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Computational Modeling of Thermodynamic Effects in Cryogenic Cavitation

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COMPUTATIONAL MODELING OF THERMODYNAMIC EFFECTS IN CRYOGENIC CAVITATION By YOGEN UTTURKAR A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Yogen Utturkar

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To my wife and parents

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iv ACKNOWLEDGMENTS I am grateful to several individuals for their support in my dissertation work. The greater part of this work was made possibl e by the instruction of my teachers, and the love and support of my family and friends. It is with my heartfelt gratitude that I acknowledge each of them. Firstly, I would like to express sincere tha nks and appreciation to my advisor, Dr. Wei Shyy, for his excellent guidance, support, trust, and patience th roughout my doctoral studies. I am very grateful for his rema rkable wisdom, thought-provoking ideas, and critical questions duri ng the course of my research wo rk. I thank him for encouraging, motivating, and always prodding me to perform beyond my own limits. Secondly, I would like to express sincere gratitude towards my co-advisor, Dr. Nagaraj Arakere, for his firm support and caring attitude during some difficult times in my graduate studies. I also would like to express my appreciation to the members of my dissertation committee Dr. Louis Cattafesta, Dr. James Klausner and Dr. Don Slinn, for their valuable comments and expertise to better my work. I deeply thank Dr. Siddha rth Thakur (ST) for providing me substantial assistance with the STREAM code and for his cordial suggestions on research wo rk and career planning. My thanks go to all the members of our lab, with whom I have had the privilege to work. Due to the presence of all these wonderf ul people, work is more enjoyable. In particular, it was a delightful experience to collaborate with Jiongyang Wu, Tushar Goel, and Baoning Zhang on vari ous research topics.

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v I would like to express my deepest grat itude towards my family members. My parents have always provide d me unconditional love. They have always given top priority to my education, whic h made it possible for me to pursue graduate studies in the United States. I would like to thank my gr andparents for their selfless affection and loving attitude during my early years. My wife’s parents a nd her sister’s family have been extremely supportive throughout my gra duate education. I grea tly appreciate their trust in my abilities. Last but never least, I am thankful beyond words to my wife, Neeti Pathare. Together, we have walked through this memora ble, cherishable, and joyful journey of graduate education. Her honest and unfaltering love has been my most precious possession all these times. I thank her for standing besides me every time and every where. To Neeti and my pare nts, I dedicate this thesis!

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES.............................................................................................................ix LIST OF FIGURES.............................................................................................................x LIST OF SYMBOLS.......................................................................................................xiv ABSTRACT...................................................................................................................xvii i CHAPTER 1 INTRODUCTION AND RESEARCH SCOPE...........................................................1 1.1 Types of Cavitation............................................................................................2 1.2 Cavitation in Cryogenic Fluids – Thermal Effect..............................................4 1.3 Contributions of the Current Study....................................................................8 2 LITERATURE REVIEW.............................................................................................9 2.1 General Review of Recent Studies.........................................................................9 2.2 Modeling Thermal Effects of Cavitation..............................................................19 2.2.1 Scaling Laws..............................................................................................19 2.2.2 Experimental Studies..................................................................................24 2.2.3 Numerical Modeling of Thermal Effects...................................................27 3 STEADY STATE COMPUTATIONS.......................................................................33 3.1 Governing Equations............................................................................................33 3.1.1. Cavitation Modeling..................................................................................35 3.1.1.1 Merkle et al. Model..........................................................................35 3.1.1.2 Sharp Interfacial Dynamics Model (IDM).......................................35 3.1.1.3 Mushy Interfacial Dynamics Model (IDM).....................................37 3.1.2 Turbulence Modeling.................................................................................40 3.1.3 Speed of Sound (SoS) Modeling................................................................41 3.1.4 Thermal Modeling......................................................................................42 3.1.4.1 Fluid property update.......................................................................42 3.1.4.2 Evaporative cooling effects..............................................................43

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vii 3.1.5 Boundary Conditions..................................................................................44 3.2 Results and Discussion.........................................................................................45 3.2.1 Cavitation in Non-cryogenic Fluids...........................................................45 3.2.2 Cavitation in Cryogenic Fluids...................................................................50 3.2.2.1 Sensitivity analyses..........................................................................52 3.2.2.2 Assessment of cryogenic cavitation models over a wide range of conditions................................................................................................58 4 TIME-DEPENDENT COMPUTATIONS FOR FLOWS INVOLVING PHASE CHANGE....................................................................................................................69 4.1 Gallium Fusion.....................................................................................................71 4.1.1 Governing Equations..................................................................................72 4.1.2 Numerical Algorithm..................................................................................73 4.1.3 Results........................................................................................................78 4.1.3.1 Accuracy and grid dependence........................................................79 4.1.3.2 Stability............................................................................................82 4.1.3.3 Data analysis by redu ced-order de scription.....................................83 4.2 Turbulent Cavitating Flow under Cryogenic Conditions.....................................88 4.2.1 Governing Equations..................................................................................88 4.2.1.1 Speed of sound modeling.................................................................89 4.2.1.2 Turbulence modeling........................................................................89 4.2.1.3 Interfacial velocity model.................................................................89 4.2.1.4 Boundary conditions........................................................................91 4.2.2 Numerical Algorithm..................................................................................91 4.2.3 Results........................................................................................................94 5 SUMMARY AND FUTURE WORK.......................................................................101 5.1. Summary............................................................................................................101 5.2 Future Work........................................................................................................103 APPENDIX A BACKGROUND OF GLOB AL SENSITIVITY ANALYSIS.................................105 B REVIEW AND IMPLEMENTATION OF POD......................................................107 B.1 Review...............................................................................................................107 B.2 Mathematical Background.................................................................................110 B.3 Numerical Implementation................................................................................112 B.3.1 Singular Value Decomposition (SVD)....................................................112 B.3.2 Post-processing the SVD Output.............................................................113 B.4 Flowchart...........................................................................................................114 B.5 Computing Efforts..............................................................................................114

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viii LIST OF REFERENCES.................................................................................................116 BIOGRAPHICAL SKETCH...........................................................................................126

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ix LIST OF TABLES Table page 1. Properties of some cryogens in comp arison to water at N.B.P, 1.01 bars.......................6 2. Source terms in cavitation models.................................................................................13 3. Variants of the k model.............................................................................................16 4. Summary of studies on thermal effects in cavitation.....................................................32 5. Flow cases chosen for the hydrofoil geometry..............................................................51 6. Flow cases chosen for the ogive geometry....................................................................51 7. Location of the primary vortex for the St = 0.042, Ra = 2.2 105 and Pr = 0.0208 case........................................................................................................................... 80

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x LIST OF FIGURES Figure page 1. Different types of cavitation (a) Trave ling cavitation (b) Cloud cavitation (c) Sheet cavitation (d) Supe rcavitation (e) Vortex cavitation..................................................3 2. Saturation curves for water, Nitrogen, and Hydrogen....................................................5 3. Phasic densities along liquid-vapor satu ration line for water and liquid Nitrogen.........7 4. General classification of nu merical methods in cavitation...........................................11 5. Variation of Speed of Sound with phase fraction.........................................................14 6. Two cavitation cases for B -factor analysis...................................................................21 7. Schematic of bubble model fo r extracting speed of sound...........................................27 8. Schematic of cavity models (a) Distin ct interface with vaporous cavity (Sharp IDM) (b) Smudged interface with mushy cavity (Mushy IDM)..............................36 9. Behavior of /l and /l vs. l for the two models; /100lv and 0.09 ...................................................................................................................40 10. Pressure-density and pressure-enthalpy diagrams for liquid ni trogen in the liquidvapor saturation regime (Lemmon et al. 2002). Lines denote isotherms in Kelvin.......................................................................................................................43 11. Illustration of the computational do mains for hemispherical projectile and NACA66MOD hydrofoil (non-cryogenic cases).....................................................46 12. Pressure coefficients over the hemispherical body (0.4 ); D is the diameter of the hemispherical projectile. (a) Impact of grid refinement for Mushy IDM (b) Comparison between pressure coefficients of different models on the coarse grid.47 13. Cavity shapes and flow structure for different cavitation models on hemispherical projectile (0.4 ). (a) Merkle et al. Model (b) Sharp IDM (c) Mushy IDM........48 14. Pressure coefficient over the NACA66M OD hydrofoil at two different cavitation numbers; D is hydrofoil chord length. (a) 0.91 (b) 0.84 ............................49

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xi 15. Illustration of the comput ational domain accounting the tunnel for the hydrofoil (Hord 1973a) and 0.357-inch ogive geomet ry (Hord 1973b) (cryogenic cases)......50 16. Non-cavitating pressure distribution (a ) case ‘290C’, D re presents hydrofoil thickness and x represents distance from the circular bend (b) case ‘312D’, D represents ogive diameter and x represents distance from the leading edge............52 17. Sensitivity of Merkle et al. Model prediction (surface pr essure and temperature) to input parameters namely destC and prodC for the hydrofoil geometry......................53 18. Main contribution of each design variable to the sensitivity of Merkle et al. (1998) model prediction; case ‘290C’ (a) Surface pressure (b) Surface temperature.........55 19. Pressure and temperature prediction for Me rkle et al. Model for the case with best match with experimental pressure; ****0.85;0.85;1.1;0.9destvCtL ............56 20. Pressure and temperature prediction for Me rkle et al. Model for the case with best match with experimental temperature;****1.15;0.85;1.1;1.1destvCtL ........56 21. Sensitivity of Mushy IDM predicti on for case ‘290C’ (surface pressure and temperature) to the exponential transitioning parameter .......................................58 22. Surface pressure and temperature for 2-D hydrofoil for all cases involving liquid Nitrogen. The results referenced as ‘Mus hy IDM’ and ‘Merkle et al. Model’ are contributions of the present study............................................................................59 23. Cavitation number (2()/(0.5)vl p pTU ) based on the local vapor pressure – Merkle et al. Model. Note the values of 2()/(0.5)vl p pTU for the cases ‘290C’ and ‘296B’ are 1.7 and 1.61, respectively..........................................61 24. Cavity shape indicated by liquid phase fraction for case ‘290C’. Arrowed lines denote streamlines (a) Merkle et al. M odel – isothermal assumption (b) Merkle et al. Model with therma l effects (c) Mushy IDM – with thermal effects.............62 25 Evaporation (m) and condensation (m ) source term contour s between the two cavitation models – case ‘290C’. Refer to equations (3.8) and (3.18) for the formulations.............................................................................................................63 26. Surface pressure and temperature fo r 2-D hydrofoil for cases involving liquid Hydrogen. The results referenced as ‘Mus hy IDM’ and ‘Merkle et al. Model’ are contributions of the present study............................................................................65 27. Surface pressure and temperature for axisymmetric ogive for all the cases (Nitrogen and Hydrogen). The results refe renced as ‘Mushy IDM’ and ‘Merkle et al. Model’ are contributi ons of the present study.................................................67

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xii 28. Schematic of the 2D Gallium square geometry with the Boundary Conditions (Shyy et al. 1998).....................................................................................................71 29. 2D interface location at various instants for St = 0.042, Ra = 2.2 105 and Pr = 0.0208. White circles represent interface locations obtained by Shyy et al. (1995) on a 41 41 grid at time instants at t = 56.7s, 141.8s, & 227s respectively..79 30. Grid sensitivity for the St = 0.042, Ra = 2.2 105 and Pr = 0.0208, 2D case (a) Centerline vertical ve locity profiles at t = 227s (b) Flow stru cture in the upperleft domain at t = 57s; 41 41 grid (c) Flow structure in the upper-l eft domain at t = 57s; 81 81 grid.....................................................................................................80 31. 3D interface location at t = 57s & 227s for St = 0.042, Ra = 2.2 105 and Pr = 0.0208 case on a 41 41 41 grid. Top and bottom: adiabatic; North and West: T = 0; South and East: T = 1 (heated walls)................................................................82 32. Interface location and fl ow pattern for the 3D case, St = 0.042, Ra = 2.2 105 and Pr = 0.0208 case at various z locations, at t = 227s..................................................82 33. POD modes showing velocity streamlines ( ();1,2,3,4iri ) for St = 0.042, Ra = 2.2 103 and Pr = 0.0208 case. (,)(,)qrtVrt ......................................................84 34 Scalar coefficients ( ();1,2,...,8iti ) for St = 0.042, Ra = 2.2 103 and Pr = 0.0208 case. (,)(,)qrtVrt ....................................................................................85 35. Time instants when coefficients of respective POD modes show the first peak; St = 0.042, Ra = 2.2 103 & Pr = 0.0208 case.............................................................86 36. Horizontal-centerline vertical velocities for St = 0.042, Ra = 2.2 103 and Pr = 0.0208 case at t = 3s.................................................................................................86 37. Comparison between CFD soluti on and re-constructed solution for St = 0.042, Ra = 2.2 103 and Pr = 0.0208 case at t = 20s. Contours represent velocity magnitude and lines represent streamlines...............................................................87 38. Comparison between CFD soluti on and re-constructed solution for St = 0.042, Ra = 2.2 103 and Pr = 0.0208 case at t = 3s. Contours represent velocity magnitude and lines represent streamlines...............................................................88 39. Pressure history at a point near inlet, 0.11 D where D is the chord length of the hydrofoil (a) ‘ 283B’ (b) ‘290C’.........................................................................95 40. Cavity shape and flow structure fo r case ‘283B’. Lines denote streamlines..............95 41. Pressure history at a point near the inlet for the case ‘290C’. St denotes the nondimensional perturba tion frequency ( / f DUft )...............................................96

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xiii 42. Pressure history at a point near the inlet for the cases ‘296B’ a nd ‘290C’. The non-dimensional perturbation frequency ( / f DUft ) is 1.0..............................97 43 POD modes for the velocity field; ();1,2,3iri ; (,)(,)qrtVrt ...........................98 44 POD modes for the velocity field; ();1,2,3iri ; (,)(,)qrtVrt ...........................98

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xiv LIST OF SYMBOLS : cavitation number : boundary layer/cavity thickness L: length (of cavity or cavitating object) q: heat flux; generali zed flow variable in POD representation V: total volume t: time x, y, z: coordinate axes r: position vector : streamwise direction in the curvilinear co-ordinate system i, j, k, n: indices ,,: uvw velocity components p: pressure T: temperature : density : volume fraction f: mass fraction h: sensible enthalpy s: entropy

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xv c: speed of sound : ratio of the two specific heats for gases : stress tensor P: production of turbulent energy 12,,,,,:kCCCC constants Q: total kinetic energy (inclusive of turbulent fluctuations) k: turbulent kinetic energy : turbulent dissipation F: filter function for filter-based modeling : dynamic viscosity K: thermal conductivity : volume flow rate L: latent heat : P C specific heat :pC pressure coefficient a: thermal diffusivity m : volume conversion rate B: B-factor to gauge thermal effect : dimensional parameter to assess thermal effect : control parameter in the Mushy IDM; coefficient of thermal expansion g: gravitational acceleration U: velocity scale D: length scale (such as hydrofoil ch ord length or ogive diameter)

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xvi R: bubble radius CFL: Courant, Freidricks, and Levy Number St: Stefan Number Ra: Rayleigh Number Pr: Prandtl Number Re: Reynolds Number : difference; filter size in filter-based turbulence modeling : gradient or di vergence operator : POD mode : time-dependent coefficient in the POD series : energy content in the respective POD mode Subscripts: : reference value (typically in let conditions at the tunnel) 0: initial conditions s: solid l: liquid v: vapor m: mixture f: friction Q: discharge c: cavity L: laminar t: turbulent

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xvii I: interfacial n: normal to local gradient of phase fraction sat: saturation conditions dest: destruction of the phase prod: production of the phase +: condensation -: evaporation A, H, S, M, G, B: terms in a discretized equation nb: neighboring nodes P: at the cell of interest Superscripts/Overhead symbols: +: condensation -: evaporation ‘: fluctuating component *: normalized value; updated value in the context of PISO algorithm : vector -: average ~: Favre-averaged n: time step level k: iteration level

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xviii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COMPUTATIONAL MODELING OF THERMODYNAMIC EFFECTS IN CRYOGENIC CAVITATION By Yogen Utturkar August 2005 Chair: Wei Shyy Cochair: Nagaraj Arakere Major Department: Mechanic al and Aerospace Engineering Thermal effects substantially impact th e cavitation dynamics of cryogenic fluids. The present study strives towards developing an effective computational strategy to simulate cryogenic cavitation aimed at liquid rocket propuls ion applications. We employ previously developed cavitation and compressibi lity models, and incorporate the thermal effects via solving the enthalpy equation and dynamically updating the fluid physical properties. The physical impli cations of an existing cavit ation model are reexamined from the standpoint of cryogeni c fluids, to incorporate a mu shy formulation, to better reflect the observed “frosty” appearance w ithin the cavity. Performance of the revised cavitation model is assessed against the exis ting cavitation models and experimental data, under non-cryogenic and cr yogenic conditions. Steady state computations are performed over a 2D hydrofoil and an axisymmetric ogive by employing real fluid propertie s of liquid nitrogen and hydrogen. The thermodynamic effect is demonstrated under consistent conditions via the reduction in

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xix the cavity length as the reference temperature tends towards the critical point. Justifiable agreement between the comput ed surface pressure and temperature, and experimental data is obtained. Specifically, the predicti ons of both the models are better; for the pressure field than the temp erature field, and for liquid nitrogen than liquid hydrogen. Global sensitivity analysis is performed to examine the sensitivity of the computations to changes in model parameters and uncer tainties in material properties. The pressure-based operato r splitting method, PISO, is adapted towards typical challenges in multiphase computations such as multiple, coupled, and non-linear equations, and sudden changes in flow vari ables across phase boundaries. Performance of the multiphase variant of PISO is examined firstly for the problem of gallium fusion. A good balance between accuracy and stability is observed. Time-dependent computations for various cases of cryogeni c cavitation are further perfor med with the algorithm. The results show reasonable agreement with the experimental data. Impact of the cryogenic environment and inflow perturba tions on the flow structure a nd instabilities is explained via the simulated flow fields and the reduced order st rategy of Proper Orthogonal Decomposition (POD).

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1 CHAPTER 1 INTRODUCTION AND RESEARCH SCOPE The phenomenon by which a liquid forms ga s-filled or vapor-filled cavities under the effect of tensile stress produced by a pre ssure drop below its vapor pressure is termed cavitation (Batchelor 1967). Cavi tation is rife in fluid m achinery such as inducers, pumps, turbines, nozzles, marine propellers, hydrofoils, journal bearings, squeeze film dampers etc. due to wide ranging pressure va riations along the flow This phenomenon is largely undesirable due to its ne gative effects namely noise, vibration, material erosion etc. Detailed description of these effects can be readily obtained from literature. It is however noteworthy that the cavitation phenomenon is also associated with useful applications. Besides drag re duction efforts (Lecoffre 1999), bi omedical applications in drug delivery (Ohl et al. 2003) and shock wa ve lithotripsy (Tangua y and Colonuis 2003), environmental applications for decomposing organic compounds (Kakegawa and Kawamura 2003) and water disinfection (Kal amuck et al. 2003), a nd manufacturing and material processing applications (Soyama and Macodiyo 2003) are headed towards receiving an impetus from cavitation. Comprehe nsive studies on variety of fluids such as water, cryogens, and lubricants have provi ded significant insights on the dual impact of cavitation. Experimental research methods including some mentioned above have relied on shock waves (Ohl et al. 2003), acoustic wa ves (Chavanne et al. 2002), and laser pulses (Sato et al. 1996), in addition to hydrodynam ic pressure changes, for triggering cavitation. Clearly, research potential in term s of understanding the mechanisms and characteristics of cavitation in different flui ds, and their applica tions and innovation is

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2 tremendous. The applicability and contributio ns of the present study within the abovementioned framework are described later in this chapter. 1.1 Types of Cavitation Different types of cavitation are observ ed depending on the flow conditions and fluid properties. Each of them has distinct characteristics as compared to others. Five major types of cavitation have been described in literature. They are as follows: (a) Traveling cavitation It is characterized by indivi dual transient cavitie s or bubbles that form in the liquid, expand or shrink, and move downstream (Kna pp et al. 1970). Typically, it is observed on hydrofoils at small angles of attack. The de nsity of nuclei present in the upstream flow highly affects the geometries of the bubbl es (Lecoffre 1999). Traveling cavitation is illustrated in Figure 1(a). (b) Cloud cavitation It is produced by vortex shedding in the flow field and is associated with strong vibration, noise, and erosion (Kawanami et al. 1997). A re-entrant jet is usually the causative mechanism for this type of cavitation (Figure 1(b)). (c) Sheet cavitation It is also known as fixed, attached cavity or pocket cavitation (Figure 1(c)). Sheet cavitation is stable in quasi-steady sense (Knapp et al. 1970). Though the liquid-vapor interface is dependent on the na ture of flow, the closure regi on is usually characterized by sharp density gradients and bubble cl usters (Gopalan and Katz 2000). (d) Supercavitation Supercavitation can be considered as an extremity of sheet cavitation wherein a substantial fraction of the body surface is engu lfed by the cavity (Figure 1(d)). It is

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3 observed in case of supersonic underwater projectiles, and has interesting implications on viscous drag reduction (Kirschner 2001). (e) Vortex cavitation It is observed in the core of vortices in re gions of high shear (Figure 1(e)). It mainly occurs on the tips of rotating bl ades and in the separation zone of bluff bodies (Knapp et al. 1970). Figure 1. Different types of cavitation (a) Traveling cavitatio n (b) Cloud cavitation (c) Sheet cavitation (d) Supercavitation (e) Vortex cavitation Reproduced from Franc et al. (1995) with permission from EDP Sciences

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4 1.2 Cavitation in Cryogenic Fl uids – Thermal Effect Cryogens serve as popular fuels for th e commercial launch vehicles while petroleum, hypergolic propellants, and solid s are other options. Typically, a combination of liquid oxygen (LOX) and liqui d hydrogen (LH2) is used as rocket propellant mixture. The boiling points of LOX and LH2 under sta ndard conditions are -183 F and -423 F, respectively. By cooling and compressing thes e gases from regular conditions, they are stored into smaller storage tanks. The co mbustion of LOX and LH2 is clean since it produces water vapor as a by-pr oduct. Furthermore, the power/g allon ratio of LH2 is high as compared to other alternatives. Though st orage, safety, and extreme low temperature limits are foremost concerns for any cryogenic application, rewards of mastering the use of cryogens as rocket propellants are substantial (NASA Online Facts 1991). A turbopump is employed to supply the low temperature propellants to the combustion chamber which is under extremely hi gh pressure. An inducer is attached to the turbopump to increase its efficiency. Design of any space vehicle component is always guided by minimum size an d weight criteria. Conseque ntly, the size constraint on the turbopump solicits hi gh impellor speeds. Such high sp eeds likely result in a zone of negative static pressure (pre ssure drop below vapor pressu re) causing the propellant to cavitate around the inducer bl ades (Tokumasu et al. 2002). In view of the dire consequences, investigation of cavitation ch aracteristics in cryoge ns, specifically LOX and LH2, is an imperative task. Intuitively, physical and thermal properties of a fluid are expected to significantly affect the nature of cavitation. For ex ample, Helium-4 shows anomalous cavitation properties especially past the -point temperature mainly due to its transition to superfluidity (Daney 1988). Besides, quantum tunneling also attributes to cavity

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5 formation in Helium-4 (Lambare et al. 1998). Cavitation of Helium-4 in the presence of a glass plate (heterogenous cavitation) has lately produced some unexpected results (Chavanne et al. 2002), which are in contrast to its regular cavita ting pattern observed under homogenous conditions (without a fo reign body). Undoubtedly, a multitude of characteristics and research avenues are offered by different types of cryogenic fluids. The focus of the current study is, however, rest ricted to cryogenic fluids such as LOX, LH2, and liquid Nitrogen due to their aforesaid strong relevan ce in space applications. It is worthwhile at the outset to contrast th eir behavior/physical properties to water. Figure 2. Saturation curves for water, Nitrogen, and Hydrogen£ £ Obtained from REFPROP v 7.0 by Lemmon et al. (2002) Water Nitrogen Hydrogen

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6 Table 1. Properties of some cryogens in comparison to water at N.B.P, 1.01 bars Substance Specific heat (J/Kg.K) Liquid density (kg/m3) Liquid/Vapor density Thermal conductivity (W/mK) Vaporization heat (KJ/Kg) Water 4200 958 1603 681 2257 H2 9816 71 53 100 446 N2 2046 809 175 135 199 O2 1699 1141 255 152 213 Source: Weisend et al. (1998) Refer to Figure 2 and Table 1. The operati ng point of cryogenic liquids is generally quite close to the critical point unlike water. Furthermore, as indicated by Figure 2, the saturation pressure curves for cryogens demonstrate a much steeper slope v/s temperature, as compared to water. Conseque ntly, the vapor pressure of liquids such as LH2 and liquid Nitrogen is expect ed to show great sensitivity to small temperature drops. Since cavitation is predominantly governed by th e vapor pressure, the significance of this thermodynamic sensitivity on the flow problem is clear upfront. Further insight can be obtaine d from Table 1. Liquid-to-v apor density ratio in the case of cryogenic fluids is s ubstantially lower than water. Thus, these fluids require a greater amount of liquid, and in turn latent heat, than water to cav itate (form vapor) under similar flow conditions. Furthermore, the thermal conductivity for the low temperature fluids is consistently lower than water. Thes e facts indicate that th e sensible-latent heat conversion in cryogenic fluids is expected to develop a noticeable temperature gradient surrounding the cavitation region. The impact of this local te mperature drop is magnified by the steep saturation curv es observed in Figure 2. Su bsequently, the local vapor pressure experiences a substa ntial drop in comparison to th e freestream vapor pressure leading to suppression in the cavitation in tensity. Experimental (Hord 1973a, 1973b) and

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7 numerical results (Deshpande et al. 1997) on sheet cavitation have shown a 20-40% reduction in cavity length due to the th ermodynamic effects in cryogenic fluids. Figure 3. Phasic densities along liquid-va por saturation line for water and liquid Nitrogen Additionally, the physical proper ties of cryogenic fluids, othe r than vapor pressure, are also thermo-sensible, as illustrated in Fi gure 3. Thus, from a standpoint of numerical computations, simulating cryogenic cavitatio n implies a tight c oupling between the nonlinear energy equation, momentum equations, and the cavitation model, via the iterative update of fluid properties (suc h as vapor pressure, densities, specific heat, thermal conductivity, viscosity etc.) wit h changes in the local temp erature. Encountering these difficulties is expected to yield a numerical methodology specifically well-suited for cryogenic cavitation, and forms the key emphasis of the present study. Obtained from REFPROP v 7.0 by Lemmon et al. (2002) Water Liq. N2

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8 1.3 Contributions of the Current Study The major purpose of the present study is to develop a robust and comprehensive computational tool to simulate cavitating fl ow under cryogenic cond itions. The specific contributions of the endeavor are summarized as follows: (a) A review of the experimental and com putational studies on cryogenic cavitation (b) Coupling of energy equation to the existing cavitation framework in conjunction with iterative update of the real fluid properties with respect to the local temperature (c) Adaptation of an existing cavitation model (Senocak and Shyy 2004a, 2004b) to accommodate the physics of the mushy natu re of cavitation observed in cryogenic fluids (Hord 1973, Sarosdy and Acosta 1961) (d) Demonstration of the impact of ther modynamic effects on cavitation over wideranging temperatures, for tw o different cryogenic flui ds. Assessment of the computational framework alongside availabl e experimental and numerical data. (e) Global sensitivity analysis of the computational predictions (pressure and temperature) with respect to the cavita tion model parameters and the temperaturedependent material properties, via employing the response surface approach. (f) Adaptation of the pressure-based operator splitting method, PISO (Issa 1985), to multiphase environments typically characte rized by strong interactions between the governing equations and steep variations of flow variab les across the phase boundary (g) Assessment of the stabili ty and accuracy of the non-iterative algorithm (PISO variant) on the test problem of Gallium fusion. (h) Time-dependent computations of cryogenic cavitation (with the PISO variant) by applying perturbation to the inlet temperature (i) Employment of Proper Orthogonal Decomposition (POD) to offer a concise representation to the simulated CFD data.

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9 CHAPTER 2 LITERATURE REVIEW Cavitation has been the focal point of numerous experimental and numerical studies in the area of fluid dynamics. A review of these st udies is presented in this chapter. Since cryogenic cavitat ion remains the primary interest of this study, the general review on cavitation studies is purely restrict ed according to relevance. Specifically, the numerical approaches in term s of cavitation, compressibi lity, and turbulence modeling are briefly reviewed in the earlier section w ith reference to pertinent experiments. The later section mainly delves into the issues of thermal effects of cavitation. Current status of numerical strategies in modeling cryogenic cavitation, and their merits and limitations are reported to underscore the gap br idged by the current research study. 2.1 General Review of Recent Studies Computational modeling of cavitation has co mplemented experimental research on this topic for a long time. Some earlier st udies (Reboud et al. 1990, Deshpande et al. 1994) relied on potential flow assumption (Eul er equations) to simulate flow around the cavitating body. However, simulation strategi es by solving the Navi er-Stokes equations have gained momentum only in the last decad e. Studies in this regard can be broadly classified based on their in terface capturing method. Ch en and Heister (1996) and Deshpande et al. (1997) adopted the interface tracking Marker and Cell approach in their respective studies, which were characterized by time-wise grid regeneration and the constant cavity-pressure assumption. The liquid-vapor interface in these studies was explicitly updated at each time step by m onitoring the surface pressure, followed by its

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10 reattachment with an appropria te wake model. This effort was mainly well-suited for only sheet cavitation. The second category, wh ich is the homogeneous flow model, has been a more popular approach, wherein the modeling for both phases is adopted via a single-fluid approach. The density change over the interface is simply modeled by a liquid mass fraction (l f ) or a liquid volume fraction (l ) that assumes values between 0 and 1. The mixture density can be expresse d in terms of either fraction as follows: (1)mlllv (2.1) 1lv m ll f f (2.2) The precise role of various cavitation models, which are reviewed later, is prediction of this volume/mass fraction as a function of space and time. Both, density-based and pressure-based me thods have been successfully adopted in conjunction with the single-fluid method in numerous studies. Due to unfeasibility of LES or DNS methods for multiphase flow, RANS approach through kturbulence model has been mostly employed in the past studies. The main limitation of density-based methods (Merkle et al. 1998, Ahuja et al. 2001, Lindau et al. 2002, Iga et al. 2003) is requirement of pre-conditioning (Kunz et al. 2000) or the artificial density approach for flows which may be largely incompressible. Pressure-based methods (Ventikos and Tzabiras 2000, Athavale et al. 2001, Seno cak and Shyy 2002, Singha l et al. 2002) on the other hand are applicable over a wide range of Mach numbers. Modeling the speed of sound in the mixture region and computationa l efficiency for unsteady calculations are the main issues for pressure-based solver s. A broad classification of the numerical methods is illustrated in Figure 4.

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11 Interface Capturing Strategy Explicit Tracking Methods (Marker and Cell) Homogenous Flow Model Euler Equations (Potential Flow Assumption) Navier Stokes Solver Artificial Compressibility Based or Pressure Based Methods Figure 4. General classification of numerical methods in cavitation Within the broad framework of above met hods, several cavitation, compressibility, and turbulence models have been deployed, whic h are reviewed next as per the aforesaid order. As mentioned earlier, the role of cavitati on modeling is basically determination of the phase fraction. Different ideas have been proposed to generate the variable density field. Some studies solved the energy e quation and obtained the density either by employing an equation of state (EOS) (Dela nnoy and Kueny 1990, Edwards et al. 2000) or from thermodynamic tables (Ventikos and Tzabiras 2000). Saurel et al. 1999 developed a cavitation model fo r hypervelocity underwater pr ojectiles based on separate EOS for the liquid, vapor and mixture zone s. EOS approach in first place does not capture the essential cavitation dynamics due to its equilibrium assumption in the phase change process. Furthermore, prevalen ce of isothermal condition in case of non-

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12 thermosensible fluids such as water imparts a barotropic form, ()m f p to the EOS. Thus, cavitation modeling via employing the EOS is devoid of the capability to capture barotropic vorticity generation in the wake region as demonstrated by Gopalan and Katz (2000). Transport-equation based cavitation models in contrast overcome the above limitations and are more popular in research studies. Typically this approach determines the liquid volume fraction (l ) or the vapor mass fraction (v f ) by solving its transport equation as shown below. .()l lumm t (2.3) .()mv mvf f umm t (2.4) Formulation of the source terms shown in above equation(s) constitutes the major effort in model development. Singhal et al. (1997), Me rkle et al. (1998), Kunz et al. (2000), and Singhal et al. (2002) formulated these so urce terms strongly based on empirical judgment. However, Senocak and Shyy ( 2002, 2004a) developed a cavitation model fundamentally relying on interfacial mass and momentum transf er. Though their model was not completely empiricism-free, it transfor med the empirical coefficients used in the earlier models into a physically explicable form. The ability of the model to capture the barotropic vorticity, 1 p in the closure region was al so clearly demonstrated by Senocak and Shyy (2002, 2004a). The source te rms of each of the above models along with value of empirical consta nts are tabulated in Table 2.

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13 Table 2 Source terms in cavitation models Authors m Production Term m Destruction term Singhal et al.(1997) Merkle et al. (1998) 2 1(,0)(1) (0.5) 810prodvl l prodCMAXpp Ut C 2(,0) (0.5) 1destlvl lv destCMINpp Ut C Kunz et al. (2000) 2 4(1) 310 p rodll l prodC t C 2(,0) (0.5) 1destvvl ll destCMINpp Ut C Singhal et al.(2002) 1/2 3(,0) 2 [] 3 /3.67510v prodlv l prodMAXpp U C C 1/2 3(,0) 2 [] 3 /1.22510v destlv l destMINpp U C C Senocak and Shyy (2002, 2004a) 2 22 ,,(,0)(1) (0.5) 1 0.5()()prodvl l prod llvvnInCMAXpp Ut C UUU 2 22 ,,(,0) (0.5) 1 0.5()()destlvl lv dest llvvnInCMINpp Ut C UUU As clearly seen from the above table, the cav itation model of Senocak and Shyy (2002) is consistent with that of Singha l et al. (1997) and Merkle et al. (1998). However, the model constants have assumed physicality. The terms ,,&InvnUU which represent the liquidvapor interface velocity and the normal comp onent of interfacial vapor velocity are calculated by suitable approximations (Senocak and Shyy 2002, 2004a). Numerous experimental studies (Leight on et al. 1990, Clarke and Leighton 2000) addressing compressible bubble oscillations are available. Muzio et al. (1998) developed a numerical compressibility model for ac oustic cavitation in a bubble inclusive of viscosity and surface tension effects. However, these models are difficult to deploy for practical multiphase computations.

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14 Figure 5 (adapted from Hosanga di et al. 2002) illustrates th e modeled behavior of speed of sound v/s the phase fraction. As observed from the figure, the speed of sound in the biphasic mixture can be 2-3 orders of magn itude lower than the individual phases. Figure 5. Variation of Speed of Sound with phase fraction Consequently, a bulk incompressible flow may transform into a transonic or supersonic stage locally in the mixture region. Due to lack of dependable equation of state for multiphase mixtures, modeling sound propagati on, which is an imperative issue in the numerical computations, is stil l an open question. A closed form expression for the speed of sound in the mixture region may be obtai ned by eigenvalue analysis on the strongly conservative form of the governing equations (Hosangadi et al. 2002), as shown below. 221 []vl m mvvllcaa (2.5) Venkateswaran et al. (2002) pr oposed use of perturbation theo ry to obtain an efficient pre-conditioned system of equa tions, which were consistent in the incompressible as well as compressible regime. Improvement in cavitation dynamics by accounting the compressibility effects was reported. The in corporation of Speed of Sound (SoS) model Gasfraction SpeedofSound 0 0.25 0.5 0.75 1 0 200 400 600 800 1000 1200 1400Pureliquid Purevapor

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15 into pressure-based cavitation computations was significantly adva nced by Senocak and Shyy (2002, 2003). In pressure-based solvers, the SoS model affects the solution mainly through the pressure correction equation. Th e following relationship is adopted between the density correction and the pressure correction terms, while enforcing the massconservation treatment through the pressure correction equation: ''mCp (2.6) The implementation of equation (2.6) imparts a convective-diffusive form to the pressure equation. Two SoS models were propos ed by Senocak and Shyy (2003, 2004a, 2004b). SoS-1:()(1) s lCC p (2.7) 11 11SoS-2:()()ii s iiC p ppp (2.8) While SoS-1 is a suitable approximation to the curve shown in Figure 5, SoS-2 approximates the fundamental definition of speed of sound by adopting a centraldifference spatial derivative along the streamline direction ( ) instead of differentiating along the isentropic curve. Computations by Senocak and Shyy (2003) on convergentdivergent nozzle demonstrated far better capability of SoS-2 to mimic the transient behavior observed in experiments. Wu et al. (2003b) extended the model assessment by pointing out the dramatic time scale differences between the two models. This points the fact that compressibility mode ling is a sensitivity issue and must be handled carefully. RANS-based approaches in form of two-equation k models have been actively pursued to model turbulent cavitating flows. The k and transport equati ons along with the definition of turbulent visc osity are summarized below.

