Improved Modeling of Turbulent Transport

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ISBN 13 :
Total Pages : 238 pages
Book Rating : 4.:/5 (644 download)

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Book Synopsis Improved Modeling of Turbulent Transport by : Debasmita Samaddar

Download or read book Improved Modeling of Turbulent Transport written by Debasmita Samaddar and published by . This book was released on 2010 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Turbulence and turbulent transport are ubiquitous in nature and are of fundamental importance in everything from the spread of pollution to confinement in fusion plasmas. In order to study this, turbulence models need to be as realistic as possible and one must also be able to evolve the turbulence and the profiles of the quantities of interest on transport (long) time scales. Improving turbulence simulations by the introduction of new techniques forms the basis of this research. One part of this work involved improving the performance of a ID transport model by the addition of noise. On a more fundamental level, studying long time dynamics for turbulence simulations is very difficult even with the fastest computers available now or in the near future. To help overcome this difficulty, a new way of simulating turbulence has been presented, namely parallelizing in time. Time parallelization of a fully developed turbulent system is a new application. parallelizing the space domain to computationally solve partial differential equations has been extensively used and is one of the most common forms of parallelization. In contrast, the Parareal Algorithm parallelizes the time domain and has been found to significantly reduce the computational wall time in many simpler systems. Despite its success in other less complex problems, it has not yet been successfully applied to a turbulent system (to the best of our knowledge). If efficiently applied, this algorithm will allow study of the turbulent transport dynamics on transport time scales - something that has heretofore been very difficult. In this work, the results of applying the Parareal Algorithm to simulations of drift wave turbulence in slab geometry in which the relative dominance of the polarization and E x B nonlinearities are tuned artificially, are presented. These turbulent systems are in many ways similar to neutral fluid turbulence models, so success of the Parareal scheme in them expands the prospect of a broader range of application to many other turbulent problems. This thesis also presents the results of a modification to the algorithm. A model to study and predict the parameters governing the convergence of the scheme is also explored"--Leaf iii.

Research Directions in Computational Mechanics

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Publisher : National Academies Press
ISBN 13 : 0309046483
Total Pages : 145 pages
Book Rating : 4.3/5 (9 download)

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Book Synopsis Research Directions in Computational Mechanics by : National Research Council

Download or read book Research Directions in Computational Mechanics written by National Research Council and published by National Academies Press. This book was released on 1991-02-01 with total page 145 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational mechanics is a scientific discipline that marries physics, computers, and mathematics to emulate natural physical phenomena. It is a technology that allows scientists to study and predict the performance of various productsâ€"important for research and development in the industrialized world. This book describes current trends and future research directions in computational mechanics in areas where gaps exist in current knowledge and where major advances are crucial to continued technological developments in the United States.

Advances in Quasilinear Gyrokinetic Modeling of Turbulent Transport

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ISBN 13 :
Total Pages : 312 pages
Book Rating : 4.:/5 (124 download)

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Book Synopsis Advances in Quasilinear Gyrokinetic Modeling of Turbulent Transport by : Cole Darin Stephens

