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Fluid Mechanics: Fourth Edition
... The viscous kick includes a viscous dissipation term that alters the thermal pressure. It is given by (Kundu and Cohen, 2008) ...
... This results in an infinitely thin vortex sheet along y = 0 which then decays. Such a flow is described by the following two equations for the fluid velocity in thex-direction, υ, and the thermal energy, u th (Kundu and Cohen, 2008), η is the dynamic viscosity, which we assume to be constant. The first equation is decoupled from the energy equation and can, thus, be solved independently. ...
... The corresponding solutions for the velocity, υ, and the vorticity, ω, are (Kundu and Cohen, 2008) υ(y,t) = V erf y 2 √ νt and ω(y,t) = ∂υ ∂y = V 1 √ πνt e −y 2 /4νt . (10.11) ν = η/ρ denotes the kinematic viscosity and erf () denotes the error function. ...
Feedback processes play an important role in galaxy formation since they regulate star formation both in low mass galaxies and in massive galaxy clusters. Which mechanisms dominate and how the feedback couples to the surrounding medium, are still open questions. In this thesis, we study the feedback from cosmic rays in different environments in more detail. We develop steady state models for a sample of galaxy clusters, in which cosmic ray heating together with thermal conduction prevents large cooling flows. Observational constraints reveal that cosmic ray heating is only viable in clusters that do not show signatures of enhanced cooling. This might indicate a self-regulated feedback cycle. On galactic scales, cosmic rays can drive winds if they are allowed to diffuse or stream out of the galaxy. We demonstrate in simulations of isolated galaxies that cosmic rays are able to regulate star formation in low mass galaxies but the wind efficiency drops rapidly with increasing galaxy mass. Furthermore, almost all astrophysical flows are highly turbulent. This is a challenge for numerical simulations, which cannot resolve all scales of the turbulent cascade. Therefore, we implement a model for turbulence on subgrid scales into the hydrodynamics code AREPO. We validate our model in idealized test cases and apply it to simulations of turbulent boxes.
... If the vortices remain attached then this is known as an unsteady bubble (UB) wake (Figure 4-B). With increasing swiftness, the sinusoidal instabilities which form in the far wake become more intense and move closer to the cylinder (Kundu and Cohen, 2008). If the instability becomes sufficient that the two shear layers interact, the far wake takes form of a von Karman vortex street (Chen and Jirka, 1997). ...
... If the instability becomes sufficient that the two shear layers interact, the far wake takes form of a von Karman vortex street (Chen and Jirka, 1997). When these instabilities move sufficiently close to the object, they begin to interact with the attached vortices causing them to be periodically shed (Kundu and Cohen, 2008). This is known as a vortex shedding (VS) wake (Figure 4-C). ...
... In the absence of lengthwise gradients, particle path is dictated by the balance between inertial forces related to flow curvature, and viscous forces related to friction at the cylinder wall (Tomczak, 1988). It is well known that this force balance, and thus the resulting wake form is governed by the diameter Reynolds number (Re D ) (Tomczak, 1988, Kundu and Cohen, 2008, Batchelor, 1967). ...
... By using the PIV, we get access to local velocities near the interface. These velocity fields are used to compute corresponding shear stress fields by using equation [2]: (Kundu & Cohen, 2008, Csuka & Olsiak, 2016 E-proceedings of the 38 th IAHR World Congress September 1-6, 2019, Panama City, Panama with the dynamic viscosity, and the horizontal and vertical velocities respectively, and ' and ' their variation from the mean flow. Laminar and turbulent shear stresses exist in the inner and outer regions respectively. ...
... In Fig. 1, + is the dimensionless velocity computed as the ratio between the local velocity and the friction velocity * = √ (Kundu & Cohen, 2008); and + = * is a dimensionless length. The Von Karman logarithmic profile given by * = 1 log ( + ) + (with the Von Karman constant) match experimental data at the interface in the outer region. ...
The Erosion Function Apparatus (EFA) is used to characterize erodibility of a soil subjected to a fluid flow. It is mainly used in north America to predict scour phenomena around structures such as bridge piers. In a first approximation, EFA gives directly the erosion rate depending on the mean water velocity or on the shear stress. Both quantities are relied by using the Colebrook law which is a very global equation. In the scope of this study, we want to get more precise knowledge on the erosion process. In particular, we aim to characterize the turbulence induced by the flow/soil interaction. These turbulences are responsible for high local velocities in the directions parallel and perpendicular to the interface. Furthermore, it imposes to consider both the shear and normal stresses. To access these parameters, we use the Particle Image Velocimetry (PIV) into the EFA. We use both a global visualization to get access to the water column profile and a zoomed visualization on the fluid/soil interface. Simulations are done by varying the flow rate in the EFA. We use three known materials (PVC, sand, clay) to compare results given by the EFA and by the PIV. Finally, we aim at better understanding of the erosion process in the EFA flow section in order to improve results accuracy.
... En ella E(k) es la densidad espectral de energía, es decir la energía por unidad de masa y número de onda (m 3 · s −2 ), es el flujo de energía de unas escalas a otras, con unidades de energía por unidad de masa y unidad de tiempo (m 2 · s−3) y ζ es una constante adimensional cuyo valor (ζ ≈ 1.5) se determina experimentalmente (Vallis, 2006). Como puede verse en la Figura 4.1, dentro del espectro de potencias podemos distinguir varias regiones que se corresponden con diferentes escalas de interés (Kundu and Cohen, 2008). Debido a que la energía se inyecta en escalas grandes (L) para k L −1 tenemos que E(k) se maximiza. ...
... En dicha teoría se demuestra que, en ausencia de estiramiento de vórtices, la energía inyectada en una escala determinada se transporta de escalas pequeñas a grandes 2 (cascada inversa de energía) acompañada de un transporte de enstrofía en sentido inverso (cascada directa de enstrofía). La teoría de Kraichnan predice dos rangos inerciales caracterizados por las leyes (Kundu and Cohen, 2008): ...
