Shear velocity is a critical variable used in many hydrodynamic and aeolian applications. The Law of the Wall is commonly used to derive shear velocity as the product of the slope of a measured velocity profile and the von Kármán constant, κ = 0.4. However, a number of hydrodynamic experiments show that there is a substantial apparent decrease of κ in sediment-laden flow, which was explained by: 1) The energy loss to support the sediment particle suspension in the fluid and 2) The buoyancy effect due to stratification. The energy loss is associated with sediment concentration and grain size, and the stratification can be characterized by sedimentological flux Richardson number or gradient Richardson number. Since there is an apparent change of κ, the term “apparent von Kármán parameter”, or κa, was adopted from Wright and Parker to replace κ in sediment-laden flow. There has been no study to attempt to detect and to evaluate the variability of κa during aeolian saltation, which is the purpose of this dissertation research. Two “clear air” runs and fifteen “sediment-laden” runs were conducted at the northeast coast of Brazil. Wind profile data were collected by a stack of cup anemometers; “true” shear velocity was estimated by an ultrasonic anemometer; and sediment mass flux profile and grain size were estimated from the sand samples collected in a stack of vertical hose-style traps. With these estimates, κa, sediment concentration and sedimentlogical Richardson numbers were derived. Regression analysis indicates that there is a statistically insignificant relationship between κa and grain size, which may be caused by small range of grain size in the study site. However, there is strong statistical relationship between κa and bulk, volumetric concentration below 25 mm, S25, and between κa and sediment transport rate Q (kg/m/s) as: ka = -2088.4S25 0.3964 and ka = -3.134Q 0.4011 A strong relationship was also found between κa and sedimentological Richardson numbers in the lower saltation layer, which can be well explained by the stratification theory.

Most heat transfer tubes are designed for either fully uniform wall temperature or fully uniform wall heat flux boundary conditions under forced convection. Several applications, including but not limited to the solar collectors of renewable energy systems, do however operate with non-uniform boundary conditions. Limited research has been conducted on non-uniform wall heat flux heat transfer coefficients in circular tubes, especially for mixed convection conditions. Such works are normally numerical in nature and little experimental work is available. In this experimental investigation the effects of the circumferential heat flux distribution and heat flux intensity on the single phase (liquid) internal heat transfer coefficient were considered for a horizontal circular tube. Focus was placed on the laminar flow regime of water within a stainless steel tube with an inner diameter of 27.8 mm and a length to diameter ratio of 72. Different outer wall heat flux conditions, including fully uniform and partially uniform heat fluxes were studied for Reynolds numbers ranging from 650 to 2 600 and a Prandtl number range of 4 to 7. The heat flux conditions included 360˚ (uniform) heating, lower 180˚ heating, upper 180˚ heating, 180˚ left and right hemispherical heating, lower 90˚ heating, upper 90˚ heating and slanted 180˚ heating. Depending on the angle span of the heating, local heat fluxes of 6 631 W/m2 , 4 421 W/m2 , 3 316 W/m2 , 2 210 W/m2 and 1 658 W/m2 were applied. Results indicate that the local and average steady state Nusselt numbers are greatly influenced by the applied heat flux position and intensity. Highest average heat transfer coefficients were achieved for case where the applied heat flux was positioned on the lower half (in terms of gravity) of the tubes circumference, while the lowest heat transfer coefficients were achieved when the heating was applied to the upper half of the tube. Variations in the heat transfer coefficient were found to be due to the secondary buoyancy induced flow effect. The relative thermal performance of the different heating scenarios where characterised and described by means of newly developed heat transfer coefficient correlations for fully uniform heating, lower 180° heating, and upper 180° heating. / Dissertation (MEng)--University of Pretoria, 2018. / Mechanical and Aeronautical Engineering / MEng / Unrestricted

Bayou St. John, an urban water body extending south from Lake Pontchartrain, has two anthropogenic structures that regulate flow from the Lake . The City of New Orleans has plans to remove the inner control structure to improve water quality. Field and numerical methods used in this study show removing this structure increased water elevations throughout the Bayou but resulted in lower water elevation signal amplitudes that caused a lower tidal flow exchange from north to south. Bulk Richardson numbers showed mixing was inversely related to flow and the Bayou generally remains stratified. Resuspension of contaminated sediment could negatively impact the local ecology but predicted shear stress values did not reach a critical value (0.1 N/m2) for resuspension. Removal of the waterfall structure will benefit Bayou St. John by decreasing energy losses from the Lake, however a more pronounced tidal signal from Lake Pontchartrain is required to flush the Bayou.