Download or read book Advances in Quasilinear Gyrokinetic Modeling of Turbulent Transport written by Cole Darin Stephens and published by . This book was released on 2021 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: The quest to harness fusion energy requires the successful modeling of plasma turbulence and transport in magnetic confinement devices. For such modeling, the requisite length and time scales span many orders of magnitude. Integrated modeling approaches are constructed to account for the wide range of physics involved in turbulent transport by coupling separate physical models together. The primary physical models used in this work are kinetic and designed to simulate microturbulence on the smallest scales associated with turbulent transport. However, high precision nonlinear kinetic simulations often cannot be easily coupled to integrated modeling suites due to the extreme computational costs that would be involved. Model reduction which drastically reduces the computational complexity of the problem is therefore necessary. One must of course ensure that the reduced model does not severely diminish the accuracy of the calculation; the model reduction itself must be founded on more exact computational approaches as well as fundamental theoretical principles. One of the most successful approaches in model reduction is quasilinear gyrokinetics. There are two fundamental assumptions for the quasilinear model examined in this work. First, the three adiabatic invariants (the magnetic moment, the longitudinal invariant, and the poloidal flux) must be appropriately conserved and their associated single charged particle motions (the gyromotion, the bounce-transit motion, and the toroidal drift motion) must be characterized accurately. Second, the quasilinear approximation must hold such that the coherent linear response is adequate enough to compute the quasilinear fluxes without full calculation of the nonlinear physics. The particular model used, QuaLiKiz, has been proven successful in reproducing local gyrokinetic fluxes in the tokamak core while remaining computationally tractable. There are three primary goals of this dissertation project. The first is to examine the fundamental physics underlying gyrokinetic and reduced model approaches at the single charged particle scale. To achieve this goal, we examine the assumption of magnetic moment invariance in a wide variety of electromagnetic fields. We successfully identify the dimensionless parameters that determine magnetic moment conservation in each scenario and then proceed to quantify the degree to which magnetic moment conservation is broken. In doing so, we confirm that the magnetic moment is sufficiently conserved for a wide range of regimes relevant to tokamak plasmas. In addition, we derive new analytic formulas for quantities associated with bounce-transit motion in circular tokamak fields. We compare these new, more exact calculations to approximations commonly used in reduced models (including QuaLiKiz) and determine the conditions such that the approximations break down. We then also confirm that the approximations are valid in the tokamak core for conventional, large aspect ratio devices. The second goal of this dissertation project is to rederive and compile the model equations for QuaLiKiz from first principles. Over the years of QuaLiKiz's development, there has never been a complete manuscript that sketches the derivation of QuaLiKiz from start to finish. The lack of such a document makes it difficult to extend the physics of QuaLiKiz to new parameter regimes of interest. Various possible extensions such as including electromagnetic effects or more realistic tokamak geometries require the adjustment of several different assumptions that would affect the derivation in key ways. As such, correct implementations of new physics would require an existing derivation as a reference point lest the implementation be handled in an incoherent fashion. In addition, a step-by-step outline of how each assumption of QuaLiKiz affects the derivation can be helpful in determining which assumptions can be relaxed for a more accurate model. The successful completion of this derivation, included in this dissertation, will be immensely useful for future QuaLiKiz improvement and validation. With the derivation in hand, we proceed to the third goal of this project: improving the collisional model of QuaLiKiz. Collisions play an essential role in characterizing the transport associated with trapped electron modes. It has become evident in recent studies that the collisional model in QuaLiKiz requires improvement; in integrated modeling, the imprecise treatment of collisional trapped electron modes leads to incorrect density profile predictions near the tokamak core for highly collisional regimes. We revisit the collision model implemented in QuaLiKiz and use the more exact gyrokinetic code GENE (Gyrokinetic Electromagnetic Numerical Experiment) to make improvements to QuaLiKiz's collision operator. We then use the new version of QuaLiKiz in integrated modeling to compare density profiles predicted by the old and new collision operators. We confirm that the new collision operator leads to density profiles that more accurately match the experimental profiles.

Turbulent Combustion Modeling

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Publisher : Springer Science & Business Media
ISBN 13 : 9400704127
Total Pages : 496 pages
Book Rating : 4.4/5 (7 download)

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Book Synopsis Turbulent Combustion Modeling by : Tarek Echekki

Download or read book Turbulent Combustion Modeling written by Tarek Echekki and published by Springer Science & Business Media. This book was released on 2010-12-25 with total page 496 pages. Available in PDF, EPUB and Kindle. Book excerpt: Turbulent combustion sits at the interface of two important nonlinear, multiscale phenomena: chemistry and turbulence. Its study is extremely timely in view of the need to develop new combustion technologies in order to address challenges associated with climate change, energy source uncertainty, and air pollution. Despite the fact that modeling of turbulent combustion is a subject that has been researched for a number of years, its complexity implies that key issues are still eluding, and a theoretical description that is accurate enough to make turbulent combustion models rigorous and quantitative for industrial use is still lacking. In this book, prominent experts review most of the available approaches in modeling turbulent combustion, with particular focus on the exploding increase in computational resources that has allowed the simulation of increasingly detailed phenomena. The relevant algorithms are presented, the theoretical methods are explained, and various application examples are given. The book is intended for a relatively broad audience, including seasoned researchers and graduate students in engineering, applied mathematics and computational science, engine designers and computational fluid dynamics (CFD) practitioners, scientists at funding agencies, and anyone wishing to understand the state-of-the-art and the future directions of this scientifically challenging and practically important field.

