Dynamics of Compressible Fluids: A Textbook

316 Pages·2022·5.191 MB·English

by  Oleksandr Girin

Oleksandr Girin
Dynamics
of Compressible
Fluids
A Textbook
Dynamics of Compressible Fluids
Oleksandr Girin
Dynamics of Compressible
Fluids
A Textbook
OleksandrGirin
DepartmentofTheoreticalMechanics
OdesaI.I.MechnikovNationalUniversity
Odesa,OdesaRegion,Ukraine
ISBN 978-3-031-11261-4 ISBN 978-3-031-11262-1 (eBook)
https://doi.org/10.1007/978-3-031-11262-1
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Inlovingmemoryofmyparents,
CATHERINEANDGEORGE,
whoselovehasbeenmylifeblood
throughout mylife.
Preface
The purpose of this book is to present the main theoretical points of gas-dynamic
science, starting from its origins. A course on compressible fluid dynamics is a
compulsorypartoftheAerospaceEngineeringcurriculumatmanyuniversities.The
coursepresented,DynamicsofCompressibleFluids(DCF),canservebothasabasic
textbookandasanadditionalreadingaid,dependingonthespecificcurriculumat
aparticularuniversity.Themostcompressiblemediaaregases,sothecompressible
mediadynamicsisbetterknownasGasDynamics,bothnameswewilluseequally.
Unfortunately, theentirecourse on DCF isoften broken upintoseparate parts,
whichareincludedinvariousotherrelatedcourses,forthesakeofpracticalneeds.
However,knowingthebasicsandunderstandingtherootsineachspecificsituation
isalwaysnecessarytoensurethereliabilityoftheresult.Quitesimilarly,havinga
propereducationin“pure”GasDynamics,theauthor,likemanyotherresearchers,
spenthiswholelifedealingwithitsapplicationstovariouspracticalproblems,where
compressiblefluiddynamicswasinvolvedasthisorthatelementofresearch.
TheauthordevelopedhisinterestinGasDynamicsatNovosibirskStateUniversity
(formerUSSR),where hestudiedthecourseonthelectures ofAcademician L.V.
Ovsiannikov. The presented textbook is based on the experience that the author
gainedwhileworkingasalecturerattheOdesaI.I.MechnikovNationalUniversity.
The textbook is intended not only to help in studying the course in Dynamics of
CompressibleFluidsbutalsoserveasabasisforthesuccessfulsolutionofmodern
practicalproblemsinvariousbranchesoftechnology.
Thiscoursewastestedinthecurriculaofthespecialty“Mechanics”oftheOdesa
NationalUniversityinthe1990–2000sandapprovedbytheMinistryofEducation
and Science of Ukraine as a textbook for students of mechanical and mathemat-
ical faculties of universities. This textbook contains the main chapters of classical
Gas Dynamics, and its content corresponds to the requirements for the programs
of lecture courses of universities. The theoretical study of gas motion is based on
generallyacceptedapproachesofContinuumMechanics,andthematerialpresented
isgroundedonitselementaryprinciplesandconceptsandassumesthatthereader
isfamiliarwithgeneraluniversitycourses:TheoreticalMechanics,Hydrodynamics,
vii
viii Preface
Aerodynamics, Mathematical Analysis, Differential Equations, Algebra, Thermo-
dynamics,andNumericalMethods.Thetextbookisdesignedtofamiliarizevarious
specialistswiththemathematicalfoundationsoftheoreticalGasDynamics.In-depth
assimilation of the basics of the theory and methods of investigating the motions
of compressible media contributes to a better understanding of the essence of the
specificproblemsunderconsideration.
Adescriptionofthestructureandcontentofthevariousmathematicalmodelsof
motionofcompressiblemediaiscarriedthroughtheentirecourse,fromtheapplica-
tionofthelawsofconservationtospecificformulaeforoneoranotherelementary
problem.Thetextisprovidedwithalargenumber(≈130)ofillustrations.
In preparing the text, the author sought to ensure the mathematical rigor of the
presentationofthematerial.Themathematicalessenceofthesubjectispresentedin
aformcomprehensiblenotonlytoteachersandstudentsspecializingintheproblems
ofMechanics,butalsotoeveryonewhoisinterestedinapplicationsofmathematics.
Thetextbookwillbeusefulforteachers,researchers,universitystudents,graduate
and postgraduate students studying Fluid Mechanics, Aerospace Engineering, and
MechanicalEngineering,aswellasforallresearcherswithaneedforapplications
ofthetheoryofcompressiblefluidmotions.
This book is intended as a Dynamics of Compressible Fluids textbook suitable
forbothbasicandsupportingundergraduateandgraduatelecturecourses.However,
theAppendixsectionadditionallymakesabridgefromDCFtoComputationalFluid
Dynamics(CFD)andconsistsofadescriptionofthoseclassicalnumericalmethods
which are inherent to Dynamics of Compressible Fluids and have it as a parent
discipline(MethodofCharacteristics,Godunov’sMethod),sothatAppendixservesto
applytheDynamicsofCompressibleFluidsconceptstothedevelopmentofnumerical
methods.Therefore,thisadditionalmaterialmaybeusefulaspartofanintroductory
courseinCFDforbeginners.
Asaresultofthepopularityofthemethodofnumericalsimulation,thenumber
ofengineersandscientistsusingCFDhasgrownsignificantly.Manyoftheseusers
donothaveacollegedegreeinCFD,sothefieldisincreasinglycoveredinunder-
graduatecurricula.ItisimportanttounderstandthatevenaCFDuser(asopposed
toadeveloper)mustbefamiliarwiththeunderlyingtheoryofFluidDynamicsand
rootnumericalalgorithmsinordertoworkcompetentlywithCFD.
The Godunov method and themethod of characteristics aredescribed indetail,
