Dissertations / Theses on the topic 'Second moment closure model'

The present thesis deals with the successful introduction of a second-moment closure turbulence model into a computer program using the Finite Element Method to solve the Navier-Stokes equations. The implementation presented has the advantage of using an equal interpolation for all the variables. It is also very economical in terms of the amount of memory required from the computer, since a fully decoupled formulation has been adopted, along with an iterative solver which permits to store in memory only the non-zero coefficients of the linear system of equations to be solved. Specialized elements are used to avoid resolving the near-wall region of the flow. The apparent viscosity concept is derived for the finite element formulation, along with a correction factor which permits a better representation of the Reynolds stresses. The RSM is compared to the older $k - epsilon$ model in two test cases where experimental data was available. The conclusion drawn from this work is that the RSM is able to reproduce more phenomenon occurring in turbulent flows than the $k - epsilon$ model. It is thought that the $k - epsilon$ model will gradually be supplanted by more complex models, as more computing power become available.

In this work, a three-dimensional turbulent boundary layer experiment was set up with alternating stream-wise and span-wise pressure gradients. The pressure gradients are generated as a result of the test section wavy side wall shape. Each side had six sine waves with a trough to peak magnitude to wavelength ratio of 0.25. Boundary layer control was used so that the flow over the side walls remains attached. The mean flow velocity components, static and total pressures were measured at six plane along the stream-wise direction. The alternating mean span-wise and stream-wise pressure gradients created alternating stream-wise and span-wise vorticity fluxes, respectively, along the test section. As the flow developed downstream the vorticity created at the tunnel floor and ceiling diffused away from the wall. The vorticity components in the stream-wise and span-wise directions are strengthened due to stretching and tilting terms in the vorticity transport equations. The positive-z half of the test section contains large areas that generate positive vorticity flux in the trough region and smaller areas generating negative vorticity around the wave peak. The opposite is true for the negative-z half of the test-section. This results in a large positive stream-wise vorticity in the positive-z half and negative stream-wise vorticity in the negative-z half of the test-section. The smaller regions of opposite sign vorticity in each half tend to mix the flow such that as they diffuse away from the wall, the turbulent stresses are more uniform. Turbulent fluctuating velocity components were measured using Laser Doppler Velocimetery. Mean velocities as well as Reynolds stresses and triple velocity component correlations were measured at thirty stations along the last wave in the test section. Profiles at the center of the test section showed three dimensionality, but exhibited high turbulence intensities in the outer layer. Profiles off the test section centerline are highly three dimensional with multiple peaks in the normal stress profiles. The flow also reaches a state where all the normal stresses have equal magnitudes while the shear stresses are non-zero. Flow angles, flow gradient angles and shear stress angles show very large differences between wall values and outer layer vlaues. The shear stress angle lagged the flow gradient angle indicating non-equilibrium. A turbulent kinetic energy transport budget is performed for all profiles and the turbulence kinetic energy dissipation rate is estimated. Spectral measurements were also made and an independent estimate of the kinetic energy dissipation rate is made. These estimates agree very well with those estimates made by balancing the turbulence kinetic energy transport equation. Multiple turbulent diffusion models are compared to measured quantities. The models varied in agreement with experimental data. However, fair agreement with turbulence kinetic energy turbulent diffusion is observed. A model for the dissipation rate tensor anisotropy is used to extract estimates of the pressure-strain tensor from the Reynolds stress transport equations. The pressure-strain estimates are compared with some of the models in the literature. The comparison showed poor agreement with estimated pressure-strain values extracted from experimental data. A tentative model for the turbulent Reynolds shear stress angle is developed that captures the shear stress angle near wall behavior to a very good extent. The model contains one constant that is related to mean flow variables. However, the developed expression needs modification so that the prediction is improved along the entire boundary layer thickness.
Ph. D.

