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published_or_final_version / abstract / toc / Mechanical Engineering / Master / Master of Philosophy

Numerical modeling has become a major tool for managing water quality in surface waterbodies such as rivers, lakes, reservoirs, and estuaries. Since the two-dimensional longitudinal/vertical model CE-QUAL-W2 is a well-known model and it has been applied to thousands of waterbodies around the world successfully, its numerical scheme was adapted to develop a new three-dimensional numerical model for simulating hydrodynamics, temperature, and water quality in surface waterbodies. Finite difference approximations were used to solve the fluid dynamic governing equations of continuity, free water surface, momentums, and mass transport. No coordinate transformations were performed and the z-coordinate system has been used. Higher-order schemes (QUICK, QUICKEST, and ULTIMATE QUICKEST) in addition to the UPWIND scheme were used for the advective temperature and mass transport. A novel numerical approach was used for the numerical formulation of the three-dimensional scheme. This approach forced the numerical solution of the free surface equation to be a tri-diagonal matrix form rather than a more computationally intensive penta-diagonal matrix solution. This new approach was done by linking a method called line-by-line with the free water surface numerical solution. Another new approach was that the three-dimensional numerical scheme involved a simultaneous solution of hydrodynamics, temperature, and water quality at every model time level instead of saving the hydrodynamic results to be used later for water quality simulation. Hence, this scheme allowed feedback between the hydrodynamics and water quality every time step. In addition, various unique numerical algorithms were employed from CE-QUAL-W2 such as the W2 turbulence model, selective withdrawal theory, surface heat fluxes, and water quality sources and sinks, making the three-dimensional model built on well-tested algorithms. To test the model structure and assumptions, an analytical verification was performed by comparing model predictions to known analytical exact solutions test cases. Good agreement was showed by the model for all of these tests. A computation of the volume balance over the simulation period was also incorporated within the model to assess how well the code performed. Sensitivity tests were also made varying bed and wind shear. The model was also applied to three reservoirs in the USA as field case studies: Lake Chaplain in WA, Laurance Lake in OR, and Cooper Creek Reservoir in OR. The model was validated by comparing the model predictions of water levels, velocities, vertical temperature profiles, and dissolved oxygen with field data. Through these real applications, the numerical predictions of the 3D model showed good agreement with field data based on error statistics. The model results of each field case study were discussed separately. In the Lake Chaplain model application, the study was focused on the importance of the higher-order schemes compared to the first-order UPWIND scheme. The model predictions of temperature were determined by using the UPWIND, QUICK, and QUICKEST scheme and compared with field data. The Error statistics of the model predictions compared to field data were an absolute mean error (AME) of 0.065 m for the water level predictions and an overall AME of 1.62 °C, 1.09 °C, and 1.23 °C for the temperature predictions by using the UPWIND, QUICK, and QUICKEST scheme, respectively. In the Laurance Lake model application, a comparison was performed between the present 3D model and the 2D CE-QUAL-W2. Since the 3D model was build based on CE-QUAL-W2, differences between the two models were evaluated. Error statistics between the model predictions of water level and temperature compared to field data showed that both models were in good agreement with field data. However, the 3D model AME (0.30 m for the water level predictions and 0.48 °C for the temperature predictions) was higher than the 2D model (0.03 m for the water level predictions and 0.42 °C for the temperature predictions). Finally, the Cooper Creek Reservoir case study was done to show the model predictions of temperature and dissolved oxygen. In this application, vertical temperature profiles were covered the entire simulation period in order to show how the model transfer heat between stratification and non- stratification conditions. The model showed good agreement with field data (0.12 m AME for the water level predictions, 1.00 °C overall AME for the temperature predictions, and 1.32 g/m3 overall AME for the dissolved oxygen predictions). Finally, comparisons were made between CE-QUAL-W2 and the 3D model. The 2D model generally performed better in the tests cases if the model user is unconcerned about lateral impacts. The 3D model is important to use when lateral currents and variation in the lateral dimension are important.

Graduation date: 2006

Lake Kinneret is Israel's only fresh water lake (unless you count the Dead Sea). It spans roughly 20km from north to south, and about 12km at its widest east west extent. It is not quite 50m deep at its deepest point. In late spring, the lake stratifies significantly and remains stratified throughout the fall. During the time the lake is stratified, it exhibits low horizontal mode semi-diurnal inertial motions in response to surface forcing from diurnal winds. This internal motion is known to be important in the ecological and chemical balances of the lake, and is suspected to be responsible for episodes in which large numbers of fish are killed. The physical response of the lake to wind forcing is studied. The lake hydrodynamics is approximated by a (x,y,t) two and three layer model on the f-plane (rotating frame) with detailed bathymetry. The numerical method for the integration of the nonlinear partial differential equation is presented, as well as, the generation of the elliptic grid used in the spatial discretization of the Kinneret domain. A suite of numerical simulations are compared to the available data in the northwestern part of the lake. The nonlinear effects, as well as, the sloping beach problem are discussed in the appendixes. / Graduation date: 2000

In this dissertation, the quantitative and qualitative aspects of modeling shallow water flow driven mainly by gravitational forces and dominated by shear stress, using an effective equation often referred to in the literature as the diffusive wave approximation of the shallow water equations (DSW) are presented. These flow conditions arise for example in overland flow and water flow in vegetated areas such as wetlands. The DSWequation arises in shallow water flow models when special assumptions are used to simplify the shallow water equations and contains as particular cases: the Porous Medium equation and the time evolution of the p-Laplacian. It has been successfully applied as a suitable model to simulate overland flow and water flow in vegetated areas such as wetlands; yet, no formal mathematical analysis has been carried out addressing, for example, conditions for which weak solutions may exist, and conditions for which a numerical scheme can be successful in approximating them. This thesis represents a first step in that direction. The outline of the thesis is as follows. First, a survey of relevant results coming from the studies of doubly nonlinear diffusion equations that can be applied to the DSWequation when topographic effects are ignored, is presented. Furthermore, an original proof of existence of weak solutions using constructive techniques that directly lead to the implementation of numerical algorithms to obtain approximate solutions is shown. Some regularity results about weak solutions are presented as well. Second, a numerical approach is proposed as a means to understand some properties of solutions to the DSW equation, when topographic effects are considered, and conditions for which the continuous and discontinuous Galerkin methods will succeed in approximating these weak solutions are established. / text

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Smoothed Particle Hydrodynamics (SPH) is a meshless numerical method which has long been put into practice for scientific and engineering applications. It arises as a numerical discretization of convolution-like integral operators that approximate local differential operators. There have been many studies on the SPH with an emphasis on its role as a numerical scheme for partial differential equations while little attention is paid to the underlying continuum nonlocal models that lie intermediate between the two. The main goal of this thesis is to provide mathematical understanding of the SPH-like meshless methods by means of ongoing developments in studies of nonlocal models with a finite range of nonlocal interactions. It is timely for such a work to be initiated with growing interests in the nonlocal models. The thesis touches on numerical, theoretical and modeling aspects of the nonlocal integro-differential equations pertaining to the SPH-like schemes. As illustrative examples of each aspect it presents robust SPH-like schemes for advection-convection equations, discusses the stabilities of nonsymmetric nonlocal gradient operators, and proposes a new formulation of nonlocal Dirichlet-like type boundary conditions.

published_or_final_version / abstract / Mechanical Engineering / Doctoral / Doctor of Philosophy