of the total irreversible mixing that has occurred in the flow is made. However the results show that the change in density of the fluid elements is only weakly correlated to the density fluctuations during the time when most of the mixing occurs, which contradicts a key modeling assumption of the PPH theory. Improvements to the parameterization of this mixing are investigated. Flow structures in stably stratified turbulence were examined using flow visualization software. The turbulence structure for strong stratification resembles randomly scattered pancakes that are flattened in the horizontal plane. It appears that overturning motions are the main mechanism by which mixing occurs in these flows. / Thesis (M.Sc.Eng.)-University of Natal, Durban, 2002.

Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Civil Engineering. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Includes bibliographical references. / Ph.D.

Thermal stratification occurs in shallow water bodies because solar energy separates the water column into an upper warm layer, a lower cold layer, and an intermediate layer between the upper and lower layers. In general the intermediate layer exhibits a significant thermal gradient over depth. Because cold water is heavier than warm water, this temperature structure produces a stable stratification, thereby inhibiting circulation from the bottom to the surface. This stable stratification results in a deficit of dissolved oxygen in the lower layer leading to water quality problems. Hence understanding the thermal structure and vertical circulation in shallow water bodies is important for water quality and its management. In this research, a numerical code is developed to examine the three-dimensional flow structure in shallow water bodies. This numerical code is used to solve the governing equations : the Reynolds averaged Navier-Stokes equations for three velocities and pressure, the depth-averaged continuity equation for free surface movement, the equations for turbulence closure, the scalar transport equation for temperature, and the international equation of state for density variation due to temperature. These equations are solved simultaneously using a finite difference method. The mathematical equations are transformed into a generalised coordinate system which allows flexibility for irregular boundaries and the allocation of vertical grid points every time step depending on free surface movements. In order to overcome possible numerical instabilities because of the small vertical length scale in shallow water bodies, an implicit method is used in the vertical direction. Several test cases involving free surface movement are used to verify the numerical code, and numerical solutions compare favourably against analytical solutions and measured data. The numerical code has been applied to the Torrens Lake in Adelaide, South Australia, where algal blooms occur frequently in summer due to thermal stratification. Typical thermal structures have been obtained from the model and these are compared with field data. The current code has been developed to improve upon existing commercial models which may not adequately address shallow water flows because of the high computational burden required to resolve free surface movements and consequential difficulties encountered for models with a small vertical length scale. / Thesis (Ph.D.)--School of Mathematical Sciences, 2007.

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (p. 147-153).

Thesis (Ph. D.)--University of Washington, 2001. / Vita. Includes bibliographical references (p. 104-108).