High-Order Numerical Methods in Lake Modelling

The physical processes in lakes remain only partially understood despite successful data collection from a variety of sources spanning several decades. Although numerical models are already frequently employed to simulate the physics of lakes, especially in the context of water quality management, improved methods are necessary to better capture the wide array of dynamically important physical processes, spanning length scales from ~ 10 km (basin-scale oscillations) - 1 m (short internal waves). In this thesis, high-order numerical methods are explored for specialized model equations of lakes, so that their use can be taken into consideration in the next generation of more sophisticated models that will better capture important small scale features than their present day counterparts. The full three-dimensional incompressible density-stratified Navier-Stokes equations remain too computationally expensive to be solved for situations that involve both complicated geometries and require resolution of features at length-scales spanning four orders of magnitude. The main source of computational expense lay with the requirement of having to solve a three-dimensional Poisson equation for pressure at every time-step. Simplified model equations are thus the only way that numerical lake modelling can be carried out at present time, and progress can be made by seeking intelligent parameterizations as a means of capturing more physics within the framework of such simplified equation sets. In this thesis, we employ the long-accepted practice of sub-dividing the lake into vertical layers of different constant densities as an approximation to continuous vertical stratification. We build on this approach by including weakly non-hydrostatic dispersive correction terms in the model equations in order to parameterize the effects of small vertical accelerations that are often disregarded by operational models. Favouring the inclusion of weakly non-hydrostatic effects over the more popular hydrostatic approximation allows these models to capture the emergence of small-scale internal wave phenomena, such as internal solitary waves and undular bores, that are missed by purely hydrostatic models. The Fourier and Chebyshev pseudospectral methods are employed for these weakly non-hydrostatic layered models in simple idealized lake geometries, e.g., doubly periodic domains, periodic channels, and annular domains, for a set of test problems relevant to lake dynamics since they offer excellent resolution characteristics at minimal memory costs. This feature makes them an excellent benchmark to compare other methods against. The Discontinuous Galerkin Finite Element Method (DG-FEM) is then explored as a mid- to high-order method that can be used in arbitrary lake geometries. The DG-FEM can be interpreted as a domain-decomposition extension of a polynomial pseudospectral method and shares many of the same attractive features, such as fast convergence rates and the ability to resolve small-scale features with a relatively low number of grid points when compared to a low-order method. The DG-FEM is further complemented by certain desirable attributes it shares with the finite volume method, such as the freedom to specify upwind-biased numerical flux functions for advection-dominated flows, the flexibility to deal with complicated geometries, and the notion that each element (or cell) can be regarded as a control volume for conserved fluid quantities. Practical implementation details of the numerical methods used in this thesis are discussed, and the various modelling and methodology choices that have been made in the course of this work are justified as the difficulties that these choices address are revealed to the reader. Theoretical calculations are intermittently carried out throughout the thesis to help improve intuition in situations where numerical methods alone fall short of giving complete explanations of the physical processes under consideration. The utility of the DG-FEM method beyond purely hyperbolic systems is also a recurring theme in this thesis. The DG-FEM method is applied to dispersive shallow water type systems as well as incompressible flow situations. Furthermore, it is employed for eigenvalue problems where orthogonal bases must be constructed from the eigenspaces of elliptic operators. The technique is applied to the problem calculating the free modes of oscillation in rotating basins with irregular geometries where the corresponding linear operator is not self-adjoint

Direct Numerical Simulations of the Degeneration and Shear Instability of Large and Small Amplitude Basin Scale Internal Waves at Varied Aspect Ratios

This thesis presents high resolution simulations of the degeneration and shear instability of standing waves, or seiches, of varying amplitudes and aspect ratios in a continuously stratified fluid. It is well known that such waves evolve to form non–linear, dispersive wave trains under certain conditions. When the initial amplitude scaled by the upper layer depth (the dimensionless amplitude) is sufficiently large, it is possible that stratified shear instability develops, possibly at the same time as the formation of wave trains early in the evolution of the flow. While both of these physical phenomena serve to move energy from large to small scales, they are fundamentally different. The development into wave trains is non-dissipative in nature, and in the asymptotic limit of small, but finite amplitude seiches may be described by variants of the Korteweg–de–Vries (KdV) equation. Shear instability, on the other hand yields Kelvin-Helmholtz billows which in turn provide one of the basic archetypes of transition to turbulence, with greatly increased rates of mixing and viscous dissipation. Discussed is how the two phenomena vary as the aspect ratio of the tank and the height of the interface between lighter and denser fluid are changed, finding examples of cases where the two phenomena co-exist. Beginning with an expository set of examples of small amplitude seiches, the process by which a seiche changes from a traditional standing wave to a more complicated small scale set of dynamics is discussed. The results demonstrate that when the initial dimensionless amplitude is small, the seiche takes more than one oscillation period for non–linear effects to become obviously present in the flow. The small amplitude results put into context the cases where the dimensionless amplitude becomes large enough such that non–linear process occur at much earlier times and there is a competition between the formation of wave trains and stratified shear instability. A quantitative accounting for the evolution of the horizontal modewise decomposition of the kinetic energy of the system is presented along with a semi-analytical model of the evolution of the fundamental mode of the seiche. Using two well known methodologies from the literature, the evolution of the mixing dynamics of the seiche is compared from an energetic perspective and a density variability perspective which illustrates a fundamental transition that occurs as the aspect ratio is decreased. Finally, the seiche degeneration and the mixing dynamics are summarized and the most likely future directions of study are highlighted

