Mathematical models of fluiddynamics: modelling, theory, basic numerical facts: an introduction

181 Pages·2003·0.953 MB·English

by  Rainer Ansorge

Rainer Ansorge
Mathematical Models of Fluiddynamics
Modelling,Theory,Basic Numerical Facts –
An Introduction
Rainer Ansorge
Mathematical Models
of Fluiddynamics
Modelling,Theory, Basic Numerical Facts –
An Introduction
WILEY-VCH GmbH & Co.KGaA
Author This book was carefully produced.Nevertheless,
authors,editors and publisher do not warrant the
Prof.Dr.Rainer Ansorge information contained therein to be free of
University of Hamburg errors.Readers are advised to keep in mind thar
Institute for Applied Mathematics statements,data,illustrations,procedural details
or other items may inadvertently be inaccurate.
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ISBN 3-527-40397-3
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Preface
Mathematicalmodellingistheprocessofreplacingproblemsfromoutsidemathemat-
icsbymathematicalproblems. Thesubsequentmathematicaltreatmentofthismodel
bytheoreticaland/ornumericalproceduresworksasfollows:
1. Transition from the outer-mathematics phenomenon to a mathematical descrip-
tion,whatatthesametimeleadstoatranslationofproblemsformulatedinterms
oftheoriginalproblemintomathematicalproblems.
This task forces the scientistor engineer who intends to use mathematicaltools
to
– cooperatewithexpertsworkinginthefieldtheoriginalproblembelongsto.
Thus, he has to learn the language of these experts and to understand their
wayofthinking(teamwork).
– createortoacceptanidealizeddescriptionoftheoriginalphenomena,i.e. an
a-priori-neglectionofpropertiesoftheoriginalproblemswhichareexpected
tobeofnogreatrelevancewithrespecttothequestionsunderconsideration.
Thesesimplificationsareusefulinordertoreducethedegreeofcomplexity
ofthemodelaswellasofitsmathematicaltreatment.
– identify structures within the idealized problem and to replace these struc-
turesbysuitablemathematicalstructures.
2. Treatmentofthemathematicalsubstitute.
Thistasknormallyrequires
– independentactivityofthetheoreticallyworkingperson.
– treatmentoftheproblembytoolsofmathematicaltheory.
– thesolutionoftheparticularmathematicalproblemsoccuringbytheuseof
these theoretical tools, i.e differential equations or integral equations, op-
MathematicalModelsofFluiddynamics.RainerAnsorge
Copyright(cid:1)c 2003Wiley-VCHVerlagGmbH&Co.KGaA,Weinheim
ISBN:3-527-40397-3
8 Preface
timal control problems or systems of algebraic equations etc. have to be
solved. Oftennumericalproceduresaretheonlywaytodothisandtoanswer
the particularquestionsunder considerationat leastapproximately. The er-
roroftheapproximatesolutioncomparedwiththeunknownso-calledexact
solutiondoes normally not reallyaffect the answer to the originalproblem,
providedthatthenumericalmethodaswellasthenumericaltoolsareofsuf-
ficientlyhighaccuracy. Inthiscontext,itshouldberealizedthatthequestion
for an exact solution does not make sense because of the idelizations men-
tioned above and since the initial data presented with the original problem
normallyoriginatefromstatisticsorfromexperimentalmeasurements.
3. Retranslationoftheresults.
Thequalitativeandquantitativestatementsreceivedfromthemathematicalmod-
el need now to be retranslated into the language by which the original problem
was formulated. This means that the results have now to be interpreted with
respecttotheirreal-world-meaning. Thisprocessagainrequiresteamworkwith
theexpertsfromthebeginning.
4. Modelcheckup.
Afterretranslation,theresultshavetobecheckedwithrespecttotheirrelevance
and accuracy, e.g. by experimental measurements. This work has to be done
by the experts of the original problem. If the mathematical results coincide
sufficiently well with the results of experiments stimulated by the theoretical
forecasts, the mathematical part is over and a new tool to help the physicists,
engineersetc. insimilarsituationsisborn.
Otherwise–ifnotriviallogicalorcomputationalerrorscanbefound–themodel
hastoberevised. Inthissituation,thegapbetweenmathematicalandrealresults
canonlyoriginatefromtooextensiveidealizationsinthemodellingprocess.
The development of mathematical models does not only stimulate new experiments
anddoesnotonlyleadtoconstructive-prognostic–hencetechnical–toolsforphysi-
cists or engineers but is also important from the point of view of the theory of cog-
nition: Itallowstounderstandconnectionsbetweendifferentelementsoutoftheun-
structuredsetofobservationsor–inotherwords–tocreatetheories.
Fieldsofapplicationsofmathematicaldescriptionsarewellknownsincecenturiesin
physics, engineering, also in music etc. In modern biology, medicine, philology and
economymathematicalmodelsareused,too,andthisevenholdsforcertainfieldsof
artslikeorientalornaments.
Preface 9
Thisbookpresentsanintroductionintomodelsoffluidmechanics,leadstoimportant