This PhD thesis consists on a study of the atmospheric dynamics of the planet Venus with data from two space missions separated in time: the Galileo mission and Venus Express. Concretely, images obtained with different wavelengths have been used to study the motions of the clouds at different vertical levels of the atmosphere, enabling to track the global atmospheric dynamics of the planet. In this thesis simultaneous measurements of the 3-dimensional structure of the winds are presented for the first time, finding and analyzing different sources of temporal variability such as the thermal waves produced by the solar insolation, global oscillations of the wind speeds and variability for periods covering from months to years. Furthermore, a study of the spatial scales of the Venusian clouds and the atmospheric turbulence have been undertaken. Frequent periodic patterns can be also seen on the lower clouds of Venus in high spatial resolution images from the instrument VIRTIS onboard Venus Express. These correspond to a type of atmospheric waves called gravity waves, and their characteristics and role in te atmospheric dynamics of Venus are analyzed at the end of the thesis.
... We start by dening what kind of uid dynamics is considered to be GFD and making some comments on the domain. The dierent sections are based on the classical GFD book by Pedlosky (1987) as well as from Roisin & Beckers (2011), Kundu & Cohen (2008, Majda (2003) and others mentioned in the text. Geophysical uid dynamics is the study of uid dynamics on a rotating sphere (our planet is close enough to be represented as a sphere). ...
... It is can be briey explained as an approximation where variations in density are neglected except where multiplied by gravity. More details can be found in Vallis (2006) or Kundu & Cohen (2008). We will be using the Boussinesq approximation throughout this paper. ...
... Blasius, apud White [14] Blasius, apud White [14] Churchill [15] (9,5 m) Churchill [15] ...
... Foi realizada a validação do aparato para medições de perda de pressão e transferência de calor através de ensaios com escoamento monofásico, em condições diabáticas e adiabáticas. A Fig. 3 apresenta comparação entre os fatores de atrito monofásicos experimentais e estimados através das correlações de Blasius apud White [14], Churchill [15] ...
Este artigo apresenta resultados experimentais de transferência de calor durante escoamento bifásico de R134a em tubo horizontal com fitas retorcidas. Foram realizados experimentos em tubo com 15.9 mm de diâmetro interno, com fitas com razões de retorcimento iguais a 3, 4, 9 e 14, com fluxos de calor de 10 e 20 kW/m² e velocidades mássicas de 75 e 150 kg/m²s para temperaturas de saturação de 5 e 15 °C. Foi verificado o incremento do coeficiente de transferência de calor com a redução da razão de retorcimento, sendo mais pronunciada para títulos de vapor reduzidos. Com o título de vapor tendendo a unidade, os incrementos dos coeficientes de transferência de calor tendem a unidade para todas as razões de retorcimento. Verificou-se incrementos do coeficiente de transferência de calor de até três vezes em relação ao tubo sem fita.
... and σ 2 w are the variances of the along stream, cross-stream and vertical velocities] (Kundu and Cohen, 2008). ...
... We can also conclude that the upper boundary layer is much thinner than 8 cm, since the lower boundary layer is less than 2.5 cm. A laminar boundary layer typically scales as Re − 1/2 (Kundu and Cohen, 2008), [where Re =UL / ν, where U is the fluid velocity relative to the boundary, L is the characteristic length scale of the annular flume, and ν is the kinematic viscosity of the fluid]. As U relative to the boundary is larger for the upper than the lower layer (e.g. 3 vs. 1 cm s − 1 for high flow), the upper boundary layer should be 1 / √3 of the lower boundary layer. ...
Most meroplanktonic larvae have been considered to behave as passive particles in the water column, and their dispersal determined by advection. However, larvae may influence their horizontal transport by sinking or swimming between overlying water masses. The flow conditions under which larvae influence their vertical distribution through depth regulation are presently unclear. Using an annular flume, we examined the effect of increasing flow, repeated exposure to flow, and acceleration and deceleration on the vertical distribution of 4-arm stage echinoplutei of Strongylocentrotus droebachiensis. Specifically, we generated different levels of vertical velocity and shear strengths by manipulating horizontal velocity (u). We increased and decreased flow speed incrementally from no flow (u=0cms−1) to intermediate flow (u=0.48cms−1) to high flow (u=1.02cms−1) for each of 3cycles within each of 2 independent trials. We used a high resolution digital camera to record, and image-analysis to quantify, larval distribution. In the absence of flow, larvae swam upwards and aggregated near the surface of the flume. With increasing flow, increasing numbers of larvae were observed in the mid to low water column indicating a negative influence on larval ability to aggregate near the surface. No differences were observed between distributions in acceleration and deceleration phases of the cycles; however, results suggest that increased exposure can decrease the ability of larvae to regulate their vertical position over time. Vertical shear can result in the re-orientation of swimming larvae and likely compromised larval ability for directed swimming in our study. The threshold shear level beyond which larvae cannot regulate their vertical position is >2s−1, suggesting that echinoid larvae may be more vulnerable to shear than other weak swimmers, most likely because of their shape. However, echinoid larvae can likely influence their vertical distribution within many areas in the ocean, since shears >2s−1 are present only in highly turbulent regions such as fronts.
... We will look at the viscous flow of an incompressible fluide. this is an exact solution of Navier-Stokes equations (see Paterson [19] and Kundu as well [15]). This flow is not very realistic but allows a complete instructive resolution. ...
... A similar process is done for the right hand side. With p = q = 1: (15) so that A = 1 and A 1 = 0. ...
