Applied Science, Faculty of
Mechanical Engineering, Department of
DSpace
UBCV
Darbhamulla, Nihar Bhardwaj
Master of Applied Science - MASc
University of British Columbia
In the current work, we present a finite-element computational framework for Fluid-Structure Interaction problems subjected to the coupling of unsteady cavitating flows with flexible structures. We build upon a previously developed stabilized variational framework for multiphase FSI by incorporating a dynamical structural solver based on the modal decomposition of the structure. In the first part of the work, we present the validation of the current framework for conducting Large Eddy Simulations (LES) of cavitating flows past rigid structures. We identify the resolution criteria for dynamic subgrid-scale LES based on the re-entrant jet momentum - a prominent phe- nomenon associated with cavity-shedding in the wake of immersed bodies. The validated framework is then used to elucidate the features of cavitating flow past a rigid hydrofoil. In particular, we iden- tify features of sheet-cavitating flow which enable the transition to cloud cavitation. Further, we evaluate the instabilities driving sheet-cavity breakdown and establish the vortical structures which drive cloud cavity collapse, and quantify the frequencies observed over the course of a cavitation cycle. In the second part of the work, we present the validation of the framework for LES of cavitating flows past flexible structures. Based on the validation study conducted over a flexible NACA66 rectangular hydrofoil, we elucidate the role of cavity and vortex shedding in the structural dynamics at three different cavitation numbers. We identify a broad spectrum frequency band whose central peak does not correlate to the frequency content of the cavitation dynamics or the natural fre- quencies of the structure, indicating the induction of unsteady flow patterns around the hydrofoil. Finally, we discuss the coupled fluid-structure dynamics during a cavitation cycle associated with the promotion and mitigation of cavitation.
A Computational Framework forFlow-Induced Vibrations of PropellerBlades in Cavitating FlowsbyNihar Bhardwaj DarbhamullaB.Tech., Indian Institute of Technology Guwahati, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2024© Nihar Bhardwaj Darbhamulla 2024The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the thesis entitled:A Computational Framework for Flow-Induced Vibrations of Propeller Blades in Cav-itating Flowssubmitted by Nihar Bhardwaj Darbhamulla in partial fulfillment of the requirements for thedegree of Master of Applied Science in Mechanical EngineeringExamining Committee:Dr. Rajeev Jaiman, Associate Professor, Department of Mechanical Engineering, UBCSupervisorDr. Carl Ollivier-Gooch, Professor, Department of Mechanical Engineering, UBCSupervisory Committee MemberDr. Anthony Wachs, Professor, Department of Chemical and Biological Engineering, UBCSupervisory Committee MemberiiAbstractIn the current work, we present a finite-element computational framework for Fluid-StructureInteraction problems subjected to the coupling of unsteady cavitating flows with flexible structures.We build upon a previously developed stabilized variational framework for multiphase FSI byincorporating a dynamical structural solver based on the modal decomposition of the structure.In the first part of the work, we present the validation of the current framework for conductingLarge Eddy Simulations (LES) of cavitating flows past rigid structures. We identify the resolutioncriteria for dynamic subgrid-scale LES based on the re-entrant jet momentum - a prominent phe-nomenon associated with cavity-shedding in the wake of immersed bodies. The validated frameworkis then used to elucidate the features of cavitating flow past a rigid hydrofoil. In particular, we iden-tify features of sheet-cavitating flow which enable the transition to cloud cavitation. Further, weevaluate the instabilities driving sheet-cavity breakdown and establish the vortical structures whichdrive cloud cavity collapse, and quantify the frequencies observed over the course of a cavitationcycle.In the second part of the work, we present the validation of the framework for LES of cavitatingflows past flexible structures. Based on the validation study conducted over a flexible NACA66rectangular hydrofoil, we elucidate the role of cavity and vortex shedding in the structural dynamicsat three different cavitation numbers. We identify a broad spectrum frequency band whose centralpeak does not correlate to the frequency content of the cavitation dynamics or the natural fre-quencies of the structure, indicating the induction of unsteady flow patterns around the hydrofoil.Finally, we discuss the coupled fluid-structure dynamics during a cavitation cycle associated withthe promotion and mitigation of cavitation.iiiLay SummaryMarine propellers operate through the creation of a differential pressure across the blades. In certainoperating conditions, the pressure over the blades drops lower than the liquid vapor pressure leadingto the rupture of water in the liquid phase. This phenomenon termed as cavitation results in theformation of pockets of vapor dispersed through the liquid phase. The dynamics of cavity growthand shedding leads to enhanced noise characteristics of the marine vessel, prominent among whichare propeller singing arising as a consequence of propeller blade vibrations, and cavity collapsedue to unfavourable pressure conditions. In the current study, we extend our in-house high-fidelitycomputational framework involving the interaction of cavitating fluid flow with deforming propellerblades. The framework is rigorously validated against experimental studies, and is employed toinvestigate the cavitation dynamics around a rigid and flexible propeller blade section in a turbulentcavitating flow. The underlying mechanisms of cavity growth and breakdown are systematicallyexplored, and strategies for evaluating mitigation are proposed.ivPrefaceThis M.A.Sc. thesis entitled “A Computational Framework for Flow-Induced Vibrations ofPropeller Blades in Cavitating Flows ” presents the research conducted by Nihar Darbhamullaunder the supervision of Dr. Rajeev Jaiman.Chapters 1,2,3,4,5,6 : Parts of the presented introduction and literature review in Chapter 1,methodology in Chapter 2, results and discussions presented in Chapters 3, 4, 5, and parts of theconclusion in Chapter 6 have been published or are under review for publication in the followingjournal articles:N. B. Darbhamulla and R.K. Jaiman. “A finite element framework for fluid-structure interactionof turbulent cavitating flows with flexible structures.” arXiv preprint arXiv:2312.17434 (2023).Accepted for publication to Computers & Fluids.The credit authorship contribution corresponding to authors are as follows: Nihar Darbhamulla:writing - original draft, code development, investigation, post-processing, data analysis. RajeevJaiman - conceptualization, methodology, editing, supervision.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii1 Introduction and Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Mechanics of Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Review of Partial Cavitation on Hydrofoils . . . . . . . . . . . . . . . . . . . . . . . 41.3 Review of Fluid-Structure Interaction Studies on Hydrofoils and Marine Propellers . 61.4 Numerical Strategies for Cavity-FSI Studies of Hydrofoils . . . . . . . . . . . . . . . 81.5 Transport-equation based Modeling of Cavitating Flows . . . . . . . . . . . . . . . . 101.6 Objectives and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Mathematical model and computational framework . . . . . . . . . . . . . . . . . 132.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.1 Navier-Stokes Equations in the Deforming Domain . . . . . . . . . . . . . . 132.1.2 Cavitation Transport Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.3 Modeling of Subgrid-Scale Stresses . . . . . . . . . . . . . . . . . . . . . . . . 152.1.4 Modeling of Structural Deformation . . . . . . . . . . . . . . . . . . . . . . . 162.1.5 Exchange of Information at Fluid-Structure Interface . . . . . . . . . . . . . 182.2 Numerical Formulation and Implementation . . . . . . . . . . . . . . . . . . . . . . 182.2.1 Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2.2 Spatial Discretization and Variational Statement . . . . . . . . . . . . . . . . 20vi2.2.3 Modal Decomposition and Spatio-Temporal Discretization of the Linear Struc-ture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.4 Unified ALE FSI framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 Validation of LES for Cavitating Flow over Rigid NACA66 Hydrofoil . . . . . 283.1 Problem Description and Mesh Characteristics . . . . . . . . . . . . . . . . . . . . . 283.1.1 Domain definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.2 Meshing Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.1.3 LES Quality Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Hydrodynamic Coefficients and Cavitation Characteristics . . . . . . . . . . . . . . 323.3 Energy spectra surrounding the hydrofoil . . . . . . . . . . . . . . . . . . . . . . . . 373.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 Cavitating Flow over Rigid NACA66 Hydrofoil . . . . . . . . . . . . . . . . . . . . 414.1 Flow Features Induced by Cavitation . . . . . . . . . . . . . . . . . . . . . . . . . . 414.2 Cavity Inception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.3 Sheet Cavity Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4 Sheet Cavity Breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.5 Cloud Cavitation Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.6 Re-laminarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575 Validation of LES for Cavitating Flow over Flexible NACA66 Hydrofoil . . . . 595.1 Problem Description and Mesh Characteristics . . . . . . . . . . . . . . . . . . . . . 595.1.1 Definition of Structural Parameters . . . . . . . . . . . . . . . . . . . . . . . 595.1.2 Domain Definition and Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Fully-wetted response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 Cavitating Response at σ = 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676 Conclusions and Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74viiList of Tables3.1 Mesh parameters for the LES validation of flow past NACA66 hydrofoil. Here Nchordand Nspan denote the number of points in the chord-wise and span-wise directionson the hydrofoil surface. NBL denotes the number of points in the boundary layerand rBL denotes the growth ratio of element sizes normal to the boundary. Nnodesand N elem. denote the total number of nodes and elements in the mesh respectively.For the current study, only 8 node tri-linear hexahedral elements are used. . . . . . . 313.2 Summary of the hydrodynamic force coefficients, and the corresponding Strouhalnumber observed during LES of cavitating flow. . . . . . . . . . . . . . . . . . . . . . 374.1 Instability propagation velocity of different right and left traveling perturbationmodes driving vorticity transport and triggering the instability of the cavity. Thenegative sign indicates a left traveling perturbation mode. . . . . . . . . . . . . . . 484.2 Evolution of the sheet cavity and re-entrant jet over a cavitation cycle. We measurethe cavity length lcav and location of cavity closure xcav from the leading edge,whereas the re-entrant jet thickness δj is measured normal to the hydrofoil at x/C =0.1 downstream of cavity closure. The mean re-entrant jet velocity Vj is estimatedbased on the Strouhal number of the flow [12, 76]. The vapor-liquid interface isconstructed using φ = 0.585, and the characteristics are evaluated at z/S = 0.5. . . . 515.1 Mesh parameters for the LES validation of flow past flexible cantilevered NACA66hydrofoil. Only 8 node tri-linear hexahedral elements are used. . . . . . . . . . . . . 615.2 Comparison of hydrodynamic load coefficients and motion characteristics for fully-wetted conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.3 Comparison of response characteristics for cavitating conditions at σ = 1.4. . . . . . 625.4 Comparison of the dominant frequencies observed in the displacement response ofthe structure with experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . 65viiiList of Figures1.1 Illustration of the key objectives and tools of the IGMV research program. Courtesy:Computational Multiphysics Laboratory UBC . . . . . . . . . . . . . . . . . . . . . 11.2 Prominent sources of underwater radiated noise (URN). Courtesy: ComputationalMultiphysics Laboratory UBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Types of cavitation around propeller blades. Sheet, cloud and tip vortex cavitationhave been observed to have a prominent influence on propeller noise [4, 13]. Imagefrom [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.1 A schematic of the flow past the NACA66 hydrofoil at Re = 750000. The com-putational setup and boundary conditions are shown for the filtered Navier–Stokesequations. Here, u = (u, v, w) denotes the components of the fluid velocity and Cdenotes the chord length. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Cross-sectional view of mesh M2 at (a) boundary layer and near wake, (b) close tothe trailing edge. (c,d) Resolution of the hydrofoil span at the leading and trailingedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Index of quality for LES in the region of interest around the hydrofoil for three meshes(a) M1, (b) M2, (c) M3, and (d) Comparison of LES IQν at (x, z) = (C, 0.15C). . . 333.4 Comparison of pressure at different chord-wise locations along the hydrofoil. . . . . . 343.5 Comparison of pressure frequency spectrum at chord-wise locations x/C ∈ {0.3, 0.5, 0.7}with the dominant frequency modes observed in the studies of Chen et al. [18], Kjeld-sen et al. [57], Leroux et al. [65]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.6 Comparison of cavity volume and force coefficients computed in the current studywith the LES studies of Chen et al. [18]. . . . . . . . . . . . . . . . . . . . . . . . . . 363.7 Evolution of the cavity and vorticity over a single cycle. The first column illustratesthe experimental results obtained by Leroux et al. [65]. The second column depictsthe cavity developed in the current numerical framework. The third column showsthe contours of Q-criterion (Q∗ = 8) colored by Z-vorticity. . . . . . . . . . . . . . . 38ix3.8 Arrangement of probes to assess turbulent flow characteristics: Probes P1-P3 areplaced at a distance of 0.02C from the hydrofoil, at x/C ∈ {0.2, 0.5, 1.0} respectively.Probes P4-P5 are placed downstream of the trailing edge at x/C ∈ {1.25, 1.5}.The origin is located at the leading edge of the hydrofoil, with the angle of attackcreated by rotating the hydrofoil about mid-chord point. Contour lines of Z-vorticityω∗z = ωzC/U∞ ∈ [−5,−2] ∪ [2, 5] are presented to indicate unsteady flow-structuresin the vicinity of the hydrofoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.9 FFT Spectra of (a) pressure fluctuations and (b)turbulent kinetic energy at locations(see Fig. 3.8) in the cavitating region and near the wake of the hydrofoil. For the sakeof clarity, the spectra of points from the same group are shifted, and the magnitudeof the shift is indicated in the legend. . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.1 Variation in CL and CD over one cycle of cavitation illustrating the duration ofsheet-cavity growth, cloud-cavity separation and collapse, residual cavity growthand collapse. The points of change in flow behavior are labeled as (A) Instantiationof the re-entrant jet, (B) Separation of cloud cavity, (C) Shedding of Trailing EdgeVortex Cavities. C1 and C2 are vortex cavities that are shed from the trailing edgeand contribute to cloud cavity collapse (D) Complete collapse of cloud cavity (E)Collapse of residual sheet cavity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Evolution of (a) vorticity (b) viscous diffusion of vorticity (c) vortex stretching and(d) x-velocity fluctuations over the streamline originating at (−0.167, 0.013, 0.039)C. 444.3 (a)A narrow layer of counter-clockwise viscous diffusion (Dz) of vorticity is createdin the vicinity of the cavity, which acts to direct the flow away from the cavity. . . . 454.3 (b)The narrow layer of counter-clockwise viscous diffusion of vorticity (Dz) is lo-cated close to the cavity, and continues to direct the flow away from the cavity. Atthe trailing edge, the baroclinic torque (Cz) begins to dominate vorticity transport,forcing the streamlines to curve back towards the cavity. The red line indicates thevapor-liquid interface constructed at φ = 0.585, and the contours on the left illus-trate viscous diffusion of vorticity (Dz), while the contours on the right illustratebaroclinic torque (Cz). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.4 Contours of material derivative of Vorticity, Dilatation, Viscous Diffusion, Velocityand Pressure Gradient during the cavity inception phase. In the figures, the whiteline indicates the isobar corresponding to the vapor pressure (Cp = −1.25), and thered line denotes the vapor-liquid interface. We observe the cavity grow to occupythe isobar of vapor pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.5 Evolution of (a) vorticity (b) viscous diffusion of vorticity (c) baroclinic torque and(d) dilatation over the streamline originating at (0.0055, 0.0603, 0.039)C for tcavf ∈[0.17, 0.37]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49x4.6 Evolution of (a) vorticity (b) viscous diffusion of vorticity (c) vortex stretching and(d) x-velocity fluctuations over the streamline originating at (−0.167, 0.013, 0.039)C.The line segments indicate the motion of perturbations in vorticity transport mech-anisms, with Lcav representing the trailing edge of the cavity. . . . . . . . . . . . . 494.7 Evolution of the re-entrant jet, and the corresponding induction of fluctuations invorticity transport for tcavf ∈ [0.43, 0.56]. The columns illustrate the z-componentof vorticity, vorticity transport, baroclinic torque, and vortex-stretching from left toright. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.8 Visualization of the coherent structures around the hydrofoil based on a non-dimensionalQ-criterion Q∗ = 8, colored by Z-vorticity and viewed from the leading edge. Thecoherent structures are overlaid on the cavity contours delineated by φ = 0.585. . . 524.8 Visualization of the coherent structures around the hydrofoil based on a non-dimensionalQ-criterion Q∗ = 8, colored by Z-vorticity and viewed from the leading edge. Thecoherent structures are overlaid on the cavity contours delineated by φ = 0.585. . . 534.9 Visualization of the coherent structures around the hydrofoil based on a non-dimensionalQ-criterion Q∗ = 8, colored by Z-vorticity and viewed from the trailing edge at dif-ferent time instances. The coherent structures are overlaid on the cavity contoursdelineated by φ = 0.585. