The aim of this work is the study of the existence and multiplicity of sign changing nonradial solutions to elliptic boundary value problems on annular domains.

Thesis advisor: Julia E. Grigsby / We use the Birman-Ko-Lee presentation of the braid group to show that all closures of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element δ are fibered. We classify links which admit such a braid representative in geometric terms as boundaries of plumbings of positive Hopf bands to a disk. Rudolph constructed fibered strongly quasipositive links as closures of positive words on certain generating sets of Bₙ and we prove that Rudolph’s condition is equivalent to ours. We compute the sutured Khovanov homology groups of positive braid closures in homological degrees i = 0,1 as sl₂(ℂ)-modules. Given a condition on the sutured Khovanov homology of strongly quasipositive braids, we show that the sutured Khovanov homology of the closure of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element agrees with that of positive braid closures in homological degrees i ≤ 1 and show this holds for the class of such braids on three strands. / Thesis (PhD) — Boston College, 2017. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

The very first study of the development of the turbulent isothermal and incompressible air jet which issues at a constant velocity from a slender annular slot, circumnavigating an open core, into an otherwise quiescent and unbounded environment of the same density, is presented. The geometry of this source is defined by three diameters: the outer diameter of the slot $D_o$; the inner diameter of the slot $D_i$; and the diameter of the (circular) open core $D_v$. `Slender' refers to a slot for which the inner and outer diameters are approximately equal, i.e. $D_i/D_o\approx 1$. Our focus lies in understanding the development of the time-averaged flow with distance downstream and the influence of the source geometry on the development of the jet. Given the absence of information on jets issuing from the sources of interest, the investigation follows an approach reminiscent of the classic investigations into round jets. That is, it begins with the development of a nozzle and experimental set-up which are suitable for studying the slender open-core annular jet. In addition to the experimental measurements, a complementary mathematical model was developed to describe the unique near-field behaviour of the open-core jet. Measurements were acquired using flow visualisation and Particle Image Velocimetry. On examining the streamwise development of the flow, the slender almost fully open-core jet was delineated into four key regions and the characteristic scalings identified. The regions were as follows: a bounded induced-flow region; a near-source planar-jet region; a transitional region; and a far-field round-jet region. Fluid induced through the open core of the nozzle and subsequently entrained into the jet significantly enhanced the near-field dilution of the jet. Following on from this, the influence of the diameter ratio $D_i/D_o$ and ventilation ratio $D_v/D_i$ on jet coalescence was examined. Over the range of diameter ratios examined ($0.845 \leq D_i/D_o\leq 0.981$), experimental measurements and the predictions from mathematical modelling indicated that $D_i/D_o$ significantly influenced the volume flux induced through the core while the coalescing behaviour of the jet and the far-field region remained largely unchanged. Over the range of ventilation ratios examined ($0 \leq D_v/D_i\leq 0.90$), experimental measurements demonstrated that $D_v/D_i$ controlled the restriction experienced by fluid induced through the open core and significantly influenced the far-field behaviour of the jet. Our findings suggest that jet of interest is then uniquely characterised by the momentum flux $M_0$, the diameter ratio $D_i/D_o$, and the ventilation ratio $D_v/D_i$.

A multi-decadal survey program undertaken at the Collaroy-Narrabeen embayment in SE Australia identifies medium-term (~2-7 year) cycles of both erosion and accretion across the entire embayment ('beach oscillation') and at its two extremities ('beach rotation'). These cycles have been observed to respond to phase shifts in the El Ni??o/Southern Oscillation (ENSO). To investigate wave and climate controls of embayment variability in finer detail, this study combines historical surveys with 45 years of wave data from the ERA-40 reanalysis and four years of high-resolution beach measurements using RTK-GPS and image-derived survey techniques. ENSO and Southern Annular Mode (SAM) controls of wave variability in the Sydney region are first explored. In general, wave heights increase/decrease and wave directions become more easterly/southerly during La Ni??a/El Ni??o phases. A positive correlation is observed between the SAM and summer wave heights, and a negative correlation between the SAM and winter wave directions. Storm variability is observed to be modified by the ENSO, but not the SAM. In particular, La Ni??a phases are generally associated with longer duration, higher energy events from a more easterly direction when compared to those during El Ni??o phases. Wave controls of embayment variability are subsequently investigated. In the short-term (days - months), beach oscillation/rotation is observed to be the most dominant process, accounting for approx. 60%/20% of overall embayment variability. Beach oscillation is related to changes in wave height and storms, whereas beach rotation is related to changes in wave direction and/or wave period. An empirical model that estimates the beach response to individual storm events is developed. In the longer-term (months - years), beach rotation is observed to respond to both wave heights and directions. Larger waves are sheltered somewhat at the southern end, creating an apparent clockwise rotation under energetic wave conditions. Clockwise/anticlockwise rotations are also observed to follow southerly/easterly wave shifts at lags of up to 12 months. Comparisons between the ENSO and beach oscillation/rotation agree with previous observations that El Ni??o/La Ni??a phases are associated with an overall accretion/erosion and clockwise/anticlockwise rotation of the embayment. In general, the SAM shows little influence on embayment variability. While it is clear that beach oscillation is driven by cross-shore processes, to what extent beach rotation is a longshore and/or cross-shore phenomena requires further investigation.