Thesis advisor: Gail C. Kineke / Experiments were conducted in the Petitcodiac River in New Brunswick, Canada during June and August 2006 to study high-concentrations of suspended sediment in a turbulent system. This study will evaluate the conditions necessary for fluid mud formation by investigating 1) the suppression of turbulence at gradient Richardson numbers greater than 0.25; 2) a threshold condition for the amount of sediment a flow can maintain in a turbulent suspension; and 3) the influence of flocculation on vertical suspended-sediment transport. Direct measurements of salinity, temperature, current velocity, and suspended-sediment concentration were collected during accelerating and decelerating flows and when fluid mud formed. In June, current velocities were typically above 1 m s<super>-1</super> and suspended-sediment concentrations were generally less than 10 g l <super>-1</super>. In August, current velocities were typically less than 1.5 m s<super>-1</super>, suspended-sediment concentrations were greater than 10 g l <super>-1</super>, and a high-concentration bottom layer formed rapidly during decelerating flood currents. Gradient Richardson numbers for concentrations greater than 10 g l <super>-1</super> were generally greater than 0.25, suggesting strong density gradients have the ability to suppress turbulence. Results from the Petitcodiac suggest a carrying capacity threshold might exist, but is based on a critical gradient Richardson number between 1.0 and 2.0 rather than the previously accepted value of 0.25. Differences in the evolution of disaggregated grain size distributions for settling suspensions suggest flocculation plays an important role for fluid mud formation by enhancing settling of fine sediments. / Thesis (MS) — Boston College, 2009. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Geology and Geophysics.

Vertical mixing is an important process that relates surface fluxes to concentrations of pollutants and other chemical species in the atmosphere. Errors in vertical mixing have been identified as a major source of uncertainties in various atmospheric modeling efforts including tracer transport, weather forecasting, and regional climate simulation. This thesis aims to quantify uncertainties in model-derived mixed layer heights (zi) over North America through direct comparisons between radiosonde observations and four models at different months of the year 2004 through the bulk Richardson number method. Results of this study suggest that considerable errors in zi exist throughout the region with the spatial and temporal variations of the errors differ significantly among the selected models. Over all, errors in zi were larger in global models than in the limited area mesoscale models, and the magnitude of the random error was two times larger than the bias. Notably, spatial regions of with extremely large positive biases correspond to those with especially large random errors. The biases and random errors, however, were not correlated linearly nor can be easily used to predict each other. Uncertainties in model-derived zi were attributed, through errors in the bulk Richardson number, to temperature and horizontal winds. Errors in both horizontal winds and temperatures were found contributing more or less the same to uncertainties in zi, with relative errors in both variables being the greatest in the lowest part of the troposphere. Lastly, independent observations from the cooperative profiler network suggest that data assimilation did not add qualitative advantages for the comparisons presented in this study. The mixed layer height uncertainties demonstrated in this study may provide a guide for selecting a model to simulate regional scale atmospheric transport and for interpreting flux estimation and inversions studies.