propertiesoffluidflowswhichcantheoreticallybederivedfromthemodelsandshows
some basic ideas for the construction of effective numerical procedures. Hence, all
aspectsoftheoreticalfluiddynamicsareaddressed,namelymodelling,mathematical
theoryandnumericalmethods.
We do not expect the reader to be familiar with a lot of experimental experiences.
Theknowledgeofsomefundamentalprinciplesofphysicslikeconservationofmass,
conservationofenergyetc. issufficient. Themostimportantidealizationis–contrary
tothemolecularstructureofmaterials–theassumptionoffluid-continua.
Concerning mathematics, it can help to understand the text more easily if the reader
isacquaintedwithsomebasicelementsof
– LinearAlgebra
– Calculus
– PartialDifferentialEquations
– NumericalAnalysis
– TheoryofComplexFunctions
– FunctionalAnalysis.
Functional Analysis does only play a role in the somewhat general theory of dis-
cretizationalgorithmsin chapter6. Inthischapter,alsothequestionoftheexistence
ofweakentropysolutionsoftheproblemsunderconsiderationisdiscussed. Physicsts
and engineers are normally not so much interested in the treatment of this problem.
Nevertheless, it had to be included into this presentation in order not to leave this
questionopen. Thenon-existenceofasolutionshowsimmediatelythatamodeldoes
not fit the reality if there is a measurable course of physical events. Existence theo-
remsarethereforeimportantnotonlyfromthepointofviewofmathematicians. But,
ofcourse,readerswhoarenotacquaintedwithsomefunctionalanalyticterminology
mayskipthischapter.
With respect to models and their theoretical treatment as well as with respect to nu-
mericalproceduresoccuringinsections4.1,5.3,6.4andchapter7,abriefintroduction
intomathematicalfluidmechanicslikethisbookcanonlypresentbasicfacts. Butthe
authorhopesthatthisoverviewwillmakeyoungscientistsinterestedinthisfieldand
that it can help people working in institutes and industries to become more familiar
withsomefundamentalmathematicalaspects.
10 Preface
Finally,Iwishtothankseveralcolleaguesforsuggestions,particularlyThomasSonar,
whocontributedtochapter7whenweorganizedajointcourseforgraduatestudents1,
andDr. MichaelBreuss,whoreadthemanuscriptcarefully. LastbutnotleastIthank
thepublishers,especiallyDr. AlexanderGrossmann,fortheirencouragement.
RainerAnsorge
Hamburg,September2002
1Partsofchapters1and5aretranslationsfrompartsofsections25.1,25.2,29.9of: Ansorgeand
Oberle:Mathematikfu¨rIngenieure,vol.2,2nded.,Berlin,Wiley-VCH2000.
Contents
Preface 7
1 IdealFluids 13
1.1 ModellingbyEuler’sEquations . . . . . . . . . . . . . . . . . . . . 13
1.2 CharacteristicsandSingularities . . . . . . . . . . . . . . . . . . . . 24
1.3 PotentialFlowsand(Dynamic)Buoyancy . . . . . . . . . . . . . . . 30
1.4 MotionlessFluidsandSoundPropagation . . . . . . . . . . . . . . . 47
2 WeakSolutionsofConservationLaws 51
2.1 GeneralizationofwhatwillbecalledaSolution . . . . . . . . . . . . 51
2.2 TrafficFlowExamplewithLossofUniqueness . . . . . . . . . . . . 56
2.3 TheRankine-HugoniotCondition . . . . . . . . . . . . . . . . . . . 62
3 EntropyConditions 69
3.1 EntropyinCaseofIdealFluids . . . . . . . . . . . . . . . . . . . . . 69
3.2 GeneralizationoftheEntropyCondition . . . . . . . . . . . . . . . . 74
3.3 UniquenessofEntropySolutions . . . . . . . . . . . . . . . . . . . . 80
3.4 TheAnsatzduetoKruzkov . . . . . . . . . . . . . . . . . . . . . . . 92
4 TheRiemannProblem 97
4.1 NumericalImportanceoftheRiemannProblem . . . . . . . . . . . . 97
4.2 TheRiemannProblemintheCaseofLinearSystems . . . . . . . . . 99
5 RealFluids 103
5.1 TheNavier-StokesEquationsModel . . . . . . . . . . . . . . . . . . 103
5.2 DragForceandtheHagen-PoiseuilleLaw . . . . . . . . . . . . . . . 111
5.3 StokesApproximationandArtificialTime . . . . . . . . . . . . . . . 116
5.4 FoundationsoftheBoundaryLayerTheory;FlowSeparation . . . . . 122
5.5 StabilityofLaminarFlows . . . . . . . . . . . . . . . . . . . . . . . 131
12 Contents
6 ExistenceProofforEntropySolutionsbyMeans ofDiscretization
Procedures 135
6.1 SomeHistoricalRemarks . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2 ReductiontoPropertiesofOperatorSequences . . . . . . . . . . . . 136
6.3 ConvergenceTheorems . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7 TypesofDiscretizationPrinciples 151
7.1 SomeGeneralRemarks . . . . . . . . . . . . . . . . . . . . . . . . . 151
7.2 TheFiniteDifferenceCalculus . . . . . . . . . . . . . . . . . . . . . 156
7.3 TheCFLCondition . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
7.4 Lax-RichtmyerTheory . . . . . . . . . . . . . . . . . . . . . . . . . 162
7.5 ThevonNeumannStabilityCriterion. . . . . . . . . . . . . . . . . . 168
7.6 TheModifiedEquation . . . . . . . . . . . . . . . . . . . . . . . . . 171
7.7 DifferenceSchemesinConservationForm . . . . . . . . . . . . . . . 173
7.8 TheFiniteVolumeMethodonUnstructuredGrids . . . . . . . . . . . 176
SomeExtensiveMonographs 181
Index 183