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544.9 Visualization of the coherent structures around the hydrofoil based on a non-dimensionalQ-criterion Q∗ = 8, colored by Z-vorticity and viewed from the trailing edge. Thecoherent structures are overlaid on the cavity contours delineated by φ = 0.585. . . 554.10 Evolution of the flow around the hydrofoil during complete collapse of cloud cavityat tcavf = 0.90 and subsequent re-laminzarization. The non-dimensional x-vorticityand z-vorticity (ω∗z = ωzC/U∞) components are presented at Z/S = 0.5. The redlines correspond to iso-contours of φ = 0.585 to indicate the cavity structure(s) atdifferent instances of the cycle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.1 A schematic of the flow past flexible cantilevered NACA66 hydrofoil at Re = 750000.The computational setup and boundary conditions are shown for the filtered Navier-Stokes equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.2 Top and side views of the flexible hydrofoil, illustrating the angle of attack and thegap between the hydrofoil tip and front wall. . . . . . . . . . . . . . . . . . . . . . . 615.3 Transition to turbulence and evolution of the vortical structures over the hydrofoilin non-cavitating conditions. Some key vortical structures are the hairpin vorticesemerging from the hydrofoil span and the tip-leakage vortex. The vorticity field(ω∗x = ωxC/U∞) and the velocity field (red-arrows) are indicated at different chord-wise locations, indicating the motion of the tip-leakage vortex away from the hydrofoil. 63xi5.4 FFT Spectra of (a) turbulent kinetic energy at the spanwise location z/S = 1 and(b)turbulent kinetic energy at the spanwise location z/S = 1 at the chordwise loca-tions defined by Fig. 3.8 for the hydrofoil under fully-wetted conditions. For sake ofclarity, spectra of points from the same group are shifted. . . . . . . . . . . . . . . . 645.5 (a) Comparison of the bending displacement at the leading and trailing edges atthe hydrofoil tip, with the Coefficient of Lift (CL). (b) Comparison of the angulardisplacement at the hydrofoil tip, with the Coefficient of Moment (CM ). . . . . . . 655.6 Standing wave response of the cantilevered hydrofoil (a) bending and (b)twistingresponse of the hydrofoil along the span with time. . . . . . . . . . . . . . . . . . . . 665.7 FFT Spectra of (a) bending displacement (yLE) and (b) angular displacement (θ)of the hydrofoil at the tip (z/L = 1). The arrows indicate sub-harmonics of fun-damental bending (nbi) and torsional modes (nθi ) which appear in the displacementspectra. Note that the bending (f bN ) and torsional (fθN ) frequencies have been non-dimensionalized here w.r.t the time-scale C/U∞. . . . . . . . . . . . . . . . . . . . . 675.8 Evolution of (a) lift coefficient (CL), (b) cavity Volume (Vc), and (c) Cavity andVortex structures from t1 to t5 as labeled in (a). The liquid vapor interface isdemarcated using φ = 0.8 for the flexible hydrofoil. . . . . . . . . . . . . . . . . . . . 695.8 (d) Cavity and Vortex structures from t6 to t9 as labeled in (a). . . . . . . . . . . . . 70xiiAcknowledgementsI would like to take this opportunity to acknowledge the people who have helped me through thisjourney.I would like to express my gratitude to my supervisor Dr. Rajeev Jaiman for his supportand motivation. His vision regarding the direction of research, ability to mould ideas, rigour inpresentation and unending dedication to the advancement of science are qualities I hope to emulatein my future endeavours.Research is a beautiful endeavour driven by unending curiosity and reinforced by rigor. Forkeeping me on this path of discovery, I thank my exceptionally brilliant colleagues at ComputationalMultiphysics Laboratory. I greatly enjoyed the endless conversations I have had about FiniteElements, Numerical Methods, Fluid Mechanics and Life with Xiaoyu, Biswajeet and Saman. FromIndu, Akash and Rui I have tried to absorb the qualities of optimism, dedication and immense thirstfor knowledge. I thank Farnoosh, Aarshana, Shayan, Sabiha and Ishan for sharing the CFD journeywith me, and for the immense help they have provided during different junctures in my research.Every discussion has helped me understand something new about the art of applying numericalmethods, and for this I will remain forever indebted to CML. I thank Suraj, Amir and Rachit whohelped me settle in during my early days in Vancouver and their guidance towards research.While life within research is sometimes a struggle, life outside it often gives one the chance toforge amazing friendships. I was fortunate to have Sidhant, Indu and Biswajeet as my immediatepillars of support during this journey, and their company has been filled with enjoyable moments andgreat lessons on skill-building. I thank Akash, Gautam, Musanna, Saman and Niki for enhancing myperspectives on the role of hardwork, dedication and balance. I thank my friends Vaibhavi, Vishak,Sharma, Sasank and Ankur who allowed me to be in touch with the past, and have motivated mydrive into the future.Above all, gratitude to my family, whose unconditional support and continuous motivation haveguided me to this juncture.xiiiChapter 1Introduction and Literature ReviewThe marine industry forms the backbone of international trade, accounting for nearly 75% of freighttransport. An increase in marine vessels associated with freight transport has led to continuouslyrising shipping tonnage, and a nearly four-fold increase in shipping traffic [41]. The rising operationof marine vessels has been accompanied by an increase in anthropogenic noise levels in oceans andcoastal areas [24]. While specific vessel noise emission characteristics vary, they depend primarilyon factors such as vessel speed, hull shape, load and size [81]. The consequence of these noiseemissions has been a wide range of noise frequencies and amplitude propagating in water, whichhas resulted in a significant impact on all marine species [24]. At the same time, there is a demandfor enhanced vessel efficiency rising out of the need to minimize carbon emissions.Figure 1.1: Illustration of the key objectives and tools of the IGMV research program. Courtesy:Computational Multiphysics Laboratory UBCFor the development of sustainable marine activities, the ship-building industry has adoptedstrategies which are commensurate with the Industry 4.0 revolution. This interest has drivenresearch on Intelligent and Green Marine Vessels (IGMVs) with the objectives of (i) enhancingenergy efficiency of marine propulsion systems and (ii) characterizing adverse underwater radiated1noise and develop noise mitigation strategies. In line with these objectives lie the development[48, 51, 69, 79] and utilization [37, 52, 60] of multi-physics computational tools and the use of novelphysics-based data-driven methods [21, 22, 98]. Fig. 1.1 highlights the key research objectives ofthe IGMV research program.Propeller NoiseThe quest for developing quieter IGMVs requires an understanding of the prominent mechanisms ofnoise generation during vessel operation. Fig. 1.2 illustrates some noise sources from ships, whichare broadly delineated into propeller noise, hull vibration noise and machinery noise. Prominentamong these is the noise emission due to propeller rotation which is the focus of the current work.The rotational work done by the propeller shaft acts to produce relative motion between waterand the propeller blades. As a consequence, a differential pressure arises across the blade resultingin the generation of thrust. When the absolute pressure over the suction side (side with lowerpressure) drops lower than the vapor pressure of water, local vaporization begins to occur leadingto the onset of cavitation. The presence of cavitation can lead to undesired outcomes on threefronts. The inception and collapse of small bubbles have the propensity to cause material erosionon the blade surface [13, 56]. Extensive cycles of cavity growth and decay (sheet/cloud cavitation)on the blade surface can result in unsteady and impulsive fluid loads on the hydrofoil leading tovibrations [4, 13]. The collapse of cavities over the hydrofoil surface or in its wake results in thecreation of pressure waves in water, which lead to noise emissions. Fig. 1.3 depicts various formsFigure 1.2: Prominent sources of underwater radiated noise (URN). Courtesy: ComputationalMultiphysics Laboratory UBCof cavitation around propeller blades. Unsteady sheet cavitation is commonly found over propellerblades, wherein a sheet of vapor develops from the leading edge of the blade, and extends over2some fraction of the blade surface. The growth and breakdown of sheet cavities are associatedwith low frequencies, within which multiple cavity collapse events are detected, each leading to adistinct pressure signature and induction of fluctuations in the flow [84]. The interaction of thesecavities with intricate vortical structures also influences the broadband and tonal noise emissioncharacteristics of the propeller. Thus, a characterization of these flow structures is essential andrequires the study of a complex multiphase flow problem. To get a physical insight into the key flowfeatures exhibited by a propeller blade, a rectangular hydrofoil can be used for a prototypical study.In this thesis, we develop the computational framework specifically to resolve the hydrodynamicinteractions observed in cavitating flows past hydrofoils.Figure 1.3: Types of cavitation around propeller blades. Sheet, cloud and tip vortex cavitationhave been observed to have a prominent influence on propeller noise [4, 13]. Image from [4].1.1 Mechanics of CavitationThe phenomenon of cavitation involves the phase transition of liquid into vapor. Often, the flowingliquid is not a pure substance, and contains impurities in the form of suspended solid particles,entrained gas bubbles, or has regions of interaction with solid surfaces. The locations of thesefeatures are identified as points of weakness in the liquid. Brennen [10] indicates the ability ofliquids to withstand tensile loads within an elastic regime, which lead to vacancies coalescing to form3a finite vapor pocket. This rupture of the liquid phase occurs isothermally at the tensile strength ofthe liquid (defined as pv−pc, where pc is the liquid pressure at which vapor pocket forms). The vaporgrowth is driven by the spatial variation in liquid pressure, with new pockets forming and potentiallycoalescing in regions where the liquid tensile strength is reached. Furthermore, fluctuations in thelocal liquid pressure can affect the spatial dimensions of the vapor pockets formed, and they aretransported by the flowing liquid, till they encounter a region of high pressure, where they havethe propensity to undergo a violent collapse.The flow of liquid water over marine propeller blades generates a low-pressure region in accor-dance with the mechanism driving thrust generation. Under certain flow conditions, the pressurecan drop close to pc, leading to the formation of vapor nuclei and their subsequent growth [81]into different cavity structures. This behaviour is termed partial cavitation - wherein the cavitiesgrow to occupy a fraction of the blade surface. This phenomenon has received extensive attentionin literature, with focus on its inception [3, 54, 80, 99], classification of partial cavitation regimes[26, 55, 58, 59, 62] and analysis of the instabilities and mechanisms driving the transition betweenregimes [12, 34, 55, 75, 99] in the context of both attached and separated flows. Partial cavitationsurrounding a propeller blade alters the performance characteristics of the propeller (affecting thethrust, and drag), while introducing undesirable effects such as vibrations and noise emissions. Re-cent studies have made efforts to investigate these phenomena experimentally [2, 5, 8, 20, 93, 94, 101]and computationally [2, 18, 45, 52, 100]. However, in the context of FSI, the approaches primarilyinvolve modelling the flexible structure as a two-dimensional elastically mounted body. In thiscontext, it becomes important to model the strongly coupled fluid-structure dynamics using eithera full-order or reduced-order representation of the three-dimensional structure, in order to eluci-date the features of cavity dynamics governing the structural response, and vice-versa. In the nextsection, we provide a review of partial cavitation on hydrofoils, particularly directed towards sheetcavitation and the instabilities driving the transition to cloud cavitation.1.2 Review of Partial Cavitation on HydrofoilsPartial cavitation interacts in different ways with the freestream flow, primarily through misalignedpressure and density gradients resulting in baroclinic torque on the fluid, leading to the produc-tion of vorticity. Around hydrofoils, partial cavitation results in the vaporization of liquid at theleading edge, which can induce local compressibility effects leading to the dilatation of vorticity.Turbulent fluctuations in the flow have been found to impact partial cavity stability, leading totheir collapse resulting in the production of a highly unsteady wake, in conjunction with pressurewaves propagating through the domain.Franc and Michel [26] experimentally studied sheet cavitation on curved walls with an emphasison the NACA 16012 hydrofoil. They observed the cavity detachment point to be behind a laminarseparation instead of the minimum pressure point. They indicate the presence of an adverse pressure4gradient to generate laminar separation and propose a criterion for locating the cavity detachmentpoint. Furthermore, they indicate the incompatibility of turbulence with cavitation and illustratethe development of cavitation is infeasible in the vicinity of a turbulent boundary layer. Furtherinvestigations on partial cavities by Le et al. [62] indicated the onset of a re-entrant jet around thickcavities, which drives the detachment of vapor pockets from the sheet cavities, and highlighted thejet’s propensity to produce vorticity in the flow. However, this production of vorticity is onlyfeasible if the re-entrant jet meets the liquid phase at the opposite end of the cavity. Furtherinvestigations by Kawanami et al. [55] indicate the re-entrant jet velocity is of the order of free-stream flow, which propagates up to the mid-chord location of the hydrofoil. They indicate thesuppression of the re-entrant jet prevents the transition to cloud cavitation. However Pham et al.[76] observed the onset of perturbations on the cavity surface which can also be attributed to thecavity’s destabilization.Callenaere et al. [12] investigated the instability in partial cavities induced by the re-entrantjet. They observed a periodic re-entrant jet in the case of thin cavities leading to a sequence ofsmall shedding events. In the case of thick cavities, there is negligible interaction between the free-stream flow and the re-entrant jet, and in cases where the re-entrant jet reaches the leading edgeof the cavity, a cloud-shedding event is observed. In their study, they illustrate the re-entrant jet’sdependence on cavity thickness and the adverse pressure gradient developing at the trailing edge ofthe cavity. In their case, the investigations were of cavitation in the wake of a backward-facing step.Their investigations also highlighted methods to investigate the re-entrant jet, with the jet having50% the freestream velocity in the case of a thick cavity. Studies by Kjeldsen et al. [57] studied theregimes of partial cavitation around a NACA0015 hydrofoil by varying the cavitation number andangle of attack. At higher angles of attack and lower cavitation numbers, they observed a nearlyinvariant chord-based Strouhal number of 0.15 associated with cavity shedding. Investigationsby Leroux et al. [65] correlated key features of the temporal pressure variations with the flowcharacteristics, and attributed a minor pressure spike to the re-entrant jet before the onset of sheetcavity breakdown. They attribute the re-entrant jet to cause the cleavage of the sheet cavity andobserve a growth of the residual sheet and its disappearance upon the collapse of the cloud cavity.Pelz et al. [75] investigated the transition from sheet to cloud cavitation based on a viscous filmmodel and constructed stability maps by varying Reynolds and cavitation numbers. However theirinvestigations are in the context of flat surfaces, and similar analysis for curved surfaces has beenwanting.More recently, cavitation has been investigated numerically with Ji et al. [45] developing a one-dimensional model to study the correlation of pressure fluctuations with variations in global cavityvolume. Subsequent studies by Chen et al. [18] have attempted to characterize the role of turbulencein driving sheet cavitation breakdown. However, further investigations are necessary to concretelyestablish the validity of this hypothesis. Kashyap and Jaiman [52] investigated the cavitation aroundrigid and elastically-mounted hydrofoil sections and proposed an adverse pressure gradient-based5vorticity generation mechanism to explain the transition from sheet to cloud cavitation. Whilemultiphase flow interactions due to cavitation alter the dynamics exhibited on rigid hydrofoils,these interactions have the propensity to produce a variety of response characteristics in flexiblehydrofoils, which is usually the case with hydrodynamic control and propulsion devices. With theincreasing development of composite structures for hydrodynamic applications, it becomes essentialto understand the nature of response elucidated in structures immersed in cavitating flows, and theconsequent alteration of flow dynamics. In the next section, we present a review of FSI studies onhydrofoils and marine propellers.1.3 Review of Fluid-Structure Interaction Studies on Hydrofoilsand Marine PropellersAs highlighted above, marine propellers operate in a broad spectrum of operating conditions andexhibit a range of vibration characteristics owing to unsteady forces experienced as a consequenceof flow structures emerging in a cavitating environment [102]. In this regard, propeller blades aremodeled as a hydrofoil and the corresponding flow characteristics are investigated in different partialcavitation regimes. Within these regimes, a flexible hydrofoil is found to experience attenuationand periodic fluctuations of loads leading to marked differences in structural response relative tothe flexible hydrofoil in non-cavitating conditions [2, 5, 27, 53, 92].Experimental investigations of Fluid-Structure Interaction (FSI) of hydrofoils in a cavitatingenvironment have illustrated the rich interplay of the cavity-flow-structure dynamics. Franc andMichel [27] demonstrated the incompatibility of attached cavities with a turbulent boundary layerin the flow past an oscillating hydrofoil, and attached cavities were found to be swept away by anupstream turbulent front. Furthermore, the leading edge roughness and the frequency of hydrofoiloscillations influence the thickness of the attached cavity [14]. Studies by Matsudaira and Okazaki[71] delineated the torsional flutter boundary of a pitching hydrofoil in the subcavitation and non-cavitating regimes, wherein subcavitation is considered to occur when the attached cavity does notextend to the hydrofoil’s trailing edge. They indicated