Dynamically Coupled Rigid Body-Fluid Flow Systems

192 Pages·2021·4.332 MB·English

by  Banavara N. Shashikanth

Banavara N. Shashikanth
Dynamically
Coupled Rigid
Body-Fluid
Flow Systems
Dynamically Coupled Rigid Body-Fluid Flow
Systems
Banavara N. Shashikanth
Dynamically Coupled Rigid
Body-Fluid Flow Systems
BanavaraN.Shashikanth
MechanicalandAerospaceEngineering
NewMexicoStateUniversity
LasCruces
NM,USA
ISBN978-3-030-82645-1 ISBN978-3-030-82646-8 (eBook)
https://doi.org/10.1007/978-3-030-82646-8
Mathematics Subject Classification: 76B47, 76M60, 37K05, 37J15, 53D20, 37N05, 70H03, 70H05,
76B07,97M50
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Preface
Thetime-honoredmethodofstudyingthedynamicsoffluid-solidinteractionprob-
lemshasbeentoeitherfixthesolidortoprescribeitsmotion,forexample,asteady
translation or a regular oscillation. Investigations then focus primarily on the dy-
namicsofthefluidflowandthestudyofthevariousfeaturesthatdevelop,suchas,
for example, boundary layers, vortices, shock waves, and turbulence. An equally
important focus, especially in engineering applications, is on computing the fluid
dynamic loads exerted on the solid with a view to designing it to withstand these
loads as it executes the prescribed motions—some familiar engineering examples
being the aerodynamic loads on an aircraft and the hydrodynamic loads on a sub-
marine.Thedynamics ofsolids—rigidanddeformable,intheabsenceofanyam-
bientfluid—isalsoawell-researchedtopic.Ineachofthesetwoprinciplebranches
of mechanics, many fundamental resultshave been obtained and many fundamen-
tal principles elucidated, starting from the very birth of the subject of theoretical
mechanics.
Relativelyfarlessattentionhasbeenpaidtothedynamicsoffluid-solidinterac-
tionproblemswherethesolidisnotconstrainedtostayputormoveinaprescribed
manner, and is free to move in response to the instantaneous fluid stress field on
its surface. The focus in such a setting is as much on the dynamics of the solid as
on the dynamics of the fluid. Moreover, the motion of the solid in turn influences
the flow of the fluid around it, thereby intricately coupling the fluid and the solid
dynamics.Thisisthedefinitionofthedynamicallycoupledsetting,andthetopic,in
the author’s opinion, merits consideration as an independent branch of theoretical
mechanics.
DespitethefactthatKirchhoffinvestigatedperhapsthefirstprobleminthisset-
tingasfarbackasthenineteenthcentury,anddespitethefactthatthereareexamples
ofsuchinteractionsallaroundusinnature—objectsliftedupbyastrongwind,ob-
jectstossedinorfloatingonwater—thetopicdidnotattractmuchresearchinterest
for a long time. The reasons are not difficult to understand. From a mathematical
pointofview,coupleddynamicsisinvariablynonlinear.Inadditiontothecomplex,
inherent nonlinearities of a fluid flow, one must now also deal with the nonlinear
responseofthebody,andthedynamiccoupling.Thetopicthereforeremainedmore
v
vi Preface
orlessdormanttillthebirthanddevelopmentofthesubjectsofdynamicalsystems
and nonlinear dynamics. Theoretical investigations in the meanwhile were mainly
restrictedtocaseswherelinearizationtechniquescouldbeused,namelyinthestudy
ofasoliddynamically coupled withwater waves (“floatingobjects”) whoseequa-
tions are amenable to linearization. The case of a solid dynamically coupled with
vortices, whose equations are not linearizable, had to wait for almost a 120years
after Kirchhoff’s work before it was investigated. And from the point of view of
applications,traditionalengineeringvehiclesdesignedforlocomotioninafluiden-
vironment (aircraft, ships, submarines, etc.) rarely required knowledge of the dy-
namicsofanuncontrolledsolidobject.
Athirdreasonmaybepostulatedastowhythesubjectdidnotgarnermuchat-
tention. The fact is dynamically coupled systems with buoyant solid objects fully
immersedinafluidenvironmenthavebeenstudied,thoughagainnotsomuchthe
nonlinearities. This is essentially the subject of bodies falling or rising freely in a
fluidenvironment.Thenotionofabodyfreetorespondtofluidstresseswasperhaps
more often associated with such problems. Here gravity typically is the dominant
feature,andevenwheninterestingdynamicalfeaturesareseen,suchas,forexam-
ple,thedeformationsofrisingbubblesortheoscillationsofafallingsheetofpaper,
gravityisanessentialingredientofsuchdynamics.Foralongtime,therewasprob-
ablynotenoughmotivation,theoreticallyorintherealworld,toconsiderthecase
ofneutrallybuoyantbodiesfullyimmersedinafluidenvironment.
