For more than 100 years, the Navier-Stokes equations and various linearizations have been used as a model to study fluid dynamics. Recently, attention has been directed toward studying the nonnormality of linearized problems and developing convergent numerical schemes for simulation of these sytems. Numerical schemes for optimal control problems often require additional properties that may not be necessary for simulation; these properties can be critical when studying nonnormal problems. This research is concerned with approximating infinite dimensional optimal control problems with nonnormal system operators. We examine three different finite element methods for a specific convection-diffusion equation and prove convergence of the infinitesimal generators. Additionally, for two of these schemes, we prove convergence of the associated feedback gains. We apply these three schemes to control problems and compare the performance of all three methods. / Ph. D.

There are still many open questions regarding spatiotemporal chaos although many well developed theories exist for chaos in time. Rayleigh-B'enard convection is a paradigmatic example of spatiotemporal chaos that is also experimentally accessible. Discoveries uncovered using numerics can often be compared with experiments which can provide new physical insights. Lyapunov diagnostics can provide important information about the dynamics of small perturbations for chaotic systems. Covariant Lyapunov vectors reveal the true direction of perturbation growth and decay. The degree of hyperbolicity can also be quantified by the covariant Lyapunov vectors. To know whether a dynamical system is hyperbolic is important for the development of a theoretical understanding. In this thesis, the degree of hyperbolicity is calculated for chaotic Rayleigh-B'enard convection. For the values of the Rayleigh number explored, it is shown that the dynamics are non-hyperbolic. The spatial distribution of the covariant Lyapunov vectors is different for the different Lyapunov vectors. Localization is used to quantify this variation. The spatial localization of the covariant Lyapunov vectors has a decreasing trend as the order of the Lyapunov vector increases. The spatial localization of the covariant Lyapunov vectors are found to be related to the instantaneous Lyapunov exponents. The correlation is stronger as the order of the Lyapunov vector decreases. The covariant Lyapunov vectors are also computed using a spectral element approach. This allows an exploration of the covariant Lyapunov vectors in larger domains and for experimental conditions. The finite conductivity and finite thickness of the lateral boundaries of an experimental convection domain is also studied. Results are presented for the variation of the Nusselt number and fractal dimension for different boundary conditions. The fractal dimension changes dramatically with the variation of the finite conductivity. / Ph. D.

yes / The coupling between natural convection and conduction within rectangular enclosure was investigated numerically. Three separate heat sources flush mounted on a vertical wall and an isoflux condition was applied at the back of heat sources. Continuity, momentum and energy conservation equations were solved by using control volume formulation and the coupling of velocity and pressure was treated by using the “SIMPLE” algorithm. The modified Rayleigh number and the substrate/fluid thermal conductivity ratio were used in the range 𝑹𝒂𝒍𝒛∗=𝟏𝟎^𝟒−𝟏𝟎^𝟕 and 𝑹𝒔=𝟏𝟎−𝟏𝟎𝟎𝟎 respectively. The investigation was extended to compare results of FC-77 with Air and also for high values of 𝑹𝒔>𝟏𝟎𝟎𝟎. The results illustrated that, when the modified Rayleigh number increases, dimensionless heat flux and local Nusselt number increases for both fluids. Opposite behaviour for the thermal spreading in the substrate and the dimensionless temperature 𝜽, they were decreased when 𝑹𝒂𝒍𝒛∗ is increased. Also with increasing the substrate/fluid thermal conductivity ratio for a given value of the modified Rayleigh number the thermal spreading in the substrate increased which is the reason of the decrease in the maximum temperature value. The present study concluded that, for high values of 𝑹𝒔>𝟏𝟓𝟎𝟎, the effect of the substrate is negligible.