the flutter boundaries depend strongly onthe degree and location of cavity separation and flow-stall conditions. However, at lower meanangles of attack, although the cavity adheres to the sub-cavitation regime, cloud cavitation sets inat larger cavitation numbers. It is this regime of operation that is commonly encountered in marinepropellers.Further investigations in this regime of operation were conducted by Kato et al. [53], whereinthe hydrofoil’s pitching frequency was found to synchronize with the cavity collapse frequency.Within the unsynchronized sub-cavitating range, they observed the growth of a thick sheet cavityand vorticity layer, which leads to small-scale bubble clusters detaching from the hydrofoil surface,and the cluster is observed to behave nearly identically to the freestream. Within the lock-in range6of the transition region, the cavitation breakdown process involves sheet cavity growth, followedby detachment and shedding of a bubble cluster from the trailing edge. This cluster evolves intovortex cavitation (cavity contained within a vortex), downstream of the hydrofoil. They observedthis process to be closely synchronized with the pitching oscillations. Ducoin et al. [25] investigatedthe effect of elastic deformations induced in a pitching hydrofoil, with cavitation increasing thevibration levels and altered modal response at specific frequencies. In particular, at a fixed angle ofattack, cavitation was found to excite the natural frequency associated with the torsional mode ofthe hydrofoil. At high pitching velocities, a long sheet cavity was found to develop on the hydrofoil’ssuction side in conjunction with high displacements. At lower pitching velocities, vortex cavitationis observed with smaller displacements. Thus, under driven structural motion in a cavitatingenvironment, the rate of pitching is intricately connected to the flutter instability and the cavitystructures generated. The presence of turbulence in the boundary layer also plays a significant rolein influencing the effects of cavitation w.r.t its development on the hydrofoil surface.While the studies discussed above consider the flow response as a consequence of structuralmotion, similar investigations have been conducted into the free-vibration response of cantileveredhydrofoils in cavitating fluid flows. Ausoni et al. [5] studied the von-Ka´rma´n vortex street in thewake of a cantilevered rectangular hydrofoil, with a truncated trailing edge. In the non-cavitatingregime, the von-Ka´rma´n vortex street was observed to lock-in with the hydrofoil eigenfrequency cor-responding to the first torsional mode. Within the cavitating regime, the vortex-shedding frequencywas found to increase, along with smaller vortex spacing attributed to cavitation-driven vorticityenhancement. The vibration amplitudes were found to be reduced under cavitating conditions ac-companied by the synchronization of vortex-shedding with the structural frequency. Correspond-ingly, they hypothesize that the unsteady flow field and vortex-shedding are driven by structuralvibrations.Investigations by De La Torre et al. [20] studied the influence of stable partial cavities on theadded mass effects experienced by a cantilevered rectangular hydrofoil, and quantified the fre-quencies of bending and torsional modes of the hydrofoil in both cavitating and non-cavitatingenvironments. They observed a reduction in added-mass effects for the first torsional and secondbending modes with decreasing cavitation number owing to a larger cavity on the suction surface.Within the context of flexible hydrofoils, Akcabay and Young [2], Benaouicha et al. [8] experimen-tally investigated the interaction of sheet and cloud cavitation with the hydrofoil. They observedexcitations in subharmonics of the foil’s wetted natural frequencies and the broadening of frequencycontent in the cloud cavitation regime. Wu et al. [101] further studied the hydroelastic responseof flexible hydrofoils in a cavitating environment and delineated the behavior of vibration velocitywith the state of surface cavitation in the cloud cavitation regime. They indicate the cloud cavity’scollapse leads to a substantial increase in the vibration velocity fluctuations. Further, they indi-cate variations observed concerning the cavitation development in the case of a highly complianthydrofoil, particularly regarding the collapse of the large-scale cloud cavity into smaller structures.7Simultaneously, Smith et al. [92] investigated the effects of cloud cavitation on a flexible trape-zoidal hydrofoil, with the dual objective of correlating hydrofoil flexibility with the frequencycontent of the force fluctuations and establishing three-dimensional flow regimes. They observedbending deformations to damp the higher frequency fluctuations in the normal force measurements.Further, they also attributed the cavity-shedding primarily to the re-entrant jet mechanism, withthe thin cavity leading to the shedding of small-scale vapor pockets. Subsequent investigations bySmith et al. [94],[93] characterized the instabilities driving cavity shedding in the vicinity of a stiffand compliant hydrofoil, and prominently identified a regime of cavity shedding associated withlock-in. They illustrate that compliant hydrofoils exhibit a prominent bend-twist coupling whichcauses an early transition from sheet to cloud cavitation, which also influences the formation of alonger sheet cavity. The time scale of sheet cavity formation leads to a reduced shedding frequency.The compliant hydrofoil attenuates the high-frequency oscillations in the force, while simultane-ously operating in lock-in conditions with the first bending mode. The studies considered aboveillustrate the intricacies of flow physics associated with flexible hydrofoils in a turbulent-cavitatingenvironment.1.4 Numerical Strategies for Cavity-FSI Studies of HydrofoilsIn the light of physics observed in experiments on the response of flexible hydrofoils in a turbulent-cavitating environment, it becomes essential to develop high-fidelity models for each of the physicalphenomena observed. It is equally essential to develop stable, accurate, and robust numericaldiscretization strategies to ensure the fidelity of numerical results to the underlying model andthe corresponding physical observations. Within the context of numerical FSI studies for marinepropellers and hydrofoils, the primary numerical challenge is to accurately capture the cavity andflow structures, to predict the forces experienced by the structure.Approaches based on potential flow theories have been extensively used to evaluate the hy-droelastic response of flexible marine propellers. Works by Lee et al. [64], Maljaars et al. [68]use a coupled potential theory-based boundary element method (BEM) and finite element method(FEM) for investigating the hydroelastic response of flexible marine propellers. The FSI of fluidand propeller blades has also been investigated with the use of panel methods, through a looselycoupled interaction between the hull wake and propeller [46]. While inviscid flow models are ef-fective in making general design decisions for propellers, they are unable to capture the range ofvortical structures that are associated with the unsteady loads experienced by the propeller. Inthe context of viscous flow modeling, [63] used a tightly coupled CFD-FEM solver to study highReynolds number flows over a flexible blade undergoing flow-induced vibrations. The authors illus-trated a requirement to deploy tightly-coupled FSI solvers for studying large amplitude vibrationsof flexible blades in the flow regimes considered.The numerical modeling of cavitating flows has received extensive attention in the literature,8particularly in the domain of flows around propellers. A majority of these studies have been aimed atcapturing the turbulence-cavitation interaction, and an accurate estimate of the loads on hydrofoilsfeasible within the available scope of cavitation and turbulence models. Dang [19] presents acomprehensive review of the early approaches towards numerically simulating cavitating flows, andadopted a potential flow theory approach in conjunction with defining a cavity detachment pointwithin a panel method framework. Schnerr et al. [84] further investigated the cloud cavitation andcollapse-induced shock dynamics through a compressible formulation of the governing equations,with thermodynamic closure. They indicated the need to have a high temporal resolution withinthis framework, owing to wave dynamics associated with acoustic cavitation, and to capture regionsof instantaneous high pressure. Further studies by Seo and Lele [88] used a density-based mixture-theory model for cavitation in conjunction with the Spalart-Allmaras turbulence model. Within thisframework, they captured the unsteady mode of the unstable partial sheet cavity and subsequentcloud cavitation and illustrated the RANS turbulence model’s capability to capture the shear-layerinstability formed by the re-entrant jet mechanism.Ji et al. [45] numerically studied the cavitation structures and vortex shedding dynamics us-ing large eddy simulations (LES) and a homogeneous mixture theory-based model to compute thepressure, velocity, and vapor volume fraction in the vicinity of a NACA66 hydrofoil. They investi-gated the turbulence-cavitation interaction through an analysis of the vorticity transport equationfor variable density flows. Chen et al. [18] extended this work further, presented a comprehensiveevaluation of the forces and moments experienced by the hydrofoil, and conducted a preliminaryanalysis of the turbulent structures generated in the hydrofoil’s vicinity. Sedlar et al. [85] inves-tigated the interactions between re-entrant flow and cavitation structures as well as cavitationexcited pressure fluctuations. They compared the performance of three different turbulence modelsto evaluate the fidelity of numerically captured flow features to physical data. They observed thebest description of vortical structures around the hydrofoil is predicted by LES, at the cost ofoverestimating the dominant frequencies of cavity oscillation. These frequencies were observed tobe better captured by the SAS-SST turbulence model and Detached Eddy Simulations (DES).While the above studies have primarily aimed at capturing the re-entrant jet dynamics andtheir role in driving the cavity breakdown process, they also establish a comparison between nu-merical approaches for evaluating the forces experienced by a structure which forms the backboneof studying the response of flexible structures within a cavitating environment. Akcabay and Young[2] employed a loose hybrid coupling to couple a commercial two-dimensional URANS solver with a2-DOF hydrofoil model to study the effect of cavitation on the hydroelastic stability of hydrofoils.Further work by Wu et al. [100] investigated both numerically and experimentally the cavitatingflow past a NACA66 hydrofoil. They attributed the hydrofoil’s displacement response to flow-induced flutter, a hypothesis that warrants closer investigation. They solved the URANS equationswith a k − ω SST turbulence model, with compressibility-based turbulent-viscosity modifications.For the structure, they used a 2-DOF hydrofoil model to represent the bending and torsional modes9of deformation. The studies however depict significant deviation quantitatively from experimen-tal observations, and thus reinforce the need for a robust and accurate unified framework for thestrongly-coupled FSI studies of cavitating flows over propellers.Kashyap and Jaiman [51] established a variational framework for cavitating flows with movingfluid-structure interfaces, with appropriate stabilization and the use of a homogeneous mixturemodel theory for capturing cavitation, and the use of DES with the SA turbulence model. Theframework depicted high fidelity with the studies of Leroux et al. [65], Senocak and Shyy [86]and demonstrated the method’s robustness in capturing flows with highly unsteady cavitationdynamics. The stabilization involves a positivity-preserving variational (PPV) scheme proposedby Joshi and Jaiman [50] for capturing the two-phase flow of immiscible fluids. The use of a localdiscrete upwind operator in an interface region with oscillations acts to diffuse the overshoots insolution and ensures the positivity of underlying element-level matrices. The scheme is effective indamping spurious pressure oscillations in multiphase flow with fluids having a high-density ratio.Correspondingly, this approach lends itself well to transport-equation modeling (TEM) of cavitatingflows. The current work extends this framework by introducing an eigenmode representation ofa three-dimensional structure, allowing for accurate numerical capture of the FSI of multi-modalstructures in cavitating flows.1.5 Transport-equation based Modeling of Cavitating FlowsIn the context of marine propellers, cavitation manifests in the form of structures evolving overmultiple orders of spatial and temporal scales [4, 10]. In this regard, an appropriate cavitationmodel requires the identification of spatio-temporal scales of significance. At macroscopic spatialscales, an extensively used strategy is the treatment of fluid as a homogeneous mixture of liquidand vapor phases, which differ primarily in the treatment of mixture density.One approach to compute density involves using the equations of state under the assumptionof equilibrium flow theory[84]. For isothermal flows, the density has a barotropic equation ofstate [17, 23]. An advantage of this approach is the absence of empirical coefficients for modeling,and the use of well-established equations of state. Correspondingly these models are difficult todeploy in an incompressible fluid framework. Furthermore, in an isothermal flow, the gradientsof density and pressure are parallel, leading to the baroclinic torque (∇ρ × ∇p) being zero. Thebaroclinic torque plays an important role in cavitating flows for vorticity generation owing to thehigh gradient of density, particularly in the closure region of attached cavities[34]. An alternateapproach assumes the individual phases of the fluid to be incompressible. A phase indicator variableis used to interpolate the density. A transport equation model (TEM) has to be constructed for thephase indicator variable, the source term of which captures the finite mass transfer rate betweenphases[72, 83, 90, 103].Owing to the phenomenological nature of the source terms for the phase indicator TEM, different10approaches have been employed for modeling the source term. Merkle [72] related the source termto the local pressure and phase fraction of the liquid based on dimensional arguments for bubbleclusters. Schnerr and Sauer [83], Singhal et al. [90], Zwart et al. [103] developed models that assumethe cavities to be present in the form of clusters of small spherical bubbles, and directly used asimplified form of the Rayleigh-Plesset equation [10] for spherical bubble dynamics to model themass transfer rate. The phenomenological arguments for modeling the bubble-bubble interactionin these models are different, which is what differentiates them. Despite the applicability of theTEM approach for capturing cavitation, they are constrained by problem-dependent semi-empiricalcoefficients in the source term. Furthermore, the phase indicator variable can have large gradients,which can force unphysical spikes in pressure near the liquid-vapor interface. In incompressibleflow simulations, these spikes can diffuse through the computational domain leading to numericalinstabilities. Senocak and Shyy [87] indicated the necessity of handling these spurious pressurespikes as a requirement for robustly capturing cavitating flows.However, the versatility of the TEMs allows for their use in modeling cavitation across differentregimes of cavitation, and have been employed extensively for studying hydrodynamic cavitationover geometries of marine interest such as hydrofoils and circular cylinders [2, 9, 18, 33, 45, 86].1.6 Objectives and MethodologyThe primary objective of this work is to augment a previously developed computational frameworkfor cavitating fluid flows to study the fluid-structure interaction of flexible cavitating propellerblades. To achieve this objective, we divide the current work into two major parts. The first partof the work involves validation of the existing framework for conducting Large Eddy Simulations(LES) of cavitating flows past rigid structures. This involves the development of mesh resolutioncriteria for conducting well-resolved LES of cavitating flows. The mathematical details of thecomputational framework are discussed in Chapter2. Using this validated framework, we elucidatethe features of cavitating flow past a rigid hydrofoil and aim to answer the following questions: What are the mechanisms driving the separation of the sheet cavity, and how can theseinstabilities be characterized? Upon the formation of the cloud cavity, what are the characteristics of flow structures devel-oping around the region of separation? What are the vortical structures driving the cloud cavity collapse, and what phenomenonleads to the development of an impulsive spike in hydrofoil forces?In the second part of the work, we incorporate a linear-elastic structural model to evaluate theresponse of a flexible structure in a cavitating flow. This development allows for studying highlyunsteady fluid-structure interaction dynamics, and driving the current framework closer to the11goal of studying cavitation dynamics around a rotating propeller in a variety of flow conditions.We conclude the work with the presentation of validation for LES of fully wetted and cavitatingflows past a flexible NACA66 hydrofoil. To validate the computational framework, we evaluate thefollowing key features: How well does the coupled framework capture the dynamics of the structure under the highlytransient structural loading? How well are the turbulence spectra being captured, particularly in the context of non-cavitating flows? Is the cavity collapse observed showing fidelity to experiments involving flexible hydrofoilsexperiencing the same cavitation characteristics?1.7 Organization of the ThesisThis thesis is organized into six chapters. In the current chapter, we have provided a detailedliterature review of the physics of cavitation, the mechanisms governing the onset and dynamics ofpartial cavitation, FSI studies involving cavitating fluid flows past hydrofoils and marine propellers,and the numerical strategies employed to approach these problems. In Chapter 2, we present themathematical formulation and implementation details of the coupled multi-physics solver involvinga partitioned framework. The implementation and methodology are utilized in subsequent chap-ters. In Chapter 3, we develop novel meshing criteria for conducting LES of cavitating turbulentflows and present a detailed validation against previous experimental and numerical studies. Chap-ter 4 discusses the dynamics of cavitation, elucidates the flow characteristics during sheet-cloudcavitation transition, and identifies the flow structures involved in cloud cavity collapse and sub-sequent re-stabilization