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Filter-Based Modeling of Unsteady Turbulent Cavitating Flow Computations

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FILTER-BASED MODELING OF UNSTEA DY TURBULENT CAVITATING FLOW COMPUTATIONS By JIONGYANG WU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005

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Copyright 2005 by Jiongyang Wu

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To my Parents; my wife, Weishu; and my son, Andy, for their love and support

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iv ACKNOWLEDGMENTS I would like to express my si ncere thanks and appreciati on to my advisor Professor Wei Shyy. His thorough guidance and support be nefited me tremendously in exploring and pursuing my research interest. I thank him for his boundless pati ence and generous attitude, and his enduring enthusiasm in e ducating me both in research and personal development. I have benefited much from the collaboration with Professor Stein T. Johansen. I would like to thank Professo r Renwei Mei for providing some helpful suggestions and serving on my committee. Add itionally, I would like to thank Professors Louis N. Cattafesta, David W. Mikolaitis, and Jacob N. Chung for serving on my committee, and Dr. Siddharth Thakur for generously sharing his experience. I have had the privilege to work with all the individuals of the Computational Thermo-Fluids group, to whom I also give my thanks. I would like to deeply thank my parents. Their encouragement, support, trust, and love have given me power and strength th rough these years. My wife, Weishu Bu, has been with me and supported me all the time. My son, Andy Wu, is another source of my invaluable wealth. No words can possibly express my gratitude and love to them. Finally, I would like to thank the NASA Cons tellation University Institute Program (CUIP) for financial support.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES.........................................................................................................viii LIST OF SYMBOLS.........................................................................................................xi ABSTRACT.....................................................................................................................xi v CHAPTER 1 INTRODUCTION........................................................................................................1 1.1 Background of Cavitation.......................................................................................1 1.1.1 Cavitation Types in Fluids............................................................................1 1.1.2 Cavitation Inception and Parameter.............................................................2 1.2 Research Motivations and Objectives....................................................................3 1.2.1 Challenges and Motivations.........................................................................3 1.2.2 Research Objectives.....................................................................................8 2 NAVIER-STOKES EQUATIONS AND TURBULENCE MODELS........................9 2.1 Governing Equations..............................................................................................9 2.2 Turbulence Model.................................................................................................11 2.3 Filter-Based RANS Model...................................................................................12 2.3.1 Literature Review.......................................................................................14 2.3.2 Filter-Based Model Concept (FBM)..........................................................16 2.3.3 Modeling Implementation..........................................................................17 2.4 Assessing FBM in Time-Dep endent Single-Phase Flow......................................18 3 CAVITATION MODELS..........................................................................................32 3.1 Governing Equations............................................................................................32 3.2 Literature Review of Cavitation Studies...............................................................33 3.2.1 Cavitation Compressibility Studies............................................................35 3.2.2 Cavitation Studies on Fluid Mach inery Components and Systems............38 3.3 Transport Equation-based Cavitation Model (TEM)............................................41

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vi 4 NUMERICAL METHODS........................................................................................45 4.1 Pressure-Based Algorithm....................................................................................46 4.2 Pressure Implicit Splitting of Oper ators (PISO) Algorithm for Unsteady Computations......................................................................................................47 4.3 Speed-of-Sound (SoS) Numerical Modeling........................................................50 5 ASSESSING TIME-DEPENDENT TURBULENT CAVITATION MODELS.......53 5.1 Cavitating Flow through a Hollow-Jet Valve.......................................................53 5.1.1 Steady and Unsteady Turbulent Cavitating Flows.....................................55 5.1.2 Impact of Speed-of-sound Modeling..........................................................60 5.2 Turbulent Cavitating Flow through a Convergent-Divergent Nozzle..................65 5.3 Turbulent Cavitating Flow over a Clark-Y Hydrofoil..........................................73 6 CONCLUSIONS AND FUTURE RESEACH...........................................................92 6.1 Conclusions of Present Research..........................................................................92 6.2 Future Research Directions...................................................................................94 LIST OF REFERENCES...................................................................................................95 BIOGRAPHICAL SKETCH...........................................................................................103

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vii LIST OF TABLES Table page 2-1 Summary of the hybrid RANS/LES studies.............................................................15 2-2 Parameters used in the computation.........................................................................19 2-3 Comparisons of diffe rent turbulence models...........................................................24 3-1 Overview of the comp ressible cavitation studies.....................................................37 3-2 Overview of cavitation on pumps, propellers, inducers and turbine blades............40 5-1 Time-averaged liquid volume fraction v/s pressure-density correlation at multiple points inside the cavity, original IDM with SoS-1, LSM..........................63 5-2 Time-averaged liquid volume fraction v/s pressure-density correlation at multiple points inside the cavity, original IDM with SoS-2, LSM..........................63 5-3 Comparisons of Strouhal number of original and m odified IDM with SoS-2A......72 5-4 Time-averaged cavity leading and trai ling positions of different turbulence models, modified IDM with SoS-2A, AOA=5 degrees...........................................78 5-5 Time-averaged cavity leading and trai ling positions of different turbulence models, modified IDM with SoS-2A, AoA=8 degrees............................................78 5-6 Comparison of mean CL and CD, LSM, Cloud cavitation 0.80 AoA=8 degrees......................................................................................................................91

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viii LIST OF FIGURES Figure page 1-1 Different types of cavitation visualization.................................................................3 1-2 Isothermal harmonic speed-ofsound in the two-phase mixture................................6 2-1 Geometry configuration of square cylinder..............................................................19 2-2 Streamline snap-shot on fine grid with 0.15D ...................................................20 2-3 Pressure behaviors at the re ference point on the fine grid.......................................21 2-4 Time-averaged U-velocity along the hor izontal centerline be hind the cylinder......23 2-5 Snap-shots of velocity contour. Colo r raster plot of axial velocities.......................23 2-6 Streamline snap-shots of two turbulence models on fine grid.................................24 2-7 Time-averaged U-velo city along y at x/D=0.0.........................................................26 2-8 Time-averaged velocities along y at x/D=1.0..........................................................26 2-9 Time-averaged u-velocity along y direction............................................................26 2-10 Mean kinetic energy on different grids....................................................................28 2-11 Comparisons of different filter sizes on time-averaged v-velocity along y at x/D=1.0.....................................................................................................................29 2-12 Comparisons of different filter sizes on kinetic energy...........................................29 2-13 Time-averaged viscosity contours of di fferent turbulence models on fine grid.......30 3-1 Sketch of a cavity in homogeneous flow.................................................................42 3-2 Interface vector sketch in a CV................................................................................43 5-1 Valve geometry........................................................................................................54 5-2 Computational domains and boundary conditions...................................................55

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ix 5-3 Density contour lines of the steady state solution, origin al IDM with SoS-1, LSM..........................................................................................................................56 5-4 Middle section density c ontours at different time inst ants, original IDM with SoS-1, LSM..............................................................................................................57 5-5 Time evolutions at different loca tions, original IDM with SoS-1, LSM.................58 5-6 Projected 2-D streamlines at middle plane and experimental observation..............59 5-7 Middle section density contours of di fferent SoS at different instant time, original IDM, LSM..................................................................................................61 5-8 Two different SoS in hollow-jet valve flow, original IDM, LSM...........................63 5-9 Pressure time evolutions of di fferent SoS, original IDM, LSM...............................64 5-10 Time evolution and spectrum of pressure and density of two SoS definitions at a point at the cavitation vici nity, original IDM, LSM................................................64 5-11 Time-averaged eddy viscosity contours of different grids, original IDM with SoS-2, 1.98 ........................................................................................................67 5-12 Time-averaged vapor volume fraction comp arisons of different grids, original IDM with SoS-2, 1.98 ......................................................................................68 5-13 Time-averaged comparisons of different turbulence models on fine grid with 0.25cavL original IDM with SoS-2, 1.98 ...................................................68 5-14 Instantaneous profile s on fine grid with 0.25cavL original IDM with SoS-2, 1.98 ....................................................................................................................69 5-15 Pressure contours and streamlines comp arison of two turbulence models on fine grid, original IDM with SoS-2, 1.98 .................................................................70 5-16 Pressure evolutions of original and modified IDM with SoS-2A at a reference point.......................................................................................................................... 72 5-17 Cavity shape and recirculation zone during cycling of original and modified IDM..........................................................................................................................72 5-18 Time-averaged volume fraction and ve locity comparisons of original and modified IDM...........................................................................................................73 5-19 Clark-Y geometry sk etch and Grid blocks...............................................................76 5-20 Grid sensitivity of the time-averag ed uand v-velocity, LSM, AoA=5.................77

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x 5-21 Time-averaged volume fraction contours and streamlines of different turbulence models, AoA=5.......................................................................................................78 5-22 Time-averaged volume fraction contours and streamlines of different turbulence models, AoA=8.......................................................................................................79 5-23 Time evolutions of cloud cavitation 0.55 AoA=5..........................................80 5-24 Time evolutions of cloud cavitation 0.80 AoA=8..........................................81 5-25 Cavity stage comparisons, cloud cavitation 0.80 AoA=8...............................83 5-26 Time-averaged uand v-velocities of two turbulence models, AoA=5..................84 5-27 Time-averaged uand v-velocities of two turbulence models, AoA=8..................86 5-28 Time-averaged lift and drag coefficients comparisons............................................88 5-29 Time-averaged viscosity contours, AoA=5............................................................90 5-30 Time-averaged u-velocity of diffe rent SoS treatments, cloud cavitation 0.80 AoA=8....................................................................................................91

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xi LIST OF SYMBOLS c speed-of-sound C arbitrary O(1) constant C pressure coefficient 12,, CCC turbulence model constants destC empirical constant in the evaporation term p rodC empirical constant in the condensation term k turbulent kinetic energy -m evaporation rate +m condensation rate P pressure q source term iu velocity components in Cartesian coordinates u+ non-dimensional velocity U magnitude of the horizontal component of velocity t time, mean flow time scale T temperature xi Cartesian coordinates y+ non-dimensional normal distance from the wall yp normal distance of the first grid point away from the wall

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xii volume fraction ij Kronecker delta function turbulent dissipation rate laminar viscosity t turbulent viscosity , curvilinear coordinates generalized dependent variable Reynolds stress w wall shear stress t kinematic viscosity m mixture density cavitation parameter ,k turbulence model constants Subscripts, Superscripts I interface L, l liquid phase V, v vapor phase m mixture phase n normal direction s tangential direction t tangential direction x component in x coordinate direction

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xiii y component in y coordinate direction z component in z coordinate direction free stream predicted value

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xiv Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy FILTER-BASED MODELING OF UNSTEA DY TURBULENT CA VITATING FLOW COMPUTATIONS By Jiongyang Wu August 2005 Chair: Wei Shyy Major Department: Mechanic al and Aerospace Engineering Cavitation plays an important role in th e design and operation of fluid machinery and devices because it causes performance degradation, nois e, vibration, and erosion. Cavitation involves complex phase-change dynami cs, large density ratio between phases, and multiple time scales. Noticeable achievements have been made in employing homogeneous two-phase Navier-Stokes equa tions for cavitating computations in computational and modeling strategies. Howe ver, these issues pose challenges with respect to accuracy, stability, efficiency a nd robustness because of the complex unsteady interaction associated with cavit ation dynamics and turbulence. The present study focuses on developing and assessing computational modeling techniques to provide better insight into unsteady turbulent cav itation dynamics. The ensemble-averaged Navier-Stokes equations along with a volume fraction transport equation for cavitation and turbulence closur e, are employed. To ensure stability and convergence with good efficiency and accuracy the pressure-based Pressure Implicit

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xv Splitting of Operators (PISO) algorithm is adopted for timedependent computations. The merits of recent transport equation-based cav itation models are first re-examined. To account for the liquid-vapor mixture compressi bility, different nume rical approximations of speed-of-sound are further i nvestigated and generalized. To enhance the generality and capability of the recent interfacial dynamics-b ased cavitation model (IDM), we present an improved approximation for the interfacial velocity via phase transformation. In turbulence modeling, a filter-based model (FBM) derived from the k two-equation model, an easily deployabl e conditional averaging method aimed at improving unsteady simulations, is introduced. The cavitation and turbulence models are assessed by unsteady simulations in various geometries including square cylinde r, convergent-divergent nozzle, Clark-Y hydrofoil, and hollow-jet valv e. The FBM reduces eddy visc osity and captures better unsteady features in singlephase flow, and yields stronge r time-dependency in cavitating flows, than the original k model. Various cavitation m odels show comparable steadystate pressure distribu tions but exhibit substantial vari ations in unstea dy computations. The influence of speed-of-sound treatments on the outcome of unsteady cavitating flows is documented. By assessing lift and drag coe fficients, pressure and velocity distributions from inception to cloud cavitati on regimes, the present appr oach can pred ict the major flow features with reasonable agreement to experimental data.

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1 CHAPTER 1 INTRODUCTION 1.1 Background of Cavitation In liquid flow, if the pressu re drops below the vapor pre ssure, the liquid is unable to withstand the tensile stress and then adjust s its thermodynamic state by forming vaporfilled two-phase mixed cavities. This phe nomenon is known as cavitation (Batchelor 1967). It can occur in a wide variety of fluid machinery components, such as nozzles, injectors, marine propellers and hydrofoils (Knapp et al. 1970). Cavitation is commonly associated with undesired effects, such as pressure fluctuation, noise, vibration and erosion, which can lead to performance reduction, and can damage the hydrodynamic surface. Apart from these negative effects, one known benefit of cavitation is that it can reduce friction drag (Lecoffre 1999, Wang et al. 2001). For example, in high-speed hydrofoil boats, supercavitating propellers or vehicles the gaseous cavity enveloping the external body surface can provide a shield isol ated from the liquid, which helps to cut down the friction. 1.1.1 Cavitation Types in Fluids Different kinds of cavitation can be obser ved depending on flow conditions, fluid properties and different geometries. Each kind of cavitation has characteristics that distinguish it from the othe rs. Major types of cavitation are briefly described below: Traveling cavitation: Indivi dual transient cavities or bubbles form in the liquid, expand or shrink, and then move downs tream (Knapp et al. 1970). Typically, traveling cavitation is observed on hydrof oils at a small angle of attack. The number of nuclei present in the upcomi ng flow highly affects the geometries of the bubbles (Lecoffre 1999) (Figure 1-1A).

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2 Cloud cavitation: Cloud cavita tion is periodically cause d by vorticity shed into the flow field. It can associate with strong vibration, noise and erosion (Kawanami et al. 1997 ) (Figure 1-1B). Sheet cavitation: Sheet cavitation is a fixed, attached cavity or pocket cavitation in a quasi-steady sense (Knapp et al 1970) (Figure 1-1C). The interface between the liquid and the va por varies with flows. Supercavitation: In Supercavitation the cavity appears as the envelope of the whole solid body, and can be observed wh en underwater vehi cles operate at very high speeds or in projectiles wi th a speed of 500 m/s to 1500 m/s (Kirschner 2001). A typical supercavitating hydr ofoil is shown in Figure 1-1D. Vortex cavitation: Vortex cavities form in the cores of vortices in regions of high shear. They can occur on th e tips of rotating blades and can also appear in the separation zones of bluff bodies (Knapp et al. 1970) (Figure 1-1E). 1.1.2 Cavitation Inception and Parameter The criterion for cavitation inception based on a static approach can be formulated as Eq. (1.1), with P as local pressure and Pv as vapor pressure vPP (1.1) A parameter is needed to describe the fl ow condition relative to those for cavitating flows and to obtain a unique value for each set of dynamically similar cavitation conditions. In cavitation terminology denotes the parameter of similitude 2/2rv rPP V (1.2) In Eq. (1.2),,rrPV are the reference pressure and velocity, respectively. Usually, they take the infinity values. Unfortunately, is not a stringent parameter of similitude for cavitating flows. It is a necessary but not a sufficient condition. One reason is that nuclei in free stream flow can impact th e two-phase flow stru ctures (Lecoffre 1999).

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3 Figure 1-1. Different types of cavitation visualiza tion (Franc et al. 1995). A) Traveling cavitation. B) Cloud cavitation. C) Sheet Cavitation. D) Supercavitation. E) Vortex cavitation 1.2 Research Motivations and Objectives 1.2.1 Challenges and Motivations Cavitation occurs in various engineering systems, such as pumps, hydrofoils and underwater bodies. It is especi ally prone to occur when fluid machinery operates at offdesign conditions. Cavitation may cause not only the degradation of the machine performance but flow instability, no ise, vibration, and surface damage.

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4 Phenomenologically, cavitation of ten involves complex intera ctions of turbulence and phase-change dynamics, large density varia tion between phases up to a ratio of 1000, multiple time scales, and pressure fluctuations. These physical mechanisms are not well understood because of the complex unsteady fl ow structures associated with cavitation dynamics and turbulence. There are signifi cant computati onal issues regarding to accuracy, stability, efficiency and robustne ss of numerical algorithms for turbulent unsteady cavitating flows. Therefore, turbulent cavitating flow comput ations need to address both turbulence and cavitation modeling issues. Noticeable effo rts have been made in employing NavierStokes equations for simulations in this fi eld. Among the various modeling approaches, the transport equation-based cavitation models (TEM) have received more interest, and both steady and unsteady flow co mputations have been repo rted (Singhal et al. 1997, Merkle et al. 1998, Kunz et al. 2000, Ahuja et al. 2001, Venkateswaran et al. 2002, Senocak and Shyy 2002a,b, 2003, 2004a,b). Senocak and Shyy (2002b, 2004a) developed an interfacial dynamics-based cavitati on model (IDM) accounting for cavitation dynamics. Also, Senocak and Shyy (2002b, 2004a ) and Wu et al. (2003b) assessed the merits of alternative TEM models. They s howed that for steady-state computations, various cavitation models produce comparable pressure distributions. Despite the good agreement, noticeable differences have been observed in the predicted density field, especially in time-dependent computations. In IDM, an empirical factor is used to construct the interfacial velocity in time -dependent computations. Then the cavity interfacial velocity is linked to the local flui d velocity. Such an approach lacks generality because the interfacial velocity is supposed to be a function of the phase-change process.