In this work the predictive capability of a number of Reynolds stress transport(RST) models was first tested in a range of non-equilibrium homogeneous flows, comparisons being drawn with existing direct numerical simulation (DNS) results and physical measurements. The cases considered include both shear and normally strained flows, in some cases with a constant applied strain rate, and in others where this varied with time. Models were generally found to perform well in homogeneous shear at low shear rates, but their performance increasingly deteriorated at higher shear rates. This was attributed mainly to weaknesses in the pressure-strain rate models, leading to over-prediction of the shear stress component of the stress anisotropy tensor at high shear rates. Performance in irrotational homogeneous strains was generally good, and was more consistent over a much wider range of strain rates. In the experimental plane strain and axisymmetric contraction cases, with time-varying strain rates, there was evidence of an accelerated dissipation rate generation. Significant improvement was achieved through the use of an alternative dissipation rate generation term, Pε , in these cases, suggesting a possible route for future modelling investigation. Subsequently, the models were also tested in the inhomogeneous case of pulsating channel flow over a wide range of frequencies, the reference for these cases being the LES of Scotti and Piomelli (2001). A particularly challenging feature in this problem set was the partial laminarisation and re-transition that occurred cyclically at low and, to a lesser extent, intermediate frequencies. None of the models tested were able to reproduce correctly all of the observed flow features, and none returned consistently superior results in all the cases examined. Finally, models were tested in the case of a plane jet interacting with a rectangular dead-end enclosure. Two geometric configurations are examined, corresponding a steady regime, and an intrinsically unsteady regime in which periodic flow oscillations are experimentally observed (Mataoui et al., 2003). In the steady case generally similar flow patterns were returned by the models tested, with some differences arising in the degree of downward deflection of the impinging jet, which in turn affected the level of turbulence energy developing in the lower part of the cavity. In the unsteady case, only two of the models tested, a two-equation k-ε model and an advanced RST model, correctly returned purely periodic solutions. The other two RST models, based on linear pressure-strain rate terms, returned unsteady flow patterns that exhibited complex oscillations with significant cycle-to-cycle variations. Unfortunately, the limited availability of reliable experimental data did not allow a detailed quantitative examination of model performance.

Cette thèse propose une modélisation avancée des transferts thermiques dans les écoulements turbulents pour tous les types de conditions aux limites sur la température aux parois. Ces travaux reposent sur un double constat : d'une part, les modèles de turbulence traitant la thermique de l'écoulement dans la plupart des applications industrielles sont basés sur de simples relations algébriques incapables de représenter des physiques complexes, comme la convection naturelle et la thermique de la zone proche-paroi. D'autre part, la condition aux limites sur la température à la paroi (température fixée, flux de chaleur imposé ou transfert thermique conjugué) influence le comportement proche-paroi des variables thermiques turbulents. La formulation du modèle bas-Reynolds du second ordre des flux thermiques turbulents EBDFM (Elliptic Blending Differential Flux Model), développée à l'origine pour traiter des cas où une température est fixée à la paroi, a été étendue à des cas de flux de chaleur imposé et de transfert thermique conjugué. Cette nouvelle formulation se fonde sur des analyses asymptotiques rigoureuses des termes des équations de transport des flux thermiques turbulents pour chaque condition aux limites sur la température. Un des éléments essentiels de la nouvelle formulation de l'EBDFM est le ratio des échelles de temps thermique et dynamique R. Le comportement asymptotique de ce ratio dépend fortement de la condition aux limites : R tend vers le nombre de Prandtl à la paroi lorsqu'une température est imposée, et vers l'infini sinon. Ainsi, dans le but de reproduire fidèlement ce comportement, il s'est avéré nécessaire de résoudre des équations de transport pour la variance de température ¯(θ^'2 )et pour son taux de dissipation ε_θ puisque ces deux variables pilotent le comportement asymptotique de R. Par conséquent, cette thèse propose des modèles bas-Reynolds pour les variables ¯(θ^'2 )et ε_θ valables pour toutes les conditions aux limites thermiques. La nouvelle formulation du modèle EBDFM ainsi que les modèles de ¯(θ^'2 )et ε_θ ont été validées par des simulations réalisées avec le logiciel de CFD Code_Saturne pour des écoulements dans un canal plan en convection forcée
Advanced modeling of turbulent heat transfer for all thermal boundary conditions is proposed. This work was motivated by two facts: first, the thermal turbulent models used in most of the industrial computations are based on eddy-viscosity models which cannot deal with complex physics such as natural convection or heat transfer in the near-wall region. Then, the thermal boundary condition at the wall (imposed temperature, imposed heat flux, conjugate heat transfer) influences the near-wall behavior of the turbulent thermal variables. The formulation of the low-Reynolds number second moment closure EBDFM (Elliptic Blending Differential Flux Model), which was originally developed for an imposed temperature at the wall, has been extended to an imposed heat flux and a conjugate heat transfer condition. This new formulation is based on rigorous asymptotic analysis of the terms of the transport equation of the turbulent heat flux for all thermal boundary conditions. One of the key elements is the thermal-to-mechanical time-scale ratio R. Its asymptotic behavior highly depends on the thermal boundary condition: R goes to the Prandtl number at the wall for an imposed temperature and tends to infinity otherwise. Thus, solving a transport equation for the temperature variance ¯(θ^'2 ) and for its dissipation rate ε_θ is necessary to reproduce the asymptotic behavior of R. Indeed, these two variables drive the behavior of R in the near-wall region. Therefore, low-Reynolds number models for ¯(θ^'2 ) and ε_θ, valid for all thermal boundary conditions, are proposed. The new formulation of the EBDFM and the models for ¯(θ^'2 ) and ε_θ have been validated by performing Code_Saturne computations of channel flows in the forced convection regime