of the flow. The developed coupled multi-physics solver is validated in 5,with a brief overview of the flow structures and lock-in features observed in cavitating flow. Someconcluding remarks and recommendations are presented in Chapter 6.12Chapter 2Mathematical model andcomputational frameworkIn this section, we present a three-dimensional framework for the coupled fluid-structure solverbased on the Navier-Stokes equations for fluid flow, the scalar transport model for cavitation trans-port, and the linear structural equation for the flexible body.2.1 Governing EquationsWe present the strong forms of the governing equation before establishing the variational formu-lation. We consider a physical domain occupied by the fluid (Ωf(xf , t)) confined within the fluidboundary (Γf (t)), where xf and t denote the spatial and temporal co-ordinates respectively. Thefluid within the domain is assumed to be a homogeneous mixture of liquid and vapor phases. Aphase indicator variable φ(xf , t)is employed to represent the phase fraction of the liquid phasewithin the fluid domain. The fluid density (ρf) and viscosity (µf) are treated as linear functions ofthe phase indicator variable as follows:ρf = ρlφ+ ρv (1− φ) , (2.1)µf = µlφ+ µv (1− φ) , (2.2)where ρl and ρv are the densities of the pure liquid and vapor phases, and µl and µv are thedynamic viscosities of the liquid and vapor phases respectively. In the current study we treat thevapor phase density to be constant, since isothermal phase change occurs close to the saturatedliquid conditions.2.1.1 Navier-Stokes Equations in the Deforming DomainOwing to the requirement of simulating the interaction of the fluid flow with a compliant structure,a body-fitted moving boundary-based approach is adopted. The influence of the deforming domainon the fluid kinematics is captured through formulating the equations governing the fluid flow andcavitation transport in an Arbitrary Lagrangian-Eulerian framework (ALE) [40]. The Navier-Stokes13equations in the ALE frame are:ρf∂uf∂t∣∣∣∣χ+ ρf((uf − um)· ∇)uf −∇ · σ = f f , on (xf , t) ∈ Ωf , (2.3)∂ρf∂t∣∣∣∣χ+ ρf∇ · uf +(uf − um)· ∇ρf = 0, on (xf , t) ∈ Ωf , (2.4)where χ denotes the referential coordinate system, uf = u(xf , t)is the fluid velocity at each spatiallocation xf ∈ Ωf and um is the relative velocity of the spatial coordinates xf with respect to thereferential coordinate system χ. f f is the body force applied on the fluid and σ is the stress tensor.The stress tensor σ is further decomposed asσ = σf + σsgs, (2.5)where σf and σsgs denote the Cauchy stress tensor for a Newtonian fluid and the subgrid-scaleturbulent stress tensor respectively. The Cauchy stress tensor for the Newtonian fluid is written asσf = −pfI + µf(∇uf + (∇uf)T ), (2.6)where pf denotes the fluid pressure. The subgrid-scale turbulent stresses are captured using aspatial-filtering-based LES approach. The details of the LES approach adopted are briefly discussedin Sec. 2.1.3.2.1.2 Cavitation Transport EquationThe phase indicator φ is obtained as the solution of a scalar transport equation, which in the ALEframework is:∂φ∂t∣∣∣∣χ+ φ∇ · uf +((uf − um)· ∇)φ =m˙ρl, on (xf , t) ∈ Ωf . (2.7)The source term m˙ in the transport equation models the finite mass transfer rate which governs theproduction and destruction of vapor under conditions conducive to cavitation. The phase transportequation couples to the Navier-Stokes equations through this mass transfer source term appearingin the continuity equation. By invoking Eq. 2.7 in Eq. 2.4, the continuity equation takes the form:∇ · uf =(1ρl− 1ρv)m˙ (2.8)14In the current study, we adopt the mass transfer term proposed by Schnerr and Sauer [83], whichhas a non-linear dependence on φ and pf , given by:m˙ =3ρlρvρfRB√23ρl |pf − pv|[Ccφ(1− φ) max(pf − pv, 0)+ Cvφ(1 + φnuc − φ) min(pf − pv, 0)]. (2.9)The model relates the finite mass transfer rate to the rate of growth/collapse of an equivalentspherical bubble under an external pressure field. Cavitation is assumed to initiate from nucleationsites present in the flow [10]. The concentration of nuclei per unit volume (n0) and the nucleusdiameter (dnuc) are taken to be constant parameters in the model. RB(xf , t)in Eq. 2.9 representsthe equivalent spherical radius of the vapor volume at a given spatial location. φnuc denotes thephase fraction of the initial nucleation sites in a unit volume. These are calculated as:RB =(34pin01 + φnuc − φφ)1/3and φnuc =pin0d3nuc61 +pin0d3nuc6. (2.10)To maintain numerical control over the model, the parameters Cc and Cv were introduced, andtermed as the condensation and evaporation coefficients respectively [15, 32]. The cavitation modelunder consideration has been applied extensively to the study of cavitating flow over hydrofoils[18, 45, 52].2.1.3 Modeling of Subgrid-Scale StressesFor modeling the turbulent structures in the fluid flow at high Reynolds numbers, a dynamicsubgrid-scale (SGS) model is utilized to model the subgrid-scale stresses σsgs as described in Eq.2.5. The dynamic subgrid model applies a filter to the incompressible Navier-Stokes equationsto model the scales smaller than the spatial resolution ∆. The subgrid-scale stress is defined asσsgsij = ufiufj − ufiufj where the overline indicates the filtered flow variables. The need for modelingarises as ufiufj involves the unknown SGS quantities ufi and ufj . The nonlinear SGS stress tensor[29] is expressed as:σsgsij −δij3σsgskk ≈ −2µtSij − 6CNLµ2tσsgskk(SikΩkj + SjkΩki − 2SikSkj + 23SnkSknδij), (2.11)where µt is the dynamic eddy viscosity given by µt = ρf(Cs∆)2 ∣∣S∣∣, the resolved strain-rate tensorSij =12(∂ufi∂xj+∂ufj∂xi). The assumption of local equilibrium between production and dissipation ofenergy allows for deriving the value of Cs in the Smagorinsky model [91].15In the dynamic SGS model[30], two filters are defined: the grid filter with scale dimension ∆and the test filter with scale dimension ∆̂. The first filter is dependent on the grid resolution andthe test filter, denoted by the operator (̂¯), which can be any coarser level filter. An identity relatingthe subgrid-scale stresses generated by different filters is given byLij = Tij − σsgsij , (2.12)where the Leonard tensor, Lij , is the stress generated by performing the test filter on the grid-filtered data, given byLij = ûfiufj − ûifûjf, (2.13)and the stress at test level Tij isTij = ûfiufj − ûif ûj f . (2.14)As no assumption is considered about modeling the turbulent stresses in Eqs. 2.13 and 2.14,any dynamic procedure can be adopted for computing the subgrid-scale stresses. Through usingthe Smagorinsky eddy-viscosity model for the unknown stress tensors σsgs and T, the followingrelation for the Leonard stress is developed:Lij = −2C2s(∆̂2 |̂S|Ŝij −∆2|S|Sij). (2.15)To aid the resolution of the viscous sublayer, an algebraic eddy viscosity model is employedto ensure reasonable behavior along the wall: µtµf = κy+w (1− e−y+w/A), where y+ = ywuτ/ν is thedistance to the wall in wall units based on the friction velocity uτ . κ is the model coefficient andA = 19 is a constant [6]. Further details of the implementation can be found in Jaiman et al. [42].The resolution of the turbulent stresses, in conjunction with the cavitation transport model andthe Navier-Stokes equations, allows for investigating the behavior of the fluid flow. Subsequently,we develop the evolution of the structure and briefly discuss the coupling of the fluid-structureinterface.2.1.4 Modeling of Structural DeformationIn the current study, the flexible cantilevered hydrofoil primarily experiences temporally fluctuatingpressure forces. The hydrofoil is modeled using eigenmodes obtained from solving the eigenvalueproblem derived from the linear elastic structural model. The structural domain Ωs is geometricallydefined by the co-ordinates xs, with displacements ws(xs, t). Neglecting the effects of damping and16body forces, the equations governing the linear elastic deformation of a structure are:ρs∂2ws∂t2= ∇ · σs + ρsb, (2.16)where ρs is the solid density, b is the body force experienced by the solid per unit volume, and σsis the Cauchy stress tensor of the solid defined as:σs = λstr(s) + 2µss, (2.17)where λs and µs are the Lame´ parameters and  is the infinitesimal strain tensor, which is definedass =12(∇ws + (∇ws)T). (2.18)To obtain the eigenvalue problem, we separate the spatial and temporal components of thedisplacement response as:ws(xs, t) =∞∑n=1wˆsn(xs)eiωnt, (2.19)where wˆsn(xs) denotes an eigenvector or mode shape of the structure, and ωn represents the cor-responding eigenvalue or modal frequency. Substituting Eq. 2.19 in Eq. 2.16 and neglecting bodyforces, we obtain the following representation:−ρsω2nwˆsneiωnt = ∇ · σs(wˆs)eiωnt, (2.20)which owing to linear elastic modeling can be simplified to:(L+ ρsω2nI) wˆsn = 0, (2.21)where L denotes the linear differential operator associated with Eqs. 2.16 and 2.18, completing thedefinition of the structural eigenvalue problem. The boundary conditions of the cantilevered beamarrangement are given by:ws(xs, t)|z=0 = 0, (2.22)σsns|z 6=0 = 0. (2.23)The solution to Eq. 2.21 is obtained through a finite element discretization of the structural domain,and the mode shapes and modal frequencies are obtained from the finite-element software CalculiX.The equivalent finite-element problem is briefly discussed in section 2.2.3.172.1.5 Exchange of Information at Fluid-Structure InterfaceThe fluid and structure are coupled through the imposition of continuity in velocity and tractionat their interface. Let Γfs(0) = ∂Ωf(0) ∩ ∂Ωs denote the fluid-structure interface at t = 0 andΓfs(t) = ψs(Γfs(0), t) be the interface at time t. The interface boundary conditions are given by:uf (ψs(x0s, t)) = w˙s(x0s, t), (2.24)∫ψs(γ,t)σf · nfdΓ = −∫γσs · nsdΓ, (2.25)where ψs denotes the position vector mapping the initial position xs0 of the flexible body to itsposition at time t, i.e. ψs(x0s, t) = x0s +ws(xs, t). ts is the fluid traction which relates to the fluidload as f sext(z, t) =∫Γfs tsdΓ. w˙s is the structural velocity at time t given by w˙s = ∂ψs/∂t. nf andns denote the outward normals to the fluid-structure interface and γ is any part of the interface Γfsin the reference configuration.2.2 Numerical Formulation and ImplementationIn this section, we present the framework within which the equations governing the FSI problemare numerically formulated. We start with a description of the temporal discretization adoptedfor the fluid-flow and cavitation transport equations in Sec. 2.2.1, and subsequently, we discussthe space of finite elements within which the variational statements for the governing equationsare formulated in Sec. 2.2.2. We conclude the section with a discussion of the spatio-temporalevolution of the linear structure in Sec. 2.2.3.2.2.1 Temporal DiscretizationThe generalized-α method follows a predictor-corrector approach to advance the solution in time.It has been shown to be second-order accurate and unconditionally stable for linear problems [44].Further, a single parameter ρ∞ allows the user to achieve high-frequency damping. For a givenspatio-temporally evolving set of variables Φ, the generalized-α method progresses according to thefollowing equations:Φn+1 = Φn + ∆t∂tΦn + γ∆t(∂tΦn+1 − ∂tΦn),∂tΦn+αm = ∂tΦn + αm(∂tΦn+1 − ∂tΦn), (2.26)Φn+α = Φn + α(Φn+1 −Φn) ,where ∆t is the time step size and ∂tΦn+αm is the temporal derivative of Φ at the n+αm time levelsuch that tn+αm ∈ [tn, tn+1]. αm, α and γ are generalized-α parameters based on ρ∞ and defined18as:αm =12(3− ρ∞1 + ρ∞), α =11 + ρ∞, γ =12+ αm − α. (2.27)For a non-linear PDE system G(Φ) = 0, the predictor-corrector approach solves the followinglinearized equation for ∆Φn+α:G(Φ˙n+αm ,Φn+α)+∂G(Φ˙n+αm ,Φn+α)∂Φn+α∆Φn+α = 0. (2.28)The complete time integration algorithm is presented in Alg. 1 as follows:Algorithm 1 Generalized-α method for time integrationn← 1while n ≤ ntime doCompute PredictorΦn+1 ← ΦnΦ˙n+1 ← γ−1γ Φ˙nk ← 0Φn+α(k) ← Φn + α(Φn+1 −Φn)Φ˙n+αm(k) ← Φ˙n + αm(Φ˙n+1 − Φ˙n)Non-linear iterations to construct correctorwhile k ≤ kmax doSolve Eq. 2.28 for ∆Φn+α(k)Φn+α(k) ← Φn+α(k−1) + ∆Φn+α(k)Φ˙n+α(k) ←(1− αmγ)Φ˙n(k−1) +αmαγ∆t(Φn+α(k) −Φn)if ||∆Φn+α(k) ||2 ≤ ||Φn+α(k) ||2 thenbreakend ifend whileUpdate solution using correctorΦn+1 ← Φn + 1α(Φn+α(k) −Φn)Φ˙n+1 ← Φ˙n + 1αm(Φ˙n+αm(k) − Φ˙n)end whileIn Eq. 2.28, the first term is the residual evaluated at n +αm and the derivative represents theJacobian of the system, obtained after linearization of the PDE system. Based on Eqs. 2.26 and192.28, we construct the semi-discrete form of the Navier-Stokes equations asρf∂tuf,n+αm∣∣∣χ+ ρf((uf,n+α − un+αm)· ∇)uf,n+α −∇ · σn+α − fn+α = 0, (2.29)∇ · uf,n+α −(1ρl− 1ρv)m˙ = 0, (2.30)and the cavitation transport equation as∂tφf,n+αm +((uf,n+α − un+αm)· ∇)φf,n+α + sφf,n+α = 0, (2.31)where the reaction coefficient s for the Schnerr-Sauer model is given by:s = − 3RB√23ρl |pf − pv|[Cc(1− φf) max(pf − pv, 0)+ Cv(1 + φnuc − φf) min(pf − pv, 0)].(2.32)2.2.2 Spatial Discretization and Variational StatementFor the spatial discretization of the flow and cavitation equations, we use a stabilized Petrov-Galerkin finite element framework. The domain Ωf is discretized into N elements such that Ωf =∪Ne=1Ωfe, and ∅ = ∩Ne=1Ωfe. For the finite element framework, we allow Sh to be the space of trialsolutions, which satisfy the Dirichlet boundary conditions, and Vh to be the space of test functionswhich vanish on the Dirichlet boundaries. The stabilized variational form for the Navier-Stokesequations has been discussed extensively in the works of Jaiman et al. [42], Joshi and Jaiman[48, 49], Kashyap and Jaiman [51], and is discussed here briefly. The variational statement for thefluid flow equations involves finding [un+αh , pn+αh ] ∈ Sh, such that ∀[Ψh, qh] ∈ Vh,∫ΩfΨh ·(ρf∂tuf,n+αmh∣∣∣χ+ ρf((uf,n+αh − un+αm)· ∇)uf,n+αh)dΩ+∫Ωf∇Ψh : σn+αh dΩ +∫Ωf(tn+1)qh(∇ · uf,n+αh)dΩ+N∑e=1∫Ωfτmρf(ρf((uf,n+αh − un+αm)· ∇)Ψh +∇qh)·RmdΩe +N∑e=1∫Ωe∇ ·ΨhτcρfRcdΩe (2.33)−N∑e=1∫ΩeτmΨh · ((Rm · ∇)un+αh )dΩe −N∑e=1∫Ωe∇Ψhρ(φ): (τmRm ⊗ τmRm)dΩe=∫ΩfΨh · f f,n+α dΩ +∫ΓfNΨh · hf,n+α dΓ +∫Ωfqh(1ρl− 1ρv)m˙f,n+α dΩ,where the first and second lines contain the standard Galerkin finite element terms of the momentumand continuity equations. The third line contains the Galerkin Least Squares stabilization terms20for the momentum and mass continuity equations. The fourth line contains stabilization termsbased on the multi-scale argument [38, 39]. The fifth line contains the Galerkin terms for the bodyforce and the Neumann boundary in the momentum equation. The element-wise residuals of themomentum and continuity equations are denoted by Rm and Rc respectively, and are given byRm(uf , pf) = ρf∂tun+αmh + ρf(uf,n+α − un+αm)· ∇uf,n+αh −∇ · σf,n+αh − f f,n+αh , (2.34)Rc(uf , pf , φf) = ∇ · uf,n+αh −(1ρl− 1ρv)m˙f,n+α. (2.35)Furthermore, τm and τc are stabilization parameters [11, 28, 97] defined asτm =[(2∆t)2+ uh ·Guh + CI(µ(φ)ρ(φ))2G : G]−1/2, (2.36)τc =1tr(G)τm, (2.37)where CI is a constant derived from the element-wise inverse estimate [35] and tr(G) denotes thetrace of the element contravariant metric tensor G, which is defined asG =∂ξT∂xf∂ξ∂xf, (2.38)where xf and ξ are the physical and parametric co-ordinates respectively. The stabilization in thevariational form provides stability to the velocity field in the convection-dominated regions of thefluid domain and circumvents the Babusˇka-Brezzi condition which is required to satisfy any mixedGalerkin method [47].In a similar manner, the variational statement for the cavitation transport equation involvesfinding φn+αh ∈ Sh such that ∀wh ∈ Vh,∫Ωfwh(∂tφn+αmh +((uf,n+α − um,n+α)· ∇)φn+αh + sφn+αh)dΩf+N∑e=1∫Ωf,e((uf,n+α − um)· ∇wh)Rφ+N∑e=1∫Ωf,eζ|Rφ||∇φn+αh |kadds ∇wh ·((uf,n+α − um)⊗ (uf,n+α − um)| (uf,n+α − um) |2)· ∇φn+αh dΩf,e (2.39)+N∑e=1∫Ωf,eζ|Rφ||∇φn+αh |kaddc ∇wh ·(I−(uf,n+α − um)⊗ (uf,n+α − um)| (uf,n+α − um) |2)· ∇φf,n+αh dΩf,e= 0,where the first line represents the standard Galerkin finite element terms. The second line consists21of linear stabilization terms with the stabilization parameter τφ given by [89]τφ =[(2∆t)2+(uf − um)·G(uf − um)+ s2]−1/2. (2.40)Rφ is the residual of the phase indicator transport equation given byRφ = ∂tφf,n+αh +((uf,n+α − um)· ∇)φf,n+αh + sφf,n+αh . (2.41)The phase indicator variable φh, takes values between [0, 1], and is associated with high gradients inregions of phase transition. Consequently, additional treatment of the variational form is essentialto address oscillations in the solution near regions of high gradients [50]. The positivity-preservingproperty is essential to prevent the introduction of numerical instability through non-physical in-terpolations of density and viscosity. Additional non-linear stabilization terms are added to impartthe positivity property to the underlying element level matrix. The third and fourth lines of Eq.2.39 contain the positivity preserving nonlinear stabilization terms in the streamwise and crosswinddirections respectively [50]. The PPV parameters ζ,kadds and ,kaddc for the phase indicator transportequation are obtained as:ζ =2|s|h+ 2| (uf − um) | , (2.42)kadds = max{ || (uf − um) | − τφ| (uf − um) |s|h2− τφ|(uf − um)|2 + sh26, 0}, (2.43)kaddc = max{ | (uf − um) |h2+sh26, 0}, (2.44)where | (uf − um) | is the magnitude of the convection velocity and h is the characteristic elementlength [48].2.2.3 Modal Decomposition and Spatio-Temporal Discretization of the LinearStructureWe define the discrete solution to the linear-elastic problem as wsh ∈ Shs , where Shs is the spaceof trial solutions for the structure and satisfy the Dirichlet boundary conditions, and Vhs is thespace of test functions for the structure which vanish on the Dirichlet boundaries. The variationalstatement involves finding wsh ∈ Shs , such that ∀ Nhs ∈ Vhs :∫ΩsNsh∂2wsh∂t2dΩs +∫Ωs∇Nsh : σshdΩs = 0. (2.45)22Using the decomposition indicated in Eq. 2.19 and simplifying we can show the discrete eigenvalueproblem takes the form: (Ksh − ω2nMsh)wˆsnh = 0, (2.46)where Msh and Ksh are the mass and stiffness matrices obtained from the finite element discretiza-tion, and wˆsnh is the discrete mode shape. The modal frequencies ωn and mode shapes wˆsnh of thestructure are computed in the pre-processing step. In the current study, we consider Nm modeswhose effective modal mass cumulatively constitutes 90% of the total structural mass.From Eq. 2.16, we can consider the equation for structural motion to take the form:Msw¨s + Ksws = f sext, (2.47)where Ms and Ks are the mass and stiffness matrices per unit length respectively. w¨s denotesthe acceleration of the structure, and f sext is the external force vector per unit length. The forcevector f sext is evaluated by computing the hydrodynamic forces generated on the structure as aconsequence of the fluid pressure and viscous stresses. As the structure is assumed to be linear,its motion is assumed to be a linear combination of the vibration modes. Therefore, the modaldecomposition of the structure at the discrete level can be expressed aswsh =Nm∑i=1ξsi wˆsih, (2.48)where wˆih is the mode shape, Nm is the number of modes and ξsi is the modal displacement of theith mode. The transformation given in Eq. 2.48 allows for decomposing the equation of structuralmotion into Nm