Vertical mixing is an important process that relates surface fluxes to concentrations of pollutants and other chemical species in the atmosphere. Errors in vertical mixing have been identified as a major source of uncertainties in various atmospheric modeling efforts including tracer transport, weather forecasting, and regional climate simulation. This thesis aims to quantify uncertainties in model-derived mixed layer heights (zi) over North America through direct comparisons between radiosonde observations and four models at different months of the year 2004 through the bulk Richardson number method. Results of this study suggest that considerable errors in zi exist throughout the region with the spatial and temporal variations of the errors differ significantly among the selected models. Over all, errors in zi were larger in global models than in the limited area mesoscale models, and the magnitude of the random error was two times larger than the bias. Notably, spatial regions of with extremely large positive biases correspond to those with especially large random errors. The biases and random errors, however, were not correlated linearly nor can be easily used to predict each other. Uncertainties in model-derived zi were attributed, through errors in the bulk Richardson number, to temperature and horizontal winds. Errors in both horizontal winds and temperatures were found contributing more or less the same to uncertainties in zi, with relative errors in both variables being the greatest in the lowest part of the troposphere. Lastly, independent observations from the cooperative profiler network suggest that data assimilation did not add qualitative advantages for the comparisons presented in this study. The mixed layer height uncertainties demonstrated in this study may provide a guide for selecting a model to simulate regional scale atmospheric transport and for interpreting flux estimation and inversions studies.

In this PhD thesis we study some weighted regular Sturm-Liouville problems in which the weight function takes on both positive and negative signs in an appropriate interval [a,b]. With such  problems there is the possible existence of non-real eigenvalues, unlike in the definite case (i.e. left or right definite) in which only real eigenvalues exist. This PhD thesis consists of five papers (papers A-E) and an introduction to this area, which puts these papers into a more general frame. In paper A we give some precise estimates on the Richardson number for the two turning point case, thereby complementing the work of Jabon and Atkinson from 1984 in an essential way. We also give a corrected version of their result since there seems to be a typographical error in their paper. In paper B we show that the interlacing property, which holds in the one turning point case, does not hold in the two turning point case. The paper consists of a detailed presentation of numerical results of the case in which the weight function is allowed to change its sign twice in the interval (-1, 2). We also present some theoretical results which support the numerical results. Moreover, a number of new open questions are raised. We also observe that the real and imaginary parts of a non-real eigenfunction either have the same number of zeros in the interval (-1,2) or the numbers of zeros differ by two. In paper C, we obtain bounds on real and imaginary parts of non-real eigenvalues of a non-definite Sturm-Liouville problem, with Dirichlet boundary conditions, thus complementing the results obtained in a paper byBehrndt et.al. from 2013 in an essential way. In paper D we obtain a lower bound on the eigenvalue of the smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. In paper E we expand upon the basic oscillation theory for general boundary problems of the form   -y''+q(x)y=λw(x)y,  on  I = [a,b], where  q(x) and w(x) are real-valued continuous functions on [a,b] and y is required to satisfy a pair of homogeneous separated boundary conditions at the end-points. Already in 1918 Richardson proved that, in the case of the Dirichlet problem,  if w(x) changes its sign exactly once and the boundary problem is  non-definite, then the zeros of the real and imaginary parts of any non-real eigenfunction interlace. We show that, unfortunately, this result is false in the case of two turning points, thus removing any hope for a general separation theorem for the zeros of the non-real eigenfunctions. Furthermore, we show that when a non-real eigenfunction vanishes inside I, then the absolute value of the difference between the total number of zeros of its real and imaginary parts is exactly 2.