Advanced Approaches in Turbulence

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Publisher : Elsevier
ISBN 13 : 0128208902
Total Pages : 554 pages
Book Rating : 4.1/5 (282 download)

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Book Synopsis Advanced Approaches in Turbulence by : Paul Durbin

Download or read book Advanced Approaches in Turbulence written by Paul Durbin and published by Elsevier. This book was released on 2021-07-24 with total page 554 pages. Available in PDF, EPUB and Kindle. Book excerpt: Advanced Approaches in Turbulence: Theory, Modeling, Simulation and Data Analysis for Turbulent Flows focuses on the updated theory, simulation and data analysis of turbulence dealing mainly with turbulence modeling instead of the physics of turbulence. Beginning with the basics of turbulence, the book discusses closure modeling, direct simulation, large eddy simulation and hybrid simulation. The book also covers the entire spectrum of turbulence models for both single-phase and multi-phase flows, as well as turbulence in compressible flow. Turbulence modeling is very extensive and continuously updated with new achievements and improvements of the models. Modern advances in computer speed offer the potential for elaborate numerical analysis of turbulent fluid flow while advances in instrumentation are creating large amounts of data. This book covers these topics in great detail. Covers the fundamentals of turbulence updated with recent developments Focuses on hybrid methods such as DES and wall-modeled LES Gives an updated treatment of numerical simulation and data analysis

Modeling of Turbulent Transport in the Surface Layer

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Publisher :
ISBN 13 :
Total Pages : 44 pages
Book Rating : 4.:/5 (31 download)

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Book Synopsis Modeling of Turbulent Transport in the Surface Layer by : George Louis Smith

Download or read book Modeling of Turbulent Transport in the Surface Layer written by George Louis Smith and published by . This book was released on 1973 with total page 44 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Turbulent Transport Modelling of Separating and Reattaching Shear Flows

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Publisher : Createspace Independent Publishing Platform
ISBN 13 : 9781723418426
Total Pages : 24 pages
Book Rating : 4.4/5 (184 download)

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Book Synopsis Turbulent Transport Modelling of Separating and Reattaching Shear Flows by : National Aeronautics and Space Administration (NASA)

Download or read book Turbulent Transport Modelling of Separating and Reattaching Shear Flows written by National Aeronautics and Space Administration (NASA) and published by Createspace Independent Publishing Platform. This book was released on 2018-07-20 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt: The improvement of capabilities for computer simulation of turbulent recirculating flows was investigated. Attention has been limited to two dimensional flows and principally to statistically stationary motion. Improvement of turbulence modeling explored the treatment of the near wall sublayer and of the exterior fully turbulent region, working within the framework of turbulence closures requiring the solution of transport equations for the turbulence energy and its dissipation rate. The work on the numerical procedure, based on the Gosman-Pun program TEACH, addressed the problems of incorporating the turbulence model as well as the extension to time dependent flows, the incorporation of a third order approximation of convective transport, and the treatment of non-orthogonal boundaries. Launder, B. E. Unspecified Center NASA-CR-169368, NAS 1.26:169368 NSG-2256

Improved Modeling of Finite-Rate Turbulent Combustion Processes in Research Combustors

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Publisher : Createspace Independent Publishing Platform
ISBN 13 : 9781724245670
Total Pages : 290 pages
Book Rating : 4.2/5 (456 download)

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Book Synopsis Improved Modeling of Finite-Rate Turbulent Combustion Processes in Research Combustors by : National Aeronautics and Space Administration (NASA)