ordinary differential equations in time given by:Mξsξ¨s+ Kξsξs = Fξsext. (2.49)Here, Mξs = WˆThMsWˆ and Kξs = WˆThKsWˆh denote the projected mass and stiffness matrices inthe eigenspace respectively. Wˆh is a matrix whose ith column is the ith mode shape wˆih. The forcevector is similarly projected into the eigenspace as Fξsext = WˆTh fsext. Using the flow field(uf , p)at time level n, and the predictor at time level n + 1, the force vectors are computed. The systemof decoupled ODE’s thus obtained is advanced in time using the Newmark-β method, where themodal velocities and displacements are updated as:ξ˙s,n+1= ξ˙s,n+((1− γ) ξ¨s,n + γξ¨s,n+1)∆t, (2.50)ξs,n+1 = ξs,n + ξ˙s,n∆t+∆t22((1− 2β) ξ¨s,n + 2βξ¨s,n+1). (2.51)Here, γ and β are parameters governing the relative bias of integration based on Cauchy’s mean23value theorem [74]. In the current case, we set (γ, β) = (0.5, 0.25), which correspond to uncon-ditional stability of the method. Upon integration of the system of ODEs over a time-step, thestructural displacement ws,n+1h is computed using Eq. 2.48.2.2.4 Unified ALE FSI frameworkThe coupling between the fluid and structure is carried out through a partitioned iterative ap-proach using the Nonlinear Iterative Force Correction (NIFC) scheme described in [42, 43]. Thedisplacements obtained through solving the structural equations have to be incorporated into thefluid-flow framework, which is accomplished by controlling the motion of each node in the meshwhile maintaining the kinematic consistency of the discretized interface.To advance the solution from tn to tn+1, a sequence of predictor-corrector steps are taken, withthe dual objectives of enforcing convergence of the individual numerical methods and coupling themultiphysics phenomena. Using the flow information(uf(xf , tn), p), the structural displacement iscomputed [A]. Next, ALE compatibility and velocity continuity at the fluid-structure interface Γfs isexplicitly maintained by equating the computed structural displacements to the mesh displacementon the wetted boundaries. The fluid mesh is represented using a hyperelastic solid model, and usinga Lagrangian finite element framework allows for constructing the equations to adapt the fluid meshto the deformed geometry of the structure. Upon deformation of the mesh, the interfacial velocityis the same as the mesh velocity, and is imposed on the fluid flow at time tn+α [B]. These conditionscan be expressed as:dn+1m = ws,n+1 on Γfs, (2.52)uf,n+α = un+αm on Γfs. (2.53)In Eqs. 2.52 and 2.53, dn+1m denotes the mesh displacement at time tn+1 and un+αm represents themesh velocity at time tn+α, and is defined as:un+αm =dn+1m − dnm∆ton Γfs. (2.54)Once the fluid mesh and structure satisfy the compatibility constraints, the fluid-flow equationsare solved first [C], followed by the cavitation transport equation [D] to a reasonable level of con-vergence. This allows for the computation of the forces acting on the structure, and the Non-LinearIterative Force Correction further filters the forces to provide stability to the overall partitionedfluid-structure coupling. The NIFC scheme acts to construct the cross-coupling effect of strongfluid-structure interaction along the interface without forming the off-diagonal Jacobian term in anexplicit manner [42, 43]. Algorithm 2 presents the staggered solution process employed to solve the24FSI systems. The sequence of equations [A], [B], [C], [D] are presented in matrix-vector form as :[A] : [Ksξ] {∆ξ} = {Rξ}, (2.55)[B] : [Ksd] {dn+1} = {0}, (2.56)[C] :[KΩf GΩf−GTΩf CΩf]{∆uf,n+α∆pf,n+1}={RmRc}, (2.57)[D] : [Kφ] {∆φf,n+α} = {Rφ} (2.58)In this numerical study, the linearized systems for incremental pressure, velocity and cavitationfraction arising from the finite element discretization are solved using the Generalized MinimumRESidual (GMRES) [82] with a diagonal preconditioner. For all computations presented, a Krylovspace of 25 orthonormal vectors has been used. In the current implementation, only the blockmatrix is explicitly constructed, and the residual for the linear system is constructed throughmatrix-vector products. The mesh motion is solved for using a conjugate gradient method owingto the symmetric structure of the left-hand side matrix. For the flow and cavitation fields, equal-order interpolations are considered, with linearly interpolated hexahedral and prism elements beingused at the required precision. The solver is written in a combination of C and Fortran languagesfor scalability to high-performance computing. The solver uses communication protocols based onstandard message passing interface [1] for parallel computing on distributed memory clusters.2.3 SummaryWe have presented a partitioned framework for the numerical modeling of fluid-structure-cavitationinteraction. The flexible structure is modeled using a linear-elastic structural model, which isfurther decomposed into dynamical modes. An eigenvalue problem is numerically solved to extractthe modal stiffnesses and mode shapes, whose time-varying coefficients are determined throughthe solution of decoupled ODEs where the fluid-forces are projected onto the dominant structuralmodes. A homogeneous mixture assumption represents the liquid-vapor phases in the domain usinga continuous phase-fraction indicator variable. The cavitation transport is modeled using a scalartransport equation involving a non-linear mass-transfer term, with appropriate stabilization appliedduring linearization. The salient features of the presented formulation are: A partitioned coupling algorithm is deployed allowing for modularity and integration of thestructural ODE solvers into the existing flow and cavitation solvers. The structure is decomposed into its constituent modes during the pre-processing step, andis represented by a set of lower-order modes. This allows for savings in computational costassociated with solving the full-order structural equations and minimizes force-displacementtransfer errors associated with non-matching meshes.25 The framework is scalable to high performance computing.The features of this framework are evaluated through an extensive validation study with respectto numerical and experimental studies. The efficacy of the LES framework for rigid structures iscovered in the next chapter.26Algorithm 2 Partitioned coupling of structural deformation, Navier-Stokes and cavitation solversInput: uf,0, pf,0, φf,0,ξs,0for itime ← 0 to ntime dofor k ← 0 to convergence/kmax doPredict solution:[uf,n+1(0) pf,n+1(0) φf,n+1(0)]←[uf,n pf,n φf,n]Interpolate density and viscosity fields: ρf(φf,n+1(0) ), µf(φf,n+1(0) )1. Project forces onto modal spaceFξsext = WˆTh fsext2. Advance system of ODEsξ˙s,n+1 ← ξ˙s,n + fξ˙(ξ¨s,n, ξ¨s,n+1)∆tξs,n+1 ← ξs,n + fξ(ξ˙s,n, ξ¨s,n, ξ¨s,n+1)∆t3. Update structural displacementsws,n+1h =∑Nmi=1 ξsi wˆsih[A] Structural Solver1. Solvedn+1m in Eq. (2.56)2. Update mesh velocityd˙n+αm in Eq. (2.54)[B] Mesh Deformation[C] Navier-Stokes Solver1. Interpolate solution2. Solve3. Correct solution4. Update solutionuf,n+α(k+1) ← uf,n + α(uf,n+1(k) − uf,n)pf,n+1(k+1) ← pf,n+1(k)∆uf,n+α and ∆pf,n+1 in Eq. (2.57)uf,n+α(k+1) ← uf,n+α(k+1) + ∆uf,n+αpf,n+1(k+1) ← pf,n+1(k+1) + ∆pf,n+1uf,n+1(k+1) ← uf,n +1α(uf,n+α(k+1) − uf,n)pf,n+1(k+1) ← pf,n+1(k+1) + ∆pf,n+1[D] Cavitation Solver1. Interpolate solution2. Solve3. Correct solution4. Update solutionφf,n+α(k+1) ← φf,n + α(φf,n+1(k) − φf,n)∆φf,n+α in Eq. (2.58)φf,n+α(k+1) ← φf,n+α(k+1) + ∆φf,n+αφf,n+1(k+1) ← φf,n +1α(φf,n+α(k+1) − φf,n)un+1(k+1), pn+1(k+1)Non-lineariterationsφn+1(k+1),ρn+1k+1 , µn+1k+11. Compute Interface forcef s,n+1ext = F(un+1(k+1), pn+1(k+1))2. Apply NIFC [42]f s,n+1ext = fs,0ext + γk+1(f s,n+1ext − f s,next)[E] FSI Couplingend doend do27Chapter 3Validation of LES for Cavitating Flowover Rigid NACA66 HydrofoilIn this chapter, we present the validation of the cavitating flow past a rigid NACA66 hydrofoilvalidated against the experimental studies of Leroux et al. [65] and the large eddy simulationspresented in Chen et al. [18]. The principle objective of the current validation study is to establishthe mesh characteristics necessary to capture the cavity-fluid flow interactions in both sheet andcloud cavitating regimes. Accordingly, we begin by describing the problem of cavitating flow pasta rigid hydrofoil, and establish a meshing criteria based on the re-entrant jet momentum. Subse-quently, we evaluate the quality of LES based on a mono-phase criteria, and present the validationstudy and indicate key parameters of interest in the cavitation cycle. For the validation study, wecompare the pressure at different locations on the hydrofoil with experimental observations and thetime varying force coefficients relative to those reported in previous LES studies. To conclude, wepresent the turbulence spectra observed over the hydrofoil surface, and near wake of the hydrofoil,wherein we establish agreement with turbulence spectra observed in bubbly multiphase flows.3.1 Problem Description and Mesh CharacteristicsWe validate the cavitating flow past a rigid NACA66 hydrofoil at a chord-based Reynolds numberRe = 7.50× 105, cavitation number σ = 1.25 for a hydrofoil oriented at an angle of attack α = 6◦.We define the cavitation number as:σ =2 (p∞ − pv)ρlU2∞, (3.1)where pv denotes the vapor pressure, and p∞ is the freestream pressure, which is taken to be theaverage pressure observed at the domain outflow. For all the cases considered in the current study,we set ρl = 998.1 kg m−3, ρv = 0.023 kg m−3, µl = 0.0011 kg m−1 s−1, µv = 9.95×10−6 kg m−1 s−1for the liquid and vapor phase properties. For the cavitation model, the parameters are Cc = 10−3,Cv = 5 × 10−3, dnuc = 2.5 µm, n0 = 1013. The freestream velocity U∞ is calculated based on theliquid phase density and viscosity, in conjunction with the chord-based Reynolds number defined28ZXY1.28CInletWall OutletSymmetric Boundary2C4.7C0.3CWallFigure 3.1: A schematic of the flow past the NACA66 hydrofoil at Re = 750000. The computationalsetup and boundary conditions are shown for the filtered Navier–Stokes equations. Here, u =(u, v, w) denotes the components of the fluid velocity and C denotes the chord length.asRe =ρlU∞Cµl, (3.2)where C denotes the hydrofoil’s chord length and is taken to be 0.15 m in the current study.At these conditions, the non-cavitating flow experiences a laminar to turbulent transition overthe hydrofoil surface. Under cavitating conditions, the flow experiences highly unsteady cavitation,with periods of rapid cavity growth, cavity separation and complete collapse, owing to a combinationof a re-entrant jet instability and interfacial instabilities observed on the upper surface of the cavity.Upon collapse, an intricate vortex-cavity field develops over the suction side of the hydrofoil withvortex-cavities convecting into the wake. Experimental studies have observed clusters of bubblesover the hydrofoil, which mixture theory approximates as clouds of vapor. [18, 45, 52, 100]3.1.1 Domain definitionTo investigate the flow as described above, we define the hydrofoil by coordinates prescribed inLeroux et al. [65]. We denote the hydrofoil’s chord length by C and truncate the hydrofoil 0.02Cfrom the trailing edge. Fig. 3.1 illustrates the computational domain used for conducting thenumerical validation. The inlet and outlet boundaries are positioned 2C and 4.7C upstream anddownstream of the leading edge (LE) respectively. The channel’s width is set to be 1.28C bythe channel height prescribed in Leroux et al. [65]. The span S of the hydrofoil is taken to be0.3C, and symmetric boundary conditions are imposed along the span-wise domain limits. This iscommensurate with the largest span-wise coherent structure observed in previous numerical works[18, 45, 52]. A uniform flow is imposed at the inlet, and the freestream velocity is set as the initialcondition for the non-cavitating flow. The non-cavitating flow is allowed to fully develop up to the29non-dimensional time t∗ = tU∞/C = 18, after which the flow is allowed to enter the cavitatingregime. A uniform time-step of ∆t∗ = ∆tU∞/C = 9.434×10−3 is used for the rigid hydrofoil study.(a) (b)(c) (d)Figure 3.2: Cross-sectional view of mesh M2 at (a) boundary layer and near wake, (b) close to thetrailing edge. (c,d) Resolution of the hydrofoil span at the leading and trailing edges3.1.2 Meshing CriteriaTo accurately capture cavity inception and evolution of re-entrant jet dynamics in the laminar sub-layer, we set y+ < 1 along the hydrofoil surface. In the case of cavitating flow, the fluctuation in y+can be substantial owing to regions covered in the vapor phase. Under unsteady partial cavitationconditions, the re-entrant jet velocity has been observed to be nearly 50% of the freestream velocity,with a thickness smaller than 30% of the cavity thickness [12]. As a consequence, the re-entrantjet is associated with high-velocity gradients normal to the hydrofoil, and the mesh resolution canbe established based on estimates of re-entrant jet momentum. To get a preliminary estimate formesh size, we use the data obtained from Detached Eddy Simulations of 2-D cavitating flow byKashyap and Jaiman [51] to estimate the re-entrant jet momentum for a rigid cavitating hydrofoil30and evaluate the magnitude of wall shear stress τw as follows:Tv = µl(∇uf +(∇uf)T)n, (3.3)Tv‖ = Tv − (Tv · n) n, (3.4)τw = ||Tv‖||2, (3.5)where Tv represents the traction vector due to viscous stresses on the hydrofoil surface, and Tv‖is the shearing component of traction arising due to viscous stresses. We use the wall-shear stressto compute the friction-velocity uτ in the re-entrant jet region as follows:uτ =(τwρl) 12. (3.6)Property M1 M2 M3y+ 0.6 0.3 0.15(∆x+,∆z+) (65, 75) (45, 60) (35, 45)(Nchord, Nspan) (200, 90) (300, 120) (400, 150)(NBL, rBL) (51, 1.136) (66, 1.103) (81, 1.096)Nnodes 3.59M 8.03M 15.25MN elem. 3.68M 8.18M 15.51MTable 3.1: Mesh parameters for the LES validation of flow past NACA66 hydrofoil. Here Nchordand Nspan denote the number of points in the chord-wise and span-wise directions on the hydrofoilsurface. NBL denotes the number of points in the boundary layer and rBL denotes the growth ratioof element sizes normal to the boundary. Nnodes and N elem. denote the total number of nodes andelements in the mesh respectively. For the current study, only 8 node tri-linear hexahedral elementsare used.This allows us to evaluate the wall-normal distance y, based on a target y+ value as y+ = uτy/ν.This method of estimation has been used to construct three successively refined meshes, labeledM1 (coarsest), M2, and M3 (finest). Further, LES requires the mesh to be isotropic in the wakeregion, to homogeneously resolve the turbulent structures. We refine the mesh by resolving thehydrofoil along chord-wise (∆x+) and span-wise (∆z+) directions, increasing the number of pointsin the boundary layer and wake regions following the guidelines prescribed by Spalart [95]. Themeshes are constructed in Gmsh [31], and the parameters associated with the meshing process arepresented in table 3.1. Fig. 3.2 presents the resolution offered by mesh M2 around the suction sideand wake of the hydrofoil.313.1.3 LES Quality EvaluationOwing to the filtering of governing equations based on the spatial resolution offered by the mesh,LES does not lend itself to a mesh convergence analysis. Theoretically, LES on a sufficiently refinedmesh can resolve flow patterns down to the Kolmogorov length scale. However, taking into accountthe computational expense involved, it is necessary to identify the coarsest level of flow resolutionacceptable for the problem under consideration. Along with the preliminary criteria established formeshing, LES quality has to be satisfactory on the constructed mesh. Some of the criteria proposedto evaluate LES quality are (a)LES IQη based on the Kolmogorov length scale η [16], (b)LES IQνbased on the turbulent eddy viscosity νt [16] and (c) turbulent kinetic energy resolution [77]. Forthe current validation study, we adopt the turbulent eddy viscosity criteria as defined by Celiket al. [16] as:LES IQν =11 + αν(〈νt,eff 〉ν)n . (3.7)αν and n are parameters governing the Index of Quality (IQ) derived based on the spatial resolutionof LES relative to DNS, and are set to αν = 0.526 and n = 0.52. 〈νt,eff 〉 is the time-averaged eddyviscosity, and ν is the kinematic viscosity. For LES to be considered adequate, we require LES IQνto be greater than 0.75 in regions of interest. In this regard, we consider the wall-normal distanceof 0.2C over the hydrofoil’s suction side, and the immediate wake (x/C < 2.0) downstream of thehydrofoil to be our regions of interest. We evaluate LES IQν , with averaging conducted overtwo cavitation cycles (discussed in section 3.2). We compare the contours of LES IQν in Figs.3.3(a-c), for each of the meshes considered, with Fig. 3.3d indicating the comparison at x/C = 1.We observe mesh M2 offers adequate resolution for verification purposes, and the correspondingmeshing parameters will be employed for further validation studies.3.2 Hydrodynamic Coefficients and Cavitation CharacteristicsThe hydrodynamic parameters obtained from the LES of cavitating flow past the NACA66 hydrofoilat Re = 7.5×105 are compared against experimental and numerical results in table 3.2. We evaluatethe fluid loads on the hydrofoil by integrating the surface traction over the first layer of elementsin contact with the hydrofoil. In this context, we compute the instantaneous coefficients of lift anddrag usingCL =2ρlU2∞CS∫Γ(n · σ)nydΓ, (3.8)CD =2ρlU2∞CS∫Γ(n · σ)nxdΓ, (3.9)32(a) (b)(c) (d)Figure 3.3: Index of quality for LES in the region of interest around the hydrofoil for three meshes(a) M1, (b) M2, (c) M3, and (d) Comparison of LES IQν at (x, z) = (C, 0.15C).where n is the unit normal to the hydrofoil surface, where nx and ny are its Cartesian components.σ is the fluid stress tensor and Γ is the surface boundary of the hydrofoil. The pressure coefficientat a location in the domain is computed usingCp =2 (p− p∞)ρlU2∞, (3.10)where p is the fluid pressure at the point of interest and p∞ is the far-field pressure. Owing tothe multiphase nature of the flow, it is essential to characterize the cavity growth-collapse cycleand co-relate it with the pressure fluctuations observed close to the cavitating regions around thehydrofoil. In this regard, we integrate the cavitation fraction over the flow domain using∫Ωf(1− φh)dΩ, (3.11)to obtain the instantaneous global vapor volume. To evaluate the spectral characteristics in thecurrent study, we define the non-dimensional frequency f∗, and the chord-based Strouhal number33St asf∗ =fCU∞, St =f cavCU∞, (3.12)where f denotes a general frequency observed during the analysis and f cav is the dominant cavityshedding frequency.Fig. 3.4 presents the comparison of pressure at different chord-wise locations on the hydrofoilwith experimental data of Leroux et al. [65]. We observe the emergence of minor pressure spikesat x/C = 0.5 and x/C = 0.7 occurring in the interval t∗ ∈ [27.5, 28.5], which co-relate with theLeroux et al. (exp.)LES (Curr.)Figure 3.4: Comparison of pressure at different chord-wise locations along the hydrofoil.34Figure 3.5: Comparison of pressure frequency spectrum at chord-wise locations x/C ∈ {0.3, 0.5, 0.7}with the dominant frequency modes observed in the studies of Chen et al. [18], Kjeldsen et al.