In dieser Arbeit entwickeln und untersuchen wir ein numerisches Schema für die eindimensionale Wellengleichung mit Hysterese für unterschiedliche Arten von Randbedingungen. Diese Gleichung ist ein Modell für die Longitudinal- oder Torsionsschwingungen eines homogenen Stabes unter dem Einfluß einer uniaxialen äußeren Kraftdichte, wobei wir ein elastoplastisches Materialgesetz annehmen. Hysterese-Operatoren sind ratenunabhängige Volterra-Operatoren, die Zeitfunktionen in Zeitfunktionen abbilden. Mit ihnen lassen sich Gedächtniseffekte modellieren, wie sie zum Beispiel in der Elastoplastizität oder im Ferromagnetismus auftauchen. Zunächst führen wir Hysterese-Operatoren allgemein ein und analysieren dann eine spezielle Klasse von Hysterese-Operatoren, die Prandtl-Ishlinskii-Operatoren. Wir untersuchen ihre Gedächtnisstruktur und erklären, wie sich die Operatoren numerisch auswerten lassen. Dazu stellen wir zwei verschiedene Approximationsansätze vor. Wir führen aus, wie sich die approximierenden Operatoren implementieren lassen und leiten lineare und quadratische Fehlerabschätzungen her. Zur numerischen Lösung des gekoppelten Systems aus der Wellengleichung mit einem Hysterese-Operator führen wir ein implizites Differenzenschema mit Gedächtnis ein. Für eine Klasse von Hysterese-Operatoren zeigen wir die Existenz und Eindeutigkeit der Lösung des numerischen Schemas, beweisen mit Hilfe von Kompaktheitsschlüssen und einem Monotonieargument die Konvergenz des Verfahrens und leiten eine Fehlerabschätzung der Ordnung 1/2 her. Wir diskutieren, wie das vorgestellte Verfahren auf die Prandtl-Ishlinskii-Operatoren angewendet werden kann. / In this thesis we develop and investigate a numerical scheme for the one-dimensional wave equation with hysteresis for different kinds of boundary conditions. This equation can be regarded as a model for the longitudinal or torsional oscillations of a homogeneous bar under the influence of an uniaxial external force density assuming an elastoplastic material law. Hysteresis operators are rate-independent Volterra operators mapping time functions to time functions. This kind of operator can be used to model memory effects as they appear in elastoplasticity or ferromagnetism, for example. We first give an introduction to the general concept of hysteresis operators before we analyze a special class of hysteresis operators called Prandtl-Ishlinskii operators. We investigate their memory structure and explain how the operators can be evaluated numerically. To that end we present two different kinds of approximation schemes. We point out how the approximating operators can be implemented and we derive linear and quadratic error estimates. For the numerical solution of the coupled system of the wave equation with a hysteresis operator we introduce an implicit difference scheme with memory. For a class of hysteresis operators we show the existence and uniqueness of the numerical solution. We prove the convergence of the scheme by compactness and monotonicity arguments. We derive an error estimate of order 1/2. We discuss the application of the method presented to Prandtl-Ishlinskii operators.

Este trabalho apresenta o estudo da influência da esteira gerada por uma hélice no arrasto induzido em um sistema que combina uma asa finita com hélice, configuração típica de aeronaves turbo-hélice. A modelagem é realizada da seguinte forma: a asa finita é representada com a teoria da linha sustentadora de Prandtl modificada para efeitos de vórtices tridimensionais e a hélice é modelada com a teoria do disco atuador. São apresentados resultados de arrasto e L/D para algumas asas simples com e sem o efeito da hélice.