It has been shown that by increasing the film coefficient "h" by means of forced convection the heat transfer rate of a finned pin can be considerably improved. For an average flow rate of air over the extended surfaces, throughout the investigation the increase in the effectiveness was as much as 4.60 times over the free convection process. However the effectiveness decreases with the increasing root excess temperatures. The heat loss through the insulation was about 1.86 percent, which will not impair the accuracy for engineering applications. / Master of Science

The onset of laminar Rayleigh-Bénard convection is investigated for a low-Prandtl number liquid metal in a cylindrical container. All surfaces are considered to be solid and no-slip. Two cases are considered for the thermal boundary conditions at the side wall: conducting and insulated surfaces. A Chebyshev Galerkin spectral model is used to reduce the governing Boussinesq system to a first-order system of ordinary differential equations. A local stability analysis using the linearized system determines the first critical Rayleigh number. The results are compared with experimental data and a numerical study. / Master of Science

Designing buildings to use energy more efficiently can lead to lower energy costs, while maintaining comfort for occupants. Computational fluid dynamics (CFD) can be utilized to visualize and simulate expected flows in buildings and structures. CFD gives architects and designers the ability to calculate the velocity, pressure, and heat transfer within a building. Previous research has not modeled natural ventilation situations that challenge common design rules of thumb used for cross-ventilation and single-sided ventilation. The current study uses a commercial code (FLUENT) to simulate cross-ventilation in simple structures and analyzes the flow patterns and heat transfer in the rooms. In the Casa Giuliana apartment and the Affleck house, this study simulates passive cooling in spaces well-designed for natural ventilation. Heat loads, human models, and electronics are included in the apartment to expand on prior research into natural ventilation in a full-scale building. Two different cases were simulated. The first had a volume flow rate similar to the ambient conditions, while the second had a much lower flow rate that had an ACH of 5, near the minimum recommended value Passive cooling in the Affleck house is simulated and has an unorthodox ventilation method; a window in the floor that opens to an exterior basement is opened along with windows and doors of the main floor to create a pressure difference. In the Affleck house, two different combinations of window and door openings are simulated to model different scenarios. Temperature contours, flow patterns, and the air changes per hour (ACH) are explored to analyze the ventilation of these structures. / Master of Science

Solar chimney power plants use the buoyancy-nature of heated air to harness the Sun's energy without using solar panels. The flow is driven by a pressure difference in the chimney system, so traditional chimneys are extremely tall to increase the pressure differential and the air's velocity. Computational fluid dynamics (CFD) was used to model the airflow through a solar chimney. Different boundary conditions were tested to find the best model that simulated the night-time operation of a solar chimney assumed to be in sub-Saharan Africa. At night, the air is heated by the energy that was stored in the ground during the day dispersing into the cooler air. It is necessary to model a solar chimney with layer of thermal storage as a porous material for FLUENT to correctly calculate the heat transfer between the ground and the air. The solar collector needs to have radiative and convective boundary conditions to accurately simulate the night-time heat transfer on the collector. To correctly calculate the heat transfer in the system, it is necessary to employ the Discrete Ordinates radiation model. Different chimney configurations were studied with the hopes of designing a shorter solar chimney without decreases the amount of airflow through the system. Clusters of four and five shorter chimneys decreased the air's maximum velocity through the system, but increased the total flow rate. Passive advections wells were added to the thermal storage and were analyzed as a way to increase the heat transfer from the ground to the air. / Master of Science

