Shallow Geophysical Mass Flows down Arbitrary Topography PDF
Preview Shallow Geophysical Mass Flows down Arbitrary Topography
Advances in Geophysical and Environmental Mechanics and Mathematics Ioana Luca Yih-Chin Tai Chih-Yu Kuo Shallow Geophysical Mass Flows down Arbitrary Topography Model Equations in Topography-fitted Coordinates, Numerical Simulation and Back-calculations of Disastrous Events Advances in Geophysical and Environmental Mechanics and Mathematics Series editors Kolumban Hutter, Zürich, Switzerland Holger Steeb, Bochum, Germany More information about this series at http://www.springer.com/series/7540 Ioana Luca Yih-Chin Tai Chih-Yu Kuo (cid:129) (cid:129) Shallow Geophysical Mass Flows down Arbitrary Topography fi Model Equations in Topography- tted Coordinates, Numerical Simulation and Back-calculations of Disastrous Events 123 IoanaLuca Chih-Yu Kuo Department ofMathematical Methods Research Centerfor Applied Sciences andModels Academia Sinica University Politehnica of Bucharest Taipei Bucharest Taiwan Romania Yih-Chin Tai Department ofHydraulic andOcean Engineering National Cheng KungUniversity Tainan Taiwan ISSN 1866-8348 ISSN 1866-8356 (electronic) Advances in GeophysicalandEnvironmental MechanicsandMathematics ISBN978-3-319-02626-8 ISBN978-3-319-02627-5 (eBook) DOI 10.1007/978-3-319-02627-5 LibraryofCongressControlNumber:2015958915 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor foranyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland To Professor Kolumban Hutter Preface The topic in the book was initiated by the long-term collaboration with Prof. Kolumban Hutter. He was consecutively invited to visit Academia Sinica, National Chi Nan University and National Cheng Kung University, Taiwan, between2006and2009.Duringhisstayandhisfrequentguidingcommunications, the research direction was planned and the ingredients were set up. The work was about the mathematical description of shallow geophysical mass flows on realistic topographies. Significant progress towards this topic has previously been made by K. Hutter and collaborators, who investigated granular flows in arbitrarily curved and twisted channels. The book, Avalanche Dynamics by S.P. Pudasaini and K. Hutter, is a comprehensive collection of their results. The paper by Bouchut and Westdickenberg [1] treated the basal topography as an arbitrary surface, and stimulatedtheinterestinthemodellingofflowsoncomplextopographies.Thepaper dealswithanidealfluidasamodelfortheflowingmassandarigidbedsurface.It wastheaimofourgrouptoaccountformoreelaboratedrheologiesinthedescription of shallow gravity-driven flows on arbitrarily curved surfaces, with the erosion/depositionprocessincluded.Weconcentratedtheeffortsintwodirections— derivation of modelling equations for various geophysical flows and numerical implementation of some modelling equations, corroborated with experimental and field observations. The objective of this monograph is twofold. First, to present, in as detailed and accessible form as possible, a way to formulate depth-averaged models for gravity-drivenshallowmassflowsonarbitrarytopographies.Second,toshowhow thesemodelscanbenumericallytreated,experimentallytestedandultimatelyused forbackcalculationsofrealisticevents.Thepresentationisbasedonearlierpapers by the authors; however, much of the text and the derived thin-layer model equations are new. The book is intended for civil engineers, geophysicists, geologists, physical scientists, applied mathematicians, engineers responsible for hazard management andforclassroomusebystudentsinterestedingeophysicalflows.Therearemodest vii viii Preface demands on the student’s mathematical background in linear algebra, calculus, geometry of a surface, and most of them are covered in various chapters. To deduce modelling equations for shallow flows on arbitrary topographies, curvilinear coordinates, suitable for numerical purposes and ordering approxima- tionseveninsteeptopographies,areintroduced.Thesecoordinateshavebeenused byseveralauthors,butthedepth-integrationalongthenormaltothebedsurfaceand thematrixformofthemodellingequations areduetoBuchutandWestdickenberg [1]. We have adopted the approach from [1]; however, unlike [1], we use vectors andtensorstoderivethemainresults,makingthederivationsmoretransparent,and formulatethemodel equationsina more general context.Tokeepthepresentation sufficiently accessible, we confined ourselves to the derivation of depth-averaged modelequationsforashallowone-layer,one-phasefluid-likematerial.Forshallow two-layer fluids or solid–fluid mixtures, references are given. Two routes, called conventional and non-conventional, are followed. In the conventional route, the intrinsic modelling equations are expressed in the topography-fitted coordinates, as it is customary in