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16 () () P[()]mj mt t m jjkjuk k k txxx (2.9) 2 11() () P[()]mj m t t m jjju CC txkkxx (2.10) The production of turbulent kinetic energy (Pt) is defined as: PR i tij ju x (2.11) while, the turbulent viscosity is defined as: 2 m tCk (2.12) The model coefficients, namely, 12,, and kCC have three known non-trivial variants which have been summarized in Table 3. Table 3. Variants of the k model Authors 1C and Relevant Details 2C k Launder and Spalding (1974) 1.44 1.92 1.3 1.0 Shyy et al. (1997) P 1.150.25t 1.9 1.15 0.89 Younis (2003) (personal communication) P (1.150.25)(10.38||/)tkQ Q t where 2 / ) (2 2 2w v u k Q 1.9 1.15 0.89 Johansen et al. (2004) 1.44 2,0.09m tCk FC 3/2Min[1,] FC k 1.92 1.3 1.0 While the Launder and Spalding (1974) model is calibrated for equilibrium shear flows, the model by Shyy et al. ( 1997) accommodates non-equilibri um effects by introducing a

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17 subtle turbulent time scale into 1C The RANS model of Younis (2003, personal communication), in comparison, accounts for the time history e ffects of the flow. Wu et al. (2003a, 2003b) assessed the above RANS m odels on turbulent cavitating flow in a valve. Experimental visuals of the valve fl ow (Wang 1999) have demonstrated cavitation instability in form of periodic cavity de tachment and shedding. However, Wu et al. (2003b) reported that the k model over predicts the turbulent viscos ity and damps such instabilities. Consequently, the RANS computat ions were unable to capture the shedding phenomenon, and also showed restrained se nsitivity to the above variants of the k model. From standpoint of an alternate approa ch, the impact of filter-based turbulence modeling on cavitating flow around a hydrof oil was reported (W u et al. 2003c). The filter-based model relies on the two-equation formulation and uses identical coefficient values as proposed by Launder and Spalding (1 974), but imposes a filter on the turbulent viscosity as seen below. 2,0.09m tCk FC (2.13) The filter function ( F ) is defined in terms of filter size ( ) as: 3/2Min[1,];1 FCC k (2.14) Note that the forms of viscosity in equations (2.12) and (2.13) are comparable barring the filter function. The proposed model recovers the Launder and Spalding (1974) model for coarse filter sizes. Furthermore, at near-wall regions, the imposed filter value F = 1 enables the use of wall functions to model the shear layer. However, in the far field zone if the filter size is able address the turbulent length scale 3/2k the solution is computed

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18 directly (0t ). The filter-based model is also characterized by the independence of the filter size from the grid size. This model enhanced the prediction of flow structure for single-phase flow across a so lid cylinder (Johansen et al 2004). Wu et al. (2003c) reported substantial unstea dy characteristics for cavitat ing flow around a hydrofoil because of this newly developed model. Furthe rmore, the time-averaged results of Wu et al. (2003c) (surface pressure, cavity morphology, lift, drag etc.) are consistent to those obtained by alternate studies (Kunz et al. 2003, Coutier-Delgosha et al. 2003, Qin et al. 2003). Further examination in the context of filter-based modeling and cavitating flows was performed over the Clark-Y aerofoil and a convergent-divergen t nozzle (Wu et al. 2004, 2005). The filter-based model produced pronounced time-dependent behavior in either case due to significan tly low levels of eddy viscos ity. While the time-averaged results showed consistency to experimental da ta, they were unable to capture the essence of unsteady phenomena in the flow-field su ch as wave propagation. In addition to implementing the two-equation m odel for turbulent viscosity, Athavale et al. (2000) also accounted for turbulent pressure fluctuations. Thus, the thre shold cavitation pressure ( Pv) was modified as: '/2vvtppp (2.15) The turbulent pressure fluctuations were modeled as follows: 0.39tm p k (2.16) Though their computations produced consistent results, the prec ise effect of incorporating the turbulent pressure fluctu ations was not discerned.

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19 In addition to the above review, Wang et al. (2001), Senocak and Shyy (2002, 2004a, 2004b), Ahuja et al. (2001), Venkateswaran et al. (2002) and Preston et al. (2001) have also reviewed the recen t efforts made in computational and modeling aspects. 2.2 Modeling Thermal Effects of Cavitation Majority of studies on cavitation have made the assumpti on of isothermal conditions since they focused on water. Howeve r, as explained earlier, these assumptions are not suitable under cryogenic conditions be cause of their low liquid-vapor density ratios, low thermal conductivities, and steep slope of pressure-temperature saturation curves. Efforts on experimental and numerical investigation of cr yogenic cavitation are dated as back as 1969. Though the number of experimental studies on this front is restricted due to the low temperature c onditions, sufficient benchmark data for the purpose of numerical validation is available. However, there is a dearth of robust numerical techniques to tack le this problem numerically The following sub-sections provide fundamental insight into the phenomen on in addition to a literature review. 2.2.1 Scaling Laws Similarity of cavitation dynamics is dict ated primarily by the cavitation number () defined as (Brennen 1994, 1995): 2() 0.5v l p pT U (2.17) Under cryogenic conditions, however, cavitation occurs at the local vapor pressure dominated by the temperature depression. Thus, the cav itation number for cryogenic fluids is modified as: 2() 0.5vc c l p pT U (2.18)

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20 where, Tc is the local temperature in the cav ity. The two cavitation numbers can be related by a first-order approximation as follows: 21 ()() 2v lccdp UTT dT (2.19) Clearly, the local temperature depression (cTT ) causes an increase in the effective cavitation number, consequently reducing the cavitation intensity. Fu rthermore, equation (2.19) underscores the effect of the steep pr essure-temperature curv es shown in Chapter 1. Quantification of the temperature dr op in cryogenic cavitation has been traditionally assessed in terms of a non-dimensional temperature drop termed as B -factor (Ruggeri and Moore 1969). A simp le heat balance between th e two phases can estimate the scale of temperature differe nce caused by thermal effect. vvllPlLCT (2.20) Here, v and l are volume flow rates for the vapo r and liquid phase respectively. The B factor can then be estimated as: *;vv llPlL T BT TC (2.21) Consider the following two flow scenarios for estimating B as shown in Figure 6 (adapted from Franc et al. 2003).

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21 Figure 6. Two cavitation cases for B -factor analysis The estimation of Bfactor for the case (a) (two-phase cavity) in above figure is expressed in following equation. 1 ~;~(1);l llcvlc lUUB (2.22) This points the fact that ex cept for the pure vapor region B has an O(1) value. For the case (b) in Figure 6 (adapted fr om Franc et al. 2003), where th e cavity is assumed to be filled with 100% vapor, Fruman et al. (1991) provided an estimate of B based on thermal boundary layer effect as: cc T cB aL U (2.23) Here, a is the thermal or eddy diffusivity, cL is the cavity length, and c and T represent the thickness of the cavity and the thermal boundary layer, respectively. It is evident from equation (2.23) that the temper ature depression is also strongly dependent on thermal diffusivity and flow pr operties. The temperature scale (*T ) in case of water and LH2 has a value of 0.01 and 1.2 K, respectively (Franc et al. 2003). The difference in these values provides an assessment of the pr onounced thermal effect s in LH2. Thus, by Thermal boundary

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22 knowing the values of *T and B (from equations (2.22) and (2.23)), the actual temperature drop can be estimated. The Bfactor, however, fails to consider timedependent or transient thermal effects due to its dependence on a steady heat balance equation. Furthermore, the sensitivity of vapor pressure to the temperature drop, which is largely responsible in a ltering cavity morphology (Deshpande et al. 1997), is not accounted by it. As a result, though the B -factor may estimate the temperature drop reasonably, it is inadequate to evaluate th e impact of the thermal drop on the cavity structure and the overall flow. Brennen (1994, 1995) developed a more appropriate parameter to assess the thermodynamic effect by incorporating it into the Rayleigh-Plesset equation (equation(2.24)) for bubble dynamics. 2 2 23 [()]() 2lvcdRdR R pTp dtdt (2.24) With help of equation(2.19), we can re-write above equation as: 2 2 23 [()]() 2v lvdp dRdR R TpTp dtdtdT (2.25) From standpoint of a transiently evolving bubble, the heat flux q at any time t can be expressed as: T qK at (2.26) The denominator in equation(2.26), at represents the thic kness of the evolving thermal boundary layer at time t. The heat balance across the bubble interface is expressed as follows. 234 4[] 3vd qRLR dt (2.27)

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23 Combining equations (2.26) and (2.27) we obtain: v lPlL Rt T C a (2.28) Introducing this temperature differen ce into equation(2.25) we obtain: 2 2 2() 3 [()] 2v l p Tp dRdR RRt dtdt (2.29) A close observation yields the fact that th e impact of thermal effect on bubble dynamics depends on which is defined as: 2 vv lPlLdP dT Ca (2.30) The units of are m/s3/2 and it proposes a crite rion to determine if cavitation process is thermally controlled or not. Franc et al. (2003) have recently extended the above analysis to pose a criterion for dynamic similarity between two thermally controlled cavitating flows. They replaced the time dependency in equation(2.29) by spatial-dependency through a simple transformation x Ut Here, x is the distance traversed by a bubble in the flow-field in time t If D is chosen as a characteristic length sc ale of the problem, the Rayleigh-Plesset equation can be recast in the following form: 2 33 [] 22pC D RRRRt U (2.31) In equation (2.31), all the quantitie s with a bar are non -dimensional, and Cp is the pressure coefficient. All the de rivatives are with respect to / x xD The above equation points out the fact that, besides two thermally dominated flows can be dynamically

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24 similar if they have a consistent value of the non-dimensional quantity 3/ DU It is important to note the important role of velocity scale U in the quantification of thermal effect at this juncture. Franc et al. (2003) suggested that though thorough scaling laws for thermosensible cavitation ar e difficult to develop, a rough assessment may be gained from above equation. However, it is imperative to highlight that all the above scaling laws have been developed using either steady-state heat bala nce or single bubble dynamics. As a consequence, their applicability to general engineer ing environments and complex flow cases is questionable. The following sections will discuss the expe rimental and numerical investigations of cavitation with thermal consideration. Particularly, emphasis is laid on the limitations of currently known numerical t echniques. Experimental studies are cited solely according to their relevance to the current study. 2.2.2 Experimental Studies Sarosdy and Acosta (1961) detected signi ficant difference between water cavitation and Freon cavitation. Their apparatus compri sed a hydraulic loop with an investigative window. While water cavitation wa s clear and more intense, th ey reported that cavitation in Freon, under similar condi tions, was frothy with greater entrainment rates and lower intensity. Though their observat ions clearly unveiled the dominance of thermal effects in Freon, they were not corroborated with phys ical understanding or numerical data. The thermodynamic effects in cavitation were expe rimentally quantified as early as 1969. Ruggeri et al. (1969) inves tigated methods to predict performance of pumps under cavitating conditions for different temper atures, fluids, and operating conditions. Typically, strategies to predict the Net Po sitive Suction Head (N PSH) were developed.

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25 The slope of pressure-temperature satura tion curve was approximated by the ClausiusClapeyron equation. Pump performance under va rious flow conditions such as discharge coefficient and impellor frequenc y was assessed for variety of fl uids such as water, LH2, and butane. Hord (1973a, 1973b) published comprehensive experimental data on cryogenic cavitation in ogives and hydrofoils. Th ese geometries were mounted inside a tunnel with a glass window to capture visu als of the cavitation zone. Pressure and temperature were measured at five pr obe location over the geometries. Several experiments were performed under varying in let conditions and their results were documented along with the instrume ntation error. As a result, Hord’s data are considered benchmark results for validating numerical techniques for thermodynamic effects in cavitation. From standpoint of latest i nvestigations, Fruman et al. (1991) proposed that thermal effect of cavitation can be estimated by attrib uting a rough wall beha vior to the cavity interface. Thus, heat transfer equations for a boundary layer flow ove r a flat plate were applied to the problem. The vol ume flow rate in the cavity was estimated by producing an air-ventilated cavity of similar shape and size. An intrin sic limitation of the above method is its applicability to only sheet-type cavitation. Larrarte et al. ( 1995) used high-speed photography and video imaging to observe natural as well as ventilated cavities on a hydrofoil. They also examined the eff ect of buoyancy on inte rfacial stability by conducting experiments at negative and pos itive AOA. They reported that vapor production rate for a growing cavity may differ substantia lly from the vapor production rate of a steady cavity. Furthermore, there may not be any vapor production during the detachment stage of the cavity. They also noti ced that cavity interface under the effect of

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26 gravity may demonstrate greater stability. Fruman et al. (1999) inves tigated cavitation in R-114 on a venturi section employing assumpti ons similar to their previous work (Fruman et al. 1991). The temperature on th e cavity surface was estimated using the following flat plate equation. 0.12.1 [1(1Pr)] 0.5Replate lPlfxq TT CUC (2.32) Note that f C is the coefficient of friction and plateT is assumed to be equal to the local cavity surface temperature. The heat flux q on cavity surface was estimated as: vQqLUC (2.33) The discharge coefficient QC was obtained from an air-ventilated cavity of similar size. They estimated the temperature drop via the flat plate equation and measured it experimentally as well. These two correspond ing results showed reasonable agreement. Franc et al. (2001) employed pr essure spectra to investig ate R-114 cavitation on inducer blades. The impact of thermodynamic effect was examined at three reference fluid temperatures. They reported a delay in the onset of blade cavitatio n at higher reference fluid temperatures, which was attributed to suppression of cavitation by thermal effects. Franc et al. (2003) further i nvestigated thermal effects in a cavitating inducer. By employing pressure spectra they observed shift in nature of cavitation from alternate blade cavitation to rotating cavitation with decrease in cavitation number. The earlier is characterized by a frequency 2fb ( fb is the rotor frequency with 4 blades) while the later is characterized by a resonant frequency fb. Furthermore, R-114 was employed as the test fluid with the view of extendi ng its results to predicting ca vitating in LH2. The scaling

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27 analysis provided in section 2.2.1 was devel oped by Franc et al. (2003) mainly to ensure dynamic similarity of cavitation in their experiments. 2.2.3 Numerical Modeling of Thermal Effects Numerical modeling has been implemented in cavitation studies broadly for two thermodynamic aspects. Firstly, attempts to model the compressible/pressure work in bubble oscillations have been made. Lertnuwat et al. (2001) modele d bubble oscillations by applying thermodynamic considerations to the Rayleigh-Plesset equation, and compared the solutions to full DNS cal culations. The bubble model showed good agreement with the DNS results. The mode led behavior, however, deviated from the DNS solutions under isothermal and adiabatic assumptions. bubble liquid(1)vvVvvVllV(1)llV Figure 7. Schematic of bubble m odel for extracting speed of sound Rachid (2003) developed a theoretical model for accounti ng compressive effects of a liquid-vapor mixture. The actual behavior of the mixture along wi th the dissipative effects associated with phase transforma tion was found to lie between two limiting reversible cases. One in which phase change occurs under equilibrium at a constant pressure, and the other in which the vapor expands and contracts reversibly in the

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28 mixture without undergoing phase change. Rapposelli and Agostino (2003) recently extracted the speed of sound for various fluids such as water, LOX, LH2 etc. employing a bubble model and rigorous thermodynamic relationships. The control volume ( V ) of the bubble is illustrated in Figure 7 (adapted from Rapposelli and Agostino 2003), and can be expressed as: (1)()(1)()llllvvvvVVVVV (2.34) The model assumed that thermodynamic equ ilibrium between the two phases is only achieved amidst fractions l and v of the total volume. Subsequently, the remaining fractions of the two phases were assumed to behave isentropically. If ml and mv are masses associated with the respective phases, the differential volume change dV can be expressed as: (1)()(1)() ()()()() ()lvllvv lsvs lv llvv lsatvsat lv vl vldd dV Vdpdp dd dpdp dm mm (2.35) A close observation of above equation yi elds that the modeling extremities of ,0 and ,1lvlv also correspond to the thermodynamic extremities mentioned in above-mentioned analyses by Lertnuwat et al. (2001) and Rachid (2003) (italicized in the above description). Finally, substitution of various therm odynamic relations into equation (2.35) yields a thermally consistent sp eed of sound in the medium. Rapposelli and Agostino (2003) reported th at their developed model was able to capture most features of bubble dynamics reasonably well.

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29 The second thermodynamic aspect of cavitation, which also forms the focal point of this study, is the effect of latent heat tr ansfer. The number of nu merical studies, at least in open literature, in this regard is highly restricted. Reboud et al. (1990) proposed a partial cavitation model for cryogenic cavitation. The model comprised three steps, which were closely adapted for sheet cavitati on, in form of an iterative loop. (a) Potential flow equations were utilized to compute the liquid flow field. The pressure distribution on the hydrofoil surface was imposed as a boundary condition based on the experimental data. Actually, this fact led to the model being called ‘ partial ’. The interface was tracked explicitly based on the local pressure. The wake was represented by imposing a reattachment law. (b) The vapor flow inside the cavity wa s solved by parabolized Navier Stokes equations. The change in cavity thickness yi elded the increase in vapor volume and thus the heat flux at each section. (c) The temperature drop over the cavity was evaluated with the following equation: |tcT qK y (2.36) The value of turbulent diffusivity Kt in the computations was arbitrarily chosen to yield best agreement to the experimental results, and q was calculated from step (b). The iterative implementation of steps (a) – (c) yielded the appropriate cavity shape in conjunction with the thermal effect. Simila r 3-step approach was adopted by Delannoy (1993) to numerically reproduce the test data with R-113 on a convergent-divergent tunnel section. The main drawback of bot h the above methods is their predictive capability is severely limited. This is ma inly because these studies do not solve the

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30 energy equation and depend greatly on simp listic assumptions for calculating heat transfer rates. Deshpande et al. (1997) developed an improved methodology for cryogenic cavitation. A pre-conditioned density-based formulation was employed along with adequate modeling assumptions for the vapo r flow inside the cav ity and the boundary conditions for temperature. The interface was captured with explicit tracking strategies. The temperature equation was solved only in the liquid domain by applying Neumann boundary conditions on the cavity surface. The te mperature gradient on the cavity surface was derived from a local heat balance simila r to equation(2.36). The bulk velocity inside the vapor cavity was assumed equal to the fr ee stream velocity. Tokumasu et al. (2002, 2003) effectively enhanced the model of Deshpande et al. ( 1997) by improving the modeling of vapor flow inside the cavity. Despite the improvements in the original approach (Deshpande et al. 1997), it is impor tant to underscore th e limitation that both the above studies did not solve the en ergy equation inside the cavity region. Hosangadi and Ahuja (2003, 2005), and Hosa ngadi et al. (2003) recently reported numerical studies on cavitation using LOX, LH2, and liquid nitrogen. Their numerical approach was primarily density-based. Their pressure and temperat ure predictions over a hydrofoil geometry (Hord 1973a) showed incons istent agreement with the experimental data, especially (Hord 1973a) at the cavity closure region. Furthermore, Hosangadi and Ahuja (2005), who employed th e Merkle et al. (1998) mode l in their computations, suggested significantly lower values of the cavitation model parameters for the cryogenic cases as compared to their previous calibrations (Ahuja et al. 2001) fo r non-cryogenic fluids.

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31 The above review (summarized in Table 4) points out the limited effort and the wide scope for improving numerical modeling of thermal effects in cryogenic cavitation. A broadly applicable and more robust num erical methodology is expected to be a significant asset to prediction and critical investigation of cr yogenic cavitation.

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32 Table 4. Summary of studies on thermal effects in cavitation Author and Year Method (Experim ental/Numerical) Main Findings Sarosdy and Acosta 1961 Experimental Cavitation in hydraulic loop Freon cavitates w ith a suppressed intensity as compared to water Ruggeri et al. 1969 Experimental Cavitation in pumps Provided assessment of pump NPSH under cryogenic conditions Hord 1973 Experimental Published comprehensive test data on ogives and hydrofoil Fruman et al. 1991 Experimental Natural and ventilated cavities Estimated temperature dr op using flat plate boundary layer equations Larrarte et al. 1995 Experimental High speed photography Vapor production for growing cavities in cryogenic fluids may not be estimated by ventilated cavities Fruman et al. 1999 Experimental Venturi section and R-114 Flat plate equations to model sheet cavitation can give good results under certain restrictions Franc et al. 2001 Experimental Cavitation in pump inducers Reported delay in onset of cavitation due to thermal effect Franc et al. 2003 Experimental Cavitation in pump inducers Provided scaling analysis to ensure dynamic similarity Reboud et al. 1990 Numerical Potential flow equations Semi empirical numerical model Did not solve energy equation Suitable for sheet cavitation Delannoy 1993 Numerical Potential flow equations Semi-empirical numerical model Did not solve energy equation Suitable for sheet cavitation Deshpande et al. 1997 Numerical Explicit interface tracking Density based formulation Simplistic approximation for cavity vapor flow Did not solve energy equation in the vapor phase Tokumasu et al. 2002, 2003 Similar to Deshpande et al. (1997) Improved flow model for cavity Applicable to only sheet cavitation Did not solve energy equation in the vapor phase Lertnuwat et al. 2001 Numerical Model for bubble oscillations Incorporated energy balance for bubble oscillations Good agreement w ith DNS simulation Rachid 2003 Theoretical Compression model liquid vapor mixture Dissipative effects in phase transformation intermediate between two extreme reversible thermodynamic phenomena Rapposelli & Agostino 2003 Numerical Model for bubble oscillations Employed thermodynamic relations to extract speed of sound for various liquids Hosangadi & Ahuja 2003, 2005 Numerical Density-based approach Solved energy equation in the entire domain with dynamic update of material properties Some inconsistency with experimental results noted Significant change in the cavitation model parameters between non-cryogenic and cryogenic conditions

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33 CHAPTER 3 STEADY STATE COMPUTATIONS This chapter firstly delin eates the governing equations that are employed in obtaining steady-state solutions to various cases on cryogen ic cavitation. Theoretical formulation/derivation and computational im plementation of various models, namely cavitation, turbulence, compressibility, an d thermal modeling aspects, are further highlighted. The boundary conditions and steady-state results yielded by the computational procedure are discussed in de tail following the desc ription of the basic framework. 3.1 Governing Equations The set of governing equations for cryoge nic cavitation under the single-fluid modeling strategy comprises the conservative form of the Favre-averaged Navier-Stokes equations, the enthalpy equation, the k two-equation turbulence closure, and a transport equation for the liquid volume fraction. The mass-continuity, momentum, enthalpy, and cavitation model equations are given below: () 0mj m ju tx (3.1) () () 2 [()()] 3mijj miik tij jijjikuuu uuu p txxxxxx (3.2) [()][()][()] PrPrt mvmjv jjLtjh hfLuhfL txxx (3.3)

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34 ()lj l ju mm tx (3.4) We neglect the effects of compressible wo rk and viscous dissipation from the energy equation because the temperature field in cryogenic cavitation is mainly dictated by the phenomenon of evaporative cooli ng. Here, the mixture density, sensible enthalpy, and the vapor mass fraction are respectively expressed as: (1)mllvl (3.5) PhCT (3.6) (1)vl v mf (3.7) The general framework of the Navier-S tokes solver (Senocak and Shyy 2002, Thakur et al. 2002) employs a pressure-based algorithm and the finite-volume approach. The governing equations are solved on multi-block, structured, curvilinear grids. The viscous terms are discretized by second-or der accurate central differencing while the convective terms are approximated by the seco nd-order accurate Controlled Variations Scheme (CVS) (Shyy and Thakur 1994). The use of CVS scheme prevents oscillations under shock-like expansion caused by the evap oration source term in the cavitation model, while retaining sec ond order of formal accuracy. Steady-state computations are performe d by discounting the time-derivative terms in the governing equations and relaxing each equation to ensu re a stable convergence to a steady state. The pressure-v elocity coupling is implem ented through the SIMPLEC (Versteeg and Malalasekera 1995) type of algorithm, cast in a combined Cartesiancontravariant formulation (T hakur et al. 2002) for the de pendent and flux variables, respectively, followed by adequate relaxatio n for each governing equation, to obtain

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35 steady-state results. The physical properties are updated, as explained earlie r, after every iteration. 3.1.1. Cavitation Modeling Physically, the cavita tion process is governed by th ermodynamics and kinetics of the phase change process. The liquid-vapor conversion associated with the cavitation process is modeled through m and m terms in Eq. (3.4), which respectively represent, condensation and evaporation. The particular form of these phase transformation rates, which in case of cryogenic fluids also dictates the heat transfer process, forms the basis of the cavitation model. Given below are the three modeling approaches probed in the present study. 3.1.1.1 Merkle et al. Model The liquid-vapor condensation rates for this particular model ar e given as (Merkle et al. 1998): 2 2Min(0,) (0.5) Max(0,)(1) (0.5)destlvl vl prodvl lCpp m Ut Cpp m Ut (3.8) Here, 1.0destC and 80.0prodC are empirical constants tuned by validating the numerical results with experimental data. The time scale ( t ) in the equation is defined as the ratio of the characteristic length scale to the reference velocity scale (/ tDU ). 3.1.1.2 Sharp Interfacial Dynamics Model (IDM) The source terms of this model are de rived by applying mass and momentum balance across the cavity interface and appropr iately eliminating the unquantifiable terms (Senocak and Shyy 2002, 2004a, 2004b).

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36 Figure 8. Schematic of cavity models (a) Dis tinct interface with vaporous cavity (Sharp IDM) (b) Smudged interface with mushy cavity (Mushy IDM) A schematic of the cavity model is illustrated in Figure 8(a). The model relies on the assumption of a distinctly vaporous cavity with a thin biphasic zone separating it from the pure liquid region. The physical form of the source terms is given below: 2 ,, 2 ,,Min(0,) ()() Max(0,)(1) ()()lvl vvnInlv vl vnInlvpp m UUt pp m UUt (3.9) Here, the normal component of the velocity is calculated as (Senocak and Shyy 2002, 2004a): ,; l vn lUunn (3.10) Due to implicit tracking of the in terface, the interfacial velocity, ,InU in unsteady computations needs modeling efforts. Prev ious studies simplistically expressed the interfacial velocity (,InU ) in terms of the vapor normal velocity (Senocak and Shyy 2002, 2004a) (, vnU ). Alternate methods of modeling InU are discussed in the Chapter 4 in the context of time-dependent simulations. In comp arison, for the steady co mputations in this chapter, we impose ,0InU ,,,mVmnpV ,,,LLLnpV IV(a) (b)

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37 3.1.1.3 Mushy Interfacial Dynamics Model (IDM) Experimental visualizations of cryoge nic cavitation (Saros dy and Acosta 1961; Hord 1973a, 1973b) have clearly indicated a mushy nature of the cavity. This salient characteristic of cryogenic cavitation solicits an adaptation of th e existing cavitation model to reflect the same. We choose the Sharp IDM (Senocak and Shyy 2004a) in our analysis because of its stronger physical r easoning. The following discussion serves to revise the above cav itation model by re-examining its derivation and assumptions, to appropriately accommodate the features of cr yogenic cavitation. Although we refer the reader to literature (Senocak and Shyy 2004a) for the detailed derivation of the earlier model, we underscore the differences betw een the two approaches in the proceeding description. Figure 8(b) – representing Mushy IDM de picts a cavity where the vapor pressure is a function of the local temperature, and the nature of cavitation demonstrates a weak intensity and consequently less probability fo r existence of pure va por phase inside the cavitation zone. In comparison, Figure 8(a) – representing Sharp ID M depicts a cavity typically produced under regular conditions (no temperature effects), characterized by a thin biphasic region separati ng the two phases. We initiate our approach similar to Senocak and Shyy (2004a) by formulating th e mass and momentum balance condition at the cavity interface, which is assumed to separate the liquid and mixture regions. We neglect the viscous terms and surface tension effects from equation (3.12) assuming high Re flows and large cavity sizes respectively. ,,,,()()llnInmmnInUUUU (3.11) 22 ,,,,()()lvmmnInllnInppUVUV (3.12)

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38 It is noteworthy that Senocak and Shyy (2004a ) formulated the above equations midst the pure vapor and mixture regions while hypothes izing an interface between the two. The two models differ in terms of their interpretation of the cavity attributes, namely, the density and the velocity field within the cavitation zone, while adhering to the fundamental idea of mass/momentum balance. Si nce we attribute a “frothy” nature to the cavity, the term unis largely expected to represen t the mixture no rmal velocity. Accounting for this fact, we eliminate the ,lnU term from equation (3.12) using equation (3.11) as shown below. 2 22 ,,,,()()m lvmmnInmnIn lppUUUU (3.13) Further re-arrangement progressive ly yields the fo llowing equations: 22 ,,,,()[1]()mlv lvmmnInmmnInv llppUUUU (3.14) 22 ,,,,()() () ()()()()llvllv v vl mlvmnInmlvmnInpppp UUUU (3.15) 22 ,,,,()()(1) ()()()()llvlllvl v mlvmnInmlvmnInpppp UUUU (3.16) From standpoint of formul ating source terms for the l transport equation, we firstly normalize the above equation by the overall convective timescale (/ tDU ). Secondly, we apply conditional statements on the pressure terms to invoke either evaporation or condensation de pending on the local pressure a nd vapor pressure (refer to Senocak and Shyy 2002 & 2004a). Lastly, we assume that the volume rates for the individual phases are interchangeable barr ing the sign convention. For instance, the

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39 evaporation term, m, in the equationl or equationv would bear the same magnitude, but negative or po sitive sign, respectively. We thus obtain the following source terms from the above analysis: 2 ,, 2 ,,Min(0,) ()() Max(0,)(1) ()()lvl mmnInlv lvl mmnInlvpp m UUt pp m UUt (3.17) The normal component of the mi xture velocity is expressed as shown in equation (3.10) and consistent with the norma l component of the vapor velocity employed by the Sharp IDM. A comparison with equation (3.9 ) shows that a factor of /vm weakens the evaporation term while a factor of /lm strengthens the condensation term of the Sharp IDM to yield the Mushy IDM. For instance, at a nominal density ratio of 100 and 0.5l the value of /vm and /lm is 21.8210 and 1.82, respectively. In terms of order of magnitude, the difference between th e evaporation terms of the two models is relatively more pronounced than the condensatio n terms, especially for high liquid-vapor density ratios. Given this fact, we attempt to make the Mushy IDM consistent with the Sharp IDM by providing an exponential transi tion of the evaporati on source term from one model to the other as a function of the phase fraction (l ). Specifically, we employ the evaporation term from equation (3.17) as we get closer to the liquid region, and that from equation (3.9) as we get farther fr om the liquid region. In comparison, the condensation term from equati on (3.17), which is mainly e xpected to be active in the region outlining the ‘vaporous’ portion of th e cavity, is unaltered. In summary, the Mushy IDM is eventually formulated as shown below.

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40 2 ,, 2 ,, (1)/Min(0,) ()() Max(0,)(1) ()() (1.0)llvl mnInlv lvl mnInlv lll vv ll mpp m UUt pp m UUt e (3.18) Note that is a free parameter, which regulates the switch between the evaporation terms and warrants calibration for different fluids. Its typical value could be (0.1) O. 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 10 20 30 40 50 60 70 80 90 100 ll/ Sharp IDM Mushy IDM 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ll/ + Sharp IDM Mushy IDM Figure 9. Behavior of /l and /l vs. l for the two models; /100lv and 0.09 Figure 9 contrasts the density-ratio terms (/l and /l ) of the Sharp IDM and Mushy IDM for the chosen parameter values. 3.1.2 Turbulence Modeling For the system closure, the original two-equation turbulence model with wall functions is presented as fo llows (Thakur et al. 2002): () () [()]mj mt tm jjkjuk k k txxx (3.19)

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41 2 12() () P[()]mj m t tm jjju CC txkkxx (3.20) The turbulence production (Pt) and the Reynolds stress tensor is defined as: P; 2 () 3i tijijmij j mijj i mijt jiu uu x ku u uu x x (3.21) The parameters for this model, namely, 11.44 C ,21.92 C ,1.3 ,1.0k are adopted from the equilibrium shear flow calibration (Launder and Spalding 1974). The turbulent viscosity is defined as: 2,0.09m tCk C (3.22) It should be noted that the tu rbulence closure and the eddy vi scosity levels can affect the outcome of the simulated cavitation dynamics es pecially in case of unsteady simulations (as reviewed in Chapter 2). In this aspect, pa rallel efforts are being made in the context of filter-based turbulence modeling (Johansen et al. 2004; Wu et al. 2003c, 2004, 2005), which has shown to significantly increase the time-dependency in cavitating flows. We do not explore these techniques in the inte rest of steady-state simulations, which disregard time-dependent phenomena. 3.1.3 Speed of Sound (SoS) Modeling Due to lack of dependable equation of state for liquid-vapor multiphase mixture, numerical modeling of sound propagation is still a topic of research. We refer the reader to past studies (Senocak and Shyy 2003, 2004a, 2004b, Wu et al. 2003b) for modeling

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42 options, their impact and issues, and just outline the currently employed SoS model below. SoS(1)lCC (3.23) The density correction term in the continuity equation is t hus coupled to the pressure correction term as shown below. ''mCp (3.24) Senocak and Shyy (2002 & 2004a) suggested an O (1) value for the constant C to expedite the convergence of the iterative computational algorithm. However, their recommendation is valid under normalized values for inlet velocity and liquid density. Since we employ dimensional form of equa tions for cryogenic fluids, we suggest an O (21/ U) value for C which is consistent with the abov e suggestion in terms of the Mach number regime. The speed of sound affects the numerical calculation via the pressure correction equation by conditionally endowing it with a convective-diffusive form in the mixture region. In the pure liquid region, we recover the diffusive form of the pressure equation. 3.1.4 Thermal Modeling The thermal effects are mainly regulated by the evaporative cooling process, which is further manifested by the temperature dependence of physical properties and vapor pressure. 3.1.4.1 Fluid property update In the present study, we subject all the physical properties, namely, ,,,,,, and lvvP p CKL to temperature dependence.

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43 Figure 10. Pressure-density and pressure-ent halpy diagrams for liquid nitrogen in the liquid-vapor saturation regime (Lemmon et al. 2002). Lines denote isotherms in Kelvin. As indicated by Figure 10, the physical prop erties are much stronger functions of temperature than pressure, and can fairly assume the respective values on the liquidvapor saturation curve at a given temperatur e. We update these properties from a NIST database (Lemmon et al. 2002) at the end of a com putational iteration. Thus, we generate a look-up table of physical properties for a particular temperat ure range as a preprocessing step. Subsequently, for any temperat ure-based update, the table is searched by an efficient bisection algorithm (Press et al 1992) and the required physical property is obtained by interpolating between th e appropriate ta bular entries. 3.1.4.2 Evaporative cooling effects The energy equation, (Eq.(3.3)), is reca st into the following temperature-based form, by separating the latent heat terms onto the right-hand-side. energy source/sink term[][][()] PrPr {[()][()]}t mPmjPP jjLtj mvmjv jT CTuCTC txxx f LufL tx (3.25)

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44 As seen from equation (3.25), the ‘lumped’ latent heat te rms manifest as a non-linear source term into the energy equa tion and physically represent the latent heat transfer rate. The spatial variation of thermodynamic propert ies and the evaporative cooling effect are intrinsically embedded into this transport-ba sed source term. We calculate the source term by discretizing the associ ated derivatives in concert with the numerical schemes applied to the terms on the left-hand-side of the equation. 3.1.5 Boundary Conditions The boundary conditions are implemented by stipulating the values of the velocity components (obtained from the experimental data), phase fraction, temperature, and turbulence quantities at the inlet. Furthermore, at the walls, pressure, phase fraction, and turbulence quantities are extr apolated, along with applying the no-slip and adiabatic condition on the velocity and te mperature, respectively. Pres sure and other variables are extrapolated at the outlet boundaries, while enforcing global mass conservation by rectifying of the outlet velocity components. In addition, we also hol d the pressure at the reference pressure point constantly at the re ference value (specified by the experiments). This is simply done by adjusting the linear coefficients of the pressure correction equation at that point, to yield zero correction, at every iteration. Symbolically, this is achieved by substituting 1,0ppp PnbPAAB into the following linear equation at the point of interest: '' pp P PnbnbPApApB (3.26) We observe that this adjustment imparts r obustness and stability to the computation.