Annular flow is an important flow regime in many industrial applications. The need for a better understanding of this flow regime is driven by the desire to improve the design of many terrestrial and space-based systems. Annular two-phase flow is frequently present in the drilling, production and transportation of oil and natural gas, boilers and condensers, and in heating and refrigeration systems. The flow regime is also important for the refueling of space vehicles, and heating and refrigeration systems for space use. Past studies on annular flow have dealt with varying the gas or liquid Reynolds numbers and studying the effect of such changes on the flow regimes and pressure drops. The effect of two other relevant dimensionless groups, namely the gas-to-liquid density ratio and the gas-to-liquid viscosity ratio, on the film characteristics are noticeably absent. As well, with the increased interest in the space environment, studies on the effect of the gravitational acceleration on two-phase flow would be beneficial. The effect of the gas density and the gravitational acceleration on the annular flow average film thickness and frictional pressure drop are examined. The film thickness was measured using two-wire conductance probes. Experimental data were collected in microgravity and hypergravity aboard the Novespace Zero-G Airbus microgravity simulator and normal gravity data were collected at the University of Saskatchewan. Data were collected for a range of annular flow set points by changing the liquid and gas mass flow rates. The liquid-to-gas density ratio was examined by collecting annular flow data using helium-water and air-water. The gravitational effect on the film thickness characteristics was examined by collecting the data during the microgravity and pull-up (hypergravity) portions of each parabolic flight. A direct comparison is possible between the normal gravity data and the microgravity data, due to the matching of the liquid and gas mass flow rates and the flow regime. The reduction in gravity causes the average film thickness to increase between two and four times from the normal gravity values. The microgravity average frictional pressure drop is within approximately 20% of the normal gravity pressure drop for the same flow conditions. For all gravity levels, the air-water and the helium-water flows give similar results, for both average film thickness and frictional pressure drop, when based on the specific energy of the gas. The hypergravity average film thickness results are larger than at normal gravity for the same flow conditions. However, no flow regime map exists for the hypergravity condition, so the similarity of the flow regime cannot be confirmed. The hypergravity flow appears more chaotic, and may be in the transition from a churn type flow. The average frictional pressure drop is increased by approximately 20% due to the increase in the gravitational acceleration. New non-dimensional equations, which include the effect of the gas density, are presented for each gravity level to predict the average film thickness and the average frictional pressure drop.