Jetströmmar betyder mycket för meteorologer och fungerar som ett hjälpmedel för prognostisering av lågtryck. Kvadrantmodellen för jetströmsstråk beskriver hur det kan bildas lågtryck och konvergens vid marknivå på olika sidor om jetströmmen. Jetströmmar är också av betydelse för flygtrafiken, eftersom man kan spara tid och bränsle genom att flyga in i – eller genom att inte flyga in i – en jetström. Det är känt att jetströmmar ibland kan ge upphov till klarluftsturbulens (CAT), och studier har visat att turbulensen ofta är associerade med stabilt stratifierade zoner i närheten av jetströmmen. Denna studie börjar med en teoridel där jetströmmars uppkomst och kvadrantmodellen redogörs. Teoridelen fortsätter med en kort beskrivning av hur jetströmmar påverkar flygtrafiken och vad det finns för system som varnar för farliga fenomen som är kopplade till jetströmmarna. Det tas även upp att flygturbulensen som upplevs av flygplan inte nödvändigtvis orsakas av termisk eller mekanisk turbulens, utan även kan orsakas av stående vågor. I arbetets senare del görs en fallstudie på två jetströmmar som gav upphov till kraftig flygturbulens under januari månad 2011. Från sonderingsdata beräknades Scorer-parametern och Richardsons tal för att se om dessa värden kunde relateras till flygturbulensen. Resultatet visade att Richardsons tal aldrig föll under 0.25 och antyder därför att det inte fanns någon fullt utvecklad turbulens. Vågor i luften kan därför vara en förklaring till flygturbulensen. Resultatet visade också att Scorer-parametern antog lokala minima och hade tydligare relation till flygturbulensen än Richardsons tal vad gäller flygturbulensens position i höjdled. / Jet streams are useful as a tool for meteorologist to make forecasts of low pressure areas. The four quadrant model describing jet streaks reveals how lows and convergence could form over ground levels on different sides of a jet streak. Jet streams are also of great importance for the aviation industry, as one could save time and fuel by flying into – or by not flying into – a jet stream. It is known that the jet streams sometimes can induce Clear Air Turbulence (CAT), and studies have shown that the turbulence is often associated with stable stratified layers in the vicinity of the jet stream. This study starts with a theory section describing the cause of jet streams and the four quadrant model. The theory section continues with a description of why jet streams could be a hazard for airplanes, and also a description of existing reporting and warning systems that are used to prevent pilots from flying into dangerous phenomena. In this section, it is also emphasized that in-flight turbulence not necessarily is a result of thermal or mechanical turbulence, but could also be induced by standing waves in the atmosphere. The latter part of this report is a case study of two jet streaks that was the cause of severe in-flight turbulence in January 2011. The Scorer parameter and the Richardson number were computed using data from atmospheric soundings. The results show that the Richardson number never fell below 0.25, suggesting that there were no fully-developed turbulence. Waves in the air could therefore be an alternative explanation to the reported in-flight turbulence. The results also show that the Scorer parameter had local minima on the levels where the in-flight turbulence occurred. In contrast to this, the Richardson number had no apparent relation to the position of the reported turbulence.

In this study, the local similarity theory functions were calculated with two different approaches and on the other hand the mean hub height wind speed was compared with the rotor equivalent wind speed. Both calculations are based on two independent data-sets from measurement campaigns Hornamossen and Ryningsnäs which were conducted in the south of Sweden between May 2015 and June 2017, and November 2010 and February 2012. The first campaign includes measurements between 100 and 173 m and the second 98 and 138 m. In general, the aims were to validate if the results with reference functions and to compare the results from both approaches. The local similarity theory was used, because well above the ground, the assumption of a constant flux layer typically does not hold especially in the stable boundary layer. The used approaches are the flux-gradient and Richardson number formulation. Based on those, the non-dimensional universal functions for momentum and heat could be calculated and those could be presented as functions of the stability parameter. As shown in this study, the scatter the Richardson number formulation results are significant smaller compared to the flux-gradient formulation. One reason can be that the stability parameter and the universal functions