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45 3.2 Results and Discussion In this section, we firstly observe the nuances of the Mushy IDM on non-cryogenic cases. The purpose is to distill the impact so lely caused due to the source term change between the Sharp and Mushy IDM, and c ontrast it against commonly referred experimental and computational results. Late r, we extend our computations to the variable temperature environm ent of cryogenic fluids. Specifi cally, we observe the effect of the temperature field on the nature of cavitation under similar conditions. Furthermore, we assess the Mushy IDM with available experi mental data (pressure and temperature) and compare it with the alternative cavitatio n models. In the pro cess of calibrating the cavitation models for cryogeni c fluids, we perform a globa l sensitivity analysis to evaluate the sensitivity of the prediction to changes in material properties and model parameters. We offer discussion about optim izing the model performance based on our sensitivity study. 3.2.1 Cavitation in Non-cryogenic Fluids As mentioned above, we register fi rst the impact of the Mushy IDM on noncryogenic cases. In this initial exercise, we consider the influence of the mushy formulation only to the liqui d boundary layer encompassing the cavity. Symbolically, we implement the following formulation for the non-cryogenic cases. 2 ,, 2 ,,Min(0,) ()() Max(0,)(1) ()() if 0.99 else if 0.99 1 else lvl mnInlv lvl mnInlv llll l vm lll l mpp m UUt pp m UUt (3.27)

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46 To prevent sudden discontinuity in the volum e transfer rates, we perform a geometric smoothing operation over the source terms. Tw o flow configurations are discussed here, namely, cavitating flow over a hemispherical pr ojectile (time-averaged experimental data by Rouse and McNown (1948) at 5Re1.3610 ) and cavitating flow over the NACA66MOD hydrofoil (time-averaged experi mental data by Shen and Dimotakis (1989) at 6Re210 ). NO-SLIP OUTLET INLET SYMMETRY SYMMETRY NO-SLIPhemisphericalprojectile INLET OUTLET NO-SLIP NO-SLIP hydrofoil(NO-SLIP) Figure 11. Illustration of the computationa l domains for hemispherical projectile and NACA66MOD hydrofoil (non-cryogenic cases)

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47 The computational domains for the two geom etries are depicted in Figure 11. We consider two grids with 15866 and 292121 points for the hemispherical geometry, which is axisymmetric. In case of the NAC A66MOD hydrofoil, we defer to the judgment of a previous study (Senocak and Shyy 2004a), and employ the finer grid from that study. s/D Cp 0 1 2 3 4 -0.5 0 0.5 1 MushyIDM-158x66grid MushyIDM-292x121grid Exp.data s/D Cp 0 1 2 3 4 -0.5 0 0.5 1 SharpIDM Merkleetal.Model MushyIDM Exp.data Figure 12. Pressure coefficients over the hemispherical body (0.4 ); D is the diameter of the hemispherical projectile. (a) Im pact of grid refinement for Mushy IDM (b) Comparison between pressure coeffi cients of different models on the coarse grid We investigate the sensitivity of Mushy IDM to grid refinement for the two hemispherical body grids through the su rface pressure plots in Figure 12. Note that the refinement factor in the vi cinity of cavitating region is roughly 2.5-3. Figure 12(a) indicates a noticeable, though modest, effect of the grid refinement on the surface pressure. This can be mainly attributed to the grid-dependent geometric smoothing operation, which smears the mushy formulation over a larger portion in the coarser grid. Nonetheless, both the grids produce soluti ons that match the experimental data reasonably well. Furthermore, it is important to mention that the near-wall nodes of the coarser grid lie in the log-la yer of the turbulent boundary la yer enabling appropriate use of the wall function (Versteeg and Malalasekera 1995), while the fine grid is over-refined (a) (b)

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48 for the wall spacings to assume values in th e suitable range. As pointed out by Senocak and Shyy (2004a), in view of the wall functi on treatment (Thakur et al. 2002; Versteeg and Malalasekera 1995), the presence of vapor in the cavity may substantially reduce the appropriate range of the wall node spacings. As a consequence, we employ the coarser grid for further computations. Figure 12(b) contrasts the performance of the Mushy IDM with the Merkle et al. Model (1998) and Sharp IDM (2004a) at 0.4 The pressure coefficient predicted by the three models illustrates noticeable variations in the condensation region of the cavity. Specifically the Mushy IDM, as seen from Figure 12 (b), tends to produce a sharper recovery of the surface pressure in the cavity closure zone. Additionally, the Mushy IDM pr oduces a small pressure dip (below the vapor pressure value) at the cavity outset due to the lower evaporation rate and hi gher condensation rate. 1.0 0.8 0.7 0.5 0.4 0.2 LiquidVolumeFraction 1.0 0.8 0.7 0.5 0.4 0.2 LiquidVolumeFraction 1.0 0.8 0.7 0.5 0.4 0.2 LiquidVolumeFraction Figure 13. Cavity shapes and flow stru cture for different cavitation models on hemispherical projectile (0.4 ). (a) Merkle et al. Model (b) Sharp IDM (c) Mushy IDM (a) (b) (c)

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49 These features are reflected by the cavity si zes depicted in Figure 13. Note that the Mushy IDM not only shrinks the cavity length, in comparison with the Sharp IDM, but also impacts the flow structure in the recirc ulation zone of the cavity closure region. The above findings support our approach in provid ing an appropriate transition between the two models and a careful cal ibration to the parameter x/D -Cp 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 SharpIDM Merkleetal.Model MushyIDM Exp.data x/D -Cp 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 SharpIDM Merkleetal.Model MushyIDM Exp.data Figure 14. Pressure coefficient over th e NACA66MOD hydrofoil at two different cavitation numbers; D is hydrofoil chord length. (a) 0.91 (b) 0.84 We further simulate cavitating flow around the NACA66MOD hydrofoil at two cavitation numbers, namely, 0.84 and 0.91, while maintaining the mushy formulation only for the cavity boundary. Again, consistent behavior in terms of surface pressure is induced by the Mushy IDM for this geometry, at both the cavitation numbers (Figure 14). In summary, the appropriate source term modulation imposed by the Mushy IDM tends to influence the prediction of surface pressure and cavity size in manners consistent with the experimental observation. The above assessment motivates the further implementation and enhancement of the Mushy IDM (w ith respect to ) on cryogenic flow cases. (a) (b)

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50 3.2.2 Cavitation in Cryogenic Fluids In this section, we inve stigate the Mushy IDM for th e cryogenic situation. We perform computations on two geometries experimentally investigated by Hord (1973a, 1973b), namely, a 2D quarter caliber hydrofo il (Hord 1973a) and an axisymmetric 0.357inch ogive (Hord 1973b). These geometries were mounted inside suitably designed tunnels in the experimental setup. INLET OUTLET NO-SLIP SYMMETRYhydrofoilsurface(NO-SLIP) INLET OUTLET SYMMETRY NO-SLIPogivesurface(NO-SLIP) Figure 15. Illustration of the computati onal domain accounting the tunnel for the hydrofoil (Hord 1973a) and 0.357-inch ogive geometry (Hord 1973b) (cryogenic cases). Figure 15 illustrates the computational domains employed for these cases. Note that the figure shows only planar slices of the domai ns, which also model the respective tunnel shapes to account for the significant blockage effects. The mesh for the hydrofoil and the ogive geometry respectively comprises 32070 and 34070 points. The mesh distribution is chosen to facilitate ade quate resolution of the cavitation zone.

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51 Furthermore, the near-wall resolution over a ll the no-slip planes (cavitating geometries and tunnel walls) accounts for deployment of wall functions (Thakur et al. 2002; Versteeg and Malalasekera 1995). Table 5. Flow cases chosen for the hydrofoil geometry. Fluid: Case name: Inlet temperature: Freestream Re: Cav. No. ( ): Liq. N2 283B 77.65 K 64.710 1.73 Liq. N2 290C 83.06 K 69.110 1.70 Liq. N2 296B 88.54 K 71.110 1.61 Liq. H2 248C 20.46 K 71.810 1.60 Liq. H2 249D 20.70 K 72.010 1.57 Liq. H2 255C 22.20 K 72.510 1.49 Source: Hord (1973a) Table 6. Flow cases chosen for the ogive geometry. Fluid: Case name: Inlet temperature: Freestream Re: Cav. No. ( ): Liq. N2 312D 83.00 K 69.010 0.46 Liq. N2 322E 88.56 K 71.210 0.44 Liq. H2 349B 21.33 K 72.310 0.38 Source: Hord (1973b) Statistically-averaged pressure and temperatur e data are available for the two geometries at five probe locations over the body surfaces The experimental findings report varying amounts of unsteady behavior in the cavity closur e regions, although no case-specific information or data/visuals ar e available in that context. Hord conducted a series of experiments over both the geometries, using liquid nitrogen and hydrogen, by varying the inlet velocity, temperature, and pressure. In our computations, we have selected several

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52 cases, referenced alphanumerically in the reports by Hord (1973a, 1973b), with different freestream temperatures and cavitation numbers (see Table 5 and Table 6). x/D Cp -1 0 1 2 3 4 5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 Computedpressure Hord,1973(incipientdata) x/D Cp 0 2 4 -2 -1.5 -1 -0.5 0 0.5 1 1.5 Computedpressure Hord,1973(incipientdata) Figure 16. Non-cavitating pressure distribu tion (a) case ‘290C’, D represents hydrofoil thickness and x represents distance from the circular bend (b) case ‘312D’, D represents ogive diameter and x represents distance from the leading edge To validate the use of re al fluid properties, we obtain the si ngle-phase flow solution to cases ‘290C’ (hydrofoil; Re=69.110 1.7 ) and ‘312D’ (ogive; Re=69.010, 0.46 ), and compare the computed surface pressure with the experimentally measured pressure under inci pient (virtually non-cavitating) conditions. Figure 16 demonstrates good agr eement between the numerical and experimental data for both the geometries, and corroborates the correct input of physical properties. 3.2.2.1 Sensitivity analyses In general, cryogenic computations are pr one to uncertainty due to a multitude of inputs, in contrast to the non-cryogenic conditions. The cavitation model parameters, namely, ,,and destprodCCt, have been selected larg ely based on the non-cryogenic conditions. Furthermore, the solutions can exhi bit substantial sensitivity with respect to minor changes in the flow environment. For example, the uncertainties involved in the (a) (b)

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53 temperature-dependent material properties may also cause noticeable differences in predictions. To address relevant issues in this context and confirm/improve our calibration of the Merk le et al. model constants, we perform a comprehensive Global Sensitivity Analysis (GSA) over the chosen ca se ‘290C’. We initiate our computations on the case ‘290C’ (6Re9.110;1.7 ) which is centrally located in the temperature range. We note that the previously calibra ted values of the Merkle et al. Model (1.0 destC and 80.0 prodC ) are inadequate to provi de a good match with the experimental data under the cryogenic cond ition. This fact was also lately noted by Hosangadi and Ahuja (2005), who suggested lo wer values of cavitation model parameters in case of cryogenic fluids. x(inchesfrombend) p-pv(N/cm2) -0.5 0 0.5 1 1.5 -20 -10 0 10 20 30 Cdest=0.68,Cprod=54.4 Cdest=1.0,Cprod=80.0 Hord,1973;p x(inchesfrombend) T 0 0.5 1 1.5 2 80 80.5 81 81.5 82 82.5 83 83.5 84 84.5 Cdest=0.68,Cprod=54.4 Cdest=1.0,Cprod=80.0 Hord,1973;T Figure 17. Sensitivity of Merkle et al Model prediction (surface pressure and temperature) to input parameters namely destC and prodC for the hydrofoil geometry Consistently, in the pres ent study, we elicit 0.68destC and 54.4prodC via numerical experimentation, as more appropriate mode l parameters. Of course, such choices are empirically supported and need to be evaluated more systematically.

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54 To initiate such evaluations, Figure 17 portr ays the response of the surface pressure and temperature to the revision of th e model parameters. We choose , and destvCtL as design variables for the GSA, while holding the Re and constant for the given case. The chosen cavitation model parameters, namely and destCt, are perturbed on either side of their reference values (0.68destC ;54.4prodC ) by 15%. In comparison, the material properties are perturbed within 10% of the value they assume from the NIST database (Lemmon et al. 2002), at every iter ation. Given these ranges on the variables, we generate a design of experiments with 50 cases using a combination of Orthogonal Arrays (Owen 1992) and Face Centered Cubic Design (JMP 2002). RMS values of the hydrofoil surface pressure coefficient (2()/(0.5)plCppU ) and temperature, which are post-processed from the CFD data, ar e selected as the objective functions. Subsequently, the two objectives are modele d by a reduced-quadratic response surface through a least-squares regression approach (JMP 2002). The coefficient of multiple regression (Myers and Montgomery 1995) in case of the pressu re and temperature fit is 0.992 and 0.993 respectively, while the standard error is less than 1%. The fidelity of the response surfaces is also confirme d against 4 test data points. ***** ***2***2*21.6750.0770.0610.0820.007 0.0120.0110.0090.0040.013RMSpdestvdestv destvvCCtC CttLt (3.28) **** *2****** ****82.5370.2430.1070.0480.220 0.0240.0160.0150.013 0.0180.019RMSdestv destdestvdestv vTCLt CCCLL tLt (3.29) Here, R MSpC and RMST are the RMS values of the surf ace pressure coefficient and the surface temperature, respectively. The superscript, *, represents the normalized values of

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55 the design variables. Equations (3.28) and (3.29) represent the respective response surfaces expressed in terms of the normalized design variables. The surface coefficients upfront indicate the importance of th e cavitation model parameters ( and destCt ) and vapor density, and the insubstantial c ontribution of latent heat ( L ), to the variability in our objectives. We quantify these overall contri butions by employing the variance-based, non-parametric global sensitivity method proposed by Sobol (1993). This method essentially comprises decomposition of the re sponse surface into a dditive functions of increasing dimensionality. This allows the tota l variance in the data to be expressed as a combination of the main effect of each variab le and its interactions with other variables (refer to Appendix A for a brief mathematical review). 0% 20% 38% 42% Latent heat Vapor density C_dest t_infinity 1% 10% 43% 46% Latent heat Vapor density C_dest t_infinity Figure 18. Main contribution of each design vari able to the sensitivity of Merkle et al. (1998) model prediction; case ‘290C’ (a) Surface pressure (b) Surface temperature We implement the procedure expounded by Sobol (1993) on the response surfaces in equations (3.28) and (3.29) to yi eld the plots in Figure 18. The pie-charts in the figure illustrate the percentage contributi on of the main effect of each variable; since we find negligible variability due to the variable interactions. The charts firstly underscore the sensitivity of pressure and temp erature predictions to the cavitation model parameters ( and destCt). Secondly, the impact of v is noticeable, while that of L is (a) (b)

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56 insubstantial. These observations indi cate that the design variables, unlike L which appear either in m or m may tend to register greater influence on the computed results. Thus, intuitively, U and l which are omitted from the present GSA, are expected to induce large variability in the computation, as compared to other omitted properties such as K and pC x(inchesfrombend) p-pv(N/cm2) -0.5 0 0.5 1 1.5 -20 -10 0 10 20 30Merkleetal.Model Hord,1973;p x(inchesfrombend) T 0 0.5 1 1.5 2 80 81 82 83 84Merkleetal.Model Hord,1973;T Figure 19. Pressure and temperature prediction for Merkle et al. Model for the case with best match with experimental pressure; ****0.85;0.85;1.1;0.9destvCtL x(inchesfrombend) p-pv(N/cm2) -0.5 0 0.5 1 1.5 -20 -10 0 10 20 30Merkleetal.Model Hord,1973;p x(inchesfrombend) T 0 0.5 1 1.5 2 80 81 82 83 84Merkleetal.Model Hord,1973;T Figure 20. Pressure and temperature prediction for Merkle et al. Model for the case with best match with experimental temperature;****1.15;0.85;1.1;1.1destvCtL

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57 Furthermore, the impact is expected to be consistent on pressure and temperature, as depicted by Figure 18, due to the tight coupling betwee n various flow variables. However, it is important to mention that our observation is meant to illustrate the relative impact of several parameters. We also utilize the available data from our 50 cases to seek possible improvement of the Merkle at al. model parameters. Fi gure 19 illustrates the cas e which produces the least RMS error between the computed surf ace pressure and the experimental data. Conversely, Figure 20 portrays the case whic h produces the least RMS error between the computed surface temperature and the experime ntal data. These figures demonstrate that a single set of parameters may not provide optimal results for both pressure and temperature within the framework of curre nt cavitation models and cryogenic conditions. This effort solicits multi-objective optimization strategies and deserves a separate study. However, we do report from the calculated error norms of th e 50 cases that 0.68destC and 54.4prodC provide the best balance between the temperature and pressure predictions for the chosen case. As a result we hereafter employ these values for the Merkle et al. Model. We perform a simpler sensitivity analys is over the Mushy IDM since it has only one control parameter ( ).

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58 x(inchesfrombend) p-pv(N/cm2) -0.5 0 0.5 1 1.5 -20 0 20=0.07=0.09=0.11 Hord,1973;p x(inchesfrombend) T 0 0.5 1 1.5 80 80.5 81 81.5 82 82.5 83 83.5 84 84.5=0.07=0.09=0.11 Hord,1973;T Figure 21. Sensitivity of Mushy IDM predic tion for case ‘290C’ (surface pressure and temperature) to the exponential transitioning parameter Figure 21 depicts the results obta ined over various values of We calibrate 0.09 from the observed results similarly based on a reasonable balance between pressure and temperature. Furthermore, Figure 21 suggests that the limiting case of 0 which essentially recovers the evaporation term of the Sharp IDM, would substantially overpredict the cavity size. This f act endorses the need to regu late the mass transfer rates modeled by the Sharp IDM, and subsequently the purported employment of the Mushy IDM. It is worthwhile to emphasize that th e Mushy IDM manifests the regulation of mass transfer rates in cryogenic conditions – incor porated empirically into the Merkle et al. Model (section 3.2.1; Ahuja et al. 2001; Hosangadi and Ahuja 2005) largely under physical pretext. 3.2.2.2 Assessment of cryogenic cavitation mo dels over a wide range of conditions We further perform computations for all the other cases by employing both the Merkle et al.’s and the present Mushy IDM cavitation models.

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59 x(inchesfrombend) p-pv(N/cm2) -0.5 0 0.5 1 1.5 -5 0 5 10 MushyIDM Merkleetal.Model Hord,1973;p x(inchesfrombend) T 0 0.5 1 1.5 2 2.5 3 75 76 77 78 79 MushyIDM Merkleetal.Model Hord,1973;T x(inchesfrombend) p-pv(N/cm2) -0.5 0 0.5 1 1.5 -20 -10 0 10 20 MushyIDM Merkleetal.Model Hord,1973;p Hosangadi&Ahuja,2005 x(inchesfrombend) T 0 0.5 1 1.5 2 2.5 3 80 81 82 83 84 MushyIDM Merkleetal.Model Hord,1973;T Hosangadi&Ahuja,2005 x(inchesfrombend) p-pv(N/cm2) -0.5 0 0.5 1 1.5 -20 -10 0 10 20 30 MushyIDM Merkleetal.Model Hord,1973;p x(inchesfrombend) T 0 0.5 1 1.5 2 2.5 3 85 86 87 88 89 90 MushyIDM Merkleetal.Model Hord,1973;T Figure 22. Surface pressure and temperature for 2-D hydrofoil for all cases involving liquid Nitrogen. The results referenced as ‘Mushy IDM’ and ‘Merkle et al. Model’ are contributions of the present study. 283B 283B 290C 290C 296B 296B

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60 Figure 22 contrasts the resultant surface pressure and temperature obtained with the hydrofoil geometry for the cases involving liqui d Nitrogen. We firstly note that both the models are able to provide a reasonable ba lance between pressure and temperature from standpoint of their predictive cap abilities. The differences be tween the experimental data and the two models are more pronounced than the mutual differences between the two models. Furthermore, the agreement with experi mental data is better in case of pressure than in case of temperature, which is genera lly under-predicted at the leading probe point by both the models. It is also observed that both models prod uce a slight temperature rise above the reference fluid temper ature at the cavity rear end, which is attributed to the release of latent heat duri ng the condensation process. Th e Merkle et al. Model also produces a steeper recovery of pressure, as compared to the Mushy IDM, in the condensation region of the cavity, for the cases shown. This suggests higher/faster condensation rates for the Merk le et al. Model than the Mu shy IDM. Secondly, we assess our results for the ca se ‘290C’ along with latest com putational data (Hosangadi and Ahuja 2005). Note that Hosangadi and Ahuja (2005) employed the Merkle et al. Model, adapted in terms of the vapor volume fraction (v ), with substantially higher values of the model coefficients (100destprodCC ). The impact of these higher source term values is evident from the steep gradient s observed in their surface temperature and surface pressure profiles. As a consequence, the temperature prediction of the present study appears better in the cavity closure re gion, while that of Hosangadi and Ahuja (2005) shows better agreement at the cavity l eading edge. Thus, we emphasize again that the choice of model parameters poses a trade-off, as we noticed in the global sensitivity analysis, in the prediction of pressure and/or temperature for cryogenic cases. Lastly, it is

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61 important to highlight that the inlet temperature gets closer to the critical temperature and the cavitation number decreases, as we proceed from case ‘283B’ to ‘296B’. In comparison, the inlet velocity and conseque ntly the Reynolds number assume values within the same order of magnitude for th e depicted cases. Thus, under isothermal conditions, an increase in the cavity length is expected from case ‘283B’ to ‘296B’. On the contrary, the surface pressure plots in Fi gure 22 clearly indicate a decrease in cavity length from case ‘283B’ to ‘296B’, despite the decrease in the freestream cavitation number. This fact clearly distills the significan t impact of the thermal effect in cryogenic fluids, especially under working conditions that are close to the thermodynamic critical point. Cavitationnumber1.78 1.76 1.75 1.73 1.71 Cavitationnumber1.78 1.74 1.70 1.67 1.63 1.62 Figure 23. Cavitation number (2()/(0.5)vl p pTU ) based on the local vapor pressure – Merkle et al. M odel. Note the values of 2()/(0.5)vl p pTU for the cases ‘290C’ and ‘296B’ are 1.7 and 1.61, respectively. 296B 290C

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62 Our contention on the thermal effect is corroborated by the cavitation number (2[()]/(0.5)vcl p pTU ) contours depicted in Figure 23. The freestream cavitation number ( ) of the case ‘296B ’ is smaller than the case ‘ 290C’ (Table 5). However, the combination of evaporative cooling and its resultant impact over the vapor pressure causes a sharp increase in the effective cavit ation number close to the cavitation zone. This increase is more substantial for the case ‘296B’ and eventually leads to comparable levels of effective cavitation number between the two cases, as seen in Figure 23. l 0.89 0.69 0.50 0.30 l 0.89 0.69 0.50 0.30 l 0.89 0.69 0.50 0.30 Figure 24. Cavity shape indicated by liquid ph ase fraction for case ‘ 290C’. Arrowed lines denote streamlines (a) Merkle et al Model – isothermal assumption (b) Merkle et al. Model with thermal effects (c) Mushy IDM – with thermal effects (a) (b) (c)

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63 Figure 24 reveals the phase fraction distributi on and the streamlines for the computational cases of ‘290C’. The two models differ noticea bly at the rear end of the cavitation zone. The cavity of the Mushy IDM consistently indi cates lower condensation rates in contrast to the Merkle et al. Model, because of its longer length. We also note the gradual variation in density and the less extent of vapor phase in th e cavities, which highlight the mushy/soggy nature of cavitation in cryogenic flui ds. This is unlike our previous findings on regular fluids such as water (Seno cak and Shyy 2004a, 2004b; Wu et al. 2003c). Under non-cryogenic conditions, the flow structur e in the cavitation vicinity is generally characterized by large streamline curvatures and formation of recirculation zones (Senocak and Shyy 2002, 2004a, 2004b; Wu et al. 2003c) (also seen in Figure 13). However, the weak intensity of cavitation in cryogenic fluids has a modest impact over the flow structure, as indicated by the st reamline patterns in Figure 24. Figure 24(a) illustrates a ‘ special’ solution to the case ‘290C’ with th e Merkle et al. Model assuming isothermal assumptions (energy equation not solved). m--5.36E+02 -3.69E+03 -6.85E+03 -1.00E+04 m+2.37E+03 3.95E+02 2.50E+01 1.36E+00 m--5.36E+02 -3.69E+03 -6.85E+03 -1.00E+04 m+2.37E+03 3.95E+02 2.50E+01 1.36E+00 Figure 25 Evaporation ( m) and condensation ( m ) source term contours between the two cavitation models – case ‘290C’. Refe r to equations (3.8) and (3.18) for the formulations. Merkle et al. Model Mushy IDM

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64 Note that discounting the thermodynamic behavi or yields a substantia lly large cavity size under similar conditions. Based on these ob servations, we emphasize that, from a modeling standpoint, the therma l effect is manifested via a combination of the cavitation model adaptations and the temperature depende nce of physical propert ies. We bolster our argument on the condensation rates between the two models in Figure 25. Note that the evaporation and condensation co ntour plots demonstrate a mu tually exclusive behavior. The condensation region of the Merkle et al Model illustrates shar per gradients and is effective over a much larger region, as compared to the Mushy IDM. Following our assessment with liquid Nitrog en, we extend our focus to the cases with liquid Hydrogen, which has a density ratio of (30) O unlike the (100) O value for Nitrogen. We experience the need to re-calibra te our cavitation models for this different fluid because of the discernible ro le played by the density terms (l v and m ) in determining the volume transfer rates. Ou r numerical experime ntation yields 0.82destC and 54.4prodC as appropriate values for the Merkle et al. Model, in case of liquid Hydrogen. Consistently, we choose 0.065 for the Mushy IDM in context of liquid Hydrogen. Our case selection for liquid Hydrogen as seen from Table 6, follows similar trends as in case of liquid Nitrogen. The inlet velocity for all the cases with liquid Hydrogen is greater than 50 m/ s, and it increases from th e case ‘248C’ to ‘255C’ (66.4 m/s for ‘255C’). As a result, we upfront unders core these substantially higher values of inlet velocities that are em ployed for Hydrogen. Figure 26 depicts reasonable balance between the temperature and pressure pred ictions for the case ‘248C’; however, the agreement with the experimental data deteri orates equally for both the models as we

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65 proceed from the case ‘248C’ to the case ‘255 C’. Especially, the temperature is highly under-predicted as the inlet velocity in creases between the two limiting cases. x(inchesfrombend) p-pv(N/cm2) -0.5 0 0.5 1 1.5 -15 -10 -5 0 5 10 15 MushyIDM Merkleetal.Model Hord,1973;p x(inchesfrombend) T 0 0.5 1 1.5 2 19 20 21 MushyIDM Merkleetal.Model Hord,1973;T x(inchesfrombend) p-pv(N/cm2) -0.5 0 0.5 1 1.5 -15 -10 -5 0 5 10 15 MushyIDM Merkleetal.Model Hord,1973;p Hosangadi&Ahuja,2005 x(inchesfrombend) T 0 0.5 1 1.5 2 19 20 21 MushyIDM Merkleetal.Model Hord,1973;T Hosangadi&Ahuja,2005 x(inchesfrombend) p-pv(N/cm2) -0.5 0 0.5 1 1.5 -15 -10 -5 0 5 10 MushyIDM Merkleetal.Model Hord,1973;p x(inchesfrombend) T 0 0.5 1 1.5 2 20 21 22 23 MushyIDM Merkleetal.Model Hord,1973;T Figure 26. Surface pressure and temperature for 2-D hydrofoil for cases involving liquid Hydrogen. The results referenced as ‘M ushy IDM’ and ‘Merkle et al. Model’ are contributions of the present study. 249D 249D 255C 255C 248C 248C

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66 The results of Hosangadi and Ahuja (2005), whil e depicting the consistent aspect of sharp gradients that we noticed earlier, also por tray significant discrepancies between the surface temperature and the experimental data at the rear region of the cavity. This finding could be either attributed to inadequ acies in the cavitation model parameters or the representation of temperature-dependent physical properties. However, the observed sensitivity of predictions to phys ical properties, the relatively better agreement in the case ‘248C’, and the clear trend of disagreement with increasing velocities indicate the likelihood of the latter reason. Our extrac tion of physical proper ties from the NIST database (Lemmon et al. 2002) is base d on models which assume thermodynamic equilibrium conditions. However, at such high fluid velocities, the timescales for thermodynamic equilibrium may be much larger than the overall flow timescales. This may introduce substantial discrepancies in th e values of various physical properties and subsequently in the predictions; refer to Hosangadi and Ahuja (2005) for similar reporting. Of course, confirmation of the a bove possibility and development of rigorous non-equilibrium strategies is a challenging pr oposition for future research, and requires additional experimental insight. Nonetheless, we note the consistent performance of the Mushy IDM, especially in terms of the pre ssure prediction, over the chosen range of reference temperatures and velocities. We finally attempt to re-assess our above calibrations (destC ,prodC and ) for both the fluids and instill confidence into our predictive capabilities by performing computations over the axisymmetric 0.357inch ogive geometry. The ogive surface pressure and temperature plots for all the three cases are displayed in Figure 27. The shrinkage of the cavity length between cases ‘3 12D’ and ‘322E’, despite the decrease in

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67 the reference cavitation number, unequivocal ly re-emphasizes the influence of the thermo-sensible working conditions. xininchesfrombend p-pv(N/cm2) 0 0.5 1 1.5 -10 -5 0 5 10 15 20 MushyIDM Merkleetal.Model Hord,1973;p xininchesfrombend TinK 0 0.5 1 1.5 80 81 82 83 84 MushyIDM Merkleetal.Model Hord,1973;T xininchesfrombend p-pv(N/cm2) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -15 -10 -5 0 5 10 15 20 25 MushyIDM Merkleetal.Model Hord,1973;p xininchesfrombend TinK 0 0.5 1 1.5 85 86 87 88 89 90 MushyIDM Merkleetal.Model Hord,1973;T xininchesfrombend p-pv(N/cm2) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -15 -10 -5 0 5 10 15 20 25 MushyIDM Merkleetal.Model Hord,1973;p xininchesfrombend TinK 0 0.5 1 1.5 19 19.5 20 20.5 21 21.5 22 MushyIDM Merkleetal.Model Hord,1973;T Figure 27. Surface pressure and temperature for axisymmetric ogive for all the cases (Nitrogen and Hydrogen). The results referenced as ‘Mushy IDM’ and ‘Merkle et al. Model’ are cont ributions of the present study. 312D 312D 322E 322E 349B 349B

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68 Furthermore, the discrepancy in our predicti ons for liquid Hydrogen, when subjected to higher velocities, is also evidenced by th e plots for the case ‘349B’. Overall, the agreement with experimental data in all the cas es is better in terms of the surface pressure than temperature. But, unlike the hydrofo il geometry, the temperature at the first (leading) probe point is ove r-predicted for all the ogive cases. As expected, this discrepancy is most in the liquid Hydroge n case (‘349B’). Thus, the two geometries do not produce a consistent pattern of disagree ment midst the surface temperature and the experimental measurements. This inconsiste nt behavior warrants further experimental and numerical probing from a standpoint of developing more precise cavitation models for cryogenic cavitation. Nonetheless, we obser ve that our analyses/c alibrations are able to yield a justifiable range of results for the ogive geometry as well.

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69 CHAPTER 4 TIME-DEPENDENT COMPUTATIONS FOR FLOWS INVOLVING PHASE CHANGE Researchers have striven to develop efficient and accurate methodologies to simulate time-dependent cavitating flow s. The multiphase nature of the flow accompanied by complex flow physics yields a system of tightly-coupled governing equations. Furthermore, interfacial dynamics, compressibility effects in mixture region, and turbulence entail deployment of numerical models to represent these physical phenomena. The formulation of these models substantially impacts the solution procedure. As a result, evolving efficient algorithms for unsteady cavitating flows is certainly a non-trivial task. Formulation of implicit procedures for pr essure-based methods is impeded mainly by the strong linkage between the flow vari ables such as velocity and pressure. Furthermore, the dynamics of the variable density field in cavitating flows imposes supplementary equations on the existing sy stem, which add to the computational challenges. As a result, iterative algorith ms such as SIMPLE, SIMPLER, and SIMPLEC (Versteeg and Malalasekera 1995), which are comm only employed for a wide variety of problems, may be computationally expens ive for solution of cavitating flows with pronounced unsteady behavior. Senocak and Shyy (2002, 2004b) circumvented this difficulty by incorporating the Pressure Im plicit with Splitting of Operators (PISO) algorithm suitably with the model equation of cavitation dynamics and the sound propagation model for the mixture region. This endeavor resulted in a non-iterative

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70 methodology for time-dependent computation of cavitating flow through a series of predictor-corrector steps. Senocak and S hyy (2002, 2004b) demonstrated the merits of this efficient approach with a series of cav itating flow computati ons on geometries such as Convergent-Divergent Nozzl e and hemispherical solids. However, all the past efforts were in the context of isothermal cavitation sans the energy implications and real fluid properties. From st andpoint of formulating a noniterative algorithm for cryogenic cases, th e inclusion of the non-linear energy equation and temperature dependence of physical pr operties into the existing methodology is expected to pose multitude of adversities to the solution efficiency and accuracy. Direct development of an algorithm for cryogenic cavitation may be a presumptuous preliminary approach. As a result, we ini tiate a multiphase non-it erative procedure on a simpler test problem with similar nature of challenges, and, with an objective of monitoring the accuracy and stab ility of the computations. In this chapter, we elucidate the newly initiated algorithm and illustrate its results on a chosen test case. The truly unsteady problem of Gallium fusion (with natural convection effects) is adopted for the purpose of validat ion. Discussion on the grid sensitivity, accuracy of results, and the stab ility criterion is provided. The above primary objective is complimented by a reduced-order description of the ga llium fusion problem by Proper Orthogonal Decomposition (POD). The flowfield in the problem is characterized by a solid-liquid front movement with a continuous change in the overall flow length scale. The ability of POD, to accommodate these varying flow scales, is mainly emphasized.

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71 The insights gained from this study on gallium fusion are later extended to improving the algorithm for cryogenic cavitati on. The algorithm in context of cryogenic cavitation is described to point out the ch anges implemented in order to address the temperature effects. POD is employed to probe the time-dependent data, and offer it a succinct representation. 4.1 Gallium Fusion Experimental studies (Gau and Viskanta 1986) have investigated the physics of Gallium fusion due to its low fusion temperature and ease of handling. The availability of 2D experimental data has motivated numerical studies (Lacroix 1989, Lacroix and Voller 1990, Shyy et al. 1995) to adopt it as a test case. Figure 28. Schematic of the 2D Gallium squa re geometry with the Boundary Conditions (Shyy et al. 1998) A schematic of the square geometry along wi th the boundary conditions can be viewed in Figure 28.

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72 4.1.1 Governing Equations The single-fluid modeling approach is adopted by employing a liquid phase fraction f The density is assumed constant th rough the Buossinesq approximation. The governing equations for the problem are as follows. 0i iu x (4.1) 2 02 3() () 1 ()()()ji ii ii jijjuu uu pf CugTT txxxxfq (4.2) () ()()j jjjhT uhk txxx (4.3) Here, represents the coefficient of therma l expansion for the fluid. The term 2 31 ()if Cu f q in the momentum equations is the Darcy source term (Shyy et al. 1998). Through their functional dependence on f they are modeled to retard the velocity to insignificant values in solid region. The constants C and q in compliance, are tuned to yield a negative source term, which is at le ast seven orders of magnitude higher than other terms, in the solid region ( f = 0). The enthalpy in equation (4.3) is expressed as: PhCTfL (4.4) The phase fraction f is modeled com putationally by the h -based method (Shyy et al. 1998). The enthalpy at any given iteration is computed as shown below followed by an update of f (explained in a later section). 1kkk PhCTfL (4.5) Note that f is always bounded by 0 and 1. Thus, equation (4.5) can be recast into a temperature equation as shown.

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73 () () ()()[()]P Pjj jjjjCT TLf CuTkLuf txxxtx (4.6) The iterative update of the liquid phase fraction, as mentioned above, renders the energy/temperature equation non-linear. Furt hermore, the buoyancy term in equation (4.2) leads to a pressure-vel ocity-temperature coupling. Th e non-linearity in the energy equation and the strong coupling between vari ous flow variables justify the use of the gallium fusion problem as a test case pre ceding the cryogenic problem. The physical properties such as viscosity, latent heat, thermal conductivity, and specific heat are, however, assumed to be constant during the fusion process. 4.1.2 Numerical Algorithm The previous computations on similar mu ltiphase problems employed an iterative solution strategy (Chuan et al. 1991, Khodada di and Zhang 2001). Conversely, the PISO method (Issa 1985, Thakur et al. 2002, Th akur and Wright 2004), which forms the backbone of the current algorithm, essentially is a series of predictor corrector steps to yield a non-iterative solution of flow equati ons through operator splitting procedure. The present methodology, however, closely follows a slightly modified version of PISO designed for buoyancy driven si ngle-phase flows (Oliveira and Issa 2001). Kim et al. (2000) proposed a series of steps to acceler ate the simple heat c onduction equation with a solid-liquid phase boundary. The following met hodology attempts to blend the merits of the modified PISO (Oliveira and Issa 2001) w ith the propositions of Kim et al. (2000), to design an efficient and accurate algorithm fo r phase change problems. The sequence of calculations for the algorithm, based on the above governing equations, is expounded below.

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74 The strongly implicit form of the disc retized governing equati ons with a finite volume formulation, at any node P is as follows: 1 1111 1111 1 111()0 ()() ()() () ()()n ii ununnunun PPnbnbupPP vnvnnvnvn PPnbnbvpPP vn P TnTnTnfnn PPnbnbPPPPu AuAupHuSu AvAvpHvSv BT ATATHTHfMf (4.7) Here, A represents the terms at the current time level, H represents the terms at older time step, M represents the terms involving the phase fraction f B represents the buoyancy term, and S represents the Darcy source term. The Darcy source terms can be absorbed into the left hand side term to yield to following equations. 1 111 1111 111()0 () ()() ()()n ii ununnun PPnbnbupP vnvnnvnvn PPnbnbvpPP TnTnTnfnn PPnbnbPPPPu GuAupHu GvAvpHvBT ATATHTHfMf (4.8) Here, uuu P PPGAS and so on. The sequence of st eps to solve equation (4.8) noniteratively is elaborated next. (a) Momentum predictor ** **() ()()uunun PPnbnbupP vvnvnvn PPnbnbvpPPGuAupHu GvAvpHvBT (4.9) These velocities *u and *v are obtained by using the pr essure value at the previous time step and hence are not divergence free.