A modelagem matemática do escoamento sobre asas através da Teoria da Linha Sustentadora possibilita a compreensão do fenômeno físico e serve como estimativa preliminar dos coeficientes aerodinâmicos da asa. O modelo clássico de Prandtl da linha sustentadora ainda é empregado atualmente devido à relativa simplicidade e por apresentar bons resultados para asas retas, perfis variados e de área em planta generalizada. Entretanto, para asas com enflechamento o modelo de Prandtl não possibilita a predição correta da distribuição de circulação, resultando em erros nos coeficientes aerodinâmicos cada vez maiores na medida em que o enflechamento aumenta. Neste trabalho, realiza-se uma revisão acerca dos modelos baseados na Teoria da Linha Sustentadora presentes na literatura e implementam-se adaptações do modelo clássico a fim de predizer a distribuição de circulação de uma asa enflechada. Foram propostos três modelos: duas formulações lineares com diferentes condições de contorno e um modelo não linear. Através dos modelos foi possível observar a distribuição de circulação ao longo da envergadura da asa. Utilizando o modelo fundamentado na Linha Sustentadora de Prandtl, foi possível obter resultados condizentes com a literatura para enflechamentos pequenos e/ou com elevados valores de alongamento. O modelo linear da Linha Sustentadora Estendida de Weissinger possibilitou a obtenção de resultados mais coerentes para asas com enflechamentos e/ou alongamentos pequenos. O modelo não linear para a Linha Sustentadora de Prandtl mostrou resultados coerentes com a literatura para casos específicos, porém requereu maior esforço para calibração dos parâmetros numéricos de simulação.

After nearly 50 years of development, Computational Fluid Dynamics (CFD) has become an indispensable component of research, forecasting, design, prototyping and testing for a very broad spectrum of fields including geophysics, and most engineering fields (mechanical, aerospace, biomedical, chemical and civil engineering). The fastest and most affordable CFD approach, called Reynolds-Average-Navier-Stokes (RANS) can predict the drag around a car in just a few minutes of simulation. This feat is possible thanks to simplifying assumptions, semi-empirical models and empirical models that render the flow governing equations solvable at low computational costs. The fidelity of RANS model is good to excellent for the prediction of flow rate in pipes or ducts, drag, and lift of solid objects in Newtonian flows (e.g. air, water). RANS solutions for the prediction of scalar (e.g. temperature, pollutants, combustable chemical species) transport do not generally achieve the same level of fidelity. The main culprit is an assumption, called Reynolds analogy, which assumes analogy between the transport of momentum and scalar. This assumption is found to be somewhat valid in simple flows but fails for flows in complex geometries and/or in complex fluids. This research explores optimization methods to improve upon existing RANS models for scalar transport. Using high fidelity direct numerical simulations (numerical solutions in time and space of the exact transport equations), the most common RANS model is a-priori tested and investigated for the transport of temperature (as a passive scalar) in a turbulent channel flow. This one constant model is then modified to improve the prediction of the temperature distribution profile and the wall heat flux. The resulting modifications provide insights in the model’s missing physics and opens new areas of investigation for the improvement of the modeling of turbulent scalar transport.

<p> An experimental study of free convection heat transfer from a heated horizontal plate facing downwards in air is reported in this thesis. The results of this study are in good agreement with the results obtained by Fishenden and Saunders. This study also investigates the effects of restraining the development of the thermal boundary layer with 1/2" and 1" edge strips around the edges of the test plate. This study led to the conclusion that edge restrains tended to decrease the heat transfer from the plate. </p> <p> The range of Grashof Prandtl Number product investigated is between 4 x 10⁸ and 8 x 10⁹ resulting in the heat flux range of 0.7 Btu/hrft² to 102 Btu/hrft². Correlations are presented relating heat flux and temperature difference between plate surface temperature and ambient temperature. </p> / Thesis / Master of Engineering (ME)

No description available.

Ziel der Arbeit ist es, auf der Basis von Messungen ein rheologisches Materialmodell für technische Gummiwerkstoffe zu erstellen, welches deren Eigenschaften nachbildet, insbesondere vorhandene komplexe Zusammenhänge zwischen Relaxation, Erholung, Versuchsgeschwindigkeit und Belastungsamplitude. Dabei wird sich auf die Simulation von großen einfachen, aber beliebigen Scherverformungen beschränkt, woraus ein skalarwertiges Modell resultiert. Anwendung finden generalisierte Maxwell-Elemente und generalisierte kontinuierliche Prandtl-Elemente. Verschiedene Modellvarianten werden diskutiert. Es wird ein Berechnungsprogramm unter MATLAB erstellt.