We aim to introduce on simple examples the Method of Matched Asymptotic Expansions ("Méthode des Développements Asympto-tiques Raccordés"). We first introduce the concept of "singular prob-lem" on a problem depending on a small parameter (the famous "ε"). We will define "singular problems" to be not "regular problems". : Problems depending on a small parameter are said regular problems if the solution of the problem does not change so much if the small parameter ε is changed a bit. We will see that at first on a simple second order equation. Then we will show that in singular problems we have to "change the scale" of the variables to focus on the local solution. The apparition of multiple scale is then natural. We show on examples that we have to rescale the equations in order to obtain the relevant solution of the problem. Hence we will the the "Dominant Balance" principle which is a principle to avoid over-simplification of a problem. The "matching principle" allows to re connect the simpli-fied problem at the new scale to the previous at normal scale. The Friedrich problem (a toy model for Navier Stokes) will be discussed, it contains all the principles of the method: rescaling, dominant balance and matching. As the examples are rather simple, they have a simple analytic solution, so that we can check the approximations.
... Le forçage de cette rotation différentielle avec les conditions aux limites rigides entraîne également un écoulement poloïdal axisymétrique, que l'on appelle la circulation méridienne. En effet, la variation spatiale rapide de la vitesse proche des conditions aux limites rend important le terme visqueux dans l'équation de Navier-Stokes (voir une dérivation dans Kundu et Cohen, 2008). Cet écoulement est donc lié à la viscosité et, plus précisément au nombre d'Ekman , un nombre sans dimension qui correspond au rapport de la viscosité sur la force de Coriolis défini par où est l'angle de l'axe de rotation avec la normale à la surface sphérique (Pedlosky, 1979 ...
La fin de vie des étoiles massives donne lieu à une explosion, appelée supernova. Ces explosions sont provoquées par l'effondrement de leur cœur de fer et la formation d'une étoile à neutrons. Les observations des supernovae montrent que certaines d'entre elles ont des caractéristiques extrêmes comme leur énergie cinétique pour les hypernovae ou leur luminosité pour les supernovae superlumineuses. Un moteur central de ces explosions différent du mécanisme des neutrinos pour les supernovae standards est souvent invoqué pour expliquer ces caractéristiques extrêmes : une explosion magnétorotationnelle. Ce mécanisme suppose la formation d'une proto-étoile à neutrons (PNS) en rotation rapide et avec un fort champ magnétique qui permet d'extraire l'énergie de rotation et obtenir une explosion plus énergétique ou lumineuse. Cette PNS, une fois refroidie en étoile à neutrons, fait partie de la classe des magnétars, qui se distingue par toute une diversité d'émissions à haute énergie dues à la dissipation de leur intense champ magnétique interne. Les observations de ces objets permettent d'inférer que la composante dipolaire de leur champ magnétique est de l'ordre de 10¹⁴-10¹⁵ G.L'origine des magnétars et de leur fort champ magnétique à grande échelle, particulièrement en présence de rotation rapide, reste une question ouverte. Deux mécanismes ont été invoqués pour amplifier le champ magnétique dans les PNS : la dynamo convective ou l'instabilité magnétorotationnelle (MRI). Cette thèse se propose d'étudier en détail le scénario de formation par la MRI. Celle-ci a déjà été étudiée de manière analytique ou dans des simulations numériques locales dans une boite représentant une partie de la PNS. Pour la première fois, cette thèse présente des modèles 3D sphériques simplifiés, ce qui permet d'étudier l'origine du dipôle. Une première étude a été menée pour étudier la génération d'un champ magnétique à grande échelle dans l'approximation incompressible, ce qui permet une plus vaste exploration des paramètres et des simulations plus longues. Nos simulations montrent la présence d'une dynamo auto-entretenue, dont l'état saturé ne dépend pas des conditions initiales du champ magnétique. Bien que cet état soit dominé par le champ magnétique turbulent (≥ 10¹⁵ G), un dipôle représentant 5% du champ magnétique moyen est généré dans toutes les simulations. De manière inédite, ce dipôle est orienté vers le plan équatorial plutôt que vers l'axe de rotation. De plus, la comparaison de ces modèles sphériques avec les modèles locaux montre que l'état turbulent de la MRI a des propriétés similaires, bien que le champ magnétique soit légèrement plus faible dans les modèles globaux. Un modèle basé sur l'approximation anélastique a ensuite été développé afin de prendre en compte les profils de densité et d'entropie d'une structure réaliste de PNS. Les simulations montrent également une dynamo auto-entrenue avec un champ magnétique moyen ≥ 10¹⁴ G et un dipôle équatorial de l'ordre de 4.3% du champ magnétique. De plus, un nouveau comportement à grande échelle apparaît avec ce modèle réaliste : une dynamo de champ moyen qui peut être décrite comme une dynamo αΩ. La comparaison de ce modèle avec des modèles idéalisés montre que la stratification en densité favorise l'apparition d'une dynamo de champ moyen. La force de flottaison limite la turbulence dans le plan équatorial mais a une influence assez faible dans l'ensemble du fait de la forte diffusion thermique due aux neutrinos. Dans l'ensemble, les résultats présentés dans cette thèse confirment la capacité de la MRI de former des magnétars dans le cas d'une rotation rapide.
... Considérons un fluide parfait, incompressible, stratifié linéairement en densité (N (z) = N = cste), sans rotation, infini et caractérisé par l'état de base U = 0, ρ = ρ(z), p = p(z), où U représente le champ de vitesse, ρ le champ de densité, p le champ de pression et tel qu'à l'équilibre hydrostatique dp/dz = −gρ. Si le fluide est soumis à des petites perturbations en vitesse u = (u, v, w), en pression p et en densité ρ , il est possible de dériver l'équation d'onde suivante (voir Lighthill [1978] p. 288, Kundu et Cohen [2008] p. 268) dans un repère cartésien (x, y, z) où z correspond à la direction verticale : ...
