\\n\\n
More than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\\n\\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\\n\\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\\n\\nAdditionally, each book published by IntechOpen contains original content and research findings.
\\n\\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Simba Information has released its Open Access Book Publishing 2020 - 2024 report and has again identified IntechOpen as the world’s largest Open Access book publisher by title count.
\n\nSimba Information is a leading provider for market intelligence and forecasts in the media and publishing industry. The report, published every year, provides an overview and financial outlook for the global professional e-book publishing market.
\n\nIntechOpen, De Gruyter, and Frontiers are the largest OA book publishers by title count, with IntechOpen coming in at first place with 5,101 OA books published, a good 1,782 titles ahead of the nearest competitor.
\n\nSince the first Open Access Book Publishing report published in 2016, IntechOpen has held the top stop each year.
\n\n\n\nMore than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\n\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\n\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\n\nAdditionally, each book published by IntechOpen contains original content and research findings.
\n\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\n\n\n\n
\n'}],latestNews:[{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"},{slug:"oiv-awards-recognizes-intechopen-s-editors-20201127",title:"OIV Awards Recognizes IntechOpen's Editors"},{slug:"intechopen-joins-crossref-s-initiative-for-open-abstracts-i4oa-to-boost-the-discovery-of-research-20201005",title:"IntechOpen joins Crossref's Initiative for Open Abstracts (I4OA) to Boost the Discovery of Research"},{slug:"intechopen-hits-milestone-5-000-open-access-books-published-20200908",title:"IntechOpen hits milestone: 5,000 Open Access books published!"},{slug:"intechopen-books-hosted-on-the-mathworks-book-program-20200819",title:"IntechOpen Books Hosted on the MathWorks Book Program"}]},book:{item:{type:"book",id:"6304",leadTitle:null,fullTitle:"Forest Fire",title:"Forest Fire",subtitle:null,reviewType:"peer-reviewed",abstract:"This book concerns the different aspects of forest fires, the impact of fire on both forest resources (e.g. forest cover) and communities that use different forest functions. 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These aspects of forest fire are the subject of this book. I realize, however, that the contents in it can only be an incentive for the reader to learn more, in an interesting aspect. I assume that this book will be valuable to researchers as well as students who are interested in different aspects connected to forest fires, not only from the ecological point of view but also from the social one. 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Paltauf",authors:[{id:"36294",title:"Dr.",name:"Peter",middleName:null,surname:"Burgholzer",fullName:"Peter Burgholzer",slug:"peter-burgholzer"},{id:"51111",title:"Dr",name:"Heinz",middleName:null,surname:"Roitner",fullName:"Heinz Roitner",slug:"heinz-roitner"},{id:"51112",title:"Mr.",name:"Johannes",middleName:null,surname:"Bauer-Marschallinger",fullName:"Johannes Bauer-Marschallinger",slug:"johannes-bauer-marschallinger"},{id:"51113",title:"MSc",name:"Hubert",middleName:null,surname:"Grün",fullName:"Hubert Grün",slug:"hubert-grun"},{id:"51114",title:"Dr.",name:"Thomas",middleName:null,surname:"Berer",fullName:"Thomas Berer",slug:"thomas-berer"},{id:"51115",title:"Mr.",name:"Günther",middleName:null,surname:"Paltauf",fullName:"Günther Paltauf",slug:"gunther-paltauf"}]},{id:"23411",title:"Low Frequency Acoustic Devices for Viscoelastic Complex Media