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75 (b) First pressure corrector ** *()0 (')()iii iiii u Pu p u G (4.10) Here, *' p pp is calculated. The pressure field p can now be employed for correcting the divergence error in velocity. Note that the pressure co rrection at this stage is limited by several approximations and is no t adequate to produce an accurate velocity field. (c) First momentum corrector **** ****() ()()uuun PPnbnbupP vvvnn PPnbnbvpPvGuAupHu GvAvpHvBT (4.11) The velocities are corrected to yield **u and**v explicitly using the intermediate pressure field obtained in the previous step. (d) First temperature corrector ***()()TTTnfn PPnbnbPPPP A TATHTHfMf (4.12) The above temperature equation utilizes the latest value of f However, in view of a noniterative strategy, solution of temperature, merely by the above equation, may not be sufficient for rapid convergence due to the delayed update of f As a remedy, a series of explicit steps are implemented following the above equation to significantly improve the prediction of temperature and, subsequently, f These steps are adapted from the algorithm proposed by Kim et al. (2000) for pur e conduction equation. They are enlisted below. (i) Update f by h -based method as shown.

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76 11 11 11 1 10 if 1 if elsewhere where, ,kkk PPPP kk Pps kk Ppl k k Ps P ls sPmlPmhCTfL fhh fhh hh f hh hCThCTL (4.13) Note that at the outset 1* k P PTT (ii) If 1 kk P P f f correct temperature using the e xplicit form of equation (4.12) with updated coefficients. This step yields the new temperature field 2 k PT. (iii) Compute residual of the temp erature equation as shown further. 221()()TkTkTnfnk PPnbnbPPPPATATHTHfMf (4.14) The above residual equation can be furt her employed to improve temperature prediction by Newton-Raphson approach as follows (Kim et al. 2000): 32 2 kk Pp k PTT T (4.15) where, the Jacobian is approximated as 2T P k pA T In summary, steps (i) – (iii) are performe d at least 10-12 times following the equation (4.12). Let the temperature and phase fraction field at the end of this stage be denoted as ** and Tf.

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77 (e) Second pressure corrector *** ** *** ***()0 (")({()() [()()]})iiii vvn iiiPP uu PP ii nbnb ii PPPu pBTBT GG AuAu SSu (4.16) This step is a powerful characteristic of the PISO algorithm. The ‘differential Darcy term’, ***()()ii P PPSSu is an addition to the terms al ready suggested by Issa (1985) and Oliveira and Issa (200 1). This term arises out of the update of iu P G after the first temperature corrector. Note that the coefficient iu P G unlike single-phase flow cases, also includes the implicit coefficient of the Darcy term (refer to the discussion in step (a)). The Darcy term undergoes a sudden change ove r several orders of magnitude between the liquid and the solid region. Since the first temperature corrector tends to change the phase fraction distribution, it is impe rative to update the coefficient iu P G post that step. The second pressure corrector, which is obtained via subtracting equation (4.11) from (4.17), is thus able to account for phase fr ont movement because of the ‘differential Darcy term’. In summary, the second pressure corrector attempts to couple the non-linear terms in the momentum equations, the temper ature field, and the phase front movement to the pressure field. (f) Second momentum corrector ******** *********() ()()uuun pPnbnbupP vvvnv pPnbnbvpPPGuAupHu GvAvpHvBT (4.17) The velocity field at this stage is derived from the more accurate prediction of pressure ** p obtained from the previous step.

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78 (g) Second temperature corrector *******()()TTTnfn PPnbnbPPPPATATHTHfMf (4.18) The prime purpose of this equation is to correct the temperature field with updated velocities. This step is followed by an update of f similar to that in equation (4.13). It was reported that repeating the corrector steps (steps (b) – (g)) for an extra time improves coupling between various equations (Thakur and Wright 2004). However, due to the large magnitude of non-linearity expect ed for the fusion problem, at least 3 more repetitions of steps (b)-(g) are recommende d (totally 8 corrector steps). The above modification to the PISO, similar to the origin al algorithm, incurs a splitting error due to the operator splitting approach. However, the or der of this splitting error is higher than the formal order of temporal accuracy (Is sa 1985), thus justifyi ng the computational accuracy. 4.1.3 Results The multiphase algorithm is implemented in conjunction with a pressure-based solver having multi-block capability (Thakur et al. 2002). The diffusive terms are handled by central differencing, while the first order upwind scheme is employed for the convective terms. Computations are mainly performed on a 2D square domain of size D =1. The range of values for St and Ra in the computations is [1 – 0.042] and [104 – 2.2 106] respectively. Sample results obtained from current algorithm are compared to those obtained by Shyy et al. (1995) in the fo llowing discussion. It is important to note that results obtained by Shyy et al. (1995) have greater fide lity than those published in similar studies, due to the use of a finer grid.

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79 4.1.3.1 Accuracy and grid dependence X Y 0 0.5 1 0 0.2 0.4 0.6 0.8 1 X Y 0 0.5 1 0 0.2 0.4 0.6 0.8 1 X Y 0 0.5 1 0 0.2 0.4 0.6 0.8 1 X Y 0 0.5 1 0 0.2 0.4 0.6 0.8 1 X Y 0 0.5 1 0 0.2 0.4 0.6 0.8 1 X Y 0 0.5 1 0 0.2 0.4 0.6 0.8 1 X Y 0 0.5 1 0 0.2 0.4 0.6 0.8 1 X Y 0 0.5 1 0 0.2 0.4 0.6 0.8 1 X Y 0 0.5 1 0 0.2 0.4 0.6 0.8 1 Figure 29. 2D interface location at various instants for St = 0.042, Ra = 2.2 105 and Pr = 0.0208. White circles represent interface locations obtained by Shyy et al. (1995) on a 41 41 grid at time instants at t = 56.7s, 141.8s, & 227s respectively. Figure 29 illustrates the 2D interface locati ons obtained with the current algorithm, with three different grid resolutions, for the parameters shown. Note that some results by Shyy et al. (1995) are at a sli ghtly earlier time instant. This contributes to the modest difference observed in the interface location, especially during the early stages, when the interface velocity is significantly higher. Cons idering this fact, the present results show reasonable agreement with the earlier com putation. Furthermore, the time-dependent movement of the interface is consistent on all the three grids, although there is a qualitative impact of the resolution on the interface profile. 21 21 grid 81 81 grid 41 41 grid

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80 X v(x,0.5) 0 0.25 0.5 0.75 1 -0.25 0 0.25 0.521x21grid 41x41grid 81x81grid X Y 0 0.1 0.2 0.8 0.9 1 X Y 0 0.1 0.2 0.8 0.9 1 Figure 30. Grid sensitivity for the St = 0.042, Ra = 2.2 105 and Pr = 0.0208, 2D case (a) Centerline vertical ve locity profiles at t = 227s (b) Flow stru cture in the upperleft domain at t = 57s; 41 41 grid (c) Flow structur e in the upper-left domain at t = 57s; 81 81 grid. Table 7. Location of the primary vortex for the St = 0.042, Ra = 2.2 105 and Pr = 0.0208 case Grid size x -location at t = 57s y -location at t = 57s x -location at t = 227s y -location at t = 227s 21 21 0.130 0.760 0.350 0.674 41 41 0.135 0.755 0.365 0.661 81 81 0.146 0.710 0.377 0.658 The influence of the grid quality on the flow structure can be assessed from the centerline velocity profiles and the locations of the cen ter of the primary vortex depicted in Figure 30(a) and Table 7 respectively. The even nu mber of nodes (cell centers) in each grid creates a slight offset between the center grid line and the geometric centerline. This offset varies inversely with the grid quality, and is expected to contribute to the reasonable (a) (c) (b)

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81 discrepancy in the centerline velocity profile s, shown in Figure 30(a). In comparison, the center of the primary vortex, from the findings reported in Table 7, indicates a gradual, downward shift with grid refinement, particular ly during the initial stages of fusion. In fact, this movement, although moderate, is conspicuous at t = 57s, when an 81-point grid is used instead of a 41-point grid. This obser vation has a physical relevance, as noticed from Figure 30(b) and Figure 30(c). The up per-left part of the domain develops a secondary vortex after some initial time laps e. This vortex is expected to induce a downward motion of the primary vortex structure. As clearly seen, the 41 41 grid, unlike the finer grid, is unable to capture this small-scale circulation, which justify the data in Table 7. The overall flow convection pattern and, consequently, the interface movement are, however, weakly affected by the seco ndary vortex structure. Furthermore, the solution demonstrates a fairly consistent impr ovement with the grid quality, in addition to a restrained grid-dep endence. These findings are enc ouraging from the standpoint of stability restrictions, which are el ucidated in the following section. Although the multiphase algorithm is elucid ated/illustrated in context of 2D calculations, its generality for 3D computations is examined by extending the case in Figure 29 to square box geometry. The fusion pr ocess is initiated by two adjacent, heated walls to yield spanwise flow variations.

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82 Figure 31. 3D interface location at t = 57s & 227s for St = 0.042, Ra = 2.2 105 and Pr = 0.0208 case on a 41 41 41 grid. Top and bottom: adiabatic; North and West: T = 0; South and East: T = 1 (heated walls) X Y 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1z=0.24 X Y 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1z=0.5 X Y 0 0.5 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1z=0.75 Figure 32. Interface location and flow pattern for the 3D case, St = 0.042, Ra = 2.2 105 and Pr = 0.0208 case at various z locations, at t = 227s Figure 31 and Figure 32 depict the interface movement and spanwise dependence of the flow structure respectively. Note that the timestep size for the 3D case is equal to that for 41 41 2D grid. Accounting the dual heating surfaces, the results are qualitatively consistent to those in Figur e 29 and instill confidence in 3D capability of the methodology. 4.1.3.2 Stability As seen clearly, the basic framework of PISO incorporates several explicit corrector steps. As a result, th e stable performance of the al gorithm is restrained by an inevitable restriction on the time-step size. Numerical experimentation over the Z

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83 previously-mentioned parametric range yielded the following heuristic stability criterion for the current problem (valid for 3D calculations). 2CFL0.0018 U x t St D UgT (4.19) All the results presented in this study employ the maximum permissible value of t Any grid refinement, as seen from above, will ha ve a direct impact on the time-step size. However, the use of small time-step sizes is ju stifiable in case of truly unsteady flows in the interest of accuracy. Furthermore, the addition of extra corrector steps seems to increases the computational costs. But, it is worthwhile noting that these correctors steps are () ON expensive, while the pre ssure solver is usually ()mON expensive (for instance, m = 1.5 for Gauss-Seidel solvers), where N represents the total number of computational nodes. Thus, instead of employing iterative procedures and solving the expensive pressure equation numerous times, adding few smart () ON corrector steps is much reasonable with the prospect of achieving th e solution non-iteratively. In summary, the efficiency of the above algorithm is ensured by keeping the costs of the pressure solver low. 4.1.3.3 Data analysis by reduced-order description Reduced-order examination of time-dependent flow problems by POD has been routinely pursued by researchers (Lumle y 1967, Podvin 2001, Lucia et al. 2002, Ahlman et al. 2002, Zhang et al. 2003). A brief review on the latest developments and issues in the technique of POD is presented in U tturkar et al. (2005) and appendix B. POD essentially projects key features of a flowfield derived from a data ensemble in form of

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84 orthogonal eigenmodes. Any flow variable at each instant can be expressed as a linear combination of the resultant eigenmodes as shown below: 1(,,)()(,)N ii iqxyttxy (4.20) Here, ()(,,)(,)T iitqxytxy and 2()it is the amount of energy of (,,) x yt in the ‘direction’ of (,)i x y (Sirovich 1987, Utturk ar et al. 2005). 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1 Figure 33. POD modes showing velocity streamlines (();1,2,3,4iri ) for St = 0.042, Ra = 2.2 103 and Pr = 0.0208 case. (,)(,) qrtVrt Useful information of the flowfield is op timally unfolded in order of the rapidly decreasing energy content of each mode. This f acilitates accurate assessment of important issues through a reduced-ord er model by allowing trunca tion of equation (4.20) with fewer modes. Despite the proven capability of POD, in case of incompressible and compressible, and, laminar and turbulent flows, the present test case is expected to pose interesting issues to the technique. The flow field in the fusion problem is truly unsteady without a time-invariant mean component. Th ere is a continuous growth in the flow 4thmode – 99%total ener gy 3r d mode – 98%total ener gy solid solid solid solid 1st mode – 84% total energy

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85 domain and flow scale as th e phase boundary traverses the do main. The ability of POD to accommodate these factors is the focal point of this brief investigation. Details on POD with regards to its mathematical backgr ound and numerical implementation can be found in Appendix B. t(s) Scalarcoefficient 10 20 30 40 50 -40 -20 01234 t(s) Scalarcoefficient 10 20 30 40 50 -8 -4 0 4 85678 Figure 34 Scalar coefficients (();1,2,...,8iti ) for St = 0.042, Ra = 2.2 103 and Pr = 0.0208 case. (,)(,) qrtVrt Figure 33 and Figure 34 illustrate the POD eigenmodes and scalar coefficients for the velocity field, fo r the flow case of St = 0.042, Ra = 2.2 103 and Pr = 0.0208. 250 time-steps of the solution, which allow about 75% of the solid to fuse, are employed for the analysis. The energy fractions denote th e cumulative kinetic energy recovered by the respective eigenmode. The first POD mode clos ely resembles the flowfield at the latest time instant. This mode, due to the truly unst eady behavior, is inconsistent with the timeaveraged data, as in the previous cases, and ju st represents the flow structure with highest overall energy content. The following eigenm odes progressively re present the smaller structures in the flowfield. The structur es shown by these su ccessive modes are nonphysical and are comparable to the harmonics in Fourier series decomposition. The first four modes of the chosen case recover almo st 99% of the kinetic energy. However, as

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86 explained further, this fact may be misleading to offer a concise representation under certain conditions. The scalar coefficients (Figure 34) too s how clear trends in their time-dependent behavior. They all initiate from zero and peak gradually in decr easing order of their ‘mode number’. Thus, the scalar coefficient of the first mode peaks latest. While the first peak in any coefficient denotes the dominan ce of the flow scales depicted by the respective eigenmode, the following peaks are re latively insignificant. This is because they are superseded by the coeffici ents of the preceding eigenmodes. t(s) PODModenumber 0 10 20 30 40 50 0 5 10 15 Figure 35. Time instants when coefficients of respective POD modes show the first peak; St = 0.042, Ra = 2.2 103 & Pr = 0.0208 case x v(x,0.5) 0 0.2 0.4 0.6 0.8 1 -0.1 0 0.1Solid Liquid x v(x,0.5) 0 0.2 0.4 0.6 0.8 1 -0.1 0 0.1Solid Liquid x v(x,0.5) 0 0.2 0.4 0.6 0.8 1 -0.1 0 0.1Solid Liquid Figure 36. Horizontal-centerline vertical velocities for St = 0.042, Ra = 2.2 103 and Pr = 0.0208 case at t = 3s CFD solution 10 modes 15 modes t > 3s t > 20s

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87 Figure 35 plots the ‘eigenmode number’ versus the time instance when the scalar coefficient shows its first peak. For in stance, the second eigenmode peaks at t = 20s, whereas the twelfth mode peaks as early as t = 3s. Thus, at least two modes are required to represent the flow scales from any time 20 t s. In comparison, a reduced-order description for the flow from 3 t s solicits at least twelve modes. This argument can be strengthened by observing the centerline vertical velocity profiles at time t = 3s, from Figure 36. The solid region, as seen from the plot s, is manifested as a sharp discontinuity in the velocity fiel d. Although the velocity profile using ten eigenmodes tends to represent the liquid region reasonably, it produces ripples in the solid region. The ripples are gradually attenuated by incorporating sma ller scales of the higher-order POD modes; for instance, fifteen modes provide an appr eciable representation of the profile at t = 3s for both the phases, as seen from the figure. While the first twelve modes construct the flow scales in the liquid region, the additional modes suppress the oscillations in the solid zone. 0 0.5 1 0 0.5 10.5000 0.4500 0.4000 0.3500 0.3000 0.2500 0.2001 0.1501 0.1001 0.0501 0.0001 VelocityMagnitude 0 0.5 1 0 0.5 10.5000 0.4500 0.4000 0.3500 0.3000 0.2500 0.2001 0.1501 0.1001 0.0501 0.0001 VelocityMagnitude 0 0.5 1 0 0.5 10.5000 0.4500 0.4000 0.3500 0.3000 0.2500 0.2001 0.1501 0.1001 0.0501 0.0001 VelocityMagnitude Figure 37. Comparison between CFD solu tion and re-construc ted solution for St = 0.042, Ra = 2.2 103 and Pr = 0.0208 case at t = 20s. Contours represent velocity magnitude and lines represent streamlines. t = 20, CFD solution t = 20, series with 2 modes t = 20, series with 5 modes

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88 0 0.5 1 0 0.5 10.0820 0.0738 0.0656 0.0574 0.0492 0.0411 0.0329 0.0247 0.0165 0.0083 0.0001 VelocityMagnitude 0 0.5 1 0 0.5 10.0820 0.0738 0.0656 0.0574 0.0492 0.0411 0.0329 0.0247 0.0165 0.0083 0.0001 VelocityMagnitude 0 0.5 1 0 0.5 10.0820 0.0738 0.0656 0.0574 0.0492 0.0411 0.0329 0.0247 0.0165 0.0083 0.0001 VelocityMagnitude Figure 38. Comparison between CFD solu tion and re-construc ted solution for St = 0.042, Ra = 2.2 103 and Pr = 0.0208 case at t = 3s. Contours represent velocity magnitude and lines represent streamlines. The above facts are corroborated by the CFD data and the re-constructed data, observed in Figure 37 and Figure 38, for two widely sepa rated time instants. Impact of POD on the thermal field is not discussed here, because it is expected to be of similar nature. In general, the efficacy of a POD representati on may strongly depend on the time interval of interest for truly unsteady problems, and ma y not merely rely on the energy content. 4.2 Turbulent Cavitating Flow under Cryogenic Conditions The previous study pointed out the importance of extra co rrector steps in case of strong non-linearity in an equa tion, and the requirement of additional source term(s) in the second pressure corrector in case of highly sensitive linear equation coefficients. These insights are applied to solving time-dependent cases of cryogenic cavitation in this section. 4.2.1 Governing Equations The governing equations are exactly simila r to those delineated in section (3.1). Some salient aspects of computational mode ling involved in time-dependent simulations are discussed below. t = 3, CFD solution t = 3, series with 10 modes t = 3, series with 15 modes

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89 4.2.1.1 Speed of sound modeling While the speed of sound model in case of steady-state computations is chosen based on the criterion of fast convergence, for time-dependent computations we employ the following form (Senocak and Shyy 2002, 2004b; also reviewed in chapter 2). 11 11SoS-2:()()ii s iiC pppp (4.21) The ability of the above formulation to produce a time-dependent behavior in close agreement with experimental observations ha s been previously demonstrated (Senocak and Shyy 2004b). 4.2.1.2 Turbulence modeling Similar to the steady-state part, the tw o-equation based tur bulence closure is adopted for time-dependent computations. In addition to the Launder and Spalding (1974) approach, we also employ the filterbased modeling (Johansen et al. 2004, Wu et al. 2004, Wu et al. 2005; also reviewed in chapte r 2). The objective is to distill the impact of the turbulent viscosity field over the uns teady tendencies in th e flow. For ease of reading, the filter-based formulation of eddy viscosity is again mention below. 2,0.09m tCk FC (4.22) where, the filter function, F is expressed in terms of a filter size ( ) as: 3/2Min[1,];1 FCC k (4.23) 4.2.1.3 Interfacial ve locity model In the context of either Sharp IDM or Mushy IDM, the interfacial velocity (, InU ) in time-dependent computations is non-zero, unl ike steady-state cases, and needs modeling

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90 efforts. Senocak and Shyy (2004b) simply correlated the interf acial velocity (, InU ) to the vapor normal velocity (, vnU ) using the mass conservation formulation at the cavity interface. In the present study, we s eek improvement to the modeling of InU Consider the volume integral of the cavitation equati on over the entire volum e of the cavity, as shown: cavity(cavity)cavity[().]()l ldVVndammdV t (4.24) The second term on the left-hand-side represents the total flux of th e liquid phase fraction over the cavity interface. In terms of the computational domain and a finite volume formulation, the above in tegral can be cast as: ,, all cellsinterface area all cells(* cell volume)()*interfacial area [()* cell volume]l lvnInUU t mm (4.25) At this juncture, we defer to an assumption that the unsteadiness of l is solely due to the movement of the interface. Thus, if there were no inte rface movement, we would get: interface areaall cells()*interfacial area[()* cell volume]lvnUmm (4.26) Subtracting equation (4.26) from (4.25) and discounting the summation, we obtain a formulation for calculating the interfacial velocity at every node as follows: ,(* cell volume) *interfacial areal In lt U (4.27)

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91 The interfacial area at every cell is computed based on the areas of the cell faces and local normal direction (/||lln ). Note that the above formulation yields ,0InU when /0lt and is thus consistent with steady-state behavior. 4.2.1.4 Boundary conditions The boundary conditions for time-dependent cavitating computations are similar to those in the steady-state pa rt barring the outlet cond ition. In case of unsteady calculations, the global mass conservation condition needs to account for the timedependent mass accumulation inside the domain. This treatment may render the computation unstable, especially in context of the PISO algorithm. To circumvent this numerical difficulty, we impose a ‘mean’ (averaged over outlet plane) value for the pressure at the outlet plane in our simulati ons. Thus, the pressure correction is imposed zero at the outlet boundary. This condition is reas onable given the fact that the tunnels in the experimental setup (Hord 1973a, 1973b) were mounted between two cryogenic reservoirs (large Dewars). 4.2.2 Numerical Algorithm The PISO algorithm is similarly adapted in case of cryogenic cavitation to handle the large density jumps (due to cavitation), the strong linka ge between the temperature and l equations, and the temperature-dependen ce of physical properties. The key steps of the algorithm in context of cryogenic cav itation are mentioned below. The changes made in the present context are specifically underscored while the identical steps are referred to the section (4.1.2).

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92 The strongly implicit forms of the continuity, momentum, cavitation, and temperature equations are as follo ws (discretized forms of the & k equations are omitted due to their straightforward nature): 1 1 111 111 11 11()0 () () () ()nn n ii ununnun PPnbnbupP vnvnnvn PPnbnbvpP nnn PPPnbnbpPP TnTnTnT PPnbnbPPPu t AuAupHu AvAvpHv A GAHS ATATHTS (4.28) Here, P S denotes the explicitly treated source terms in the cavitation and energy equation, while P G represents the implicitly treated source term of the cavitation equation. (a) Momentum predictor Logically identical to previous study (b) First pressure corrector This step is also similar to previous study except for the convective-diffusive form of the pressure corrector, as shown below ( C is modeled via the speed of sound). * *(')()(')iinn i n iiiii uu PPCu p up tAA (4.29) (c) First momentum corrector Logically identical to previous study (d) First scalar predictor

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93 This step constitutes the prediction of ,,,and lTk Note that the temperature and cavitation source terms are highly inter-linked. As a result, we resort to adding explicit corrector steps, as we implemented for the gallium problem. The phase fraction and temperature is firstly predicted as: ** **() ()n P PPnbnbpPP TTTnT PPnbnbpPPAGAHS ATATHTS (4.30) Following this prediction, we perform th e following series of steps 2-5 times. (i) Update physical prope rties, namely, ,,and lvv p based on the predicted temperature ( T*). (ii) Correct l explicitly by the pseudo Ne wton-Raphson technique, as per the previous study. (iii) Correct temperature base d on the corrected value of l obtained from (ii) We note that as many as 5 repetitions are need ed for cryogenic cases cl oser to the critical point, because of the steep variation of physi cal properties in that regime. However, 1-2 iterations of the above series may suffice for working conditions (temperature) away from the critical point. (e) Update of physical properties Values of all the physical properties, density, turbulent viscosity (t ), and C are updated at this point. Since cavitation is ch aracterized by large changes in the density field, we also update the coefficients and uv P PAA in the momentum equations to account these changes. (f) Second pressure corrector

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94 ** ***** ****** ***(")([][])(")i ii ii u ii iiinbnbPi uuu PPPCu p AuAuAup tAAA (4.31) Due to the update on and uv P P A A in the previous step, the derivation of the second pressure corrector entails the inclusion of an extra term ('* ****()iiiuuu PPP A uAAu ) into its source terms. This term relates to the ‘di fferential Darcy term’ of the previous study, and couples the modifications/corre ctions in the density eff ects to the pressure field. (g) Second momentum corrector Logically identical to previous study (h) First scalar corrector Here, all the scalar equations (,,,and lTk ) undergo explicit corrections, as shown previously. (i) Update of physical properties Identical to step (e) To enhance the coupling between various fl ow variables, we re peat all the above steps ((a) – (i)) one more time. 4.2.3 Results Time-dependent experimental data in fo rm of pressure signals and/or cavity snapshots are unavailable for cavitation in cryogenic fluids. As a consequence, we perform time-dependent computations ove r the 2D hydrofoil (Hord 1973a) that we employed for our steady-state cases. The deta ils about the geometry and grid can be referred from chapter 3. Specifically, cases ‘ 283B’ and ‘290C’ are sele cted in conjunction with the filter-based model to preclude any undesirable damp ening effects (filter size,

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95 0.11 D D is the chord length). The time-step si ze is chosen in ac cordance with the CFL=/1 Utx criterion. t(s) Cp 0 0.005 0.01 0.015 0.1 0.2 0.3 0.4 t(s) Cp 0 0.002 0.004 0.006 0.008 0 0.1 0.2 0.3 0.4 Figure 39. Pressure history at a point near inlet, 0.11 D where D is the chord length of the hydrofoil (a) ‘283B’ (b) ‘290C’ Figure 39 illustrates the timedependent behavior of norma lized pressure at a point near the tunnel inlet for the two cases. The pl ots indicate that, desp ite the deployment of filter-based turbulence model, the computati on stabilizes to a steady state eventually. l0.89 0.74 0.60 0.45 0.30 l0.89 0.74 0.60 0.45 0.30 Figure 40. Cavity shape and flow structure for case ‘283B’. Lines denote streamlines (a) (b) single-phase cavitation

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96 The surface pressure and temperature obtained via the unsteady computations agree well with the steady state results in both cases (not shown here to avoid repetition of figures). The cause of this stable behavior can be ex plained based on the fl ow structure observed in Figure 40. Due to the remarkable weak intensity and mushy nature of cavitation in cryogenic fluids, the streamlin es do not depict sharp defl ections or formation of recirculation zones, unlike cavitation in regular fluids (Senocak and Shyy 2004a, 2004b; Wu et al. 2003c, 2004, 3005). In fact, the flow structure around the cavity does not seem to be noticeably different than the single-phase flow struct ure. As a consequence, the phenomenon of cavity auto-oscilla tions, which is mainly caused due to the re-entrant jet effects/instabilities (Senocak and Shyy 2004b), fa ils to occur in the above situation of cryogenic cavitation. t(sec) Cp 0.015 0.02 0.025 0.00 0.01 0.02 0.03 St=0.5 St=1 St=2 Figure 41. Pressure history at a point near th e inlet for the case ‘290C’. St denotes the non-dimensional perturbation frequency (/ f DUft ) The time-dependent simulation strategy e xpounded in the previous section is further employed to investigate the impact of inlet temperature perturbation on the overall flow field. Unsteady comput ations are performed on cases ‘290C’ and ‘296B’, as

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97 described in Chapter 3, by applying a sinusoidal boundary condition on the inlet temperature. The amplitude of pertur bation is chosen 0.5 K. A filter-size 0.11 D is selected for the filter-based turbulence model. t/T Cp 4 6 8 10 -0.02 -0.01 0 0.01 0.02 0.03Case'296B' Case'290C' Figure 42. Pressure history at a point near th e inlet for the cases ‘296B’ and ‘290C’. The non-dimensional perturbation frequency (/ f DUft ) is 1.0. Figure 41 depicts the impact of the temperatur e perturbation on the pr essure at an inlet point for three different excita tion frequencies. Note that the variation in vapor pressure and phasic densities is about 5% gi ven the temperature variation of 83.060.5 K. We note that the amplitude of the pressure re sponse is lowered at higher perturbation frequencies. This mainly due to the timescale, t associated with the cavitation source terms. At higher inlet frequencies, the response of the cavitation model to the rapidly changing temperature environment is relatively slower. Due to the lack of this unison, we observe a restrained oscillatory behavior at St =2. Figure 42 illustrates the influence of operating conditions on the tendency to respond to temperature fluctuations. The mean temperature for case ‘290C’ and ‘296B’ is 83.06 K and 88.54 K, respectively. Since the mean temperature in case ‘296B ’ is closer to the critical point, the phys ical properties are

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98 more temperature-sensitive as compared to the case ‘290C’. This fact is evident from Figure 42, which demonstrates larger amplit ude of oscillation for the former case. Figure 43 POD modes for the velocity field; ();1,2,3iri ; (,)(,) qrtVrt Figure 44 POD modes for the velocity field; ();1,2,3iri ; (,)(,) qrtVrt 1 s t mode 1 s t mode 2 n d mode 2 n d mode 3 r d mode 3 r d mode St = 1 St = 2 1 s t mode 1 s t mode 2 n d mode 2 n d mode ‘290C’ ‘296B’

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99 Figure 44. Continued Since the degree of perturbation in the phys ical properties is a bout 5%, we fail to observe noticeable impact on the flow structure via instantaneous snap shots of the flow field. As a consequence, POD is employed to unearth the various leve ls of impact caused by the temperature perturbation on the flow stru cture. Figure 43 indicates that the first POD mode, which represents the mean/overall flow structure, is consistent between the two perturbation frequencies. However, we notice increasing amount of discrepancy between the corresponding plots of the highe r order POD modes. Conversely, the POD modes in Figure 44, which contrasts cases ‘290 C’ and ‘296B’, depict consistency in the flow structures of the two cases to a larg e extent. These trends can be explained by referring to Figure 41 and Figure 42. The dyna mic behavior of the flow field under varying perturbation frequencies, as seen from Figure 41, shows different oscillatory amplitudes as well as dissimilar time-dependent patterns. In comparis on, similar nature of perturbation in the two flow cases (‘290C’ and ‘296B’) seems to produce a major impact only on the amplitude of the re sponse; the time-dependent pattern is consistent between the two flow configurations as observed from Figure 42. These facts are manifested likewise in the flow structure. Thus, the differences in fl ow cases ‘290C’ and ‘296B’ are largely absorbed by the time-dependent POD coefficients (()it ) rendering consistentlooking modes of the flow structure (()ir ). In contrast, the effects of different perturbation frequencies are re gistered over the POD modes of the flow structure as well 3 r d mode 3 r d mode

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100 as the POD coefficients. The above observations may tend to vary at higher amplitudes of the temperature perturbation, a phenomenon wh ich may not be predicted upfront by the above investigation. Neverthele ss, the POD analysis serves to effectively discern the effects of varying inlet temperature on the fl ow field by offering a su ccinct representation to the time-dependent CFD data.

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101 CHAPTER 5 SUMMARY AND FUTURE WORK 5.1. Summary An effective computational procedure is formulated for simulating turbulent cavitating flow under cryogenic conditions. Th e existing framework of equations for turbulent cavitating flows is coupled with the energy equation to account for the evaporative cooling effect. The material pr operties are dynamically updated, as functions of temperature, to induce the thermo-sensible behavior into the cavitation characteristics. The derivation of the interfacial dynamics-based cavitation model is revised to accommodate the mushy features of cavitati on typically observed for cryogenic fluids. The resultant model causes a non-linear decr ease and increase in the evaporation and condensation source term of the original model, respectively. The cavitation model parameters, which seem to be dependent on the fluid type, in the newly revised model (referred to as the Mushy IDM) and the Merkle et al. model are calibrated for liquid Nitrogen and Hydrogen via numerical experimentation. The sensitivity of predictions to these empirica l choices and the uncertainties in material properties is systematically investigated th rough a global sensitivity analysis. The results indicate that the parameters and physical properties, which especially occur in the cavitation model source terms (,,,,,,,,and destprodvlvmCCtUp ), are more likely to have a significant impact on the results than others (,,,and P LKC ). Furthermore, the pressure and temperature fields pose a tradeoff in terms of optimizing their prediction,

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102 which is achieved by balancing their agreement with the experimental data. In comparison with the Merkle et al. Model, the Mushy IDM offers ease in calibration due to the presence of a single control parameter ( ). The performance and tendencies of the Mushy IDM depict consistency to the alternate models in contex t of the non-cryogenic cases. Subsequently, cryogenic computations are performed over two geometri es (2D hydrofoil and axisymmetric ogive) with liquid Nitrogen and Hydrogen to exte nsively assess the framework over wideranging combinations of flow conditions a nd geometries. The thermodynamic effect is demonstrated under consistent conditions via the reduction in cavity length and the increase in effective cavitation number as th e working conditions tend towards the critical point. Reasonable agreement between the com puted surface pressure and temperature, and experimental data is obtai ned. Specifically, the agreemen t between the two models is closer than their respective agreement with the experimental data. In addition, the predictions of both the models ar e better; for the pressure fiel d than the temperature field, and for liquid Nitrogen than liquid Hydroge n. The Hydrogen cases are characterized by very high inlet velocities, which seemingly ha ve an adverse effect on the predictions. The deviation of temperature from the experiment al values is not c onsistent between the hydrofoil and ogive geometry. This inconsiste nt behavior and the above-mentioned high velocity conditions for liquid Hydrogen warr ant further experimental and numerical investigation for developing more precise m odels. Nevertheless, the Mushy IDM, whose derivation is based on the physical interpretation of cryogenic cavities, is ab le to offer reasonable results over all the flow cases.