The near-wall modelling of turbulent flows has been an active field of research due to the computational cost associated with the direct numerical simulations of such flow, which are characterized by a wide range of length and time scales. With the recent advancements in technological capabilities, the availability of high-fidelity data has enabled the construction of data-driven approaches to model turbulence. In this thesis, deep-learning models are used to model the dynamically important near-wall region in a turbulent boundary layer. As a first step, a direct numerical simulation (DNS) of an incompressible zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) over a flat plate is performed using a pseudo-spectral code, SIMSON (Chevalier et al., 2007). The Reynolds number based on free-stream velocity and inlet displacement thickness is 450 and the passive scalars are simulated at Prandtl numbers of 1, 2, 4 and 6. Turbulence statistics for the flow and thermal fields are computed and compared against the numerical simulations at a similar Reynolds number. To generate the training, validation and test datasets for the neural network, the turbulent velocity fluctuation fields are sampled at various wall-normal locations, y+ = 15, 30, 50, 100 at a constant sampling time of ∆t+ = 0.99, in addition to the streamwise and spanwise wall-shear-stress fields, pressure field and heat flux fields at the wall. A fully convolutional network (FCN) based model is proposed for the prediction of two-dimensional velocity-fluctuation fields farther from the wall using the sampled fields at the wall. The quality of predictions from the network is assessed based on (i) the mean-squared error (MSE) between the predictions and the DNS fields, (ii) the relative percentage error in prediction of root-mean-squared (RMS) of fluctuations or fluctuation intensity and (iii) the correlation coefficient between the predicted and the DNS fields. Different types of predictions are performed, where the three components of the velocity-fluctuation fields are predicted simultaneously by the FCN, and these predictions are classified based on the input fields to the FCN. Three different types of predictions are presented in this study, and an auxiliary-loss-function approach is also introduced to improve the performance of the FCN. The results from the proposed data-driven model for ZPG TBL shows a good capability in the prediction of both the instantaneous fluctuation fields and the turbulent statistics like fluctuation intensity. In particular, the prediction of velocity-fluctuation fields at y+ = 30 using only the heat-flux field at Pr = 6 exhibits less than 12% error in the prediction of streamwise fluctuation intensity. The results obtained in this study indicate the potential of FCN in serving as a computationally effective tool to predict turbulent-velocity-fluctuation fields close to the wall using the inputs from the wall and finds useful application in flow-control problems. / Nära väggmodelleringen av turbulenta flöden har varit ett aktivt forskningsfält på grund av beräkningskostnaderna i samband med de direkta numeriska simuleringarna av sådant flöde, som kännetecknas av ett brett spektrum av längd- och tidsskalor. Med de senaste tekniska framstegen har tillgången på data i hög kvalitet möjliggjort konstruktion av datadrivna metoder för modellturbulens. I denna avhandling används djupinlärningsmodeller för att modellera det dynamiskt viktiga området nära väggen i ett turbulent gränsskikt. Som ett första steg utförs en direkt numerisk simulering (DNS) av ett inkomprimerbart nolltryck-gradient (ZPG) turbulent gränsskikt (TBL) över en platt platta med hjälp av en pseudo-spektral kod, SIMSON (Chevalier et al., 2007). Reynolds-talet baserat på friströmshastighet och inloppsförskjutningstjocklek är 450 och de passiva skalarna simuleras vid Prandtlnumbers på 1, 2, 4 och 6. Turbulensstatistik för flödet och termiska fält beräknas och jämförs med de numeriska simuleringarna vid ett liknande Reynolds -nummer. För att generera utbildnings-, validerings- och testdatauppsättningar för det neuralanätverket samplas turbulenta hastighetsfluktuationsfält på olika väggnormala platser, y+ = 15, 30, 50, 100 vid en konstant provtagningstid på ∆t+ ≈ 0, 99, dessutom till strömmande och spanvisa väggskjuvspänningsfält, tryckfält och värmeflödesfält vid väggen. En helt konvolutionsnät (FCN) baserad modell föreslås för förutsägelse av tvådimensionella hastighetsfluktuationsfält längre från väggen med hjälp av de samplade fälten vid väggen. Kvaliteten påförutsägelser från nätverket bedöms baserat på (i) medelkvadratfelet (MSE) mellan förutsägelserna och DNS-fälten, (ii) det relativa procentuella felet vid förutsägelse av rot-medelkvadrat (RMS) för fluktuationer eller fluktuationsintensitet och (iii) korrelationskoefficienten mellan de förutsagda och DNS fälten. Olika typer av förutsägelser utförs, där de tre komponenterna i hastighetsfluktuationsfälten förutspås samtidigt av FCN, och dessa förutsägelser klassificeras baserat på inmatningsfälten till FCN. Tre olika typer av förutsägelser presenteras i denna studie, och en metod för hjälp-förlustfunktion introduceras också för att förbättra prestanda för FCN. Resultaten från den föreslagna datadrivna modellen för ZPG TBL visar en god förmåga i förutsägelsen av både momentana fluktuationsfält och den turbulenta statistiken som fluktuationsintensitet. I synnerhet uppvisar förutsägelsen av hastighetsfluktuationsfält at y+ = 30 med endast värmeflödesfältet vid Pr = 6 mindre än 12% fel i förutsägelsen av strömningsvis fluktuationsintensitet. Resultaten som erhållits i denna studie indikerar FCN: s potential att fungera som ett beräkningsmässigt effektivt verktyg för att förutsäga turbulenta hastighetsfluktuationsfält nära väggen med hjälp av ingångarna från väggen och finner användbar tillämpning i flödeskontroll -problem.