Characterization",slug:"low-frequency-acoustic-devices-for-viscoelastic-complex-media-characterization",signatures:"Georges Nassar",authors:[{id:"55634",title:"Prof.",name:"Nassar",middleName:null,surname:"Georges",fullName:"Nassar Georges",slug:"nassar-georges"}]},{id:"23412",title:"Modeling of Biological Interfacial Processes Using Thickness–Shear Mode Sensors",slug:"modeling-of-biological-interfacial-processes-using-thickness-shear-mode-sensors",signatures:"Ertan Ergezen, Johann Desa, Matias Hochman, Robert Weisbein Hart, Qiliang Zhang, Sun Kwoun, Piyush Shah and Ryszard Lec",authors:[{id:"35928",title:"Dr.",name:"Ertan",middleName:null,surname:"Ergezen",fullName:"Ertan Ergezen",slug:"ertan-ergezen"},{id:"50492",title:"Mr",name:"Johann",middleName:null,surname:"Desa",fullName:"Johann Desa",slug:"johann-desa"},{id:"50494",title:"Dr.",name:"Qiliang",middleName:null,surname:"Zhang",fullName:"Qiliang Zhang",slug:"qiliang-zhang"},{id:"50495",title:"Dr.",name:"Sun",middleName:null,surname:"Kwoun",fullName:"Sun Kwoun",slug:"sun-kwoun"},{id:"50496",title:"Mr",name:"Piyush",middleName:null,surname:"Shah",fullName:"Piyush Shah",slug:"piyush-shah"},{id:"50497",title:"MSc",name:"Matias",middleName:"Gabriel",surname:"Hochman",fullName:"Matias Hochman",slug:"matias-hochman"},{id:"50498",title:"Prof.",name:"Ryszard",middleName:null,surname:"Lec",fullName:"Ryszard Lec",slug:"ryszard-lec"},{id:"110042",title:"Dr.",name:"Robert",middleName:null,surname:"Weisbein Hart",fullName:"Robert Weisbein Hart",slug:"robert-weisbein-hart"}]},{id:"23413",title:"Analysis of Biological Acoustic Waves by Means of the Phase–Sensitivity Technique",slug:"analysis-of-biological-acoustic-waves-by-means-of-the-phase-sensitivity-technique",signatures:"Wojciech Michalski, Wojciech Dziewiszek and Marek Bochnia",authors:[{id:"40516",title:"PhD.",name:"Wojciech",middleName:"Tomasz",surname:"Michalski",fullName:"Wojciech Michalski",slug:"wojciech-michalski"},{id:"46192",title:"Prof.",name:"Marek",middleName:null,surname:"Bochnia",fullName:"Marek Bochnia",slug:"marek-bochnia"},{id:"46193",title:"Dr.",name:"Wojciech",middleName:null,surname:"Dziewiszek",fullName:"Wojciech Dziewiszek",slug:"wojciech-dziewiszek"}]},{id:"23414",title:"Photoacoustic Technique Applied to Skin Research: Characterization of Tissue, Topically Applied Products and Transdermal Drug Delivery",slug:"photoacoustic-technique-applied-to-skin-research-characterization-of-tissue-topically-applied-produc",signatures:"Jociely P. Mota, Jorge L.C. Carvalho, Sérgio S. Carvalho and Paulo R. Barja",authors:[{id:"32237",title:"Dr.",name:"Paulo",middleName:null,surname:"Barja",fullName:"Paulo Barja",slug:"paulo-barja"},{id:"87448",title:"MSc.",name:"Jociely",middleName:null,surname:"Mota",fullName:"Jociely Mota",slug:"jociely-mota"},{id:"87449",title:"BSc",name:"Jorge",middleName:"Luis",surname:"Carvalho",fullName:"Jorge Carvalho",slug:"jorge-carvalho"},{id:"87450",title:"MSc.",name:"Sérgio",middleName:null,surname:"Carvalho",fullName:"Sérgio Carvalho",slug:"sergio-carvalho"}]},{id:"23415",title:"Acoustic–Gravity Waves in the Ionosphere During Solar Eclipse Events",slug:"acoustic-gravity-waves-in-the-ionosphere-during-solar-eclipse-events",signatures:"Petra Koucká Knížová and Zbyšek Mošna",authors:[{id:"35967",title:"Dr.",name:"Petra",middleName:null,surname:"Koucká Knížová",fullName:"Petra Koucká Knížová",slug:"petra-koucka-knizova"},{id:"89765",title:"MSc.",name:"Zbyšek",middleName:null,surname:"Mošna",fullName:"Zbyšek Mošna",slug:"zbysek-mosna"}]},{id:"23416",title:"Use of Acoustic Waves for Pulsating Water Jet Generation",slug:"use-of-acoustic-waves-for-pulsating-water-jet-generation",signatures:"Josef Foldyna",authors:[{id:"32826",title:"Dr.",name:"Josef",middleName:null,surname:"Foldyna",fullName:"Josef Foldyna",slug:"josef-foldyna"}]},{id:"23417",title:"Molecular Desorption by