for momentum and heat depend solely on the Richardson number. Despite the higher scatter, the medians of the universal function for momentum based on the flux-gradient formulations for both data-sets agree also with the references. Furthermore, for the results of the universal function for heat based on the flux-gradient formulation agree with the references if the minimum limit for the kinetic heat flux is significantly higher than for the universal function for momentum. Furthermore, in the publication from England &amp; McNider 1995, who derived the Richardson number formulation, includes two erroneous equations for stable stratification. One of them has a tipping error and the other was incorrectly derived. Thus, the corrected equations are presented in this work. This work also presents new equations which are not based on the assumption that the constants of the empirical formulation for the universal function for momentum and heat with the same value. A comparison of the old and new equations show for a generated Richardson number vales a agreement of the results over the defined Richardson number range. Finally, in the wind industry it is a common practice to use the mean wind speed at the hub height as the representative mean wind speed over the entire rotor swept area. However, this assumption differs increasingly from the reality, because turbine sizes increase constantly. Thus, in this study, this common method is compared with another averaging concept. Hence, the work focuses on a area-weighted mean wind speed which is called the rotor equivalent wind speed. This average gives a better estimation of the existing wind field because it is based on multiple measurements at various heights. Since the wind gradient changes with height, those two velocities are plotted as functions of the same stability parameter as above. The main results in unstable stratification are that the hub height wind speed underestimates the rotor equivalent wind speed by about 1 to 1.5 %. In stable stratification the results vary: Two calculations show a overestimates by about 1 % and another shows no difference between those averages. Hence, the conclusion based on those findings are that the hub height wind speed is a source for a higher modelling uncertainty. On the contrary, the rotor equivalent wind speed gives more accurate modelling results. / I denna studie beräknades de lokala likhetsteorifunktionerna med två olika tillvägagångssätt och å andra sidan jämförs genomsnittliga vindhastigheten vid navhöjden med rotorekvivalent vindhastigheten. Beräkningarna är baserad på två oberoende datamängder från mätningskampanjer Hornamossen och Ryningsnäs som genomfördes i södra Sverige mellan maj 2015 och juni 2017 och november 2010 och februari 2012. Första kampanj innehåller mätningar mellan 100 och 173 m och den andra 98 och 138 m. Generellt var målet att validera resultaten med referensfunktioner och jämföra både tillvägagångssätt med varandra. Lokala likhetsteorinen används eftersom för mätningar långt över marken håller antagandet om ett konstant flödesskikt vanligtvis inte. Det gäller särskilt i det stabila gränsskiktet. De två tillvägagångssätten är flödesgradientförhållandet och Richardson-talformuleringen. Baserade på de formuleringarna kan de icke-dimensionella universella funktionerna för momentum och värme beräknas och de visas som en funktion av stabilitetsparametern. I denna studie visas att spridningen av Richardsons talformuleringsresultat är signifikant mindre jämfört med andra metoden. En anledning är att stabilitetsparameter och både universella funktioner beror endast på Richardson tal. Trots den högre spridningen överensstämmer medianerna för den universella funktionen för momentum baserat på flödesgradientformuleringarna med referenserna. Detsamma gäller för resultaten av den universella funktionen för värme baserat på flödesgradientformuleringen om minimigränsen om kinetiska värmeflödet är betydligt högre än för den universella funktionen för momentum. Dessutom innehåller publikationen från England &amp; McNider 1995, som innehåller härledning av Richardson talformulering, två felaktiga ekvationer för stabila gränsskiktet. En av dem har ett tippfel och den andra var felaktigt härledd. Detta arbete presenteras de korrigerade ekvationerna. Dessutom presenteras en uppsättning nya ekvationer där de konstanterna av den empiriska formuleringen för den universella funktionen för momentum och värme inte antas att har samma värde. Slutligen är det i vindindustrin en vanlig praxis att använda den genomsnittliga vindhastigheten vid navhöjden som den representativa medelvindhastigheten för hela "rotor swept area". Turbinstorlekarna ökar dock ständigt och därför får skillnaden mellan realitet och beräkningen alltid