continuum mechanics, and the tangential linear momentum balance equation is depth-integrated. In the non-conventional route, a hybrid form (i.e. using both Cartesian and contravariant components of vectors and tensors as functions of the terrain-fitted coordinates) of the horizontal–vertical linear momentum balance equation is depth-integrated. The depth-averaged modelling equations emerging from these two routes are not equivalent.Thoseinthenon-conventionalmethodseemtobemoreappealing,since severaltermsstemmingfromtheChristoffelsymbolsdonotarisehereastheydoin the modelling equations derived in the conventional route. In this monograph, the numerical implementation, experimental validation and back calculations of real- istic eventsareperformedfor caseswherethemodellingequationsasderivedwith thetwomethodscoincide.Therefore,thequestionofthedepth-averagedequations (derived by the two methods here and in fact by any other method), best suited to describe a realistic flow on arbitrary topography, is an open problem. The non-conventionalroutecombinestheapproachbyBouchutandWestdickenberg[1] andHui’sunifiedcoordinatesmethod[2].Theunifiedcoordinatesmethodisamesh velocity approach in computational fluid dynamics, based on a hybrid form (i.e. using both Cartesian and contravariant components of vectors and tensors as functions of curvilinear coordinates) of the mass and linear momentum balance equations. For the case where erosion/deposition takes place, it inspired a certain choice for the time-dependent terrain-fitted coordinates, which simplifies the modelling equations (in both routes) and avoids complicated calculations in numerical simulations. We have endeavoured to use the same notation, style and spirit throughout. To maintain clarity in exposure, we have relegated sophisticated proofs to appendices and proposed a few exercises at the end of Chap. 2. We have also refrained from giving a list of symbols. Instead, we have collected definitions of some quantities and important rules which they satisfy at the end of a few sections. Preface ix Acknowledgements All the authors are deeply indebted to Prof. Kolumban Hutter for bringing them together, for his motivating enthusiasm and involvement along the research pro- gress of the group and for the invitation to prepare this book within the Springer series, Advances in Geophysical and Environmental Mechanics and Mathematics. We dedicate our most sincere acknowledgement to him. I. Luca takes the oppor- tunitytoexpresshergreatintellectualdebtstoF.BouchutandM.Westdickenberg, whosepaper[1]inComm.Math.Sci.inspiredherworkinmodellinggravity-driven shallow mass flows. Sincere gratitude goes also to all of those who made her researchstaysoenjoyableinPuli(NationalChiNanUniversity),Taipei(Academia Sinica) and Tainan (National Cheng Kung University), ending as a fascinating life experience. Y.C. Tai and C.Y. Kuo express their sincere gratitude to Prof. Wai-How Hui for lecturing on the unified coordinate method during his stay in Academia Sinica, Taipei, in 2005, which inspired them to the concept of the non-conventional route. During the preparation of the manuscript, we received tremendous help from our colleagues, research assistants and students: Profs. Jia-Jyun Dong, Chien-Chih Chen, Chyi-Tyi Lee, Rou-Fei Chen, Kuo-Jen Chang, Yu-Chang Chan, Toshihiko Shimamoto, Mr. Yang-Chen Lin and Mr. Che-Ming Yang, for their efforts in rock/soil laboratory experiments and field studies; Mr. Ying-Tsao Li, Shen-En Lin, Ms. Ruo-Ying Wu, for their help in numerical simulations and results presentation; Profs. Ruey-Der Hwang, Pi-Wen TsaiandMr.Shao-KuanWei,forseismicmotionanalysis;andMs.Chia-ChiShen, for typing drafts of iterated versions of different chapters. We thank all those who havesupportedusinthisendeavour;theformerandpresentstudentsandscientific researchers for the fruitful collaboration that made this book possible. The grants received from the various Taiwanese research organizations and agents: Academia Sinica, Ministry of Science and Technology (former National Science Council), National Chi Nan University and National Cheng Kung University are greatly appreciated. Without their support the research would not have been possible. Last but not least, our special thanks go to the staff, especially to Ms. Abirami PurushothamanofSpringer-Verlag,foralltheirsupportinpreparingthemanuscript for publication. Romania Ioana Luca Taiwan Yih-Chin Tai Taiwan Chih-Yu Kuo November 2015 x Preface References [1] F. Bouchut, M. Westdickenberg, Gravity driven shallow water models for arbitrary topography.Comm.Math.Sci.,2(3),359–389(2004) [2] W.H. Hui, K. Xu, Computational Fluid Dynamics Based on the Unified Coordinates. (Springer-Verlag,BerlinHeidelbergNewYork,2012)
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