Laser–Driven Acoustic Waves: Analytical Applications and Physical Mechanisms",slug:"molecular-desorption-by-laser-driven-acoustic-waves-analytical-applications-and-physical-mechanisms",signatures:"Alexander Zinovev, Igor Veryovkin and Michael Pellin",authors:[{id:"30284",title:"Dr.",name:"Alexander",middleName:null,surname:"Zinovev",fullName:"Alexander Zinovev",slug:"alexander-zinovev"},{id:"32489",title:"Dr.",name:"Igor",middleName:null,surname:"Veryovkin",fullName:"Igor Veryovkin",slug:"igor-veryovkin"},{id:"32490",title:"Dr.",name:"Michael",middleName:null,surname:"Pellin",fullName:"Michael Pellin",slug:"michael-pellin"}]},{id:"23418",title:"Excitation of Periodical Shock Waves in Solid–State Optical Media (Yb:YAG, Glass) at SBS of Focused Low–Coherent Pump Radiation: Structure Changes, Features of Lasing",slug:"excitation-of-periodical-shock-waves-in-solid-state-optical-media-yb-yag-glass-at-sbs-of-focused-low",signatures:"N.E. Bykovsky and Yu.V. Senatsky",authors:[{id:"38239",title:"Dr.",name:"N.E.",middleName:null,surname:"Bykovsky",fullName:"N.E. Bykovsky",slug:"n.e.-bykovsky"},{id:"44089",title:"Mr",name:"Yuri",middleName:null,surname:"Senatsky",fullName:"Yuri Senatsky",slug:"yuri-senatsky"}]},{id:"23419",title:"An Optimal Distribution of Actuatorsin Active Beam Vibration – Some Aspects, Theoretical Considerations",slug:"an-optimal-distribution-of-actuatorsin-active-beam-vibration-some-aspects-theoretical-considerations",signatures:"Adam Brański",authors:[{id:"36857",title:"Prof.",name:"Adam",middleName:null,surname:"BRAŃSKI",fullName:"Adam BRAŃSKI",slug:"adam-branski"}]},{id:"23420",title:"Multilayered Structure as a Novel Material for Surface Acoustic Wave Devices: Physical Insight",slug:"multilayered-structure-as-a-novel-material-for-surface-acoustic-wave-devices-physical-insight",signatures:"Natalya Naumenko",authors:[{id:"30794",title:"Dr.",name:"Natalya",middleName:"F.",surname:"Naumenko",fullName:"Natalya Naumenko",slug:"natalya-naumenko"}]},{id:"23421",title:"SAW Parameters Analysis and Equivalent Circuit of SAW Device",slug:"saw-parameters-analysis-and-equivalent-circuit-of-saw-device",signatures:"Trang Hoang",authors:[{id:"36754",title:"Dr.",name:"Trang",middleName:null,surname:"Hoang",fullName:"Trang Hoang",slug:"trang-hoang"}]},{id:"23422",title:"Sources of Third–Order Intermodulation Distortion in Bulk Acoustic Wave Devices: A Phenomenological Approach",slug:"sources-of-third-order-intermodulation-distortion-in-bulk-acoustic-wave-devices-a-phenomenological-a",signatures:"Eduard Rocas and Carlos Collado",authors:[{id:"34260",title:"Prof.",name:"Carlos",middleName:null,surname:"Collado",fullName:"Carlos Collado",slug:"carlos-collado"},{id:"34385",title:"Mr",name:"Eduard",middleName:null,surname:"Rocas",fullName:"Eduard Rocas",slug:"eduard-rocas"}]},{id:"23423",title:"Shear Mode Piezoelectric Thin Film Resonators",slug:"shear-mode-piezoelectric-thin-film-resonators",signatures:"Takahiko Yanagitani",authors:[{id:"32602",title:"Dr.",name:"Takahiko",middleName:null,surname:"Yanagitani",fullName:"Takahiko Yanagitani",slug:"takahiko-yanagitani"}]},{id:"23424",title:"Polymer Coated Rayleigh SAW and STW Resonators for Gas Sensor Applications",slug:"polymer-coated-rayleigh-saw-and-stw-resonators-for-gas-sensor-applications",signatures:"Ivan D. Avramov",authors:[{id:"40655",title:"Dr.",name:"Ivan",middleName:null,surname:"Avramov",fullName:"Ivan Avramov",slug:"ivan-avramov"}]},{id:"23425",title:"Ultrananocrystalline Diamond as Material for Surface Acoustic Wave Devices",slug:"ultrananocrystalline-diamond-as-material-for-surface-acoustic-wave-devices",signatures:"Nicolas Woehrl and Volker Buck",authors:[{id:"36963",title:"Dr.",name:"Volker",middleName:null,surname:"Buck",fullName:"Volker Buck",slug:"volker-buck"}]},{id:"23426",title:"Aluminum Nitride (AlN) Film Based Acoustic Devices: Material Synthesis and Device Fabrication",slug:"aluminum-nitride-aln-film-based-acoustic-devices-material-synthesis-and-device-fabrication",signatures:"Jyoti Prakash Kar and Gouranga Bose",authors:[{id:"40705",title:"Dr.",name:"Jyoti Prakash",middleName:null,surname:"Kar",fullName:"Jyoti Prakash Kar",slug:"jyoti-prakash-kar"},{id:"47379",title:"Prof.",name:"Gouranga",middleName:null,surname:"Bose",fullName:"Gouranga Bose",slug:"gouranga-bose"}]},{id:"23427",title:"Surface Acoustic Wave Devices for Harsh Environment",slug:"surface-acoustic-wave-devices-for-harsh-environment",signatures:"Cinzia Caliendo",authors:[{id:"44168",title:"Dr.",name:"Cinzia",middleName:null,surname:"Caliendo",fullName:"Cinzia Caliendo",slug:"cinzia-caliendo"}]},{id:"23428",title:"Applications of In–Fiber Acousto–Optic Devices",slug:"applications-of-in-fiber-acousto-optic-devices",signatures:"C. Cuadrado-Laborde, A. Díez, M. V. Andrés, J. L. Cruz, M. Bello-Jimenez, I. L. Villegas, A. Martínez-Gámez and Y. O. Barmenkov",authors:[{id:"29543",title:"Dr.",name:"Christian",middleName:"A",surname:"Cuadrado-Laborde",fullName:"Christian Cuadrado-Laborde",slug:"christian-cuadrado-laborde"},{id:"46578",title:"Dr.",name:"Miguel V.",middleName:null,surname:"Andrés",fullName:"Miguel V. Andrés",slug:"miguel-v.-andres"},{id:"46579",title:"Dr.",name:"Antonio",middleName:null,surname:"Diez",fullName:"Antonio Diez",slug:"antonio-diez"},{id:"46580",title:"Dr.",name:"José L.",middleName:null,surname:"Cruz",fullName:"José L. Cruz",slug:"jose-l.-cruz"},{id:"46581",title:"Dr.",name:"Miguel",middleName:null,surname:"Bello-Jimenez",fullName:"Miguel Bello-Jimenez",slug:"miguel-bello-jimenez"},{id:"46582",title:"MSc",name:"Irma",middleName:null,surname:"Villegas",fullName:"Irma Villegas",slug:"irma-villegas"},{id:"46583",title:"Dr.",name:"M.A.",middleName:null,surname:"Martinez-Gamez",fullName:"M.A. Martinez-Gamez",slug:"m.a.-martinez-gamez"},{id:"46584",title:"Dr.",name:"Yuri",middleName:null,surname:"Barmenkov",fullName:"Yuri Barmenkov",slug:"yuri-barmenkov"}]},{id:"23429",title:"Surface Acoustic Waves and Nano–Electromechanical Systems",slug:"surface-acoustic-waves-and-nano-electromechanical-systems",signatures:"Dustin J. Kreft and Robert H. Blick",authors:[{id:"36802",title:"Dr.",name:"Robert",middleName:null,surname:"Blick",fullName:"Robert Blick",slug:"robert-blick"},{id:"90631",title:"Mr.",name:"Dustin",middleName:null,surname:"Kreft",fullName:"Dustin Kreft",slug:"dustin-kreft"}]}]}]},onlineFirst:{chapter:{type:"chapter",id:"64860",title:"Modeling of Fluid-Solid Two-Phase Geophysical Flows",doi:"10.5772/intechopen.81449",slug:"modeling-of-fluid-solid-two-phase-geophysical-flows",body:'\n
Fluid-solid two-phase flows are important in many geophysical problems such as sediment erosion, transport and deposition in rivers or coastal environment, debris flows, scour at river or marine structures, and submarine landslides. Behaviors of fluid-solid two-phase flows are very different from those of liquid-gas two-phase flows where bubbles are dispersed in the liquid or droplets dispersed in the gas. Vast numbers of experiments on various scales have been carried out for different applications of fluid-solid two-phase flows; these experiments have advanced our understanding of bulk behaviors of some important flow characteristics. However, development of measurement techniques suitable for collecting data that contribute to understanding important physics involved in fluid-solid two-phase flows is a still-evolving science. With the modern computer technology, many data that are not obtainable currently in the experiment can be easily produced by performing time-dependent, multidimensional numerical simulations. Of course, empirical closure models required to close the governing equations still need high-quality experimental data for model validation.