större. Således fokusera denna studien en areaviktad medelvindhastighet som heter rotorekvivalent vindhastighet. Den beräknar medelvindhastigheten med ett mindre osäkerhet eftersom den är baserad på flera vindmätningar på olika höjder. På grund av ett ojämt vindgradient i gränsskiktet visas resulten som funktion av densamma stailitetsparameter från likhetsteorien. Huvudresultaten för instabil gränskiktet är att navhöjdens vindhastigheten underskattar rotorekvivalent vindhastigheten med cirka 1 till 1,5 %. För det stabila gränskiktet finns olika resultaten: Två beräkningar visar att navhöjdens vindhastigheten överskattningar rotorekvivalent vindhastigheten med ungefär 1 % och en beräkning visa inget skillnad mellan medelvärdarna. Slutsatsen är att navhöjdens vindhastigheten är ett källa till ett högre modelleringsosäkerhet. Däremot visades att användningen av rotorekvivalent vindhastigheten leda till ett bättre prognosresultat. / In dieser Studie wurde einerseits die lokale ähnlichkeitstheorie mit zwei unterschiedlichen Ansätzen berechnet und andererseits die durchschnittliche Nabenhöhen- mit der rotor-äquivalenten Windgeschwin- digkeit verglichen. Dafür standen zwei unabhängige Datensätze zur Verfügung, welche Messwerte zwischen 98 und 173 m beinhalteten. Die Messungen wurden in Südschweden durch die Messtürme Hornamossen und Ryningsnäs in den Zeiträumen von Mai 2015 bis Juni 2017 und von November 2012 bis Februar 2012 erhoben. Das Ziel dieser Studie war es, die Ergebnisse aus den Berechnungen mit Referenzfunktionen zu validieren und den ausgewählten Ansätzen zu vergleichen. Für Messungen mit mehr als 100 m über der Erdoberfläche ist die lokale ähnlichkeitstheorie anstelle der ähnlichkeitstheorie basierend auf der konstanten Flussschicht besser geeignet, da letztere von einem konstanten Wert ausgeht. Die Ansätze, die hierfür verwendet wurden, sind die „Flux-Gradient Formulation“ und die „Gradient Richardson Number Formulation“. Mit beiden kann die universelle Impuls- und Wärmefunktion berechnet und als Funktion des Stabilitätsparameters dargestellt werden. Wie diese Studie zeigt, ist die Streuung um die Referenzkurven sehr klein für die „Gradient Richardson Number“ Ergebnisse im Vergleich zur „Flux-Gradient Formulation“. Dies liegt daran, dass sowohl der Stabilitätsparameter und die universellen Impuls- und Wärmefunktion nur von der Richardson-Zahl abhängen. Trotz der höheren Streuung für die „Flux-Gradient Formulation“ stimmen die Mediane der universellen Impulsfunktionen von beiden Datensätzen und die Referenzkurven überein. Das Gleiche gilt für die Ergebnisse der universellen Wärmefunktion basierend auf der „Flux-Gradient Formulation“, wenn der Mindestwert für den kinetischen Wärmefluss signifikant größer ist als für die universelle Impulsfunktion. Ausgangspunkt ist die Veröffentlichung von England &amp; McNider 1995, welche die „Gradient Richardson Number Formulation“ dargestellt haben. Diese enthält zwei fehlerhafte Gleichungen, wovon eine einen Tippfehler aufweist und die andere auf einer fehlerhaften Herleitung basiert. Diese Arbeit stellt daher ebenso die korrigierten Gleichungen dar. Zusätzlich werden neue Gleichungen vorgestellt, bei denen nicht angenommen wird, dass die von England &amp; McNider angenommenen Konstanten der empirischen Funktionen für die universelle Impuls- und Wärmefunktion den gleichen Wert haben. In der Windindustrie ist es üblich, dass die durchschnittliche Nabenhöhen-Windgeschwindigkeit als Durchschnittsgeschwindigkeit für die gesamte „rotor swept area“ angenommen wird. Diese Annahme weicht immer mehr von der Realität ab, weil Windkraftwerke kontinuierlich größer werden. Daher wird in dieser Arbeit auf eine flächengewichtete mittlere Windgeschwindigkeit mit der Bezeichnung Rotor-äquivalente Windgeschwindigkeit gearbeitet. Diese produziert realistischere Durchschnittswerte, weil sie Messungen von mehreren vertikal verteilten Messpunkten einbezieht. Aufgrund des sich verändernden Windgradientens in der Grenzschicht wird die Abweichung zwischen diesen beiden Geschwindig- keiten als Funktion der Schichtungsstabilität dargestellt. Die zentralen Ergebnisse dieser Studie zeigen zusammenfassend, dass bei labiler Schichtung die Nabenhöhen-Windgeschwindigkeit die Rotor-äquivalente Windgeschwindigkeit um etwa 1 % bis 1.5 % unterschätzt. Für die stabile Schichtung unterscheiden sich die Ergebnisse: Zwei Berechnungen zeigen, dass die Nabenhöhen-Windgeschwindigkeit die Rotor-äquivalente Windgeschwindigkeit um ca. 1 % überschätzt und eine andere Berechnung zeigt keinen Unterschied zwischen den Mittelwerten. Daraus kann gefolgert werden, dass die Nabenhöhen-Windgeschwindigkeit eine höhere Fehlerquelle aufweist. Im Gegensatz dazu liefert die rotor-äquivalente Windgeschwindigkeit genauere Prognosewerte.