[57], Leroux et al. [65].experimental observations of a re-entrant jet’s presence. Subsequently, we observe the emergenceof a sharp pressure spike at t∗ ≈ 29, which is associated with the sheet cavity cleaving, and theresultant cloud cavity reaching the hydrofoil’s trailing edge. While the current LES simulationsaccurately capture the instantaneous pressure development over the cavity cycle well, the pressurespikes are over-predicted relative to experimental studies. For t∗ ∈ [29.5, 31.5], the cloud cavityexperiences a sequence of collapse events leading to the local pressure spikes observed in the pressurespectra. These events are discussed further in Sec. 4.5. At t∗ = 32.5, a complete collapse of thecloud cavitation occurs, and the hydrofoil enters a new cycle of cavitation.For the periodicity of pressure oscillations, we observe the frequency spectra of the wall pressureat the chord-wise locations x/C ∈ {0.3, 0.5, 0.7} in Fig. 3.5. In the current study, we observethe dominant frequency of the pressure oscillation to be f cav = 5.33 Hz, which is equivalent toSt = 0.15. This result is in agreement with the experimental studies of Kjeldsen et al. [57], whereinthe authors observed St = 0.15 for cavitation cycles of a NACA0015 hydrofoil. While Leroux et al.[65] observed St ≈ 0.11 for the cavitation cycles of a NACA66 hydrofoil, they indicate an agreementof their results with the earlier work of Kjeldsen et al. [57]. This difference in the frequency responseof the pressure spikes can be used to explain the offset observed in the pressure evolution time seriesrelative to experimental data.35LES - Current LES - (Chen et. al.)Figure 3.6: Comparison of cavity volume and force coefficients computed in the current study withthe LES studies of Chen et al. [18].As illustrated by Callenaere et al. [12], the extent of confinement alters the cavity’s periodicitywith respect to growth, shedding, and collapse. In this regard, the nature of boundary conditionsimposed in a numerical study can alter the periodicity observed in cavitation dynamics, leading toa shorter cycle when the lateral boundaries are no-slip walls. In the case of symmetric boundaryconditions at the lateral boundaries, reasonable agreement with St = 0.11 is observed [45, 52],whereas the use of no-slip walls at the lateral boundaries indicates agreement with St = 0.15[18]. The effects of confinement and their role in influencing cavitation dynamics require furtherinvestigation and the analysis is beyond the scope of the current work.Further, in Fig. 3.6 we compare the time-series of cavity volume, the lift and drag coefficients36CL CrmsL CD CrmsD StCurrent 0.950 0.324 0.072 0.046 0.15Num.-4[18] 0.953 0.311 0.074 0.041 0.168Table 3.2: Summary of the hydrodynamic force coefficients, and the corresponding Strouhal numberobserved during LES of cavitating flow.over two cavitation cycles with the LES data of Chen et al. [18].The mean and rms lift (CL, CrmsL )and drag (CD, CrmsD ) coefficients are summarized along with the Strouhal number (St) in table 3.2.We observe the current study shows good agreement with the observed lift and drag coefficientsin previous numerical studies [18]. We further investigate the dynamics of cavity growth andbreakdown in Chapter 4. To conclude the validation with experiments, we present a comparisonof the cavitation structures observed during a cavitation cycle between the current study and thework of Leroux et al. [65] in Fig. 3.7.3.3 Energy spectra surrounding the hydrofoilIn this section, we assess the turbulence power spectra at the points illustrated in Fig. 3.8 tofurther validate the mesh resolution offered by mesh M2. For locally homogeneous turbulence,Kolmogorov’s −5/3 power law relates the turbulent kinetic energy (E ≡ u′iu′i/2) to the wavenumber of integral scale (κ) eddies as E(κ) ∝ κ−5/3 in the inertial range. Taylor [96] in their frozenturbulence hypothesis indicated −5/3 decay of the kinetic energy versus frequency is a reasonableindicator of the turbulence spectrum. While anisotropic and highly unsteady mono-phase turbulentflows show good agreement with Kolmogorov’s turbulent kinetic energy decay law, multi-phaseflows have been observed to deviate significantly. Lance and Bataille [61] studied the turbulenceexhibited by the liquid phase in a turbulent bubbly flow field and observed the high-frequency decayto observe a −8/3 power law. Further works investigating the turbulence decay characteristics ofspectra [70, 78] have depicted agreement with the deviation of turbulence in multi-phase flows fromthe −5/3 power law.In the current study, we expect the integral scale eddies to lie in the frequency band of 0.1 ≤fC/U∞ ≤ 10. In this range, Fig. 3.9b indicates the spectrum of turbulent kinetic energy decay tobe approximately parallel to a −3 slope in the high-frequency region, and parallel to a −5/3 slopein the mid-range frequency region. Furthermore, we observe a variation in the decay spectra withthe spatial location at which turbulent characteristics are being monitored. The non-local effectsof phase transitions and their contributions to turbulence are observed in the form of TKE spectra,where the decay is observed to deviate beyond the −3 slope. From these observations, and theirvalidation with the studies of turbulence spectra in multi-phase flows, we can confirm the currentSGS turbulence model is adequate for capturing cavitating flows at high Reynolds numbers.37Figure 3.7: Evolution of the cavity and vorticity over a single cycle. The first column illustratesthe experimental results obtained by Leroux et al. [65]. The second column depicts the cavitydeveloped in the current numerical framework. The third column shows the contours of Q-criterion(Q∗ = 8) colored by Z-vorticity.38XYFigure 3.8: Arrangement of probes to assess turbulent flow characteristics: Probes P1-P3 are placedat a distance of 0.02C from the hydrofoil, at x/C ∈ {0.2, 0.5, 1.0} respectively. Probes P4-P5 areplaced downstream of the trailing edge at x/C ∈ {1.25, 1.5}. The origin is located at the leadingedge of the hydrofoil, with the angle of attack created by rotating the hydrofoil about mid-chordpoint. Contour lines of Z-vorticity ω∗z = ωzC/U∞ ∈ [−5,−2] ∪ [2, 5] are presented to indicateunsteady flow-structures in the vicinity of the hydrofoil.3.4 SummaryIn this chapter, we have presented a numerical validation study of Large Eddy Simulations (LES) ofcavitating flow past a rigid NACA66 hydrofoil. The current numerical studies have been observedto accurately capture the key characteristics of cavitating flows over hydrofoils based on certainresolution criteria demanded by LES. We summarise the results of the validation studies as follows: We begin by establishing mesh resolution criteria based on the shear stress generated as a con-sequence of the re-entrant jet developing on the hydrofoil surface under cavitating conditions.We estimate this shear-stress distribution through Detached Eddy Simulations of cavitatingflow over a two-dimensional NACA66 hydrofoil. Based on this estimate, we construct thefriction velocity and use it in conjunction with y+ criteria to generate different boundarylayer resolutions. In the near-wake of the hydrofoil, an isotropic resolution corresponding to(∆x+,∆y+,∆z+) = (45, 45, 60) is adopted up to a downstream location of 1.5C. Based on the constructed meshes, we evaluate the eddy-viscosity-based LES quality criteriain regions of interest in the hydrofoil’s wake. Subsequently, we pick the mesh which indicatesa quality index exceeding 0.75. However this criteria has been established in the context ofmono-phase flows, and a criteria needs to be developed for multiphase flows. To further vali-date the adequacy of resolution for LES, we compare the turbulence spectra with experimentalobservations of bubbly-multiphase flows. We compare pressure fluctuations on the hydrofoil surface, and observe good agreement in theamplitude and frequency content of suction-side pressure evolution. With regard to previousnumerical studies, we observe good agreement with the magnitude and dominant frequenciescorresponding to the force coefficients. The identification of these dominant frequencies is important in the analysis of the cavitation39(a) (b)Figure 3.9: FFT Spectra of (a) pressure fluctuations and (b)turbulent kinetic energy at locations(see Fig. 3.8) in the cavitating region and near the wake of the hydrofoil. For the sake of clarity,the spectra of points from the same group are shifted, and the magnitude of the shift is indicatedin the legend.cycle, and to elucidate mechanisms governing sheet-to-cloud transition, growth of vortexcavities, and complete breakdown of the cloud cavitation. These features are analyzed anddiscussed in the next chapter.40Chapter 4Cavitating Flow over Rigid NACA66HydrofoilIn the current chapter, we evaluate cavitating flow over a rigid NACA66 hydrofoil at a cavitationnumber σ = 1.25. We begin by discussing the effects induced by cavitation on the fluid flowthrough the vorticity transport equation and briefly elucidate the characteristics of sheet and cloudcavitating flow. Over a cycle of cavitation, we observe the following sequence of events: (i) inception(4.2) (ii) sheet cavity growth (4.3) (iii) destabilization of sheet cavity (4.4) (iv) cloud cavity evolution(4.5) (vi) re-laminarization (4.6). We discuss the dominant mechanisms of vorticity production andtransport in each of these stages during the cycle and propose a mechanism driving the transitionfrom sheet to cloud cavitation.4.1 Flow Features Induced by CavitationSheet cavitating flow is characterized by the presence of a large pocket of vapor, across whoseinterface a large density gradient exists ( ρlρv ∼ 105). Upon misalignment with the pressure gradient,a baroclinic torque develops on the fluid elements in the cavity’s vicinity, leading to an enhancementin the flow’s vorticity. The added vorticity can potentially destabilize the flow-field around thecavity leading to the breakdown and separation of the cavity. Furthermore, the cavity’s growth isassociated with a rapid vaporization of the liquid phase at the cavity’s leading edge, which createslocal compressibility effects in the surrounding fluid, leading to the dilatation of vorticity. Thedilatation effects at the leading edge can induce fluctuations on the cavity surface, which can travelalong the interface and create fluctuations in the baroclinic torque.Through Fig. 4.1, we delineate the phases of cavitation development over a cycle. All the flowvariables analyzed in this chapter have been non-dimensionalized based on the physical parametersdefined in Sec. 3.1. To aid the analysis of the cycle, we assume the cavitation cycle begins at thenon-dimensional time tcav0 , and the cavity cycle’s non-dimensional period to be Tcav = 1/St = 5.333.Based on this we define a fractional cavitation cycle time astcavf =t∗ − tcav0T cav. (4.1)To understand the effects of cavitation on fluid flow, and the interaction of cavitation with41Figure 4.1: Variation in CL and CD over one cycle of cavitation illustrating the duration of sheet-cavity growth, cloud-cavity separation and collapse, residual cavity growth and collapse. The pointsof change in flow behavior are labeled as (A) Instantiation of the re-entrant jet, (B) Separation ofcloud cavity, (C) Shedding of Trailing Edge Vortex Cavities. C1 and C2 are vortex cavities that areshed from the trailing edge and contribute to cloud cavity collapse (D) Complete collapse of cloudcavity (E) Collapse of residual sheet cavity.turbulence in the flow, we utilize the vorticity transport equations, which can be obtained bytaking the curl of the momentum equations (Eq. 2.3). The vorticity transport equation can beexpressed as:DωDt= (ω · ∇) uf︸ ︷︷ ︸vortex stretching (A)−ω(∇ · uf)︸ ︷︷ ︸dilatation (B)+1ρf2(∇ρf ×∇pf)︸ ︷︷ ︸baroclinic torque (C)+ ∇×(∇ · τ vρf)︸ ︷︷ ︸viscous diffusion (D), (4.2)where ω denotes the vorticity of the fluid defined asω = ∇× uf , (4.3)42and τ v is the viscous stress tensor defined asτ v = µf(∇uf + (∇uf)T). (4.4)The interplay of vorticity transport mechanisms governs the flow behavior around the cavity, par-ticularly with regard to the inception and growth of instabilities along the cavity surface, re-entrantjet development, and partial collapse and transport of cloud cavities. In the subsequent sections,we will elucidate the kinematics and dynamics of the cavitating flow through the analysis of thesetransport mechanisms.4.2 Cavity InceptionBetween tcavf ∈ [0, 0.14], the flow past the hydrofoil continues to recover to the mono-phase state,with the dissipation of the turbulent structures created during the previous cavitation cycle, re-laminarization of the flow close to the hydrofoil’s leading edge, and subsequent establishment of theturbulence transition point as observed in non-cavitating flow. While laminar flow is establishedover the hydrofoil’s leading edge, the pressure fluctuations are dissipated, and a low-pressure regionis established over the suction side of the hydrofoil. Owing to the low cavitation number of 1.25, acavity develops close to the leading edge, while the trailing edge continues to recover mono-phaseflow conditions. In the vicinity of the trailing edge, the turbulent structures from the previouscavity breakdown cycle are prevalent, as highlighted by the temporal evolution of the velocityfluctuations illustrated in Fig4.2(d). Correspondingly, the vortex-stretching effects transport thevorticity away from the hydrofoil surface, whereas viscous diffusion effects dissipate these turbulentstructures over the hydrofoil surface. Furthermore, we observe a large spike in viscous diffusion ofvorticity into the streamline close to the leading edge of the hydrofoil.43Figure 4.2: Evolution of (a) vorticity (b) viscous diffusion of vorticity (c) vortex stretching and (d)x-velocity fluctuations over the streamline originating at (−0.167, 0.013, 0.039)C.At the location of cavitation inception, viscous stresses act to diffuse counter-clockwise vorticityinto the oncoming flow. This effect is prominently illustrated in the spatio-temporal evolution of44vorticity transport along a streamline arising as a consequence of viscous diffusion in Fig. 4.2(b).Owing to the inception of a cavity, a density gradient is induced close to the leading edge, which ismisaligned with the favorable pressure gradient, leading to the development of a clockwise baroclinictorque. As illustrated in Figs. 4.3a and 4.3b, the viscous diffusion of vorticity is the dominant effectat the leading edge of the cavity, driving a counter-clockwise deflection of the streamline aroundthe cavity. As the cavity develops, the baroclinic torque becomes the dominant effect close tothe trailing edge of the cavity, with a clockwise torque deflecting the streamline back towards itsoriginal path. Simultaneously, the liquid around the cavity shows a large propensity to undergophase transition to vapor, owing to the significantly small size of the cavity relative to the regionenclosed by the isobar of vapor pressure. This phase of cavity inception and growth is illustratedin Fig. 4.4(a-d).(a)Figure 4.3: (a)A narrow layer of counter-clockwise viscous diffusion (Dz) of vorticity is created inthe vicinity of the cavity, which acts to direct the flow away from the cavity.4.3 Sheet Cavity GrowthWe can consider the cavity inception phase to be complete once the cavity attains a state as shownin Fig. 4.4 at tcavf = 0.313. In this state, the vapor-liquid interface is adjacent to the isobar of vaporpressure, creating a surface over which the oncoming fluid can rapidly undergo vaporization. Close45(b)Figure 4.3: (b)The narrow layer of counter-clockwise viscous diffusion of vorticity (Dz) is locatedclose to the cavity, and continues to direct the flow away from the cavity. At the trailing edge, thebaroclinic torque (Cz) begins to dominate vorticity transport, forcing the streamlines to curve backtowards the cavity. The red line indicates the vapor-liquid interface constructed at φ = 0.585, andthe contours on the left illustrate viscous diffusion of vorticity (Dz), while the contours on the rightillustrate baroclinic torque (Cz).to the leading edge of the cavity, we observe a drop in viscous stresses owing to the stabilization offlow velocities and flow gradients, and local dilatation associated with mass transfer develops. Asa consequence, the dominant mode of vorticity transport transitions from the diffusion of viscousstresses to the dilatation of vorticity. This is illustrated in Fig. 4.5 which indicates the transitionin vorticity transport along the streamline close to the leading edge of the cavity.46Figure 4.4: Contours of material derivative of Vorticity, Dilatation, Viscous Diffusion, Velocity and Pressure Gradient during thecavity inception phase. In the figures, the white line indicates the isobar corresponding to the vapor pressure (Cp = −1.25), andthe red line denotes the vapor-liquid interface. We observe the cavity grow to occupy the isobar of vapor pressure.47As illustrated in the contours of pressure gradient in Fig. 4.4, an adverse pressure gradientdevelops downstream of the cavity and plays a role in directing the flow towards the cavity trailingedge and force the cavity closure point to separate from the hydrofoil’s surface. Simultaneously,we observe the evolution of a density gradient misaligned with the pressure gradient at the trailingedge of the cavity, leading to the production of a clockwise baroclinic torque on the top surfaceof the trailing edge, and a counter-clockwise torque on the bottom surface of the cavity’s trailingedge. The clockwise baroclinic torque acts to deviate streamlines closer to the hydrofoil surfacealbeit unsuccessfully, owing to the strong convective effects in the flow. As a consequence, ahigh-pressure stagnation point is created downstream of the cavity’s trailing edge. In conjunctionwith the adverse pressure gradient, the entraining fluid restrains the cavity attachment point frompropagating further downstream, and results in the formation of a re-entrant jet.L u∗L = uL/U∞Lcav 0.451LR1 0.968LR2 1.310LL1 −0.056LL2 −0.114Table 4.1: Instability propagation velocity of different right and left traveling perturbation modesdriving vorticity transport and triggering the instability of the cavity. The negative sign indicatesa left traveling perturbation mode.While the trailing edge baroclinic torque leads to the development of the trailing edge, the fullyincepted cavity rapidly expands owing to the leading edge interface. Furthermore, the trailing edgeof the cavity (Lcav) is observed to propagate with a nearly constant velocity as illustrated in Fig.4.6. The mass transfer at the leading edge also potentially induces disturbances on the cavity surfaceSmith et al. [94], which result in geometrical fluctuations of the evolving cavity. These geometricalfluctuations can create misaligned pressure and density gradients, leading to varying modes offluctuating baroclinic torque on the fluid elements in the cavity’s vicinity. A characterization ofthe velocities of these perturbation modes is presented in table 4.1. As can be observed in Fig. 4.6,the interactions of perturbations at the trailing edge of the cavity intensify, leading to a high rateof vorticity growth. In conjunction with the re-entrant jet instability, the cavity’s breakdown canbe attributed to the growth of these perturbations. From the characterization of perturbations,we observe a sequence of left traveling perturbations arise at the trailing edge of the cavity andtravel upstream at a slow speed, whereas the perturbations induced upstream rapidly advect to thetrailing edge of the cavity. Subsequently, the right-traveling perturbations slow down, whereas theleft-traveling perturbations accelerate. The interaction of perturbations close to the trailing edgeof the cavity requires further investigation with regard to the energy transfer mechanism at play.48Figure 4.5: Evolution of (a) vorticity (b) viscous diffusion of vorticity (c) baroclinic torque and (d)dilatation over the streamline originating at (0.0055, 0.0603, 0.039)C for tcavf ∈ [0.17, 0.37].Figure 4.6: Evolution of (a) vorticity (b) viscous diffusion of vorticity (c) vortex stretching and (d)x-velocity fluctuations over the streamline originating at (−0.167, 0.013, 0.039)C. The line segmentsindicate the motion of perturbations in vorticity transport mechanisms, with Lcav representing thetrailing edge of the cavity.494.4 Sheet Cavity BreakdownThe formation of the sheet cavity over the suction side of the hydrofoil leads to an enhanced pressuredifferential across the pressure and suction sides of the hydrofoil, leading to increasing lift force.As highlighted in Sec. 4.3, the fluid entraining below the cavity enhances the re-entrant jet andretards the downstream advection of the cavity closure point. For thick cavities, Callenaere et al.