\nNumerical approaches to two-phase flows include Eulerian-Eulerian approach, direct numerical simulations (DNS) based on Eulerian-Lagrangian formulations (Lagrangian point-particle approach), and fully resolved DNS approach [1]. Fully resolved DNS can resolve all important scales of the fluid and particles, but these simulations are currently limited to about 10 k uniform-size spheres on a Cray XE6 with 2048 cores [2], and it is not practical to use this method to model large-scale geophysical flow problems in the foreseeable future [1]. Lagrangian point-particle approach uses Eulerian formulation for the fluid phase and Lagrangian formulation for tracking the instantaneous positions of the particles. Lagrangian point-particle simulations make use of semiempirical relationships to provide both hydrodynamic force and torque acting on each particle and thus avoid modeling processes on scales smaller than Kolmogorov scale [1], making it possible to include more particles and run in a domain larger than that for fully resolved DNS. The application of Lagrangian point-particle approach is crucially dependent on the availability and accuracy of such semiempirical relationships. A recent study shows that good results can be obtained for about 100k uniform-size spherical particles in a vertical channel flow [3]; however, using this approach to investigate large-scale two-phase flow problems is still beyond the current computing capacity. Two-phase Eulerian-Eulerian approach treats both the fluid and particle phases as continuum media and is suitable for solving large-scale two-phase flow problems.
\nEulerian-Eulerian two-phase flow models based on large-eddy-simulations solve a separate set of equations describing conservation of mass, momentum, and kinetic energy for each phase [4, 5, 6, 7] and thus have the potential to consider all important processes involved in the interactions between the two phases through parameterization of particle-scale processes. This chapter introduces the basics of Eulerian-Eulerian two-phase flow modeling, its implementation in the finite-volume framework of OpenFOAM®, and two applications in geophysical flow problems.
\nLet us consider a mixture of fluid and solid particles. Fluid can be gas, water, or a mixture of water and gas. In DNS and Lagrangian point-particle approaches to two-phase flows, the flow field is solved by solving the Navier-Stokes equations, and the motion of each particle is determined by the Newton’s equation of motion. In Eulerian-Eulerian two-phase flow approaches, however, the motions of individual particles are not of the interest, and the focus is on the macroscopic motion of the fluid and solid particles instead. For this purpose, the solid particles are modeled as a continuum mass through an ensemble averaging operation, which is based on the existence of possible equivalent realizations. After taking ensemble average, the mixture of fluid and particles consists of two continuous phases: the fluid (water, gas, or a mixture of water and gas) is the fluid phase, and the solid particle is the solid phase. Both phases are incompressible. The motions of the fluid and solid phases are governed by their own equations, which are obtained by taking ensemble average of the microscopic governing equations for each phase [8]. Even though some aspects of fluid-solid interaction can be considered through the ensemble average, the ensemble averaging operation itself, however, does not explicitly introduce any turbulent dispersion in the resulting equations. To consider the turbulent dispersion in the Eulerian-Eulerian description of the fluid-solid two-phase flows, another averaging operation (usually a Favre average) is needed to consider the correlations of turbulent components [5, 9].
\nAt the microscopic scale, the fluid-solid mixture is a discrete system. The purpose of performing an ensemble averaging operation is to derive a set of equations describing this discrete system as a continuous system at the macroscopic scale, where the typical length scale should be much larger than one particle diameter.
