Burgers' equation is a one-dimensional simple model for convection-diffusion phenomena such as shock waves, supersonic flow about airfoils, traffic flows, acoustic transmission, etc. For high Reynolds number, the open-loop system (no control) produces steep gradients due to the nonlinear nature of the convection. The steep gradients are stabilized by feedback control laws. In this phase, sufficient conditions for the control input functions and the location of sensors are obtained. Also, explicit exponential decay rates for open-loop and closed-loop systems are obtained. Numerical experiments are given to illustrate some of typical results on convergence and stability. / Ph. D.

This thesis is a numerical study of the coupled Burgers equation. The coupled Burgers equation is motivated by the Boussinesq equations that are often used to model the thermal-fluid dynamics of air in buildings. We apply Finite Element Methods to the coupled Burgers equation and conduct several numerical experiments. Based on these results, the Group Finite Element method (GFE) appears to be more stable than the standard Finite Element Method. The design and implementation of controllers heavily relies on rapid solutions to complex models such as the Boussinesq equations. Thus, we further examine the feasibility and efficiency of the Proper Orthogonal Decomposition (POD) for the coupled Burgers equation. Using POD, we reduce the system to a "minimal" number of ODE's and conduct numerous numerical studies comparing the POD and GFE method. Further numerical experiments consider an application where the dynamics are projected on a POD basis and then the governing parameters of the system are varied. / Master of Science

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy / In this thesis we used the following four types of optimal control problems: (i) Problems governed by systems of ordinary differential equations; (ii) Problems governed by systems of ordinary differential equations with time-delayed arguments appearing in both the state and the control variables; (iii) Problems governed by linear systems subject to sudden jumps in parameter values; (iv) A chemical reactor problem governed by a couple of nonlinear diffusion equations. • The aim of this thesis is to devise computational algorithms for solving the optimal control problems under consideration. However, our main emphasis are on the mathematical theory underlying the techniques, the convergence properties of the algorithms and the efficiency of the algorithms. Chapters II and III deal with problems of the first type, Chapters IV and V deal with problems of the second type, and Chapters VI and VII deal with problems of the third and fourth type respectively. A few numerical problems have been included in each of these Chapters to demonstrate the efficiency of the algorithms involved. The class of optimal control problems considered in Chapter II consists of a nonlinear system, a nonlinear cost functional, initial equality constraints, and terminal equality constraints. A Sequential Gradient-Restoration Algorithm is used to devise an iterative algorithm for solving this class of problems. 'I'he convergence properties of the algorithm are investigated. The class of optimal control problems considered in Chapter III consists of a nonlinear system, a nonlinear cost functional, and terminal as well as interior points equality constraints. The technique of control parameterization and Liapunov concepts are used to solve this class of problems, A computational algorithm for solving a class of optimal control problems involving terminal and continuous state constraints or inequality type was developed by Rei. 103 in 1989. In Chapter IV, we extend the results of Ref. 103 to a more general class of constrained time-delayed optimal control problems, which involves terminal state equality constraints, as well as terminal state inequality constraints and continuous state inequality constraints. In Ref. 104, a computational scheme using the technique of control parameterization was developed for solving a class of optimal control problems in which the cost functional includes the full variation of control. Chapter V is a straightforward extension of Ref. 104 to the time-delayed case. However the main contribution of this chapter is that many numerical examples have been solved. In Chapter VI, a class of linear systems subject to sudden jumps in parameter values is considered. To solve this class of stochastic control problem, we try to seek for the best feedback control law depending only on the measurable output. Based on this idea, we convert the original problem into an approximate constrained deterministic optimization problem, which can be easily solved by any existing nonlinear programming technique. In Chapter VII, a chemical reactor problem and its control to achieve a desired output temperature is considered. A finite element Galerkin method is used to convert the original distributed optimal control problem into a quadratic programming problem with linear constraints, which can he solved by any standard quadratic programming software . / Andrew Chakane 2018