[12] observed the re-entrant jet to travel up to the leading edge and cut off the cavity resultingin shedding. The experiments of Leroux et al. [65] observe a high-pressure peak at the middle ofthe cavity surface, to which they correlate the presence of the re-entrant jet. This observation ofa non-piercing re-entrant jet has also been corroborated by previous numerical studies[18, 85]. Aspresented in table 4.2, we observe the re-entrant jet’s development up to x/C = 0.515, beyondwhich the sheet-cavity begins to de-stabilize. However, as illustrated in Fig. 4.7, we do not observethe growth of flow fluctuations till the de-stabilization of the cavity.Figure 4.7: Evolution of the re-entrant jet, and the corresponding induction of fluctuations invorticity transport for tcavf ∈ [0.43, 0.56]. The columns illustrate the z-component of vorticity,vorticity transport, baroclinic torque, and vortex-stretching from left to right.While the re-entrant jet develops, the sheet cavity also expands downstream, with an increasingseparation between the cavity closure point and the cavity’s trailing edge. Within this period, weobserve the growth of surface perturbations on the cavity surface, which are in accordance with the50observations of Pham et al. [76]. These perturbations lead to locally misaligned pressure and densitygradients, which lead to the exertion of baroclinic torque on the fluid elements in the vicinity of thecavity. These fluctuations are accentuated as the cavity expands and correspondingly destabilizesthe cavity’s geometric deformations. From a careful observation of the contours of baroclinic torquepresented in Fig. 4.7, we can observe the onset of these perturbations close to the cavity’s pointof inflection (tcavf = 0.463). This observation is reinforced through the spatio-temporal evolutionof baroclinic torque along a streamline presented in Fig. 4.6(b), wherein these fluctuations can beobserved to propagate in both upstream and downstream directions. This contrasts the observationsof Sedlar et al. [85], wherein they indicate cavity de-stabilization to occur at the instant where itattains the maximum length (lcav).tcavf =t∗−tcav0T cav lcav/C xcav/C 100δj/C Vj/U∞0.16 0.021 0.021 − −0.21 0.061 0.061 − −0.26 0.080 0.080 − −0.32 0.240 0.219 − −0.37 0.421 0.372 0.122 0.5430.433 0.670 0.515 0.436 0.5300.463 0.732 0.553 0.690 0.6110.505 0.826 0.579 0.677 0.7130.552 0.906 0.515 1.000 0.759Table 4.2: Evolution of the sheet cavity and re-entrant jet over a cavitation cycle. We measure thecavity length lcav and location of cavity closure xcav from the leading edge, whereas the re-entrantjet thickness δj is measured normal to the hydrofoil at x/C = 0.1 downstream of cavity closure.The mean re-entrant jet velocity Vj is estimated based on the Strouhal number of the flow [12, 76].The vapor-liquid interface is constructed using φ = 0.585, and the characteristics are evaluated atz/S = 0.5.The studies of Chen et al. [18] attribute the cavity’s de-stabilization to the flow’s transition toturbulence over the cavity surface. However, as indicated in Figs. 4.7 and 4.6(c-d), instabilities inthe flow, along with the dominance of vortex-stretching transport of vorticity is primarily observedafter tcavf = 0.55. This illustrates that transition is potentially caused by the propagation ofperturbations on the cavity surface, and not vice-versa. The emergence of these fluctuations alsoinfluences the effect of baroclinic torque at the trailing edge, which can be detrimental to flowentrainment into the re-entrant jet. Eventually, the cavity breaks down leading to the shedding ofa cloud cavity at tcavf = 0.62. The cavity breakdown process results in the liquid phase impingingback on the hydrofoil surface, while simultaneously interacting with the cloud cavity. Furthermore,the obstruction effect imposed by the smaller residual cavity reduces, leading to a reduction in thedrag force experienced by the hydrofoil.514.5 Cloud Cavitation EvolutionThe evolution of the cloud cavity interplays closely with the onset and growth of turbulent structureson the cavity surface. As illustrated in Sec. 4.4, the growth of instabilities in the baroclinic torqueleads to the onset of fluctuations in the flow around the cavity (Fig. 4.6). The interaction ofturbulent flow structures with the residual cloud cavity drives its collapse, leading to the formationof smaller vapor pockets and vortex cavities. This interaction primarily occurs during tcavf ∈[0.55, 0.86].Horseshoe OnsetCylindrical Coherent StructuresTrailing Edge Roll upChord-wise Stretching : Cylinder - HorseshoeRidge Cavity InductionRib-Vortex SheddingLeading Edge TransitionCloud Cavity Elongation: Gap InductionCollapse of Intermediate CavityLaminar ShearLayerFigure 4.8: Visualization of the coherent structures around the hydrofoil based on a non-dimensionalQ-criterion Q∗ = 8, colored by Z-vorticity and viewed from the leading edge. The coherent struc-tures are overlaid on the cavity contours delineated by φ = 0.585.During destabilization of the sheet cavity, the laminar shear layer at the leading edge beginsto separate and form cylindrical coherent structures which begin to experience vortex-stretching52which acts to transport the vorticity being produced in the liquid due to baroclinic torque. Thebaroclinic torque destabilization forces a clockwise rotation of the cavity structure locally leadingto the formation of chord-wise ridge structures in the sheet cavity. Furthermore, span-wise asym-metrical perturbations tend to enhance the ridge-like structures along the span-wise direction. Asa consequence, the flow field in the vicinity of the deforming sheet cavity enhances the stretchingof the coherent structures in the chord-wise direction, whereas spanwise variations in pressure canlead to the fluid experiencing baroclinic torques tangential to the cavity interface. This results inthe deformation of the spanwise coherent structures leading to the formation of horseshoe vorticesaligning with the ridge-like cavity structures. These chord-wise evolving vortical structures furtherenhance the span-wise asymmetry of the cavity ridges, and these structures carry over into thecloud cavity. This behavior is illustrated in Fig. 4.8(a-b). Close to the trailing edge, a small trail-ing edge vortex rolls up leading to the ejection of a region of clockwise vorticity into the hydrofoilwake from the developing gap region between the cavity and hydrofoil. This vortex takes a flathorseshoe shape and is depicted in Fig. 4.9(b).Residual Sheet CavityCloud Cavity Vortex InteracitonsRelaminarizationCloud Cavity CollapseFigure 4.8: Visualization of the coherent structures around the hydrofoil based on a non-dimensionalQ-criterion Q∗ = 8, colored by Z-vorticity and viewed from the leading edge. The coherent struc-tures are overlaid on the cavity contours delineated by φ = 0.585.Upon separation of the cloud cavity, a part of the sheet cavity remains close to the leading edge,wherein the fluctuations in flow velocity interact continuously with the remaining sheet cavity, anda transition to turbulence is observed at the leading edge, with the formation of several small-scale turbulent structures. Furthermore, the cleavage of the sheet cavity, results in the reversedflow in the re-entrant jet interacting with the downstream advecting freestream flow. Under theseconditions, the entrainment of fluid around the trailing edge of the cavity becomes even strongerleading to the cloud cavity moving away from the hydrofoil’s trailing edge, resulting in the formation53of a gap-region. The low pressure of the gap region forces the roll-up of a trailing edge vortex andleads to the formation of a vortex cavity, which has been labeled C1 in Fig. 4.1. The shedding ofthe trailing edge vortex cavity is accompanied by the shedding of a rib-like vortex with oppositevorticity from the gap region. The formation and advection of these vortices are presented in Fig.4.8(c-d) and Fig. 4.9(c-d).Rib-Vortex SheddingCollapse of Intermediate CavityC1 Roll upCloud Cavity Elongation: Gap InductionHorseshoe Shedding, ElongationClockwise Horseshoe Shed over detached CavityRidge Cavity InductionHorseshoe OnsetTrailing Edge Roll upVortex-Stretching into RidgesFigure 4.9: Visualization of the coherent structures around the hydrofoil based on a non-dimensionalQ-criterion Q∗ = 8, colored by Z-vorticity and viewed from the trailing edge at different timeinstances. The coherent structures are overlaid on the cavity contours delineated by φ = 0.585.The breakdown of C1 leads to the emergence of smaller vortices in the wake, while the largercloud cavity continues to progress downstream. Within this period, cavity growth close to theleading edge subsides completely, with flow up to x/C = 0.5 being highly unsteady and populatedwith multiple small-scale vortices [27]. Once the cloud cavity encounters the trailing edge, thepressure differential across the hydrofoil forces the roll-up of a second vortex around a fragment ofthe cloud cavity. This event is labeled C2 in Fig. 4.1, and is depicted in Fig. 4.9(e). A consequenceof this fragmentation is the formation of multiple vapor clouds of smaller sizes. This breakdownalso allows for the oncoming flow to stabilize again, leading to the development of the residual sheet54cavity on the hydrofoil surface (Fig. 4.9(e-f)).Cloud Cavity Vortex InteractionsC2 Roll upClockwise Vorticity Ejection post C1C2 Ejection into wakeUnsteady Wake post cavity collapseFigure 4.9: Visualization of the coherent structures around the hydrofoil based on a non-dimensionalQ-criterion Q∗ = 8, colored by Z-vorticity and viewed from the trailing edge. The coherent struc-tures are overlaid on the cavity contours delineated by φ = 0.585.The ejection of C2 along with the collapse of vapor clouds up to the instant tcavf = 0.86 leadsto the development of highly unsteady fluctuations upstream of the hydrofoil. This results in thecollapse of the residual sheet cavity, resulting in the impulsive loads experienced by the hydrofoiland is discussed further in Sec. 4.6. Further investigation of C1, C2 and C events in the cavitationcycle are necessary, particularly concerning the content of turbulent kinetic energy in the liquidphase and the corresponding turbulence spectra for this interval of the cavitation cycle. Thisanalysis aids the determination of the spatio-temporal scales of eddies governing the dynamics ofcavity-turbulence interactions.4.6 Re-laminarizationThe process of re-laminarization of flow begins upon the complete collapse of the cloud cavity inthe wake (tcavf ≈ 0.85). As the cavities propagate into the hydrofoil’s wake, they encounter a highpressure region created at the start of the cycle due to the streamline nature of the hydrofoil. Asa consequence, the cloud cavities begin to break down into smaller vapor pockets, resulting in theformation of a highly turbulent wake. A sheet cavity remains close to the leading edge of thehydrofoil which interacts with the turbulent structures in its wake and sheds small cavity clouds.Furthermore, the turbulent structures over the suction side of the hydrofoil are observed in a rangeof length scales, and their relative interaction acts to dissipate them leading to turbulence beingextinguished.The remaining sheet cavity is highly deformed and the flow around it continues to exhibitsignificant fluctuations, resulting in a complete cavity collapse [26], and a corresponding spike55Figure 4.10: Evolution of the flow around the hydrofoil during complete collapse of cloud cavityat tcavf = 0.90 and subsequent re-laminzarization. The non-dimensional x-vorticity and z-vorticity(ω∗z = ωzC/U∞) components are presented at Z/S = 0.5. The red lines correspond to iso-contoursof φ = 0.585 to indicate the cavity structure(s) at different instances of the cycle.56in pressure, which leads to a sudden drop in the lift coefficient. Subsequently, the laminar flowdevelops from the leading edge and the hydrofoil regains the non-cavitating flow state. This processis illustrated in Fig. 4.10 for the section located at z/C = 0.5, and is identified as C in Fig. 4.1.The collapse of this cavity and re-laminarization of the flow initiates a new cycle of cavity growth.4.7 SummaryIn this chapter, we have presented the numerical results obtained from the Large Eddy Simulationsof cavitating flow past a rigid NACA66 hydrofoil. The flow characteristics have been evaluated inextensive detail, and observations have been made about the cavity growth and breakdown cycle.We summarise the observations and hypotheses as follows: We begin by recognizing the presence of a sharp density gradient across the vapor-liquidinterface, which can induce vorticity in the flow owing to the density gradient’s misalignmentwith the pressure gradient. As a consequence, we evaluate different terms of the vorticitytransport equation to identify the dominant vorticity transport mechanisms at play duringthe cavity cycle. Before the cavity’s inception, the suction side of the hydrofoil is rendered highly turbulentowing to cavity collapse observed in the previous cycle. In the initial phase of the cycle, theviscous stresses dominate the vorticity transport and they act to dissipate these turbulentstructures on the hydrofoil’s surface. The flow over the suction-side re-laminarizes resultingin the recovery of the transition point as would be observed in the case of a non-cavitatingflow. Upon cavity inception, the velocity vectors are re-directed around the cavity surface by theviscous stresses. The inception phase occurs until the cavity expands to fill the isobar ofthe vapor-pressure contour. The expanding cavity allows for the creation of a vapor-liquidinterface at the leading edge across which mass transfer occurs. The consequence of masstransfer is a dilating velocity field, which becomes the dominant vorticity transport mechanismat the leading edge of the cavity. A region of adverse pressure gradient and counter-actingbaroclinic torque is created at the trailing edge of the cavity. The vaporization of the fluid drives a rapid expansion of the cavity, while the baroclinictorque in conjunction with the adverse pressure gradient hinders the downstream advectionof the cavity attachment point relative to the trailing edge of the cavity. This results in theformation of a re-entrant jet under the cavity surface at the trailing edge. While the extent of the re-entrant jet is limited, the presence of geometric perturbations onthe hydrofoil surface forces a misalignment in density and pressure gradients leading to the57onset of perturbations in baroclinic torque at the vapor-liquid interface. These instabilitiespropagate along the cavity and rapidly de-stabilize the interface. Consequently, the sheetcavity is cleaved and a large cloud cavity is shed into the hydrofoil’s wake. Upon the de-stabilization of the cavity, cylindrical coherent structures form on the surfaceof the cavity, which begin to interact with the irregular cavity surface. While vorticity isproduced at the interface through baroclinic torque, the vorticity stretching term dominatesthe vorticity transport, leading to the deformation of the cylindrical coherent structures. Thedeforming coherent structures alter the flow field, leading to the deformation of the cloudcavity. The deforming cloud cavity takes the form of ridge-like structures, which result in a span-wise asymmetry in the baroclinic torque, leading to the cylindrical structures transforminginto horseshoe-like structures on the cavity surface. Vortex stretching along the chord-wisedirection leads to the elongation of the horseshoe legs, resulting in the formation of ribstructures and the creation of hairpin vortices. In the gap between the cavity and hydrofoil suction side, a recirculating region forms owingto the remnant of the re-entrant jet, which slowly propagates downstream. The interactionof this recirculating region with the flow along the hydrofoil pressure side rapidly drags thevortex from beneath the cloud cavity resulting in a highly elongated hairpin vortex. Thisliquid contained within this hairpin vortex has the propensity to experience cavitation. The adverse pressure encountered in the hydrofoil’s wake creates conditions unfavorable tosustain cavitation resulting in the successive collapse of cavities. The remnant of sheet cavita-tion at the leading edge experiences growth up to the instant of cloud cavitation collapse[55],at which the cavity breaks down completely, and the process of re-laminarization begins.58Chapter 5Validation of LES for Cavitating Flowover Flexible NACA66 HydrofoilIn this chapter, we present the validation of a cantilevered flexible NACA66 hydrofoil under fully-wetted and cavitating conditions. We begin with a description of the problem for validation,and establish the structural parameters for characterizing the cantilevered hydrofoil. We utilizethe meshing criteria developed for the rigid hydrofoil to construct the meshes used for the currentstudy. For the purposes of validation, we compare the amplitude and frequency content of deforma-tion against previous experimental studies, and present the turbulence spectra and flow structuresobserved during the cavitation cycle.5.1 Problem Description and Mesh Characteristics5.1.1 Definition of Structural ParametersWe allow both non-cavitating and cavitating flows to develop past a flexible NACA66 hydrofoil ata chord-based Reynolds number Re = 7.50 × 105, cavitation number σ = 1.4 