Annular flow is an important flow regime in many industrial applications. The need for a better understanding of this flow regime is driven by the desire to improve the design of many terrestrial and space-based systems. Annular two-phase flow is frequently present in the drilling, production and transportation of oil and natural gas, boilers and condensers, and in heating and refrigeration systems. The flow regime is also important for the refueling of space vehicles, and heating and refrigeration systems for space use. Past studies on annular flow have dealt with varying the gas or liquid Reynolds numbers and studying the effect of such changes on the flow regimes and pressure drops. The effect of two other relevant dimensionless groups, namely the gas-to-liquid density ratio and the gas-to-liquid viscosity ratio, on the film characteristics are noticeably absent. As well, with the increased interest in the space environment, studies on the effect of the gravitational acceleration on two-phase flow would be beneficial. The effect of the gas density and the gravitational acceleration on the annular flow average film thickness and frictional pressure drop are examined. The film thickness was measured using two-wire conductance probes. Experimental data were collected in microgravity and hypergravity aboard the Novespace Zero-G Airbus microgravity simulator and normal gravity data were collected at the University of Saskatchewan. Data were collected for a range of annular flow set points by changing the liquid and gas mass flow rates. The liquid-to-gas density ratio was examined by collecting annular flow data using helium-water and air-water. The gravitational effect on the film thickness characteristics was examined by collecting the data during the microgravity and pull-up (hypergravity) portions of each parabolic flight. A direct comparison is possible between the normal gravity data and the microgravity data, due to the matching of the liquid and gas mass flow rates and the flow regime. The reduction in gravity causes the average film thickness to increase between two and four times from the normal gravity values. The microgravity average frictional pressure drop is within approximately 20% of the normal gravity pressure drop for the same flow conditions. For all gravity levels, the air-water and the helium-water flows give similar results, for both average film thickness and frictional pressure drop, when based on the specific energy of the gas. The hypergravity average film thickness results are larger than at normal gravity for the same flow conditions. However, no flow regime map exists for the hypergravity condition, so the similarity of the flow regime cannot be confirmed. The hypergravity flow appears more chaotic, and may be in the transition from a churn type flow. The average frictional pressure drop is increased by approximately 20% due to the increase in the gravitational acceleration. New non-dimensional equations, which include the effect of the gas density, are presented for each gravity level to predict the average film thickness and the average frictional pressure drop.

Recent observational and modeling studies indicate a robust dynamical coupling between the stratosphere and troposphere during boreal winter. This coupling occurs in association with the Northern Annular Mode (NAM), which itself accounts for a significant fraction of the variability of the extratropical circulation. While monthly NAM dynamics have been studied previously, the mechanisms that give rise to NAM variability on short intraseasonal timescale are still unclear. We perform regression analyses, case studies, and composites based on periods of dynamical growth/decay to investigate the roles of the different proposed mechanisms in driving the atmospheric variability observed in association with the NAM on short intraseasonal timescales. More specifically, lag-regression analyses are used to identify the mean canonical structures present during the evolution of a typical NAM event. Illustrative case studies of robust stratospheric NAM events but with different tropospheric signals are contrasted in order to identify the underlying dynamical reasons for the observed differences. Finally, composite analyses of NAM tendencies are performed to isolate the structural and dynamical evolution of NAM events. Zonal-mean and three-dimensional eddy-flux diagnoses are used to examine the role of eddy-mean flow interaction in driving the wind tendencies characteristic of the NAM. In particular, Plumb flux analyses are employed to quantify the contribution of regional stationary wave anomalies toward the zonal mean wind tendency field. Potential vorticity inversions are also used to determine the role of stratospheric anomalies in inducing tropospheric circulations. The case study analyses indicate that preexisting tropospheric PV anomalies can mask the downward penetration of an initial stratospheric NAM signal into the troposphere. PV inversions further suggest that a minimum requirement for a direct downward stratospheric influence is that the stratospheric NAM signal be robust in the lower stratosphere. The dynamical composites show a remarkable degree of reverse symmetry between the zonal-mean dynamical evolution of positive and negative NAM events. Anomalous Eliassen-Palm fluxes are observed in the troposphere and stratosphere, consistent with index of refraction considerations and an indirect downward influence of the stratosphere on the troposphere. The patterns of anomalous wave driving, primarily due to low-frequency planetary scale waves, provide the main forcing of the zonal mean wind tendency field. Regional wave activity fluxes indicate that the wave driving pattern represents the manifestation of planetary scale anomalies over the North Atlantic.

Windback seals work similarly to labyrinth seals except for the effect of helical groove. These seals are essentially a tooth on stator or tooth on rotor labyrinth seal where the grooves are a continuous helical cut like a thread. Windback seals are used in centrifugal gas compressor to keep oil out of the gas face seal area. These face seals cannot be contaminated by oil. A purge gas is applied to the seal to help force the oil back into the bearing area. The windback seal should be designed to prevent any oil contamination into the supply plenum and to reduce purge gas leakage. The CFD simulations have been performed with the effect of clearance, tooth width, cavity shape, shaft rotation, eccentricity, and tooth location on the seal leakage performance and the flow field inside the seal. The leakage flow rate increases with increasing the pressure differential, rotor speed, radial clearance, cavity size, and shaft diameter and with decreasing the tooth width. The eccentricity has a minimal effect for the windback seal. From oil simulations, the windback seal with 25% rotor eccentricity has some of the journal bearing action and drives back flow into the gas plenum. However the windback seal can be used to force the oil back into the bearing side before starting the compressor by applying a purge gas flow since the positive axial velocity inside the cavity is larger than the negative axial velocity. m A Rw cav & / ? is constant for varying shaft rotation since the leakage flow rate for the windback seal increases linearly as the the rotor speed increases. The leakage flow rate for the windback seal increases as the groove size increases due to the pumping action of the windback seal. A windback seal design based upon the numerical simulations that minimize gas leakage and help prevent gas face seal oil contamination was optimized. The windback seal has two leakage flow paths. Since the leakage flow rate under teeth of windback seals is the same as for a similar geometry labyrinth seal, the flow under the teeth can be predicted by two-dimensional labyrinth seal analysis. An empirical model for the leakage rate through the cavity has been developed which fits the data with a standard deviation of 0.12.