Cette thèse porte sur l'interaction entre des structures tourbillonnaires modèles et observées dans de nombreux types d'écoulement, les anneaux tourbillonnaires, et un milieu linéairement stratifié en densité. Un anneau tourbillonnaire constitué d'un fluide homogène léger pénètre sous l'effet de son inertie dans une stratification. Les paramètres de contrôle sont les dimensions de l'anneau, sa vitesse de propagation, son orientation par rapport à l'axe vertical, le gradient de densité de la stratification linéaire et la densité du fluide constituant l'anneau tourbillonnaire. Aux temps courts, lors de la phase de pénétration, la distorsion des isopycnes par l'anneau tourbillonnaire est responsable de la génération de vorticité barocline. Aux temps longs, la stratification se relaxe et des ondes internes sont générées. La dynamique de l'anneau est considérablement affectée par son interaction avec le milieu stratifié et la vorticité est amenée à se réorganiser sur des échelles de temps et d'espace qui dépendent des paramètres de contrôle. Différents verrous scientifiques sont abordés au fil des chapitres. Quelle est la profondeur maximale de pénétration de l'anneau et quels sont les mécanismes précurseurs de l'arrêt de la structure tourbillonnaire avant retour en arrière ? Quelles sont les échelles de temps associées à la réorganisation tourbillonnaire et à la génération d'ondes internes ? Quelles sont les caractéristiques des ondes internes forcées par l'impact ponctuel d'une structure tourbillonnaire localisée et comment les décrire ? Quelle est l'influence de l'angle de propagation de l'anneau tourbillonnaire sur la réorganisation de l'écoulement ? Comment est-ce que l'écoulement est modifié lorsque deux anneaux tourbillonnaires sont générés consécutivement ? Existe-t-il une influence du délai de génération entre les deux anneaux tourbillonnaires sur la profondeur de pénétration et sur les ondes internes générées ? Des campagnes expérimentales où des mesures qualitatives (visualisations) et quantitatives (vélocimétrie par image de particule 2D et vélocimétrie par suivi de particules 4D) ont été réalisées pour répondre à ces questions.
... Interestingly, there composition gradient could also be negative. This is reminiscent of diffusive regime (Kundu, Cohen & Dowling 2012) of double diffusive instabilities leading to oscillatory instability in the corresponding saline water. Such oscillatory instability eventually results in the formation of a number of horizontal convecting layers. ...
... Interestingly, there composition gradient could also be negative. This is reminiscent of diffusive regime (Kundu, Cohen & Dowling 2012) of double diffusive instabilities leading to oscillatory instability in the corresponding saline water. Such oscillatory instability eventually results in the formation of a number of horizontal convecting layers. ...
In this paper we investigate the convection phenomenon in the intracluster medium (the weakly-collisional magnetized inhomogeneous plasma permeating galaxy clusters) where the concentration gradient of the Helium ions is not ignorable. To this end, we build upon the general machinery employed to study the salt finger instability found in the oceans. The salt finger instability is a form of double diffusive convection where the diffusions of two physi- cal quantities—heat and salt concentrations—occur with different diffusion rates. The analo- gous instability in the intracluster medium may result owing to the magnetic field mediated anisotropic diffusions of the heat and the Helium ions (in the sea of the Hydrogen ions and the free electrons). These two diffusions have inherently different diffusion rates. Hence the convection caused by the onset of this instability is an example of double diffusive convection in the astrophysical settings. A consequence of this instability is the formation of the verti- cal filamentary structures having more concentration of the Helium ions with respect to the immediate neighbourhoods of the filaments. We term these structures as Helium fingers in analogy with the salt fingers found in the case of the salt finger instability. Here we show that the width of a Helium finger scales as one-fourth power of the radius of the inner region of the intracluster medium in the supercritical regime. We also determine the explicit mathematical expression of the criterion for the onset of the heat-flux-driven buoyancy instability modified by the presence of inhomogeneously distributed Helium ions.
... To simplify the determination of values, we made several assumptions in the calculations. One of the assumptions we made was that the liquid inside the tracker was already at its boiling point [9]. Additionally, in reality, as the liquid evaporate in the tracker, the pressure would increase. ...
Sun tracking system is used for efficiency improvement. To solve problems that presents in existing solar-tracking system such as it cannot work without electricity, this report puts forward an automatic sun tracker driven by differential pressure Controlled system. Main work of this system is to collect the maximum solar energy and convert it into electricity. Due to gravity it turns the tracker, solar ray is used to heat canister to move liquid from one side to the other side in the container to reliably track the suns path from east to west. There is no motor, no gears and no controls to fail. The present system provides apparatus for tracking the sun which reorients itself immediately in the absence of sunlight. Large and small canisters are provided at the respective ends of a pivot-able frame. When the sun is not normal to the plane containing the canister, the near canister is shaded from direct sunlight and the far canister is exposed. A conduit is provided between the canisters, and a quantity of volatile fluid is located in the canister and conduit. The liquid volume of the volatile fluid is greater than that of the small canister plus the conduit, but less than the volume of the large canister. A gas spring fluid is located in the large canister, which has a vapor pressure sufficient to force the volatile fluid into the small canister in the absence of sunlight on the east canister.
... In the following sections the main mathematical principles useful for the statistical analysis will be introduced, following mainly the text books by Pope (2000), Kundu & Cohen (2007) and Tropea et al. (2007). ...
Fully developed turbulent pipe flows have been studied experimentally for more than a century and for more than two decades by means of Direct Numerical Simulations, nonetheless there are still unresolved and of fundamental nature issues. Among those are the scaling of the mean velocity profile or the question whether the near-wall peak in the variance profile is Reynolds number invariant.
In this thesis new experimental results on high Reynolds number turbulent pipe flows, obtained by means of hot-wire anemometry, are carefully documented and results are presented, thereby extending the Reynolds number range of an available in-house experimental database (Sattarzadeh 2011). The main threads of this thesis are the spatial resolution effects and the Reynolds number scaling of wall-bounded flows and were investigated acquiring the measurements with probes of four different wire-lengths at different Reynolds numbers covering the friction Reynolds number range of 550 < R+ < 2 500.
The small viscous length-scales encountered required a high accuracy in the wall-position. Therefore, a vibration analysis of the probe exposed to the flow was performed on two different traversing systems and on several probe-holder/probe configurations, proving that the vibrations of the probe can be large and should be taken into account when choosing the traverse system and probe-holder geometry.