Donaldson's single-point second-order model [13] is used to close the Reynolds stress transport equations in cylindrical coordinates. A reduced set of equations are then solved for the decay of axisymmetric vortices and jets. A self-similar solution to the axisymmetric vortices is obtained numerically. The characteristics of the mean flow variables as well as the Reynolds stresses in this solution are discussed. Comparisons of the current results with Donaldson[13J and Donaldson and Sullivan[16] are also presented.

The results show that the vortex core is free from turbulent shear stresses. The turbulent kinetic energy is also found to be relatively weak within the core region. The overshoot of the circulation is found to be 5% of the circulation at infinity over a wide range of Reynolds numbers.

The effects of Reynolds number on the decay of the vortices are computed and discussed. Some of the quantities, such as mean flow circulation and turbulent kinetic energy, are found to be sensitive to the Reynolds number. However, the overshoot is found to be insensitive to the Reynolds number but its location does.

A set of suitable model constants for the axisymmetric jets is also found and a self similar solution for the jet case is obtained. Comparisons of the computed results with some recent experimental data are presented.
Master of Science

Dans certains écosystèmes et plus particulièrement dans certaines forêts tropicales, différentes espèces ayant les mêmes exigences écologiques cohabitent sur un même milieu. Par exemple, certaines forêts présentent plus de cent espèces d’arbres différentes sur un hectare. Pour expliquer cette étonnante diversité, les scientifiques ont construit des modèles dans lesquels la composition de la communauté est uniquement due à la dispersion stochastique des individus. Le modèle mathématique étudié dan thèse s’inscrit dans cette lignée. Il a été suggéré par M.Kalyuzhni dans un article [9] où il justifie sa pertinence. Il est connu sous le nom de modèle de Moran en environnement aléatoire. Il s'agit donc d'étudier un processus de naissance et mort qui tient compte des aléas environnementaux (climats, maladies etc...) qui favorisent ou défavorisent de façon aléatoire certaines espèces. Pour étudier cette dynamique, on s’intéresse à l’approximation par une diffusion du modèle de Moran dans l’ échelle classique où l’accélération en temps est donnée par le carré de la taille de la population, l’avantage sélectif et l'immigration sont inversement proportionnels à cette taille qui tend vers l’infini. L’avantage sélectif varie aléatoirement et est modélisé par un processus markovien de sauts. On étudie la convergence en loi de la suite de processus, et donnons une estimation quantitative de l’erreur commise pour une population donnée. On s’intéresse ensuite à l’estimation des moments des fréquences au sein de la population, motivé en particulier par les indices de biodiversité comme l’indice de Simpson et s’appuyant sur les approximations obtenues. Dans le cas d’une sélection non nulle, l’équation différentielle stochastique régissant un moment fait appel au moment d’ordre supérieure. Pour surmonter cette difficulté, on met en place une méthode de fermeture des équations pour ramener l’ étude des premiers moments à un système d’ équations différentielles finies. Elle nécessite de contrôler l’erreur commise en négligeant les termes de degrés supérieur. Enfin, dans le cas de deux espèces et toujours avec des coefficients constants, on donne une estimation de la vitesse de convergence de la diffusion limite vers la probabilité stationnaire. Dans un