In a counter-current two-phase flow system, flooding can be defined as the onset of flow reversal of the liquid component which results in an upward co-current flow. Flooding in the surge line of pressurized water reactors poses a significant technical challenge in the analysis of several postulated nuclear reactor accident scenarios. Despite the importance of flooding in these analyses, previous work does not identify a universally accepted rigorous physics-based model of flooding, even for the simple case of flooding in adiabatic, vertical tubes. This can be attributed to a lack of conclusive understanding of the physics of two-phase counter-current flow, specifically the mechanism of flooding, and the large amount of uncertainty among data from various flooding experiments. This deficiency in phenomenological and experimental knowledge has led to the use of many empirical and semi-empirical correlations for specific system conditions and geometries. The goal of this work is the development of a model for flooding in vertical, adiabatic tubes from first principles. To address a source of uncertainty in the analysis of flooding, a model for the prediction of average film thickness in annular co- and counter-current flows has been developed by considering the conservation of momentum of the liquid and gas flows. This model is shown to be a quantitative improvement over the most commonly used models, those of Nusselt and Belkin, Macleod, Monrad, and Rothfus. The new model better considers the effects of interfacial shear and tube curvature by using closure relations known to represent forces appropriately in co- and counter-current flow. Previous work based on semi-empirical flooding models has been analyzed to develop a new theory on the hydrodynamic mechanism which causes flooding. It is postulated that the growth of an interfacial wave due to interfacial instability results in a flow reversal to ensure that momentum is conserved in the counter-current flow system by causing a partial or complete co-current flow. A model for the stability of interfacial waves in a counter-current flow system is proposed and has been developed herein. This model accurately represents the geometric and flow conditions in vertical adiabatic tubes and has been shown to have limits that are consistent with the physical basis of the system. The theory of waves of finite amplitude was employed to provide closure to an unknown parameter in the new model, the wave number of the wave that generates the interfacial instabil- ity. While this model underpredicts the flooding superficial gas velocity, the result is a conservative estimate of what conditions will generate flooding for a system. In the context of the analysis of a nuclear reactor, specifically a pressurized water reactor, conservatism means that the gas flow rate predicted to cause flooding for a fixed liquid flow rate will be less than the flow rate found experimentally, mean- ing that liquid delivery to the core would be safely underestimated. Future work includes the improvement of the closure relation for the limiting wave number that will cause unstable interfacial waves, as well as an assessment of the applicability of the stability-based model to flooding in the presence of phase change and flooding in complex geometries.

Friction factor data are important for better prediction of leakage and rotordynamic coefficients of gas annular seals. A flat-plate test rig is used to determine friction factor of hole-pattern/honeycomb flat-plate surfaces representing annular seals. Three flat-plates, having a hole-pattern with hole diameter of 12.15 mm and hole depths of 0.9 mm, 1.9 mm, and 2.9 mm, are tested with air as the working medium. Air flow is produced between two surfaces, one having the hole-pattern roughness representing the hole-pattern seal and the other smooth, at the following three clearances of 0.254, 0.381, and 0.635 mm and three inlet pressures of 56, 70, and 84 bar with all possible pressure ratios at each configuration. The friction factor data are presented for all tested configurations, with description of the test rig and theory behind the calculations. The effect of hole diameter, hole depth, clearance, Reynolds number, and inlet pressure are analyzed, and friction factor models based on these parameters are calculated. Friction factor upset (an undesirable phenomenon making the test data non repeatable) is also explained. Dynamic pressure data are presented, measured from dynamic pressure probes located at both the hole-pattern plate and the smooth plates at different locations.