since they are essentially gas-dynamic numerical methods. The (u,p)-diagram
method (which isvery usefulfor the analysis of wave interactions and its features
areillustratedbymanyexamplesinChap.7)isalsogivenineverydetail,sinceitis
oneofthemaintoolsintheGodunovmethod.
Not every section of the material presented will be included by the lecturer in
the main course, but in general, the content is consistent with the syllabus of the
universitycourseonDynamicsofCompressibleFluids.
Preface ix
Wepresentthisbookinthehopethatitwillcontributetoadeeperunderstanding
of the dynamics of compressible fluids and an intelligent and creative approach to
thedevelopmentandapplicationofCFD.
Odesa,Ukraine OleksandrGirin
March2022
Contents
1 GeneralEquationsofGasMotion ................................ 1
1.1 TheThermodynamic Model ofaPerfectGas;Adiabatic
Formulae .................................................. 2
1.1.1 Internal State of a Gas Particle; Thermodynamic
Variables ........................................... 2
1.1.2 PerfectGasModel;PolytropicGas ..................... 3
1.1.3 AdiabaticFormulae .................................. 6
1.2 GoverningEquationsofGasMotion;MathematicalModel
ofanIdeal,Non-heat-conductiveGas .......................... 8
1.3 SpeedofPropagationofSmallDisturbancesinIdealGas;
SoundSpeed ............................................... 13
1.4 ThermodynamicsofaMovingGas ............................ 17
1.4.1 Bernoulli—Saint-VenantEquation;Enthalpy ............. 17
1.4.2 StagnationGasState;IsentropicFormulae ............... 19
1.4.3 Laval’s Number; Other Characteristic States
ofaMovingGas ..................................... 21
References ..................................................... 23
PartI One-DimensionalSteadyGasFlow
2 ContinuousFlows .............................................. 27
2.1 Equations of One-Dimensional Steady Gas Flow; Rule
ofaStreamReversal ........................................ 27
2.2 GasOutflowfromReservoir;Saint-Venant—VantzelFormula ..... 30
2.3 SupersonicOutflowMode;Laval’sNozzle ..................... 34
References ..................................................... 36
3 DiscontinuityinaGasFlow ..................................... 37
3.1 ConservationLawsataStrongDiscontinuitySurface ............ 38
3.2 ClassificationofStrongDiscontinuities;Shocks ................ 42
3.3 NormalShockTheory ....................................... 45
3.4 NormalShockRegularities .................................. 49
3.4.1 VelocityJump ....................................... 49
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xii Contents
3.4.2 PressureJump ....................................... 50
3.4.3 DensityJump ........................................ 51
3.4.4 EntropyJump ....................................... 51
3.5 ShockAdiabaticCurveandItsProperties ...................... 54
3.5.1 EquationofShockAdiabaticCurve ..................... 54
3.5.2 “Asterisk”Property ................................... 55
3.5.3 LimitingDegreeofGasCompressioninShockWaves ..... 56
3.5.4 ApproximationofStrongShocks ....................... 57
3.5.5 ApproximationofWeakShocks ........................ 57
References ..................................................... 58
PartII One-DimensionalUnsteadyGasFlow
4 GoverningEquationsandInitial-Boundary-ValueProblems ........ 61
4.1 GeometryofOne-DimensionalFlows ......................... 62
4.2 EquationsofMotioninEuler’sForm;InitialandBoundary
Conditions ................................................ 63
4.2.1 Euler’sEquationsofMotion ........................... 63
4.2.2 InitialConditions .................................... 64
4.2.3 BoundaryConditions ................................. 64
4.3 EquationsofMotioninLagrange’sForm ...................... 66
4.4 EquationsofMotioninCharacteristicForm;theCharacteristic
Curves;CompatibilityRelationsAlongtheCharacteristics ....... 67
4.5 TheMethodofCharacteristics ................................ 71
4.6 GeneralizedCauchyProblem(TypeI Problem);theDomain
ofDeterminacy;DomainofDependence;RangeofInfluence ..... 74
4.7 TheGoursatProblem(TypeII Problem) ....................... 79
4.8 CombinedProblemofaSpecialType(TypeIII Problem) ......... 80
4.9 Characteristics as Trajectories of a Possible Weak
DiscontinuityofaSolution .................................. 82
4.9.1 Relationships Along the Weak Discontinuity
Trajectory ........................................... 82
4.9.2 BreakupofArbitraryWeakDiscontinuity ................ 85
References ..................................................... 87
5 IsentropicGasFlowswithPlaneWaves ........................... 89
5.1 RiemannMethod ........................................... 90
5.1.1 RiemannInvariants ................................... 90
5.1.2 RiemannVariables;RiemannMethod ................... 90
5.1.3 TheEuler–PoissonEquation ........................... 91
5.1.4 TheRemarkableCaseγ =3 .......................... 93
5.2 TheRiemannWaves ........................................ 93
5.2.1 SimpleWaves ....................................... 93
5.2.2 AdjoiningTheorem .................................. 95
5.2.3 SimpleWaveEquations ............................... 96
5.2.4 PropertiesofSimpleWaves ............................ 97