Document Zbl 1226.76001

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(2011). </div> <div class="abstract"><div class="pre">Gas-liquid multiphase flows occur in many industrial applications, such as heat transfer by boiling, bubbly flows, chemical and mechanical industries, etc. The book under review presents a careful and detailed development of numerical methods for flows containing sharp interfaces, such as fluids consisting of two or more immiscible components. The appearance of fast and powerful computers has led to the development of new numerical methods for direct numerical simulations (DNS) of multiphase flows. The authors focus mostly on two specific classes of numerical methods: volume-of-fluid (VOF) and front-tracking methods. Precise results in this area involve an exciting mixture of fluid mechanics, physics, chemistry, analytical and related numerical methods. <br class="zbmathjax-paragraph">The book consists of eleven chapters, a preface, four appendices, a large list of references and an index. The description of these chapters is, in short, as follows:<br class="zbmathjax-paragraph">Chapter 1, “Introduction”, presents examples of multiphase flows; computation modeling; simple flows as measured by the Reynolds number; finite Reynolds number flows. In the limit of either very large or very small viscosity, it is sometimes possible to simplify considerably the flow description by ignoring the inertia terms completely (Stokes flow) or by ignoring viscous effects completely (inviscid flow). For inviscid flows it is usually to assume that the flow is irrotational, except at the fluid interface.<br class="zbmathjax-paragraph">Chapter 2, “Fluid mechanics with interfaces”, is devoted to the equations governing multiphase flows, where a sharp interface separates immiscible fluids or phases. It contains: general principles; basic equations (mass conservation, momentum conservation, energy conservation, incompressible flow, boundary conditions); interfaces: description and definitions; fluid mechanics with interfaces: the one-fluid formulation; nondimensional numbers; thin films; intermolecular forces; contact lines; notes.<br class="zbmathjax-paragraph">Chapter 3, “Numerical solutions of the Navier-Stokes equations”, discusses the time integration; spatial discretization; discretization of advection terms; viscous terms; pressure equation; velocity boundary conditions; outflow boundary conditions; adaptive mesh refinement; summary. The authors state that for any numerical solution of time-dependent Navier-Stokes equations it is necessary to decide: (i) how the grid points, where various discrete approximations are stored, are arranged; (ii) how the velocity field is integrated in time; (iii) how advection and viscous terms are discretized; (iv) how the pressure equation, resulting from the incompressibility condition, is solved; and (v) how boundary conditions are implemented.<br class="zbmathjax-paragraph">Chapter 4, “Advecting a fluid interface”, contains the notations; advecting the color function; the volume-of-fluid method; front tracking method; the level-set method; phase-field methods; the CIP methods; summary. The abbreviation CIP has stayed constant since the introduction of the method, but the method has evolved. CIP initially stood for cubic interpolated pseudo-particle, then for cubic interpolated propagation, and most recently for the constrained interpolation profile method. <br class="zbmathjax-paragraph">Chapter 5 “The volume-of-fluid method (VOF)”. The VOF method has been historically used for two-fluid flows. In the VOF method, the marker function is represented by a fraction of computational grid cell which is occupied by the fluid assumed to be the reference phase. The following questions are treated in this chapter: basic properties; interface reconstruction; tests of reconstruction method; interface advection; tests of reconstruction and advection methods; hybrid methods.<br class="zbmathjax-paragraph">Chapter 6, “Advecting marker points: front tracking”, describes the basic idea briefly and presents a short historical overview of the term “front tracking”. The front refers to the complete set of computational objects used to represent the interface. The topics presented here are: the structure of the front (structured two-dimensional fronts; unstructured fronts); restructuring the fronts; front-grid communications; advection of the front; constructing the marker function; changes in the front topology; notes.<br class="zbmathjax-paragraph">Chapter 7, “Surface tension”, describes how to find surface tension of both VOF and front-tracking methods, and deals with the computing surface tension from marker functions (continuous surface force (CSF) method; continuous surface stress method; direct addition and elementary smoothing in the VOF method; weighted distribution in the VOF method: kernel smoothing; axisymmetric interfaces); computing the surface tension of a tracked front (two-dimensional interfaces; three-dimensional interfaces; smoothing the surface tension on the fixed grid); testing the surface tension methods; more sophisticated surface tension method (direct addition with pressure correction; CSF method with better curvature: PROST; numerical estimate of the curvature from the volume fractions: the HF method); conclusion on numerical methods.