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Compressible dynamics of cavitating 3-D multi-phase flows [Elektronische Ressource] / İsmail Hakkı Sezal

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¨ ¨Technische Universitat Munchen¨Lehrstuhl fur FluidmechanikFachgebiet GasdynamikCompressible Dynamics of Cavitating3-D Multi-Phase Flows˙Ismail Hakkı SezalVollst¨ andiger Abdruck der von der Fakult¨ at fur¨ Maschinenwesen der TechnischenUniversit¨ at Munc¨ hen zur Erlangung des akademischen Grades einesDoktor-Ingenieursgenehmigten Dissertation.Vorsitzender: Univ.-Prof. Dr.-Ing. habil. R. SchillingPrufer¨ der Dissertation:1. Univ.-Prof. Dr.-Ing. habil. G.H. Schnerr2. habil. N.A. AdamsDie Dissertation wurde am 03.03.2009 bei der Technischen Universit¨ at Munc¨ hen ein-gereicht und durch die Fakult¨ at fur¨ Maschinenwesen am 24.06.2009 angenommen.AcknowledgementsThis Ph.D. thesis is the outcome of my research concluded at the Fachgebiet Gasdy-namik of Lehrstuhl fur¨ Fluidmechanik at the Technische Universit¨ at Munchen¨ .Duringmy employment at the university from 2004 until April 2008 and after I left my as-sistantship position many people have contributed in the work presented here andsupported me throughout these years, whom I would like to address here.First of all I want to express my sincere thanks to my supervisor, Prof. Dr.-Ing. habil.G.H. Schnerr for his guidance, insight and patience throughout this study. I shall alsostate my deepest appreciation and admiration about his exceptional enthusiasm andinterest in my work. Our fruitful discussions on the subject made enormous contribu-tionstothisthesis.

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Published 01 January 2009
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¨ ¨Technische Universitat Munchen
¨Lehrstuhl fur Fluidmechanik
Fachgebiet Gasdynamik
Compressible Dynamics of Cavitating
3-D Multi-Phase Flows
˙Ismail Hakkı Sezal
Vollst¨ andiger Abdruck der von der Fakult¨ at fur¨ Maschinenwesen der Technischen
Universit¨ at Munc¨ hen zur Erlangung des akademischen Grades eines
Doktor-Ingenieurs
genehmigten Dissertation.
Vorsitzender: Univ.-Prof. Dr.-Ing. habil. R. Schilling
Prufer¨ der Dissertation:
1. Univ.-Prof. Dr.-Ing. habil. G.H. Schnerr
2. habil. N.A. Adams
Die Dissertation wurde am 03.03.2009 bei der Technischen Universit¨ at Munc¨ hen ein-
gereicht und durch die Fakult¨ at fur¨ Maschinenwesen am 24.06.2009 angenommen.Acknowledgements
This Ph.D. thesis is the outcome of my research concluded at the Fachgebiet Gasdy-
namik of Lehrstuhl fur¨ Fluidmechanik at the Technische Universit¨ at Munchen¨ .During
my employment at the university from 2004 until April 2008 and after I left my as-
sistantship position many people have contributed in the work presented here and
supported me throughout these years, whom I would like to address here.
First of all I want to express my sincere thanks to my supervisor, Prof. Dr.-Ing. habil.
G.H. Schnerr for his guidance, insight and patience throughout this study. I shall also
state my deepest appreciation and admiration about his exceptional enthusiasm and
interest in my work. Our fruitful discussions on the subject made enormous contribu-
tionstothisthesis.
Deepest gratitude are also due to the members of the examination committee; Prof.
Dr.-Ing. habil. R. Schilling has willingly accepted the chairperson position and Prof. habil. N.A. Adams has acted as my co-examiner with a great interest in the
subject.
Next I would like to mention our remarkable team at the Fachgebiet Gasdynamik that
I miss even today. In particular, my good old roommate and friend Steffen Schmidt;
more than four years we shared not only the same office but most of the time the same
challenges both at work and outside. His excellent knowledge in mathematics helped
us a lot to build our scheme and understand the underlying theory in numerics. He
was also always on my side and calmed all of us at the most needed and stressful times
during the past years. Without his help this work could never have been completed.
Thank you Steffen, I owe you so much! Then comes my good friend and colleague
Nisar Al-Hasan; as soon as he joined our group we instantly started to work together
and supported each other almost on everything. He was always the first person that
I consulted whenever I needed assistance on any technical or official complication.
Moreover, we shared lots of common interest that made us good friends. His driving
force and encouragements helped me a lot to finish my thesis.
I also had the pleasure to work with Matthias Thalhamer and Marcus Giglmaier during
their student theses. Matze parallelized our code and handled all the technical details to
use the LRZ clusters. He also worked intensively on visualization efforts and processed
huge amount of data, including the wonderful cover picture used in this thesis. Marcus
helped us a lot in preparing the workshops and also printed my endless draft versions
without any complaints.