This dissertation aims to use theory and numerical simulations to quantify the propagation of fronts, which consist of autocatalytic reaction fronts, fronts with feedback and pattern forming fronts in Rayleigh-Bénard convection. The velocity and geometry of fronts are quantified for fronts traveling through straight parallel convection rolls, spatiotemporally chaotic rolls, and weakly turbulent rolls. The front velocity is found to be dependent on the competing influence of the orientation of the convection rolls and the geometry of the wrinkled front interface which is quantified as a fractal having a non-integer box-counting dimension. Front induced solutal and thermal feedback to the convective flow field is then studied by solving an exothermic autocatalytic reaction where the products and the reactants can vary in density. A single self-organized fluid roll propagating with the front is created by the solutal feedback while a pair of propagating counterrotating convection rolls are formed due to heat release from the reaction. Depending on the relative change in density induced by the solutal and thermal feedback, cooperative and antagonistic feedback scenarios are quantified. It is found that front induced feedback enhances the front velocity and reactive mixing length and induces spatiotemporal oscillations in the front and fluid dynamics. Using perturbation expansions, a transition in symmetry and scaling behavior of the front and fluid dynamics for larger values of feedback is studied. The front velocity, flow structure, front geometry and reactive mixing length scales for a range of solutal and thermal feedback are quantified. Lastly, pattern forming fronts of convection rolls are studied and the wavelength and velocity selected by the front near the onset of convective instability are investigated. This research was partially supported by DARPA Grant No. HR0011-16-2-0033. The numerical computations were done using the resources of the Advanced Research Computing center at Virginia Tech. / Doctor of Philosophy / Quantification of transport of reacting species in the presence of a flow field is important in many problems of engineering and science. A front is described as a moving interface between two different states of a system such as between the products and reactants in a chemical reaction. An example is a line of wildfire which separates burnt and fresh vegetation and propagates until all the fresh vegetation is consumed. In this dissertation the propagation of reacting fronts in the presence of convective flow fields of varying complexity is studied. It is found that the spatial variations in a convective flow field affects the burning and propagation of fronts by reorienting the geometry of the front interface. The velocity of the propagating fronts and its dependence on the spatial variation of the flow field is quantified. In certain scenarios the propagating front feeds back to the flow by inducing a local flow that interacts with the background convection. The rich and emergent dynamics resulting from this front induced feedback is quantified and it is found that feedback enhances the burning and propagation of fronts. Finally, the properties of pattern forming fronts are studied for fronts which leave a trail of spatial structures behind as they propagate for example in dendritic solidification and crystal growth. Pattern forming fronts of convection rolls are studied and the velocity of the front and spatial distribution of the patterns left behind by the front is quantified. This research was partially supported by DARPA Grant No. HR0011-16-2-0033. The numerical computations were done using the resources of the Advanced Research Computing center at Virginia Tech.

This work reports the convection calibration of Schmidt-Boelter heat flux gages in shear and stagnation air flow. The gages were provided by Sandia National Laboratories and included two one-inch diameter and two one-and-one-half-inch diameter Schmidt-Boelter heat flux gages. In order to calibrate the sensors a convection calibration facility has been designed, including a shear test stand, a stagnation test stand, an air heater and a data acquisition system. The current physical model for a combined radiation and convection heat transfer environment uses an additional thermal resistance around the heat flux gage. This model clearly predicts a non-linear dependency of the gage sensitivity over a range of heat transfer coefficients. A major scope of this work was to experimentally verify the relation found by the model assumptions. Since the actual heat sink temperature is not known and cannot be measured, three different cases have been examined resulting in three different sensitivities for one pressure value, which is the gage sensitivity for the not cooled case and the gage sensitivity for the cooled case, based on the plate temperature or on the cooling water temperature. All of the measured sensitivities for shear as well as for stagnation flow fit well in the theory and show the non-linear decay for increasing heat transfer coefficient values. However, the obtained data shows an offset in the intersection with the sensitivity at zero heat transfer coefficient. This offset might arise from different radiation calibration techniques and different surface coatings of test gage and reference standard. / Master of Science

Heat flux gages are important in applications where measurement of the transfer of energy is more important than measurement of the temperature itself. There is a need for a heat flux sensor that can perform reliably for long periods of time in high temperature and high heat flux environment. The primary objective is to design and build a heat flux sensor that is capable of operating for extended periods of time in a high heat flux and high temperature environment. A High Temperature Heat Flux Sensor (HTHFS) was made by connecting 10 brass and steel thermocouple junctions in a thermopile circuit. This gage does not have a separate thermal resistance layer making it easier to fabricate. The HTHFS was calibrated in a custom-made convection calibration facility using a commercial Heat Flux Microsensor (HFM) as the calibration standard. The measured sensitivity of the HTHFS was 20.4 ±2.0ìV/(W/cm2). The measured sensitivity value matched with the theoretically calculated value of 20.5 ìV/(W/cm2). The average sensitivity of the HTHFS prototype was one-fifth of the sensitivity of a commercially available HFM. Better ways of mounting the HTHFS in the calibration stand have been recommended for future tests on the HTHFS for better testing. The HTHFS has the potential to be made into a microsensor with thousands of junctions added together in a thermopile circuit. This could lead to a heat flux sensor that could generate large signals (~few mV) and also be capable of operating in high heat flux and high temperature conditions. / Master of Science