and the stationaryangle of attack α0 = 8◦. The liquid and vapor properties are carried over from those used in thestudies of the rigid hydrofoil. The hydrofoil’s flexibility is characterized by a Polyacetate material(POM) whose structural properties are defined by Young’s modulus Es = 3 GPa, Poisson’s ratioνs = 0.35 and density ρs = 1480 kg/m3. The added mass effects are accounted for in the modalmass, based on the coefficients in still-water derived from the studies of De La Torre et al. [20].Based on the structural parameters and estimates of modal added mass coefficients obtained in stillwater, we obtain the dominant bending and torsional modes with natural frequencies f1 = 55Hzand f2 = 177Hz respectively. Correspondingly, the non-dimensional parameters of the structurearem∗ =ρsρf= 1.48, U∗ =U∞f1C= 0.682, St1 =f1CU∞= 1.466, St2 =f2CU∞= 4.69, (5.1)where m∗ and U∗ denote the mass ratio and reduced velocity respectively. St1 and St2 denote thenon-dimensional structural frequencies relative to the flow characteristic time scale. Due to thedynamic nature of cavitation observed at the current cavitation number, the added mass fluctuates59with the extent of vapor formation over the hydrofoil surface, and requires further investigation incontext of the current framework. Furthermore, as highlighted in the studies of Benaouicha andAstolfi [7], the hydrofoil in a cavitating flow experiences both positive and negative damping effects,which can lead to instabilities in the dynamical system.ZXY1.28CInletWallOutletSymmetric Boundary3C6CFlexible Wall1.28CFigure 5.1: A schematic of the flow past flexible cantilevered NACA66 hydrofoil at Re = 750000.The computational setup and boundary conditions are shown for the filtered Navier-Stokes equa-tions.5.1.2 Domain Definition and MeshThe domain for studying the flow past the cantilevered hydrofoil is presented in Figs. 5.1 and5.2. The inlet and outlet boundaries are positioned 3C and 6C upstream and downstream of theleading edge (LE) respectively. The channel’s width is set to be 1.28C, with side walls equidistantfrom the half-chord point of the hydrofoil, with no-slip boundary conditions. The aspect ratio ofthe cantilevered hydrofoil is taken to be 1.27, with symmetric boundary conditions taken at thehydrofoil root. At the hydrofoil tip, the domain is extended to give the channel width of 1.28Calong the Z-axis, to give a gap-to-span ratio of 0.00523. A no-slip boundary condition is imposedat the tip wall of the channel. A uniform flow field is imposed at the inlet and is also set as theinitial condition for the non-cavitating flow. For conducting LES of the cantilevered hydrofoil, weuse the meshing requirements established by M2 in section 3.1. Owing to the large aspect ratio ofthe flexible hydrofoil, a uniform mesh-sizing of ∆z+ = 60, requires 495 mesh layers in the span-wisedirection, leading to a mesh with 32M nodes. In order to mitigate the computational expense, wedecrease mesh-sizing along the hydrofoil span, maintaining a ratio of rz = 0.995, from root to tip,leading to a mesh with 13M nodes. The mesh information is summarized in table 5.1.60Property M21-Flex M22-Flexy+ 0.3 0.3(∆x+,∆z+) (65, 120) (48, 60)Nchord 200 280(NBL, rBL) (60, 1.15) (60, 1.15)(Nspan, rspan) (240, 0.988) (495, 0.995)Nnodes 12.12M 32.29MN elem. 15.81M 42.33MTable 5.1: Mesh parameters for the LES validation of flow past flexible cantilevered NACA66hydrofoil. Only 8 node tri-linear hexahedral elements are used.XZS =1.27C H=1.28CRootWall Tip0.01CαC 1.28CXYWallWallUFigure 5.2: Top and side views of the flexible hydrofoil, illustrating the angle of attack and the gapbetween the hydrofoil tip and front wall.615.2 Fully-wetted responseIn the current section, we present some representative results for rigid and flexible hydrofoils infully-wetted flow. Table 5.2 presents the LES results of the mean drag and lift coefficient ofthe rigid hydrofoil, along with the comparison against experimental data of Leroux et al. [65].Furthermore, we also observe good agreement of the mean transverse displacement and tip-rotationof the hydrofoil with experimental data of Ducoin et al. [25]. The performance of the dynamic SGSmodel is further evaluated through the wake characteristic data for the flexible hydrofoil as shown inFig. 5.4. Based on the data presented in table 5.2, the current LES simulations accurately predictthe forces and their corresponding fluctuations, which in turn influence the fidelity of structuralresponse. With respect to the flow features, we observe a transition to turbulence on the hydrofoil’ssurface, along with the formation of a tip-leakage vortex in the gap region between the tip and frontwall. This phenomenon is illustrated in Fig. 5.3.ParametersRigid FlexibleExp.[65] Num.-1 [2] Num.-2[36] Current Exp. [25] Num.-3[2] CurrentCL 1.031 1.19 1.16 1.122 − 1.22 1.112CD 0.041 0.022 0.036 0.044 − 0.022 0.051hmax/C − − − − 0.024 0.01 0.031θmax (deg) − − − − −0.39 −0.18 −0.27Table 5.2: Comparison of hydrodynamic load coefficients and motion characteristics for fully-wettedconditions.5.3 Cavitating Response at σ = 1.4ParametersFlexibleExp. [8] Num.-3[2] Currenthmax/C 0.032 0.016 0.091θmax (deg) − −0.39 −0.18f∗ = fC/U∞ 0.3054 0.2391 0.3132Table 5.3: Comparison of response characteristics for cavitating conditions at σ = 1.4.We next evaluate the dynamic response of the hydrofoil under unsteady cavitating conditionsat σ = 1.4. With regard to both the amplitude and frequency response of the hydrofoil, we observegood agreement in measurements relative to experimental and numerical studies. However, weobserve some over-prediction in the bending response of the hydrofoil. We attribute this to thelimited data available with regard to the added-mass coefficient to account for its time-varyingnature within the modal response model. Added mass oscillations have been found to broaden the62Figure 5.3: Transition to turbulence and evolution of the vortical structures over the hydrofoil innon-cavitating conditions. Some key vortical structures are the hairpin vortices emerging from thehydrofoil span and the tip-leakage vortex. The vorticity field (ω∗x = ωxC/U∞) and the velocity field(red-arrows) are indicated at different chord-wise locations, indicating the motion of the tip-leakagevortex away from the hydrofoil.frequency response of the hydrofoil, owing to variations in the natural frequency of the structure[2].Figure 5.5a presents the hydrofoil’s tip displacements at the leading and trailing edges along63(a) (b)Figure 5.4: FFT Spectra of (a) turbulent kinetic energy at the spanwise location z/S = 1 and(b)turbulent kinetic energy at the spanwise location z/S = 1 at the chordwise locations defined byFig. 3.8 for the hydrofoil under fully-wetted conditions. For sake of clarity, spectra of points fromthe same group are shifted.with the coefficient of lift. We observe the lift to be in phase with the tip displacement, withnearly identical spectral content. The presence of low amplitude, high-frequency fluctuations inthe lift signal indicates the presence of cavity or vortex shedding events as was observed in thecase of a rigid hydrofoil. As a result, we observe the absence of high-frequency impulsive loadsobserved in the case of a rigid hydrofoil (Fig. 4.1) over the flexible hydrofoil, indicating the absence64(a)(b)Figure 5.5: (a) Comparison of the bending displacement at the leading and trailing edges at thehydrofoil tip, with the Coefficient of Lift (CL). (b) Comparison of the angular displacement at thehydrofoil tip, with the Coefficient of Moment (CM ).of abrupt cavity collapse events. Fig. 5.5b illustrates the tip angular rotation with the momentcoefficient, wherein we observe the moment coefficient is out of phase with the angular rotation ofthe hydrofoil. The pitching moment acts to increase the hydrofoil’s effective angle of attack closeto the tip, resulting in higher lift coefficients relative to a rigid hydrofoil [2].f∗ = fC/U∞ (Current) f∗ = fC/U∞ (Experimental) [2, 8]0.15 0.17290.3 0.30540.45 0.52610.73 0.74870.93 1.2969Table 5.4: Comparison of the dominant frequencies observed in the displacement response of thestructure with experimental data.65(a) (b)Figure 5.6: Standing wave response of the cantilevered hydrofoil (a) bending and (b)twisting re-sponse of the hydrofoil along the span with time.With regard to the spectral characteristics exhibited by the flexible hydrofoil, we observe goodagreement with experimental observations of Benaouicha et al. [8] and Akcabay and Young [2].The frequencies from the respective studies are non-dimensionalized w.r.t the time-scale C/U∞.Through Fig. 5.7a, we illustrate the power spectrum of the bending displacement, lift coefficient andglobal cavity volume. We notice the emergence of peaks in the spectral response which are integralmultiples of the Strouhal number St = 0.15, indicating that for the current structural parameters,the cavity dynamics govern the hydrofoil’s response. Therefore, we conclude the presence of a three-way lock-in between cavitation, displacement and fluid forces for U∗ = 0.678 which is governed bythe cavitation number. This observation is consistent with the results of Kashyap and Jaiman[52], who observed the emergence of this Strouhal number at U∗ = 1 for an elastically mountedcavitating hydrofoil.A similar power spectrum is presented for the torsional response of the hydrofoil in Fig. 5.7b.While the dominant frequencies are the integral multiples of St, we also observe the emergenceof a broadband frequency spectrum whose central peak occurs at f∗ = 3.35, which is absentin the spectrum of the global vapor volume, and does not correlate with the evaluated naturalfrequencies of the structure or their integral sub-harmonics. We attribute the emergence of thisfrequency to unsteady flow characteristics induced as a consequence of the intricate interactionbetween cavitation, turbulence and structural deformation.With the flow parameters considered, the flexible hydrofoil’s deformation perturbs the flow66(a) (b)Figure 5.7: FFT Spectra of (a) bending displacement (yLE) and (b) angular displacement (θ) ofthe hydrofoil at the tip (z/L = 1). The arrows indicate sub-harmonics of fundamental bending (nbi)and torsional modes (nθi ) which appear in the displacement spectra. Note that the bending (fbN )and torsional (fθN ) frequencies have been non-dimensionalized here w.r.t the time-scale C/U∞.within the boundary layer and inhibits the formation of a sheet cavity. This renders the developingcavity unstable, leading to the emergence of unsteady flow structures in the wake. Furthermore, ashighlighted by Fig. 5.5a, there is a significant variation in the forces experienced by the hydrofoilfrom cycle to cycle. This indicates that, in certain cycles, the flow has the propensity to re-stabilizeupstream of the leading edge, creating conditions conducive for sheet cavitation. However, in suchcycles, the hydrofoil experiences large deformations owing to extensive pressure-side loading. Thisdeformation leads to the perturbation of the re-entrant jet, resulting in early sheet cavity collapse.Consequently, only the region close to the hydrofoil tip retains conditions conducive to cavityformation. These observations are further discussed in the next chapter.5.4 SummaryIn this chapter, we have presented a numerical validation study of Large Eddy Simulations (LES)of cavitating flow past a flexible NACA66 hydrofoil. The numerical studies have been observedto accurately capture the key characteristics of cavitating flows over hydrofoils based on certainresolution criteria demanded by LES. We summarise the results of the validation studies as follows: The mesh resolution criteria developed for the rigid hydrofoil have been extended for the study67of fully-wetted and cavitating flow of the hydrofoil. Based on the turbulent kinetic energyspectra developed, we can establish the adequacy of the mesh employed for the current study. We observe good agreement in the amplitude and frequency content of the hydrofoil’s responsecharacteristics with experimental studies pertaining to cantilevered flexible hydrofoils We observe the spectral content indicates a three-way coupling between cavitation, forcingand displacement response of the hydrofoil at the current cavitation number. We hypothesizethat68(a) (b)(c)Figure 5.8: Evolution of (a) lift coefficient (CL), (b) cavity Volume (Vc), and (c) Cavity and Vortexstructures from t1 to t5 as labeled in (a). The liquid vapor interface is demarcated using φ = 0.8for the flexible hydrofoil.69(d)Figure 5.8: (d) Cavity and Vortex structures from t6 to t9 as labeled in (a).70Chapter 6Conclusions and RecommendationsIn this chapter, we present the conclusions of this dissertation and provide recommendations forfuture investigations.6.1 ConclusionsThrough this dissertation, we aimed to develop a high-fidelity computational framework for coupledfluid-structure interaction of flexible structures with turbulent cavitating flows. This is in linewith understanding and controlling cavitation over marine propellers to achieve the objective ofmitigating underwater noise emission.In the development of this framework, we evaluated the performance of a stabilized cavitationtransport equation in conjunction with large-eddy simulations of flows involving the transition fromsheet to cloud cavitation. The cavitation model is proficient in predicting cavitation physics at thespatio-temporal resolutions demanded by LES. Key highlights include (i) accurate near and far-field pressure solutions, (ii) accurate prediction of vapor-liquid interface and corresponding localcompressibility effects, and (iii) stability at a range of empirical models used for calibrating todifferent flow conditions. With regard to performance during validation, good agreement withexperiments was observed between the unsteady pressure field over the hydrofoil surface. Thehomogeneous mixture theory accurately captures the geometrical features of sheet cavities andperforms well in predicting the locations of bubble clusters. Furthermore, numerical instabilitiesare mitigated completely, and subject to sufficient grid resolution, the framework allows for anevaluation of the growth of physical instabilities. Within this framework, we have incorporated alinear-elastic structural model discretized in a finite element framework to integrate with the fluid-cavitation solver. In this regard, a modal decomposition of the structure can be used, with thecorresponding mode shapes projected onto a subset of the fluid domain. This framework accuratelypredicted the amplitude and frequency response of a highly flexible hydrofoil (E = 3GPa) and canbe employed to study the effects of flexibility in influencing cavitation growth and breakdowncharacteristics.Using this framework, we have focused on studying turbulent cavitating flows over a rigid andcantilevered flexible hydrofoil. Under partial cavitating conditions, the rigid hydrofoil experiencesa re-entrant jet, along with surface perturbations arising as a consequence of leading edge phasechange. The phase change process misaligns the density gradient with the pressure gradient, leading71to fluctuations in the baroclinic torque. The combination of these two effects has been observedto destabilize the cavity and drive its breakdown. In contrast to suggestions in the literature, non-turbulent flow conditions have been identified during the cavity’s destabilization, illustrating thatthe instability is a consequence of the interaction between the geometry and the vapor cavity.In the case of the flexible hydrofoil, we observe the presence of a three-way lock-in betweencavity-shedding frequency, displacement, and force experienced by the hydrofoil, which is governedby the cavitation dynamics. This can be attributed to the large time scale of the cavitation cyclerelative to that of the structure, and the cavity dynamics is altered significantly relative to therigid hydrofoil. The flexible hydrofoil’s motion induces span-wise velocity fluctuations around thehydrofoil’s point of inflection, leading to sustained cavity breakdown, albeit at the same frequenciesas that of cavitation over rigid hydrofoils. Simultaneously, the region close to the hydrofoil tipexperiences extensive sheet and tip-leakage vortex cavitation owing to the absence of a re-entrantjet and changes in the topology of the hydrofoil. The far-field pressure signatures of both hydrofoilsare significantly different, with the rigid hydrofoil presenting a mellower spectrum, which can beattributed to the absence of sudden cavity collapse events.The current state of the framework allows for numerically studying flow past arbitrarily shapedflexible structures with response characteristics lying within the linear elastic regime. Thus, flowpast flexible propellers in stationary conditions can be explored allowing for developing and evalu-ating new design strategies aimed at cavitation and noise reduction.6.2 RecommendationsThe present dissertation can be extended to explore newer computational strategies and tackledifferent problems involving cavitation dynamics of rigid and flexible structures. Some recommen-dations are: The hydroelastic cavitation FSI framework currently operates the fluid-flow solver using adynamic subgrid-scale model, which requires isotropic grid resolution. The in-house solver canbe augmented with a Detached Eddy Simulation framework employing a k−ω SST model forcapturing cavitating flows interacting with flexible structures. This would allow less stringentmeshing requirements, and allow for conducting parametric studies for evaluating structuraldynamics. The effects of morphing rigid hydrofoils can be explored as a passive mitigation strategy forsheet and cloud cavitation. Further investigation of propeller blades with full hydroelasticdeformation is necessary to identify the pressure field created as a consequence of cavitation,and noise characteristics need to be evaluated.In the current work, cavitation has been phenomenologically modeled using the homogeneousmixture assumption. While these models are adequate for identifying vapor-cavity interfaces, and72studying the effects on the liquid phase due to the presence of the cavity and breakdown charac-teristics of cavities, there are some practical limitations such as (i) absence of mass and momentatransfer associated with the liquid and vapor phase, constraining analysis of flow dynamics involv-ing bulk vapor structures, (ii) multi-scale nature of cavitation cannot be studied, i.e. situationsinvolving bulk vapor structures and discrete bubbles.A linear-elastic model is employed for the structure, with modal decomposition for reducingstructural evaluation time. While this approach is useful for small-strain problems and integrateswell into morphing-based approaches, it is limited by: (i) the inability to account for large strains- necessitating the employment of a hyperelastic model. 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Thesis/Dissertation
eng
Mechanical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Jaiman, Rajeev
A computational framework for flow-induced vibrations of propeller blades in cavitating flows
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