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5 A much closer investigation is needed to further understand differences in model performance, especially in unsteady cavitating flows. The differences mentioned above also impl y that the compressibility characteristics embodied in each cavitation model are different. This aspect can be significant in the unsteady flow computations because the speed -of-sound directly and substantially affects the cavity time-dependent features (Seno cak and Shyy 2003, 2004b). Actually, the local speed-of-sound of the two-phase mixture can reduce by an order of magnitude from either value of pure liquid or vapor. It then leads to large enough M ach number variations around the cavity to make the compressibility effect substantial. The harmonic expression for speed-of-sound in an isothermal two-pha se mixture (Venkateswaran et al. 2002, Ahuja et al. 2003) can be presented as ) 1 ( 12 2 2 l l g g g g m mC C C (1.3) where l g m , are mixture density, vapor dens ity, liquid density, respectively, g is the vapor fraction, and l g mC C C , are speed-of-sound in mixture, in pure vapor, and in pure liquid, separately The behavior of mC is plotted in Figure 1-2 as a function of the vapor volume fraction. The speed-of-sound drops dramatically over a wide range of the mixture regime. Even though the bulk flow is of a very low Mach number flow, the local Mach number in the interface region may beco me large. Because of the lack of a dependable equation of state for the mi xture sound propagation, Senocak and Shyy (2003) presented two different numerical tr eatments of speed-of-sound, namely SoS-1 and SoS-2, and found that they had an impact on the unsteady cavitating behavior in the convergent-divergent nozzles. Th ey also suggested that SoS-2 was more likely to produce

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6 the correct unsteady behaviors. This underscore s that careful and accurate handling of the speed-of-sound is important. Currently, a r obust compressible model for cavitating flows is still the subject of research. Vaporvolumefraction Speed-of-Sound 0 0.25 0.5 0.75 1 0 200 400 600 800 1000 1200 1400Pureliquid Purevapor Flowinthisrangecanbe transonicevensupersonic Figure 1-2. Isothermal harmonic speed -of-sound in the two-phase mixture Besides cavitation modeling, the turbulen ce model can significantly influence the cavitating flow structures. Se rious implications of tur bulence modeling on cavitating flows were recently revealed by researchers (Senocak and Shyy 2002a, Kunz et al. 2003, Coutier-Delgosha et al. 2003, Wu et al. 2003b) They reported that high eddy viscosity of the original Launder-Spalding version of the k Reynolds-averaged Navier-Stokes (RANS) model (Launder and Spalding 1974) can dampen the vortex shedding motion and excessively attenuate th e cavitation inst abilities. Wu et al. (2003a) conducted the non-equilibrium modification, in cluding stationary and non-st ationary, in the cavitating flows over a hollow-jet valve, and observed no striking impact on the incipient cavitation. Consequently, simulation of phenomena su ch as periodic cavity inception and detachment requires improved modeling approaches. The Large Eddy Simulation (LES)

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7 approach, originally proposed by Smagorin sky (1963) and refined by many researchers (Piomelli 1999, Moin 2002, Sagaut 2003) is an actively pursued route to simulate turbulent flows. However, it is fundamenta lly difficult to find a grid-independent LES solution unless one explicitly assigns a filter scale (Moin 2002), making the state-of-theart immature for cavitating fl ow computations. On the other hand, attempts have also been made to employ the information obt ained from Direct Numerical Simulations (DNS) to supplement lower order models (e.g. Sandham et al. 2001). To our knowledge, no efforts have been reported to employ LES or DNS for turbulent cavitating flows of practical interest. Recently, efforts have been made to combine the filter concept and the RANS model in single-phase (Mavridis et al. 1998, Batten and Goldberg 2002, Nichols and Nelson 2003, Nakayama 2002, Breuer et al. 2003) and, recently and even more preliminary, cavitating flow com putations (Kunz et al. 2003). Hence, it is useful to identify ways to improve the predictive capability of the updated cavitation models and the current R ANS-based engineering turbulence closures, which can keep the advantages of RANS approaches and can be easily implemented in practical engineering applications with a clea r physical concept. This can help better capture unsteady characters (e.g. the sheddi ng hardly achieved in current turbulent cavitating flow simulations), and can provide better insight into the interactions between cavitation and turbulence models. This can al so help establish a dependable, robust and accurate computational CFD tool to analyze and minimize the cavitation effects in the design stage of fluid machin ery components or systems.

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8 1.2.2 Research Objectives Our goal is to develop and assess comput ational modeling techniques to provide better insight into unsteady turbulent cav itation dynamics, aimed at improving the handling of the above-mentioned issues in practical engineering problems. Objective 1: Investigate alternative turbulence mode ling strategies in the context of the Favre-averaged Navier-Stokes approach by doing the following: Develop a filter-based turbulence model (F BM) as the alternative to the original Launder-Spalding Model, both employing the k two-equation closure. Use the FBM in the cavitation simulations to provide insight into the dynamic interactions between tu rbulence and cavitation. Objective 2: Enhance the cavitation model prediction capability based on the recent transport equation-based model (TEM) by doing the following: Further examine the merits of the re cent achievements in the TEM, focusing on the interplay of the two-phase compressibi lity and turbulence by using correlation and spectral analysis with the direct ev aluation by experimental measurements, and generalize the speed-of-s ound numerical treatment. Adopt an improved approximation to c onstruct the interf acial velocity by accounting for phase transformation based on the recently developed interfacial dynamics-based cavitation model (IDM).

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9 CHAPTER 2 NAVIER-STOKES EQUATIONS AND TURBULENCE MODELS 2.1 Governing Equations The Navier-Stokes equations in their conservative form governing a Newtonian fluid without body forces and heat transfer are presented below in the Cartesian coordinates 0j ju tx (2.1) 2 ()() 3j il iijij jijjilu uu p uuu txxxxxx (2.2) The viscous stress tensor is given by 2 3j il ijij jilu uu xxx (2.3) In theory, direct numerical simulation (DNS) can be used to solve Equations (2.1)(2.3) (Rogallo and Moin 1984). However, in practice the limited computing resources prevent us from pursuing such endeavors for high-Reynolds-number flows. Since we are most interested in predicting the mean prope rties of the turbulent flow, we can first conduct an averaging procedure to simplif y the content of the equations. For timedependent flow computations, ensemble av eraging is an appropriate conceptual framework. Specifically, to avoid the a dditional terms involving the products of fluctuations between density and other variab les in variable density flows in Reynolds

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10 time averaging, Favre-averagi ng is preferred (Favre 1965) in the following form. Further details can be obtained from many refere nces (e.g. Shyy et al. 1997, Wilcox 1998) ;;0 (2.4) Then the Favre-averaged N-S equations become ()0ju t (2.5) ()()ijijijuuPuu t (2.6) 22 ()() 33jj ikik ijijij jikjikuu uuuu xxxxxx (2.7) Note that the viscosity fluctuation is neglected. The nonlinear terms ( ijuu ), namely the Reynolds stresses, need addi tional modeling. The Boussinesq’s eddyviscosity hypothesis for turbulence cl osure leads to the following form 2 ()() 3j R ik ijijtijt jiku uu uuk xxx (2.8) Finally, the Favre-averaged N-S e quations reach the following form ()0ju t (2.9) ()()R ijijijuuP t (2.10) 2 ()() 3j R ik ijijtij jiku uu xxx (2.11) Compared with the original N-S equations Favre-averaged N-S equations have the same apparent structure. The above equati ons can be cast in ge neralized curvilinear

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11 coordinates. The procedure for the transfor mation of these governing equations is well established (Shyy 1994, Thakur et al. 2002). 2.2 Turbulence Model There are several types of turbulence m odels. Commonly the classification can be presented as: (a) Algebraic (zero-equation) models (Baldwin and Lomax 1978); (b) Oneequation models (Bradshaw et al. 1967); (c) Two-equation model, such as k twoequation (Launder and Spalding 1974), k two-equation (Saffm an 1970); (d) Secondmoment closure models (Launder et al. 1975). Among the above models, the k turbulence model (Launder and Spalding 1974) has been popular because it is computationally tractable with deficiencies reasonably well documented (Shyy et al. 1997, Wilcox 1998). In this model, two partia l differential equations accounting for the transport of turbulent kinetic energy k and for dissipation rate are solved. The following transport equations follow the concep t of Launder and Spalding (1974) and are commonly adopted as 1 ; 2i ijij ju kuu x (2.12) () () [()]j t t jjkjuk kk txxx (2.13) 2 11() () P[()]j t t jjju CC txkkxx (2.14) where k and are the turbulence model constants, C1 and C2 are the turbulence model parameters that regulate the produc tion and destruction of dissipation of turbulence kinetic energy, respectively. The turbulent production term Pt and the turbulent viscosity are defined as

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12 PR i tij ju x ; 2tCk (2.15) Wall functions are used to address the effect of wall boundaries along with the k turbulence model. The empirical coeffici ents originally proposed by Launder and Spalding (1974), assuming local equilibrium between production and dissipation of turbulent kinetic energy, as given below 44 11C 92 12 C 3 1 0 1 k (2.16) In the following, we call it the Launder-Spalding model (LSM). 2.3 Filter-Based RANS Model Turbulence plays a very impor tant role in flow phenomena, especially since the Reynolds numbers are high in practical e ngineering problems. The Reynolds-averaged Navier-Stokes (RANS) and, for variable dens ity flows, the corres ponding Favre-averaged Navier-Stokes (FANS) models, such as the k two-equation closure, have been very popular in providing good predictio n for a wide variety of flows with presently available computational resources. However, RANS mode ls describe flows in a statistical sense typically leading to time-averaged pressu re and velocity fields. Generally these approaches are not able to di stinguish between quasi-period ic large-scale and turbulent chaotic small-scale features of the flow fi eld. The representation may lose the unsteady characteristics when the flow field is governed by both phenome na, even with the help of the non-equilibrium modifications on the set of em pirical constants (Wu et al. 2003a). It is clear that the statisti cal turbulence models have difficulties with the complex phenomena, such as flows past bluff bodies which involve separation and reattachment, unsteady vortex shedding and bimodal behavi ors, high turbulence, large-scale turbulent

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13 structures as well as curved shear layers (Frank and Rodi 1993, Rodi 1997, Shyy et al. 1997). On the other hand, Large Eddy Simulation (LES) operates with unsteady fields of physical values. Spatial filtering is applied in stead of averaging in time or ensemble, and turbulent stresses are divided into resolved and modeled parts, su ch as subgrid-scale models (SGS) with Smagorinsky’s hypothesi s (Breuer et al. 2003) Only the large energy-containing eddies are numerically re solved, accomplished by filtering out the high frequency component of the flow fields and using the low-pa ss-filtered form of the N-S equations to solve for the larg e-scale component (Kosovic 1997). In recent years, attempts have been made to remedy the gap between RANS and LES, called the Hybrid RANS/L ES methods (Nakayma and Vengadesan 2001, Breuer et al. 2003, Nelson and Nichols 2003). Various st rategies have been investigated as summarized below Ad-hoc models (Kato and Launder 1 993, Bosch and Rodi 1996,1998): using a rotation parameter to replace the original production term. Filter RANS/LES model (Koutmos and Mavirdis 1997): combining elements from both LES and standard eddy-visc osity approaches, by comparing the characteristic length with the mesh size—spatial filter to reconstruct the viscosity. Multiple Time-scale (MTS) Method (Hanja lic et al. 1980, Nichols and Nelson 2003): dividing the turbulent energy spect rum as two parts and breaking the standard k equations into two sets of equations. Detached Eddy Simulation (DES) (Spalart 1997, Roy et al. 2003): keeping the whole boundary layer (attached eddies ) to a RANS model and only the separated regions (detached eddies) to LES.

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14 2.3.1 Literature Review Firstly, we will review the recent studie s related to the hybrid RANS/LES studies according to the above categories with a summary in Table 2-1. Bosch and Rodi (1996) applied the ad-hoc model, which used a rotation term to reduce the turbulent production, to simulate the vortex shed ding past a square cylinder near a wall. Compared with the standard turbulence model, this modification produced better unsteady behavior a nd obtained good agreement with the experimental data. Following their previous study, Bosch and R odi (1998) adopted a 2D ensemble-averaged unsteady Navier-Stokes equation, with the ad -hoc model to compute the vortex shedding past a square cylinder. The numerical results agreed well with expe rimental measurement and other similar numerical results. They also carried out other versions of turbulence models in the same configurations for comparisons. They concluded the present modification was better. Rodi (1997) simulated tur bulent flows over two basic bluff bodies, 2-D square cylinder and 3-D surface-mounted-cube, using di fferent Reynolds numbers and a variety of LES and RANS methods. The various calc ulation results gene rally agreed with detailed experimental data. Assessment was given based on performance, cost and the potential of the various methods based on the comparison with the measurement. Koutmos and Mavridis (1997) combined LES and the standard k models to formulate the eddy-viscosity by comparing a mesh size with the characteristic length in the flow field. Then the turbulent viscosity was constructed in two different ways. The calculation of unsteady separa ted flows of square cylinder wake and backward-facing

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15 step re-circulating flows under low(Re= 5000) and high-Reynolds number (Re=37000) conditions agreed well with th e experimental measurements. Nagano et al. (1997) developed a low Reynolds number multiple time-scale turbulence model (MTS) which separated the turbulent energy spectrum into two parts, namely, production and transfer according to the wave-numbe r. Then the eddy-viscosity was approximated by solving both wall and hom ogeneous shear flows. The test results compared well with the DNS and experiment al data. They concluded for homogeneous shear flow the difference between the new model and the standard k model from the estimation of the characteristic time-scal es, not from the discrepancy in the eddy approximation. Nichols and Nelson (2003) employed di fferent turbulence models, including RANS, MTS and DES to simulate several uns teady flows. Based on the comparison with experimental data, they made the assessment of different models, and suggested that the MTS hybrid RANS/LES model need ed more investigations in term of grid and time-step sensitivities. Roy et al. (2003) examined Detach ed Eddy Simulation (DES) and RANS turbulence modeling in incompre ssible flow over a square cy linder. They found that the 2-D and 3-D simulations using DES are almost identical, and the results also compared well with experimental data, while the stead y-state RANS significantly over-predicted the recirculating vorte x behind the cylinder. Table 2-1. Summary of the hybrid RANS/LES studies Author and year Methods Conclusions Bosch and Rodi 1996, 1998 Ensemble-averaged N-S with ad-hoc modification in turbulence closure Capture better unsteady shedding than RANS and agreed well with experimental data

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16 Table 2-1. Continued Rodi 1997 RANS with LES as turbulence closure Generally agreed with experiment measurements and found 2-D could not resolve the 3-D effect Koutmos and Mavridis 1997 Ensemble-averaged N-S with a spatial filtering form LES Resolved good unsteady features in square cylinder and backward facing step under lowand highReynolds numbers Nagano et al. 1997 A low-Reynolds number MTS turbulence model in homogeneous shear flow Agreed with the DNS and experiment data Nichols and Nelson 2003 RANS, Hybrid RANS/LES, DES Hybrid models is better than RANS with comparable accuracy with LES Roy et al. 2003 N-S equation with a DES turbulence closure Compared well with experimental data 2.3.2 Filter-Based Model Concept (FBM) In the RANS models developed for steady st ate flows, the turbulent length scales predicted by the model extend over a large part of the flow domain. By imposing a filter on the flow, the turbulent scales smaller than the filter will not be resolved. When the filter size is smaller than the length scales re turned by the RANS models, this will allow the development of flow stru ctures that are not dissipate d by the modeled effective viscosity. The sub-filt er flow may be characterized by transport equations for turbulent energy, dissipation, and Reynolds stresses. In the present example, we choose to apply the LSM (Launder and Spalding 1974) as the corresponding RANS mo del. The filtering operation will be controlled by the size of the imposed filter and the size of the RANS length scale RANSl More detailed can be obtained fr om Johansen et al. (2004). The brief concept is given below. We st art from the RANS length scale 3 3/21/RANSlC k (2.17) We may now construct filtered eddy viscosity in the following general form

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17 33 3/23/222 ()() 33RANS teffRANStCklCklfCfC kk (2.18) where effl is the turbulent length scale survived dur ing the filtering ope ration. The scaling function f must impose the filter, and have limiting properties such as 33 3/23/2()1exp() fCC kk (2.19) Here and k are the non-resolved turbulent ener gy and dissipation rate separately, corresponding to unresolved turbulent length scales effl 3/2 3 3/2 31exp()effk lC Ck (2.20) The proposed model becomes identical to th e RANS model in the extremely coarse filter. In the case of a fine filter, the tur bulent length scale is controlled only by the imposed filter, and the LES type of model is obtained under this condition. The model is expected to have the following properties If filter size is identical to cell size a nd grid Reynolds numbers are sufficiently small (Kolmogorov scales ) the DNS model is recovered. If filter size becomes large, the RANS model is recovered. The statistical understanding of the adva ncement of the averaged flow during one time-step implies that the time step itself should be a part of the model. The filter should be almost independent to the grid in a specific computation, though it may vary with different geometries (different characteristic lengths). 2.3.3 Modeling Implementation The k two-equation is adopted in similar formula as Eq.(2.13)-(2.14). In Eq.(2.15), the turbulent viscosity modeled with a filter by the scaling function of Eq.(2.19) leads to the following

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18 2 3 3/20.09Min[1,]tk c k (2.21) with C3 ~1, here we choose 31.0 C This choice helped to assure that in near wall nodes the scaling function 3 3/2Min[1,] fc k will always return 1.0 f Then the use of standard wall functions is fu lly justified as the standard k model. In the limit of a very large filter size the viscosity becomes 20.09/tk and the standard k model is recovered. In the limit of a filter much smaller than the turbulent scale 3/2/ k the viscosity model becomes 1/2 30.09tCk (2.22) Then the FBM is identical to the one-equation LES mode ls of Schumann (1975) and Yoshizawa (1993). 2.4 Assessing FBM in Time-Dependent Single-Phase Flow For the filter-based model (FBM), we first carry out the studies on the single-phase simulations of vortex shedding past a square cylinder, and compare with experimental measurements (Lyn and Rodi 1994, Lyn et al. 1995). The parameters for the computations are gi ven in Table 2-2. The square cylinder, with a height D, is located at the center of a water channel. Figure 2-1 shows the geometry of the experimental set-up of Lyn and Rodi (1994) and Lyn et al. (1995), which is selected to guide the investigation, and th e non-dimensionalized coor dinates. It consists of a 2D square cylinder inside a channel. To reduce numerical errors in the vicinity of the cylinder, we adopt uniform grid spacing within the 4D3D domain surrounding the cylinder. Outside this block, the grid is slightly expanded toward the edges of the

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19 computational domain. All variables are non-di mensionalized by the free stream velocity and the cylinder height. The fluid properties are held unchanged and the Mach number is zero. At the inlet, the mean velocity Uin is uniform and follows the horizontal direction. The inlet turbulent intensity is 2%. Based on the definition of the turbulent viscosity formula adopted in the LSM and by assigning the turbulent Reynolds number to be 200, we determine the inlet dissipation rate. The flow variables are extrapolated at the outlet. The wall function (Shyy et al. 1997) is empl oyed for the solid boundary treatment. Table 2-2. Parameters us ed in the computation Uin (m/s) (kg/m3) (kg/m.s) Re D (m) k (m2/s2) (m2/s3) 0.535 1000 31.00210 21357 0.04 41.717410 62.510 1.0 0.56D=0.04 0.20 6.5DInlet Outlet Solidboundary SolidboundaryA x/D y/D 0 1 2 3 4 5 6 0 1 2 3 4CeterlineydirectionB Figure 2-1. Geometry configuration of squa re cylinder. A) Computational domain. B) Non-dimensional coordinates To investigate the effect of grid reso lution on numerical accuracy, we used three levels of grids including fine, intermediate and coarse grids, which have 25, 20 and 10 nodes on each side of the cylinder respectiv ely. For comparison, the Launder-Spalding model (LSM) is also carried out on coarse and fine grids. To invest igate the sensitivity, we use four different filter sizes: 0.15D, 0.3D, 0.6D and 0.9D on coarse grid (10 intervals), and 0.15D, 0.25D, 0.6D and 0.9D on fine grid (25 intervals). Unless explicitly mentioned, 0.15D is used as the reference filter size.