<br class="zbmathjax-paragraph">Chapter 8 “Disperse bubbly flows”. Bubbly flows is of critical importance in a large number of industrial applications, including boiling heat transfer in power plants, various metallurgical processes, and in bubble columns in the chemical industry. The chapter contains an introduction; homogeneous bubbly flows; bubbly flows in vertical channels, and a discussion. It is stated that the simulations of bubbly flows discussed in this chapter are good examples of opportunities and challenges in applying DNS to understand the complex multiphase flows. <br class="zbmathjax-paragraph">Chapter 9, “Atomization and breakup” describes a striking process in which finely divided sprays or droplet clouds are produced. The chapter refers to thread, sheet and rim breakup (Plateau-Rayleigh jet instability; film and thread breakup; Taylor-Culick rim; rims leading to droplets and fingers); high-speed jets (structure of the atomizing jet; mechanism of droplet formation; stability theory; Orr-Sommerfeld analysis); atomization simulations (two-dimensional, temporal simulations; two-dimensional spatially developing simulations; three-dimensional calculations).<br class="zbmathjax-paragraph">Chapter 10 “Droplet collision, impact, and splashing”. Droplet impacts are of major industrial interest in fuel droplets impact on walls of pipes and combustion chambers. The authors focus on numerical work and cite relevant theoretical and experimental works. The chapter contains an introduction; early simulations; low-velocity impacts and collisions; more complex slow impacts; corolla, crowns, and splashing impacts (impacts on thin liquid layers; three-dimensional impacts).<br class="zbmathjax-paragraph">Chapter 11, “Extensions”, refers to additional fields and surface physics; (electrohydrodynamics; boiling; cavitation); imbedded boundaries (the immersed boundary method of Peskin; solid boundaries; solidification); multiscale issues; summary. It is shown that multiphase flows are important for energy production, manufacturing and chemical processes, wastewater treatment, agriculture, and many others processes critical for our current way of living.<br class="zbmathjax-paragraph">The book contains also four appendices: Appendix A – Interfaces: description and definitions; Appendix B – Distributions concentrated on the interfaces; Appendix C – Cube-chopping algorithm; and Appendix D – The dynamics of liquid sheets: linearized theory. A large list of 447 references and an Index end the book. <br class="zbmathjax-paragraph">In the reviewer’s opinion this book provides a solid fundamental and comprehensive presentation of mathematical principles of multiphase flows, pointing out the most important practical applications. The book is very well written and readable. Results of numerical solutions of the considered problems are given graphically and in tabular form. The book will be of great interest to researchers and graduate students. In fact, it could be also useful to a wide range of specialists working in the area of applied mathematics, fluid mechanics, numerical method for partial differential equations, such as design engineers, physicists, chemical engineers, and also to researchers interested in the mathematical theory of fluid mechanics and connected topics. It can be also recommended as a text for seminars and courses, as well as for independent study. Some chapters of the book provide a solid background for future research. I believe that the concepts presented in this book will stimulate new research in the area of multiphase flows in complex fluids and their applications.</div><div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A4074">Ioan Pop (Cluj-Napoca)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A5853342">Cited in <strong>251</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A76-01" title="MSC2020">76-01</a> </td> <td class="space"> Introductory exposition (textbooks, tutorial papers, etc.) pertaining to fluid mechanics </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76T10" title="MSC2020">76T10</a> </td> <td class="space"> Liquid-gas two-phase flows, bubbly flows </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A76T30" title="MSC2020">76T30</a> </td> <td class="space"> Three or more component flows </td> </tr><tr> <td class="mono"> 76M25 </td> <td class="space"> Other numerical methods (fluid mechanics) (MSC2010) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AStokes+flow">Stokes flow</a>; <a href="/?q=ut%3ANavier-Stokes+equations">Navier-Stokes equations</a>; <a href="/?q=ut%3Avolume-of-fluid+method">volume-of-fluid method</a>; <a href="/?q=ut%3Alevel-set+method">level-set method</a>; <a href="/?q=ut%3ACIP+methods">CIP methods</a>; <a href="/?q=ut%3Afront+tracking+method">front tracking method</a></div> <div class="software"> <h3>Software:</h3><a href="/software/677">PARAMESH</a>; <a href="/software/2561">KIVA-4</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1226.76001" data-ciurl="/ci/05853342" data-biburl="/bibtex/05853342.bib" data-amsurl="/amsrefs/05853342.bib" data-xmlurl="/xml/05853342.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05853342.pdf" title="Zbl 1226.76001 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1017/CBO9780511975264" aria-label="DOI for “Direct numerical simulations of gas-liquid multiphase flows.”" title="10.1017/CBO9780511975264">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> </article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; 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