The subject of dynamical systems/nonlinear systems has grown enormously in
thepastfourdecades orso,whileatthesametimetherehasbeendevelopment of
new”green”paradigmsforthedesignofengineeringdevicesdynamicallyinteract-
ing with fluids. For example, underwater biomimetic vehicles that typically have
limited propulsive power, unlike traditional vehicles, and need to maneuver opti-
mallyinthefluidenvironment.Understandingofthecoupleddynamicsoffluidand
vehicle therefore becomes paramount. There are also other engineering examples,
wherethedynamiccouplingispartial,forexample,floatingoffshorewindturbines,
and it is important when dealing with such problems to have a knowledge of the
fullydynamicallycoupledproblem.
Theoretical investigations on the topic of dynamically coupled systems there-
forehavebeengrowingataslowbutsteadyrate.Itseemstimelyandappropriateto
presentacompilationofimportantworkdonesofarinthearea,andtopresentthisin
aframeworkwhosefoundationsareKirchhoff’sequations.Thisalsoallowsbring-
ing together under a common banner topics which have traditionally been studied
in separate research groups, such as, for example, the system of a rigid body and
waves,andthesystemofarigidbodyandvortices.ThefocusisonLagrangianand
Hamiltonianmethods.Functionalanalyticapproachesthatinvestigateexistenceand
smoothness of solutions in Eulerian and Navier-Stokes settings are not covered.1
Following an introduction to Kirchhoff’s equations of motion, the book discusses
severalextensionsofKirchhoff’swork,withthefocusontheextensionsinvolving
1Theinterestedreadermaylookupjournalssuchas,forexample,JournalofMathematicalFluid
MechanicsandArchiveforRationalMechanicsandAnalysisforpapersthatusesuchapproaches,
andreferencestherein.
Preface vii
vortices.TheequationsofmotionsofthesesystemsandtheirLagrangianandHamil-
tonianformulationsarepresentedanddiscussed.Insomecases,detailedderivations
are included, especially on the Euler-Lagrange equations for some of the models
which, to the best of my knowledge, have not been published before. In addition,
whereverappropriate,Ihavealsodiscussedapplications.
Inwritingthisbook,Ihavetriedtostrikeabalancebetweenthemoderntheories
ofreductionofLagrangianandHamiltoniansystemswithsymmetry—orgeometric
mechanics, as the subject is more commonly known—and classical fluid mechan-
ics.Ihavenotwrittenthisasapurelygeometricmechanicsbookandthereasonsare
twofold:firstly,thereremainsmuchtobedoneinapplyingtheseverytechnicalthe-
ories towhatisessentiallyadifficult problem, especially theextensions discussed
inthelaterchapters,andthesearetasksnoteasilydonesinglehandedly.Secondly,
I want to emphasize that this topic is as much about applications as about theory,
anditisforthisreasonthatonechapterofthisbookisdevotedtoexperimentaland
numericalworkdonebymycollaborators.
Myintentionistomakethebookunderstandabletoanyonewithbasicknowledge
of Lagrangian and Hamiltonian mechanics, and of theoretical fluid mechanics—
regardlessofwhethertheyaremathematicians,physicists,orengineers.Toaidthe
readerunfamiliarwiththesubject,anappendixwhichbrieflyintroducesbasiccon-
ceptsandterminologyingeometricmechanicshasbeenincluded.Ibelievethebook
is written at a level which would make it comprehensible to an advanced doctoral
candidatewiththisbackgroundknowledge.
Finally,itgoeswithoutsayingthatthisisanongoingproject.Fromthepointof
viewofapplicationssuchas,forexample,biomimeticlocomotion,thereareatleast
twomoresequelsthatonecouldthinkof:oneonthetopicofdynamicallycoupled
elasticbodiesandfluids,andthesecondonthetopicofaddingcontrolactuationto
thesemodelsanddevelopingcontroltheoreticmodels.