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20 If 3/2()/ k exceeds 1.0, the filter scaling function 3/2Min[1.0,()/] f k will take a value of 1.0. This treatment enables one to apply the wall f unction of LSM for the solid wall treatment, which is confirmed by the outcome presented in Figure 2-2. A pressure reference point was located at position x/D=0.0 and y/D=0.50, which is the midpoint of the cylinder upper wall. For th e fine grid we show predicted pressure development at the wall reference point in Figure 2-3A. We see a modulated and inexact periodic signal that qualitatively agrees with the experiments (Lyn and Rodi, 1994). The FBM produces pressure oscillations correspond ing to amplitudes of pressure coefficient of approximately 2.5, while pressure coeffi cient amplitude predicted by the LSM varies from approximately 0.1 to less than 0.05. We also found that the LSM tends to be more time independent on the finer grid and the osc illations die out eventually. The FFT of the Cp by filter-based model (Figure 2-3A) shows multiple frequencies with a dominant one at around 2.15 (Figure 2-3B). A B Figure 2-2. Streamline snap -shot on fine grid with 0.15D Red shadow indicates 3 3/2(1.0,)1.0 C fMin k for recovering the LSM. At other regions the filter function is employed. A) Zoom in. B) Full view

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21 t/s Cp 5 10 15 20 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 Launder-Spalding Filter-based A Frequency Power 1 2 3 4 5 200 400 600 800 1000 1200 1400 1600 1800B Figure 2-3. Pressure behavior s at the reference point on the fine grid. A) Pressure coefficients Cp time evolutions LSM and FBM. B) Cp FFT, FBM Figure 2-4 shows the time-averaged hor izontal velocities along the centerline behind the cylinder. The LSM pred icts too long wake lengths rl behind the cylinder at about 3.0 D for both grids (coarse and fine grids) which are almost identical to the numerical results by Rodi (1997) Further results of the FBM, using a constant filter size of 0.15D are shown in the same figure. For the intermediate and fine grids, the FBM results quantitatively agree with the experimental data of L yn et al. (1995) including the asymmetric behavior. Even for the coarse gr id, the size of the sepa ration zone is well reproduced. However, the reverse velocity in the wake as well as the velocity defect in the remaining part of the wake is slightly under-pre dicted. In the case of the coarse grid the resolution of the shear layer at the cyli nder wall is sub-critical and this affects both the onset of each shedding cycle as well as th e magnitude of the vorticity in the wake. Flow structures of the solutions at a given time instant on coar se and fine grids, and with LSM and FBM are highlighted in Figure 2-5. For the FBM, as the grid resolution is refined, the vortex structure becomes more dispersed and less confined in the wake region. In contrast, for the LSM, the impact of the turbulent viscos ity is dominant, and

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22 the fine grid solution exhibits less fluctuati on in time. The streamlines at one time instant for both models are presented in Figure 2-6. The LSM only catches two separated pockets at the cylinder upper and lower shoul ders, and two wakes behind the cylinder. The shedding motion is almost gone. The FBM is able to capture th e sharp separation in the shear layer, which agrees well with th e experimental observation from Lyn and Rodi (1994) and Lyn et al. (1995). From Table 2-3 we see that the predic ted Strouhal numbers are about 20% higher than the St=0.135 from experiment by Lyn and Rodi (1994). By comparing the present solutions on different grids using the FBM with the same filter size, the variations of the Strouhal number are less than 4%. Hence, the only significant result is the Strouhal number being approximately 20% larger. This over-prediction may be due to the 2D effect, compared with the 3D measurement in the experiment. The large-scale structures of the flow have a three-dimensional natu re and we do not expect to reproduce all features of this flow correctly in two dimens ions. This will be more pressing as the filter size and grid is reduced and we depart more and more from the LSM. This seems to be consistent with the results from Yu and Kareem (1997) who needed to use a larger Smagorinsky coefficient to reproduce the Str ouhal number in 2D compared to their full 3D simulations. Another interesting finding is that the LSM required much smaller time steps, compared with the FBM in order to get stable and co nvergent solutions using the PISO algorithm. From Table 23 it is seen that the time steps have to be reduced substantially in order to empl oy the LSM. Hence, the time consumption with the FBM is around half smaller in the present study.

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23 x/D uavg/Uin 5 10 15 -0.2 0 0.2 0.4 0.6 0.8 1 LSM,coarse LSM,fine =0.15D,coarse =0.15D,intermediate =0.15D,fine Experiment Figure 2-4. Time-averaged U-ve locity along the horizontal ce nterline behind the cylinder. Experimental data are from Lyn a nd Rodi (1994) and Lyn et al. (1995) A B Figure 2-5. Snap-shots of veloc ity contour. Color raster plot of axial velocities (red is largest, blue lowest). A) FBM: coarse grid, velocity range is -0.31~1.53. B) FBM: fine grid, velocity range is -0.6 6~1.88. C) LSM: coarse grid, velocity range is -0.09 to 1.25. D) LSM: fine grid, velocity range is -0.31 to 1.20 Coarse grid Fine grid Coarse grid Fine grid

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24 A B Figure 2-6. Streamline snap-shots of two tur bulence models on fine grid. A) LSM. B) FBM Table 2-3. Comparisons of di fferent turbulence models. /RlD is the relative position of the reattachment, measured from the cy linder center (coarse: 10 intervals, intermediate: 20 intervals, fine: 25 intervals on each cylinder face) Model Grid (nx ny) Filter Size Time Step dt/(D/Uin) /RlD St 0.15 D 0.0134 1.22 0.155 0.30 D 0.0134 1.40 0.151 0.60 D 0.0134 2.12 0.143 Coarse (16292) 0.90 D 0.0134 2.73 0.137 Intermediate (290190) 0.15 D 0.0134 1.25 0.163 0.15 D 0.0669 1.23 0.161 0.25 D 0.0669 1.25 0.148 0.60 D 0.0669 1.98 0.140 FBM Fine (300195) 0.90 D 0.0669 2.64 0.134 Coarse (16292) 0.00268 3.03 0.124 LSM Fine (300195) 0.000803 2.80 0.125 Exp. Lyn et al. (1995) 1.38 0.135 The predicted velocity profile s are given in Figure 2-7 to Figure 2-10. One thing to be noted is that the experiments of Lyn and Rodi (1994) and L yn et al. (1995) were recorded on a single side, assuming that th e ensemble and time averaged flow was symmetrical across the axial symmetry line. As a result, data were recorded only at one side of the symmetry line. Hence, we have mirrored the data to enable the visual

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25 comparison with the current numerical results. By close inspection of the experimental data, we find that the transversal time av eraged velocity on th e symmetry line is nonvanishing. This indicates that the data are not completely symmetrical. Along the vertical centerlin e (Figure 2-7), we see th at the LSM is unable to correctly reproduce the separation in the sh ear layer. The shoulders on the velocity profiles are not captured, presumably due to the high effective viscosity in the incoming flow. The FBM solution on the coarse grid is similar to the fine grid solution, but only as a result of poor resolution of the shear layer. The profiles for the FBM match the experimental data of Lyn and Rodi (1994). Th e time-averaged transversal velocity in the wake, one cylinder diameter behind the center of the cylinder, is shown in Figure 2-8B, where the LSM fails to capture the correct tr ansversal velocity. For both intermediate and fine grids, the FBM again gi ves very good results. The timeaveraged axial velocity is shown in Figure 2-8A. Here we see that the FBM solutions become more asymmetrical by grid refinement. The asymmetry in the time -averaged solutions is clearly seen from Figure 2-8 to Figure 2-9, and has been confirmed independen tly by calculations using a code that employs a staggered grid arrangeme nt. The development of asymmetrical time averaged solutions seemed to be caused by the initial bias of the solution, which is pathdependent. Asymmetries in the time-averaged fields were noted by Sohankar et al. (1999). They reported that the asymmetries became more distinct as the Reynolds number increased. This is consistent with our findings. As suggested by Sohankar et al. (1999), the reason could be the two-dimensional geometry that is forced on the flow. In two dimensions wall attachment may be reinforced due to the imposed twodimensionality. This can only be resolved by comparing full 3D computations.

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26 y/D u/Uin -1 0 1 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 LSM,coarse LSM,fine =0.15D,coarse =0.15D,intermediate =0.15D,fine Experiment Figure 2-7. Time-averaged U-ve locity along y at x/D=0.0. E xperimental data are from Lyn and Rodi (1994) an d Lyn et al. (1995) y/D u/Uin -2 -1 0 1 2 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 LSM,coarse LSM,fine =0.15D,coarse =0.15D,intermediate =0.15D,fine Experiment A y/D vavg/Uin -2 -1 0 1 2 -0.2 0 0.2 LSM,coarse LSM,fine =0.15D,coarse =0.15D,intermediate =0.15D,fine Experiment B Figure 2-8. Time-averaged velocities along y at x/D=1.0. Experimental data are from Lyn and Rodi (1994) and Lyn et al. (199 5). A) U-velocity. B) V-velocity y/D u/Uin -3 -2 -1 0 1 2 3 0 0.2 0.4 0.6 0.8 1 1.2 LSM,coarse LSM,fine =0.15D,coarse =0.15D,intermediate =0.15D,fine Experiment A y/D u/Uin -4 -3 -2 -1 0 1 2 3 4 0.4 0.6 0.8 1 1.2 LSM,coarse LSM,fine =0.15D,coarse =0.15D,intermediate =0.15D,fine Experiment B Figure 2-9. Time-averaged u-ve locity along y direction. Expe rimental data are from Lyn and Rodi (1994) and Lyn et al. (1995) A) At x/D=3.0. B) At x/D=8.0

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27 The predicted kinetic energies along the horizontal centerline are compared with experiments in Figure 2-10. We find that the LSM significa ntly under-predicts the total kinetic energy, including the periodic and turb ulent parts, mainly due to poor resolution of the large scale structures in the wake. Th e FBM results agree well with the experiment, even in the coarse grid, in terms of the magnitude and the peak position (Figure 2-10A). The turbulent kinetic energy (the stochastic part only) is under-p redicted for both LSM and FBM, as shown in Figure 2-10B. Compar ed with the total and turbulent kinetic energy, the turbulent part dominates for the LSM, while the periodic part is dominant for the FBM. Since the FBM over-predicts the turb ulent kinetic energy, it slightly overpredicts the periodic kinetic energy at the peak. Next, we investigate the effect of the f ilter size. As mentioned above, we present the results on the coarse grid with four filter sizes 0.15D, 0.3D, 0.6D and 0.9D, and fine grid with four filter size 0.15D, 0.25D, 0.6D and 0.9D. Figure 2-11 shows the transversal v-velocity of both coarse and fine grids with different filter sizes. The profiles exhibit a clear trend toward the LSM when the filter size increases. Table 2-3 further demonstrates that the computed Strouhal number and the reattachment length rl also move toward the LSM as the filter size increase on the coarse grid. Also, the kinetic energy shows a similar trend with the various filter sizes on both coarse and fine grid (Figure 2-12). Overall, the solution with 0.15D agrees better with the experiment. By inspection of the results it appears that the present filter-based calculations give quite regular solutions. In many calculations of the shedding from a square cylinder, perturbations of the flow ar e induced by "numerical noise" that may be caused by a large number of different phenomena. Examples of such phenomena are unbounded convective

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28 fluxes, reduced numerical order caused by expa nded or unstructured grids, and too large time step sizes. In the filter-based mode l randomness can be added to the solution by applying a random force field, similar to what is used in Renormalization-Group analyses (Smith and Woodruff, 1998). Without i nducing randomness to the flow by inlet conditions or by random forcing we expect that the present model will produce regular oscillating solutions similar to URANS (unst eady RANS) calculations by Iaccarino et al. (2003). The viscosity contours of LSM and FB M are shown in Figure 2-13. The LSM predicts a very high viscosity in the upstream of the cyli nder, and the distribution is almost symmetric. The FBM significantly reduces the viscosity around 2-orders of magnitude. As discussed above, the high effec tive viscosity in the coming flow can damp out the unsteadiness behind the cylinder. x/D u'/Uin 1 2 3 4 5 6 7 8 0 0.2 0.4 0.6 LSM,coarse LSM,fine =0.15D,coarse =0.15D,intermediate =0.15D,fine Experiment A x/D u'/Uin 1 2 3 4 5 6 7 0 0.05 0.1 0.15 0.2 0.25 0.3 LSM,coarse LSM,fine =0.15D,coarse =0.15D,intermediate =0.15D,fine Experiment B Figure 2-10. Mean kinetic ener gy on different grids. Experime ntal data are from Lyn and Rodi (1994) and Lyn et al (1995). A) Total kinetic energy (mean + turbulent). B) Turbulent kinetic energy (stochastic)

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29 y/D v/Uin -2 0 2 -0.2 0 0.2 LSM,coarse =0.90D,coarse =0.60D,coarse =0.30D,coarse =0.15D,coarse Experiment A y/D v/Uin -2 -1 0 1 2 -0.2 0 0.2 LSM,fine =0.90D,fine =0.60D,fine =0.25D,fine =0.15D,fine Experiment B Figure 2-11. Comparisons of different filter sizes on time-averaged v-velocity along y at x/D=1.0. Experimental data are from Lyn and Rodi (1994) and Lyn et al. (1995). A) Coarse gr id. B) Fine grid x/D Totalkf/Uin 2 1 2 3 4 5 6 7 0 0.2 0.4 0.6 LSM,coarse =0.90D,coarse =0.60D,coarse =0.30D,coarse =0.15D,coarse Experiment A x/D Turbulentkt/Uin 2 1 2 3 4 5 6 7 0 0.05 0.1 0.15 0.2 0.25 0.3 LSM,coarse =0.90D,coarse =0.60D,coarse =0.30D,coarse =0.15D,coarse Experiment B x/D Totalkf/Uin 2 1 2 3 4 5 6 7 0 0.2 0.4 0.6 LSM,fine =0.90D,fine =0.60D,fine =0.25D,fine =0.15D,fine Experiment C x/D Turbulentkt/Uin 2 1 2 3 4 5 6 7 0 0.05 0.1 0.15 0.2 0.25 0.3 LSM,fine =0.90D,fine =0.60D,fine =0.25D,fine =0.15D,fine Experiment D Figure 2-12. Comparisons of different filter sizes on kinetic energy. Experimental data are from Lyn and Rodi (1994) and Lyn et al. (1995). A) Co arse grid: total kinetic energy (periodic + tu rbulent). B) Coarse grid: turbulent kinetic energy (stochastic). C) Fine grid: total kinetic energy (periodic + tu rbulent). D) Fine grid: turbulent kinetic energy (stochastic)

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30 A B Figure 2-13. Time-averaged viscosity contours of different turbulence models on fine grid. A) LSM. B) FBM In summary, a filtered-Navier-Stokes model, originated from the LSM, is applied to vortex shedding from a square cylinder. The introduction of the filter led to an effective viscosity that depends on both turbul ent quantities and the filter size itself. The method is capable of working with standard wall-functions, allowing much coarser grids in the boundary layer compared to common LES methods. Presently the use of wall functions is justified as y values for near wall nodes are greater than 20 using the fine grid (25-node). Based on the discussions above, we have the following conclusions Both coarse and fine grids reproduce th e time averaged experimental results quantitatively. However, by refining the grid we see improved results for the velocity profiles. For the investigated filter size of 0.15D the solutions on both intermediate and fine grids are in agreement with experimental data, demonstrating that the model can produ ce better resolutions based on the LSM by allowing the numerical scheme to simu late the fluid physics at the scales where numerical resolutions are satisfactory. The increase of filter size shows that the filter-based model smoothly approaches the LSM. The filter-based model is shown to produce improvement over the LSM for all grids investigated. The Strouhal number of FBM is generall y over-predicted. Further investigation is needed to investigate whether it can be resolved by full 3D simulations.

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31 Generally, the model is expected to yiel d better results if full 3D solutions are performed, since the large scale 3D cohe rent flow structures can be resolved.

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32 CHAPTER 3 CAVITATION MODELS In this chapter, the single set of the governing equations for the flow field, including the continuity and momentum equations which were introduced in Chapter2 along with a transport equation for the cavitati on, is presented firs t. Then, th e cavitation models utilized for the presen t study are provided, al ong with a review of selected recent studies. The above equations coupled with turbulence models, including LaunderSpalding model (LSM) and Filter-based model (F BM), presented in the previous chapter will complete the whole system of equa tions for the turbulent cavitating flow computations. 3.1 Governing Equations The Favre-averaged Navier-Stokes equa tions, in their conservative form, are employed for incompressible flows. The cav itation is governed by a volume fraction transport equation. The equations are pres ented below in the Ca rtesian coordinates 0mj m ju tx (3.1) () () 2 [()()] 3mijj miik tij jijjikuuu uuu p txxxxxx (3.2) ()()l lj jumm tx (3.3) where m is the mixture density, ui is the velocity component in Cartesian coordinates, t is the time, p is the pressure, l is volume fraction of liquid, m is the condensation rate

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33 and m is the evaporation rate, t is the turbulent viscosity, and is the laminar viscosity of the mixture defined as (1)llvl (3.4) with l is the laminar viscosity of the liquid and v is the viscosity of the vapor. The mixture density and turbulent viscos ity are respectively defined as below 1mllvl (3.5) 2m tCk (3.6) where l is the density of the liquid and v is the density of the vapor, C is the turbulence model constant and its value is 0.09 generally, k is the turbulence kinetic energy and is the turbulence dissipation rate. The relevant details of the different turbulence closures have been review ed in detail in Section 2.2-2.4. 3.2 Literature Review of Cavitation Studies Cavitation can produce negative effects in fluid machinery components and systems. Details of the existence, extent a nd effects of cavitation can help to minimize cavitation effects and optimize the designs. E xperiments have been conducted in the past few decades for different types of fluid m achinery devices and components. Ruggeri and Moor (1969) investigated methods that predicted the performance of pumps under cavitating condition. Typically, the strategy to predict the Net Positive Suction Head (NPSH) was developed. Stutz and Reboud ( 1997, 2000) studied the two-phase flow structure of unsteady sheet cavitation in a convergent-di vergent nozzle. Wang (1999) used high-speed camera and Laser Light Sh eet (LLS) to observe the cavitation in a hollow-jet valve under different openings. Wang et al (2001) studi ed broad cavitation

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34 regimes of turbulent cavitati ng flows, covering from inception to supercavitation, over a Clark-Y hydrofoil under two diffe rent angle-of-attacks. Besides experimentation, using CFD simu lation to analyze ca vitation phenomena has become convenient and popular with the development of computer hardware and software. A number of cavitation models have been developed. These studies can be put into two categories, namely interface tracking methods and homogeneous equilibrium flow models. A review of the representative studies is presente d by Wang et al. (2001). Taking the advantage of homoge neous equilibrium flow theo ry, the mixture concept is introduced. And a unique set of mass and mo mentum equations along with turbulence and cavitation models is solved in the whole flow field. Within the homogeneous equilibrium flow theory, two approaches can model the cavitation dynamics. The first one is the arbitrary barotr opic equation model, which s uggests that the relationship between density and pressure is () f p and the second one is the transport equationbased model (TEM). Barotopic equations were proposed by Delannoy and Kueny (1990). They assumed that density was a continuous function of pressure where both pure phases were incompressible, and the phase change could be fitted by a sine curve. They could not correctly produce the unsteady behavior s in the venturi simulation. Arbitrary barotropic equation models (density is onl y a function of pressu re) do not have the potential to capture baroclinic vorticity pr oduction because the baro clinic term of the vorticity transport equation yields zero by definition (Senocak 2002, Senocak and Shyy 2004a). In addition to agreement with the experimental study of Gopalan and Katz (2000), the above two references have demonstr ated computationally that the baroclinic vorticity generation is important in the clos ure region. In TEM, a transport equation for

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35 either mass or volume fraction, with appr opriate source terms to regulate the mass transfer between vapor and liquid phases, is adopted. The apparent advantage of this model comes from the convective character of the equation, which allows modeling of the impact of inertial forces on cavities lik e elongation, detachment and drift of cavity bubbles, especially in complex 3-D interface situations (Wang et al. 2001). Different modeling concepts embodying qualitatively simila r source terms with alternate numerical techniques have been proposed by various res earchers (Singhal et al 1997, Merkle et al. 1998, Kunz et al. 2000, Ahuja et al 2001, Senocak and Shyy 2003, 2004a,b). Numerically, Singhal et al. (1997, 2002) and Senocak and Shyy (2002a,b, 2003, 2004a,b) utilized pressure-based algorithms, while Me rkle et al. (1998) a nd Kunz et al. (2000) employed the artificial compressi bility method. In addition, Vaidyanathan et al. (2003) performed a sensitivity analysis on a tran sport equation-based cavitation model to optimize the coefficients of its source terms. More recently, attempts have been carried out on numerical simulation of cavitating flow s in turbomachines, such as pumps and inducers. Medvitz et al. (2002) utilized th e pre-conditioned two-phase N-S equations to analyze the performance of cav itating flow in a centrifugal pump. Hosangadi et al. (2004) simulated the cavitating flow in the full geom etry of a 3-blade Simplex inducer. CoutiesDelgosha et al. (2005) pres ented a 3D model for cavita ting flow through a 4-blade inducer adopting only one blad e-to-blade passage. However, a complete robust and accurate CFD framework in this fi eld is still a longstanding work. 3.2.1 Cavitation Compressibility Studies Since the flow fields are rich in complexi ty at the cavity interf ace region, as pointed out in chapter 1, compressibility effect is one of the major issues in cavitation studies.