A técnica de transformada integral generalizada é utilizada para a obtenção de soluções analíticas do problema de convecção forçada em regime laminar, na região de entrada de dutos de geometria simples, tais como canais de placas paralelas e dutos circulares. Nessa região as camadas limites hidrodinâmica e térmica estão se desenvolvendo simultaneamente. O fluido é considerado como newtoniano e suas propriedades físicas como constantes. São utilizadas distribuições de velocidades longitudinais na forma analítica, as quais estão disponíveis na literatura e foram obtidas por métodos de linearização da equação de qualidade de movimento na direção axial. A partir da distribuição de velocidade axial obtém-se a distribuição de velocidade normal e analisa-se a influência desta velocidade sobre os parâmetros de transferência de calor: temperatura média de mistura e números de Nusselt local e médio. Analisa-se também a influência de dois perfis de velocidades diferentes sobre os resultados. Obtém-se resultados numéricos para a temperatura média de mistura e números de Nusselt local e médio considerando-se a geometria de um canal de placas planas paralelas, resolvendo-se um sistema completo de equações diferenciais ordinárias acopladas, para vários números de Prandtl. Implementam-se também, soluções aproximadas para um cálculo mais rápido dos resultados, verificando-se a precisão de tais soluções. Quando possível os resultados são comparados com resultados existentes na literatura.

