Advances in Geophysical and Environmental Mechanics and Mathematics Kolumban Hutter Yongqi Wang Fluid and Thermodynamics Volume 1: Basic Fluid Mechanics Advances in Geophysical and Environmental Mechanics and Mathematics Series editors Kolumban Hutter, Zürich, Switzerland Holger Steeb, Stuttgart, Germany More information about this series at http://www.springer.com/series/7540 Kolumban Hutter Yongqi Wang (cid:129) Fluid and Thermodynamics Volume 1: Basic Fluid Mechanics 123 Kolumban Hutter YongqiWang c/o Versuchsanstalt fürWasserbau, Department ofMechanical Engineering HydrologieundGlaziologie Technische UniversitätDarmstadt ETHZürich Darmstadt, Hessen Zürich Germany Switzerland ISSN 1866-8348 ISSN 1866-8356 (electronic) Advances in GeophysicalandEnvironmental MechanicsandMathematics ISBN978-3-319-33632-9 ISBN978-3-319-33633-6 (eBook) DOI 10.1007/978-3-319-33633-6 LibraryofCongressControlNumber:2016938676 ©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 Preface Fluid and thermodynamics (FTD) are generally taught at technical universities as separate subjects and this separation can be justified simply by reasons of the assignedtime;theelementsofeachsubjectcanbeintroduced withinasemesterof *15 weeks. Most likely, these outer educational boundaries may even have well furtheredthisseparation.Intellectually,thetwosubjects,however,belongtogether, especially since for all but ideal fluids the second law of thermodynamics imposes constraint conditions on the parameters of the governing equations (generally partialdifferentialequations)thatarethenusedinthefluiddynamicpartofthejoint efforttoconstructsolutionstophysicallymotivatedinitialboundaryvalueproblems thatteachusimportantfactsofthebehaviorofthemotionofthefluidundercertain circumstances. One of the authors (K.H.) found this combination offluid and thermodynamics as an assigned one-semester course, when he started in 1987 in the Department of Mechanics at Technische Universität Darmstadt (at that time ‘Technische Hochschule’) as successor of the late Prof. Dr. rer.nat. Ernst Becker (1929–1984). WithK.H’semphasizedinterestincontinuummechanicsandthermodynamics,this dualunderstandingofthemathematicaldescriptionoffluidmatterwasidealandthe assignmenttoteachitwasawelcomechallenge,whichwasdeclaredasa‘credo’to the working environment in both teaching and research in his group. The course notes of FTD taught to upper-class electrical engineers for 18 years were quickly worked out into the book ‘Fluid und Thermodynamik – eine Einführung’ and published by Springer Verlag, Berlin etc., (ISBN 3-540-59235-0, secondedition).Allthechaptersofthisbook—someslightlyextended—havebeen translated (by K.H.) into the English language and are interwoven in this treatise withchapters,which,asawhole,shouldprovideafairlydetailedunderstandingof FTD. All subjects of this treatise of FTD have been taught in one or another form as lectures in courses to students at Technische Universität Darmstadt, Swiss Federal Institute of Technology in Zürich (ETHZ), and in guest lectures in advanced courses at other universities and research institutions worldwide. The audience in v vi Preface these courses consisted of students, doctoral candidates and postdoctoral assistants of engineering (civil, mechanical, chemical, mechanics), natural sciences (meteo- rologists, oceanographers, geophysicists), mathematics and physics. Some of the topics included are as follows: (cid:129) Fluid mechanics, (cid:129) Continuum mechanics and thermodynamics, (cid:129) Mechanics of environmentally related systems (glacier, ice-sheet mechanics, physical oceanography, lake physics, soil motion, avalanches, debris, and mud flows), (cid:129) Vorticity and angular momentum, (cid:129) Turbulence modeling (of zeroth, first and second order), (cid:129) Regular and singular perturbations, (cid:129) Continuum mechanics and thermodynamics of mixtures, (cid:129) ContinuummechanicsandthermodynamicsofCOSSERATcontinuaandCOSSERAT mixtures, (cid:129) Theoretical glaciology, (cid:129) Shallow creeping flows of landslides, glaciers, and ice sheets, andothers.Itishopedthatweweresuccessfulindesigningacoherentpictureofthe intended text FTD. Writing the book chapters also profited from books that were written earlier by us and co-authors [1–6]. Fluid and Thermodynamics Volume 1: Basic Fluid Mechanics This volume consists of 10 chapters and begins in an introductory Chap. 1 with some historical facts, definition of the subject field and lists the most important properties of liquids. ThisdescriptiveaccountisthenfollowedinChap.2bythesimplemathematical description of the fundamental hydrostatic equation and its use in analyses of equilibrium of fluid systems and stability of floating bodies, the derivation of the ARCHIMEDEan principle and determination of the pressure distribution in the atmosphere. Chapter 3 deals with hydrodynamics of ideal incompressible (density pre- serving) fluids. Streamlines, trajectories, and streaklines