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103 A non-iterative algorithm for pressure-bas ed calculation of multiphase flows is initiated from the viewpoint of simula ting cryogenic cavitation. The Gallium fusion problem, which is characteri zed by a non-linear energy equation and strong pressurevelocity-temperature coupling, is employed as a test case. The algorithm adopts the framework of the operator split ting strategy (PISO) and incor porates steps to couple the phase front movement to velocity and pressu re fields, and accelerat e the convergence of the non-linear energy equation. Accurate and fa irly grid-independent results over a wideranging parameter space are obtained. For i nherently unsteady flow problems, the noniterative formulation of the al gorithm provides substantial co mputational benefit over the iterative approach, when accurate and time-dependent information is of interest. The newly varied PISO algorithm is further adopted for time-dependent computations of cavitating flow in cryogeni c fluids. Impact of perturbing the inlet temperature on the flow field is investigated via the time-dependent simulations. A stable performance of the algorithm is observed for a ll the cases. The weak nature of cavitation in cryogenic fluids has a modest influence ove r the flow structure, and thus tends to reduce the possibility of flow instabilities caused due to re-entrant jet effects. POD is employed to offer a succinct representation to the time-dependent data. POD is effectively able to distill dominant flow st ructures from the voluminous CFD data, and reveal the impact of the temperature perturbations and flow conditions on the unsteady behavior of the flow field. 5.2 Future Work The present study may be extended in the following manner: (a) Optimization of the cryogenic cavitation models via multi-objective strategies (namely Pareto Front analyses)

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104 (b) Systematic experimental probing of the phase transformation rates involved in cavitation to enable development of more precise cavitation models (c) Critical investigation of vari ous assumptions in the case of liquid Hydrogen to reveal the cause of the observed discrepancies (d) Investigation of compressibi lity and turbulence characte ristics in cryogenic fluids (e) Deployment/extension of the cryogenic cavitation framework in turbo-machinery applications

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105 APPENDIX A BACKGROUND OF GLOBAL SENSITIVITY ANALYSIS The surrogate model (response surfaces in our case), g (x), is decomposed as the sum of functions of increasing dimensionality as: 01212,,,,iiijijnn iijgggxgxxgxxxx (A.1) Equation (A.1) is subjected to the following constraint: 111 ...... 00ssiiiigdx (A.2) Note that x denotes normalized values of the desi gn variables scaled between 0 and 1, and 0g represents the mean value (0() ggd xx ). The constraint in equation (A.2) renders the decomposition shown in equation (A.1) unique (Sobol 1993). Consequently, the total variance ( V ) in the data can be decomposed as: 1 11()n iijn iijnVyVVV (A.3) The partial variances, as shown in equa tion (A.3), are computed as follows: ([|]) ([|,]) ([|,,])ii ijijij ijkijjijikjkijkVVEyx VVEyxxVV VVEyxxxVVVVVV (A.4) Here, V and E represent the variance and ex pected value respectively, and i j k s and n are the indices. Thus, the tota l contribution of a variable (total iV ) to the variance in the data

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106 can be expressed as the sum of the main effect of the variable and its interactions with other variables, as shown below. ,,, main effect interactions...total iiijijk jjijjikkiVVVV (A.5)

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107 APPENDIX B REVIEW AND IMPLEMENTATION OF POD B.1 Review POD was first introduced by Lumley (1967) to investigate coherency of turbulent flow structures. Since this seminal contribu tion, POD has been a popul ar tool to extract systematically hidden, but deterministic, structures in turbul ent flows and can be extensively found in literature. Gradual progress in this area has established the application of this technique to laminar fl ows, as well as to flows under incompressible and compressible conditions. The latest studies and relevant issues of POD implementation are reviewed hereafter. Aubry et al (1988) built a five-mode, reducedorder model for the wall region of a fully turb ulent channel. Their effort was extended by Podvin (2001), who provided numerical valid ation of a ten-dimensional model for the wall layer. The ability of the ten-dimensio nal model to produce intermittent features, which are reminiscent of bursting process in a wall layer, was also demonstrated. Comprehensive POD studies on turbulent mixi ng layers are available (Delville et al. 1999; Ukeiley et al. 2001), which not only id entify the large-scaled structures in the layer, but also model the dynamic behavior of the lower-order modes. Prabhu et al (2001) explored the effect of various flow control mechanisms in a turbulent channel, on the flow structure in POD modes. The POD modes of various flow control mechanisms showed significant differences close to the wall but were similar in the channel core. Similarly, while Liberzon et al (2001) employed POD to stu dy vorticity characterization in a turbulent boundary layer, Kostas et al (2002) adopted the approach to probe PIV

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108 data for the backward-facing step flow. Picard and Delville (2000) investigated the effect of longitudinal pressure distri bution on velocity fluctuations in the turbulent shear layer of a subsonic round jet, using POD. Annaswam y et al. (2002) examined ‘edge-tones’ of an aircraft nozzle by analyzing the POD mode s of azimuthal pressu re distribution of a circular jet. Cizmas and Palacios (2003) gained insight on the turbine rotor-stator interaction with a lower-order POD investig ation. They effectively utilized the time history and phase-plane plots of the POD coefficients to unravel the key dynamics in the flow behavior. POD-based investigations on pulsed jet flow field (Bera et al. 2001), temperature field in flow over heated wa ves (Gunther and Rohr 2002), and many such flow cases provide further evidence on the wide applicability of the technique. As mentioned earlier, POD has been successfu lly extended to laminar flow cases for extracting the principle features from timedependent flow data (Ahlman et al. 2002). Despite the ongoing progress in POD, several issues of its applicability to variable density and compressible flows are lately being examined. As indicated before, POD essentially yields a series, which rapidly converges toward s the norm of a variable (,) qrt. Several past studies intuitively adopted scalar-valued norms for the convergence criterion. For example, each flow variable namely pressure, density, or any velocity component was separately decomposed into POD modes. However, Lumley and Poje (1997) observed that for variable density flows, such as buoyancy-driven flows, an independent POD analysis may decouple th e physical relationship between the flow variables. They suggested that the norm selection should incorporate the density variations into the velocity field, to achieve the convergence of a physically relevant quantity – mass rather than mere velocitie s through the POD procedure. However,

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109 simultaneous use of two flow variables also poses an important issue of deciding the significance of each variable in convergence process. Lumley and Poje (1997) suggested a vector form to (,) qrt as 1112(,)[,,,'] qrtCuCvCwC In addition, they provided a mathematical analysis to optimize the values of the weighing factors for expediting the convergence. Colonius et al. (2002) exte nded the above argument by examining the impact of the norm selection on POD anal ysis of compressible flow over a cavity. POD modes independently obtained by scalar-val ued norms of pressure and velocity were compared to those derived by vector-based norms. The choice of the vector, 2 (,)[,,,] 1 qrtuvwc was chosen with ingenuity so as to yield a norm as shown below: 1 22223 22 ||(,)||[()] 1 qrtuvwcdr (B.1) The above norm effectively yielded a linear series th at converged towards the stagnation enthalpy instead of mere kinetic energy. Co lonius et al. (2002) fu rther reported that scalar-valued POD modes were unable to capture key processes such as acoustic radiation, which heavily rely on coupling mechanisms between the variables. In comparison, the vector-valued POD modes were in cohort with the compressible flow dynamics. Ukeiley et al. (2002) employed 0(,)[,,,,] TT uvw qrt UUUTT to perform POD on numerical data of compressible mixi ng layer. The variables, as shown above, were normalized by their freestream values to ensure a rational weighing of their fluctuations. A slow convergence towards th e multi-variable norm was reported. POD implementation was also shown to face serious issues in case of supersonic flows due to

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110 presence of shock fronts. Lucia et al. (2002) noticed that though the bulk flow can be modeled by few eigenmodes, a larger number of eigenmodes are required to accurately capture the discontinuity in the flow. They circumvented the issue by employing domain decomposition in their POD implementation. Though reasonable success in the reducedorder representation of the shock was repor ted, issues of extending the decomposition technique to moving shock fr onts are still unresolved. B.2 Mathematical Background Consider a flow quantity, (,) qrt where r denotes spatial variables and t denotes time. The objective of POD can be simply stated as minimizing the L2 norm of the objective error function, (,)kFrt which is defined as (Lumle y 1967; Delville et al. 1999; Ukeiley et al. 2001; Ahlman et al. 2002; Arian et al. 2002): (,)(,)(,)k kFrtqrtqrt (B.2) The L2 norm in terms of volume integral can be defined as: 1 23 2||(,)||[(,)]kkFrtFrtdr (B.3) Here, (,)kqrt denotes the data projected with a linear combination of certain number ( k ) of orthogonal basis functions (() r ). Through calculus of va riations, the problem of determining the optimally converging basis modes, () r can be well-posed in form of an integral equation as shown (Lumley 1967; Delville et al. 1999): (,')(')'() R rrrdrr (B.4) where, (,') R rr is the space correlation tensor, defined as: (,')(,)(',) R rrqrtqrt (B.5)

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111 In the context of classical app lication of POD to turbulent st ructures, the angular brackets denote ensemble average of statistically stationary turbulent flow data. However, the angular brackets in context of the present st udy interchangeably represent time-averaging of either unsteady RANS or laminar soluti ons. This approach is often referred in literature as the ‘snapshot POD approach’ (Sirovich 1987; Zhang et al. 2003), where every instantaneous solution is considered as a ‘snapshot’ of the data. Solution of equation (B.4), which is also subjected to a constraint of unitary norm ||()||1 r yields the orthogonal eigenmodes ()nr with corresponding eigenvalues n Consequently, solution of the flow quantity at each time step can be exactly expressed as a linear combination of the eigenmodes (also know n as POD modes) in the following form. 1(,)()()N nn nqrttr (B.6) The time-dependent multipliers or scalar coefficients, ()nt are obtained by projecting the solution at each time step on the basis functions as shown. 3()(,)()nntqrtrdr (B.7) Merely, from a standpoint of equation (B.6), infinite choices for the basis functions ()nr are available. But, the basis functions de termined in conjunction with the minima of ||(,)||kFrt ensure optimal and sequential extrac tion of key features from the data ensemble, enabling truncation of the series (B.6) with fewer POD modes. Thus, succinct expression of a voluminous data ensemble, with focus on its salient aspects is facilitated through the lower-order modes.

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112 B.3 Numerical Implementation In the current study, the CFD solution at every time step is considered as a snapshot of the flow field, and an ensemble in form of a matrix A is generated as follows: 123[,,,...,]NAaaaa (B.8) where, ; 1,2,3,...,M naRnN denotes the solution at all M nodes in the domain at the nth time step (usually M> > N ). Thus, when (,) qrt is expressed in the above matrix form, the solution of equation (B.4) reduces to obta ining the eigenvalues and eigenvectors of the matrix AAT, which is achieved by the numeri cal technique of Singular Value Decomposition (SVD). B.3.1 Singular Value Decomposition (SVD) Any real matrix M NA can be decomposed into the form (Utturkar et al. 2005): TAUV (B.9) where, U is a (MM) matrix whose columns form left singular vectors; V is a (NN) matrix whose columns form right singular vectors, and is a pseudo-diagonal (MN) matrix whose diagonal elements are the singular values ns Furthermore, it can be easily shown that the eigenvalue decomposition of the matrix AAT can be expressed as: 2TT A AUU (B.10) From equations (B.9) and (B.10), it is evident that eigenvalues of AAT are squares of the singular values of A (2nns ), and its eigenvectors are the left singular vectors of A Thus, SVD of the ensemble matrix A can effectively yield the desired POD basis functions and their respective eigenvalues.

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113 B.3.2 Post-processing the SVD Output The SVD subroutine returns the matrix U with MM elements. The matrix U is reshaped by extracting only its first N columns. The resultant matrix U which is now an MN matrix, comprises the N eigenmodes corresponding to the original data in matrix A Note that the constraint ||()||1 r manifests itself in the following form TUUI (B.11) where, I is an NN identity matrix. The coefficient matrix for ()nt in compliance with equation (B.7), is further obtained as: TUA (B.12) Subsequently, the truncated series with k eigenmodes, in form of another matrix, ˆ () Ak is constructed as follows: ˆ ˆ ˆ () M kkNAkU (B.13) where, ˆ UU and ˆ It is thus straightforward to observe that the error/discrepancy between the actual and re constructed data is ˆ ()() kAAk (B.14) The fraction of cumulative energy captured by k eigenmodes is calculated as follows (Ahlman et al. 2002; Zhang et al. 2003): 2 11 2 11 kk nn nn k NN nn nns E s (B.15) The energy fractions and the error matrix (equations (B.14) and (B.15)) are commonly used parameters to judge a POD representation.

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114 B.4 Flowchart B.5 Computing Efforts In the first step, the algorithm to reduces A to upper bi-diagonal form UTAV=B by Householder bi-diagonalizat ion. It requires 2MN2-(2/3)*N3 flops. In the second step, diagonalization of the bi -diagonal form requires 14N flops.

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115 The SVD subroutine (shown in dotted recta ngle) is adopted from Press et al. (1992) which is essentially a Fortran-90 code. Creat ing subroutines to read the CFD data into matrix A, determine the coefficient matrix, reconstruct the truncated series, perform various error estimates and output the soluti on, constitutes a major part of the present effort. The Fast Fourier Transform (FFT) a nd least square analyses are performed on MATLAB and MATHCAD.

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116 LIST OF REFERENCES Ahlman, D., Soderlund, F., Jackson, J., Kurdilla, A. and Shyy, W., 2002, “Proper Orthogonal Decomposition for Time-dep endent Lid-driven Cavity Flows,” Numerical Heat Transfer, Part B, vol. 42, pp. 285-306. Ahuja, V., Hosangadi, A. and Arunajatesan, S., 2001, “Simulations of Cavitating Flows Using Hybrid Unstructured Meshes,” Journa l of Fluids Engineering, vol. 123, pp. 331-340. Annaswamy, A., Choi, J.J., Sahoo, D., Alvi F.S. and Lou, H., 2002, “Active Closed Loop Control of Supersonic Impinging jet Flows Using POD models,” Proc. 41st IEEE Conference on Decision and Control, pp. 3294-3299, Las Vegas, NV. Arian, E., Fahl, M. and Sachs, E.W., 2002, “Managing POD Models by Optimization Methods,” Proc. 41st IEEE Conference on Decision and Control, pp. 3300-3305, Las Vegas, NV. Athavale, M.M., Li, H.Y., Jiang, Y. and Singhal, A.K., 2000, “Application of Full Cavitation Model to Pumps and Inducers,” Presented at 8th International Symposium on Transport Phenomenon and Dynamics, Honolulu, HI. Athavale, M.M. and Singhal, A.K., 2001, “Num erical Analysis of Cavitating Flows in Rocket Turbopump Elements,” AIAA Paper 2001-3400. Aubry, N., Holmes, P., Lumley, J.L. and Ston e, E., 1988, “The Dynamics of Coherent Structures in the Wall Region of the Wall Boundary Layer,” J. Fluid Mech., vol. 192, no. 15, pp. 115-173. Batchelor, G.K., 1967, An Introduction to Fluid Dynamics, Cambridge University Press, New York. Bera, J.C., Michard, M., Grosjean, N. and Comte-Bellot, G., 2001, “Flow Analysis of Two-Dimensional Pulsed Jets by Particle Image Velocimetry,” Experiments in Fluids, vol. 31, pp. 519-532. Brennen, C.E., 1994, Hydrodynamics of Pumps, Oxford University Press, New York Brennen, C.E., 1995, Cavitation and Bubble Dynamics, Oxford University Press, New York.

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117 Chavanne, X., Balibar, S. and Caupin, F., 2002, “Heterogeneous Cavitation in liquid Helium-4 near a Glass Plate,” Journal of Low Temperature Physics, vol. 126, no. 1, pp. 615-620. Chen, Y. and Heister, S.D., 1996, “Modeling Hydrodynamic Non-equilibrium in Cavitation Flows,” Journal of Flui ds Engineering, vol. 118, pp. 172-178. Chuan, C.H., Schreiber, W. C., Kuo, C.H. and Ganesh, S., 1991, “The Effect of Convection on the Interface Location between Solid and Liquid Aluminum,” Proc. ASME HTD, vol. 163, pp. 79-84. Cizmas, P.G.A. and Palacios, A., 2003, “Proper Orthogonal Decomposition of Turbine Rotor-Stator Interaction”, J. Propulsion and Power, vol. 19, no. 2, pp. 268-281. Clarke, J.W.L. and Leighton, T.G., 2000, “A Method for Estimating Time-dependent Acoustic cross-sections of Bubbles and Bubble clouds prior to the Steady State,” J. Acoust. Soc. Am., vol. 107, no. 4, pp. 1922-1929. Colonius, T., Rowley, C.W., Freund, J.B. an d Murray, R., 2002, “On the Choice of Norm for Modeling Compressible Flow Dynamics at Reduced-order Using the POD,” Proc. 41st IEEE Conference on Decision and Control, pp. 3273-3278, Las Vegas, NV. Coutier-Delgosha, O. and Astolfi, J.A., 2 003, “Numerical Predic tion of the Cavitating Flow on a Two-Dimensional Symmetrical H ydrofoil With a Single Fluid Model,” Presented at Fifth International Symp osium on Cavitation, Osaka, Japan. Daney, D.E., 1988, “Cavitation in flowing S uperfluid Helium,” Cryogenics, vol. 28, pp. 132-136. Delannoy, Y. and Kueny, J.L., 1990, “Cavity Flow Predictions Based on the Euler Equations,” ASME Cavitation and Multip hase Flow Forum, Toronto, Canada. Delannoy, Y. and Reboud, J.L., 1993, “Heat and Mass Transfer on a Vapor Cavity,” Proc. ASME Fluids Engineering Confer ence, Washington, DC, vol. 165, pp. 209214. Delville, J., Ukeiley, L., Cordier, L., Bonnet, J.P. and Glauser, M., 1999, “Examination of Large-scale Structures in a Turbulent Plane Mixing Layer: Part 1 Proper Orthogonal Decomposition,” J. Fluid Mech., vol. 391, pp. 91-122. Deshpande, M., Feng, J. and Merkle, C.L., 1994, “Cavity Flow Predictions Based on the Euler Equations,” Journal of Fluids Engineering, vol. 116, pp. 36-44. Deshpande, M., Feng, J. and Merkle, C.L., 1997, “Numerical Modeling of the Thermodynamic Effects of Cavitation,” Journal of Fluids Engineering, vol. 119, pp. 420-427.

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118 Edwards, J.R., Franklin, R.K. and Liou, M.S., 2000, “Low-Diffusion Flux Splitting Methods for Real Fluid Flows with Phase Transitions,” AIAA Journal, vol. 38, no. 9, pp. 1624-1633. Franc, J.P., Avellan, F., Bela hadji, B., Billard, J.Y., Briancon, L., Marjollet, Frechou, D., Fruman, D. H., Karimi, A., Kueny, J.L., and Michel, J. M., 1995, La Cavitation: Mecanismes Physiques et Aspects Industriel s, Presses Universitaires de Grenoble, Grenoble (EDP Sciences), pp. 28, 141, 142, 144, 449, France. Franc, J.P., Janson, E., Morel, P., Rebattet, C. and Riondet, M., 2001, “Visualizations of Leading Edge Cavitation in an Inducer at Different Temperatures,” Presented at Fourth International Symposium on Cavitation, Pasadena, Canada. Franc, J.P., Rebattet, C. and Coulon, A., 2003, “An Experimental Investigation of Thermal Effects in a Cavitating Inducer,” Presented at Fifth International Symposium on Cavitation, Osaka, Japan. Fruman, D.H., Benmansour, I. and Sery, R., 1991, “Estimation of the Thermal Effects on Cavitation of Cryogenic Liquids,” Proc. Cavitation and Multiphase Forum, pp. 9396. Fruman, D.H., Reboud, J.L. and Stutz, B., 1999, “Estimation of Thermal Effects in Cavitation of Thermosensible Liquids,” Int. J. Heat and Mass Transfer, vol. 42, pp. 3195-3204. Gau, C. and Viskanta, R., 1986, “Melting and Solidification of a Pure Metal on a Vertical Wall,” J. Heat Transfer, vol. 108, pp. 174-181. Gopalan, S. and Katz, J., 2000, “Flow Stru cture and Modeling Issues in the Closure Region of Attached Cavitation,” Phys Fluids, vol. 12, no. 4, pp. 895-911. Gunther, A. and Rohr, P.R., 2002, “Structure of Temperature Field in a Flow over Heated Waves,” Experiments in Fluids, vol. 33, pp. 920-930. Hord, J., 1973a, “Cavitation in Liquid Cryogens, II – Hydrofoil,” NASA Contractor Report, NASA CR – 2156. Hord, J., 1973b, “Cavitation in Liquid Cryogens, III – Ogives,” NASA Contractor Report, NASA CR – 2242. Hosangadi, A., Ahuja, V. and Ungewitter, R. J., 2002, “Simulations of Cavitating Inducer Flowfields,” JANNAF 38th Combustion Subcommittee Meeting, Sandestin, FL. Hosangadi, A. and Ahuja, V., 2003, “A Generalized Multi-phase Framework for Modeling Cavitation in Cryogenic Fluids,” AIAA Paper 2003-4000. Hosangadi, A., Ahuja, V. and Ungewitte r, R.J., 2003, “Generalized Numerical Framework for Cavitation in Induce rs,” ASME Paper FEDSM2003-45408.

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119 Hosangadi, A. and Ahuja, V., 2005, “Num erical Study of Cavitation in Cryogenic Fluids,” J. Fluids Engineering, vol. 127, pp. 267-281. Iga, Y., Nohmi, M., Goto, A., Shin, B.R. and Ikohagi, T., 2003, “Numerical Study of Sheet Breakoff Phenomenon on a Cascade Hydrofoil,” Journal of Fluids Engineering, vol. 125, pp. 643-651. Issa, R.I., 1985, “Solution of the Implicitly Discretized Fluid Flow Equations by Operator Splitting,” J. Comp. Phys., vol. 62, pp. 40-65. JMP, 2002, The Statistical Discovery SoftwareTM, version 5.0, SAS Institute Inc., Cary, NC, U.S.A. Johansen, S., Wu, J. and Shyy, W., 2004, “F ilter-based Unsteady RANS Computations,” Int. J. Heat and Fluid Flow, vol. 25, pp. 10-21. Kakegawa, A. and Kawamura, T., 2003, “An Experimental Study on Oxidation of Organic Compounds by Cavitating Water-Jet ,” Presented at Fifth International Symposium on Cavitation, Osaka, Japan. Kalamuck, K.M., Chahine, G.L., Hsiao, C.T. and Choi, J.K., 2003, “Remediation and Disinfection of Water Using Jet Genera ted Cavitation,” Presented at Fifth International Symposium on Cavitation, Osaka, Japan. Kawanami, Y., Kato, H., Tanimura, M. a nd Tagaya, Y., 1997, “Mechanism and Control of Cloud Cavitation,” Journal of Flui ds Engineering, vol. 119, pp. 788-794. Khodadadi, J.M. and Zhang, Y., 2001, “Effects of Buoyancy-driven convection on melting with Spherical Containers,” Int. J. Heat and Mass Transfer, vol. 44, pp. 1605-1618. Kim, S., Lee, B. and Cho, S., 2000, “A Front Layer Predictor-Corrector Algorithm with pseudo Newton-Raphson Method for the Phase Change Heat Conduction Problems,” Int. Comm. Heat Mass Transfer, vol. 27, no. 7, pp. 1003 – 1012. Kirschner, I.N., 2001, “Results of Sel ected Experiments Involving Supercavitating Flow,” VKI Lecture Series on Supercavita ting Flow, VKI Press, Brussels, Belgium. Knapp, R.T., Daily, J.W. and Hammitt, F.G., 1970, Cavitation, McGraw-Hill, New York. Kostas, J., Soria, J. and Chong, M.S., 2002, “Particle Image Velocimetry Measurements of a Backward-facing Step Flow,” Experiments in Fluids, vol. 33, pp. 838-853. Kunz, R.F., Boger, D.A., Stinebring, D.R., C hyczewski, T.S., Lindau, J.W. and Gibeling, H.J., 2000, “A Preconditioned Navier-St okes Method for Two-phase Flows with Application to Cavitation,” Comp ut. Fluids, vol. 29, pp. 849-875.

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124 Thakur, S.S., Wright, J. and Shyy, W., 2002, “ STREAM: A Computational Fluid Dynamics and Heat Transfer Navier-Stokes Solver,” Streamline Numerics Inc. and Computational Thermo-Fluids Laboratory, Department of Mechanical and Aerospace Engineering Technical Report, Un iversity of Florida, Gainesville, FL. Tokumasu, T., Kamijo, K. and Matsumoto, Y., 2002, “A Numerical Study of Thermodynamic Effects of Sheet Cavitation,” Proceedings of ASME FEDSM’02, Montreal, Quebec, Canada. Tokumasu, T., Sekino, Y. and Kamijo, K., 2003, “A new Modeling of Sheet Cavitation Considering the Thermodynamic Effects,” Presented at Fifth International Symposium on Cavitation, Osaka, Japan. Ukeiley, L., Cordier, L., Manceau, R., Delvil le, J., Glauser, M. and Bonnet, J.P., 2001, “Examination of Large-scale Structures in a Turbulent Plane Mixing Layer: Part 2 Dynamical Systems Model,” J. Fluid Mech., vol. 441, pp. 67-108. Ukeiley, L., Kannepalli, C. and Arunaj atesan, S., 2002, “Development of Low dimensional Models for Control of Compressible Flows,” Proc. 41st IEEE Conference on Decision and Control, pp. 3282-3287, Las Vegas, NV. Utturkar, Y., Zhang, B., and Shyy, W., 2005, “Reduced-order Description of Fluid Flow with Moving Boundaries by Proper Orthog onal Decomposition,” Int. J. Heat and Fluid Flow, vol. 26, pp. 276-288. Venkateswaran, S., Lindau, J.W., Kunz, R. F. and Merkle, C., 2002, “Computation of Multiphase Mixture Flows with Compressibl ity Effects,” J. Comp. Phys., vol. 180, pp. 54-77. Ventikos, Y. and Tzabiras, G., 2000, “A Numerical methods for Simulation of Steady and Unsteady Cavitating Flows,” Co mput. Fluids, vol. 29, pp. 63-88. Versteeg, H.K. and Malalasekera, W., 1995, An Introduction to Co mputational Fluid Dynamics: The Finite Volume Method, Pearson Education Limited, England. Wang, G., 1999, “A Study on Safety assessm ent of a Hollow Jet Valve accompanied by Cavitation,” Ph.D. Dissertation, Tohuku University, Sendai, Japan. Wang, G., Senocak, I., Shyy, W., Ikohagi, T. and Cao, S., 2001, “Dynamics of Attached Turbulent Cavitating Flows,” Prog. Aero. Sci., vol. 37, pp. 551-581. Weisend, J.G., 1998, Handbook of Cryogenic Engineering, Taylor and Francis, Philadelphia, PA. Wu, J., Senocak, I., Wang, G., Wu, Y. and Shyy, W., 2003a, “Three-Dimensional Simulation of Turbulent Cavitating Flow s in a Hollow-Jet Valve,” Journal of Comp. Modeling in Eng. & Sci., vol. 4, no. 6, pp. 679-690.

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126 BIOGRAPHICAL SKETCH Yogen Utturkar was born in Mumbai (formerly Bombay), India, on June 12, 1977. He obtained his B.E. (Bachelor of Engineer ing) degree in mechan ical engineering from the University of Mumbai in May 1999. He worked as a software engineer from 19992000 before proceeding to pursue graduate studies in the United States. Yogen obtained his master’s degree in mechanical engineerin g (minor in engineering sciences) at the University of Florida in April 2002. He investig ated synthetic jet flow fields for his M.S. thesis under the tutelage of Dr. Rajat Mittal. Yogen worked towards his doctoral dissertation under the guidance of Dr. Wei Shyy (chair) and Dr. Nagaraj Arakere (coc hair). His research interests comprised numerical modeling of thermodynamic effect s in cryogenic cavitation, reduced-order description of simulated data by Pr oper Orthogonal Decomposition (POD), and development of efficient algorithms for problems involving phase change. Yogen met his wife, Neeti Pathare, during hi s graduate years. They married in May 2002 and completed their Ph.D. degree from the same institution (University of Florida).


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COMPUTATIONAL MODELING OF THERMODYNAMIC EFFECTS IN
CRYOGENIC CAVITATION















By

YOGEN UTTURKAR


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2005

































Copyright 2005

by

Yogen Utturkar

































To my wife and parents















ACKNOWLEDGMENTS

I am grateful to several individuals for their support in my dissertation work. The

greater part of this work was made possible by the instruction of my teachers, and the

love and support of my family and friends. It is with my heartfelt gratitude that I

acknowledge each of them.

Firstly, I would like to express sincere thanks and appreciation to my advisor, Dr.

Wei Shyy, for his excellent guidance, support, trust, and patience throughout my doctoral

studies. I am very grateful for his remarkable wisdom, thought-provoking ideas, and

critical questions during the course of my research work. I thank him for encouraging,

motivating, and always prodding me to perform beyond my own limits. Secondly, I

would like to express sincere gratitude towards my co-advisor, Dr. Nagaraj Arakere, for

his firm support and caring attitude during some difficult times in my graduate studies. I

also would like to express my appreciation to the members of my dissertation committee

Dr. Louis Cattafesta, Dr. James Klausner, and Dr. Don Slinn, for their valuable

comments and expertise to better my work. I deeply thank Dr. Siddharth Thakur (ST) for

providing me substantial assistance with the STREAM code and for his cordial

suggestions on research work and career planning.

My thanks go to all the members of our lab, with whom I have had the privilege to

work. Due to the presence of all these wonderful people, work is more enjoyable. In

particular, it was a delightful experience to collaborate with Jiongyang Wu, Tushar Goel,

and Baoning Zhang on various research topics.









I would like to express my deepest gratitude towards my family members. My

parents have always provided me unconditional love. They have always given top

priority to my education, which made it possible for me to pursue graduate studies in the

United States. I would like to thank my grandparents for their selfless affection and

loving attitude during my early years. My wife's parents and her sister's family have

been extremely supportive throughout my graduate education. I greatly appreciate their

trust in my abilities.

Last but never least, I am thankful beyond words to my wife, Neeti Pathare.

Together, we have walked through this memorable, cherishable, and joyful journey of

graduate education. Her honest and unfaltering love has been my most precious

possession all these times. I thank her for standing besides me every time and every

where. To Neeti and my parents, I dedicate this thesis!
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ....................................................... ............ ....... ....... ix

LIST OF FIGURES ............................... .... ...... ... ................. .x

LIST OF SY M B OL S .... .................................................. .. ....... ............. xiv

A B S T R A C T ...............x.................v.................................................. xv iii

CHAPTER

1 INTRODUCTION AND RESEARCH SCOPE ........................................................1

1.1 T ypes of C aviation ................ ...2..... ... ................ ........................ ...2
1.2 Cavitation in Cryogenic Fluids Thermal Effect................... ...............4
1.3 Contributions of the Current Study.................... ...................8

2 LITER A TU RE R EV IEW ............................................................. ....................... 9

2.1 General Review of Recent Studies ........................................ ...... ............... 9
2.2 M odeling Therm al Effects of Cavitation ......................................... .................19
2.2.1 Scaling Law s ................................. ... .. ..... ........... ... 19
2.2.2 E xperim ental Studies.............................................. ............... ... 24
2.2.3 Numerical Modeling of Thermal Effects ................................................27

3 STEADY STATE COM PUTATION S............................ ....................................... 33

3.1 G governing E quations .................................................. .............................. 33
3.1.1. C aviation M modeling ..................................................... ...................35
3.1.1.1 M erkle et al. M odel ................................. ...... .............. 35
3.1.1.2 Sharp Interfacial Dynamics Model (IDM) ..................................... 35
3.1.1.3 Mushy Interfacial Dynamics Model (IDM) ...................................37
3.1.2 Turbulence M odeling ............................................................ ...............40
3.1.3 Speed of Sound (SoS) M odeling .... .......... ...................................... 41
3.1.4 T herm al M odeling ......................................................................... ... ... 42
3.1.4.1 Fluid property update ......................... ............ ... .................42
3.1.4.2 Evaporative cooling effects............... .............................................43









3.1.5 B oundary C onditions...................................................................... .. .... 44
3.2 R results and D discussion ................................................. .............................. 45
3.2.1 Cavitation in N on-cryogenic Fluids ................................. ............... 45
3.2.2 Cavitation in Cryogenic Fluids........................................ ............... 50
3.2.2.1 Sensitivity analyses ........................ ....... ..............52
3.2.2.2 Assessment of cryogenic cavitation models over a wide range of
c o n d itio n s ...................................... ...................................... 5 8

4 TIME-DEPENDENT COMPUTATIONS FOR FLOWS INVOLVING PHASE
CH AN GE ............................................................... ..... ..... ......... 69

4 .1 G alliu m F u sion ................................................................... 7 1
4 .1.1 G governing E quations ...................................................................... .. .... 72
4.1.2 N um erical A lgorithm ...................................................................... .. .... 73
4 .1.3 R esu lts ...................... ....... ......................... ................. 7 8
4.1.3.1 A accuracy and grid dependence ................................ ............... 79
4 .1.3 .2 S tab ility ............... .... ..... ... .. .............................. 82
4.1.3.3 Data analysis by reduced-order description ...................................83
4.2 Turbulent Cavitating Flow under Cryogenic Conditions ...................................88
4.2.1 G governing Equations ........................................................ ............... 88
4.2.1.1 Speed of sound modeling .............. ............................................ 89
4.2.1.2 Turbulence m odeling.................................... ........................ 89
4.2.1.3 Interfacial velocity m odel...................................... ............... 89
4.2.1.4 B oundary conditions ............................................. ............... 91
4.2.2 N um erical A lgorithm ........................................... ........................... 91
4 .2 .3 R e su lts ................................................................9 4

5 SUMMARY AND FUTURE WORK ............................................ ............... 101

5 .1 S u m m a ry ............................................................................................1 0 1
5.2 Future W ork ................................. .............................. ........ 103

APPENDIX

A BACKGROUND OF GLOBAL SENSITIVITY ANALYSIS..............................105

B REVIEW AND IMPLEMENTATION OF POD ................................................107

B .1 R ev iew ....................................................... ................. 107
B .2 M them atical B background ...................................................... ..... .......... 110
B .3 N um erical Im plem entation ...................................... ......................... ........... 112
B.3.1 Singular Value D ecom position (SVD ) ...................................................112
B .3.2 Post-processing the SVD Output........................................................... 113
B .4 F low chart .................................................................. .... ......................... 114
B.5 Computing Efforts...... .................. ........ ........ 114









L IST O F R E FE R E N C E S ......................................................................... ................... 116

BIOGRAPHICAL SKETCH ............................................................. ..................126
















LIST OF TABLES

Table pge

1. Properties of some cryogens in comparison to water at N.B.P, 1.01 bars...................6

2. Source term s in cavitation m models ...................................................... ......... ......... 13

3. V ariants of the k m odel..................... ........................................... ............... 16

4. Summary of studies on thermal effects in cavitation................... .................32

5. Flow cases chosen for the hydrofoil geometry. .................................. .................51

6. Flow cases chosen for the ogive geometry. ...................................... ............... 51

7. Location of the primary vortex for the St = 0.042, Ra = 2.2 x 105 and Pr = 0.0208
c a se ...................................... ..................................................... 8 0















LIST OF FIGURES


Figure p

1. Different types of cavitation (a) Traveling cavitation (b) Cloud cavitation (c) Sheet
cavitation (d) Supercavitation (e) Vortex cavitation.......................................3

2. Saturation curves for water, Nitrogen, and Hydrogen............................... ...............5

3. Phasic densities along liquid-vapor saturation line for water and liquid Nitrogen.........7

4. General classification of numerical methods in cavitation....................................11

5. Variation of Speed of Sound with phase fraction ......... ............................ 14

6. Two cavitation cases for B-factor analysis ....................................... ............... 21

7. Schematic of bubble model for extracting speed of sound........................ ........27

8. Schematic of cavity models (a) Distinct interface with vaporous cavity (Sharp
IDM) (b) Smudged interface with mushy cavity (Mushy IDM).............................36

9. Behavior of p,/ p and p, /p vs. a, for the two models; pl/v = 100 and
/ = 0 .0 9 .............................................................................4 0

10. Pressure-density and pressure-enthalpy diagrams for liquid nitrogen in the liquid-
vapor saturation regime (Lemmon et al. 2002). Lines denote isotherms in
K elv in ............................................................................... 4 3

11. Illustration of the computational domains for hemispherical projectile and
NACA66MOD hydrofoil (non-cryogenic cases) ..........................................46

12. Pressure coefficients over the hemispherical body (o = 0.4); D is the diameter of
the hemispherical projectile. (a) Impact of grid refinement for Mushy IDM (b)
Comparison between pressure coefficients of different models on the coarse grid.47

13. Cavity shapes and flow structure for different cavitation models on hemispherical
projectile (o = 0.4). (a) Merkle et al. Model (b) Sharp IDM (c) Mushy IDM........48

14. Pressure coefficient over the NACA66MOD hydrofoil at two different cavitation
numbers; D is hydrofoil chord length. (a) a = 0.91(b) a = 0.84 ..........................49









15. Illustration of the computational domain accounting the tunnel for the hydrofoil
(Hord 1973a) and 0.357-inch ogive geometry (Hord 1973b) (cryogenic cases)......50

16. Non-cavitating pressure distribution (a) case '290C', D represents hydrofoil
thickness and x represents distance from the circular bend (b) case '312D', D
represents ogive diameter and x represents distance from the leading edge...........52

17. Sensitivity of Merkle et al. Model prediction (surface pressure and temperature) to
input parameters namely Cdet and Cprod for the hydrofoil geometry....................53

18. Main contribution of each design variable to the sensitivity of Merkle et al. (1998)
model prediction; case '290C' (a) Surface pressure (b) Surface temperature .........55

19. Pressure and temperature prediction for Merkle et al. Model for the case with best
match with experimental pressure; Ce,, = 0.85; t = 0.85; = 1.1; = 0.9 ...........56

20. Pressure and temperature prediction for Merkle et al. Model for the case with best
match with experimental temperature; Ct = 1.15; t ; = 0.85; p =1.1; L = 1.1 ........56

21. Sensitivity of Mushy IDM prediction for case '290C' (surface pressure and
temperature) to the exponential transitioning parameter P....................................58

22. Surface pressure and temperature for 2-D hydrofoil for all cases involving liquid
Nitrogen. The results referenced as 'Mushy IDM' and 'Merkle et al. Model' are
contributions of the present study. ........................................ ....................... 59

23. Cavitation number (o = p p py(T)/(0.5pU )) based on the local vapor pressure
Merkle et al. Model. Note the values of ao = p p p(TJ)/(0.5pU2 ) for the
cases '290C' and '296B' are 1.7 and 1.61, respectively. .............. ................... 61

24. Cavity shape indicated by liquid phase fraction for case '290C'. Arrowed lines
denote streamlines (a) Merkle et al. Model isothermal assumption (b) Merkle
et al. Model with thermal effects (c) Mushy IDM with thermal effects ............62

25 Evaporation (ti ) and condensation (i+ ) source term contours between the two
cavitation models case '290C'. Refer to equations (3.8) and (3.18) for the
form ulations. .........................................................................63

26. Surface pressure and temperature for 2-D hydrofoil for cases involving liquid
Hydrogen. The results referenced as 'Mushy IDM' and 'Merkle et al. Model' are
contributions of the present study. ........................................ ....................... 65

27. Surface pressure and temperature for axisymmetric ogive for all the cases
(Nitrogen and Hydrogen). The results referenced as 'Mushy IDM' and 'Merkle
et al. M odel' are contributions of the present study.................................................67









28. Schematic of the 2D Gallium square geometry with the Boundary Conditions
(S h y y et al. 19 9 8 ) ................................................................... 7 1

29. 2D interface location at various instants for St = 0.042, Ra = 2.2 x 105 and Pr =
0.0208. White circles represent interface locations obtained by Shyy et al.
(1995) on a 41x41 grid at time instants at t = 56.7s, 141.8s, & 227s respectively. .79

30. Grid sensitivity for the St = 0.042, Ra = 2.2 x 105 and Pr = 0.0208, 2D case (a)
Centerline vertical velocity profiles at t = 227s (b) Flow structure in the upper-
left domain at t = 57s; 41x41 grid (c) Flow structure in the upper-left domain at t
= 57s; 81x81 grid. ......................................................................80

31. 3D interface location at t = 57s & 227s for St = 0.042, Ra = 2.2 x 105 and Pr =
0.0208 case on a 41x41x41 grid. Top and bottom: adiabatic; North and West: T
= 0; South and East: T = 1 (heated walls) ..................................... ............... 82

32. Interface location and flow pattern for the 3D case, St = 0.042, Ra = 2.2 x 105 and
Pr = 0.0208 case at various z locations, at t = 227s...................... ...................82

33. POD modes showing velocity streamlines (y /(r); i = 1, 2,3,4 ) for St = 0.042, Ra =
2.2 x 103 and Pr = 0.0208 case. q(r,t) = (r,t). ......................................... 84

34 Scalar coefficients (q (t);i = 1, 2,..., 8) for St = 0.042, Ra = 2.2 x 103 and Pr =
0 .0208 case. q (r,t) = V (r,t) ..................................... ...................... ............... 85

35. Time instants when coefficients of respective POD modes show the first peak; St
= 0.042, Ra = 2.2 x 103 & Pr = 0.0208 case......................................... ..............86

36. Horizontal-centerline vertical velocities for St = 0.042, Ra = 2.2 x 103 and Pr =
0 .02 0 8 case at t = 3 s ................................................................. 86

37. Comparison between CFD solution and re-constructed solution for St = 0.042, Ra
= 2.2 x 103 and Pr = 0.0208 case at t = 20s. Contours represent velocity
magnitude and lines represent streamlines ........... .............................................87

38. Comparison between CFD solution and re-constructed solution for St = 0.042, Ra
= 2.2 x 103 and Pr = 0.0208 case at t = 3s. Contours represent velocity
magnitude and lines represent streamlines ........... .............................................88

39. Pressure history at a point near inlet, A = 0.11 D, where D is the chord length of
the hydrofoil (a) '283B (b) '290C' .......................................................................... 95

40. Cavity shape and flow structure for case '283B'. Lines denote streamlines .............95

41. Pressure history at a point near the inlet for the case '290C'. St denotes the non-
dimensional perturbation frequency (fD / U, = ft, )................................ 96









42. Pressure history at a point near the inlet for the cases '296B' and '290C'. The
non-dimensional perturbation frequency (fD/U, = ft.) is 1.0..............................97

43 POD modes for the velocity field; V, (r); i = 1, 2,3; q(r, t) = V(r, t) ...........................98

44 POD modes for the velocity field; f, (r); i = 1, 2,3; q(r, t) = V(r, t) ...........................98
















LIST OF SYMBOLS


c: cavitation number

3: boundary layer/cavity thickness

L: length (of cavity or cavitating object)

q: heat flux; generalized flow variable in POD representation

V: total volume

t: time

x, y, z: coordinate axes

r: position vector

S: streamwise direction in the curvilinear co-ordinate system

i, k, n: indices

u, v, w: velocity components

p: pressure

T: temperature

p: density

a: volume fraction

f mass fraction

h: sensible enthalpy

s: entropy









c: speed of sound

7: ratio of the two specific heats for gases

r: stress tensor

P: production of turbulent energy

C1, CE2, ok, oE, C, C: constants

Q: total kinetic energy (inclusive of turbulent fluctuations)

k: turbulent kinetic energy

E : turbulent dissipation

F: filter function for filter-based modeling

/ : dynamic viscosity

K: thermal conductivity

v: volume flow rate

L: latent heat

C : specific heat

C,: pressure coefficient

a: thermal diffusivity

ih : volume conversion rate

B: B-factor to gauge thermal effect

E: dimensional parameter to assess thermal effect

,7: control parameter in the Mushy IDM; coefficient of thermal expansion

g: gravitational acceleration

U: velocity scale

D: length scale (such as hydrofoil chord length or ogive diameter)









R: bubble radius

CFL: Courant, Freidricks, and Levy Number

St: Stefan Number

Ra: Rayleigh Number

Pr: Prandtl Number

Re: Reynolds Number

A: difference; filter size in filter-based turbulence modeling

V: gradient or divergence operator

y/: POD mode

0: time-dependent coefficient in the POD series

r : energy content in the respective POD mode

Subscripts:

oo : reference value (typically inlet conditions at the tunnel)

0: initial conditions

s: solid

1: liquid

v: vapor

m: mixture

f friction

Q: discharge

c: cavity

L: laminar

t: turbulent









I: interfacial

n: normal to local gradient of phase fraction

sat: saturation conditions

dest: destruction of the phase

prod: production of the phase

+: condensation

-: evaporation

A, H, S, M, G, B: terms in a discretized equation

nb: neighboring nodes

P: at the cell of interest

Superscripts/Overhead symbols:

+: condensation

-: evaporation

': fluctuating component

*: normalized value; updated value in the context of PISO algorithm

-: vector

-: average

: Favre-averaged

n: time step level

k: iteration level















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

COMPUTATIONAL MODELING OF THERMODYNAMIC EFFECTS IN
CRYOGENIC CAVITATION

By

Yogen Utturkar

August 2005

Chair: Wei Shyy
Cochair: Nagaraj Arakere
Major Department: Mechanical and Aerospace Engineering

Thermal effects substantially impact the cavitation dynamics of cryogenic fluids.