This thesis considers one-dimensional moving mesh (MM) methods coupled with semi-Lagrangian (SL) discretisations of partial differential equations (PDEs) for meteorological applications. We analyse a semi-Lagrangian numerical solution to the viscous Burgers’ equation when using linear interpolation. This gives expressions for the phase and shape errors of travelling wave solutions which decay slowly with increasing spatial and temporal resolution. These results are verified numerically and demonstrate qualitative agreement for high order interpolants. The semi-Lagrangian discretisation is coupled with a 1D moving mesh, resulting in a moving mesh semi-Lagrangian (MMSL) method. This is compared against two moving mesh Eulerian methods, a two-step remeshing approach, solved with the theta-method, and a coupled moving mesh PDE approach, which is solved using the MATLAB solver ODE45. At each time step of the SL method, the mesh is updated using a curvature based monitor function in order to reduce the interpolation error, and hence numerical viscosity. This MMSL method exhibits good stability properties, and captures the shape and speed of the travelling wave well. A meteorologically based 1D vertical column model is described with its SL solution procedure. Some potential benefits of adaptivity are demonstrated, with static meshes adapted to initial conditions. A moisture species is introduced into the model, although the effects are limited.

The use of efficient basis functions to model and represent flows with internal sharp velocity gradients, such as shocks or eddy microfronts, are investigated. This is achieved by analysing artificial data, observed atmospheric turbulence data and by the use of a Burgers' equation based spectral model. The concept of an efficient decomposition of a function into a basis set is presented and alternative analysis methods are investigated. The development of a spectral model using a generalized basis for the Burgers' equation is presented and simulations are performed using a modified Walsh basis and compared with the Fourier (trigonometric) basis and finite difference techniques. The wavelet transform is shown to be superior to the Fourier transform or the windowed Fourier transform in terms of defining the predominant scales in time series of turbulent shear flows and in 'zooming in' on local coherent structures associated with sharp edges. Disadvantages are found to be its inability to provide clear information on the scale of periodicity of events. Artificial time series of varying amounts of noise added to structures of different scales are analyzed using different wavelets to show that the technique is robust and capable of detecting sharp edged coherent structures such as those found in shear driven turbulence. The Haar function is used as a wavelet to detect ubiquitous zones of concentrated shear in turbulent flows sometimes referred to as microfronts. The location and organization of these shear zones suggest that they may be edges of larger scale eddies. A wavelet variance of the wavelet phase plane is defined to detect and highlight events and obtain measures of predominant scales of coherent structures. Wavelet skewness is computed as an indicator of the systematic sign preference of the gradient of the transition zone. Inverse wavelet transforms computed at the dilation corresponding to the peak wavelet variance are computed and shown to contain a significant fraction of the total energy contained in the record. The analysis of data and the numerical simulation results are combined to propose that the sharp gradients normally found in shear induced turbulence significantly affect the nature of the turbulence and hence the choice of the basis set used for the simulation of turbulence. / Graduation date: 1991

The objective of the thesis is to characterize the linear and nonlinear aspects of inertial and pressure effects in turbulent flows. In the first part of the study, computations of Navier-Stokes and 3D Burgers equations are performed in the rapid distortion (RD) limit to analyze the inviscid linear processes in homogeneous turbulence. By contrasting the results of Navier- Stokes RD equations and Burgers RD equations, the effect of pressure can be isolated. The evolution of turbulent kinetic energy and anisotropy components and invariants are examined. In the second part of the thesis, the velocity gradient dynamics in turbulent flows are studied with the help of inviscid 3D Burgers equations and restricted Euler equations. The analytical asymptotic solutions of velocity gradient tensor are obtained for both Burgers and restricted Euler equations. Numerical computations are also performed to identify the stable solutions. The results are compared and contrasted to identify the effect of pressure on nonlinear velocity gradient dynamics. Of particular interest are the sign of the intermediate principle strain-rate and tendency of vorticity to align with the intermediate principle strain-rate. These aspects of velocity gradients provide valuable insight into the role of pressure in the energy cascade process.

Thesis (Ph. D.)--Institute of Paper Chemistry, 1965. / Includes bibliographical references (p. 107-109).