second temps, on s'intéresse cette fois à un changement de temps proportionnel à la taille de la population. Ceci conduit à une convergence en loi du processus vers une limite déterministe caractérisée par une équation différentielle ordinaire. Le coefficient évoluant de façon aléatoire, à nouveau suivant un processus markovien de sauts, ce processus est un PDMP.On étudie alors la persistance des différentes espèces et les potentielles coexistences en temps long en s'appuyant un cadre développé par Benaïm et Schreiber : la persistance stochastique. Dans cette partie, on s’intéresse en particulier au cas où toutes les espèces persistent, avec deux environnements seulement : on montre que deux espèces peuvent persister mais pas trois. Avec plus d’environnements, la classification explicite est laissée ouverte mais un exemple de persistance avec trois espèces et trois environnements est donné
In some ecosystems and more particularly in virgin tropical forests, different species having the same ecological requirements coexist in the same environment. For example, some forests have over a hundred different tree species on one hectare. To explain this incrediblediversity, scientists have built models in which the community composition isonly due to the stochastic dispersion of individuals.The mathematical model studied in this thesis follows this line. It was suggested by Mr. Kalyuzhni in an article where he justifies its relevance. It is known as the Moran model in random environment. It is therefore a question of studying a birth and death process taking into account the environmental stochasticity (climates, diseases, etc.) To study this dynamic, we use an approximation by a diffusion, on the classical scale where the acceleration in time is given by the square of the population size, moreover selective advantage and immigration are inversely proportional to thethis size. The selective advantage varies randomly and is modeled by a Markov jump process. We study the convergence in law of the processes sequence and give a quantitative estimate of the error made for a given population. We are then interested in the moments estimation of the population frequencies, motivated in particular by biodiversity indices such as the Simpson's index andbased on the approximations obtained before.In the case of a non-zero selection, the stochastic differential equation governing a moment appeals to the higher order moment. To overcome this difficulty, we create a closure method to reduce the study of the first moments to a finite system of differential equations. We give an estimation of the error made by neglecting the terms of higher degrees. Finally, in the case of two species and with constant coefficients, we give an estimate of the convergence speed of the diffusion towards the stationary measure. In a second time, we are interested in a time scale proportional to the size of the population. This leads to a convergence of the process law towards a deterministic limitcharacterized by an ordinary differential equation. The selection coefficient evolving randomly, still following a Markov jump process, this process is a PDMP.We then study the persistence of the different species and the potential coexistencethanks the persistence theory, developed by Benaïm and Schreiber. In this part, we are particularly interested in the case where all the species persist. With only two environments: we show that two species can persist but not three. With more environments,the explicit classification stay an open problem but an example of persistence with three speciesand three environments is given