The present study strives towards developing an effective computational strategy to

simulate cryogenic cavitation aimed at liquid rocket propulsion applications. We employ

previously developed cavitation and compressibility models, and incorporate the thermal

effects via solving the enthalpy equation and dynamically updating the fluid physical

properties. The physical implications of an existing cavitation model are reexamined

from the standpoint of cryogenic fluids, to incorporate a mushy formulation, to better

reflect the observed "frosty" appearance within the cavity. Performance of the revised

cavitation model is assessed against the existing cavitation models and experimental data,

under non-cryogenic and cryogenic conditions.

Steady state computations are performed over a 2D hydrofoil and an axisymmetric

ogive by employing real fluid properties of liquid nitrogen and hydrogen. The

thermodynamic effect is demonstrated under consistent conditions via the reduction in


xviii









the cavity length as the reference temperature tends towards the critical point. Justifiable

agreement between the computed surface pressure and temperature, and experimental

data is obtained. Specifically, the predictions of both the models are better; for the

pressure field than the temperature field, and for liquid nitrogen than liquid hydrogen.

Global sensitivity analysis is performed to examine the sensitivity of the computations to

changes in model parameters and uncertainties in material properties.

The pressure-based operator splitting method, PISO, is adapted towards typical

challenges in multiphase computations such as multiple, coupled, and non-linear

equations, and sudden changes in flow variables across phase boundaries. Performance of

the multiphase variant of PISO is examined firstly for the problem of gallium fusion. A

good balance between accuracy and stability is observed. Time-dependent computations

for various cases of cryogenic cavitation are further performed with the algorithm. The

results show reasonable agreement with the experimental data. Impact of the cryogenic

environment and inflow perturbations on the flow structure and instabilities is explained

via the simulated flow fields and the reduced order strategy of Proper Orthogonal

Decomposition (POD).














CHAPTER 1
INTRODUCTION AND RESEARCH SCOPE

The phenomenon by which a liquid forms gas-filled or vapor-filled cavities under

the effect of tensile stress produced by a pressure drop below its vapor pressure is termed

cavitation (Batchelor 1967). Cavitation is rife in fluid machinery such as inducers,

pumps, turbines, nozzles, marine propellers, hydrofoils, journal bearings, squeeze film

dampers etc. due to wide ranging pressure variations along the flow. This phenomenon is

largely undesirable due to its negative effects namely noise, vibration, material erosion

etc. Detailed description of these effects can be readily obtained from literature. It is

however noteworthy that the cavitation phenomenon is also associated with useful

applications. Besides drag reduction efforts (Lecoffre 1999), biomedical applications in

drug delivery (Ohl et al. 2003) and shock wave lithotripsy (Tanguay and Colonuis 2003),

environmental applications for decomposing organic compounds (Kakegawa and

Kawamura 2003) and water disinfection (Kalamuck et al. 2003), and manufacturing and

material processing applications (Soyama and Macodiyo 2003) are headed towards

receiving an impetus from cavitation. Comprehensive studies on variety of fluids such as

water, cryogens, and lubricants have provided significant insights on the dual impact of

cavitation. Experimental research methods including some mentioned above have relied

on shock waves (Ohl et al. 2003), acoustic waves (Chavanne et al. 2002), and laser pulses

(Sato et al. 1996), in addition to hydrodynamic pressure changes, for triggering

cavitation. Clearly, research potential in terms of understanding the mechanisms and

characteristics of cavitation in different fluids, and their applications and innovation is









tremendous. The applicability and contributions of the present study within the above-

mentioned framework are described later in this chapter.

1.1 Types of Cavitation

Different types of cavitation are observed depending on the flow conditions and

fluid properties. Each of them has distinct characteristics as compared to others. Five

major types of cavitation have been described in literature. They are as follows:

(a) Traveling cavitation

It is characterized by individual transient cavities or bubbles that form in the liquid,

expand or shrink, and move downstream (Knapp et al. 1970). Typically, it is observed on

hydrofoils at small angles of attack. The density of nuclei present in the upstream flow

highly affects the geometries of the bubbles (Lecoffre 1999). Traveling cavitation is

illustrated in Figure 1(a).

(b) Cloud cavitation

It is produced by vortex shedding in the flow field and is associated with strong

vibration, noise, and erosion (Kawanami et al. 1997). A re-entrant jet is usually the

causative mechanism for this type of cavitation (Figure l(b)).

(c) Sheet cavitation

It is also known as fixed, attached cavity, or pocket cavitation (Figure l(c)). Sheet

cavitation is stable in quasi-steady sense (Knapp et al. 1970). Though the liquid-vapor

interface is dependent on the nature of flow, the closure region is usually characterized by

sharp density gradients and bubble clusters (Gopalan and Katz 2000).

(d) Supercavitation

Supercavitation can be considered as an extremity of sheet cavitation wherein a

substantial fraction of the body surface is engulfed by the cavity (Figure l(d)). It is









observed in case of supersonic underwater projectiles, and has interesting implications on

viscous drag reduction (Kirschner 2001).

(e) Vortex cavitation

It is observed in the core of vortices in regions of high shear (Figure 1(e)). It mainly

occurs on the tips of rotating blades and in the separation zone of bluff bodies (Knapp et

al. 1970).


Figure 1. Different types of cavitation* (a) Traveling cavitation (b) Cloud cavitation (c)
Sheet cavitation (d) Supercavitation (e) Vortex cavitation


Reproduced from Franc et al. (1995) with permission from EDP Sciences


(b)









1.2 Cavitation in Cryogenic Fluids Thermal Effect

Cryogens serve as popular fuels for the commercial launch vehicles while

petroleum, hypergolic propellants, and solids are other options. Typically, a combination

of liquid oxygen (LOX) and liquid hydrogen (LH2) is used as rocket propellant mixture.

The boiling points of LOX and LH2 under standard conditions are -183 F and -423 F,

respectively. By cooling and compressing these gases from regular conditions, they are

stored into smaller storage tanks. The combustion of LOX and LH2 is clean since it

produces water vapor as a by-product. Furthermore, the power/gallon ratio of LH2 is high

as compared to other alternatives. Though storage, safety, and extreme low temperature

limits are foremost concerns for any cryogenic application, rewards of mastering the use

of cryogens as rocket propellants are substantial (NASA Online Facts 1991).

A turbopump is employed to supply the low temperature propellants to the

combustion chamber which is under extremely high pressure. An inducer is attached to

the turbopump to increase its efficiency. Design of any space vehicle component is

always guided by minimum size and weight criteria. Consequently, the size constraint on

the turbopump solicits high impellor speeds. Such high speeds likely result in a zone of

negative static pressure (pressure drop below vapor pressure) causing the propellant to

cavitate around the inducer blades (Tokumasu et al. 2002). In view of the dire

consequences, investigation of cavitation characteristics in cryogens, specifically LOX

and LH2, is an imperative task.

Intuitively, physical and thermal properties of a fluid are expected to significantly

affect the nature of cavitation. For example, Helium-4 shows anomalous cavitation

properties especially past the X-point temperature mainly due to its transition to

superfluidity (Daney 1988). Besides, quantum tunneling also attributes to cavity










formation in Helium-4 (Lambare et al. 1998). Cavitation of Helium-4 in the presence of a

glass plate (heterogenous cavitation) has lately produced some unexpected results

(Chavanne et al. 2002), which are in contrast to its regular cavitating pattern observed

under homogenous conditions (without a foreign body). Undoubtedly, a multitude of

characteristics and research avenues are offered by different types of cryogenic fluids.

The focus of the current study is, however, restricted to cryogenic fluids such as LOX,

LH2, and liquid Nitrogen due to their aforesaid strong relevance in space applications. It

is worthwhile at the outset to contrast their behavior/physical properties to water.




Water Nitrogen

"50 i




Temperture (K) Tempewatre (K)





lo Hydrogen
I





7 225 5 "250' 27S" 0 5 3
Temperature (K)



Figure 2. Saturation curves for water, Nitrogen, and Hydrogen


Obtained from REFPROP v 7.0 by Lemmon et al. (2002)










Table 1. Properties of some cryogens in comparison to water at N.B.P, 1.01 bars
Substance Specific heat Liquid Liquid/Vapor Thermal Vaporization
(J/Kg.K) density density conductivity heat
(kg/m3) (W/mK) (KJ/Kg)
Water 4200 958 1603 681 2257
H2 9816 71 53 100 446
N2 2046 809 175 135 199
02 1699 1141 255 152 213
Source: Weisend et al. (1998)

Refer to Figure 2 and Table 1. The operating point of cryogenic liquids is generally

quite close to the critical point unlike water. Furthermore, as indicated by Figure 2, the

saturation pressure curves for cryogens demonstrate a much steeper slope v/s

temperature, as compared to water. Consequently, the vapor pressure of liquids such as

LH2 and liquid Nitrogen is expected to show great sensitivity to small temperature drops.

Since cavitation is predominantly governed by the vapor pressure, the significance of this

thermodynamic sensitivity on the flow problem is clear upfront.

Further insight can be obtained from Table 1. Liquid-to-vapor density ratio in the

case of cryogenic fluids is substantially lower than water. Thus, these fluids require a

greater amount of liquid, and in turn latent heat, than water to cavitate (form vapor) under

similar flow conditions. Furthermore, the thermal conductivity for the low temperature

fluids is consistently lower than water. These facts indicate that the sensible-latent heat

conversion in cryogenic fluids is expected to develop a noticeable temperature gradient

surrounding the cavitation region. The impact of this local temperature drop is magnified

by the steep saturation curves observed in Figure 2. Subsequently, the local vapor

pressure experiences a substantial drop in comparison to the freestream vapor pressure

leading to suppression in the cavitation intensity. Experimental (Hord 1973a, 1973b) and











numerical results (Deshpande et al. 1997) on sheet cavitation have shown a 20-40%


reduction in cavity length due to the thermodynamic effects in cryogenic fluids.


Water


Liq. N2


....... ............ ... ... .. ..
TemperaluM wK)


g
g

S~ i
E
J1 s
0


A
i asa ,, 7
J s :-


Figure 3. Phasic densities along liquid-vapor saturation line for water and liquid
Nitrogen*

Additionally, the physical properties of cryogenic fluids, other than vapor pressure, are


also thermo-sensible, as illustrated in Figure 3. Thus, from a standpoint of numerical


computations, simulating cryogenic cavitation implies a tight coupling between the non-


linear energy equation, momentum equations, and the cavitation model, via the iterative


update of fluid properties (such as vapor pressure, densities, specific heat, thermal


conductivity, viscosity etc.) with changes in the local temperature. Encountering these


difficulties is expected to yield a numerical methodology specifically well-suited for


cryogenic cavitation, and forms the key emphasis of the present study.


Obtained from REFPROP v 7.0 by Lemmon et al. (2002)


1


i


Tomi ~ u i~ K)
Trmp~rfrtur (K)


Tmparalur (K)


Twmprtaluf (K)









1.3 Contributions of the Current Study

The major purpose of the present study is to develop a robust and comprehensive

computational tool to simulate cavitating flow under cryogenic conditions. The specific

contributions of the endeavor are summarized as follows:

(a) A review of the experimental and computational studies on cryogenic cavitation

(b) Coupling of energy equation to the existing cavitation framework in conjunction with
iterative update of the real fluid properties with respect to the local temperature

(c) Adaptation of an existing cavitation model (Senocak and Shyy 2004a, 2004b) to
accommodate the physics of the mushy nature of cavitation observed in cryogenic
fluids (Hord 1973, Sarosdy and Acosta 1961)

(d) Demonstration of the impact of thermodynamic effects on cavitation over wide-
ranging temperatures, for two different cryogenic fluids. Assessment of the
computational framework alongside available experimental and numerical data.

(e) Global sensitivity analysis of the computational predictions (pressure and
temperature) with respect to the cavitation model parameters and the temperature-
dependent material properties, via employing the response surface approach.

(f) Adaptation of the pressure-based operator splitting method, PISO (Issa 1985), to
multiphase environments typically characterized by strong interactions between the
governing equations and steep variations of flow variables across the phase boundary

(g) Assessment of the stability and accuracy of the non-iterative algorithm (PISO
variant) on the test problem of Gallium fusion.

(h) Time-dependent computations of cryogenic cavitation (with the PISO variant) by
applying perturbation to the inlet temperature

(i) Employment of Proper Orthogonal Decomposition (POD) to offer a concise
representation to the simulated CFD data.














CHAPTER 2
LITERATURE REVIEW

Cavitation has been the focal point of numerous experimental and numerical

studies in the area of fluid dynamics. A review of these studies is presented in this

chapter. Since cryogenic cavitation remains the primary interest of this study, the general

review on cavitation studies is purely restricted according to relevance. Specifically, the

numerical approaches in terms of cavitation, compressibility, and turbulence modeling

are briefly reviewed in the earlier section with reference to pertinent experiments. The

later section mainly delves into the issues of thermal effects of cavitation. Current status

of numerical strategies in modeling cryogenic cavitation, and their merits and limitations

are reported to underscore the gap bridged by the current research study.

2.1 General Review of Recent Studies

Computational modeling of cavitation has complemented experimental research on

this topic for a long time. Some earlier studies (Reboud et al. 1990, Deshpande et al.

1994) relied on potential flow assumption (Euler equations) to simulate flow around the

cavitating body. However, simulation strategies by solving the Navier-Stokes equations

have gained momentum only in the last decade. Studies in this regard can be broadly

classified based on their interface capturing method. Chen and Heister (1996) and

Deshpande et al. (1997) adopted the interface tracking Marker and Cell approach in their

respective studies, which were characterized by time-wise grid regeneration and the

constant cavity-pressure assumption. The liquid-vapor interface in these studies was

explicitly updated at each time step by monitoring the surface pressure, followed by its









reattachment with an appropriate wake model. This effort was mainly well-suited for

only sheet cavitation. The second category, which is the homogeneous flow model, has

been a more popular approach, wherein the modeling for both phases is adopted via a

single-fluid approach. The density change over the interface is simply modeled by a

liquid mass fraction (f) or a liquid volume fraction (a,) that assumes values between 0

and 1. The mixture density can be expressed in terms of either fraction as follows:

Pm =ap, +(1-a)p, (2.1)


= P + (2.2)


The precise role of various cavitation models, which are reviewed later, is prediction of

this volume/mass fraction as a function of space and time.

Both, density-based and pressure-based methods have been successfully adopted in

conjunction with the single-fluid method in numerous studies. Due to unfeasibility of

LES or DNS methods for multiphase flow, RANS approach through k-e turbulence model

has been mostly employed in the past studies. The main limitation of density-based

methods (Merkle et al. 1998, Ahuja et al. 2001, Lindau et al. 2002, Iga et al. 2003) is

requirement of pre-conditioning (Kunz et al. 2000) or the artificial density approach for

flows which may be largely incompressible. Pressure-based methods (Ventikos and

Tzabiras 2000, Athavale et al. 2001, Senocak and Shyy 2002, Singhal et al. 2002) on the

other hand are applicable over a wide range of Mach numbers. Modeling the speed of

sound in the mixture region and computational efficiency for unsteady calculations are

the main issues for pressure-based solvers. A broad classification of the numerical

methods is illustrated in Figure 4.












Interface Capturing Strategy




Explicit Tracking Methods Homogenous Flow
(Marker and Cell) Model



Euler Equations Navier Stokes
(Potential Flow
Assumption) Solver





Artificial Compressibility Based or Pressure
Based Methods



Figure 4. General classification of numerical methods in cavitation

Within the broad framework of above methods, several cavitation, compressibility, and

turbulence models have been deployed, which are reviewed next as per the aforesaid

order.

As mentioned earlier, the role of cavitation modeling is basically determination of

the phase fraction. Different ideas have been proposed to generate the variable density

field. Some studies solved the energy equation and obtained the density either by

employing an equation of state (EOS) (Delannoy and Kueny 1990, Edwards et al. 2000)

or from thermodynamic tables (Ventikos and Tzabiras 2000). Saurel et al. 1999

developed a cavitation model for hypervelocity underwater projectiles based on separate

EOS for the liquid, vapor and mixture zones. EOS approach in first place does not

capture the essential cavitation dynamics due to its equilibrium assumption in the phase

change process. Furthermore, prevalence of isothermal condition in case of non-









thermosensible fluids such as water imparts a barotropic form, Pm = f(p), to the EOS.

Thus, cavitation modeling via employing the EOS is devoid of the capability to capture

barotropic vorticity generation in the wake region as demonstrated by Gopalan and Katz

(2000). Transport-equation based cavitation models in contrast overcome the above

limitations and are more popular in research studies. Typically, this approach determines

the liquid volume fraction (a,) or the vapor mass fraction (f,) by solving its transport

equation as shown below.

S+ V.(cai) = i+ + i (2.3)
at

a + V.(p, fii) = ih + (2.4)
at

Formulation of the source terms shown in above equation(s) constitutes the major effort

in model development. Singhal et al. (1997), Merkle et al. (1998), Kunz et al. (2000), and

Singhal et al. (2002) formulated these source terms strongly based on empirical

judgment. However, Senocak and Shyy (2002, 2004a) developed a cavitation model

fundamentally relying on interfacial mass and momentum transfer. Though their model

was not completely empiricism-free, it transformed the empirical coefficients used in the

earlier models into a physically explicable form. The ability of the model to capture the

1
barotropic vorticity, Vp xV in the closure region was also clearly demonstrated by
P

Senocak and Shyy (2002, 2004a). The source terms of each of the above models along

with value of empirical constants are tabulated in Table 2.











Table 2 Source terms in cavitation models
Authors m' Production Term m Destruction term

Singhal et al.(1997) CprodAX(p p,O0)(l- a,) CdspMIN(p p,,0)a,
(0.5p,U~)t. (0.5p,/.)pt
Merkle et al. (1998) Cr = 8x101 Cds =1


Kunz et al. (2000) Cprodc a (1- a,) CppMIN(p p,,0)a,
pit (0.5pU )Pit,
Cprod = 3x 104 Cd, =1

Singhal et al.(2002) U 2 2AX( p,,0) 1/2 U 2MIN(p- p, 0)1/2
pr 1 v dest r I-v
7 3 PI 3
Cprod = 3.675 x 103 Cds / = 1.225 x 103

Senocak and Shyy CprodAX (p p,0)(1- al) CdtpMIN(p- p,,0)a,
(0.5pU )t (0.5P/U )pt
(2002, 2004a) Cro 1 Cd 1
0.5p,U, (P- )U -,)2 0.5pU (P PV)(U,, U,)2




As clearly seen from the above table, the cavitation model of Senocak and Shyy (2002) is

consistent with that of Singhal et al. (1997) and Merkle et al. (1998). However, the model

constants have assumed physicality. The terms U,,, & U, ,, which represent the liquid-

vapor interface velocity and the normal component of interfacial vapor velocity are

calculated by suitable approximations (Senocak and Shyy 2002, 2004a).

Numerous experimental studies (Leighton et al. 1990, Clarke and Leighton 2000)

addressing compressible bubble oscillations are available. Muzio et al. (1998) developed

a numerical compressibility model for acoustic cavitation in a bubble inclusive of

viscosity and surface tension effects. However, these models are difficult to deploy for

practical multiphase computations.











Figure 5 (adapted from Hosangadi et al. 2002) illustrates the modeled behavior of speed

of sound v/s the phase fraction. As observed from the figure, the speed of sound in the

biphasic mixture can be 2-3 orders of magnitude lower than the individual phases.



Pure liquid

1400

1200

S1000

800

I 600 -

400
Pure vapor
200 -

0 0.25 0.5 0.75 1
Gas fraction


Figure 5. Variation of Speed of Sound with phase fraction

Consequently, a bulk incompressible flow may transform into a transonic or supersonic

stage locally in the mixture region. Due to lack of dependable equation of state for

multiphase mixtures, modeling sound propagation, which is an imperative issue in the

numerical computations, is still an open question. A closed form expression for the speed

of sound in the mixture region may be obtained by eigenvalue analysis on the strongly

conservative form of the governing equations (Hosangadi et al. 2002), as shown below.


m= P a + a2] (2.5)
Cm pva pa,


Venkateswaran et al. (2002) proposed use of perturbation theory to obtain an efficient

pre-conditioned system of equations, which were consistent in the incompressible as well

as compressible regime. Improvement in cavitation dynamics by accounting the

compressibility effects was reported. The incorporation of Speed of Sound (SoS) model









into pressure-based cavitation computations was significantly advanced by Senocak and

Shyy (2002, 2003). In pressure-based solvers, the SoS model affects the solution mainly

through the pressure correction equation. The following relationship is adopted between

the density correction and the pressure correction terms, while enforcing the mass-

conservation treatment through the pressure correction equation:

P'm = Cp' (2.6)

The implementation of equation (2.6) imparts a convective-diffusive form to the pressure

equation. Two SoS models were proposed by Senocak and Shyy (2003, 2004a, 2004b).


SoS-1: C = =C(1- ) (2.7)



SoS-2: C = ( ) p) p1 (2.8)
Ap Op P, /-Pr,

While SoS-1 is a suitable approximation to the curve shown in Figure 5, SoS-2

approximates the fundamental definition of speed of sound by adopting a central-

difference spatial derivative along the streamline direction (0 instead of differentiating

along the isentropic curve. Computations by Senocak and Shyy (2003) on convergent-

divergent nozzle demonstrated far better capability of SoS-2 to mimic the transient

behavior observed in experiments. Wu et al. (2003b) extended the model assessment by

pointing out the dramatic time scale differences between the two models. This points the

fact that compressibility modeling is a sensitivity issue and must be handled carefully.

RANS-based approaches in form of two-equation k e models have been actively

pursued to model turbulent cavitating flows. The k and E transport equations along with

the definition of turbulent viscosity are summarized below.









a(pk) (piuk) [ u, tk
+ =P, PE+ [(+ )
at x, xa 0 xa
a(P (P ) + E2 a U au
+ = c,, P -PtCp +-[(p+ t -]
at ax, k k [Ox x]

The production of turbulent kinetic energy (Pt) is defined as:

p = 1R a'
Pr=r

while, the turbulent viscosity is defined as:

t = p "k 2


(2.9)


(2.10)


(2.11)


(2.12)


The model coefficients, namely, C,1,C, ok, and oa have three known non-trivial
variants which have been summarized in Table 3.
Table 3. Variants of the k ; model
Authors Cg, and Relevant Details C2, 0k C,

Launder and 1.44 1.92 1.3 1.0

Spalding (1974)

Shyy et al. (1997) 115 + 0.25 P, 1.9 1.15 0.89


Younis (2003) PQ k )( 1.9 1.15 0.89
u(1.15+0.25 )x(1+0.38 |-I/Q)
(personal
where Q = k + (u2 +v2 + w2)/2
communication)

Johansen et al. 1.44 1.92 1.3 1.0

(2004) p C,, k2 A
F, -C F,C = 0.09,F = Min[1, C -
s k



While the Launder and Spalding (1974) model is calibrated for equilibrium shear flows,

the model by Shyy et al. (1997) accommodates non-equilibrium effects by introducing a









subtle turbulent time scale into C,1. The RANS model of Younis (2003, personal

communication), in comparison, accounts for the time history effects of the flow. Wu et

al. (2003a, 2003b) assessed the above RANS models on turbulent cavitating flow in a

valve. Experimental visuals of the valve flow (Wang 1999) have demonstrated cavitation

instability in form of periodic cavity detachment and shedding. However, Wu et al.

(2003b) reported that the k e model over predicts the turbulent viscosity and damps such

instabilities. Consequently, the RANS computations were unable to capture the shedding

phenomenon, and also showed restrained sensitivity to the above variants of the k E

model. From standpoint of an alternate approach, the impact of filter-based turbulence

modeling on cavitating flow around a hydrofoil was reported (Wu et al. 2003c). The

filter-based model relies on the two-equation formulation and uses identical coefficient

values as proposed by Launder and Spalding (1974), but imposes a filter on the turbulent

viscosity as seen below.

pC, k2
pt = F, C = 0.09 (2.13)


The filter function (F1 is defined in terms of filter size (A) as:

Ac
F = Min[1,CA ]; CA = 1 (2.14)


Note that the forms of viscosity in equations (2.12) and (2.13) are comparable barring the

filter function. The proposed model recovers the Launder and Spalding (1974) model for

coarse filter sizes. Furthermore, at near-wall regions, the imposed filter value F = 1

enables the use of wall functions to model the shear layer. However, in the far field zone


if the filter size is able address the turbulent length scale


, the solution is computed
s









directly (/u, = 0 ). The filter-based model is also characterized by the independence of the

filter size from the grid size. This model enhanced the prediction of flow structure for

single-phase flow across a solid cylinder (Johansen et al. 2004). Wu et al. (2003c)

reported substantial unsteady characteristics for cavitating flow around a hydrofoil

because of this newly developed model. Furthermore, the time-averaged results of Wu et

al. (2003c) (surface pressure, cavity morphology, lift, drag etc.) are consistent to those

obtained by alternate studies (Kunz et al. 2003, Coutier-Delgosha et al. 2003, Qin et al.

2003). Further examination in the context of filter-based modeling and cavitating flows

was performed over the Clark-Y aerofoil and a convergent-divergent nozzle (Wu et al.

2004, 2005). The filter-based model produced pronounced time-dependent behavior in

either case due to significantly low levels of eddy viscosity. While the time-averaged

results showed consistency to experimental data, they were unable to capture the essence

of unsteady phenomena in the flow-field such as wave propagation. In addition to

implementing the two-equation model for turbulent viscosity, Athavale et al. (2000) also

accounted for turbulent pressure fluctuations. Thus, the threshold cavitation pressure (P,)

was modified as:

P' = P, + /2 (2.15)

The turbulent pressure fluctuations were modeled as follows:

p, = 0.39pk (2.16)

Though their computations produced consistent results, the precise effect of incorporating

the turbulent pressure fluctuations was not discerned.









In addition to the above review, Wang et al. (2001), Senocak and Shyy (2002,

2004a, 2004b), Ahuja et al. (2001), Venkateswaran et al. (2002), and Preston et al. (2001)

have also reviewed the recent efforts made in computational and modeling aspects.

2.2 Modeling Thermal Effects of Cavitation

Majority of studies on cavitation have made the assumption of isothermal

conditions since they focused on water. However, as explained earlier, these assumptions

are not suitable under cryogenic conditions because of their low liquid-vapor density

ratios, low thermal conductivities, and steep slope of pressure-temperature saturation

curves. Efforts on experimental and numerical investigation of cryogenic cavitation are

dated as back as 1969. Though the number of experimental studies on this front is

restricted due to the low temperature conditions, sufficient benchmark data for the

purpose of numerical validation is available. However, there is a dearth of robust

numerical techniques to tackle this problem numerically. The following sub-sections

provide fundamental insight into the phenomenon in addition to a literature review.

2.2.1 Scaling Laws

Similarity of cavitation dynamics is dictated primarily by the cavitation number (o)

defined as (Brennen 1994, 1995):

a P -ZPT) (2.17)
0.5 p,U

Under cryogenic conditions, however, cavitation occurs at the local vapor pressure

dominated by the temperature depression. Thus, the cavitation number for cryogenic

fluids is modified as:

P-P,(T) (2.18)
S 5p(2.18)
0.5pU









where, Tc is the local temperature in the cavity. The two cavitation numbers can be

related by a first-order approximation as follows:

1 pU (o -)= dPv(T- T.) (2.19)
2 dT

Clearly, the local temperature depression (T -T.) causes an increase in the effective

cavitation number, consequently reducing the cavitation intensity. Furthermore, equation

(2.19) underscores the effect of the steep pressure-temperature curves shown in Chapter

1.

Quantification of the temperature drop in cryogenic cavitation has been

traditionally assessed in terms of a non-dimensional temperature drop termed as B-factor

(Ruggeri and Moore 1969). A simple heat balance between the two phases can estimate

the scale of temperature difference caused by thermal effect.

pt)L = pltlCPIAT (2.20)

Here, vt and vu are volume flow rates for the vapor and liquid phase respectively. The B-

factor can then be estimated as:


B=- ;AAT pL (2.21)
v, AT* P1CP

Consider the following two flow scenarios for estimating B, as shown in Figure 6

(adapted from Franc et al. 2003).











U0

(a) Two-phase cavity


U0


(b) 100% vapor cavity
Thermal boundary


Figure 6. Two cavitation cases for B-factor analysis

The estimation ofB-factor for the case (a) (two-phase cavity) in above figure is expressed

in following equation.


-a
v0 a, U.9; v, (1- a,)U. 9; B = 1 (2.22)


This points the fact that except for the pure vapor region B has an 0(1) value. For the

case (b) in Figure 6 (adapted from Franc et al. 2003), where the cavity is assumed to be

filled with 100% vapor, Fruman et al. (1991) provided an estimate of B based on thermal

boundary layer effect as:


B = (2.23)
gT raL
SU0

Here, a is the thermal or eddy diffusivity, Le is the cavity length, and '5 and ,5

represent the thickness of the cavity and the thermal boundary layer, respectively. It is

evident from equation (2.23) that the temperature depression is also strongly dependent

on thermal diffusivity and flow properties. The temperature scale (AT*) in case of water

and LH2 has a value of 0.01 and 1.2 K, respectively (Franc et al. 2003). The difference in

these values provides an assessment of the pronounced thermal effects in LH2. Thus, by









knowing the values of AT* and B (from equations (2.22) and (2.23)), the actual

temperature drop can be estimated. The B-factor, however, fails to consider time-

dependent or transient thermal effects due to its dependence on a steady heat balance

equation. Furthermore, the sensitivity of vapor pressure to the temperature drop, which is

largely responsible in altering cavity morphology (Deshpande et al. 1997), is not

accounted by it. As a result, though the B-factor may estimate the temperature drop

reasonably, it is inadequate to evaluate the impact of the thermal drop on the cavity

structure and the overall flow.

Brennen (1994, 1995) developed a more appropriate parameter to assess the

thermodynamic effect by incorporating it into the Rayleigh-Plesset equation

(equation(2.24)) for bubble dynamics.

d2R 3 dR
P1[R + 2 (R) = pv(T)- p. (2.24)
dt 2 dt

With help of equation(2.19), we can re-write above equation as:

d2R 3 dR dp
p [R d + 3 )2]+ d AT=p(To)- p (2.25)
dt 2 dt dT

From standpoint of a transiently evolving bubble, the heat flux q at any time t can be

expressed as:

AT
q=K K (2.26)
VaZ

The denominator in equation(2.26), -vt, represents the thickness of the evolving

thermal boundary layer at time t. The heat balance across the bubble interface is

expressed as follows.

d4
q4rR2 = pL d[ 4 rR3] (2.27)
dt 3









Combining equations (2.26) and (2.27) we obtain:


AT Rv pL (2.28)
'Fa pCPl

Introducing this temperature difference into equation(2.25) we obtain:

d2R 3 dR2 P,(T.)- P
[Rd +2 ( ) 2]+ jXR (T= (2.29)
dt2 2 dt pl

A close observation yields the fact that the impact of thermal effect on bubble dynamics

depends on Y which is defined as:

Y= pL d= (2.30)
p2Cp, a dT

The units of are m/s3/2 and it proposes a criterion to determine if cavitation process is

thermally controlled or not.

Franc et al. (2003) have recently extended the above analysis to pose a criterion for

dynamic similarity between two thermally controlled cavitating flows. They replaced the

time dependency in equation(2.29) by spatial-dependency through a simple

transformation x = Ut Here, x is the distance traversed by a bubble in the flow-field in

time t. If D is chosen as a characteristic length scale of the problem, the Rayleigh-Plesset

equation can be recast in the following form:

3 _V C +U
[RR+3-RR2]+ DR!=- (2.31)
2 U 2

In equation (2.31), all the quantities with a bar are non-dimensional, and Cp is the

pressure coefficient. All the derivatives are with respect to x = x/D The above equation

points out the fact that, besides o., two thermally dominated flows can be dynamically









similar if they have a consistent value of the non-dimensional quantity X D/UU It is

important to note the important role of velocity scale U. in the quantification of thermal

effect at this juncture. Franc et al. (2003) suggested that though thorough scaling laws for

thermosensible cavitation are difficult to develop, a rough assessment may be gained

from above equation. However, it is imperative to highlight that all the above scaling

laws have been developed using either steady-state heat balance or single bubble

dynamics. As a consequence, their applicability to general engineering environments and

complex flow cases is questionable.