Dehulled black beans were micronized at 90⁰C, 100⁰C, 110⁰C, 120⁰C, 130⁰C and 140⁰C; milled to flour and tested for lipoxygenase activity. Non micronized black bean flour was higher in lipoxygenase activity than flours at ≥120ºC (p=≤0.05). Micronized (100⁰C, 110⁰C, 120⁰C) and non micronized black bean flour was added to low fat beef burgers (6%). C18:3 was significantly higher in the black bean flour samples (raw and cooked). Whole wheat flour had the highest amount of C18:2 in all samples (p= ≤0.05). The all beef control was significantly higher in Newton value, drip loss, cook loss and percent shrinkage compared to burgers with binders (p= ≤0.05). Ninety-three participants participated in an consumer sensory panel. Results showed higher acceptability of micronized burgers compared to all beef or whole wheat flour controls. This study demonstrated incorporation of black bean flour into low fat beef burgers can improve their physical, chemical and sensory properties.

This study investigated the effect of seed moisture level of lentil and surface temperature of micronization (infrared heat treatment) on the physico-chemical and functional properties of resulting flours and how these flours affected colour and unsaturated lipid oxidation when incorporated into ground beef products. Flour from raw seed (non-tempered and non-micronized) was used as the control. Whole seeds of small green lentil (Lens culinaris L., var. Eston) without tempering (8% moisture) and tempered to 16% or 23% moisture was infrared heat treated (micronized) to 115, 130, 150 or 165 °C surface temperature. The decreased protein solubility (2-60%) and lipoxygenase (70-100%), peroxidase (32-100%) and trypsin inhibitory (up to 54%) activities of resulting flours indicated changes in the protein fraction due to heat-moisture treatment. Starch gelatinization was observed at the 23% moisture level and changes in pasting properties, and water and oil absorption capacities varied with treatment. The heat-moisture combinations modified properties of starch and protein to different degrees and, consequently, lentil flour functionalities. Incorporation of lentil flour as a binder in low fat (<10%) beef burgers at 6% (w/w) showed that flours from micronized lentil seeds enhanced retention of redness and suppression of lipid oxidation as indicated by Hunter a* values and thiobarbituric acid reactive substances values, respectively, in a retail display setting. Investigation of total phenolics in aqueous salt extracts of lentil flours showed a decrease in content with increased micronization temperature. The antioxidant assays showed no changes in the ferric ion reducing power or reduction of hydroxyl radical scavenging and superoxide radical scavenging activities with heat-moisture treatment. Reduction of lipoxygenase and peroxidase activities was evident in lentil flour aqueous salt extracts, and the enzyme activities were localized to seed cotyledons. The myoglobin-liposome model study showed that a flour extract from the 16% moisture and 150 °C treatment resulted in a slower rate of oxymyoglobin oxidation initiation than other treatments which had different levels of lipoxygenase and peroxidase activities. Unsaturated lipids accelerated oxymyoglobin degradation irrespective of the presence of lentil extract. The extended fresh red colour retention of ground beef due to addition of flours from micronized seed compared to that from non-micronized seed may be related to suppression of pro-oxidant activities and the activity of potential antioxidants. The putative antioxidative compounds in lentil that are available for meat components may include compounds other than lentil seed phenolics.

Dehulled black beans were micronized at 90⁰C, 100⁰C, 110⁰C, 120⁰C, 130⁰C and 140⁰C; milled to flour and tested for lipoxygenase activity. Non micronized black bean flour was higher in lipoxygenase activity than flours at ≥120ºC (p=≤0.05). Micronized (100⁰C, 110⁰C, 120⁰C) and non micronized black bean flour was added to low fat beef burgers (6%). C18:3 was significantly higher in the black bean flour samples (raw and cooked). Whole wheat flour had the highest amount of C18:2 in all samples (p= ≤0.05). The all beef control was significantly higher in Newton value, drip loss, cook loss and percent shrinkage compared to burgers with binders (p= ≤0.05). Ninety-three participants participated in an consumer sensory panel. Results showed higher acceptability of micronized burgers compared to all beef or whole wheat flour controls. This study demonstrated incorporation of black bean flour into low fat beef burgers can improve their physical, chemical and sensory properties.