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36 Nishiyama (1977) developed a linearized subsonic theory for supercavitating hydrofoils to clarify the compre ssibility correction of Mach number effect in pure water. The 2-D & 3-D steady characteristics of s uper-cavitating hydrofoils in subsonic flow were compared to those in incompressible flow. The essential diffe rences, including the compressibility effect between vapor and liq uid and the co-relati on between cavitation and Mach number, were shown in detail. Saurel and Cocchi (1999) focused on cavit ation in the wake of a high-velocity underwater projectile. They pr esented a physical model base d on the Euler equations in terms of two-phase mixture properties. Th e mathematical closure was achieved by providing state equations for the possible thermodynamic states: compressible liquid, two-phase mixture and pure vapor. The m odel was then solved using a hybrid computational scheme to accurately main tain the property profiles across the discontinuities. The results demonstrated a reasonable agreement compared with the known analytical solutions. Saurel and Lemetayer (2001) proposed a compressible multiphase unconditionally hyperbolic model to deal w ith a wide range of applic ation: interfaces between compressible materials, shock wave in c ondensed multiphase mixtures, homogeneous two-phase flows and cavitation in liquids. The model did no t require a mixture equation of state and was able to pr ovide themodynamic variables fo r each phase. The results and validations with analytical solutions were provided. Kunz et al. (2000), followed by Venkate swaran et al. ( 2002), developed a preconditioned time-marching algorithm for the computation of multiphase mixture flows based on carrying out perturba tion expansions of the underl ying time-dependent system.

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37 The method was efficient and accurate in bot h incompressible and compressible flow regimes. However, it was not a fully compre ssible formulation for the flow fields. It could not be accurate if the bulk flows were supersonic, which indicated that compressibility should be considered even in pure liquid phase. Senocak and Shyy (2004a,b) used the pressure-based method to simulate the cavitating flows through conve rgent-divergent no zzles. They presented two different treatments of speed-of-sound (SoS), incl uding SoS-1 and SoS-2, to build up the relationship between the pressure and density for steady and unsteady cavitating flows. The comparison with the experimental data and other simulation results displayed that SoS-2 was more likely better in predicting the unsteady characteristics. Table 3-1. Overview of the compressible cavitation studies Author and year Methods (Analytic/Num erical) Conclusions(Analytic/Numerical) Nishiyama 1977 Proposed linearized theory linearized relationships between velocity, pressure and sound speed Saurel and Cocchi 1999 A physical model based on the Euler equations, with state equations Reasonable agreed with analytic solutions and could reliably deal with strong shock wave and complex EOS Saurel and Lemetayer 2001 An unconditional hyperbolic model, with accurate treatment for nonconservative form to ensure mass conservation Compared with analytic solutions And clouded it could deal with a wide range of application, such as shock wave, homogeneous twophase flow and cavitation. Kunz et al. 2000, Venkateswaran et al. 2002 A preconditioned time-marching CFD method for isothermal multiphase mixture flows, associated artificial compressibility Agreed with experiment data under good precondition, and suggested the compressible treatment could improve the dynamics description than incompressible computations Senocak and Shyy 2004a,b Two numerical models for of speed-of-sound in steady and unsteady cavitating flows Agreed well with experimental data and other numerical simulations, and demonstrated the capability in unsteady computation.

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38 3.2.2 Cavitation Studies on Fluid Ma chinery Components and Systems Ruggeri and Moor (1969) investigated similarity methods to predict the performance of pumps under cavitating condition for different temperatures, fluids and operating conditions. Typically, the strategy to predict the NPSH was developed based on two sets of available test data for each pump at the concerned range of operating conditions. Pumps performance under various flow conditions such as discharge coefficient, impellor frequency as assessed fo r different fluids, such as water, LH2 and butane. Wang (1999) studied the cavitating phe nomena in a hollow-jet valve under different openings and cavitation conditions using high speed camera and LLS. Further more, he also studied the indu ced vibration mechanism, cavitation damage characteristics and ventilation effect in the valve. Athavale and Singhal (2001) presented a homogeneous two-phase approach with a transport equation for vapor, and the redu ced Rayleigh-Plesset equations for bubble dynamics based on local pressure and velo city. Compared with the experimental measurements, they obtained reasonable pred ictions of cavitating flows in two typical rocket turbo-pump elements: i nducer and centrifugal impeller. Lee et al. (2001a,b) analyzed the cavitation of the pump inducer sequentially. They found that the forward-swept blade demonstr ated more resistance to vortex cavitation than the conventional one. They also observe d that the cavity length of surface cavitation at various conditions was closely related with the cavitating number and event duration, but inconsistent with the m ode predicted by linear cascade analysis in the cross-flow plane at far off-design points.

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39 Medvitz et al. (2002) used the homogeneous two-phase RANS e quations to analyze the performance of 7-blade centrifugal pumps under cavitating conditions. By using the quasi-3D analysis, the numerical results were found to be qu alitatively comparable with the experimental measurements across a wide range of flow coeffi cients and cavitation numbers, including off-design flow blade cavitation and breakdown. Duttweiler and Brennen (2002) experime ntally investigat ed a previously unrecognized instability on a cav itating propeller in a wate r tunnel. The cavitation on blades and in the tip vortices was explored through visual obse rvation. The cyclic behavior of the attached blad e cavities had strong similarities to that of partial cavity oscillation on single hydrofoils. Furthermore, the reduced frequency of the instability was consistent with the partial cavit y instability on a single hydrofoil. Friedrichs and Kosyna (2002) described an experimental investigation of two similar centrifugal pump impellers at low sp ecific speed. The high-speed-film displayed rotating cavitation over a wide range of part loaded operati ng points, and i llustrated the development of this instability mechanism, which was mainly driven by an iteration of the cavity closure region and the subsequent blades. Hosangadi et al. (2004) numerically simulated the perf ormance characteristics of an inducer using an N-S methodol ogy coupled with a two-equati on turbulence model. The simulations were performed at a fixed flow rate with different Net suction Specific Speeds (NSS). The numerical results showed the head loss was related to the extent of cavitation blockage. The breakdown NSS and th e head loss prediction agreed well with experimental data. The insights provided a sequence of traveling and alternate cavitation phenomena in blade passages.

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40 Coutier-Delgosha et al. (2005) presented a 3D model for a cavitating flow in 2-D venture and in a 4-blade inducer with the co mparison to experimental data and visuals. They assumed symmetry in the inducer and on ly considered one bl ade-to-blade channel in the inducer. The quasi-static results s howed a consistent agreement with the experiment, but did not catc h the performance breakdown. Table 3-2. Overview of cav itation on pumps, propellers, i nducers and turbine blades Author and year Problem study (Experimental) Methods (Numerical) Findings(Experimental) Conclusions(Numerical) Ruggeri and Moore 1969 Similarity to predict the pumps performance under cavitation conditions Provide strategies to predict the pump NPSH Wang 1999 High-speed camera and LLS in a hollow-jet valve Cavitation, induced vibration, damage characteristics and the ventilated cavitation effect Athavale and Singhal 2001 A homogeneous two-phase approach, with Rayleigh-Plesset equations. Plausibly agreed the experiment, with robustness and stability in numerical convergence behaviors Lee et al. 2001a,b Tested the inducer performance, inlet pressure signals and event characteristics Forward-swept blade was more resistive to vortex cavitation than the conventional one Medvitz et al. 2002 Homogeneous two-phase RANS model Obtained qualitative performance trends with experimental data in off-design flow, blade cavitation and breakdown Duttweiler and Brennen 2002 Surge instability on a cavitation propeller in water tunnel Cyclic oscillation of the attached blade cavities Friedrichs and Kosyna 2002 High-speed camera in a centrifugal pump impeller cavitation Rotating cavitation was mainly driven by an interaction of cavity and the leading edge of the following blades. Hosangadi et al. 2004 N-S equations with an compressible multiphase model Well agreed with the experimental data in terms of the head loss and breakdown NSS, and pointed out the performance loss was strongly correlated with cavitation blockage. Coutier-Delgosha et al. 2005 3-d code FINETURBOTM, with a time-marching algorithm and lowspeed preconditioner for low Mach number flows Obtained good agreement in 2-D venturi computation, And qualitatively agreed with the experimental measurements of efficiency and cavity visuals in 3-D inducer.

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41 3.3 Transport Equation-based Cavitation Model (TEM) Cavitation process is governed by ther modynamics and kinetics of the phase change dynamics in the system. These issues are modeled with th e aid of a transport equation with source terms regulating the evap oration and condensati on of the phases. In the present study, two different cavitation mode ls, with similar source terms for Eq.(3.3), are presented: Heuristic model (HM). The liquid volume fraction is chosen as the dependent variable in the transport equation (Kunz et al. 2000). The evaporation term m is a function of pressure whereas the condensation term m is a function of the volume fraction t C m t U p p C ml l l v prod l l v l v dest ) 1 ( ) 2 / ( ) 0 (2 2 Min (3.7) where 510 0 9 destC and 410 0 3 prodC are empirical constant values, t is the ratio of the characteristic length scale to the reference velocity scale (/) L U Interfacial dynamics-based cavitation model (IDM). The interfacial dynamicsbased cavitation model (IDM) was deve loped by Senocak and Shyy (2004a,b). A hypothetical interface is assumed to lie in the liquid-vapor mixture region ( Figure 3-1). By applying the mass and normal momentum co nservation equations at the interface, and normalizing with a characteristic time scale /tLU the evaporation m and condensation m are given by

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42 t V V p p m t V V p p mv l n I n v v l v l n I n v v v l l) ( ) ( ) 0 ( 1 ( ) ( ) ( ) 0 (2 , 2 , )Max Min (3.8) with vapor phase normal velocity, the inte rface normal, and the in terfacial velocity 9 0 1 1 ,, , f V f V n n u Vn v v l v l n I l l n v (3.9) where f is found by computational sa tisfaction and a value of -0.9 is used. For steady state, we have ,0.0InV This approach will be called original IDM. Figure 3-1. Sketch of a cav ity in homogeneous flow Regarding the above original IDM, the cavit y interfacial velocity was linked to the local fluid velocity in the time-dependent computations. Such an approach lacks generality because the interfaci al velocity is supposed to be a function of the phase change process. Fundamentally, the interfacial velocity can possibly be estimated more accurately based on the moving boundary comput ational techniques (Shyy et al. 2004).

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43 Here, we estimate the interfacial velocity via an approximate procedure by accounting for the phase transformation process in each co mputational cell. By integrating Eq. (3.8) through the control volume, we have mV and mV The net interface velocity (the interface velocity rela tive to the local flow field) becomes ,()()net In net InVAm mabsmVabsmV V AA (3.10) where A is the interface ar ea between vapor and liquid phases. Practically, the control volume face area, CV A rea, is projected to the interface normal direction, which can be obtained by ta king the gradient of the volume fraction, denoted as S, as shown in Figure 3-2 for the 2-D situation 22/L xy L CV xxynninj Area S nnn (3.11) n AreaCVInterface VI,nVapor Liquid Figure 3-2. Interface vector sketch in a CV Substituting S into Eq. (3.10) leads to ,()()net InmabsmVabsmV V SS (3.12)

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44 The interface velocity in cludes two parts: flow field local velocity ,, Local InVnVV and net velocity net InV. Then we can have the following derivation 222 ,,,,,,()[()]()netLocalnet vnInvnInInInVVVVVV (3.13) Finally, the source terms assume the format, which we call modified IDM 2 2 ,Min(0,) ()() (1)Max(0,) ()()LLV net VInLV LV net InLVpp m Vt p p m Vt (3.14) Recalled that for steady-state condition, we have the relationships of ,,,0netLocal InInInVVV and ,,, netLocal InInvnVVV In this case, the modified approach is identical to the original IDM.

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45 CHAPTER 4 NUMERICAL METHODS The governing equations, presented in the pr evious chapters, are discretized using a finite volume approach in the present study. In this approach, the fl ow domain is divided into control volume cells and the governing eq uations are integrated over each control volume. The main advantage of the finite vo lume method is that the conservation laws are satisfied locally for each control volume. A non-staggere d grid system is defined at the center of the cell. The decoupling of velocity-pressu re can be handled by the momentum interpolation method, originally proposed by Rhie and Chow (1983). While, the original momentum interpolation had some problems, such as under-relaxation factordependent, time step size-dependent, and even checkerboard pressure field, which can be solved using linear interpolat ion of the two neighbor nodes in the cell-face velocity evaluation. A detailed review can be obtained in Yu et al. (2002). In cavitating flow computation, the c onventional computati onal algorithm of single-phase incompressible flow meets seve re convergence and stability problems. The situation is improved by using either density -based method or pressure-based method. Both have been successful to compute turbulent cavitating flows in different configurations with comparable accuracy (Wang et al. 2001, Senocak and Shyy 2004a). However, generally the density-based method needs pre-conditioning or artificial density for flows which are largely incompressible (M erkle et al. 1998, K unz et al. 2000, Ahuja et al. 2001). Hence we choose the pressu re-based method (Shyy 1994, Senocak and Shyy 2002a). To take the advantage of non-iterati on in the time-dependent computations, we

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46 use the Pressure-Implicit Splitting Operator (PISO) algorithm other than the SemiImplicit Method for Pressure-Linked (SIMP LE) method (Patankar 1980). The original PISO method was introduced by Issa (1985) a nd was modified later to be suitable for large density variation (Tha kur and Wright 2002, Thakur et al. 2004). Senocak and Shyy (2002a) further extend it to cavitating fl ows by addressing the large density jump between phases. 4.1 Pressure-Based Algorithm The pressure-based algorithm for steadystate computation follows the SIMPLE algorithm (Patankar 1980). The momentum equations can be discretized as ()uuu ppnbnbpdppAuAuVPb (4.1) where ,uu pnb A A are the coefficients of the ce ll center and neighboring nodes from convection and diffusion terms, and ,u ppVb are the cell volume and source term separately. d is the discrete form of the gradient operator. When there is no source term, the above equation turns into H()D()pnbpdpuuP (4.2) with / 00 D0/0 00 /u pp v ppp w ppVA VA VA The pressure-correction equation in the pr essure-based method has been revised to achieve successful solutions for highly compressible flows (Shyy 1994, Senocak and Shyy 2002a). Generally, the mixture density at the phases interface region has high

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47 variations. Here we will illu strate some key computationa l issues by focusing on the flux terms in the continuity equation, which can be decomposed as u u u u u u u * * *) )( ( (4.3) where the starred variables represent the pred icted values and primed variables represent the correction terms. And it leads to P Cp (4.4) ) ( ) ( ) ( ) (* * *P P C u P u C Pd p d d d d d D D (4.5) The relative importance of th e first and second terms in Eq. (4.3) depends on the local Mach number (Shyy 1994). For low Mach number flows, only the first term prevails; for high Mach number flows, the s econd term dominates. The fourth term is a nonlinear second-order term and can be either neglected or included in the source term for stability in early iterations. In the pr esent algorithm, the following relation between density and pressure correction is taken p p l pP C ) 1 ( (4.6) where C is an arbitrary consta nt and it does not affect the final converged solution. The further details of the model can be obta ined from Senocak and Shyy (2002a, 2004a) and Senocak (2002). 4.2 Pressure Implicit Splitting of Op erators (PISO) Algorithm for Unsteady Computations In the SIMPLE-type of the pressure-b ased methods (Patankar 1980, Shyy 1994), the equations are solved successively by employing iterations. In cavitating flow computations, the typical relaxation factors used in the iterative solution process are smaller than the ones used in single-phase fl ows, and hence smaller time steps are needed

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48 to study the cavitation dynamics. Issa (1985) developed the PISO me thod for the solution of unsteady flows. The splitting of pressure and velocity makes the solution procedure sequential in time domain and enables the accura cy at each time step. It also eliminates the need for severe under-relaxation as in SIMPLE type algorithm. Bressloff (2001) extended the PISO method for high-speed flows by adopting the pressure-density coupling procedure in all-speed SIMPLE type of methods. Oliveira and Issa (2001) followed the previous PISO work to comb ine the temperature equation to simulate buoyancy-driven flows. Thakur and Wright ( 2002) and Thakur et al. (2004) developed approaches using curvilinear coordinates with suitability to all speeds. Senocak and Shyy (2002a) further extended this PISO algorithm to enhance the coupli ng of cavitation and turbulence models and to handle the large de nsity ratio associated with cavitation. The PISO algorithm contains predictor and correction steps. In the predictor step the discretized momentum equati ons are solved implicitly usi ng the old time pressure to obtain an intermediate velocity field. A backward Euler scheme is used for the discretization of the time derivative term t u P D u un p n d p p p 1 1 *) ( ) ( ] H[ (4.7) The intermediate velocity field does not sati sfy continuity and needs to be corrected using the continuity equation as a constraint. In the first corrector step, a new velocity field, *u and pressure field P are expected. The discretized momentum equation at this step is written as t u P D u un p p d p p p 1 * *) ( ) ( ] H[ (4.8) Subtracting Eq. (4.7) from Eq. (4.8), we have

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49 p d p pP D u u) (* * (4.9) If the density field depends on the pressure field, such as in high Mach number flows or in cavitating flows, the density is corrected using the pressure-based method p n p p 1 *, P Cp (4.10) The discretized continuity equation written for the new velocity field and density field becomes 0 ] [* * 1 p cf p n p pS n u V t (4.11) Combining Eq. (4.9), (4.10) and (4.11), a firs t pressure-correction st ep equation is obtain below 1*1*[()][][]pp nn dcfpppCP DPnSCPUU t (4.12) To satisfy the mass conservation, the sec ond corrector step is conducted to seek a new velocity field, * *u and pressure field * P t u P D u un p p d p p p 1 * * * *) ( ) ( ] H[ (4.13) Subtracting Eq. (4.9) from Eq. (4. 13) leads to the correction term p d p p p p pP D u u u u ) ( ]* * * * H[ (4.14) The corrected density field leads to p p n p p p p 1 * (4.15) The second pressure-correction step can be derived from Eq.(4.14), Eq. (4.15) and the mass continuity equation to reach the below format

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50 ****** ****[()][][] [H()]pp dcfppp cfpCP DPnSCPUU t uunS (4.16) Then, by solving the above predictor a nd correction steps coupled with the cavitation model, which was formulated in Ch apter 3, and turbulence closures given in Chapter 2, the solution procedure for turb ulent cavitating flow computations is accomplished. 4.3 Speed-of-Sound (SoS) Numerical Modeling As mentioned in chapter 1, the harmoni c speed-of-sound in the two-phase mixture is significantly attenuated. Therefore the multiphase flow fields are characterized by widely different flow regimes, such as incompressible in pure liquid phase, low Mach compressible in the pure vapor phase, and transonic or supersoni c in the mixture. Consequently, an accurate evaluation of speed-of-sound is necessary and important. From Eq. (4.6) and the definition of speed-of-sound, the relation between C and the speed-of-sound is 21 ) ( c P Cs (4.17) In high-speed flows, the exact form of the speed-of-sound can be computed easily from the equation of state. However, in cav itating flows, computation of the speed-ofsound is difficult. Each transport equationbased cavitation model defines a different speed-of-sound as a result of a more complex functional relationship. In the literature, there have been theoretical studies on defining the speed-of-sound in multiphase flows (Wallis 1969). One-dimensional assumption and certain limitations are typical in these studies. These definitions do not necessarily represent the actual speed-of-sound imposed