The following sections will discuss the experimental and numerical investigations

of cavitation with thermal consideration. Particularly, emphasis is laid on the limitations

of currently known numerical techniques. Experimental studies are cited solely according

to their relevance to the current study.

2.2.2 Experimental Studies

Sarosdy and Acosta (1961) detected significant difference between water cavitation

and Freon cavitation. Their apparatus comprised a hydraulic loop with an investigative

window. While water cavitation was clear and more intense, they reported that cavitation

in Freon, under similar conditions, was frothy with greater entrainment rates and lower

intensity. Though their observations clearly unveiled the dominance of thermal effects in

Freon, they were not corroborated with physical understanding or numerical data. The

thermodynamic effects in cavitation were experimentally quantified as early as 1969.

Ruggeri et al. (1969) investigated methods to predict performance of pumps under

cavitating conditions for different temperatures, fluids, and operating conditions.

Typically, strategies to predict the Net Positive Suction Head (NPSH) were developed.









The slope of pressure-temperature saturation curve was approximated by the Clausius-

Clapeyron equation. Pump performance under various flow conditions such as discharge

coefficient and impellor frequency was assessed for variety of fluids such as water, LH2,

and butane. Hord (1973a, 1973b) published comprehensive experimental data on

cryogenic cavitation in ogives and hydrofoils. These geometries were mounted inside a

tunnel with a glass window to capture visuals of the cavitation zone. Pressure and

temperature were measured at five probe location over the geometries. Several

experiments were performed under varying inlet conditions and their results were

documented along with the instrumentation error. As a result, Hord's data are considered

benchmark results for validating numerical techniques for thermodynamic effects in

cavitation.

From standpoint of latest investigations, Fruman et al. (1991) proposed that thermal

effect of cavitation can be estimated by attributing a rough wall behavior to the cavity

interface. Thus, heat transfer equations for a boundary layer flow over a flat plate were

applied to the problem. The volume flow rate in the cavity was estimated by producing an

air-ventilated cavity of similar shape and size. An intrinsic limitation of the above method

is its applicability to only sheet-type cavitation. Larrarte et al. (1995) used high-speed

photography and video imaging to observe natural as well as ventilated cavities on a

hydrofoil. They also examined the effect of buoyancy on interfacial stability by

conducting experiments at negative and positive AOA. They reported that vapor

production rate for a growing cavity may differ substantially from the vapor production

rate of a steady cavity. Furthermore, there may not be any vapor production during the

detachment stage of the cavity. They also noticed that cavity interface under the effect of









gravity may demonstrate greater stability. Fruman et al. (1999) investigated cavitation in

R-114 on a venturi section employing assumptions similar to their previous work

(Fruman et al. 1991). The temperature on the cavity surface was estimated using the

following flat plate equation.

q 2.1
Te = T+ [1- 2 (1- Pr)] (2.32)
ple 0.5plCUC Re X

Note that Cf is the coefficient of friction and Tpat is assumed to be equal to the local

cavity surface temperature. The heat flux q on cavity surface was estimated as:

q =-pLUoC, (2.33)

The discharge coefficient C. was obtained from an air-ventilated cavity of similar size.

They estimated the temperature drop via the flat plate equation and measured it

experimentally as well. These two corresponding results showed reasonable agreement.

Franc et al. (2001) employed pressure spectra to investigate R-114 cavitation on inducer

blades. The impact of thermodynamic effect was examined at three reference fluid

temperatures. They reported a delay in the onset of blade cavitation at higher reference

fluid temperatures, which was attributed to suppression of cavitation by thermal effects.

Franc et al. (2003) further investigated thermal effects in a cavitating inducer. By

employing pressure spectra they observed shift in nature of cavitation from alternate

blade cavitation to rotating cavitation with decrease in cavitation number. The earlier is

characterized by a frequency 2fb (fb is the rotor frequency with 4 blades) while the later is

characterized by a resonant frequency fb. Furthermore, R-114 was employed as the test

fluid with the view of extending its results to predicting cavitating in LH2. The scaling









analysis provided in section 2.2.1 was developed by Franc et al. (2003) mainly to ensure

dynamic similarity of cavitation in their experiments.

2.2.3 Numerical Modeling of Thermal Effects

Numerical modeling has been implemented in cavitation studies broadly for two

thermodynamic aspects. Firstly, attempts to model the compressible/pressure work in

bubble oscillations have been made. Lertnuwat et al. (2001) modeled bubble oscillations

by applying thermodynamic considerations to the Rayleigh-Plesset equation, and

compared the solutions to full DNS calculations. The bubble model showed good

agreement with the DNS results. The modeled behavior, however, deviated from the

DNS solutions under i\,theli imal and adiabatic assumptions.













bubble

liquid



Figure 7. Schematic of bubble model for extracting speed of sound

Rachid (2003) developed a theoretical model for accounting compressive effects of a

liquid-vapor mixture. The actual behavior of the mixture along with the dissipative

effects associated with phase transformation was found to lie between two limiting

reversible cases. One in which phase change occurs under equilibrium at a constant

pressure, and the other in which the vapor expands and contracts reversibly in the









mixture without undergoing phase change. Rapposelli and Agostino (2003) recently

extracted the speed of sound for various fluids such as water, LOX, LH2 etc. employing a

bubble model and rigorous thermodynamic relationships. The control volume (V) of the

bubble is illustrated in Figure 7 (adapted from Rapposelli and Agostino 2003), and can be

expressed as:

V = (1- El)V + (E)V + (1- E,)V + (,)V, (2.34)

The model assumed that thermodynamic equilibrium between the two phases is only

achieved amidst fractions E, and E, of the total volume. Subsequently, the remaining

fractions of the two phases were assumed to behave isentropically. If mi and m, are

masses associated with the respective phases, the differential volume change dV can be

expressed as:

dV (1 a, d)p, d p,
V p, dp' p, dp

-( ) a, dp, 8 dp )S, (2.35)
P, dp p, dp

+dm(av O)
m, m1

A close observation of above equation yields that the modeling extremities of

E1, = 0 and E1,, =1 also correspond to the thermodynamic extremities mentioned in

above-mentioned analyses by Lertnuwat et al. (2001) and Rachid (2003) (italicized in the

above description). Finally, substitution of various thermodynamic relations into equation

(2.35) yields a thermally consistent speed of sound in the medium. Rapposelli and

Agostino (2003) reported that their developed model was able to capture most features of

bubble dynamics reasonably well.









The second thermodynamic aspect of cavitation, which also forms the focal point

of this study, is the effect of latent heat transfer. The number of numerical studies, at least

in open literature, in this regard is highly restricted. Reboud et al. (1990) proposed a

partial cavitation model for cryogenic cavitation. The model comprised three steps, which

were closely adapted for sheet cavitation, in form of an iterative loop.

(a) Potential flow equations were utilized to compute the liquid flow field. The

pressure distribution on the hydrofoil surface was imposed as a boundary condition

based on the experimental data. Actually, this fact led to the model being called

'partial'. The interface was tracked explicitly based on the local pressure. The wake

was represented by imposing a reattachment law.

(b) The vapor flow inside the cavity was solved by parabolized Navier Stokes

equations. The change in cavity thickness yielded the increase in vapor volume and

thus the heat flux at each section.

(c) The temperature drop over the cavity was evaluated with the following equation:

aT
q = K, O- (2.36)


The value of turbulent diffusivity Kt in the computations was arbitrarily chosen to

yield best agreement to the experimental results, and q was calculated from step (b).

The iterative implementation of steps (a) (c) yielded the appropriate cavity shape in

conjunction with the thermal effect. Similar 3-step approach was adopted by Delannoy

(1993) to numerically reproduce the test data with R-113 on a convergent-divergent

tunnel section. The main drawback of both the above methods is their predictive

capability is severely limited. This is mainly because these studies do not solve the









energy equation and depend greatly on simplistic assumptions for calculating heat

transfer rates.

Deshpande et al. (1997) developed an improved methodology for cryogenic

cavitation. A pre-conditioned density-based formulation was employed along with

adequate modeling assumptions for the vapor flow inside the cavity and the boundary

conditions for temperature. The interface was captured with explicit tracking strategies.

The temperature equation was solved only in the liquid domain by applying Neumann

boundary conditions on the cavity surface. The temperature gradient on the cavity surface

was derived from a local heat balance similar to equation(2.36). The bulk velocity inside

the vapor cavity was assumed equal to the free stream velocity. Tokumasu et al. (2002,

2003) effectively enhanced the model of Deshpande et al. (1997) by improving the

modeling of vapor flow inside the cavity. Despite the improvements in the original

approach (Deshpande et al. 1997), it is important to underscore the limitation that both

the above studies did not solve the energy equation inside the cavity region.

Hosangadi and Ahuja (2003, 2005), and Hosangadi et al. (2003) recently reported

numerical studies on cavitation using LOX, LH2, and liquid nitrogen. Their numerical

approach was primarily density-based. Their pressure and temperature predictions over a

hydrofoil geometry (Hord 1973a) showed inconsistent agreement with the experimental

data, especially (Hord 1973a) at the cavity closure region. Furthermore, Hosangadi and

Ahuja (2005), who employed the Merkle et al. (1998) model in their computations,

suggested significantly lower values of the cavitation model parameters for the cryogenic

cases as compared to their previous calibrations (Ahuja et al. 2001) for non-cryogenic

fluids.






31


The above review (summarized in Table 4) points out the limited effort and the

wide scope for improving numerical modeling of thermal effects in cryogenic cavitation.

A broadly applicable and more robust numerical methodology is expected to be a

significant asset to prediction and critical investigation of cryogenic cavitation.










Table 4. Summary of studies on thermal effects in cavitation
Author and Year Method (Experimental/Numerical) Main Findings
Sarosdy and Acosta Experimental Freon cavitates with a suppressed intensity
1961 Cavitation in hydraulic loop as compared to water
Ruggeri et al. Experimental Provided assessment of pump NPSH under
1969 Cavitation in pumps cryogenic conditions
Hord Experimental Published comprehensive test data on
1973 ogives and hydrofoil
Fruman et al. Experimental Estimated temperature drop using flat plate
1991 Natural and ventilated cavities boundary layer equations
Larrarte et al. Experimental Vapor production for growing cavities in
1995 High speed photography cryogenic fluids may not be estimated by
ventilated cavities
Fruman et al. Experimental Flat plate equations to model sheet
1999 Venturi section and R-114 cavitation can give good results under
certain restrictions
Franc et al. Experimental Reported delay in onset of cavitation due
2001 Cavitation in pump inducers to thermal effect
Franc et al. Experimental Provided scaling analysis to ensure
2003 Cavitation in pump inducers dynamic similarity
Reboud et al. Numerical Semi empirical numerical model
1990 Potential flow equations Did not solve energy equation
Suitable for sheet cavitation
Delannoy Numerical Semi-empirical numerical model
1993 Potential flow equations Did not solve energy equation
Suitable for sheet cavitation
Deshpande et al. Numerical Density based formulation
1997 Explicit interface tracking Simplistic approximation for cavity vapor
flow
Did not solve energy equation in the vapor
phase
Tokumasu et al. Similar to Deshpande et al. (1997) Improved flow model for cavity
2002, 2003 Applicable to only sheet cavitation
Did not solve energy equation in the vapor
phase
Lertnuwat et al. Numerical Incorporated energy balance for bubble
2001 Model for bubble oscillations oscillations
Good agreement with DNS simulation
Rachid Theoretical Dissipative effects in phase
2003 Compression model liquid vapor transformation intermediate between two
mixture extreme reversible thermodynamic
phenomena
Rapposelli & Agostino Numerical Employed thermodynamic relations to
2003 Model for bubble oscillations extract speed of sound for various liquids
Hosangadi & Ahuja Numerical Solved energy equation in the entire
2003, 2005 Density-based approach domain with dynamic update of material
properties
Some inconsistency with experimental
results noted
Significant change in the cavitation model
parameters between non-cryogenic and
cryogenic conditions














CHAPTER 3
STEADY STATE COMPUTATIONS

This chapter firstly delineates the governing equations that are employed in

obtaining steady-state solutions to various cases on cryogenic cavitation. Theoretical

formulation/derivation and computational implementation of various models, namely

cavitation, turbulence, compressibility, and thermal modeling aspects, are further

highlighted. The boundary conditions and steady-state results yielded by the

computational procedure are discussed in detail following the description of the basic

framework.

3.1 Governing Equations

The set of governing equations for cryogenic cavitation under the single-fluid

modeling strategy comprises the conservative form of the Favre-averaged Navier-Stokes

equations, the enthalpy equation, the k two-equation turbulence closure, and a

transport equation for the liquid volume fraction. The mass-continuity, momentum,

enthalpy, and cavitation model equations are given below:

8pm a(pi.,)
a + =0 (3.1)
at ax

O(p.ur) 0P uzu) ap a au, au 2 al/
( + [,) ((pU +)( + 3)] (3.2)
at ax ax 9x 9x 9x( 3 (3


a[p (h+ fL)]+a[pm, (h+ fL)]= [(R + -) )- (3.3)
at 9x ix Pr Pr /x c









a,+ (a + h (3.4)
at dx

We neglect the effects of compressible work and viscous dissipation from the energy

equation because the temperature field in cryogenic cavitation is mainly dictated by the

phenomenon of evaporative cooling. Here, the mixture density, sensible enthalpy, and the

vapor mass fraction are respectively expressed as:

Pm = Pla + (l-a,) (3.5)

h=CpT (3.6)

f= (1- (3.7)
Pm

The general framework of the Navier-Stokes solver (Senocak and Shyy 2002,

Thakur et al. 2002) employs a pressure-based algorithm and the finite-volume approach.

The governing equations are solved on multi-block, structured, curvilinear grids. The

viscous terms are discretized by second-order accurate central differencing while the

convective terms are approximated by the second-order accurate Controlled Variations

Scheme (CVS) (Shyy and Thakur 1994). The use of CVS scheme prevents oscillations

under shock-like expansion caused by the evaporation source term in the cavitation

model, while retaining second order of formal accuracy.

Steady-state computations are performed by discounting the time-derivative terms

in the governing equations and relaxing each equation to ensure a stable convergence to a

steady state. The pressure-velocity coupling is implemented through the SIMPLEC

(Versteeg and Malalasekera 1995) type of algorithm, cast in a combined Cartesian-

contravariant formulation (Thakur et al. 2002) for the dependent and flux variables,

respectively, followed by adequate relaxation for each governing equation, to obtain









steady-state results. The physical properties are updated, as explained earlier, after every

iteration.

3.1.1. Cavitation Modeling

Physically, the cavitation process is governed by thermodynamics and kinetics of

the phase change process. The liquid-vapor conversion associated with the cavitation

process is modeled through &' and i- terms in Eq. (3.4), which respectively represent,

condensation and evaporation. The particular form of these phase transformation rates,

which in case of cryogenic fluids also dictates the heat transfer process, forms the basis of

the cavitation model. Given below are the three modeling approaches probed in the

present study.

3.1.1.1 Merkle et al. Model

The liquid-vapor condensation rates for this particular model are given as (Merkle

et al. 1998):

CdestplMin(O, p p,)a
p,(O.5pUU )t
SCprodMax(0, p p,)(1- a))
(0.5p,U )t.

Here, Cdet =1.0 and Cprod =80.0 are empirical constants tuned by validating the

numerical results with experimental data. The time scale (t.) in the equation is defined as

the ratio of the characteristic length scale to the reference velocity scale (to = D/U ).

3.1.1.2 Sharp Interfacial Dynamics Model (IDM)

The source terms of this model are derived by applying mass and momentum

balance across the cavity interface and appropriately eliminating the unquantifiable terms

(Senocak and Shyy 2002, 2004a, 2004b).











u (a) (b) p


I




Figure 8. Schematic of cavity models (a) Distinct interface with vaporous cavity (Sharp
IDM) (b) Smudged interface with mushy cavity (Mushy IDM)

A schematic of the cavity model is illustrated in Figure 8(a). The model relies on the

assumption of a distinctly vaporous cavity with a thin biphasic zone separating it from the

pure liquid region. The physical form of the source terms is given below:

lh- pMin(0, p p, )a,
P,(U,,n U1,n)2(pI P)()t9
Max(0, p p,)(l- a,)
(U, ,n- U1,n)2 (pI- Pv )t.

Here, the normal component of the velocity is calculated as (Senocak and Shyy 2002,

2004a):

Va'
U =un; n= l (3.10)
Vac,

Due to implicit tracking of the interface, the interfacial velocity, U,,,, in unsteady

computations needs modeling efforts. Previous studies simplistically expressed the

interfacial velocity (U1,,) in terms of the vapor normal velocity (Senocak and Shyy 2002,

2004a) (U,,n). Alternate methods of modeling U1,, are discussed in the Chapter 4 in the

context of time-dependent simulations. In comparison, for the steady computations in this

chapter, we impose U, = 0 .









3.1.1.3 Mushy Interfacial Dynamics Model (IDM)

Experimental visualizations of cryogenic cavitation (Sarosdy and Acosta 1961;

Hord 1973a, 1973b) have clearly indicated a mushy nature of the cavity. This salient

characteristic of cryogenic cavitation solicits an adaptation of the existing cavitation

model to reflect the same. We choose the Sharp IDM (Senocak and Shyy 2004a) in our

analysis because of its stronger physical reasoning. The following discussion serves to

revise the above cavitation model by re-examining its derivation and assumptions, to

appropriately accommodate the features of cryogenic cavitation. Although we refer the

reader to literature (Senocak and Shyy 2004a) for the detailed derivation of the earlier

model, we underscore the differences between the two approaches in the proceeding

description.

Figure 8(b) representing Mushy IDM depicts a cavity where the vapor pressure

is a function of the local temperature, and the nature of cavitation demonstrates a weak

intensity and consequently less probability for existence of pure vapor phase inside the

cavitation zone. In comparison, Figure 8(a) representing Sharp IDM depicts a cavity

typically produced under regular conditions (no temperature effects), characterized by a

thin biphasic region separating the two phases. We initiate our approach similar to

Senocak and Shyy (2004a) by formulating the mass and momentum balance condition at

the cavity interface, which is assumed to separate the liquid and mixture regions. We

neglect the viscous terms and surface tension effects from equation (3.12) assuming high

Re flows and large cavity sizes respectively.

p (U,, -(U,) = P.-(U.," VU,) (3.11)

P P, = P,(Un, V J,) p,(Up,. V,-)2 (3.12)









It is noteworthy that Senocak and Shyy (2004a) formulated the above equations midst the

pure vapor and mixture regions while hypothesizing an interface between the two. The

two models differ in terms of their interpretation of the cavity attributes, namely, the

density and the velocity field within the cavitation zone, while adhering to the

fundamental idea of mass/momentum balance. Since we attribute a "frothy" nature to the

cavity, the term fi.n is largely expected to represent the mixture normal velocity.

Accounting for this fact, we eliminate the U,,, term from equation (3.12) using equation

(3.11) as shown below.


P1 P = (Um, U,,) -Pm (Um,, U,,)2 (3.13)
P1

Further re-arrangement progressively yields the following equations:


P1- = ,(Um,, -U,)2[1 ] p(U,, U,,)2 1 a, (3.14)

a, P(P, P') P(P P')
a = =(a, + a,) (3.15)
P, (P, p)(U,,, U,1)2 p, (P p,)(U U1)2

p,(p, P)a, p+ (p,- p)(1- a1)
a = + (3.16)
p, (P P)(U,,, U~,,)2 p,(P Pv)(Um ,, U1),n)

From standpoint of formulating source terms for the a, transport equation, we firstly

normalize the above equation by the overall convective timescale (t = D/U,).

Secondly, we apply conditional statements on the pressure terms to invoke either

evaporation or condensation depending on the local pressure and vapor pressure (refer to

Senocak and Shyy 2002 & 2004a). Lastly, we assume that the volume rates for the

individual phases are interchangeable barring the sign convention. For instance, the









evaporation term, ih in the a, -equation or a, -equation would bear the same

magnitude, but negative or positive sign, respectively. We thus obtain the following

source terms from the above analysis:

pMin(0, p p-,)a
Pm (UIm, U],n)Z(P, P )t
(3.17)
p Max(O, p p,)(1- a,)
p, (U,,, U_~,)2 (pi- p,)t

The normal component of the mixture velocity is expressed as shown in equation (3.10)

and consistent with the normal component of the vapor velocity employed by the Sharp

IDM.

A comparison with equation (3.9) shows that a factor of p/Pm, weakens the

evaporation term while a factor of p, / p strengthens the condensation term of the Sharp

IDM to yield the Mushy IDM. For instance, at a nominal density ratio of 100 and

a, = 0.5, the value of p / pv and p / p is 1.82 10 2 and 1.82, respectively. In terms of

order of magnitude, the difference between the evaporation terms of the two models is

relatively more pronounced than the condensation terms, especially for high liquid-vapor

density ratios. Given this fact, we attempt to make the Mushy IDM consistent with the

Sharp IDM by providing an exponential transition of the evaporation source term from

one model to the other as a function of the phase fraction (a, ). Specifically, we employ

the evaporation term from equation (3.17) as we get closer to the liquid region, and that

from equation (3.9) as we get farther from the liquid region. In comparison, the

condensation term from equation (3.17), which is mainly expected to be active in the

region outlining the 'vaporous' portion of the cavity, is unaltered. In summary, the

Mushy IDM is eventually formulated as shown below.










p- Min(0, p p,)a,


p Max(O, p p,)(1- a,)
P (Um U- (U,n)2(p p, )t
p( (3.18)
AL _L + (1. OPl)e-(a-,)p
p pL p

P P,
P Pm

Note that / is a free parameter, which regulates the switch between the evaporation

terms and warrants calibration for different fluids. Its typical value could be 0(0.1).


4 5
"o 4









Figure 9. Behavior of p/pP and p,/P vs. at for the two models; pl/P = 100 and

/ = 0.09

Figure 9 contrasts the density-ratio terms ( pl / p and pl /p) of the Sharp IDM and

Mushy IDM for the chosen parameter values.

3.1.2 Turbulence Modeling

For the system closure, the original two-equation turbulence model with wall

functions is presented as follows (Thakur et al. 2002):

G(pOk) *(p uk) \ -, k
+ = Pt-- PmE + [(U + ) ] (3.19)
dt &x x O-k
Fi 5e90eairo ^Ipad/;/+v.a o hetomdl;p/p=0 n
:?=0.0


















O.~k O. u, k)a









O(P(E) dPuE) E E2 at UOE
+ = -P, C2Pm --+ -[(u + ) ] (3.20)
at cx k k cxj oxj

The turbulence production (P,) and the Reynolds stress tensor is defined as:

au,
Pt = 7-'j 0- =-pMu'u'
x (3.21)
2P-k- au au
Pu 'u' m lut ( a' +
3 x ax,

The parameters for this model, namely, C, = 1.44, C,2 =1.92, o- =1.3, o, =1.0 are

adopted from the equilibrium shear flow calibration (Launder and Spalding 1974). The

turbulent viscosity is defined as:

pmC k2
pt ,,C = 0.09 (3.22)

It should be noted that the turbulence closure and the eddy viscosity levels can affect the

outcome of the simulated cavitation dynamics especially in case of unsteady simulations

(as reviewed in Chapter 2). In this aspect, parallel efforts are being made in the context of

filter-based turbulence modeling (Johansen et al. 2004; Wu et al. 2003c, 2004, 2005),

which has shown to significantly increase the time-dependency in cavitating flows. We

do not explore these techniques in the interest of steady-state simulations, which

disregard time-dependent phenomena.

3.1.3 Speed of Sound (SoS) Modeling

Due to lack of dependable equation of state for liquid-vapor multiphase mixture,

numerical modeling of sound propagation is still a topic of research. We refer the reader

to past studies (Senocak and Shyy 2003, 2004a, 2004b, Wu et al. 2003b) for modeling









options, their impact and issues, and just outline the currently employed SoS model

below.

SoS= C, = C(1-a,) (3.23)

The density correction term in the continuity equation is thus coupled to the pressure

correction term as shown below.

P = Cp (3.24)

Senocak and Shyy (2002 & 2004a) suggested an 0(1) value for the constant C to

expedite the convergence of the iterative computational algorithm. However, their

recommendation is valid under normalized values for inlet velocity and liquid density.

Since we employ dimensional form of equations for cryogenic fluids, we suggest an

0(1 / U ) value for C, which is consistent with the above suggestion in terms of the Mach

number regime. The speed of sound affects the numerical calculation via the pressure

correction equation by conditionally endowing it with a convective-diffusive form in the

mixture region. In the pure liquid region, we recover the diffusive form of the pressure

equation.

3.1.4 Thermal Modeling

The thermal effects are mainly regulated by the evaporative cooling process, which

is further manifested by the temperature dependence of physical properties and vapor

pressure.

3.1.4.1 Fluid property update

In the present study, we subject all the physical properties, namely,

p1, Pv, ,v,, Cp, K, and L to temperature dependence.







43








S..........



Dsnallty kl m) Enthlpy |kJkg)



Figure 10. Pressure-density and pressure-enthalpy diagrams for liquid nitrogen in the
liquid-vapor saturation regime (Lemmon et al. 2002). Lines denote isotherms
in Kelvin.

As indicated by Figure 10, the physical properties are much stronger functions of

temperature than pressure, and can fairly assume the respective values on the liquid-

vapor saturation curve at a given temperature. We update these properties from a NIST

database (Lemmon et al. 2002) at the end of a computational iteration. Thus, we generate

a look-up table of physical properties for a particular temperature range as a pre-

processing step. Subsequently, for any temperature-based update, the table is searched by

an efficient bisection algorithm (Press et al. 1992) and the required physical property is

obtained by interpolating between the appropriate tabular entries.

3.1.4.2 Evaporative cooling effects

The energy equation, (Eq.(3.3)), is recast into the following temperature-based

form, by separating the latent heat terms onto the right-hand-side.


a[PmCpT]+ [pmu CpT]= P[CP(P + P '
et 9xg xo, Pr, Pr txe
a a (3.25)
[EP/(f;L)]+- [pu(f~L)]}
-v
t soue/s
energy source/sink term









As seen from equation (3.25), the 'lumped' latent heat terms manifest as a non-linear

source term into the energy equation and physically represent the latent heat transfer rate.

The spatial variation of thermodynamic properties and the evaporative cooling effect are

intrinsically embedded into this transport-based source term. We calculate the source

term by discretizing the associated derivatives in concert with the numerical schemes

applied to the terms on the left-hand-side of the equation.

3.1.5 Boundary Conditions

The boundary conditions are implemented by stipulating the values of the velocity

components (obtained from the experimental data), phase fraction, temperature, and

turbulence quantities at the inlet. Furthermore, at the walls, pressure, phase fraction, and

turbulence quantities are extrapolated, along with applying the no-slip and adiabatic

condition on the velocity and temperature, respectively. Pressure and other variables are

extrapolated at the outlet boundaries, while enforcing global mass conservation by

rectifying of the outlet velocity components. In addition, we also hold the pressure at the

reference pressure point constantly at the reference value (specified by the experiments).

This is simply done by adjusting the linear coefficients of the pressure correction

equation at that point, to yield zero correction, at every iteration. Symbolically, this is

achieved by substituting Ap = 1, Ap = BP = 0 into the following linear equation at the

point of interest:

Ap pp = ZAn + Bp (3.26)


We observe that this adjustment imparts robustness and stability to the computation.









3.2 Results and Discussion

In this section, we firstly observe the nuances of the Mushy IDM on non-cryogenic

cases. The purpose is to distill the impact solely caused due to the source term change

between the Sharp and Mushy IDM, and contrast it against commonly referred

experimental and computational results. Later, we extend our computations to the

variable temperature environment of cryogenic fluids. Specifically, we observe the effect

of the temperature field on the nature of cavitation under similar conditions. Furthermore,

we assess the Mushy IDM with available experimental data (pressure and temperature)

and compare it with the alternative cavitation models. In the process of calibrating the

cavitation models for cryogenic fluids, we perform a global sensitivity analysis to

evaluate the sensitivity of the prediction to changes in material properties and model

parameters. We offer discussion about optimizing the model performance based on our

sensitivity study.

3.2.1 Cavitation in Non-cryogenic Fluids

As mentioned above, we register first the impact of the Mushy IDM on non-

cryogenic cases. In this initial exercise, we consider the influence of the mushy

formulation only to the liquid boundary layer encompassing the cavity. Symbolically, we

implement the following formulation for the non-cryogenic cases.

lh- pMin(0, p p,)al
p (Um,n U,,,)2(P,- Pv)t
p Max(O, p p,)(1- a,)
P.(Un, -U,,)2 (P- P)t (3.27)
(3.27)
ifa,< 0.99 P =- else P -PA
P Pv P P,

if a,<0.99 p 1 elseP P
P+ P+ P,










To prevent sudden discontinuity in the volume transfer rates, we perform a geometric

smoothing operation over the source terms. Two flow configurations are discussed here,

namely, cavitating flow over a hemispherical projectile (time-averaged experimental data


by Rouse and McNown (1948) at Re=1.36x105) and cavitating flow over the

NACA66MOD hydrofoil (time-averaged experimental data by Shen and Dimotakis


(1989) at Re = 2x106).





NO-SLIP

OUTLET

hemispherical projectile


INLET ( NO-SLIP


NO-SLIP


INLET


OUTLET


NO-SLIP



Figure 11. Illustration of the computational domains for hemispherical projectile and
NACA66MOD hydrofoil (non-cryogenic cases)


hydrofoil (NO-SLIP)











The computational domains for the two geometries are depicted in Figure 11. We

consider two grids with 158x66 and 292x121 points for the hemispherical geometry,

which is axisymmetric. In case of the NACA66MOD hydrofoil, we defer to the judgment

of a previous study (Senocak and Shyy 2004a), and employ the finer grid from that study.


(a) -- Mushy -18x66 gd (b) ----Sharp IDM
...... ushy IDM 292x121 grid- --- Merkle et al. Model
1- Mushy IDM 292 x121 grid Mushy DM
0 Exp. data 1 Exp. dushy ID
U Exp. data


05 05


0 0
0 0 0 \



-05 -05 -

0 1 2 3 4 0 1 2 3 4
s/D s/D


Figure 12. Pressure coefficients over the hemispherical body (o = 0.4); D is the diameter
of the hemispherical projectile. (a) Impact of grid refinement for Mushy IDM
(b) Comparison between pressure coefficients of different models on the
coarse grid

We investigate the sensitivity of Mushy IDM to grid refinement for the two

hemispherical body grids through the surface pressure plots in Figure 12. Note that the

refinement factor in the vicinity of cavitating region is roughly 2.5-3. Figure 12(a)

indicates a noticeable, though modest, effect of the grid refinement on the surface

pressure. This can be mainly attributed to the grid-dependent geometric smoothing

operation, which smears the mushy formulation over a larger portion in the coarser grid.

Nonetheless, both the grids produce solutions that match the experimental data

reasonably well. Furthermore, it is important to mention that the near-wall nodes of the

coarser grid lie in the log-layer of the turbulent boundary layer enabling appropriate use

of the wall function (Versteeg and Malalasekera 1995), while the fine grid is over-refined











for the wall spacings to assume values in the suitable range. As pointed out by Senocak


and Shyy (2004a), in view of the wall function treatment (Thakur et al. 2002; Versteeg


and Malalasekera 1995), the presence of vapor in the cavity may substantially reduce the


appropriate range of the wall node spacings. As a consequence, we employ the coarser

grid for further computations. Figure 12(b) contrasts the performance of the Mushy IDM


with the Merkle et al. Model (1998) and Sharp IDM (2004a) at c = 0.4. The pressure


coefficient predicted by the three models illustrates noticeable variations in the


condensation region of the cavity. Specifically, the Mushy IDM, as seen from Figure 12


(b), tends to produce a sharper recovery of the surface pressure in the cavity closure zone.

Additionally, the Mushy IDM produces a small pressure dip (below the vapor pressure


value) at the cavity outset due to the lower evaporation rate and higher condensation rate.


(a)

Liquid Volume Fraction
1 0
08
05



(b)
Liquid Volume Fraction
S04
028
07
05



(c)
Liquid Volume Fraction
1 0
08
04
*02


Figure 13. Cavity shapes and flow structure for different cavitation models on
hemispherical projectile (o = 0.4). (a) Merkle et al. Model (b) Sharp IDM (c)
Mushy IDM







49


These features are reflected by the cavity sizes depicted in Figure 13. Note that the


Mushy IDM not only shrinks the cavity length, in comparison with the Sharp IDM, but


also impacts the flow structure in the recirculation zone of the cavity closure region. The


above findings support our approach in providing an appropriate transition between the


two models and a careful calibration to the parameter /7.

(a) (b)
(a) Sharp IDM - Sharp IDM
------ Merkle et al. Model ------- Merkle et al. Model
MushylDM Mushy lDM
1 2 Exp. data 1 2 U Exp. data

08 k08 -
06 06 -

02 02-
0 0-
-0 2 -02 -
-0 4 -04
0 01 02 03 04 05 06 07 08 09 1 0 01 02 03 04 05 06 07 08 09 1
x/D x/D


Figure 14. Pressure coefficient over the NACA66MOD hydrofoil at two different
cavitation numbers; D is hydrofoil chord length. (a) a = 0.91 (b) a = 0.84

We further simulate cavitating flow around the NACA66MOD hydrofoil at two


cavitation numbers, namely, 0.84 and 0.91, while maintaining the mushy formulation


only for the cavity boundary. Again, consistent behavior in terms of surface pressure is


induced by the Mushy IDM for this geometry, at both the cavitation numbers (Figure 14).


In summary, the appropriate source term modulation imposed by the Mushy IDM tends


to influence the prediction of surface pressure and cavity size in manners consistent with


the experimental observation. The above assessment motivates the further


implementation and enhancement of the Mushy IDM (with respect to /) on cryogenic


flow cases.









3.2.2 Cavitation in Cryogenic Fluids

In this section, we investigate the Mushy IDM for the cryogenic situation. We

perform computations on two geometries experimentally investigated by Hord (1973a,

1973b), namely, a 2D quarter caliber hydrofoil (Hord 1973a) and an axisymmetric 0.357-

inch ogive (Hord 1973b). These geometries were mounted inside suitably designed

tunnels in the experimental setup.



NO-SLIP
INLET OUTLET
SYMMETRY
hydrofoil surface (NO-SLIP)




NO-SLIP

INLET I I OUTLET
SYMMETRY
ogive surface (NO-SLIP)




Figure 15. Illustration of the computational domain accounting the tunnel for the
hydrofoil (Hord 1973a) and 0.357-inch ogive geometry (Hord 1973b)
(cryogenic cases).

Figure 15 illustrates the computational domains employed for these cases. Note that the

figure shows only planar slices of the domains, which also model the respective tunnel

shapes to account for the significant blockage effects. The mesh for the hydrofoil and the

ogive geometry respectively comprises 320x70 and 340x70 points. The mesh

distribution is chosen to facilitate adequate resolution of the cavitation zone.









Furthermore, the near-wall resolution over all the no-slip planes (cavitating geometries

and tunnel walls) accounts for deployment of wall functions (Thakur et al. 2002;

Versteeg and Malalasekera 1995).

Table 5. Flow cases chosen for the hydrofoil geometry.
Fluid: Case name: Inlet temperature: Freestream Re: Cav. No. (o.):

Liq. N2 283B 77.65 K 4.7x106 1.73

Liq. N2 290C 83.06 K 9.1x106 1.70

Liq. N2 296B 88.54 K 1.1x107 1.61

Liq. H2 248C 20.46 K 1.8x107 1.60

Liq. H2 249D 20.70 K 2.0x107 1.57

Liq. H2 255C 22.20 K 2.5x107 1.49

Source: Hord (1973a)

Table 6. Flow cases chosen for the ogive geometry.
Fluid: Case name: Inlet temperature: Freestream Re: Cav. No. ( ,):

Liq. N2 312D 83.00 K 9.0x106 0.46

Liq. N2 322E 88.56 K 1.2x 107 0.44

Liq. H2 349B 21.33 K 2.3x107 0.38

Source: Hord (1973b)

Statistically-averaged pressure and temperature data are available for the two geometries

at five probe locations over the body surfaces. The experimental findings report varying

amounts of unsteady behavior in the cavity closure regions, although no case-specific

information or data/visuals are available in that context. Hord conducted a series of

experiments over both the geometries, using liquid nitrogen and hydrogen, by varying the

inlet velocity, temperature, and pressure. In our computations, we have selected several











cases, referenced alphanumerically in the reports by Hord (1973a, 1973b), with different

freestream temperatures and cavitation numbers (see Table 5 and Table 6).