are defined. A careful derivation of the balances of mass and linear momentum is given and it is shown howtheBERNOULLIequationisderivedfromthebalancelawofmomentumandhow itisusedinapplications.Inone-dimensionalsmoothflowproblemsthemomentum and BERNOULLI equations are equivalent. For discontinuous processes with jumps this is not so. Nevertheless the BERNOULLI equation is a very useful equation in Preface vii many engineering applications. This chapter ends with the balance law of moment of momentum and its application for EULER’S turbine equation. Theconservationlawofangularmomentum,presentedinChap.4,providesthe occasiontodefinecirculationandvorticityandthevorticitytheorems,amongthem those of HELMHOLTZ and ERTEL. The goal of this chapter is to build a fundamental understanding of vorticity. InChap.5acollectionofsimpleflowproblemsinidealfluidsispresented.Itis shown how vector analytical methods are used to demonstrate the differential geometric properties of vortex-free flow fields and to evaluate the motion-induced force on a body in a potential field. The concept of virtual mass is defined and two-dimensional fluid potential flow is outlined. This almanac offlows of ideal fluids is complemented in Chap. 6 by the pre- sentation of the solution techniques of two-dimensional potential flow by complex-valued function theoretical methods using conformal mappings. Potential flows around two-dimensional air foils, laminar free jets, and the SCHWARZ– CHRYSTOFFEL transformations are employed to construct the mathematical descrip- tionsofsuchflowsthroughaslitorseveralslits,aroundairwings,freejets,andin ducts bounding an ideal fluid. The mathematical physical study of viscous flows starts in Chap. 7 with the derivation of the general stress–strain rate relation of viscous fluids, in particular NAVIER–STOKES fluids and more generally, non-NEWTONian fluids. Application of these equations to viscometric flows, liquid films, POISEUILLE flow, and the slide bearing theory due to REYNOLDS and SOMMERFELD demonstrate their use in an engineeringcontext.Creepingflowforapseudo-plasticfluidwithfreesurfacethen shows the application in the glaciological-geological context. Chapter 8 continues with the study of two-dimensional and three-dimensional simple flow of the NAVIER–STOKES equations. HAGEN–POISEUILLE flow and the EKMANtheoryofthewall-nearwall-parallelflowonarotatingframe(Earth)andits generalization are presented as solutions of the NAVIER–STOKES equations in the half-space above an oscillating wall and that of a stationary axisymmetric laminar jet. This then leads to the presentation of PRANDTL’S boundary layer theory with flows around wedges and the BLASIUS boundary layer and others. InChap.9two-andthree-dimensionalboundarylayerflowsinthevicinityofa stagnationpointarestudiedasareflowsaroundwedgesandalongwedgesidewalls. Theflow,inducedinthehalfplaneabovearotating plane,isalsodetermined.The techniqueoftheboundarylayerapproachiscommencedwiththeBLASIUSflow,but more importantly, the boundary layer solution technique for the NAVIER–STOKES equations is explained by use of the method of matched asymptotic expansions. Moreover, the global laws of the steady boundary layer theory are explained with the aid of the HOLSTEIN–BOHLEN procedure. The chapter ends with a brief study of non-stationary boundary layers, in which an impulsive start from rest, flow in the vicinity of a pulsating body, oscillation induced drift current, and non-stationary plate boundary layers are studied. In Chap. 10 pipe flow is studied for laminar (HAGEN–POISEUILLE) as well as for turbulentflows;thissituationculminatesviaadimensionalanalysistothewell-known viii Preface MOODYdiagram.Thevolumeendsinthischapterwiththeplaneboundarylayerflow alongawallduetoPRANDTLandVONKÁRMÁNwiththefamouslogarithmicvelocity profile.Thislastproblemislaterreanalyzedasthecontroversiesbetweenapowerand logarithmicvelocityprofilenearwallsisstillongoingresearchtoday. Fluid and Thermodynamics Volume 2: Advanced Fluid Mechanics and Thermodynamic Fundamentals This volume consists of 10 chapters and commences in Chap. 11 with the deter- minationofthecreepingmotionaroundspheresatrestinaNEWTONianfluid.Thisisa classical problem of singular perturbations in the form of matched asymptotic expansions. For creeping flow the acceleration terms in NEWTON’s law can be ignoredtoapproximatelycalculateflowaroundthespherebythisso-calledSTOKES approximation. It turns out that far away from the sphere the acceleration terms become larger than those in the STOKES solution, so that the latter solution violates theboundaryconditionsatinfinity.Thislowestordercorrectionoftheflowaround thesphereisduetoOSEEN(1910).Inasystematicperturbationexpansiontheouter— OSEEN—series and the inner—STOKES—series with the small REYNOLDS number as perturbation parameter must be matched together to determine all boundary and transitionconditionsofinnerandouterexpansions.Thisprocedureisrathertricky, i.e.,noteasytounderstandforbeginners.Thistheory,originallydueKAPLUNandto LAGERSTRÖM hasbeen extended,and thedrag coefficient for thesphere, whichalso can be measured is expressible in terms of a series expansion of powers of the REYNOLDSnumber.However,forREYNOLDSnumberslargerthanunity,convergence to measured values is poor. About 20–30 years ago a new mathematical approach wasdesigned—theso-calledHomotopyAnalysisMethod;itisbasedonanentirely differentexpansiontechnique,andresultsforthedragcoefficientliemuchcloserto theexperimentalvaluesthanvaluesobtainedwiththe‘classical’matchedasymptotic expansion, asshown inFig. 