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51 by the cavitation models of interest in this st udy. They can only be an approximation. On the other hand, the fundamental definition of speed-of-sound as given in Eq. (4.17) could be useful, and the path to co mpute the partial derivative is known. From these arguments, it is clear that the computat ion of the speed-of-sound in cavitating flows is an open question. Due to the lack of a dependable equation of state for multiphase mixtures modeling sound propagation, Senocak (2002) and Senocak and Shyy (2003) present two numerical forms of speed-of-sound, called SoS-1 and SoS2, and showed that the SoS-2 was more likely to produce the correct unsteady behavior in unsteady simulations. In the present study, these two different defi nitions for the speed-of-sound ar e further inve stigated in the pressure-density coupling scheme. The SoS-1 is the previous pressure-density coupling scheme with an orde r of 1 constant coefficient C as SoS-1: ) 1 ( 1 ) (2 l sC c P C (4.18) The SoS-2 is based on an approximation ma de to the fundamental definition of speed-of-sound. It is assumed that the pa th to compute the partial derivative P is the mean flow direction ( ) other than the is entropic direction ( s ), because the details of thermodynamic properties are not known and th e entropy can not be directly computed. This definition is referred to as SoS-2 in the rest of the study and given below SoS-2: | | | | ) ( 1 ) (1 1 1 1 2 i i i i sP P P c P C (4.19)

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52 The partial derivative is computed based on central differencing of the neighboring nodes. The absolute value function is intr oduced to make sure a positive value is computed. Since the above approximation in SoS-2 is based on the mean flow direction, to generalize it, an averaged form is adopted. It will be referred as SoS-2A, taking the format as SoS-2A: 1,,1,,,1,,1, 1,,1,,,1,,1, ,,1,,1 ,,1,,1 ijkijkijkijk ijkijkijkijk ijkijk ijkijkCcoffucoffv pppp coffw pp (4.20) with the velocity weight coefficients ,,,, ,,,,,,,,,,,, ,, ,,,,,,,ijkijk ijkijkijkijkijkijk ijk ijkijkijkuv coffucoffv uvwuvw w coffw uvw

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53 CHAPTER 5 ASSESSING TIME-DEPENDENT TURBULENT CAVITATION MODELS 5.1 Cavitating Flow through a Hollow-Jet Valve There are serious implications on the sa fe and sound operation in flow-control valve cavitation phenomena. A limited number of experimental studies have been published on this topic, such as those by Oba et al (1985) and Tani et al. (1991a,b). In addition, Wang (1999) used high-speed cameras and Laser Light Sheet (LLS) to observe the cavitation behavior in a hollow-jet valve under various cavitation conditions and for different valve openings. However, to the best of our knowledge, to date, no comprehensive numerical study has been done in this respect. Furthermore, complex geometries and inaccessible regions of occurrence restrain expe rimental investigations in cavitation. Hence, we investigate the capab ility of transport e quation-based cavitation models to predict incipient level cavitation. As documented in Wang (1999), Figure 51 shows the geometry and the main configurations of the valve. A key component is the needle, used to control the flow rate by moving to different location in the x-directio n. A cylindrical seal supports the needle. There are six struts supporting the cylinder in the pipe center, which are called splitters. The gear is used to control the need le position moving through the x-axis. Figure 5-2 illustrates the computational dom ain in selected planes according to the geometry. A multi-block structured curvilin ear grid is adopted to facilitate the computation. Figure 5-2A shows the confi guration on the X-Y plane, and Figure 5-2B from the Y-Z plan. Figure 5-2C shows the plane’s location, and Figure 5-2D shows the

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54 boundary conditions in the computations. In th e present study, the splitter thickness is neglected and its shape is c onsidered to be rectangular. The Reynolds number is 5105 and the cavitation number is 0.9, with the density ratio between the liquid phase and vapor phase v l being 1000 in the water, and the valve opening is 33%. Two investigations are conducted here us ing the original in terfacial dynamicsbased cavitation model (IDM) (Section 3.3), using the La under-Spalding model (LSM) (Section 2.2) as turbulence closure. We study steady and unsteady computations. In steady-state, we adopt SoS-2. In time-depende nt simulations, we further examine the two different SoS impact, including SoS-1 and SoS2 (Section 4.3), which have been firstly investigated by Senocak and Shyy (2004b) in cavitating flows. For the time-dependent computation, the steady single-phase turbulen t flow, without consid ering cavitation, is computed and then the solution is adopted as the initial condition for the unsteady cavitating turbulent flow. Th e results and discussion are presented in the following. Figure 5-1. Valve geometry. (1) Splitter. (2) Cy linder. (3) Plunger. (4) Needle. (5) Needle seal overlay. (6) Seal seat inlay. (7) Stroke. (8) Ventilation duct. (9) Gear

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55 Inlet Pipecenter Valve Upperboundary outlet Lowerboundary Splitter A X Z60oB X Y ZSplitter planes Middle planeC Inlet Slip No-slip No-slip No-slip outlet No-slipD Figure 5-2. Computational domains and boundary conditions A) Computational domain in X-Y plane. B) Computational domain in Y-Z plane. C) Planes location. D) Boundary conditions 5.1.1 Steady and Unsteady Turbulent Cavitating Flows Figure 5-3 shows the density distributions for the steady situation. The cavity is located at the valve tip. The comparison show s that the cavities on the splitter plane and the middle plane are slightly different in si ze. Figure 5-4 shows how the cavity shape and location vary with time. For time-dependent computa tions, the cavity fluctuates quasi-periodically. At nondimensional time t* = t/t =0.4, the cavity is the biggest with smallest density, see Figure 5-4A. As time passes, the cavity size reduces to the smallest in Figure 5-4(B, C) at time t*=0.6 and t*=0.8, respectively. After one cy cle it reaches the ma ximum again (Figure 54D at time t*=1.2s). Both steady and unsteady results exhibit the cavity at the needle tip

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56 (Figure 5-3 and Figure 5-4). The steady case corresponds to a single instantaneous result in the time-dependent soluti on (Figure 5-3B and Figure 5-4C). The time-dependent results are qualitatively consistent with the experiment in size and shape (Figure 5-4). Unfortunately, there is insufficient experi mental information to ascertain the timedependent characteristics in detail. The experiment observed that the cavities in cept, grow, then detach from the needle tip and transport to the downstream periodi cally, which is clearly shown in Figure 5-4E. However, as already discussed in Senocak (2002), with the current combination of turbulence and cavitation models, the detachme nt of the cavity is not captured, possibly due to the representation of the turbulence via a scalar e ddy viscosity. Henceforth, the issue is under further inve stigation in following by m eans of adopting different turbulence models. 2 3 4 5 50.99 40.98 30.96 20.9 10.8NeedletipF l o wA 2 3 4 5 50.99 40.98 30.96 20.9 10.8NeedletipF l o wB Figure 5-3. Density contour lines of the stea dy state solution (The mixture of vapor and liquid inside the outer line forms the cav ity), original IDM with SoS-1, LSM. A) At splitter plane. B) At middle plane

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57 3 2 4 5 50.99 40.98 30.96 20.9 10.8F l o wNeedleA 4 5 3 2 50.99 40.98 30.96 20.9 10.8F l o wNeedleB 4 5 3 2 50.99 40.98 30.96 20.9 10.8F l o wNeedleC 4 5 3 2 50.99 40.98 30.96 20.9 10.8F l o wNeedleD E Figure 5-4. Middle section dens ity contours at different time instants (The mixture of vapor and liquid inside the outer line forms the cavity), original IDM with SoS-2, LSM. A) Time t*=0.4. B) Time t*=0.6. C) Time t*=0.8. D) Time t*=1.2. E) Experimental observations fr om Wang (1999): cavity at needle tip and cavitation aspects around the needle Valve tip

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58 To further demonstrate the cavity-induced qua si-periodic characteristics of the flow field, Figure 5-5 highlights the time evolutions at selected locations. Figure 5-5A shows the locations of the points selected. The pressure is at the middle points, and the density is near the bottom boundary, just one point away from it. From Figure 5-5B, at the places away from the cavity, A, B and D, the dens ity is constant sin ce only the liquid phase exists there, and is quasi-periodic inside the cavity at Point C. On the other hand pressure oscillates in the whole domain except on the inlet plane, wher e the flow condition is fixed (Figure 5-5C). A B C DA t* Density 0 1 2 3 4 0.7 0.75 0.8 0.85 0.9 0.95 1 LocationA LocationB LocationC LocationD B t* Cp 0 1 2 3 4 -2 -1 0 1 2 3 4 5 6 7 8 LocationA LocationB LocationC LocationD C Figure 5-5. Time evolutions at different locations, original IDM with SoS-2, LSM. A) Samples locations. B) Density. C) Pressure Figure 5-6 presents the flow structure on the middle plane. For the steady case, the flow fields of both single phase (without i nvoking the cavitation mode l in the course of computation) (Figure 5-6A) and cavitating flow s are almost identical (Figure 5-6B). It

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59 indicates that in the pres ent case, the detailed cavita tion dynamics does not exhibit substantial influence on the overall flow pattern For the time-dependent case, at different time instants, t*=0.4 and t*=0.8 (Figure 5-6C ), the flow field around the needle remains largely the same. The comparison between th e steady and unsteady case indicates that there is not very much difference in the flow pattern. As expected, there are two recirculating zones: one behind the need le and the other on e downstream around the splitter region, which is located at about x= 3.4~4.2. Compared with the experimental observation in Figure 5-6D (the cavitating flow structures behind the needle, and in the splitter region), the present results are in general agreement. X Y 3 3.5 4 4.5 5 0 0.5 1A X Y 3 3.5 4 4.5 5 5.5 0 0.5 1B X Y 3 3.5 4 4.5 5 0 0.5 1 X Y 3 3.5 4 4.5 5 0 0.5 1C Figure 5-6. Projected 2-D str eamlines at middle plane and ex perimental observation. A) Steady single-phase flow, LS M. B) cavitating flow, original IDM with SoS-2, LSM (Right). B) Unsteady cavitating flow at time t*=0.4 (Left) and t*=0.8 (Right). D) Flow patte rn from Wang (1999), behind valve about x=3.0~3.4 (Left) and at splitter regi on about x=3.4~4.2 (Right)

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60 D Figure 5-6. Continued 5.1.2 Impact of Speed-of-sound Modeling We further examine the behavior of th e different handlings of speed-of-sound (Chapter 4), that is SoS-1 and SoS-2. Figure 5-7 shows the variation in cavity shapes and location for LSM at maximum and minimum fraction instants. Compare the half period between maximum and minimum instant, the cavity oscillation fo r SoS-1 is distinguished by a significantly higher frequency and moderately higher amplitude than for SoS-2 (Figure 5-7A, B). These observations, as also di scussed later, are a precursor to the significant impact produced by the speed-of-sound definition on th e flow time scales. The qualitative shape and location of the cavity for both SoS-1 and SoS-2, however, are similar to the experimental observations (Figure 5-6D). Senocak and Shyy (2003,2004b) pointed out that SoS-2 performed better than SoS-1 in simulating the oscillatory behavior of the cavitating flow in the convergent-diverg ent nozzles, because SoS-2 successfully produced a quasi-steady solution at high cavitation number and a periodic solution at low cavitation number. In Figure 5-8, SoS-2 and SoS-1 behave similarly in this computed Flow Flow

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61 case, and they both are nearly constant when away from the pure phase. These qualitatively agree with the theoretical analys is (Figure 1-2). Also, this helps to explain why the cavity shapes of the two different So S models are qualitatively close. One thing needs to be pointed out here that the volume fraction is very close to the pure liquid phase. Hence the linear appr oximation SoS-1 can have sim ilar behavior as SoS-2. 3 4 5 50.99 40.98 30.96 20.9 10.8 3 4 5 50.99 40.98 30.96 20.9 10.8A 3 5 4 50.99 40.98 30.96 20.9 10.8 5 4 50.99 40.98 30.96 20.9 10.8B Figure 5-7. Middle section dens ity contours of different SoS at different instants (the mixture of vapor and liquid inside the outer line forms the cavity), original IDM, LSM. A) SoS-2, maximum frac tion at t*=0.32 (Left) and minimum fraction at t*= 0.77 (Right). B) SoS-1, maximum fraction at t*=5.92 (Left) and minimum fraction at t*=5.86 (Right) Lack of substantial experimental data prevents us from making more direct comparisons at this stage. Various experi mental studies have shown that, besides oscillating, the cavities incept, grow, detach from the needle tip and get periodically

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62 transported downstream (Figure 5-4E). The pe riodic detachment and inception of cavities is, however, difficult to capture through currently known turbulence and SoS models (Wang et al. 2001, Senocak and Shyy 2003, Cou tier-Delgosha et al. 2003, Wu et al. 2003 a). Consequently, fundamental understanding on the above modeling aspects, which is discussed in the following sections, is impe rative to encounter the above limitation. The following discussions serve to probe the sensitivit y of the various modeling concepts, guided by our qualitative, but incomple te, insight into the fluid physics. The implications of speed-of-sound (SoS) definition are quantified by a correlation study (Table 5-1 and Table 52) and spectral analysis (F igure 5-10) on a series of pressure-density time history at various nodes in cavitation vicinity. The Pearson’s correlation (r) between pressure and density is calculated as 22[()()] ; ()()N ii i NN ii iiXXYY rXPY XXYY (5.1) Table 5-1 and Table 5-2 indi cate that within similar variation limits of liquid volume fraction, SoS-1 has a broader range of pressure-density correlation coefficients than SoS-2. Furthermore, SoS-2 consistently exhibits much stronger pressure-density coefficients than SoS-1. The distinct e ffects produced by the speed-of-sound definition are further corroborated by the dramatic tim e scale differences observed from the time history and FFT plots of pressu re and density (Figure 5-9 an d Figure 5-10A, B, C). SoS-1 plots clearly indicate domina nce of a single high-frequency compressibility effect unlike SoS-2, which are characterized by a wider bandwidth for pressure and density, and a lower dominant frequency. It is worth noting he re that although the pl ots in Figure 5-9(B,

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63 C) are plotted against a smaller time range we have used a convincingly long time history for our analysis. liquidfraction SoS-1 0.9 0.95 0 10 20 30 40 50 60 70 Liquidfraction SoS-2 0.9 0.95 1 0 10 20 30 40 50 60 70 Figure 5-8. Two different SoS in hollo w-jet valve flow, original IDM, LSM Table 5-1. Time-averaged liquid volume frac tion v/s pressure-density correlation at multiple points inside the cavity, original IDM with SoS-1, LSM Time averaged liquid volume fraction (l) Pearson’s correlation (r) between pressure and density (SoS-1) 0.862 -0.674 0.882 -0.09 0.936 0.093 0.970 -0.045 0.978 -0.077 0.983 -0.096 0.986 -0.103 0.988 -0.12 Mean(r) = -0.13; SD(r) = 0.9 Table 5-2. Time-averaged liquid volume frac tion v/s pressure-density correlation at multiple points inside the cavity, original IDM with SoS-2, LSM Time averaged liquid volume fraction ( l) Pearson’s correlation (r) between pressure and density (SoS-2) 0.858 -0.733 0.868 -0.106 0.934 -0.106 0.968 -0.46 0.977 -0.47 0.982 -0.412 0.985 -0.412 0.987 -0.412 Mean(r) = -0.42; SD(r) = 0.1

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64 A B C DA t* Cp 0 1 2 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 LocationA LocationB LocationC LocationD B t* Cp 0 1 2 -2 -1 0 1 2 3 4 5 6 7 8 LocationA LocationB LocationC LocationD C Figure 5-9. Pressure time evolut ions of different SoS, orig inal IDM, LSM. A) Samples locations. B)SoS-1. C) SoS-2 t/T Cp 0 0.5 1 1.5 2 2.5 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2Sos-2 Sos-1 t/T 0 0.5 1 1.5 2 2.5 0.9 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98Sos-2 Sos-1 A Figure 5-10. Time evolution and spectrum of pressure and density of two SoS definitions at a point at the cavitation vicinity, origin al IDM, LSM. A) Pressure (Left) and density (Right) time history. B) Spec tral analysis on pressure (Left) and density (Right, SoS-1. C) Spectral anal ysis on pressure (Left) and density (Right), SoS-2

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65 B C Figure 5-10. Continued 5.2 Turbulent Cavitating Flow th rough a Convergent-Divergent Nozzle To study the performance of the filter-bas ed model (FBM, Section 2.3), and the modified IDM (Section 3.3) in time-depende nt computations, the turbulent cavitating flow through a convergent-diverg ent nozzle is investigated. Stutz and Reboud (1997, 2000) have studied the unsteady cavitation formed in a convergent-divergent no zzle that has a converg ent part angle of 18 and a divergent part angle of 8 The experimental Reynolds number is 66Re10310 based on the reference velocity and cavity length. Cavita tion formed in this nozzle is described as “unsteady and vapor cloud shedding”. Senocak and Shyy (2003, 2004b) conducted the numerical simulations usi ng IDM coupled with the LS M. They found that under cavitation number 1.98 the cavity length cavL matched the experiments (Stutz and

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66 Reboud 1997, 2000), which did not specify the cavitation number. Therefore, to be consistent, we use 1.98 in the present study. First, we apply the original IDM with SoS-2 for the simulations to assess the FBM performance. To test the grid and filter size dependency two grids are adopted including a coarse grid with 0.5cavL and a fine grid with 0.25cavL The time-averaged eddy viscosity contours of two grids are given in Figure 511. The FBM gives effective viscosities for both grids about half an orde r of magnitude lower than th at produced by the LSM. The differences of the time-averaged eddy viscos ity are noticeable between the two grids. The largest viscosity on the fine grid is lower a nd its location shifts to the upstream, compared with Figure 5-11(A, B) left and right columns correspondingly. However, the timeaveraged vapor volume fraction does not show significant difference (Figure 5-12A, B). The time-averaged u-velocity profiles have the similar trend. Hereafter, all the results are ba sed on the fine grid with 0.25cavL Figure 5-13 shows the time-averaged velocity and vapor vol ume fraction profiles within the cavity at four different sections, using two differe nt turbulence models. The boundaries of the cavitating region, from experiments, LSM and FBM, are also included. Although the numerical results of the cav itating boundaries are about 5% higher than that from the experiments, the computations capture th e main cavity body and the overall trends are agreeable (Figure 5-13B). The velocity profile s qualitatively agree with the experimental data, especially in the core of the revers e flow (Figure 5-13A). Comparing the LSM and FBM, both the time-averaged velocity and volume fraction show ma rginal difference. However, the instantaneous velocity and volume fraction profile s display significant

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67 differences (Figure 5-14). The current results show an auto-oscillating cavity. Here only the maximaland minimal-volume fraction time instants are presented. The maximalvolume fraction of the FBM is bigger than that of the LSM. On the contrary, the minimalvolume fraction is much smaller by comp aring Figure 5-14A a nd Figure 5-14B. The reverse velocity profiles show the same trend. The increase in fluctuations is due to the FBM reducing the eddy viscosity (Figure 5-11) The observation reveals that the timeaveraging process can be misleading and not suitable as the only indicator for performance evaluation, which is similar to the previous study by Wu et al. (2003b). A B Figure 5-11. Time-averaged eddy vi scosity contours of different grids, original IDM with SoS-2, 1.98 A) Coarse grid with 0.50cavL B) Fine grid with 0.25cavL LSM LSM FBM FBM