Results of the hot-wire velocity measurements showed that when accounting for spatial resolution effects, a clear Reynolds number effect on the statistical and spectral quantities can be observed. The peak of velocity variance, for instance, appeared to increase with the Reynolds number and the growth seems to be justified from the increase of the low frequency modes. This result together with the appearance of an outer peak located in the low frequency range at higher Reynolds numbers suggests that the increase of the peak of the velocity variance is due to the influence that the large-scale motions have on the near-wall cycle of the velocity fluctuations.
As a side results of the velocity measurements, temperature, i.e. passive scalar, mean and variance profile were obtained by means of cold-wire anemometry. Also here, clear spatial resolution effect on the temperature variance profile could be documented.
... One of the most important equations governing the pressure in the lubricant film is Reynolds Equation, which represents the simplification of the Reynolds's paper in 1886 [2]. To apply this equation to the lubricant film, many assumptions should be considered [3]- [5]: ...
This paper illustrates evaluating performance parameters for the journal bearing design by using three different methods. In the first approach, Raimondi and Boyd Charts (Graphical) method will be used. The second approach represents the first analytical method called Reynold equations tables, which can be used by utilizing familiar equations connected by other values. The second analytical method by Reason and Narang (Combined Solution Method) requires using the empirical equation and tables. Finally, the error percentage for all performance parameters for the analytical methods will be compared with the graphical to show which method is more precise. The main goal of obtaining the precise procedure for the journal bearing performance parameters evaluation is to determine the exact temperature rise in journal bearing and the average temperature of the oil film inside the journal bearing. By knowing these temperatures, the designers can select the best type of lubricant oil and bearing material to avoid bearing failure
... The computation can not pass the separation. Most of classical text book of fluid mechanics do the same and end their course on boundary layers by this dead end, for example one can read in Kundu [9]: "the boundary layer equations are valid only far downstream as the point of separation. Beyond it the boundary layer becomes so thick that the basic underlying assumptions become invalid. ...
We present here the Interacting Boundary Layer Equations. It is called Inviscid-Viscous Interactions as well. This is a way to solve an approximation of the Navier Stokes equations at large Reynolds number using the Ideal Fluid / Boundary Layer decomposition. But, instead of solving first the ideal Fluid and second the Boundary Layer, both are solved together. This “strong coupling” (or this Viscous-Inviscid Interaction) allows to compute separated flows. This was impossible in the classical Boundary Layer framework, because in this framework, the boundary layer is constrained by the Ideal Fluid which imposes its slip velocity at the wall. This coupling is justified in the Triple Deck theory which is the rational explanation of IBL. We present some numerical experiments showing some simple academic examples of interactions such as flows over bumps or wedges in subsonic, supersonic, subcritical and supercritical external flows and in pipes. Some examples from the literature are then presented.
... The phase velocity has the value of about 0.25 m/s while Cohen & Hanratty [1] state the value of about 0.3 m/s. However, our result agrees well with the analytical formula (18), see [8], for phase velocities of surface tension depending gravity waves. ...
The article presents a liquid film instability model designed using results of the set of CFD simulations. The governing equations of the model are derived using a linear equation of motion. The stability analysis is carried out by imposing a liquid surface disturbance which growth rate is investigated in dependence on the geometrical and physical configuration. The gas effect parameters, which are decisive variables in the model, are derived using results of the set of CFD simulations of turbulent flow in channel with wavy surface. The agreement between predicted and measured critical gas velocities and wavelengths in dependence on the liquid film thickness is very good.
... When Re << 1, the flow is laminar and very smooth. When Re > 10 3 , the flow is turbulent and mainly characterised by vortices (Kundu, 2007). Flow within microstructures typically have Reynolds numbers of 10 -3 to 10 -5 and are characterised by a laminar flow. ...
In this research a CO2 IR laser was used to create reconfigurable microchannels in ice where parameters could be changed during operation. The flow of liquid acts as its own shut-off valve by freezing/melting. Concentration of dye can be achieved in ice and was observed with a reflection spectrophotometer
... Drag coefficients for each net panel reach steady values for current velocities of 0.4–0.5 m/s and stay approximately constant up to 1 m/s (highest tested velocities). This is attributed to the transition of the flow regime from viscous laminar at low currents to the developed turbulent flow at higher velocities, as discussed, for example, in Kundu and Cohen, 2008; Sumer and Fredsøe, 2006; for the case of smooth cylinders. Also note that the relative accuracy of measurements increases at higher values of drag force. ...
... B . −0.68, which agrees well with values calculated from existing data (Wieselberger 1921; Proudman & Pearson 1957; Roos & Willmarth 1971; Dennis & Walker 1971; White 1999). ...
The variation in the drag coefficient for low-Reynolds-number flow past rings orientated normal to the direction of flow is investigated numerically. An aspect ratio parameter is used for a ring, which describes at its limits a sphere and a circular cylinder. This enables a continuous range of bodies between a sphere and a circular cylinder to be studied. The computed drag coefficients for the flow past rings at t he minimum and maximum aspect ratio limits are compared with the measured and computed drag coefficients reported for the sphere and the circular cylinder. Some interesting features of the behaviour of the drag coefficients with variation of Reynolds number and aspect ratio emerge from the study. These include the decrease in the aspect ratio at which the minimum drag coefficient occurs as the Reynolds number is increased, from AR ≈ 5 at Re = 1 to AR ≈ 1 at Re = 200. In addition, a substantial decrease in the pressure component of the drag coefficient is observed after the onset of three-dimensional flow while the viscous contribution is similar to that for flow with imposed axisymmetry. Typically, the sudden reduction in drag caused by transition to Mode A shedding is 6%, which is consistent with the behaviour for flow past a circular cylinder. Power-law fits to the drag coefficient for Re ≲ 100 are provided, which are accurate within approximately 2%.
... Note that the derivation of those equations may be done with only 1D fluid mechanics arguments ([67] or [39]). One has to be very careful then to write precisely the action of the pressure on the lateral walls (for example Kundu [40] page 793 does a mistake in his text book). But, the real point of view to derive the equations is the one explained at the beginning of this section and involves the Reduced Navier-Stokes equations ([42]). ...