Many thanks go to my colleagues at the Aerothermal Technologies Laboratory in GE
Global Research - Europe; my former lab manager Dr.-Ing. Michael B. Schmitz has
always supported and encouraged me to finish my thesis. Further thanks go to myteammates and project leaders Dr. Alexander Simpson and Dr. Christian Aalburg,
they were extremely understanding and helpful in all the matters related to both thesis
and work. I would also like to mention my roommate Rodrigo Rodriguez-Erdmenger
for his willingness to help on everything and our discussions on many subjects.
I am also grateful to KSB foundation, Stuttgart, who supported this project and fi-
nanced it.
Finally, I would like to thank my family and my Meltem, for their invaluable patience,
trust and encouragement during this thesis. They never stop believing in me even the
times I have lost my confidence and hope.
˙Munich, August 2009 Ismail H. SezalContents
Nomenclature V
Abstract IX
Zusammenfassung XI
1 Introduction 1
1.1 Background................................. 1
1.2 Motivation ................................. 1
1.2.1 Fluid Compressibility . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 CavitationPhenomenon...................... 4
1.2.2.1 CavitationTypes .................... 6
1.2.2.2 Collapse Dynamics of Cavitation Bubbles . . . . . . . 7
1.3 NumericalMotivation ........................... 9
1.4 LiteratureOverview . 1
1.4.1 TheoreticalStudies ........................ 11
1.4.2 ExperimentalStudies ....................... 12
1.4.3 NumericalStudies ......................... 17
1.5 ThesisOverview .............................. 19
2 Physical Model 21
2.1 Equilibrium vs. Non-equilibrium Processes . . . . . . . . . . . . . . . 21
2.1.1 Flows with Non-equilibrium Effects . . . . . . . . . . . . . . . . 21
2.1.2 Vapor-liquid Equilibria . . . . . . . . . . . . . . . . . . . . . . 30
2.2 EffectofMolecularViscosityandTurbulence .............. 34
2.3 GoverningEquations............................ 36
2.3.1 DifferentialFormoftheEquations ................ 37
2.3.2 IntegralFormoftheEquations .................. 38
III CONTENTS
2.4 Equilibrium Two-phase Model . . . . . . . . . . . . . . . . . . . . . . 40
2.4.1 RelaxationTimeinCavitatingFlows............... 41
2.4.2 IntegralAverageFormulation................... 42
2.5 EquationofStateandSpedofSound 4
3 Numerical Method 55
3.1 GoverningEquations............................ 5
3.2 GeometryDefinition 56
3.3 ConvectiveFluxcalculation........................ 58
3.3.1 RiemannProblem . 58
3.3.2 TheHLLCRiemannsolver .................... 60
3.3.3 TheAUSMFamilyofMethods .................. 62
3.3.4 TheLowMachNumberProblem 63
3.3.5 Modification for Liquid Flows - Hybrid Formulation . . . . . . 64
3.4 HigherOrderReconstruction ....................... 65
3.5 TimeIntegration.............................. 67
3.5.1 1stOrderTimeIntegration .................... 67
3.5.2 4-StageRunge-KutaMethod................... 68
3.5.3 TimeStepCalculation....................... 69
3.6 Initial and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 70
3.6.1 InitialConditions ......................... 70
3.6.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 71
3.6.2.1 SolidWals 71
3.6.2.2 Periodic Boundaries . . . . . . . . . . . . . . . . . . . 72
3.6.2.3 Inlet and Outlet Boundaries . . . . . . . . . . . . . . 73
4 Validation 77
4.1 1-DShockTube .............................. 7
4.1.1 IdealGas.............................. 7
4.1.2 LiquidWater............................ 82
4.2 3-D Bubble Collapse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3 Discretization and Mesh Dependence of the Cavitation Regions . . . . 92
4.4 ValidationSummary 96CONTENTS III
5 Results and Discussion 97
5.1 InjectionNozzles.............................. 97
5.1.1 2-DPlaneInjectionNozle .................... 98
5.1.1.1 Symmetry Break-up of the Flow Field . . . . . . . . . 100
5.1.1.2 CavitationCycle.....................106
5.1.2 3-DInjectionNozzlewithSwirl..................109
5.1.3 3-D Injection Nozzle with Swirl and Divergence . . . . . . . . . 114
5.1.4 3-DMulti-holeInjectionNozle..................18
5.2 HydraulicMachinery............................123