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68 X(mm) Y(mm) 0 20 40 60 80 0 5 10 15 Vaporvolumefraction,coarsegrid Vaporvolumefraction,finegrid Vaporvolumefraction,Exp.data Boundaryofcavitatingflow,coarsegrid Boundaryofcavitatingflow,finegrid v(%) 25% 25% 25% 25%A X(mm) Y(mm) 0 20 40 60 80 0 5 10 15 Vaporvolumefraction,coarsegrid,=0.5LcavVaporvolumefraction,finegrid,=0.25LcavVaporvolumefraction,Exp.data Boundaryofcavitatingflow,coarsegrid Boundaryofcavitatingflow,finegrid v(%) 25% 25% 25% 25% B Figure 5-12. Time-averaged vapor volume fr action comparisons of different grids, original IDM with SoS-2, 1.98 The vertical scale is the distance from the wall. Experimental data are from Stutz and Reboud (1997&2000). A) LSM. B) FBM X(mm) Y(mm) 0 10 20 30 40 50 60 70 80 0 5 10 15 Velocity,Standard Velocity,filter-based Velocity,Exp.data Boundaryofcavitatingflow,Exp. Boundaryofcavitatingflow,Standard Boundaryofcavitatingflow,Filter-based 10 10 10u(m/s)10A X(mm) Y(mm) 0 10 20 30 40 50 60 70 80 0 5 10 15 Vaporvolumefraction,Standard Vaporvolumefraction,filter-based Vaporvolumefaction,Exp.data Boundaryofcavitatingflow,Exp. Boundaryofcavitatingflow,Standard Boundaryofcavitatingflow,Filter-based 25% 25%v(%) 25% 25%B Figure 5-13. Time-averaged comparisons of diffe rent turbulence models on fine grid with 0.25cavL original IDM with SoS-2, 1.98 The vertical scale is the distance from the wall. Experiment al data are from Stutz and Reboud (1997&2000). A) U-velocity. B) Vapor volume fraction An illustration that the FBM can induce noti ceably stronger flow oscillations is presented in Figure 5-15. It is observed that time-averaged pressure contours and streamlines are similar at the cavity zone, although a recirculation region produced by the FBM exists at downstream near the lo wer wall (Figure 5-15A). However, the

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69 instantaneous pressure contours and velocity field produced by the two turbulence models are distinctly different. The FBM yi elds wavy flow struct ures, which induce the auto-oscillation of the flow fields in Figur e 5-15(B, C). The experimental measurements (Figure 5-14) are confined in the throat region, which have missed the large scale unsteadiness of the flow field. X(mm) Y(mm) 0 10 20 30 40 50 60 70 80 0 5 10 15 Velocity,minimalinstant Velocity,maximalinstant Boundaryofcavitatingflow,Minimal Boundaryofcavitatingflow,Maximal 10 10 1010u(m/s) X(mm) Y(mm) 0 10 20 30 40 50 60 70 80 0 5 10 15 Vaporvolumefraction,minimalinstant Vaporvolumefraction,maximalinstant Boundaryofcavitatingflow,Minimal Boundaryofcavitatingflow,Maximal 25% 25% 25% 25%v(%)A X(mm) Y(mm) 0 10 20 30 40 50 60 70 80 0 5 10 15 Velocity,minimalinstant Velocity,maximalinstant Boundaryofcavitatingflow,Minimal Boundaryofcavitatingflow,Maximal 10 10 10 10u(m/s) X(mm) Y(mm) 0 10 20 30 40 50 60 70 80 0 5 10 15 Vaporvolumefraction,minimalinstant Vaporvolumefraction,maximalinstant Boundaryofcavitatingflow,Minimal Boundaryofcavitatingflow,Maximal 25% 25% 25% 25%v(%)B Figure 5-14. Instantaneous prof iles on fine grid with 0.25cavL original IDM with SoS-2, 1.98 The vertical scale is the distance from the wall. A) LSM: uvelocity and vapor volume fraction. B) FBM: u-ve locity and vapor volume fraction

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70 A B C Figure 5-15. Pressure contours and streamlin es comparison of two turbulence models on fine grid, original IDM with SoS-2, 1.98 A) Time-averaged. B) At maximal-volume fraction instant. C) At minimal-volume fraction instant LSM LSM LSM FBM FBM FBM

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71 Obviously, both turbulence and cavitation models directly affect the cavity dynamics such as the shedding pattern and fre quency. Hence in this section, we further investigate the implication of the cavitati on model in the context of the interfacial velocity estimated, as discussed in Section 3.3 which is called the modified interfacial dynamics-based cavitation model. The speed -of-sound (SoS) is based on the SoS-2A, formulated in Section 4.3. The turbulen ce closure is employed the original LaunderSpalding model (LSM) described in Section 2.2. The detailed results are presented below. Figure 5-16 gives the time history of the pre ssure at a reference point near the inlet. The pressure of the original IDM, which adopts an empirical factor to assign the interfacial velocity based on the local fluid ve locity in Section 3.3 Eq. (3.9), exhibits much smaller variation than that computed with the modified IDM. As given in table 5-3, although both IDMs under-predict the Str ouhal number, the dynamically adjusted interfacial velocity embodied by the modifi ed IDM yields better agreement compared with the experimental measurement. Selected time evolutions of the cavity shap e and the associated flow structures are presented in Figure 5-17 with the available experimental obs ervation. The original IDM demonstrates an attached periodic cavity, while the modified IDM produces cavity breakup and detachment, in manners qualitative ly consistent with the experimental observations. The cavity break-up and collapse of the modified IDM also helps to explain the Cp behavior shown Figure 5-16. Furthermor e, because of the shedding of the cavity, the modified IDM better predic ts the vapor volume fraction profiles than the original IDM, with less difference in the velocity profiles between these two cavitation models (Figure 5-18).

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72 Table 5-3. Comparisons of St rouhal number of original a nd modified IDM with SoS-2A Original IDM, SoS-2A Modified IDM, SoS-2A Experiment 0.07 0.13 0.30 Figure 5-16. Pressure evoluti ons of original and modifi ed IDM with SoS-2A at a reference point Figure 5-17. Cavity shape and recirculation z one during cycling of original and modified IDM. The experimental observation is adopted from Stutz and Rebound (1997, 2000) Original IDM t/T=0 Original IDM t/T=5.28 Modifiedl IDM t/T=0 Experimental Observation

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73 Figure 5-17. Continued X(mm) Y(mm) 20 30 40 50 60 70 80 0 5 10 15 ModifiedIDM OriginalIDM Exp.data 25% 25% 25% 25%v(%)A X(mm) Y(mm) 20 30 40 50 60 70 80 0 5 10 15 ModifiedIDM OriginalIDM Exp.data 10 10u(m/s)10 10 10B Figure 5-18. Time-averaged volume fraction a nd velocity comparisons of original and modified IDM. The vertical scale is th e distance from the wall. Experimental data are from Stutz a nd Reboud (1997, 2000). A) Vapor volume faction. B) uvelocity 5.3 Turbulent Cavitating Flow over a Clark-Y Hydrofoil The capability of the modified IDM (Secti on 3.3) and turbulence models are further investigated in unsteady cavitating flow s over a Clark-Y hydrofoil, assessed by experimental data from Wang et al. (2001). To facilitate th e performance evaluation, the Modifiedl IDM t/T=4.2 Modifiedl IDM t/T=4.5 Modifiedl IDM t/T=3.9

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74 generalized treatments of the SoS-2, includi ng SoS-2, SoS-2A described in Section 4.3, are coupled. For the computational set-up (Figure 5-19A), the computational domain and boundary conditions are given according to the experimental set-up. The Clark-Y hydrofoil is located at the wa ter tunnel center. Two angles -of-attack (AoA) considered are 5 and 8. The hydrofoil chord length is c and the hydrofoil l eading edge is 3 c away from the inlet. The two important paramete rs are the Reynolds number and the cavitation number, which is based on inlet pressure P and vapor pressure vP with inlet velocity U 5Re710 Uc (5.2) 2/2vPP U (5.3) The filter size in the present study is chosen to be larger than the largest grid scale employed in the computation, and is set to be 0.08 c Computations have been done for tw o AoA and several cavitation numbers. Specifically, for AoA=5 under four flow regimes: no-cavitation (2.02 ), inception (1.12 ), sheet cavitation (0.92 ), and cloud cavitation (0.55 ), and AoA=8 under three cavitation nu mbers: no-cavitation (2.50 ), sheet cavitation (1.40 ), cloud cavitation (0.80 ). All cases above are at the same Reynolds number 5Re710 Two different turbulence models, LSM a nd FBM, have been employed to help probe the characteristics of cavitation and turbulence modeling interactions, coupled with

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75 different SoS numerical models. The outlet pressure is fixed and adjustment of the vapor pressure is needed to be consistent with the prescribed cavitation number in each case. Grid sensitivity analysis. To investigate the grid dependency, two grids are adopted in the computation: coar se grid and fine grid. The gr id blocks and numbers of the coarse grid are shown in Fi gure 5-19B. The fine grid ha s 60% more number of nodes than that of the coarse grid in the ver tical direction while maintaining the same distribution in the horizo ntal direction. Two diffe rent cavitation numbers, 2.02 and 0.55 both at AoA=5 have been investigated. Overall, the solutions on both grids are in good agreement. The time-averaged u-ve locity and v-veloci ty profiles with 2.02 and 0.55 using the LSM are shown in Figure 520. The results based on the FBM are of similar nature and will not be repeated. Hereafter, to reduce the cost of time-de pendent computations, we use the coarse grid in the computations. Visualization of cavity and flow field. First, we focus on the LSM results to analyze the performance of the IDM. Figure 5-21 and Figure 5-22 show the timeaveraged flow structure and cavity shap e under varied cavitation numbers. With no cavitation, the flow field is attached without separation for both AoA=5 and AoA=8 (Figure 5-21A left column and Figure 5-22A left column). This is consistent with the experimental observation. When cavitation appears, the density will change by a factor of 1000 between liquid and vapor pha ses. Consequently, there is a drastic reduction in the amount of mass inside the cavity, and a cont raction of the fluid flow behind the cavity. With the reduction of the cavitation number, the cavity and recirculation zone become bigger. At cloud cavitation regime, the cavity experiences shedding, causing multiple

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76 recirculating flows (Figure 5-21D left column and Figure 5-22C left column). Compared to the experimental data for both AoAs (Tab le 5-4 and Table 5-5), the predicted cavity sizes demonstrate qualitatively consistent re nding, albeit generally over-predicted. For incipient cavitation, the experiment observed the recurring formation of hair-pin type cavitating vortex structures, which are not attached to the soli d surface (Wang et al. 2001). This type of flow struct ure is not captured in the co mputation. The time-averaged flow structures associated w ith sheet and cloud cavitation, on the other hand, seem to be reasonably captured computat ionally. Furthermore, the time-averaged outcome of employing both LSM and FBM seems compatible. Outlet InletNo-SlipHydrofoil:No-SlipNo-Slip10c2.7cA Block2:70x50 Block1:70x50 Block3:70x50 Block4:70x50Block1:80x50 Block5:80x50CoarseGridB Figure 5-19. Clark-Y geometry sketch and Gr id blocks. A) Geomet ry configuration and boundary conditions, c is the hydrofoil chord. B) Grid blocks and coarse grid numbers

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77 x/c y/c 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 Grid Experiment Finegrid Coarsegrid u/U 1.0 1.0 1.0 1.0 1.0 1.0 x/c y/c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4Airfoil Experiment Finegrid Coarsegrid v/U 1.0 -1.0 1.0 -1.0 -1.0 -1.0A x/c y/c 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4Grid Experiment Fine,LSM Coarse,LSM u/U 1.0 1.0 1.0 1.0 1.0 1.0 x/c y/c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4Airfoil Experiment Fine,LSM Coarse,LSM v/U 1.0 -1.0 1.0 -1.0 -1.0 -1.0B Figure 5-20. Grid sensitivity of the time-av eraged uand v-velocity, LSM, AoA=5. Experimental data are from Wang et al. (2001). A) No-cavitation. B) Cloud cavitation 0.55

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78 Table 5-4. Time-averaged cavity leading and trailing positions of different turbulence models (Taking 0.95L as cavity boundary), modi fied IDM with SoS-2A, AOA=5 degrees Inception 1.12 Sheet cavitation 0.92 Cloud cavitation 0.55 Position LSM FBM Exp. LSM FBM Exp. LSM FBM Exp. Leading 0.13 0.13 0.13 0.13 0.23 0.16 0.17 0.22 Trailing 0.62 0.57 0.64 0.64 0.68 1.01 1.04 0.90 Table 5-5. Time-averaged cavity leading and trailing positions of different turbulence models (Taking 0.95L as cavity boundary), modi fied IDM with SoS-2A, AoA=8 degrees. Sheet cavitation 1.40 Cloud cavitation 0.80 Position LSM FBM Exp. LSM FBM Exp. Leading 0.094 0.098 0.15 0.12 0.13 0.15 Trailing 0.53 0.52 0.56 1.07 1.10 0.84 A B C D Figure 5-21. Time-averaged volume fraction contours and streamlines of different turbulence models, AoA=5. A) No-cavit ation. B) Incipient cavitation. C) Sheet Cavitation 0.92 D) Cloud cavitation 0.55 LSM FBM LSM LSM FBM FBM FBM LSM

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79 A B C Figure 5-22. Time-averaged volume fraction contours and streamlines of different turbulence models, AoA=8. A) Nocavitation. B) Sheet cavitation 1.40 C) Cloud cavitation 0.80 The temporal evolution of the computed and experimentally observed flow structures with cloud cavitation, under two AoAs, are shown in Figure 5-23 and Figure 524. Figure 5-23A left column shows the time se quences of flow structures predicted by LSM at AoA=5. The corresponding flow pred icted by the FBM is shown in the right column. The experimental visual image is s hown in Figure 5-23B. The flow structures at AoA=8 are shown in Figure 5-24. Both the computations and the experiment indicate that as the AoA increases, the cavity exhibi ts a more pronounced r ecurrence of the size variation. The FBM predicts stronger time-dependency than the LSM. We will further investigate this aspect in the following discus sion. To help elucidate the main features of the cavity dynamics from both numerical and experimental studies, we show in Figure 525 three stages of the cav ity sizes, for cloud cavitation 0.55 at AoA=8. The figure demonstrates that the numerical simulation is capable of capturing the initiation of the cavity, growth toward traili ng edge, and subsequent shedding, in accordance with the qualitative features observed experimentally. FBM FBM LSM LSM LSM FBM

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80 t=0.00 t=3.36 ms t=10.1 ms t=11.8 ms t=15.1 ms t=20.2 ms A t=0.0 t=2.2 ms t=4.4 ms t=6.7 ms t=8.9 ms t=11.1 ms B Figure 5-23. Time evolutions of cloud cavitation 0.55 AoA=5. A) Numerical results of different turbulence models. B) Side views of the experimental visuals from Wang et al. (2001) Flow LSM LSM LSM LSM LSM FBM FBM FBM FBM FBM LSM FB M

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81 Figure 5-24. Time evolutions of cloud cavitation 0.80 AoA=8. A) Numerical results of different turbulence models. B) Side views of the experiment visuals from Wang et al. (2001) LSM t=0.00 LSM t=5.88 LSM t=11.8 LSM t=16.7 LSM t=21.6 LSM t=24.5 LSM t=26.5 LSM t=28.4 LSM t=32.3 FBM t=0.00 FBM t=5.88 FBM t=11.8 FBM t=16.7 FBM t=21.6 FBM t=24.5 FBM t=26.5 FBM t=28.4 FBM t=32.3

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82 A B Figure 5-24. Continued Flow LSM t=41.2 FBM t=41.2

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83 A B C Figure 5-25 Cavity stage co mparisons, cloud cavitation 0.80 AoA=8: FBM (Left) and Experiment (Right). Experimental visuals are from Wang et al. (2001). A) Early stage: cavity formation. B) Second stage: cavity growth toward trailing edge. C) Third stage: cavity break-up and shedding Velocity profiles and lift/drag coefficients. The mean horizontal uvelocity and vertical v-velocity of the fl ow field are illustrated in Fi gure 5-26 (AoA=5) to Figure 5-27 (AoA=8). The time-averaged velocity profile s are documented at 6 chordwise locations, 0%, 20%, 40%, 60%, 80%, and 100% of the leading edge, under different cavitation numbers. With no cavitation, the numerical results agree well with the experiment, and the results of the two turbulence models are virtually identical (Figure 5-26A and Figure 5-27A). With the cavitation number decreasing, the differences between prediction and measurement become more substantial, especi ally at the cavity closure region. In fact, recirculation breaks up the cavity and tilts th e rear portion of the cavity more upward, which makes the cavity thicker in comparison to the experiment. With the reduction in cavitation number, it becomes more distingu ished in Figure 5-23 and Figure 5-24. The cavity tilt-up results in over-shooting the u-velo city at the closure region (0.8c and 1.0c sections) (Figure 5-26 and Figure 5-27). Ov erall, considering the difficulties in experimental measurement (Wang et al 2001) the agreement is reasonable.

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84 x/c y/c 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 Grid Experiment Filter-based Launder-Spalding u/U 1.0 1.0 1.0 1.0 1.0 1.0 x/c y/c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4Airfoil Experiment Filter-based Launder-Spalding v/U 1.0 -1.0 1.0 -1.0 -1.0 -1.0A x/c y/c 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 Grid Experiment Filter-based Launder-Spalding FBM,cavityboundary LSM,cavityboundary u/U 1.0 1.0 1.0 1.0 1.0 1.0 x/c y/c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4Airfoil Experiment Filter-based Launder-Spalding FBM,cavityboundary LSM,cavityboundary v/U 1.0 -1.0 1.0 -1.0 -1.0 -1.0B Figure 5-26. Time-averaged uand v-veloci ties of two turbulence models, AoA=5. Experimental data are from Wang et al (2001). A) no-cavitation. B) Inception 1.12 C) Sheet cavitation 0.92 D) Cloud cavitation 0.55

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85 x/c y/c 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 Grid Experiment Filter-based Launder-Spalding FBM,cavityboundary LSM,cavityboundary u/U 1.0 1.0 1.0 1.0 1.0 1.0 x/c y/c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4Airfoil Experiment Filter-based Launder-Spalding FBM,cavityboundary LSM,cavityboundary v/U 1.0 -1.0 1.0 -1.0 -1.0 -1.0C x/c y/c 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4Grid Experiment Filter-based Launder-Spalding FBM,cavityboundary LSM,cavityboundary u/U 1.0 1.0 1.0 1.0 1.0 1.0 x/c y/c 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4Airfoil Experiment Filter-based Launder-Spalding FBM,cavityboundary LSM,cavityboundary v/U 1.0 -1.0 1.0 -1.0 -1.0 -1.0D Figure 5-26. Continued

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86 x/c y/c 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 Grid Experiment Filter-based Launder-Spalding u/U 1.0 1.0 1.0 1.0 1.0 1.0 x/c y/c 0 0.2 0.4 0.6 0.8 1 0 0.2Airfoil Experiment Filter-based Launder-Spalding 1.0v/U 1.0 -1.0 -1.0 -1.0 -1.0A x/c y/c 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 Grid Experiment Filter-based Launder-Spalding Exp.,cavityboundary FBM,cavityboundary LSM,cavityboundary u/U 1.0 1.0 1.0 1.0 1.0 1.0 x/c y/c 0 0.2 0.4 0.6 0.8 1 0 0.2Airfoil Experiment Filter-based Launder-Spalding Exp.,cavityboundary FBM,cavityboundary LSM,cavityboundary 1.0v/U 1.0 -1.0 -1.0 -1.0 -1.0B Figure 5-27. Time-averaged uand v-veloci ties of two turbulence models, AoA=8. Experimental data are from Wang et al (2001). A) No-cavitation. B) Sheet cavitation 1.40 C) Cloud cavitation 0.80

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87 x/c y/c 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4 Grid Experiment Filter-based Launder-Spalding Exp.,cavityboundary FBM,cavityboundary LSM,cavityboundary u/U 1.0 1.0 1.0 1.0 1.0 1.0 x/c y/c 0 0.2 0.4 0.6 0.8 1 0 0.2Airfoil Experiment Filter-based Launder-Spalding Exp.,cavityboundary FBM,cavityboundary LSM,cavityboundary 1.0v/U 1.0 -1.0 -1.0 -1.0 -1.0C Figure 5-27. Continued Figure 5-28 shows the time-averaged lift a nd drag coefficients collected from experiments and computations. The computa tional models estimate that as cavitation appears, lift decreases. However, experiment ally, such a drop doesn’t take place until the sheet cavitation regime. For sheet and cloud cavitation, both computations and experiment show consistent trends, namely marked reduction in lift as the cavitation becomes more pronounced. From the time-averag ed flow structures shown earlier, one clearly sees that the cavity changes the effective shape of the hydrofoil, causing flow to separate. Hence, the reduction in lift is expect ed. Regarding the drag coefficient, there is a marked increase from sheet to cloud cavitation, which is reflected by the computational models. Overall, the lift coe fficient is under-predicted and the drag coefficient is overpredicted by both turbulence models, but the tr ends are reasonably captured in the sheet and cloud cavitation regimes.