The present study is focused on the simulation of turbulent bubbly flows by utilizing the two-fluid model (TFM) in conjunction with advanced near-wall Reynolds-stress models (RSMs) within the Reynolds-averaged Navier-Stokes (RANS) framework. Such anisotropy-resolving turbulence models, employed in combination with the TFM, have been rarely used so far for two-phase flow computations. The presently adopted RSMs are based on the formulations initially proposed by Jakirlic and co-workers for incompressible single-phase flows. Two essentially different RSM versions are selected to be applied in the present work. One model version is formulated within the conventional RANS framework, whereas the second one resembles an instability-sensitized RSM variant, capable of adequately resolving the fluctuating turbulent motions in accordance with the scale-adaptive simulation (SAS) proposal by Menter and Egorov. The necessary modifications of both Reynolds-stress models to be used within the TFM computational framework, also in conjunction with different model formulations accounting for the bubble-induced turbulence, require an appropriate coupling algorithm, which, independent of the geometrical complexity of the flow configurations considered, represents by itself a challenging task. The three reference flow configurations, chosen for the model validation, are the turbulent bubbly flows in a straight and suddenly-expanded vertical pipe over a range of Reynolds numbers and a square cross-sectioned bubble column. In addition, due to sake of comparative evaluation, the available corresponding single-phase flows are investigated by using both RSMs. The presently realized numerical investigations demonstrate a successful employment of both Reynolds-stress models for bubbly flow computations. In all three flow configurations the results obtained with the conventional RSM exhibit a high level of qualitative and quantitative agreement with the available reference data. The novel scale-resolving RSM reveals its high potential, representing a promising approach for further investigations.

Results from numerical simulation were performed to study flow and heat transfer in two types of rotating multi-pass cooling channels. Second moment closure model was used to solve flow in domain generated from Chimera method. The first type was a four-pass channel with two different inlet settings. The main flowing channel was rectangular channel (AR=2:1) with hydraulic diameter (Dh ) equals to 2/3 inch (16.9 mm). The first and fourth channel were set as different aspect ratio (AR=2:1; AR=1:1). Reynolds number (Re) used in this part was 10,000. The rotating angle was set as 90 degrees. The density ratio was set as 0.115. The rotation number varied from 0.0 to 0.22. It was showed that inlet effect only caused influence to flow and heat transfer in first two passages. The second type was a four-pass channel with/without addition of vane in smooth turn portion. The main flowing channel was rectangular channel (AR=2:1) with hydraulic diameter (Dh) equals to 2/3 inch. The first and fourth passages were set to be square duct (AR=1:1). The Reynolds number (Re) used in this part was 20,000. Three rotation numbers were set here (Ro=0.0; Ro=0.2; Ro=0.4). The density ratio and rotating angle varied from 0.12 to 0.32 and from 45 degrees to 90 degrees respectively. According to numerical results, it was revealed that the addition of vane in smooth turn portion did not cause influence to part before it. However, it caused significant influence to flow and heat transfer in smooth turn portion and part after it.

碩士
國立交通大學
機械工程研究所
86
Numerical simulations were appied to a number of turbulent flows, including:(1) flow past a backward-facing step, (2) flow through a sudden-expansion pipe without swirl and (3) various swirling flows, using an eddy-viscosity type k-ε model and Reynolds stress transport model (RSTM). The predicted mean and turbulent results were compared with measurements. 　　For the non-swirling cases, the flow field were well represented by the two models, and the k-ε model predictions showed a slightly higher level of radial diffusive transport across the shear layer in the recirculation zone. In the case of flow past a backward-facing step the RSTM led to a larger reattachment length, but it yielded a smaller reattachment length for the flow through an axisymmetric diffuser. 　　For the swirling cases,both models gave accurate values of the mean flow in regions remote from the central vortex core, the biggest discrepancies between predicticons and measurements occurred along the centreline in which the two models failed to reproduce correctly the strength of the decay of swirl-induced deceleration of the axial velocity. Generally, the RSTM provided good agreement with measured meana velocity profiles. 　　The performance of the two turbulent closure models in the prediction of stress field is also presented. The RSTM predictions were found to be in good agreement with the experimental data, regardless of the flows.