Download or read book Improved Modeling of Finite-Rate Turbulent Combustion Processes in Research Combustors written by National Aeronautics and Space Administration (NASA) and published by Createspace Independent Publishing Platform. This book was released on 2018-07-25 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: The objective of this thesis is to further develop and test a stochastic model of turbulent combustion in recirculating flows. There is a requirement to increase the accuracy of multi-dimensional combustion predictions. As turbulence affects reaction rates, this interaction must be more accurately evaluated. In this work a more physically correct way of handling the interaction of turbulence on combustion is further developed and tested. As turbulence involves randomness, stochastic modeling is used. Averaged values such as temperature and species concentration are found by integrating the probability density function (pdf) over the range of the scalar. The model in this work does not assume the pdf type, but solves for the evolution of the pdf using the Monte Carlo solution technique. The model is further developed by including a more robust reaction solver, by using accurate thermodynamics and by more accurate transport elements. The stochastic method is used with Semi-Implicit Method for Pressure-Linked Equations. The SIMPLE method is used to solve for velocity, pressure, turbulent kinetic energy and dissipation. The pdf solver solves for temperature and species concentration. Thus, the method is partially familiar to combustor engineers. The method is compared to benchmark experimental data and baseline calculations. The baseline method was tested on isothermal flows, evaporating sprays and combusting sprays. Pdf and baseline predictions were performed for three diffusion flames and one premixed flame. The pdf method predicted lower combustion rates than the baseline method in agreement with the data, except for the premixed flame. The baseline and stochastic predictions bounded the experimental data for the premixed flame. The use of a continuous mixing model or relax to mean mixing model had little effect on the prediction of average temperature. Two grids were used in a hydrogen diffusion flame simulation. Grid density did not effect the predictions except fo...

An Improved [k-epsilon] Model for Near-wall Turbulence and Comparison with Direct Numerical Simulation

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Publisher :
ISBN 13 :
Total Pages : 26 pages
Book Rating : 4.:/5 (31 download)

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Book Synopsis An Improved [k-epsilon] Model for Near-wall Turbulence and Comparison with Direct Numerical Simulation by : T. H. Shih

Download or read book An Improved [k-epsilon] Model for Near-wall Turbulence and Comparison with Direct Numerical Simulation written by T. H. Shih and published by . This book was released on 1990 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt:

An Improved K-epsilon Model for Near-wall Turbulence and Comparison with Direct Numerical Simulation

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Publisher :
ISBN 13 :
Total Pages : 26 pages
Book Rating : 4.:/5 (317 download)

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Book Synopsis An Improved K-epsilon Model for Near-wall Turbulence and Comparison with Direct Numerical Simulation by :

Download or read book An Improved K-epsilon Model for Near-wall Turbulence and Comparison with Direct Numerical Simulation written by and published by . This book was released on 1990 with total page 26 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Improved Modeling of Finite-rate Turbulent Combustion Processes in Research Combustors

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Publisher :
ISBN 13 :
Total Pages : 576 pages
Book Rating : 4.:/5 (319 download)

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Book Synopsis Improved Modeling of Finite-rate Turbulent Combustion Processes in Research Combustors by : Thomas Julius VanOverbeke

Download or read book Improved Modeling of Finite-rate Turbulent Combustion Processes in Research Combustors written by Thomas Julius VanOverbeke and published by . This book was released on 1998 with total page 576 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Subgrid-scale Turbulence Modeling for Improved Large-eddy Simulation of the Atmospheric Boundary Layer

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ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (839 download)

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Book Synopsis Subgrid-scale Turbulence Modeling for Improved Large-eddy Simulation of the Atmospheric Boundary Layer by : Rica Mae Enriquez

Download or read book Subgrid-scale Turbulence Modeling for Improved Large-eddy Simulation of the Atmospheric Boundary Layer written by Rica Mae Enriquez and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Large-eddy simulation (LES), as the name suggests, resolves the large eddies in the flow while modeling the effects of smaller motions (turbulence) on those larger eddies. Powerful computers make LES increasingly practical for analyzing a variety of atmospheric behavior in more detail, creating a need for more realistic turbulence models. Advances in describing atmospheric turbulence can impact many disciplines, e.g., weather and climate prediction, wind energy production, ocean dynamics, and, indeed, even computational fluid dynamics itself. Although the turbulence model can significantly affect the accuracy of the LES, simple turbulence models, which are known to be less accurate, are widely used. As an alternative, the Generalized Linear Algebraic Subgrid-Scale (GLASS) model, that actively couples momentum and heat transport, was developed. This model is more complete than conventional LES turbulence models because it accounts for additional transport processes. GLASS includes production, dissipation, pressure redistribution, and buoyancy terms. With the inclusion of an actively coupled turbulent heat flux model, GLASS is applicable to a range of atmospheric stability conditions for the unsaturated atmosphere. LES at various resolutions in a neutrally stratified boundary layer flow indicated that the GLASS model is a more physically complete subgrid-scale turbulence model that provides near-wall anisotropies and yields proper velocity profiles in the logarithmic layer. LES of the moderately convective boundary layer demonstrated that GLASS predicted the evolution of resolved quantities at least as well as the LESs with simple models, while including additional physics. Additional simulations of the stable boundary layer and the transitioning boundary layer highlight that GLASS can be applied to various stability conditions without the need of tuning model coefficients.