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88 Cavitationnumber CL 1 2 3 0.2 0.4 0.6 0.8 1 Exp.,AoA=8 LSM,AoA=8 FBM,AoA=8 Exp.,AoA=5 LSM,AoA=5 FBM,AoA=5 sheet cloud inceptionA Cavitationnumber CD 1 2 3 0 0.05 0.1 0.15 Exp.,AoA=8 LSM,AoA=8 FBM,AoA=8 Exp.,AoA=5 LSM,AoA=5 FBM,AoA=5 cloud inception sheetB Figure 5-28. Time-averaged lift and drag coeffi cients comparisons. Experimental data are from Wang et al. (2001). A) Lift CL. B) Drag CD Assessing modified IDM parameters. To better understand th e IDM performance, we examine the condensation m and evaporation m source terms. A TEM cavitation model heuristically developed by Merkle et al. (1998) and extensiv ely applied by Ahuja et al. (2001) shows identical model e quations to the source terms given as 2 2Min(0,) (/2) (1)Max(0,) (/2)destLLV VL prodLV LCpp m Ut Cpp m Ut In the published applications (Merkle et al. 1998) 1.0destC and 80prodC are often adopted. To compare the IDM with Merkle et al.’s model, we can rearrange the source term in IDM (in Section 3.2.2) followi ng the above equation format, and get the equivalent model parameters destC and prodC as 2 2 ,0.5 ()()L destprod net LVInU CC V Using the solutions obtained at AoA=5 presented in Figure 5-21 and Figure 5-22, we have the following observations

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89 With the decrease of the cavitation number from no-cavitation to cloud cavitation, /destdestCC varies from O( 10) to O(100), and /prodprodCC varies from O(1) to O(10), exhibiting substantial variations in accordance with the cavitation regimes. For a given cavitating flow re gime, the variation of ratios /destdestCC and /prodprodCC for time-averaged results is not much along a constant volume fraction contour. Approximately, they can be considered as unchanged The values of destC and prodC experience temporal variations in each cavitation regime. The ratio of max_min/destdestCC tends to be O(1)~O(10) at the same location inside the cavity, with max destC denoting the value at maximum cavity size and min destC at minimum cavity size. The i nvestigation on the FBM results shows that the destC(prodC ) has larger temporal fluctuation than the LSM between the maximum and minimum cavity sizes. Venkateswaran et al. (2002) stated that the choice of destC and prodC for steady computations is not critical. However, th e same apparently doesn’t hold for timedependent computations. Comparison of turbulence models. To assess the turbulence models’ performance, we compare the results of the LSM and FBM in each matching cavitation flow regime. For no-cavitation, the LSM a nd FBM exhibit very close time-averaged solutions. As the cavitation number is redu ced, the difference becomes more noticeable. With cavitation, the FBM results give a larg er wake, especially for sheet cavitation (comparing the Left and Right columns in Figure 5-21 and Figure 5-22). Nevertheless, the performance of both turbulence models is largely consistent. The difference between the FBM and LSM can be illustrated more clearly by the time-dependent results. As already discu ssed, for cloud cavitati on, the cavity breakup phenomenon is more noticeable using the FB M than using the LSM in the cavity time evolutions, and FBM has a larger secondary cavity at the hydrofoil tailing tip which may

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90 induce the re-entrance effect (Figure 5-23 and Figure 5-24). In Figure 5-29, we further compare the time-averaged eddy viscosity distributions yielded by both turbulence models, with different cavitation numbers at AoA=5. With both turbulence models, we observe that the eddy viscosity decreases as the cavitation becomes more pronounced and fluctuating in time. Furthermore, the eddy visc osity increases in the recirculation region. It is also clear that the LSM yields cons istently higher eddy viscosity, resulting in reduced unsteadiness of the computed flow field. A B C Figure 5-29. Time-averaged viscosity contours, AoA=5. A) No-cavitation. B) Inception 1.12 C) Sheet cavitation 0.92 D) Cloud cavitation 0.55 LSM LSM LSM FBM FBM FBM

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91 D Figure 5-29. Continued Comparison of Speed-of-Sound models. The time-averaged CL and CD predicted with different speed-of-sound treatments, in cluding SoS-2 and SoS-2A (Section 4.3), are summarized in Table 5-6. It seems that the SoS-2A can marginally improve the aerodynamic predictions for two different cavita tion models, the modified and heuristic cavitation models (HM) (Section 3.3). The tim e-average u-velocity profiles by SoS-2 and SoS-2A using modified IDM are shown in Figure 5-30 with little difference. Table 5-6 Comparison of mean CL and CD, LSM, Cloud cavitation 0.80 AoA=8 degrees (Taking 0.95L as cavity boundary) Cavitation model IDM, LSM HM, LSM Exp. SoS-2 SoS-2A SoS-2 SoS-2A CL 0.643 0.665 0.682 0.681 0.804 CD 0.130 0.128 0.132 0.131 0.116 x/c y/c 0 0.2 0.4 0.6 0.8 1 1.2 -0.2 0 0.2 0.4Grid Experiment IDM,SoS-2 IDM,SoS-2A u/U 1.0 1.0 1.0 1.0 1.0 1.0 Figure 5-30. Time-averaged u-velocity of different SoS treatments, cloud cavitation 0.80 AoA=8. Experimental data are from Wang et al. (2001) LSM FBM

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92 CHAPTER 6 CONCLUSIONS AND FUTURE RESEACH 6.1 Conclusions of Present Research The Favre-averaged Navier-Stokes equations along with the turbulence closure and transport equation-based cavitation model, are employed in tur bulent cavitating flow computations. To ensure stability and c onvergence with good efficiency and accuracy, the pressure-based PISO algorithm is pres ented for time-dependent computations. The cavitation and turbulence models are assesse d by both steady and unsteady simulations in various configurations, incl uding cylinder, hydrofoils, convergent-divergent nozzle, and hollow-jet valve. In the context of turbulence modeling strategy, a filter-based model (FBM) for turbulence closure is presented for time-de pendent computations. Based on an imposed filter, conditional averaging is adopted for the Navier-Stokes equation to introduce one more parameter into the definition of the eddy viscosity. The filter can be decoupled from the grid, making it possible to ob tain grid-independent solutions with a given filter scale. The FBM is implemented in the time-dependent single-phase flow over a square cylinder. It can effectively modulate th e eddy viscosity, and produces much better resolution in capturing the unsteady features than the or iginal Launder-Spalding model (LSM). The results agree well with experimental data ex cept for an over-predicted Strouhal number. Also, the FBM shows compatibility with the LSM by changing the filter size. Subsequently, the FBM is applied to cavitati ng flows. It can signi ficantly reduce the eddy

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93 viscosity and generate much stronger time-dependency other than the LSM, which can damp out the oscillation and shedding. In the context of the cavitation modeling strategy, the various recently developed transport equation-based cavitation models demonstrate comparable pressure distributions in steady-state co mputations. However, they exhibit substantial variations for time-dependent computations. To gain more insight into the compressibility of the mixture regarding the local Mach number, th e two different approximations of speed-ofsound (SoS), namely, SoS-1 and SoS-2, are fu rther investigated in the time-dependent cavitating flows over a hollow-jet valve. The di fferent treatments have significant impact on the unsteady dynamics. Both are similar at the region close to th e pure liquid phase and theoretically agree with rational analysis. But the linea r approximation of the SoS-1 deviates from the theoretical analysis at th e region close to the gas phase. The analysis reveals that the SoS-1 is weak in pres sure-density correla tion, and the SoS-2 demonstrates a stronger pressure-density co rrelation in the cavitation vicinity. Therefore, SoS-2 is better than SoS-1 from this point of view. Furthermore, the generalization in terms of SoS-2 is introduced by averagi ng the local surrounding nodes. Generally, the SoS-2A predicts comparable result with SoS-2. In the recently developed interfacial dyna mics-based cavitaion model (IDM), an empirical factor is used to construct th e interfacial velocity in time-dependent computations, which links the interfacial veloci ty between phases to the local flow field velocity. This approach lacks generality beca use the interfacial velocity is a function of the phase change process. An enhanced a pproximation based on the net transformation between liquidand vapor-phase is presented to reconstruct the interfacial velocity. The

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94 modification can capture the cavity break-up phe nomena, which is qualitatively similar to the experimental observations. By assessing lift and drag coefficients, pressure and velocity distributions covering a wide regime from inception to cl oud cavitation, the new approximation can predict the major flow feature and reasonably agree with the experimental measurements. Meanwhile, we observe that the time-aver aged surface pressures give a reasonable estimate of mean lift and drag coefficients under various flow c onditions. While, since the average effect can smooth out some im portant instantaneous feature, the timeaveraged results alone are insufficient to assess turbulence model performance, with respect to flow physics such as wa ve propagation in particular. 6.2 Future Research Directions The present research can be further extended in the following directions Investigating the FBM in 3-D for single phase computations over the cylinder to further investigate the eff ects on the Strouhal number. Adopting flexible filter function f for the filter-based model to handle the steep variation in grid spacing, to avoid the weak filtered effect at the fine grid regions since the model currently uses a constant filter size with the criterion of max() x y in the computations. Developing a more accurate estimate of the interfacial velocity for IDM. Combining the technique developed in th e present study with suitable cryogenic modeling and numerical techni ques, as developed in pa rallel by Utturkar et al. (2005), to handle issues arisi ng from liquid rocket propulsion.

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95 LIST OF REFERENCES Ahuja, V., Hosangadi, A. and Arunajatesan, S., 2001, “Simulations of Cavitating Flows Using Hybrid Unstructured Meshes,” J. Fluids Eng., vol. 123, pp. 331-340. Ahuja, V., Hosangadi, A. and Ungewitter, R. J., 2003, “Simulations of Cavitating Flows in Turbopumps,” AIAA-2003-1261. Athavale, M.M and Singhal, A.K, 2001, “Num erical Analysis of Cavitating Flows in Rocket Turbopump Elements,” AIAA-2001-3400. Baldwin, B.S. and Lomax, H., 1978, “Thi n-Layer Approximation and Algebraic Model for Separated Turbulent Flow,” AIAA 1978-257. Batchelor, G.K., 1967, An Introduction to Fluid Dynamics, Cambridge University Press, New York. Batten, P, Goldberg, U. and Chakravarthy, S., 2002, “LNS an Approach towards Embedded LES,” AIAA-2002-0427. Bosch, G. and Rodi, W., 1996, “Simulation of Vortex Shedding past a Square Cylinder near a Wall,” Int. J. Heat and Fluid Flow, Vol. 17(3), pp. 267-275. Bosch, G. and Rodi, W., 1998, “Simulation of Vortex Shedding past a Square Cylinder with Different Turbulence Models,” Int. J. Numer. Meth. Fl., Vol. 28, pp. 601-616. Bradshaw, P., Ferriss, D.H. and Atwell, N.P., 1967, “Calculation of Boundary Layer Development Using the Turbulent Energy Equations, J. Fluid Mech., Vol. 28(3), pp. 593-616. Bressloff, N.W., 2001, “A Parallel Pressure Implicit Splitting of Operator Algorithm Applied to Flows at All Speed,” Int. J. Numer. Meth. Fl., Vol. 36(5), pp. 497-518. Breuer, M., Jovicic N. and Mazaev K., 2003, “Comparison of DES, RANS and LES for the Separated Flow around a Flat Plate at Hi gh Incidence,” Int. J. Numer. Meth. Fl., Vol. 41, pp. 357-388. Coutier-Delgosha, O., Fortes -Patella, R. and Reboud, J.L., 2003, “Evaluation of the Turbulence Model Influence on the Nu merical Simulations of Unsteady Cavitation,” J. Fluids Eng., Vol. 125, pp. 33-45.

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96 Coutier-Delgosha, O., Fortes-P atella, R., Reboud, J.L., Hakimi, N. and Hirsch, C., 2005, “Numerical Simulation of Cavitating Flow in 2D and 3D Inducer Geometries,” Int. J. Numer. Meth. Fl., Vol. 48, pp. 135-167. Delannoy, Y. and Kueny, J.L, 1990, “Two Phase Flow Appraoch in Unsteady Cavitation Modeling,” ASME Cavitation and Multiphase Forum, Vol. 98, pp. 153-158. Duttweiler, M.E. and Brennen, C.E., 2002, “S urge Instability on a Cavitating Propeller,” J. Fluid Mech., Vol. 458, pp. 133-152. Favre, A., 1965, “Equations des Gaz Turbulents Compressibles,” J. de Mecanique, Vol. 4, No. 3, pp. 361-390. Franc, J.P., Avellan, F., Belahadji, B., Billa rd, J.Y., Briancon, L., Marjollet, Frechou, D., Fruman, D. H., Karimi, A., Kueny, J.L. and Michel, J. M., 1995, La Cavitation: Mecanismes Physiques et Aspects Industriels, Presses Universitaires de Grenoble, Grenoble, France. Frank, R. and Rodi, W., 1993, “Calculation of Vortex Shedding past a Square Cylinder with Various Turbulence Models,” Turbulen t Shear Flows 8, (Durst et al., eds.), Springer, New York, pp. 189-204. Friedrichs, J. and Kosyna, G., 2002, “Rotati ng Cavitation in a Cent rifugal Pump Impeller of Low Specific Speed,” J. Fluids Eng., Vol. 124, pp.356-362 Gopalan, S. and Katz, J., 2000, “Flow Struct ure and Modeling Issues in the Closure Region of Attached Cavitation,” Phys ics of Fluids, Vol. 12, pp. 895-911. Hanjalic, K., Launder, B.E. and Schiestel, R., 1980, “Multiple-Time-Scale Concepts in Turbulent Transport Modeling,” Proc. of Turbulent Shear Flows 2, Springer, Berlin, pp. 36-49. Hosangadi, A., Ahuja, V. and Ungewitter, R. J., 2004, “Simulations of Cavitating Flows in Turbopumps,” J. Propulsion and Power, Vol. 20 (4), pp. 604-611. Iaccarino, G., Ooi A., Durbin, P.A. and Behni a, M., 2003, “Reynolds averaged simulation of unsteady separated flow,” Int. J. of Heat and Fluid Flow, Vol. 24, pp. 147–156. Issa, R.I., 1985, “Solution of the Implic itly Discretised Fluid Flow Equations by Operator-Splitting,” J. Comp. Physics, Vol. 62, pp. 40-65. Johansen, S.T., Wu, J. and Shyy, W., 2004, “F ilter-based unsteady RANS computations,” Int. J. Heat and Fluid Flow, Vol. 25 (1), pp. 10-21. Kato, M. and Launder, B.E., 1993, “The Mode ling of Turbulent Flow around Stationary and Vibrating Square Cylinders,” Proc. 9th Symposium on Turbulent Shear Flows, Kyoto, Japan.

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103 BIOGRAPHICAL SKETCH Jiongyang Wu was born in a small village in Jieyang county, Guangdong province, P.R. China. He received both his B.S. and M. S. degrees from the Hydraulic Engineering Department at Tsinghua University, in 1997 and 2000, respectively. In August 2000, he joined the Computational Thermo-Fluids gr oup at the Department of Mechanical and Aerospace Engineering, University of Florida for his Ph.D study. His research interests include turbulence and cavitation modeling, mu lti-phase flows, fluid machinery flows. Jiongyang met his wife, Weishu Bu, during his study in University of Florida. They married in March 2003. And then their son, Andy Wu, was born and became a new family member.


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Title: Filter-Based Modeling of Unsteady Turbulent Cavitating Flow Computations
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FILTER-BASED MODELING OF UNSTEADY TURBULENT CAVITATING FLOW
COMPUTATIONS













By

JIONGYANG WU


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

UNIVERSITY OF FLORIDA


2005





























Copyright 2005

by

Jiongyang Wu

































To my Parents; my wife, Weishu; and my son, Andy, for their love and support















ACKNOWLEDGMENTS

I would like to express my sincere thanks and appreciation to my advisor Professor

Wei Shyy. His thorough guidance and support benefited me tremendously in exploring

and pursuing my research interest. I thank him for his boundless patience and generous

attitude, and his enduring enthusiasm in educating me both in research and personal

development. I have benefited much from the collaboration with Professor Stein T.

Johansen. I would like to thank Professor Renwei Mei for providing some helpful

suggestions and serving on my committee. Additionally, I would like to thank Professors

Louis N. Cattafesta, David W. Mikolaitis, and Jacob N. Chung for serving on my

committee, and Dr. Siddharth Thakur for generously sharing his experience.

I have had the privilege to work with all the individuals of the Computational

Thermo-Fluids group, to whom I also give my thanks.

I would like to deeply thank my parents. Their encouragement, support, trust, and

love have given me power and strength through these years. My wife, Weishu Bu, has

been with me and supported me all the time. My son, Andy Wu, is another source of my

invaluable wealth. No words can possibly express my gratitude and love to them.

Finally, I would like to thank the NASA Constellation University Institute Program

(CUIP) for financial support.
