LasCruces,NM,USA BanavaraShashikanth
May2021
Contents
1 Kirchhoff’sInsufficiently-CelebratedEquationsofMotion.......... 1
1.1 Introduction ............................................... 1
1.2 Kirchhoff’sEquations....................................... 2
1.3 TheLegacyofKirchhoff’sEquations.......................... 7
1.4 TheGeometricMechanicsofKirchhoff’sEquations ............. 10
1.4.1 The Euler–Lagrange and Hamilton’s Equations
intheSpatially-FixedFrame........................... 12
1.5 ExtendingKirchhoff’sModel ................................ 21
1.5.1 TheSumPoissonBracket ............................. 23
2 TheAdditionofVortices ........................................ 25
2.1 TheImportanceofVorticity.................................. 26
2.2 SingularVortexModels ..................................... 28
2.2.1 TheN-Point-VortexModel ............................ 29
2.2.2 TheN VortexRingModel............................. 33
3 DynamicallyCoupledRigidBody+PointVorticesinR2 ............. 43
3.1 N-Point-VorticesandStationaryRigidBoundaries:C.C.Lin’s
Problem .................................................. 45
3.2 N-Point-Vortices Dynamically Coupled with a Single Rigid
ContourofArbitraryShape .................................. 48
3.2.1 TheEuler–LagrangeEquationsinaSpatially-Fixed
Frame.............................................. 53
3.2.2 TheVorticalMomentaandReciprocityRelations ......... 58
3.3 N-Point-Vortices Dynamically Coupled with a Single Rigid
CircularContour ........................................... 63
3.3.1 TheHalf-SpaceModel................................ 70
ix
x Contents
4 DynamicallyCoupledRigidBody+VortexRingsinR3.............. 79
4.1 N VortexRingsandaSingleStationaryRigidBoundary .......... 80
4.2 N VortexRingsDynamicallyCoupledwithaSingleRigidBody
ofArbitraryShape.......................................... 89
4.2.1 TheEuler–LagrangeEquationsinaSpatially-Fixed
Frame.............................................. 93
4.2.2 TheVorticalMomentaandReciprocityRelations .........100
4.3 N VortexRingsDynamicallyCoupledwithaRigidSphere........105
4.3.1 TheAxisymmetricModelofaSphereandN Circular
Rings ..............................................110
5 ViscousEffectsandTheirModeling ..............................121
5.1 SystemMomentumBalanceLawsintheViscousSetting .........121
5.2 Some Experimental and Numerical Work of Vortex Rings
CollidingwithRigidBodies..................................124
6 MiscellaneousExtensions .......................................133
6.1 DynamicallyCoupledRigidBody+freeSurface .................133
6.1.1 AFreeSurfaceDynamicallyCoupledwithaCompletely
SubmergedSingleRigidBodyofArbitraryShape.........138
6.2 DynamicallyCoupledN RigidBodiesintheAbsence
ofVorticity................................................145
6.2.1 TheEuler–LagrangeEquationsinaSpatially-Fixed
Frame..............................................147
6.3 ASingleBuoyantRigidBodyAboveanImpermeableFlat
Boundary .................................................152
ABriefIntroductiontoGeometricMechanics .........................159
BLeadingOrderBehaviorofVelocityandVectorPotentialFieldsofa
CurvedVortexFilament ........................................165
CHamiltonianFunctionandVectorFieldintheHalf-spaceModelfor
N =2[151]....................................................171
p
References.........................................................173
Index .............................................................185
Chapter 1
Kirchhoff’s Insufficiently-Celebrated Equations
of Motion
1.1 Introduction
GustavRobertKirchhoff(1824–1887)madepioneeringcontributionstophysicsand
engineering. His most famous contributions were in electrical circuits, black body
radiation,elasticity,andspectroscopy.InhisbiographicalsketchonKirchhoff,P.G.
Taitwrites[161]:
Whiletherearenowadayshundredsofmenthoroughlyqualifiedtoworkout,toitsdetails,
aproblemalreadycouchedinsymbols,therearebutfewwhohavethegiftofputtingan
entirelynewphysicalquestionintosuchaform.ThenamesofStokes,Thomson,andClerk-
MaxwellwillatonceoccurtoBritishreadersasinstancesofmenpossessingsuchpowerin
amarkeddegree.KirchhoffhadinthisrespectnosuperiorinGermany,excepthislifelong
friendandcolleaguev.Helmholtz.
Hiscontributionstofluidmechanicsareperhapsnotaswell-known.Indeedmost
graduatelevelfluidmechanicstextbooks,withtheobviousexceptionsofLamb[92]
and Milne-Thomson [117], contain no description of the equations of motion that
hederivedinhispaperof1869[81].Itisthereforeappropriatethatthefirstchapter
ofthisbook,inwhichtheseequationsformacentraltheme,bedevotedtothem.
Hisequationsarefirstdescribedintheirclassicalformandthenintheframework
of modern Lagrangian and Hamiltonian mechanics. The reader is referred to the
excellent text by Marsden and Ratiu [109] for many of the geometric mechanics
and variational tools used in this chapter and the remainder of the book. A brief
introductiontothesenotionsisalsopresentedinAppendixA.
©SpringerNatureSwitzerlandAG2021 1
B.N.Shashikanth,DynamicallyCoupledRigidBody-FluidFlowSystems,
https://doi.org/10.1007/978-3-030-82646-8 1