11.11. Incidentally thelaminar flow of a viscous fluid around acylinder cananalogously betreated,butisnot contained inthis treatise. Chapter12isdevotedtotheapproximatedeterminationofthevelocityfieldina shallow layer of ice or granular soil, treated as a non-NEWTONian material flowing under the action of its own weight and assuming its velocity to be so small that STOKESflowcanbeassumed.Twolimitingcasescanbeanalyzed:(i)Inthefirst,the flowingmaterialonasteepslope(whichisthecaseforcreepinglandslidesorsnow on mountain topographies with inclination angles that are large). (ii) In the second case the inclination angles are small. Situation (ii) is apt to ice flow in large ice sheets such as Greenland and Antarctica, important in climate scenarios in a warming atmosphere. We derive perturbation schemes in terms of a shallowness parameter in the two situations and discuss applications under real-world conditions. Preface ix In shallow rapid gravity driven free surface flows the acceleration terms in NEWTON’s law are no longer negligible. Chapter 13 is devoted to such granular flows in an attempt to introduce the reader to the challenging theory of the dynamical behavior of fluidized cohesionless granular materials in avalanches of snow,debrisandmud,etc.Thetheoreticaldescriptionofmovinglayersofgranular assemblies begins with the one-dimensional depth integrated MOHR–COULOMB plastic layer flows down inclines—the so-called SAVAGE–HUTTER theory—but then continues with the general formulation of the model equations referred to topog- raphy following curvilinear coordinates with all its peculiarities in the theory and the use of shock-capturing numerical integration techniques. Chapter14onuniquenessandstabilityprovidesafirstflavorintothesubjectof laminar-turbulent transition. Two different theoretical concepts are in use and both assume that the laminar–turbulent transition is a question of loss of stability of the laminar motion. With the use of the energy method one tries to find upper bound conditions for the laminar flow to be stable. More successful for pinpointing the laminar-turbulent transition has been the method of linear instability analysis, in which a lowest bound is searched for, at which the onset of deviations from the laminar flow is taking place. In Chap. 15, a detailed introduction to the modeling of turbulence is given. Filteroperationsareintroducedtoseparatethephysicalbalancelawsintoevolution equationsfor theaveraged fields ontheonehand,andinto fluctuating orpulsating fields on the other hand. This procedure generates averages of products offluctu- ating quantities, for which closure relations must be formulated. Depending upon the complexity of these closure relations, so-called zeroth, first, and higher order turbulencemodelsareobtained:simplealgebraicgradient-typerelationsfortheflux terms, oneortwoequation models, e.g.,k-ε;k-ω,inwhichevolutionequations for the averaged correlation products are formulated, etc. This is done for density preserving fluids as well as so-called BOUSSINESQ fluids and convection fluids on a rotating frame (Earth), which are important models to describe atmospheric and oceanic flows. Chapter 16 goes back one step by scrutinizing the early zeroth order closure relations as proposed by PRANDTL, VON KÁRMÁN and collaborators. The basis is BOSSINESQ’s (1872) ansatz for the shear stress in plane parallel flow, τ12, which is expressedtobeproportionaltothecorrespondingaveragedshearrateo(cid:1)v =ox with 1 2 coefficientofproportionalityρε,whereρisthedensityandεakinematicturbulent viscosity or turbulent diffusivity [m2 s−1]. In turbulence theory the flux terms of momentum, heat, and suspended mass are all parameterized as gradient-type rela- tions with turbulent diffusivities treated as constants. PRANDTL realized from data collectedinhisinstitutethatεwasnotaconstantbutdependedonhismixinglength squared and the magnitude of the shear rate (PRANDTL 1925). This proposal was later improved (PRANDTL 1942) to amend the unsatisfactory agreement at positions where shear rates disappeared. The 1942-law is still local, which means that the REYNOLDS stress tensor at a spatial point depends on spatial velocity derivatives at the same position. PRANDTL in a second proposal of his 1942-paper suggested that theturbulentdiffusivityshoulddependonthevelocitydifferenceatthepointswhere