5.2.1 2-D NACA 0015 Hydrofoil . . . . . . . . . . . . . . . . . . . . 123
5.2.2 3-D NACA 0009 Twisted Wing - Half Wing Calculation . . . . 135
5.2.3 3-D NACA 0009 Wing - Full Wing . . . . 145
6 Conclusions 161
6.1 Summary ..................................161
6.2 RecommendationsforFurtherDevelopment ...............162
References 165
A Physical Constants and Relations 175
A.1SaturationVariables............................175
A.2LiquidandGasConstants.........................176
B Viscous Flow Formulation 179
B.1Navier-StokesEquations..........................179
B.2FavreAveragedNavier-StokesEquations.................180
B.3 Single-phase Turbulence Modeling . . . . . . . . . . . . . . . . . . . . 184
B.3.1 Wilcox k-ωModel .........................184
B.4NumericalFormulation185
B.4.1 NondimensionalizationoftheVariables .............185
B.4.2 Governing Equations with k-ωTurbulenceModel........186
B.4.3 Discussion about Two-phase Turbulence Modeling . . . . . . . 187
B.4.4 DiffusiveFluxCalculation.....................18
B.4.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 189
B.5 Test Case Calculations for Single-phase Ideal Gas Flows . . . . . . . . 190
B.5.1 LaminarFlatPlate ........................190IV CONTENTS
B.5.2 TurbulentFlatPlate........................191
B.5.3 RAE 2822 Airfoil . . . . . . . . . . . . . . . . . . . . . . . . . . 192
B.6Outlook...................................194
C CATUM Manual 195
C.1 Single-phase Inviscid Ideal Gas Calculation . . . . . . . . . . . . . . . 200
C.2 S Inviscid Liquid Water . . . . . . . . . . . . . . 202
C.3 Two-phase Cavitating Hydrofoil Calculation . . . . . . . . . . . . . . . 204Nomenclature
Latin Symbols
c [m/s] Speed of sound
c [m] Chord length
c [m/s] Equilibrium speed of sounde
c [-] Skin friction coefficientf
c [-] Drag coefficientD
c [-] Pressure drag coefficientD,p
c [-] Lift coefficientL
c [-] Pressure coefficientp
c [J/kg· K] Specific heat at constant pressurep
c [J/kg· K] Specific heat att volumev
CFL [-] Courant-Friedrichs-Lewy number
e [J/kg] Mass-specific internal energy
E [J/kg] Total internal energy
f [1/s] Frequency
f [-] Favre mass-weighted operator
f [J/kg] Helmholtz free energy per unit mass
∗f , f [-] Turbulence model closure coefficientsβ β
f [-] Vector for the convective fluxes in x-direction
f [-] Vector for the viscous fluxes in xv
g [J/kg] Gibbs free energy per unit mass
g [-] Vector for the convective fluxes in y-direction
g [-] Vector for the viscous fluxes in yv
h [J/kg] Mass-specific enthalpy
H [J/kg] Total enthalpy
h [-] Vector for the convective fluxes in z-direction
h [-] Vector for the viscous fluxes in zv
k [J/kg] Turbulent kinetic energy
K [-] Runge-Kutta stage
l [m] Characteristic length
L [J/kg] Latent heat of vaporization
M [-] Mach number
n [-] Normal vector
ni [-] Maximum grid points in x-direction
nj [-] Maximum grid points in y
nk [-] Maximum grid points in z-direction
VVI NOMENCLATURE
p [Pa] Pressure
2P [N/m · s] Turbulent production term
Pr [-] Prandtl number
Pr [-] Turbulent Prandtl numberT
2q [W/m ] Heat flux
q [-] Vector of conserved variables
r [m]Radius
R [m] Bubble radius
R [m] Initial bubble radius
0
Re [-] Reynolds number
R [-] Residual vector
s [J/kg· K] Mass-specific entropy
s [-] Source vector
2S [m ] Surface area vector
S [m/s] Wave speeds in Riemann problem
S [N/m] Surface tension
S [1/s] Strain tensorij
St [-] Strouhal number
t [s]Time
T [K] Temprature
Tu [m/s] Turbulence intensity
u [m/s] Velocity component in x-direction
u [-] Vector of primitive variables
v [m/s] Velocity component in y-direction
3v [m /kg] Specific volume
3V [m]Volume
w [m/s] Velocity component in z-direction
W [-] Weighting function in Galerkin method
x,y,z [m] Cartesian coordinates
x [-] Mass fraction of vapor
Greek Symbols
α [-] Void fraction
◦α [ ] Angle of attack
α [-] Runge-Kutta coefficienti
∗β , β [-] Turbulence model closure coefficients
0
γ [-] Limiter function
Γ [-] Control volume surface
δ [-] Kronecker deltaij
2 3 [m /s ] Turbulent dissipation rate
κ [-] Specific heat capacity
λ [W/m· K] Heat conductivity
λ [-] Eigenvalues of the equation systemi
μ [kg/m· s] Dynamic viscosity