\nIn the Eulerian-Eulerian approach to two-phase flows, it is assumed that the equations governing the motion of phase \n
and
\nwhere \n
where \n
Because the fluid phase and the solid phase are immiscible, at any time \n
The volumetric concentration of phase \n
There are several methods to derive the ensemble averaged equations governing the motion of phase \n
where \n
The ensemble averaged equations governing the motion of phase \n
and
\nThe resulting equations governing the ensemble average motion of phase \n
and
\nwith
\nNote that \n
which is the density of the interfacial force [8]. Physically, \n
After using Eq. (3) for \n
and
\nwhere \n
and \n
For compressible materials \n
Now we examine the limiting case where the fluid-solid system is at its static state. Because the phase functions for the two phases satisfy \n
for the fluid phase, and
\nfor the solid phase.
\nBecause \n
which, physically, is the buoyancy acting on the solid phase. Now Eq. (18) becomes
\nwhich states that the weight of the solid particles is supported by the buoyancy and the interparticle forces. Therefore, the ensemble pressure of the solid phase can be written as \n
For brevity of the presentation, we shall denote simply \n
and
\nThe ensemble averaged equations governing the motion of the solid phase are
\nand
\nwhere \n
To close the equations for the fluid and solid phases, closure models are needed for \n
It is remarked here that the definitions of the ensemble averages given in Eq. (14) do not consider the contribution from the correlations between the fluctuations of the velocities and the fluctuations of phase functions at microscopic scale; therefore, the effects of turbulent dispersion are not directly included in the ensemble averaged equations describing the motion of the each phase. In the literature, two approaches have been used to consider the turbulent dispersion: (i) considering the correlation between the fluctuations of \n
In the absence of the turbulent dispersion from \n
This expression for \n
where \n
The volumetric concentration and the velocities can be written as
\nwhere the Favre averages are defined as
\nand the overline stands for an integration with respect to time over a time scale longer than small-scale turbulent fluctuations but shorter than the variation of the mean flow field.
\nThe averaged equations for the mean flow fields of the two phases are obtained by taking the following steps: (i) substituting Eq. (25) with Eq. (26) in Eqs. (22) and (24), (ii) substituting Eq. (27) in the equations obtained at step (i), and (iii) taking average of the equations obtained at step (ii) to obtain the following equations:
\nfor the fluid phase, with \n
and
\nfor the solid phase, with \n
It is remarked here that the terms \n
In order to close these averaged equations, closure models are required for the following terms: \n
where \n
For brevity of the presentation, the symbols representing Favre averages are dropped hereinafter, and the final equations governing the conservation of mass and momentum of each phase are
\nfor the fluid phase and
\nfor the solid phase.
\nThe stress tensor for the fluid phase \n
The viscous stress tensor \n
where \n
The stress tensor \n
where \n
with \n
The equations governing \n
and
\nwhere coefficients \n
where \n
with \n
It is remarked here that the presence of solid particles in the turbulent flow may either enhance (for large particles) or reduce (for small particles) the turbulence [18]. The \n
The closure models for \n
where \n
For solid particles in a compact bed, the formula proposed by Hsu et al. [19] can be used to compute \n
where \n
The closure models for \n
The kinematic viscosity of the solid phase \n
where \n
Based on an analysis of heavy and small particles in homogeneous steady turbulent flows, Hinze [20] suggests that \n
and
\nwhere the coefficient \n
For dense fluid-solid two-phase flows, the visco-plastic rheological characteristics depend on a dimensionless parameter \n
Following the work of Boyer et al. [22], Lee et al. [16] assumed
\nwhere \n
where \n
which considers the solid phase in its static state as a very viscous fluid and
\nwhere \n
The drag force between the two phases is modeled through the particle response time \n
The first model is based on particle sedimentation in still water, which can be simplified as a one-dimensional problem, where the steady sedimentation assures that there are no stresses in both the solid and fluid phases in the vertical direction \n
and
\nwhere \n
Because net volume flux through any horizontal plane must be zero, we have
\nCombining Eqs. (59) and (61) yields
\nSubstituting Eqs. (61) and (62) into Eq. (60) leads to
\nwhere the solid-phase velocity \n
where the coefficient \n
The terminal velocity of a single particle \n
where \n
where \n
It is remarked that Eq. (64) is validated only for \n
Another model for particle response time can be derived by examining the pressure drop in the steady flow through a porous media. For a one-dimensional problem of a horizontal, steady flow through porous media, the terms containing the stresses of the fluid phase disappear, and Eq. (38) reduces to
\nwhere the horizontal coordinate \n
For this problem, Forchheimer [29] suggested
\nwhere \n
Comparing Eqs. (69) and (70) and using Eq.(71) give
\nwhere \n
For flow in a porous media, the particle response time can also be related to its permeability \n
where \n
When the flow is very slow, Eqs. (70), (71), and (73) suggest that
\nwhich means that the particle response time can be related to the permeability.