Aeronautical engineering: A continuing bibliography with indexes (supplement 189)

This bibliography lists 579 reports, articles and other documents introduced into the NASA scientific and technical information system in June 1985

Second Microgravity Fluid Physics Conference

The conference's purpose was to inform the fluid physics community of research opportunities in reduced-gravity fluid physics, present the status of the existing and planned reduced gravity fluid physics research programs, and inform participants of the upcoming NASA Research Announcement in this area. The plenary sessions provided an overview of the Microgravity Fluid Physics Program information on NASA's ground-based and space-based flight research facilities. An international forum offered participants an opportunity to hear from French, German, and Russian speakers about the microgravity research programs in their respective countries. Two keynote speakers provided broad technical overviews on multiphase flow and complex fluids research. Presenters briefed their peers on the scientific results of their ground-based and flight research. Fifty-eight of the sixty-two technical papers are included here

Numerical Experiments of Atmospheric Boundary Layer flows:interplay between distributed drag elements and buoyancy effects

Anthropogenic emissions of greenhouse gases due to human activity is causing global warming and inducing climate change. A major implication of global warming is the decreasing ice mass in the polar regions resulting in sea-level rise. It is now known that sublimation of drifting and blowing snow is one of the dominant terms of the mass balance of Antarctica. There are various efforts underway to curtail greenhouse gas emissions and mitigate the impact of global warming. One of the most promising solutions involves using non-polluting renewable sources of electricity. Global wind energy estimates have been shown to be far in excess of current and projected energy requirements. From a fluid dynamics perspective, turbulence in the lowest region of the atmosphere, known as the Atmospheric boundary layer (ABL) exerts significant control on both wind energy extraction systems as well as drifting and blowing snow particles, both of which can be considered as distributed drag elements that act as a sink of momentum. The first part of the thesis is concerned with large-eddy simulations (LES) of the turbulent, time-varying ABL with an immersed wind farm. First, a new time-adaptive wind turbine model for LES is introduced that enables the wind turbines to yaw and realign with the incoming wind vector, similar to real wind turbines. The performance of the new model is tested with in a neutrally-stratified ABL forced with a time varying geostrophic wind as well as a synthetic time-changing thermal ABL. Next, the effect of extensive terrestrial wind farms on the spatio-temporal structure of the diurnally-evolving ABL is explored. It is shown that extensive wind farms substantially perturb the vertical structure of the stable boundary layer and the dynamics of the `morning' transition. The effect of these perturbations on the potential power output of an extensive wind farm output is also analysed. Finally, flow characteristics through finite-sized wind farms and the influence of the wind-farm configuration on modulating this evolution is explored using LES. The principal aim for this part of the thesis is to identify regions of flow-adjustment and flow equilibrium within the wind farm. Three diagnostic variables, namely, the wind-farm induced effective surface roughness, the wake viscosity and the wake-expansion coefficient are also computed using the LES-generated database and are used to characterize the flow. In the second part of the thesis, LES of drifting and blowing snow are performed with the aim of calculating sublimation of saltating snow grains. The Thorpe and Mason [1966] model for calculating the mass lost from a sublimating snow grain is the basis of all existing estimates of drifting snow sublimation. This model is revisited to test its validity for saltating snow grains. It is shown that residence times for saltating snow grains are such that using the steady-state model of sublimation losses by the Thorpe and Mason (TM) approach is questionable. Furthermore, the residence times were found to be independent of the imposed pressure gradient and buoyancy. The relaxation time needed for the unsteady mass loss rate to reconcile with the TM solution is found to be comparable to typical residence times for saltating grains and the resulting errors due to use of the TM approach are quantified