We are interested in simulating blood flow in arteries with a one dimensional
model. Thanks to recent developments in the analysis of hyperbolic system of
conservation laws (in the Saint-Venant/ shallow water equations context) we
will perform a simple finite volume scheme. We focus on conservation properties
of this scheme which were not previously considered. To emphasize the necessity
of this scheme, we present how a too simple numerical scheme may induce
spurious flows when the basic static shape of the radius changes. On contrary,
the proposed scheme is "well-balanced": it preserves equilibria of Q = 0. Then
examples of analytical or linearized solutions with and without viscous damping
are presented to validate the calculations. The influence of abrupt change of
basic radius is emphasized in the case of an aneurism.
... In both of these cases we can select a common state in which to measure the relative forces of the respective spots upon one another , as if they were point vortices. With the three spots touching end-to-end in terms of their major axis as our reference state, we calculated the relative forces that the spots would create on each other, knowing that this force is proportional to the product of the strength of the intervening spots (C % V2pr, where V is the tangential velocity defining the spots and r ¼ ffiffiffiffiffi ffi ab p is its effective radius ) and inversely proportional to the distance separating them (Faber, 1995; Kundu et al., 2008). For such configuration, we compared the attractive force between the anticyclones ðf aa Þ to that of the repulsion force exerted by the cyclone in the anticyclones ðf ca Þ for the cases with V c ¼ 25 m s À1 and V c ¼ 30 m s À1 . ...
The region in Jupiter’s atmosphere with the highest density of anticyclonic spot-like vortices is the region known as the South South Temperate Zone (SSTZ), which is located between the eastward jet at ≈−42.9° latitude and the westward jet at ≈−39.2° latitude. We present a characterization of the spots found in this region based on ground-based and Hubble Space Telescope observations from the years 1993 to 2007. Mergers have been reported between spots in this region, similar to those observed for the White Ovals in the latitudinal domain immediately equatorward (northward). We use a multilayer model to perform numerical simulations that capture the details of a well-observed merger event involving multiple interacting vortices. We find that the vertical stratification has an important effect in the outcome of the interaction between spots. In particular it can play a determining role on whether or not a cyclone embedded between two approaching anticyclones can inhibit their merging. From our simulations we conclude that the background static stability of the atmosphere in the SSTZ is better characterized by an average value of .
... The fluid velocity was set to 1.05 m/s so that the deflection of the particles submitted to the gradient strength would be half the radius of the channel (in 0.052 second) during a travel of 5.5 cm corresponding to the length of the inlet channel. In order to obtain a maximum velocity of 1.05m/s on the centerline of the channel [11], the flow was set to 0.248 ml/min using a New Era NE-1000 programmable syringe pump [12] located outside the bore of the MRI system. ...
Steering micro-carriers being tracked by an MRI system may be very attractive in oncology. Here, iron oxide microparticles have been steered in a Y-shaped microchannel placed between a Maxwell pair (dB/dz=443 mT/m) located in the center of an MRI bore. A suspension of 10.82 microm iron oxide particles was injected into the channel and a magnetic gradient generated by the Maxwell pair was used to deflect their trajectory. The experimental results based on the percentage of particles retrieved at the targeted outlet during the experiments show that magnetic gradient steering in the human cardiovascular system within an MRI bore can be envisioned.
The objective of the present work is to better understand chemical evolution of both solid and gaseous phases during torrefaction of various biomasses. Torrefaction experiments were carried out with a dynamic profile of temperatures between 200 and 300°C, under inert atmosphere, for pine, ash-wood, miscanthus and wheat straw. Mass loss and formation of condensable species were analyzed by TGA-GC-MS, and chemical evolution of solid phase was characterized by 13C CP/MAS solid-state NMR. Thirty condensable species could be detected; a half of these species were formed during the whole temperature range, and a third were formed by all biomass types. The main phenomena that occurred in solid phase were found to be decrystallization of cellulose, severe degradation of hemicellulose, devolatilization of acetyl groups, conservation of methoxyl groups and charring. It was also found that mass loss and chemical evolution of solid were not directly correlated for different biomasses. Based on the experimental results, a conceptual model was developed to describe biomass degradation during torrefaction. Thirty reactions were determined for the three major macromolecular constituents, namely cellulose, hemicellulose – represented by C5 and C6 sugars – and
lignin – represented by H, G and S units. The main innovations of this model are in the detailed approach of hemicellulose and lignin compositions, as well as in the prediction of sixteen condensable and five permanent species, and six forms of solid char, through chemically meaningful and stoichiometrically valid reactions.
Keywords: biomass, torrefaction, reaction mechanisms, Thermogravimetric Analysis–Gas Chromatography–Mass Spectrometry, solid-state Nuclear Magnetic Resonance
Soil and fresh leachate samples were collected from Erbil Landfill Site (ELS), Iraq. ELS is anaerobic landfill opened in 2001. Field permeability test for soil hydraulic conductivity was performed directly in the site. Meanwhile, other examinations for liquid limit, plastic limit, particle size distribution, etc., were conducted at the laboratory. Unprocessed leachate samples were analyzed for pH, electrical conductivity, biochemical oxygen demand, as well as chloride, solids, sodium, potassium, and calcium contents, among others. The sorptivity, steady state infiltration rate, and hydraulic conductivity of the soil samples were 0.0006 í µí± √í µí± , 0.00004 m s , and 2.17 × 10 −5 m s , respectively. Raw leachate analysis identified the leachate as mature. The soil at ELS was found to be light brown clayey gravel with sand and light brown gravelly lean clay layers with low permeability. In addition, the stabilized leachate from ELS requires treatment prior to disposal to the natural environment. ELS is considered a level 2 anaerobic landfill.