\nEquation (64) is validated only for \n
where \n
Combining Eqs. (63), (76), and (66)–(67) gives
\nWe stress that \n
where \n
For given values of \n
This section introduces how to use OpenFOAM® to solve the governing equations with the closure models presented in the previous section. OpenFOAM® is a C++ toolbox developed based on the finite-volume method; it allows CFD code developers to sidestep the discretization of derivative terms on unstructured grids.
\nTo avoid numerical noises occurring when \n
and
\nThe solutions of Eqs. (80) and (81) are expressed in the following semidiscretized forms:
\nwhere \n
If Eq. (83) is directly used to calculate \n
which is corrected by the following corrector
\nThis predictor-corrector scheme can improve the numerical stability by introducing a numerical diffusion term. To see this, we combine Eqs. (39) and (85) to obtain the following equation describing the evolution of \n
The right-hand side of Eq. (86) now has a diffusive term introduced by the numerical scheme. High sediment concentration and large \n
For the velocity-pressure coupling, Eq. (82) is similarly solved using a predictor \n
which is corrected by the following corrector
\nSubstituting Eq. (88) into Eq. (37) gives a pressure equation. However, when using this pressure equation to simulate air-water flows, numerical experiments have shown that the lighter material is poorly conserved [36]. The poor conservation of lighter material can be avoided by combining Eqs. (37) and (39) into the following Eq. (37):
\nand using Eq. (89) to correct \n
and combine Eqs. (83) and (88)–(90) to obtain the following equation
\nThe numerical diffusion term on the right-hand side of Eq. (91) can help improve the numerical stability.
\nThe prediction-correction method presented here deals with velocity-pressure coupling and avoids the numerical instability caused by high concentration. The turbulence closure \n
When \n
An iteration procedure is needed to solve the governing equations at each time step for the values of \n
Compute \n
Solve Eq. (86) for \n
Compute \n
Compute \n
Compute \n
Solve Eq. (91) for \n
Repeat Eqs. (5)–(7) for \n
Compute \n
Set \n
Repeat Eqs. (1)–(10) with the updated \n
Solve Eqs. (45) and (46) for \n
Figure 1 is a flowchart showing these 12 solution steps.
\nA flow chart showing the solution procedure using OpenFOAM®.
In the absence of the solid phase, the numerical scheme outlined here reduces to the “PIMPLE” scheme, which is a combination of the “pressure implicit with splitting of operator” (PISO) scheme and the “semi-implicit method for pressure-linked equations” (SIMPLE) scheme. Iterations need to be done separately to solve Eq. (80) for \n
To ensure the stability of the overall numerical scheme, the Courant-Friedrichs-Lewy (CFL) condition must be satisfied for each cell. The local Courant number for each cell, which is related to the ratio between the distance of a particle moving within \n
where in \n
This section briefly describes two examples that have been studied using the two-phase flow models described. The problem descriptions and numerical setups for these two problems are included here; for other relevant information, the reader is referred to Lee and Huang [35] and Lee et al. [38].
\nA sluice gate is a hydraulic structure used to control the flow in a water channel. Sluice gate structures usually have a rigid floor followed by an erodible bed. The scour downstream of a sluice gate is caused by the horizontal submerged water jet issuing from the sluice gate. It is of practical importance to understand the maximum scour depth for the safety of a sluice gate structure. Many experimental studies have been done to investigate the maximum scour depth and the evolution of scour profile (e.g., Chatterjee et al. [39]). For numerical simulations, this problem includes water (fluid phase) and sediment (solid phase) and is best modeled by a liquid-solid two-phase flow approach. In the following, the numerical setup and main conclusions used in Lee et al. [38] are briefly described. The experimental setup of Chatterjee et al. [39] is shown in Figure 2. To numerically simulate the experiment of [8], we use the same sand and dimensions to set up the numerical simulations: quartz sand with \n
A sketch of the experimental setup for scour induced by a submerged water jet.