(a) Computed pressure (b) Computed pressure
1 U- Hord, 1973 (incipient data) 1 5 Hord, 1973 (incipient data)
05 1 -
0 -
05 -
-05-
05




-25 -1 5
I I I I I I -2 I I I
1 0 1 2 3 4 5 0 2 4
x/D x/D


Figure 16. Non-cavitating pressure distribution (a) case '290C', D represents hydrofoil
thickness and x represents distance from the circular bend (b) case '312D', D
represents ogive diameter and x represents distance from the leading edge

To validate the use of real fluid properties, we obtain the single-phase flow


solution to cases '290C' (hydrofoil; Re=9.1x106, c-=1.7) and '312D' (ogive;


Re=9.0x106, o-,=0.46), and compare the computed surface pressure with the


experimentally measured pressure under incipient (virtually non-cavitating) conditions.

Figure 16 demonstrates good agreement between the numerical and experimental data for

both the geometries, and corroborates the correct input of physical properties.

3.2.2.1 Sensitivity analyses

In general, cryogenic computations are prone to uncertainty due to a multitude of

inputs, in contrast to the non-cryogenic conditions. The cavitation model parameters,

namely, Cdet, Cprod, and t, have been selected largely based on the non-cryogenic


conditions. Furthermore, the solutions can exhibit substantial sensitivity with respect to

minor changes in the flow environment. For example, the uncertainties involved in the







53


temperature-dependent material properties may also cause noticeable differences in


predictions. To address relevant issues in this context and confirm/improve our


calibration of the Merkle et al. model constants, we perform a comprehensive Global


Sensitivity Analysis (GSA) over the chosen case '290C'. We initiate our computations on


the case '290C' (Re = 9.1 x106; o- = 1.7) which is centrally located in the temperature


range. We note that the previously calibrated values of the Merkle et al. Model


(Cdet =1.0 and C prod=80.0) are inadequate to provide a good match with the


experimental data under the cryogenic condition. This fact was also lately noted by


Hosangadi and Ahuja (2005), who suggested lower values of cavitation model parameters


in case of cryogenic fluids.


C-=0 068, C d=54 4544
------- C = 0, Cp 80 0 845 C- = 68, Cprod
Hord, 1973,p -... .. dest=l 0, Cprod-80 0
30 84rd 1973 Hord, 1973, T

20 835 -- -.- -
835

20 i 825.

82

0. -0' 81 5 -
81
-10
805

0 5 0 05 1 15 0 05 1 15 2
x (inches from bend) x (inches from bend)

Figure 17. Sensitivity of Merkle et al. Model prediction (surface pressure and
temperature) to input parameters namely Cdet and Cprod for the hydrofoil

geometry

Consistently, in the present study, we elicit Cdet = 0.68 and Cprod = 54.4 via numerical


experimentation, as more appropriate model parameters. Of course, such choices are


empirically supported and need to be evaluated more systematically.









To initiate such evaluations, Figure 17 portrays the response of the surface pressure

and temperature to the revision of the model parameters. We choose Cde,, to, Pv, and L

as design variables for the GSA, while holding the Re and ro- constant for the given

case. The chosen cavitation model parameters, namely Cdet and to, are perturbed on

either side of their reference values (Cdt = 0.68; Cprod = 54.4) by 15%. In comparison,

the material properties are perturbed within 10% of the value they assume from the NIST

database (Lemmon et al. 2002), at every iteration. Given these ranges on the variables,

we generate a design of experiments with 50 cases using a combination of Orthogonal

Arrays (Owen 1992) and Face Centered Cubic Design (JMP 2002). RMS values of the

hydrofoil surface pressure coefficient (Cp = (p- pm)/(0.5p1U2)) and temperature, which

are post-processed from the CFD data, are selected as the objective functions.

Subsequently, the two objectives are modeled by a reduced-quadratic response surface

through a least-squares regression approach (JMP 2002). The coefficient of multiple

regression (Myers and Montgomery 1995) in case of the pressure and temperature fit is

0.992 and 0.993 respectively, while the standard error is less than 1%. The fidelity of the

response surfaces is also confirmed against 4 test data points.

S 1.675 + 0.077C-, -0.061p- 0.082t: 0.007C*,dp -
(3.28)
0.012C tst +0.011p*2 +0.009p9t* -0.004L2 +0.013t.2

s = 82.537 0.243Cd, + 0.107p* 0.048* + 0.220t +
0.024C*t -0.016Cd ,pv -0.015C,tL* +0.013pl L + (3.29)
0.018p tc +0.019L*t

Here, CPM and Ts are the RMS values of the surface pressure coefficient and the

surface temperature, respectively. The superscript, *, represents the normalized values of










the design variables. Equations (3.28) and (3.29) represent the respective response

surfaces expressed in terms of the normalized design variables. The surface coefficients

upfront indicate the importance of the cavitation model parameters (Cde, and t.) and

vapor density, and the insubstantial contribution of latent heat (L), to the variability in our

objectives. We quantify these overall contributions by employing the variance-based,

non-parametric global sensitivity method proposed by Sobol (1993). This method

essentially comprises decomposition of the response surface into additive functions of

increasing dimensionality. This allows the total variance in the data to be expressed as a

combination of the main effect of each variable and its interactions with other variables

(refer to Appendix A for a brief mathematical review).


(a) (b)


42% m Latent heat Latent heat
4200 1 10
m Vapor density 46% m Vapor density
oC dest oC dest
Ot infinity 43% [ t infinity
u-% I'
38%



Figure 18. Main contribution of each design variable to the sensitivity of Merkle et al.
(1998) model prediction; case '290C' (a) Surface pressure (b) Surface
temperature

We implement the procedure expounded by Sobol (1993) on the response

surfaces in equations (3.28) and (3.29) to yield the plots in Figure 18. The pie-charts in

the figure illustrate the percentage contribution of the main effect of each variable; since

we find negligible variability due to the variable interactions. The charts firstly

underscore the sensitivity of pressure and temperature predictions to the cavitation model

parameters (Cde,, and t.). Secondly, the impact of p, is noticeable, while that of L is









56



insubstantial. These observations indicate that the design variables, unlike L, which



appear either in mi or i+ may tend to register greater influence on the computed results.


Thus, intuitively, U, and p,, which are omitted from the present GSA, are expected to



induce large variability in the computation, as compared to other omitted properties such


as K and Cp.


- Merkle et al. Model
* Hord,1973;p


15


0 05 1
x (inches from bend)


Figure 19. Pressure and temperature prediction for Merkle et al. Model for the case with

best match with experimental pressure; Cde, = 0.85; t = 0.85; p =1.1; L = 0.9


-- Merkle et al. Model
Hord,1973;p


1


-- Merkle et al. Model
Hord,1973;T


I I II80I I I


U U5 1
x (inches from bend)


15 2


Figure 20. Pressure and temperature prediction for Merkle et al. Model for the case with

best match with experimental

temperature; Cet =1.15; t = 0.85; p* =1.1;L* = 1.1


30


20


E 10

0

0.

-10


-20


-- Merkle et al. Model
Hord,1973;T


-0U U 05 1
x (inches from bend)


15 2


20.9..


-Ub U Ub 1
x (inches from bend)


I I I I I I I III II I I I









Furthermore, the impact is expected to be consistent on pressure and temperature, as

depicted by Figure 18, due to the tight coupling between various flow variables.

However, it is important to mention that our observation is meant to illustrate the relative

impact of several parameters.

We also utilize the available data from our 50 cases to seek possible improvement

of the Merkle at al. model parameters. Figure 19 illustrates the case which produces the

least RMS error between the computed surface pressure and the experimental data.

Conversely, Figure 20 portrays the case which produces the least RMS error between the

computed surface temperature and the experimental data. These figures demonstrate that

a single set of parameters may not provide optimal results for both pressure and

temperature within the framework of current cavitation models and cryogenic conditions.

This effort solicits multi-objective optimization strategies and deserves a separate study.

However, we do report from the calculated error norms of the 50 cases that Cdet = 0.68

and Cprod = 54.4 provide the best balance between the temperature and pressure

predictions for the chosen case. As a result, we hereafter employ these values for the

Merkle et al. Model.

We perform a simpler sensitivity analysis over the Mushy IDM since it has only

one control parameter (/8).








58



---- = 0.07 - =0.07
---- = 0.09 ---- p=0.09
----- P= 0.11 ....... p=0.11
Hord,1973;p 84 5 Hord,1973;T
84 -
20 -
835 -

E./ / / -83
S82/ / 82/
/ II82 -
81 5

81
80 5
20 80
-05 0 05 1 15 0 05 1 15
x (inches from bend) x (inches from bend)




Figure 21. Sensitivity of Mushy IDM prediction for case '290C' (surface pressure and
temperature) to the exponential transitioning parameter 0

Figure 21 depicts the results obtained over various values of f/. We calibrate / = 0.09


from the observed results similarly based on a reasonable balance between pressure and


temperature. Furthermore, Figure 21 suggests that the limiting case of = 0, which


essentially recovers the evaporation term of the Sharp IDM, would substantially over-


predict the cavity size. This fact endorses the need to regulate the mass transfer rates


modeled by the Sharp IDM, and subsequently the purported employment of the Mushy


IDM. It is worthwhile to emphasize that the Mushy IDM manifests the regulation of mass


transfer rates in cryogenic conditions incorporated empirically into the Merkle et al.


Model (section 3.2.1; Ahuja et al. 2001; Hosangadi and Ahuja 2005) largely under


physical pretext.


3.2.2.2 Assessment of cryogenic cavitation models over a wide range of conditions

We further perform computations for all the other cases by employing both the


Merkle et al.'s and the present Mushy IDM cavitation models.














-- Mushy DM
- Merkle et al. Model
U Hord, 1973;p


283B


5 0 05 1
x (inches from bend)


15


-- Mushy DM
- Merkle et al. Model
Hord, 1973;p
---A--- Hosangadi &Ahuja, 2005


290C


5 0 05 1 15
x (inches from bend)


- MushylDM
- Merkle et al. Model
U Hord, 1973;p


296B


20 .... .


-05 U 05 1
x (inches from bend)


1


-- Mushy DM
- Merkle et al. Model
U Hord,1973;T


283B


0 05 1 15 2 25 3
x (inches from bend)


-- Mushy DM
- Merkle et al. Model


U Hord, 1973; T
-A--- Hosangadi &Ahuja,







9-


S290C
A 290C


2005


]


I . I . I . I I I . I . I
0 05 1 15 2 25 3
x (inches from bend)


- Mushy IDM
- Merkle et al. Model
Hord,1973;T


296B


0 05 1 15 2 25 3
x (inches from bend)


Figure 22. Surface pressure and temperature for 2-D hydrofoil for all cases involving

liquid Nitrogen. The results referenced as 'Mushy IDM' and 'Merkle et al.

Model' are contributions of the present study.


-I I IIII.. .I I









Figure 22 contrasts the resultant surface pressure and temperature obtained with the

hydrofoil geometry for the cases involving liquid Nitrogen. We firstly note that both the

models are able to provide a reasonable balance between pressure and temperature from

standpoint of their predictive capabilities. The differences between the experimental data

and the two models are more pronounced than the mutual differences between the two

models. Furthermore, the agreement with experimental data is better in case of pressure

than in case of temperature, which is generally under-predicted at the leading probe point

by both the models. It is also observed that both models produce a slight temperature rise

above the reference fluid temperature at the cavity rear end, which is attributed to the

release of latent heat during the condensation process. The Merkle et al. Model also

produces a steeper recovery of pressure, as compared to the Mushy IDM, in the

condensation region of the cavity, for the cases shown. This suggests higher/faster

condensation rates for the Merkle et al. Model than the Mushy IDM. Secondly, we assess

our results for the case '290C' along with latest computational data (Hosangadi and

Ahuja 2005). Note that Hosangadi and Ahuja (2005) employed the Merkle et al. Model,

adapted in terms of the vapor volume fraction (ca,), with substantially higher values of

the model coefficients (Cdet =Cprod =100). The impact of these higher source term

values is evident from the steep gradients observed in their surface temperature and

surface pressure profiles. As a consequence, the temperature prediction of the present

study appears better in the cavity closure region, while that of Hosangadi and Ahuja

(2005) shows better agreement at the cavity leading edge. Thus, we emphasize again that

the choice of model parameters poses a trade-off, as we noticed in the global sensitivity

analysis, in the prediction of pressure and/or temperature for cryogenic cases. Lastly, it is










important to highlight that the inlet temperature gets closer to the critical temperature and

the cavitation number decreases, as we proceed from case '283B' to '296B'. In

comparison, the inlet velocity and consequently the Reynolds number assume values

within the same order of magnitude for the depicted cases. Thus, under isothermal

conditions, an increase in the cavity length is expected from case '283B' to '296B'. On

the contrary, the surface pressure plots in Figure 22 clearly indicate a decrease in cavity

length from case '283B' to '296B', despite the decrease in the freestream cavitation

number. This fact clearly distills the significant impact of the thermal effect in cryogenic

fluids, especially under working conditions that are close to the thermodynamic critical

point.


290C





Cavitation number
1.78
1.76
1.75
1.73
1.71

296B





Cavitation number
1.78
1.74
1.70
1.67
1.63
1.62


Figure 23. Cavitation number (c = p py(T)/(0.5pU2)) based on the local vapor
pressure Merkle et al. Model. Note the values of
o- = P. -p,(TO)/(O.5pU2) for the cases '290C' and '296B' are 1.7 and
1.61, respectively.











Our contention on the thermal effect is corroborated by the cavitation number


(o = [p P(T)]/(0.5pU2) ) contours depicted in Figure 23. The freestream cavitation


number (o-.) of the case '296B' is smaller than the case '290C' (Table 5). However, the


combination of evaporative cooling and its resultant impact over the vapor pressure

causes a sharp increase in the effective cavitation number close to the cavitation zone.

This increase is more substantial for the case '296B' and eventually leads to comparable


levels of effective cavitation number between the two cases, as seen in Figure 23.


(a)







S089
069
050
S030

(b)







089
069
050
S030

(c)







S089
0 69
0 50
0 30
Figure 24. Cavity shape indicated by liquid phase fraction for case '290C'. Arrowed lines
denote streamlines (a) Merkle et al. Model isothermal assumption (b)
Merkle et al. Model with thermal effects (c) Mushy IDM with thermal
effects







63


Figure 24 reveals the phase fraction distribution and the streamlines for the computational

cases of '290C'. The two models differ noticeably at the rear end of the cavitation zone.

The cavity of the Mushy IDM consistently indicates lower condensation rates in contrast

to the Merkle et al. Model, because of its longer length. We also note the gradual

variation in density and the less extent of vapor phase in the cavities, which highlight the

mushy/soggy nature of cavitation in cryogenic fluids. This is unlike our previous findings

on regular fluids such as water (Senocak and Shyy 2004a, 2004b; Wu et al. 2003c).

Under non-cryogenic conditions, the flow structure in the cavitation vicinity is generally

characterized by large streamline curvatures and formation of recirculation zones

(Senocak and Shyy 2002, 2004a, 2004b; Wu et al. 2003c) (also seen in Figure 13).

However, the weak intensity of cavitation in cryogenic fluids has a modest impact over

the flow structure, as indicated by the streamline patterns in Figure 24. Figure 24(a)

illustrates a 'special' solution to the case '290C' with the Merkle et al. Model assuming

isothermal assumptions (energy equation not solved).


Merkle et al. Model



m f ( m
36E+02 2 37E+03
3 69E+03 3 95E+02
-6 85E+03 2 50E+01
-1 OOE+04 I 1 36E+00

Mushy IDM



m ( m
1 -536E+02 2 37E+03
-369E+03 3 95E+02
-685E+03 2 50E+01
-1 OOE+04 1 36E+00


Figure 25 Evaporation (ih ) and condensation (i+ ) source term contours between the
two cavitation models case '290C'. Refer to equations (3.8) and (3.18) for
the formulations.









Note that discounting the thermodynamic behavior yields a substantially large cavity size

under similar conditions. Based on these observations, we emphasize that, from a

modeling standpoint, the thermal effect is manifested via a combination of the cavitation

model adaptations and the temperature dependence of physical properties. We bolster our

argument on the condensation rates between the two models in Figure 25. Note that the

evaporation and condensation contour plots demonstrate a mutually exclusive behavior.

The condensation region of the Merkle et al. Model illustrates sharper gradients and is

effective over a much larger region, as compared to the Mushy IDM.

Following our assessment with liquid Nitrogen, we extend our focus to the cases

with liquid Hydrogen, which has a density ratio of 0(30), unlike the 0(100) value for

Nitrogen. We experience the need to re-calibrate our cavitation models for this different

fluid because of the discernible role played by the density terms (p,, p, and p,) in

determining the volume transfer rates. Our numerical experimentation yields Cd,, = 0.82

and Cprod = 54.4 as appropriate values for the Merkle et al. Model, in case of liquid

Hydrogen. Consistently, we choose / = 0.065 for the Mushy IDM in context of liquid

Hydrogen. Our case selection for liquid Hydrogen, as seen from Table 6, follows similar

trends as in case of liquid Nitrogen. The inlet velocity for all the cases with liquid

Hydrogen is greater than 50 m/s, and it increases from the case '248C' to '255C' (66.4

m/s for '255C'). As a result, we upfront underscore these substantially higher values of

inlet velocities that are employed for Hydrogen. Figure 26 depicts reasonable balance

between the temperature and pressure predictions for the case '248C'; however, the

agreement with the experimental data deteriorates equally for both the models as we














proceed from the case '248C' to the case '255C'. Especially, the temperature is highly



under-predicted as the inlet velocity increases between the two limiting cases.


- Mushy IDM
- Merkle et al. Model
U Hord,1973;p


248C


0 05 1
x (inches from bend)


- Mushy IDM
- Merkle et al. Model
U Hord,1973;p
---A--- Hosangadi &Ahuja, 2005


i


249D


05 0 05 1
x (inches from bend)


15


- MushylDM
- Merkle et al. Model
U Hord,1973;p


255C


-05 0 05 1 15
x (inches from bend)


-- Mushy DM
- Merkle et al. Model
U Hord,1973;T


248C


0 05 1
x (inches from bend)


15 2


-- Mushy DM
- Merkle et al. Model
U Hord,1973;T
---A-- Hosangadi &Ahuja,2005


249D


0 05 1
x (inches from bend)


15 2


- Mushy IDM
- Merkle et al. Model
U Hord,1973;T


S255C




0 05 1 15 2
x (inches from bend)


Figure 26. Surface pressure and temperature for 2-D hydrofoil for cases involving liquid

Hydrogen. The results referenced as 'Mushy IDM' and 'Merkle et al. Model'

are contributions of the present study.


i


I,,,,I,,,,I,,,,I


I I II,, I I I I III, ,II, ,II, I









The results of Hosangadi and Ahuja (2005), while depicting the consistent aspect of sharp

gradients that we noticed earlier, also portray significant discrepancies between the

surface temperature and the experimental data at the rear region of the cavity. This

finding could be either attributed to inadequacies in the cavitation model parameters or

the representation of temperature-dependent physical properties. However, the observed

sensitivity of predictions to physical properties, the relatively better agreement in the case

'248C', and the clear trend of disagreement with increasing velocities indicate the

likelihood of the latter reason. Our extraction of physical properties from the NIST

database (Lemmon et al. 2002) is based on models which assume thermodynamic

equilibrium conditions. However, at such high fluid velocities, the timescales for

thermodynamic equilibrium may be much larger than the overall flow timescales. This

may introduce substantial discrepancies in the values of various physical properties and

subsequently in the predictions; refer to Hosangadi and Ahuja (2005) for similar

reporting. Of course, confirmation of the above possibility and development of rigorous

non-equilibrium strategies is a challenging proposition for future research, and requires

additional experimental insight. Nonetheless, we note the consistent performance of the

Mushy IDM, especially in terms of the pressure prediction, over the chosen range of

reference temperatures and velocities.

We finally attempt to re-assess our above calibrations (Cde, Cprod, and / ) for both

the fluids and instill confidence into our predictive capabilities by performing

computations over the axisymmetric 0.357-inch ogive geometry. The ogive surface

pressure and temperature plots for all the three cases are displayed in Figure 27. The

shrinkage of the cavity length between cases '312D' and '322E', despite the decrease in









67




the reference cavitation number, unequivocally re-emphasizes the influence of the



thermo-sensible working conditions.


0 05 1
x in inches from bend


322E


0 02 04 06 08 1
x in inches from bend


349B


0 02 04 06 08 1
x in inches from bend


-- Mushy DM
- Merkle et al. Model
U Hord,1973; p


12 14


-- Mushy DM
- Merkle et al. Model
U Hord,1973; p


12 14


312D


I F-1


-- Mushy IDM
- Merkle et al. Model
U Hord,1973; T


0 05 1
x in inches from bend


322E


-- MushyDM
- Merkle et al. Model
U Hord,1973;T


4


0 05 1
x in inches from bend


349B


15


-- MushyDM
- Merkle et al. Model
U Hord,1973;T


0 05 1
x in inches from bend


15


Figure 27. Surface pressure and temperature for axisymmetric ogive for all the cases

(Nitrogen and Hydrogen). The results referenced as 'Mushy IDM' and

'Merkle et al. Model' are contributions of the present study.


i


, I I


I I I I I I I









Furthermore, the discrepancy in our predictions for liquid Hydrogen, when subjected to

higher velocities, is also evidenced by the plots for the case '349B'. Overall, the

agreement with experimental data in all the cases is better in terms of the surface pressure

than temperature. But, unlike the hydrofoil geometry, the temperature at the first

(leading) probe point is over-predicted for all the ogive cases. As expected, this

discrepancy is most in the liquid Hydrogen case ('349B'). Thus, the two geometries do

not produce a consistent pattern of disagreement midst the surface temperature and the

experimental measurements. This inconsistent behavior warrants further experimental

and numerical probing from a standpoint of developing more precise cavitation models

for cryogenic cavitation. Nonetheless, we observe that our analyses/calibrations are able

to yield a justifiable range of results for the ogive geometry as well.














CHAPTER 4
TIME-DEPENDENT COMPUTATIONS FOR FLOWS INVOLVING PHASE
CHANGE

Researchers have striven to develop efficient and accurate methodologies to

simulate time-dependent cavitating flows. The multiphase nature of the flow

accompanied by complex flow physics yields a system of tightly-coupled governing

equations. Furthermore, interfacial dynamics, compressibility effects in mixture region,

and turbulence entail deployment of numerical models to represent these physical

phenomena. The formulation of these models substantially impacts the solution

procedure. As a result, evolving efficient algorithms for unsteady cavitating flows is

certainly a non-trivial task.

Formulation of implicit procedures for pressure-based methods is impeded mainly

by the strong linkage between the flow variables such as velocity and pressure.

Furthermore, the dynamics of the variable density field in cavitating flows imposes

supplementary equations on the existing system, which add to the computational

challenges. As a result, iterative algorithms such as SIMPLE, SIMPLER, and SIMPLEC

(Versteeg and Malalasekera 1995), which are commonly employed for a wide variety of

problems, may be computationally expensive for solution of cavitating flows with

pronounced unsteady behavior. Senocak and Shyy (2002, 2004b) circumvented this

difficulty by incorporating the Pressure Implicit with Splitting of Operators (PISO)

algorithm suitably with the model equation of cavitation dynamics and the sound

propagation model for the mixture region. This endeavor resulted in a non-iterative









methodology for time-dependent computation of cavitating flow through a series of

predictor-corrector steps. Senocak and Shyy (2002, 2004b) demonstrated the merits of

this efficient approach with a series of cavitating flow computations on geometries such

as Convergent-Divergent Nozzle and hemispherical solids.

However, all the past efforts were in the context of isothermal cavitation sans the

energy implications and real fluid properties. From standpoint of formulating a non-

iterative algorithm for cryogenic cases, the inclusion of the non-linear energy equation

and temperature dependence of physical properties into the existing methodology is

expected to pose multitude of adversities to the solution efficiency and accuracy. Direct

development of an algorithm for cryogenic cavitation may be a presumptuous

preliminary approach. As a result, we initiate a multiphase non-iterative procedure on a

simpler test problem with similar nature of challenges, and, with an objective of

monitoring the accuracy and stability of the computations.

In this chapter, we elucidate the newly initiated algorithm and illustrate its results

on a chosen test case. The truly unsteady problem of Gallium fusion (with natural

convection effects) is adopted for the purpose of validation. Discussion on the grid

sensitivity, accuracy of results, and the stability criterion is provided. The above primary

objective is complimented by a reduced-order description of the gallium fusion problem

by Proper Orthogonal Decomposition (POD). The flowfield in the problem is

characterized by a solid-liquid front movement with a continuous change in the overall

flow length scale. The ability of POD, to accommodate these varying flow scales, is

mainly emphasized.









The insights gained from this study on gallium fusion are later extended to

improving the algorithm for cryogenic cavitation. The algorithm in context of cryogenic

cavitation is described to point out the changes implemented in order to address the

temperature effects. POD is employed to probe the time-dependent data, and offer it a

succinct representation.

4.1 Gallium Fusion

Experimental studies (Gau and Viskanta 1986) have investigated the physics of

Gallium fusion due to its low fusion temperature and ease of handling. The availability of

2D experimental data has motivated numerical studies (Lacroix 1989, Lacroix and Voller

1990, Shyy et al. 1995) to adopt it as a test case.

T=0
Y=l 7 0

liquid solid



T=1 T=O


Y



_X aT = 0 X= 1
X y

Figure 28. Schematic of the 2D Gallium square geometry with the Boundary Conditions
(Shyy et al. 1998)

A schematic of the square geometry along with the boundary conditions can be viewed in

Figure 28.









4.1.1 Governing Equations

The single-fluid modeling approach is adopted by employing a liquid phase

fraction f The density is assumed constant through the Buossinesq approximation. The

governing equations for the problem are as follows.

au
= 0 (4.1)


O(put) O(puuox) ap a au 1- f2
+ +- (P )-C(f )u +gpf(T T)2 (4.2)
at axi ax, Qxj cx f3 +q

a(ph) a a aT
+ (puh)= (k ) (4.3)
at )x ax ax

Here, / represents the coefficient of thermal expansion for the fluid. The term

1- f2
C( )u, in the momentum equations is the Darcy source term (Shyy et al. 1998).
f +q

Through their functional dependence on f they are modeled to retard the velocity to

insignificant values in solid region. The constants C and q, in compliance, are tuned to

yield a negative source term, which is at least seven orders of magnitude higher than

other terms, in the solid region (f= 0). The enthalpy in equation (4.3) is expressed as:

h= CT + jL (4.4)

The phase fraction f is modeled computationally by the h-based method (Shyy et al.

1998). The enthalpy at any given iteration is computed as shown below followed by an

update off(explained in a later section).

hk =CpTk +fk-L (4.5)

Note that f is always bounded by 0 and 1. Thus, equation (4.5) can be recast into a

temperature equation as shown.









a(pCPT) a a aT a(pLf) a
CT) (pCuT)= (k )-[O+ (pLujf)] (4.6)
at 8x) 8x^ 8x) at axi

The iterative update of the liquid phase fraction, as mentioned above, renders the

energy/temperature equation non-linear. Furthermore, the buoyancy term in equation

(4.2) leads to a pressure-velocity-temperature coupling. The non-linearity in the energy

equation and the strong coupling between various flow variables justify the use of the

gallium fusion problem as a test case preceding the cryogenic problem. The physical

properties such as viscosity, latent heat, thermal conductivity, and specific heat are,

however, assumed to be constant during the fusion process.

4.1.2 Numerical Algorithm

The previous computations on similar multiphase problems employed an iterative

solution strategy (Chuan et al. 1991, Khodadadi and Zhang 2001). Conversely, the PISO

method (Issa 1985, Thakur et al. 2002, Thakur and Wright 2004), which forms the

backbone of the current algorithm, essentially is a series of predictor corrector steps to

yield a non-iterative solution of flow equations through operator splitting procedure. The

present methodology, however, closely follows a slightly modified version of PISO

designed for buoyancy driven single-phase flows (Oliveira and Issa 2001). Kim et al.

(2000) proposed a series of steps to accelerate the simple heat conduction equation with a

solid-liquid phase boundary. The following methodology attempts to blend the merits of

the modified PISO (Oliveira and Issa 2001) with the propositions of Kim et al. (2000), to

design an efficient and accurate algorithm for phase change problems. The sequence of

calculations for the algorithm, based on the above governing equations, is expounded

below.









The strongly implicit form of the discretized governing equations with a finite

volume formulation, at any node P, is as follows:

A (pu, ) = 0
(A)"u =_ A1"u) -Ap1+ +H"up +Spu(un+1)
(APv +1 = A vb'1 -A pn+l + H;v + S (vn1) (4.7)
+Bp (T n+)
) TTn+1 TT;I-I T n n + M (fn
(AP)T + = A '+HT +Hfp +M(f )

Here, A represents the terms at the current time level, H represents the terms at older time

step, M represents the terms involving the phase fraction f B represents the buoyancy

term, and S represents the Darcy source term. The Darcy source terms can be absorbed

into the left hand side term to yield to following equations.

A (pul n+') 0
(G;P1 )p' = C AnInln A "n+l + Hiu"
(4.8)
(Gpv n)1 = ZAb1 -A, pn1 +Hvp + B (Tn+1)
STn+l T T +I T n n + + ( n
(AP)T' = A b +HP +H f/ +M(f"1)

Here, Gp" = Ap +Sp" and so on. The sequence of steps to solve equation (4.8) non-

iteratively is elaborated next.

(a) Momentum predictor

(GPu)uP = Au au" + H u
(4.9)
(Gp)v = ZA v A avp" + H + BPv (T+ )

These velocities u* and v* are obtained by using the pressure value at the previous

time step and hence are not divergence free.









(b) First pressure corrector

A, (pu,) = 0
p (4.10)
A A, p')= A,(pu) )
GP

Here, p'=p*-p is calculated. The pressure field p* can now be employed for

correcting the divergence error in velocity. Note that the pressure correction at this stage

is limited by several approximations and is not adequate to produce an accurate velocity

field.

(c) First momentum corrector

(GPu)u**= _Au -Ap +Hu"p
(4.11)
(GP )vP *= 'Avn A p* + Hv + B(T")

The velocities are corrected to yield u** andv** explicitly using the intermediate pressure

field obtained in the previous step.

(d) First temperature corrector

(AP )TP = Ab + +HTP" +Hf fP +M(f*) (4.12)

The above temperature equation utilizes the latest value off However, in view of a non-

iterative strategy, solution of temperature, merely by the above equation, may not be

sufficient for rapid convergence due to the delayed update off As a remedy, a series of

explicit steps are implemented following the above equation to significantly improve the

prediction of temperature and, subsequently, f These steps are adapted from the

algorithm proposed by Kim et al. (2000) for pure conduction equation. They are enlisted

below.


(i) Updatefby h-based method as shown.









hk+1 k+1 k fL
f = 0 if hk+l <

f~+l =1 if hk > h, (4.13)

f, k P elsewhere
h,-hk
where, h = CpT,h, = CpT, +L

Note that at the outset T7' = Tp.

(ii) If fpl fp correct temperature using the explicit form of equation (4.12)

with updated coefficients. This step yields the new temperature field T+2

(iii) Compute residual of the temperature equation as shown further.

O = (AP)T A, +HT +Hf f/ +M(fk+l) (4.14)


The above residual equation can be further employed to improve temperature

prediction by Newton-Raphson approach as follows (Kim et al. 2000):



aTk+2
P


where, the Jacobian is approximated as 0+ 1 Af .
P P

In summary, steps (i) (iii) are performed at least 10-12 times following the equation

(4.12). Let the temperature and phase fraction field at the end of this stage be denoted as

T* andf*.









(e) Second pressure corrector

A (pul ) = 0

SAP") = ( Bp(T*)-Bp(T")+
G"p Gp" (4.16)
SAbu** Abu*
[(SP)* (SP)]** })

This step is a powerful characteristic of the PISO algorithm. The 'differential Darcy

term', (Sp')* -(Sp)up**, is an addition to the terms already suggested by Issa (1985) and

Oliveira and Issa (2001). This term arises out of the update of G" after the first

temperature corrector. Note that the coefficient G"' unlike single-phase flow cases, also

includes the implicit coefficient of the Darcy term (refer to the discussion in step (a)).

The Darcy term undergoes a sudden change over several orders of magnitude between

the liquid and the solid region. Since the first temperature corrector tends to change the

phase fraction distribution, it is imperative to update the coefficient Gp post that step.

The second pressure corrector, which is obtained via subtracting equation (4.11) from

(4.17), is thus able to account for phase front movement because of the 'differential

Darcy term'. In summary, the second pressure corrector attempts to couple the non-linear

terms in the momentum equations, the temperature field, and the phase front movement

to the pressure field.

(f) Second momentum corrector

(Gpu*)u* = C Auu u p- Ap** + Hu
*** ** ** (4.17)
(Gpv )v = Avb Ap + Hv"+Bv (T)

The velocity field at this stage is derived from the more accurate prediction of pressure

p** obtained from the previous step.









(g) Second temperature corrector

(A)'T* T** +HT+H f, +AM*(f**) (4.18)


The prime purpose of this equation is to correct the temperature field with updated

velocities. This step is followed by an update off, similar to that in equation (4.13).

It was reported that repeating the corrector steps (steps (b) (g)) for an extra time

improves coupling between various equations (Thakur and Wright 2004). However, due

to the large magnitude of non-linearity expected for the fusion problem, at least 3 more

repetitions of steps (b)-(g) are recommended (totally 8 corrector steps). The above

modification to the PISO, similar to the original algorithm, incurs a splitting error due to

the operator splitting approach. However, the order of this splitting error is higher than

the formal order of temporal accuracy (Issa 1985), thus justifying the computational

accuracy.

4.1.3 Results

The multiphase algorithm is implemented in conjunction with a pressure-based

solver having multi-block capability (Thakur et al. 2002). The diffusive terms are handled

by central differencing, while the first order upwind scheme is employed for the

convective terms. Computations are mainly performed on a 2D square domain of size

D=1. The range of values for St and Ra in the computations is [1 0.042] and [104

2.2x106] respectively. Sample results obtained from current algorithm are compared to

those obtained by Shyy et al. (1995) in the following discussion. It is important to note

that results obtained by Shyy et al. (1995) have greater fidelity than those published in

similar studies, due to the use of a finer grid.










4.1.3.1 Accuracy and grid dependence


Figure 29. 2D interface location at various instants for St = 0.042, Ra = 2.2 x 105 and Pr =
0.0208. White circles represent interface locations obtained by Shyy et al.
(1995) on a 41x41 grid at time instants at t = 56.7s, 141.8s, & 227s
respectively.

Figure 29 illustrates the 2D interface locations obtained with the current algorithm,

with three different grid resolutions, for the parameters shown. Note that some results by

Shyy et al. (1995) are at a slightly earlier time instant. This contributes to the modest

difference observed in the interface location, especially during the early stages, when the

interface velocity is significantly higher. Considering this fact, the present results show

reasonable agreement with the earlier computation. Furthermore, the time-dependent

movement of the interface is consistent on all the three grids, although there is a

qualitative impact of the resolution on the interface profile.


0t5 0


0 x 0 x0
--------- -------------------
--------


21x2l grid











(a)
21x21 gdd
41x41 gdd
S 81x81gdd



; -;



S 025 05 075
X


(b) 1 (c)










x x


Figure 30. Grid sensitivity for the St = 0.042, Ra = 2.2 x 105 and Pr = 0.0208, 2D case (a)
Centerline vertical velocity profiles at t = 227s (b) Flow structure in the upper-
left domain at t = 57s; 41x41 grid (c) Flow structure in the upper-left domain
at t= 57s; 81x81 grid.

Table 7. Location of the primary vortex for the St = 0.042, Ra = 2.2 x 105 and Pr =
0.0208 case
Grid size x-location at t y-location at t x-location at t y-location at t
= 57s = 57s = 227s = 227s

21x21 0.130 0.760 0.350 0.674
41x41 0.135 0.755 0.365 0.661
81x81 0.146 0.710 0.377 0.658



The influence of the grid quality on the flow structure can be assessed from the centerline

velocity profiles and the locations of the center of the primary vortex depicted in Figure

30(a) and Table 7 respectively. The even number of nodes (cell centers) in each grid

creates a slight offset between the center gridline and the geometric centerline. This offset

varies inversely with the grid quality, and is expected to contribute to the reasonable









discrepancy in the centerline velocity profiles, shown in Figure 30(a). In comparison, the

center of the primary vortex, from the findings reported in Table 7, indicates a gradual,

downward shift with grid refinement, particularly during the initial stages of fusion. In

fact, this movement, although moderate, is conspicuous at t = 57s, when an 81-point grid

is used instead of a 41-point grid. This observation has a physical relevance, as noticed

from Figure 30(b) and Figure 30(c). The upper-left part of the domain develops a

secondary vortex after some initial time lapse. This vortex is expected to induce a

downward motion of the primary vortex structure. As clearly seen, the 41x41 grid, unlike

the finer grid, is unable to capture this small-scale circulation, which justify the data in

Table 7. The overall flow convection pattern and, consequently, the interface movement

are, however, weakly affected by the secondary vortex structure. Furthermore, the

solution demonstrates a fairly consistent improvement with the grid quality, in addition to

a restrained grid-dependence. These findings are encouraging from the standpoint of

stability restrictions, which are elucidated in the following section.

Although the multiphase algorithm is elucidated/illustrated in context of 2D

calculations, its generality for 3D computations is examined by extending the case in

Figure 29 to square box geometry. The fusion process is initiated by two adjacent, heated

walls to yield spanwise flow variations.