Fundamentals Of Turbulence Modelling

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Publisher : CRC Press
ISBN 13 : 9781560324058
Total Pages : 312 pages
Book Rating : 4.3/5 (24 download)

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Book Synopsis Fundamentals Of Turbulence Modelling by : Ching Jen Chen

Download or read book Fundamentals Of Turbulence Modelling written by Ching Jen Chen and published by CRC Press. This book was released on 1997-12-01 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: Focuses on the second-order turbulence-closure model and its applications to engineering problems. Topics include turbulent motion and the averaging process, near-wall turbulence, applications of turbulence models, and turbulent buoyant flows.

Hydrodynamic and Magnetohydrodynamic Turbulent Flows

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Publisher : Springer Science & Business Media
ISBN 13 : 9401718105
Total Pages : 426 pages
Book Rating : 4.4/5 (17 download)

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Book Synopsis Hydrodynamic and Magnetohydrodynamic Turbulent Flows by : A. Yoshizawa

Download or read book Hydrodynamic and Magnetohydrodynamic Turbulent Flows written by A. Yoshizawa and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 426 pages. Available in PDF, EPUB and Kindle. Book excerpt: TUrbulence modeling encounters mixed evaluation concerning its impor tance. In engineering flow, the Reynolds number is often very high, and the direct numerical simulation (DNS) based on the resolution of all spatial scales in a flow is beyond the capability of a computer available at present and in the foreseeable near future. The spatial scale of energetic parts of a turbulent flow is much larger than the energy dissipative counterpart, and they have large influence on the transport processes of momentum, heat, matters, etc. The primary subject of turbulence modeling is the proper es timate of these transport processes on the basis of a bold approximation to the energy-dissipation one. In the engineering community, the turbulence modeling is highly evaluated as a mathematical tool indispensable for the analysis of real-world turbulent flow. In the physics community, attention is paid to the study of small-scale components of turbulent flow linked with the energy-dissipation process, and much less interest is shown in the foregoing transport processes in real-world flow. This research tendency is closely related to the general belief that universal properties of turbulence can be found in small-scale phenomena. Such a study has really contributed much to the construction of statistical theoretical approaches to turbulence. The estrangement between the physics community and the turbulence modeling is further enhanced by the fact that the latter is founded on a weak theoretical basis, compared with the study of small-scale turbulence.

Numerical Modeling of Turbulent Combustion and Flame Spread

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Publisher :
ISBN 13 :
Total Pages : 60 pages
Book Rating : 4.:/5 (186 download)

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Book Synopsis Numerical Modeling of Turbulent Combustion and Flame Spread by : Zhenghua Yan