Scalar (heat or mass) transfer plays an important role in many industrial and engineering applications. Difficulties in experimental measurements means that there is limited detailed information available, especially in the near-wall region. Prediction in simple flows is well documented and the basis for development of many Computational Fluid Dynamics (CFD) models. This is, however, not the case for scalar transfer, especially when the Prandtl (Pr) or Schmidt number (Sc) is much greater than unity. In complex flows that involve separation and reattachment, the scalar transfer coefficient is significantly different to that of fully developed turbulent flow. The purpose of this Thesis is to investigate high Prandtl number (Pr ≥ 10) scalar transfer in fully developed (pipe) and disturbed (sudden pipe expansion) turbulent flow using CFD. Direct Numerical Simulation (DNS) is the most straight-forward approach to the solution of turbulent flows with scalar transfer. However, this technique is computationally intensive because all turbulent scales need to be resolved by the simulation. Large eddy simulation (LES) is a compromise compared to DNS. Instead of resolving all spatial scales, LES resolves only the large-scales with the small-scales being accounted for by a subgrid-scale model. Chapter 2 details the mathematical, numerical and computational details of LES with scalar transfer. From this, an optimized and highly scalable parallel LES solver was developed based on state-of-the-art LES subgrid-scale models and numerical techniques. Chapter 3 provides a verification of the LES solver for fully developed turbulent pipe flow. Reynolds numbers between Re = 180 and 1050 were simulated with a single Prandtl number of Pr = 0.71. Detailed turbulent statistics are provided for Re = 180, 395 and 590 with varying grid resolution for each Reynolds number. The results from these simulations were compared to established experimental and numerical databases of fully developed turbulent pipe and channel flows. The LES solver was shown to be in good agreement with the prior work with most discrepancies being accounted for by only reporting the resolved (large-scale) component directly reported from the LES results. For a Prandtl number close to unity, the mechanisms of turbulent transport and scalar transfer are similar. The near-wall region was shown to be dominated by large-scale sweeping structures that bring high momentum and scalar concentrations to the near-wall region. These are convected parallel to the wall as diffusion mechanisms act to transfer this to the wall where dissipation takes effect. An ejection structure then acts to transport the resultant low momentum, scalar depleted fluid back to the bulk to be replenished and continue the cycle. As the Prandtl number increases, molecular diffusivity decreases relative to viscosity, and the mechanisms of scalar transfer differ to those at Pr = 0.71. This is investigated in Chapter 4 using simulations at Re = 180, 395 and 590, with detailed statistics at Re = 395 for Pr = 0.71, 5, 10, 100 and 200. Where possible the results are compared to other numerical work and the LES solver was shown to accurately resolve the higher Prandtl number flows. There are marked variations in the scalar transfer with increasing Prandtl number as the turbulent scalar transfer becomes concentrated closer to the wall and dominated by large-scale turbulent structures. Sweeping structures are still responsible for bringing the high scalar concentrations towards the wall, however, high Prandtl number scalars are unable to completely diffuse to the wall in the time that the structure is convected parallel to the wall adjacent to the diffusive sublayer. Therefore, most of the high Prandtl number scalar is returned to the bulk via the ejection structure rather than being dissipated at the wall. Chapter 5 uses the sudden pipe expansion (SPE) to investigate disturbed turbulent flow for an inlet Reynolds numbers of Reb = 15600 and a diameter ratio of E = 1.6. These simulation parameters were chosen to match the experimental LDA measurements of Stieglmeier et al. (1989). The LES results for a range of grid resolutions were shown to be in very good agreement with the experimental work. From the LES results it was determined that the fluctuations in the wall shear stress are important in the near-wall turbulent transport. These are the result of eddies originating from the free shear layer down-washing and impinging upon the wall. This is a more effective sweeping mechanism than that observed for the fully developed turbulent pipe flow. Despite the down-wash structures impinging upon the wall, a viscous sublayer still exists in the reattachment region, albeit much thinner than the fully developed turbulent pipe flow further downstream. Using the same Reynolds number and diameter ratio, scalar transfer simulations were also undertaken in the SPE with Prandtl numbers of Pr = 0.71, 5, 10, 100 and 200. An applied scalar flux was used to heat the expanded pipe wall. The LES results are in agreement with experimental Nusselt numbers from Baughn et al. (1984) for Pr = 0.71. The disturbed turbulent flow enhances the scalar transfer and this is the result of down wash events transporting low (cold) scalar from the inlet pipe to the near-wall of the expanded pipe. This cools the heated wall and enhances localized scalar transfer downstream of the expansion. A diffusive sublayer still exists in the reattachment region within the viscous sublayer for Prandtl numbers greater than unity. As the Prandtl number increases the diffusivity decreases relative to viscosity and near-wall scalar transfer enhancement decreases as the diffusion time-scales increase.