SUMMARYA weighted residual collocation methodology for simulating two-dimensional shear-driven and natural convection flows has been presented. Using a dyadic mesh refinement, the methodology generates a basis and a multiresolution scheme to approximate a fluid flow. To extend the benefits of the dyadic mesh refinement approach to the field of computational fluid dynamics, this article has studied an iterative interpolation scheme for the construction and differentiation of a basis function in a two-dimensional mesh that is a finite collection of rectangular elements. We have verified that, on a given mesh, the discretization error is controlled by the order of the basis function. The potential of this novel technique has been demonstrated with some representative examples of the Poisson equation. We have also verified the technique with a dynamical core of a two-dimensional flow in primitive variables. An excellent result has been observed—on resolving a shear layer and on the conservation of the potential and the kinetic energies—with respect to previously reported benchmark simulations. In particular, the shear-driven simulation at CFL = 2.5 (Courant–Friedrichs–Lewy) and (Reynolds number) exhibits a linear speed up of CPU time with an increase of the time step, Δt. For the natural convection flow, the conversion of the potential energy to the kinetic energy and the conservation of total energy is resolved by the proposed method. The computed streamlines and the velocity fields have been demonstrated. Copyright © 2014 John Wiley & Sons, Ltd.
In membrane distillation (MD) and gas permeation (GP) processes, a general equation given by dusty gas model (DGM) is often used for gas flux through porous media. However, several discrepancies were found in fitting experimental data with the DGM equation. An assumption based on the fact that gas is a compressible viscous fluid is introduced in a modified DGM equation (MDGM). Polytropic equation of state is used to estimate the variation of gas temperature along the pores of membrane. The gas temperature influences its viscosity, which affects its flow resistance through membrane. The MDGM equations for single gases are provided by incorporating polytropic equation of state when integrating the differential DGM flux equation. The MDGM for single gases can be used in modeling the inconsistencies in using DGM in literature and it is verified by published experimental data through fitting the distinguished trends under DGM into one line. The MDGM reveals that the temperature may decrease in membrane pores and exit side, and the phase change of vapor may be one of the inducements for pore wetting in MD.
Spectacular deformations observed in lake sediments in an earthquake
prone region (Lisan Formation, pre-Dead Sea lake) appear in phases of
laminar, moderate folds, billow-like asymmetric folds, coherent
vortices, and turbulent chaotic structures. These deformations are tied
to earthquake events which are speculated to be intensified by seiche
(mini Tsunami)-induced shear at the bottom of the lake. Power spectral
analysis of the deformation indicates that the geometry robustly obeys a
power-law of -1.89, similar to the measured value of Kelvin-Helmholtz
(KH) turbulence in other environments. Numerical simulations are
performed using properties of the layer materials based on measurements
of the modern Dead Sea sediments, which are a reasonable analogue of
Lake Lisan. The simulations show that for a given induced shear, the
smaller the thickness of the layers the greater is the turbulent
deformation. This is due to the fact that although the effective
viscosity increases (the Reynolds number decreases) the bulk Richardson
number becomes smaller with decrease in the layer thickness. The latter
represents the ratio between the gravitational potential energy of the
stably stratified sediments and the shear energy generated by the
earthquake. Hence for thin layers the shear energy density is larger and
the KH instability mechanism, become more efficient. The peak ground
acceleration (PGA) is related then to the observed thickness and
geometry of the deformed layers.
The objective of the article is an introduction to the theoretical study of atomization of droplets from the surface of a thin liquid film. The overview of basic principles of atomization prediction is complemented by the comparison of the calculations performed according to the selected approaches.
The effect of hull-mounted cavitating spoilers system on ship stability is studied experimentally and numerically. The injection of air or exhaust gas stabilizes cavities behind spoilers. The spoiler system is tested experimentally to understand the parameters affecting the flow field and bubble formation around the spoiler. These parameters are the spoiler inclination angle, rise of floor angle and injected air position. The spoiler inclination angle effect is studied in this paper. The images of flow field variation and bubble formation are recorded with scientific video camera and compared with the computed flow field at different conditions and time sequence. The two-phase flow field around a ship spoiler with the free surface simulation in Piecewise Linear Interface Construction method is modeled numerically using a three-dimensional Navier-Stokes code. The bubbles shape, the three-dimensional flow field around the spoiler body and the pressure variation on the wake of the spoiler body are computed. The comparison between the numerical and experimental results shows a good matching of bubble formation in case of spoiler with 60° and 90° inclination angles except that the computational bubble shape has small dissimilarity and not flatted at bubble tail, and the computational bubble shape has slow splitting and not attached to the ship body in case of spoiler with 30° inclination angle. This may be attributed to the laminar flow computation without including turbulence effects. The moment around the 90° inclined spoiler fixation line is 1.651 times that around the 30° inclined spoiler and 1.1 times that around the 90° inclined spoiler after 1 second. Therefore, varying the spoiler inclination angle produces different bubble shapes and consequently different forces are introduced to control the roll, pitch motion and speed of the ship leading to ship stability.
The effect of hull-mounted cavitating spoilers system on ship stability is studied experimentally and numerically. The injection of air or exhaust gas stabilizes cavities behind spoilers. The spoiler system is tested experimentally to understand the parameters affecting the flow field and bubble formation around the spoiler. These parameters are the spoiler inclination angle, rise of floor angle and injected air position. The spoiler inclination angle effect is studied in this paper. The images of flow field variation and bubble formation are recorded with scientific video camera and compared with the computed flow field at different conditions and time sequence. The two-phase flow field around a ship spoiler with the free surface simulation in Piecewise Linear Interface Construction method is modeled numerically using a three-dimensional Navier-Stokes code. The bubbles shape, the three-dimensional flow field around the spoiler body and the pressure variation on the wake of the spoiler body are computed. The comparison between the numerical and experimental results shows a good matching of bubble formation in case of spoiler with 60° and 90° inclination angles except that the computational bubble shape has small dissimilarity and not flatted at bubble tail, and the computational bubble shape has slow splitting and not attached to the ship body in case of spoiler with 30° inclination angle. This may be attributed to the laminar flow computation without including turbulence effects. The moment around the 90° inclined spoiler fixation line is 1.651 times that around the 30° inclined spoiler and 1.1 times that around the 90° inclined spoiler after 1 second. Therefore, varying the spoiler inclination angle produces different bubble shapes and consequently different forces are introduced to control the roll, pitch motion and speed of the ship leading to ship stability.