Download or read book Numerical Modeling of Turbulent Combustion and Flame Spread written by Zhenghua Yan and published by . This book was released on 1999 with total page 60 pages. Available in PDF, EPUB and Kindle. Book excerpt: Theoretical models have been developed to address several important aspects of numerical modeling of turbulent combustion and flame spread. The developed models include a pyrolysis model for charring and non-charring solid materials, a fast narrow band radiation property evaluation model (FASTNB) and a turbulence model for buoyant flow and flame. In the pyrolysis model, a completely new algorithm has been proposed, where a moving dual mesh concept was developed and implemented. With this new concept, it provides proper spatial resolution for both temperature and density and automatically considers the regression of the surface of the non-charring solid material during its pyrolysis. It is simple, very efficient and applicable to both charring and non-charring materials. FASTNB speeds up significantly the evaluation of narrow band spectral radiation properties and thus provides a potential of applying narrow band model in numerical simulations of practical turbulent combustion. The turbulence model was developed to improve the consideration of buoyancy effect on turbulence and turbulent transport. It was found to be simple, promising and numerically stable. It has been tested against both plane and axisymmetric thermal plumes and an axisymmetric buoyant diffusion flame. When compared with the widely used standard buoyancy-modified k-e model, it gives significant improvement on numerical results. These developed models have been fully incorporated into CFD (Computational Fluid Dynamics) code and coupled with other CFD sub-models, including the DT (Discrete Transfer) radiation model, EDC (Eddy Dissipation Concept) combustion model, flamelet combustion model, various soot models and transpired wall function. Comprehensive numerical simulations have been carried out to study soot formation and oxidation in turbulent buoyant diffusion flames, flame heat transfer and flame spread in fires. The gas temperature and velocity, soot volume fraction, wall surface temperature, char depth, radiation and convection heat fluxes, and heat release rate were calculated and compared with experimental measurements. In addition to provide comprehensive data for comparison, experiments on room corner fire growth were undertaken, where the gas temperature, solid fuel surface temperature, radiative heat flux, char depth and heat release rate were all measured.

Engineering Turbulence Modelling and Experiments - 4

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Publisher : Elsevier
ISBN 13 : 0080530982
Total Pages : 975 pages
Book Rating : 4.0/5 (85 download)

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Book Synopsis Engineering Turbulence Modelling and Experiments - 4 by : D. Laurence

Download or read book Engineering Turbulence Modelling and Experiments - 4 written by D. Laurence and published by Elsevier. This book was released on 1999-04-14 with total page 975 pages. Available in PDF, EPUB and Kindle. Book excerpt: These proceedings contain the papers presented at the 4th International Symposium on Engineering Turbulence Modelling and Measurements held at Ajaccio, Corsica, France from 24-26 May 1999. It follows three previous conferences on the topic of engineering turbulence modelling and measurements. The purpose of this series of symposia is to provide a forum for presenting and discussing new developments in the area of turbulence modelling and measurements, with particular emphasis on engineering-related problems. Turbulence is still one of the key issues in tackling engineering flow problems. As powerful computers and accurate numerical methods are now available for solving the flow equations, and since engineering applications nearly always involve turbulence effects, the reliability of CFD analysis depends more and more on the performance of the turbulence models. Successful simulation of turbulence requires the understanding of the complex physical phenomena involved and suitable models for describing the turbulent momentum, heat and mass transfer. For the understanding of turbulence phenomena, experiments are indispensable, but they are equally important for providing data for the development and testing of turbulence models and hence for CFD software validation.

Turbulent Flows

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Publisher : Springer Science & Business Media
ISBN 13 : 3662035596
Total Pages : 767 pages
Book Rating : 4.6/5 (62 download)

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Book Synopsis Turbulent Flows by : Jean Piquet

Download or read book Turbulent Flows written by Jean Piquet and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 767 pages. Available in PDF, EPUB and Kindle. Book excerpt: obtained are still severely limited to low Reynolds numbers (about only one decade better than direct numerical simulations), and the interpretation of such calculations for complex, curved geometries is still unclear. It is evident that a lot of work (and a very significant increase in available computing power) is required before such methods can be adopted in daily's engineering practice. I hope to l"Cport on all these topics in a near future. The book is divided into six chapters, each· chapter in subchapters, sections and subsections. The first part is introduced by Chapter 1 which summarizes the equations of fluid mechanies, it is developed in C~apters 2 to 4 devoted to the construction of turbulence models. What has been called "engineering methods" is considered in Chapter 2 where the Reynolds averaged equations al"C established and the closure problem studied (§1-3). A first detailed study of homogeneous turbulent flows follows (§4). It includes a review of available experimental data and their modeling. The eddy viscosity concept is analyzed in §5 with the l"Csulting ~alar-transport equation models such as the famous K-e model. Reynolds stl"Css models (Chapter 4) require a preliminary consideration of two-point turbulence concepts which are developed in Chapter 3 devoted to homogeneous turbulence. We review the two-point moments of velocity fields and their spectral transforms (§ 1), their general dynamics (§2) with the particular case of homogeneous, isotropie turbulence (§3) whel"C the so-called Kolmogorov's assumptions are discussed at length.