Comparison of the computed scour depth with measurements of Chatterjee et al. [39].
The problem involves also an air-water surface, which can be tracked using a modified volume-of-fluid method introduced in [38]. A nonuniform mesh is used in the two-phase flow simulation because of the air-water interface, the interfacial momentum transfer at the bed, and the large velocity variation due to the water jet. The finest mesh with a vertical mesh resolution of \n
The scour process is sensitive to the model for particle response time used in the simulation. Because Eq. (72) can provide a better prediction of sediment transport rate for small values of Shields parameter, it is recommended for this problem. The two-phase flow model can reproduce well the measured scour depth and the location of sand dune downstream of the scour hole.
\nAnother application of the fluid-solid two-phase flow simulation is the simulation of the collapse of a deeply submerged granular column. The problem is best described as a granular flow problem, which involves sediment (a solid phase) and water (fluid phase). Many experimental studies have been reported in the literature on this topic. This section describes a numerical simulation using the fluid-solid two-phase flow model described in this chapter.
\nFigure 4 shows the experimental setup of Rondon et al. [40]. A 1:1 scale two-phase flow simulation was performed by Lee and Huang [35] using the fluid-solid two-phase flow model presented in this chapter. The diameter and the density of the sand grain are 0.225 mm and 2500 kg/m3, respectively. The density and the dynamic viscosity of the liquid are 1010 kg/m3 and 12 mPa s, respectively. Note that the viscosity of the liquid in the experiment is ten times larger than that for water at room temperature. For this problem, using a mesh of 1.0 × 1.0 mm and the particle response model given by Eq. (78), the fluid-solid two-phase flow model presented in this chapter can reproduce well the collapse process reported in Rondon et al. [40]. Figure 5 shows the simulated collapsing processes compared with the measurement for two initial packing conditions: initially loosely packed condition and initially densely packed condition.
\nA sketch of the experimental setup for the collapse of a deeply submerged granular column.
The simulated collapsing processes for the initially loose condition (a)–(d) and the initially dense condition (e)–(h). The lines represent contours of the computed concentrations, and the symbols were experimental data of Rondon et al. [40]. The figure is adapted from Lee and Huang [35].
The two-phase model and closure models presented in this chapter are able to deal with both initially loose packing and initially dense packing conditions and reveal the roles played by the contractancy inside the granular column with a loose packing and dilatancy inside a granular column with a dense packing. One of the conclusions of Lee and Huang [35] is that the collapse process of a densely packed granular column is more sensitive to the model used for particle response time than that of a loosely packed granular column. The particle response model given by Eq. (78) performs better than other models; this is possibly because the liquid used in Rondon et al. [40] is much viscous than water.
\nThis chapter presented a brief introduction to the equations and closure models suitable for fluid-solid two-phase flow problems such as sediment transport, submarine landslides, and scour at hydraulic structures. Two averaging operations were performed to derive the governing equations so that the turbulent dispersion, important for geophysical flow problems, can be considered. A new model for the rheological characteristics of sediment phase was used when computing the stresses of the solid phase. The \n
This material presented here is partially based upon work supported by the National Science Foundation under Grant No. 1706938 and the Ministry of Science and Technology, Taiwan [MOST 107-2221-E-032-018-MY3]. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
\nIntechOpen’s Academic Editors and Authors have received funding for their work through many well-known funders, including: the European Commission, Bill and Melinda Gates Foundation, Wellcome Trust, Chinese Academy of Sciences, Natural Science Foundation of China (NSFC), CGIAR Consortium of International Agricultural Research Centers, National Institute of Health (NIH), National Science Foundation (NSF), National Aeronautics and Space Administration (NASA), National Institute of Standards and Technology (NIST), German Research Foundation (DFG), Research Councils United Kingdom (RCUK), Oswaldo Cruz Foundation, Austrian Science Fund (FWF), Foundation for Science and Technology (FCT), Australian Research Council (ARC).
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