This thesis presents a simple, meanline analysis of the impact of blade roughness on the mass flow, work coefficient, and efficiency of a three-stage axial compressor as a function of the location of fouling. First, an extensive review is presented on the state-of-the-art of measuring compressor degradation and on the impact of roughness on loss and deviation in a compressor cascade. The performance of a baseline, three-stage compressor, which has hydrodynamically smooth blades, is predicted. Using this baseline geometry, the influence of roughness in the front, middle and rear stages is calculated using empirical data for the enhanced losses and increased deviation, with a stage stacking technique. Influence coefficients that relate percentage changes in one parameter to percentage changes in other parameters are calculated. This analysis predicts that the most sensitive parameter for predicting fouling in the front stages is the percentage change in mass flow and the most sensitive parameter for predicting fouling in the rear stages is the efficiency.
Numerical simulation of airflow around several representative high velocity fragments, originating from naturally fragmenting HE warhead 130mm M79, was performed using FLUENT ® finite element method package. FLUENT ® software contains the broad physical modeling capabilities needed to model flow, turbulence, heat transfer for different applications, ranging from air flow over an aerodynamic surface to real combustion processes. Aerodynamic drag coefficients of fragments were predicted for a wide range of Mach numbers (0.2, 0.4, 0.6, 0.8, 1, 1.1, 1.2, 1.3, 1.5, 2, 3, 4 and 5 Ma) and for eight different positions of fragments (0°, 45°, 90°, 135°, 180°, 225°, 270°, i 315°). Different positions simulate dynamic instability of fragments during their flight through the atmosphere and account for stochastic behavior of fragments, which directly influences aerodynamic drag coefficients, fragments range and warhead lethal zones. Obtained results of aerodynamic drag coefficients were compared with available experi-mental data. 1 Numerical simulation Over 500 numerical simulations of high speed external aerodynamic flow over projectile fragments were performed in Fluent ® . Simulations were performed on 4-processor computer. Aim of the research was to predict range and variations of aerodynamic drag coefficient C d for real rotating fragments with different mass and velocities. Fragments used represent real fragments (fig. 1) from detonating HE projectile 130mm M79. Using Pit tests it was determined that during natural fragmentation of projectile 130mm M79 around 4000 do 6800 fragments were formed. [2] Gurney method for different sections of projectile 130mm M79 was used to determine ini-tial velocity of fragments variations from 1,2 Ma to 3 Ma [6], where Mach number (Ma) is dimensionless number representing the ratio of speed of an object moving through air to the local speed of sound. Simulations were carried out for different fragment masses, different Ma numbers (0.2, 0.4, 0.6, 0.8, 1, 1.1, 1.2, 1.3, 1.5, 2, 3, 4 and 5 Ma) and different angles of atack (0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°) that simulated rotation of fragment and its dynamic instability during the flight (fig. 2). Verification of numerical model used was performed on a sphere and obtained results were compared to experimental data [20]. Values of drag coefficient for sphere obtained from simu-lation differed 2,6% compared to experimental data for sphere velocity of 1,2 Ma, 1,6% for 1,5 Ma, 9,3% for 2 Ma and 1,4% compared to experimental data for sphere velocity of 3 Ma.
This work deals with the cooling of high-speed electric machines, such as motors and generators, through an air gap. It consists
of numerical and experimental modeling of gas flow and heat transfer in an annular channel. Velocity and temperature profiles
are modeled in the air gap of a high-speed test machine. Friction and heat transfer coefficients are presented in a large
velocity range. The goals are reached acceptably using numerical and experimental research. The velocity field by the numerical
method does not match in every respect the estimated flow mode. The absence of secondary Taylor vortices is evident when using
time averaged numerical simulation.
Aluminum multi-port microchannels are currently utilized in automotive air conditioners for refrigerant condensation. In general, research activities are directed towards developing other air conditioning and refrigeration systems with microchannel condensers and evaporators by analyzing heat exchanger performance parameters of heat transfer, pressure drop, and void fraction. The purpose of this particular study is the experimental investigation of frictional pressure drop and void fraction in microchannels. Experiments are performed using 6-port and 14-port microchannels with hydraulic diameters of 1.54 mm and 1.02 mm, respectively. Fluids and saturation temperatures used in this experimentation include carbon dioxide at 15ºC, ammonia at 35ºC, and R245fa at 40ºC. Two-phase flow conditions include mass fluxes from 50 to 440 kg/s.m2 and qualities varying from 0 to 1. Experiments indicate that two-phase pressure drop and void fraction are dependent upon hydraulic diameter, mass flux, quality, and vapor density. Two-phase pressure drop is most significantly influenced by inertial force of the flow. Void fraction is strongly influenced by flow regime, which can be related to the vapor density of the refrigerant. In general, models exist to predict the experimental two-phase flow pressure drop and void fraction satisfactorily for specific conditions, but no comprehensive model has been formulated that encompasses the physical properties defining two-phase flow. Air Conditioning and Refrigeration Project 131
Inertial particles suspended in many natural and industrial flows undergo coagulation upon collisions and fragmentation if their size becomes too large or if they experience large shear. Here we study this coagulation-fragmentation process in time-periodic incompressible flows. We find that this process approaches an asymptotic dynamical steady state where the average number of particles of each size is roughly constant. We compare the steady-state size distributions corresponding to two fragmentation mechanisms and for different flows and find that the steady state is mostly independent of the coagulation process. While collision rates determine the transient behavior, fragmentation determines the steady state. For example, for fragmentation due to shear, flows that have very different local particle concentrations can result in similar particle size distributions if the temporal or spatial variation in shear forces is similar.
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