Autoignition of high pressure methane jets at engine relveant conditions within a shock tube is investigated using Conditional Moment Closure (CMC). The impact of two commonly used approximations applied in previous work is examined, the assumption of homogeneous turbulence in the closure of the micro-mixing term and the assumption of negligible radial variation of terms within the CMC equations. In the present work two formulations of an inhomogeneous mixing model are implemented, both utilizing the β -PDF, but differing in the respective conditional velocity closure that is applied. The common linear model for conditional velocity is considered, in addition to the gradient diffusion model. The validity of cross-stream averaging the CMC equations is examined by comparing results from two-dimensional (axial and radial) solution of the CMC equations with cross-stream averaged results. The CMC equations are presented and all terms requiring closure are discussed. So- lution of the CMC equations is decoupled from the flow field solution using the frozen mixing assumption. Detailed chemical kinetics are implemented. The CMC equations are discretized using finite differences and solved using a fractional step method. To maintain consistency between the mixing model and the mixture fraction variance equation, the scalar dissipation rate from both implementations of the inhomogeneous model are scaled. The autoignition results for five air temperatures are compared with results obtained using homogeneous mixing models and experimental data. The gradient diffusion conditional velocity model is shown to produce diverging be- haviour in low probability regions. The corresponding profiles of conditional scalar dis- sipation rate are negatively impacted with the use of the gradient model, as unphysical behaviour at lean mixtures within the core of the fuel jet is observed. The predictions of ignition delay and location from the Inhomogeneous-Linear model are very close to the homogeneous mixing model results. The Inhomogeneous-Gradient model yields longer ig- nition delays and ignition locations further downstream. This is influenced by the higher scalar dissipation rates at lean mixtures resulting from the divergent behaviour of the gradient conditional velocity model. The ignition delays obtained by solving the CMC equations in two dimensions are in excellent agreement with the cross-stream averaged values, but the ignition locations are predicted closer to the injector.

Turbulent dynamics at two sites (C and D) in a hypoxic zone on the Texas- Louisiana continental shelf were studied by investigating turbulence quantities i.e. turbulence kinetic energy (TKE), dissipation rate of TKE (E), Reynolds stress (τ ), dissipation rate of temperature variance (χ), eddy diffusivity of temperature (ν't), and eddy diffusivity of density (ν'p). Numerical models were also applied to test their capability of simulating these turbulence quantities. At site D, TKE, E, and τ were calculated from velocity measurements in the bot- tom boundary layer (BBL), using the Kolmogorov’s -5/3 law in the inertial subrange of energy spectra of vertical velocity fluctuations in each burst measurement. Four second-moment turbulence closure models were applied for turbulence simulations, and modeled turbulence quantities were found to be consistent with those observed. It was found from inter-model comparisons that models with the stability functions of Schumann and Gerz predicted higher values of turbulence quantities than those of Cheng in the mid layer, which might be due to that the former stability functions are not sensitive to buoyancy. At site C, χ, E, v’t, and ν’p were calculated from profile measurements throughout the water column, and showed high turbulence level in the surface boundary layer and BBL, as well as in the mid layer where shear stress was induced by advected non-local water above a hypoxic layer. The relatively high dissolved oxygen in the non-local water resulted in upward and downward turbulent oxygen fluxes, and the bottom hypoxia will deform due to turbulence in 7.11 days. Two of the four models in the study at site D were implemented, and results showed that turbulence energy resulting from the non-local water was not well reproduced. We attribute this to the lack of high-resolution velocity measurements for simulations. Model results agreed with observations only for χ and E simulated from the model with the stability function of Cheng in the BBL. Discrepancies between model and observational results lead to the following conclusions: 1) the stability functions of Schumann and Gerz are too simple to represent the turbulent dynamics in stratified mid layers; 2) detailed velocity profiles measurements are required for models to accurately predict turbulence quantities. Missing such observations would result in underestimation,