TABLE OF CONTENTS

page

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

LIST OF TABLES ............. ................... ......... .... ... ............... .. vii

LIST O F FIG U R E S ......................................................... ......... .. ............. viii

L IST O F SY M B O L S .... ....................................................... .. ....... .............. xi

ABSTRACT ........ .............. ............. ...... ...................... xiv

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

1.1 B background of C aviation ............................................................................ 1
1.1.1 C aviation Types in Fluids....................................................... ............ ..
1.1.2 Cavitation Inception and Param eter ........................................ ..................2
1.2 Research Motivations and Objectives .......................................... ...............3
1.2.1 Challenges and M otivations ........................................ ...... ............... 3
1.2 .2 R research O bjectiv es ........................................................................ ...... 8

2 NAVIER-STOKES EQUATIONS AND TURBULENCE MODELS ....................9

2 .1 G ov earning E qu action s .................................................................... .............. .9
2.2 Turbulence M odel ....... .... .................................. ........ .. .. .. ............ 11
2.3 Filter-Based RAN S M odel .............................................................................12
2 .3.1 L literature R review ............................................................ ....................14
2.3.2 Filter-Based M odel Concept (FBM ) .................................. ............... 16
2.3.3 Modeling Implementation ............................ ..............17
2.4 Assessing FBM in Time-Dependent Single-Phase Flow............... ..................18

3 CA V ITA TION M OD ELS ................................................ .............................. 32

3.1 Governing Equations ....................... ............................ 32
3.2 Literature Review of Cavitation Studies.................................... ............... 33
3.2.1 Cavitation Com pressibility Studies ......................... ....................... .... 35
3.2.2 Cavitation Studies on Fluid Machinery Components and Systems............38
3.3 Transport Equation-based Cavitation Model (TEM)................ .............. ....41









4 NUMERICAL METHODS .................................... ............................. 45

4.1 Pressure-B ased A lgorithm ............................................................. ................ .... 46
4.2 Pressure Implicit Splitting of Operators (PISO) Algorithm for Unsteady
Com putations ............................ ..................... 47
4.3 Speed-of-Sound (SoS) Numerical Modeling............................................50

5 ASSESSING TIME-DEPENDENT TURBULENT CAVITATION MODELS .......53

5.1 Cavitating Flow through a Hollow-Jet Valve ............... ............. ....................53
5.1.1 Steady and Unsteady Turbulent Cavitating Flows ...................................55
5.1.2 Impact of Speed-of-sound Modeling..................................... .................60
5.2 Turbulent Cavitating Flow through a Convergent-Divergent Nozzle ................65
5.3 Turbulent Cavitating Flow over a Clark-Y Hydrofoil.......................................73

6 CONCLUSIONS AND FUTURE RESEACH .....................................................92

6.1 Conclusions of Present Research...................................... ........................ 92
6.2 Future R research D irections........................................................ ............... 94

LIST OF REFEREN CES ............................................................................. 95

BIOGRAPHICAL SKETCH ............................................................. ............... 103
















LIST OF TABLES


Table page

2-1 Summary of the hybrid RANS/LES studies ........................................................ 15

2-2 Parameters used in the computation................................ ................................ 19

2-3 Comparisons of different turbulence models ................................. ............... 24

3-1 Overview of the compressible cavitation studies................. ............... ...............37

3-2 Overview of cavitation on pumps, propellers, inducers and turbine blades ............40

5-1 Time-averaged liquid volume fraction v/s pressure-density correlation at
multiple points inside the cavity, original IDM with SoS-1, LSM.......................... 63

5-2 Time-averaged liquid volume fraction v/s pressure-density correlation at
multiple points inside the cavity, original IDM with SoS-2, LSM.......................... 63

5-3 Comparisons of Strouhal number of original and modified IDM with SoS-2A ......72

5-4 Time-averaged cavity leading and trailing positions of different turbulence
models, modified IDM with SoS-2A, AOA=5 degrees ............................ 78

5-5 Time-averaged cavity leading and trailing positions of different turbulence
models, modified IDM with SoS-2A, AoA=8 degrees .............................78

5-6 Comparison of mean CL and CD, LSM, Cloud cavitation c = 0.80, AoA=8
d e g re e s ...................................... ................................... ................ 9 1
















LIST OF FIGURES


Figure p

1-1 Different types of cavitation visualization ...................................... ............... 3

1-2 Isothermal harmonic speed-of-sound in the two-phase mixture .............................6

2-1 Geom etry configuration of square cylinder ......................................... ................ 19

2-2 Streamline snap-shot on fine grid with A= 0.15D ........................................20

2-3 Pressure behaviors at the reference point on the fine grid ....................................21

2-4 Time-averaged U-velocity along the horizontal centerline behind the cylinder......23

2-5 Snap-shots of velocity contour. Color raster plot of axial velocities .....................23

2-6 Streamline snap-shots of two turbulence models on fine grid ..............................24

2-7 Time-averaged U-velocity along y at x/D=0.0................................. ... ................ 26

2-8 Time-averaged velocities along y at x/D=1.0 .................................................. 26

2-9 Tim e-averaged u-velocity along y direction ................................. ................26

2-10 M ean kinetic energy on different grids ........................................ .....................28

2-11 Comparisons of different filter sizes on time-averaged v-velocity along y at
x/D = 1.0 .............. .................................... ................ ......... 29

2-12 Comparisons of different filter sizes on kinetic energy .......................................29

2-13 Time-averaged viscosity contours of different turbulence models on fine grid.......30

3-1 Sketch of a cavity in homogeneous flow ...................................... ............... 42

3-2 Interface vector sketch in a CV ...................................................... .............. 43

5-1 Valve geom etry ................................. ... ..... ........... ......... 54

5-2 Computational domains and boundary conditions................................................55









5-3 Density contour lines of the steady state solution, original IDM with SoS-1,
L S M ................................................... ...................... ................ 5 6

5-4 Middle section density contours at different time instants, original IDM with
SoS-1, LSM .............................................................................. 57

5-5 Time evolutions at different locations, original IDM with SoS-1, LSM .................58

5-6 Projected 2-D streamlines at middle plane and experimental observation ..............59

5-7 Middle section density contours of different SoS at different instant time,
original ID M L SM ............................................ ......................... 61

5-8 Two different SoS in hollow-jet valve flow, original IDM, LSM ...........................63

5-9 Pressure time evolutions of different SoS, original IDM, LSM.............................. 64

5-10 Time evolution and spectrum of pressure and density of two SoS definitions at a
point at the cavitation vicinity, original IDM LSM ............................................. 64

5-11 Time-averaged eddy viscosity contours of different grids, original IDM with
S o S -2 c = 1.9 8 .......................................................................67

5-12 Time-averaged vapor volume fraction comparisons of different grids, original
ID M w ith S o S -2 c = 1.9 8 ........................................................... .....................6 8

5-13 Time-averaged comparisons of different turbulence models on fine grid with
A = 0.25Lc,, original IDM with SoS-2, c =1.98 ................. ............... 68

5-14 Instantaneous profiles on fine grid with A = 0.25Lc original IDM with SoS-2,
U = 1.9 8 ............................................................................. 6 9

5-15 Pressure contours and streamlines comparison of two turbulence models on fine
grid, original IDM with SoS-2, c = 1.98 ...................................... ............... 70

5-16 Pressure evolutions of original and modified IDM with SoS-2A at a reference
p o in t ..............................................................................7 2

5-17 Cavity shape and recirculation zone during cycling of original and modified
ID M ......... ............................... ... ...................................................7 2

5-18 Time-averaged volume fraction and velocity comparisons of original and
m o dified ID M .................................................................... .. 7 3

5-19 Clark-Y geometry sketch and Grid blocks.........................................................76

5-20 Grid sensitivity of the time-averaged u- and v-velocity, LSM, AoA=5 .................77









5-21 Time-averaged volume fraction contours and streamlines of different turbulence
m odels, A oA = 5 ................................................................ ................. 7 8

5-22 Time-averaged volume fraction contours and streamlines of different turbulence
models, AoA=8 ........................... .................. .......... .............79

5-23 Time evolutions of cloud cavitation a = 0.55, AoA=50.......................................80

5-24 Time evolutions of cloud cavitation a = 0.80, AoA=8 .................................. 81

5-25 Cavity stage comparisons, cloud cavitation a = 0.80, AoA=8 ..............................83

5-26 Time-averaged u- and v-velocities of two turbulence models, AoA=5 ..................84

5-27 Time-averaged u- and v-velocities of two turbulence models, AoA=80 ..................86

5-28 Time-averaged lift and drag coefficients comparisons ........................................88

5-29 Time-averaged viscosity contours, AoA=50 ................................ .............. 90

5-30 Time-averaged u-velocity of different SoS treatments, cloud cavitation
c = 0.80, A oA =8 .............. .............................................. ..... .............. 91



















c

C

C


C1 C'2 Cp


Cdest


Cprod

k

m

m

P

q

U

u

U

t

T

xi

y

yp


LIST OF SYMBOLS

speed-of-sound

arbitrary 0(1) constant

pressure coefficient


turbulence model constants


empirical constant in the evaporation term


empirical constant in the condensation term


turbulent kinetic energy

evaporation rate

condensation rate

pressure

source term

velocity components in Cartesian coordinates

non-dimensional velocity

magnitude of the horizontal component of velocity

time, mean flow time scale

temperature

Cartesian coordinates

non-dimensional normal distance from the wall

normal distance of the first grid point away from the wall









a volume fraction

5ij Kronecker delta function

F turbulent dissipation rate

t laminar viscosity

Pt turbulent viscosity

, r, curvilinear coordinates

generalized dependent variable

r Reynolds stress

r, wall shear stress

vt kinematic viscosity

Pm mixture density

a cavitation parameter

ok, 7 turbulence model constants



Subscripts, Superscripts

I interface

L, 1 liquid phase

V, v vapor phase

m mixture phase

n normal direction

s tangential direction

t tangential direction

x component in x coordinate direction









y component in y coordinate direction

z component in z coordinate direction

oo free stream

* predicted value















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

FILTER-BASED MODELING OF UNSTEADY TURBULENT CAVITATING FLOW
COMPUTATIONS

By

Jiongyang Wu

August 2005

Chair: Wei Shyy
Major Department: Mechanical and Aerospace Engineering

Cavitation plays an important role in the design and operation of fluid machinery

and devices because it causes performance degradation, noise, vibration, and erosion.

Cavitation involves complex phase-change dynamics, large density ratio between phases,

and multiple time scales. Noticeable achievements have been made in employing

homogeneous two-phase Navier-Stokes equations for cavitating computations in

computational and modeling strategies. However, these issues pose challenges with

respect to accuracy, stability, efficiency and robustness because of the complex unsteady

interaction associated with cavitation dynamics and turbulence.

The present study focuses on developing and assessing computational modeling

techniques to provide better insight into unsteady turbulent cavitation dynamics. The

ensemble-averaged Navier-Stokes equations, along with a volume fraction transport

equation for cavitation and turbulence closure, are employed. To ensure stability and

convergence with good efficiency and accuracy, the pressure-based Pressure Implicit









Splitting of Operators (PISO) algorithm is adopted for time-dependent computations. The

merits of recent transport equation-based cavitation models are first re-examined. To

account for the liquid-vapor mixture compressibility, different numerical approximations

of speed-of-sound are further investigated and generalized. To enhance the generality and

capability of the recent interfacial dynamics-based cavitation model (IDM), we present an

improved approximation for the interfacial velocity via phase transformation. In

turbulence modeling, a filter-based model (FBM) derived from the k e two-equation

model, an easily deployable conditional averaging method aimed at improving unsteady

simulations, is introduced.

The cavitation and turbulence models are assessed by unsteady simulations in

various geometries including square cylinder, convergent-divergent nozzle, Clark-Y

hydrofoil, and hollow-jet valve. The FBM reduces eddy viscosity and captures better

unsteady features in single-phase flow, and yields stronger time-dependency in cavitating

flows, than the original k model. Various cavitation models show comparable steady-

state pressure distributions but exhibit substantial variations in unsteady computations.

The influence of speed-of-sound treatments on the outcome of unsteady cavitating flows

is documented. By assessing lift and drag coefficients, pressure and velocity distributions

from inception to cloud cavitation regimes, the present approach can predict the major

flow features with reasonable agreement to experimental data.














CHAPTER 1
INTRODUCTION

1.1 Background of Cavitation

In liquid flow, if the pressure drops below the vapor pressure, the liquid is unable to

withstand the tensile stress and then adjusts its thermodynamic state by forming vapor-

filled two-phase mixed cavities. This phenomenon is known as cavitation (Batchelor

1967). It can occur in a wide variety of fluid machinery components, such as nozzles,

injectors, marine propellers and hydrofoils (Knapp et al. 1970). Cavitation is commonly

associated with undesired effects, such as pressure fluctuation, noise, vibration and

erosion, which can lead to performance reduction, and can damage the hydrodynamic

surface. Apart from these negative effects, one known benefit of cavitation is that it can

reduce friction drag (Lecoffre 1999, Wang et al. 2001). For example, in high-speed

hydrofoil boats, supercavitating propellers or vehicles the gaseous cavity enveloping the

external body surface can provide a shield isolated from the liquid, which helps to cut

down the friction.

1.1.1 Cavitation Types in Fluids

Different kinds of cavitation can be observed depending on flow conditions, fluid

properties and different geometries. Each kind of cavitation has characteristics that

distinguish it from the others. Major types of cavitation are briefly described below:

Traveling cavitation: Individual transient cavities or bubbles form in the liquid,
expand or shrink, and then move downstream (Knapp et al. 1970). Typically,
traveling cavitation is observed on hydrofoils at a small angle of attack. The
number of nuclei present in the upcoming flow highly affects the geometries of
the bubbles (Lecoffre 1999) (Figure 1-1A).










Cloud cavitation: Cloud cavitation is periodically caused by vorticity shed into
the flow field. It can associate with strong vibration, noise and erosion
(Kawanami et al. 1997) (Figure 1-1B).

Sheet cavitation: Sheet cavitation is a fixed, attached cavity or pocket cavitation
in a quasi-steady sense (Knapp et al. 1970) (Figure 1-1C). The interface
between the liquid and the vapor varies with flows.

Supercavitation: In Supercavitation the cavity appears as the envelope of the
whole solid body, and can be observed when underwater vehicles operate at
very high speeds or in projectiles with a speed of 500 m/s to 1500 m/s
(Kirschner 2001). A typical supercavitating hydrofoil is shown in Figure 1-1D.

Vortex cavitation: Vortex cavities form in the cores of vortices in regions of
high shear. They can occur on the tips of rotating blades, and can also appear in
the separation zones of bluff bodies (Knapp et al. 1970) (Figure 1-1E).

1.1.2 Cavitation Inception and Parameter

The criterion for cavitation inception based on a static approach can be formulated

as Eq. (1.1), with P as local pressure and Pv as vapor pressure

P
A parameter is needed to describe the flow condition relative to those for cavitating

flows and to obtain a unique value for each set of dynamically similar cavitation

conditions. In cavitation terminology a denotes the parameter of similitude


a-= (1.2)
p2 /2

In Eq. (1.2), P, V, are the reference pressure and velocity, respectively. Usually,

they take the infinity values. Unfortunately, a is not a stringent parameter of similitude

for cavitating flows. It is a necessary but not a sufficient condition. One reason is that

nuclei in free stream flow can impact the two-phase flow structures (Lecoffre 1999).













































Figure 1-1. Different types of cavitation visualization (Franc et al. 1995). A) Traveling
cavitation. B) Cloud cavitation. C) Sheet Cavitation. D) Supercavitation. E)
Vortex cavitation

1.2 Research Motivations and Objectives

1.2.1 Challenges and Motivations

Cavitation occurs in various engineering systems, such as pumps, hydrofoils and

underwater bodies. It is especially prone to occur when fluid machinery operates at off-

design conditions. Cavitation may cause not only the degradation of the machine

performance but flow instability, noise, vibration, and surface damage.









Phenomenologically, cavitation often involves complex interactions of turbulence and

phase-change dynamics, large density variation between phases up to a ratio of 1000,

multiple time scales, and pressure fluctuations. These physical mechanisms are not well

understood because of the complex unsteady flow structures associated with cavitation

dynamics and turbulence. There are significant computational issues regarding to

accuracy, stability, efficiency and robustness of numerical algorithms for turbulent

unsteady cavitating flows.

Therefore, turbulent cavitating flow computations need to address both turbulence

and cavitation modeling issues. Noticeable efforts have been made in employing Navier-

Stokes equations for simulations in this field. Among the various modeling approaches,

the transport equation-based cavitation models (TEM) have received more interest, and

both steady and unsteady flow computations have been reported (Singhal et al. 1997,

Merkle et al. 1998, Kunz et al. 2000, Ahuja et al. 2001, Venkateswaran et al. 2002,

Senocak and Shyy 2002a,b, 2003, 2004a,b). Senocak and Shyy (2002b, 2004a) developed

an interfacial dynamics-based cavitation model (IDM) accounting for cavitation

dynamics. Also, Senocak and Shyy (2002b, 2004a) and Wu et al. (2003b) assessed the

merits of alternative TEM models. They showed that for steady-state computations,

various cavitation models produce comparable pressure distributions. Despite the good

agreement, noticeable differences have been observed in the predicted density field,

especially in time-dependent computations. In IDM, an empirical factor is used to

construct the interfacial velocity in time-dependent computations. Then the cavity

interfacial velocity is linked to the local fluid velocity. Such an approach lacks generality

because the interfacial velocity is supposed to be a function of the phase-change process.









A much closer investigation is needed to further understand differences in model

performance, especially in unsteady cavitating flows.

The differences mentioned above also imply that the compressibility characteristics

embodied in each cavitation model are different. This aspect can be significant in the

unsteady flow computations because the speed-of-sound directly and substantially affects

the cavity time-dependent features (Senocak and Shyy 2003, 2004b). Actually, the local

speed-of-sound of the two-phase mixture can reduce by an order of magnitude from

either value of pure liquid or vapor. It then leads to large enough Mach number variations

around the cavity to make the compressibility effect substantial. The harmonic expression

for speed-of-sound in an isothermal two-phase mixture (Venkateswaran et al. 2002,

Ahuja et al. 2003) can be presented as

1 a 1-a
C= P( g + g) (1.3)
m PgCg P1CI

where p,, Pg, p, are mixture density, vapor density, liquid density, respectively, ag is

the vapor fraction, and Cm, Cg, C1 are speed-of-sound in mixture, in pure vapor, and in

pure liquid, separately. The behavior of Cm is plotted in Figure 1-2 as a function of the

vapor volume fraction. The speed-of-sound drops dramatically over a wide range of the

mixture regime. Even though the bulk flow is of a very low Mach number flow, the local

Mach number in the interface region may become large. Because of the lack of a

dependable equation of state for the mixture sound propagation, Senocak and Shyy

(2003) presented two different numerical treatments of speed-of-sound, namely SoS-1

and SoS-2, and found that they had an impact on the unsteady cavitating behavior in the

convergent-divergent nozzles. They also suggested that SoS-2 was more likely to produce











the correct unsteady behaviors. This underscores that careful and accurate handling of the


speed-of-sound is important. Currently, a robust compressible model for cavitating flows


is still the subject of research.



Pure liquid

1400

1200


0
t 1000
CO
800

g- 600

400
Pure vapor
200 Flow in this range can be
transonic even supersonic
20 Flow, i i I I iI
0 025 05 075 1
Vapor volume fraction

Figure 1-2. Isothermal harmonic speed-of-sound in the two-phase mixture

Besides cavitation modeling, the turbulence model can significantly influence the


cavitating flow structures. Serious implications of turbulence modeling on cavitating


flows were recently revealed by researchers (Senocak and Shyy 2002a, Kunz et al. 2003,


Coutier-Delgosha et al. 2003, Wu et al. 2003b). They reported that high eddy viscosity of


the original Launder-Spalding version of the k Reynolds-averaged Navier-Stokes


(RANS) model (Launder and Spalding 1974) can dampen the vortex shedding motion


and excessively attenuate the cavitation instabilities. Wu et al. (2003a) conducted the


non-equilibrium modification, including stationary and non-stationary, in the cavitating


flows over a hollow-jet valve, and observed no striking impact on the incipient cavitation.


Consequently, simulation of phenomena such as periodic cavity inception and


detachment requires improved modeling approaches. The Large Eddy Simulation (LES)









approach, originally proposed by Smagorinsky (1963) and refined by many researchers

(Piomelli 1999, Moin 2002, Sagaut 2003) is an actively pursued route to simulate

turbulent flows. However, it is fundamentally difficult to find a grid-independent LES

solution unless one explicitly assigns a filter scale (Moin 2002), making the state-of-the-

art immature for cavitating flow computations. On the other hand, attempts have also

been made to employ the information obtained from Direct Numerical Simulations

(DNS) to supplement lower order models (e.g. Sandham et al. 2001). To our knowledge,

no efforts have been reported to employ LES or DNS for turbulent cavitating flows of

practical interest. Recently, efforts have been made to combine the filter concept and the

RANS model in single-phase (Mavridis et al. 1998, Batten and Goldberg 2002, Nichols

and Nelson 2003, Nakayama 2002, Breuer et al. 2003) and, recently and even more

preliminary, cavitating flow computations (Kunz et al. 2003).

Hence, it is useful to identify ways to improve the predictive capability of the

updated cavitation models and the current RANS-based engineering turbulence closures,

which can keep the advantages of RANS approaches and can be easily implemented in

practical engineering applications with a clear physical concept. This can help better

capture unsteady characters (e.g. the shedding hardly achieved in current turbulent

cavitating flow simulations), and can provide better insight into the interactions between

cavitation and turbulence models. This can also help establish a dependable, robust and

accurate computational CFD tool to analyze and minimize the cavitation effects in the

design stage of fluid machinery components or systems.









1.2.2 Research Objectives

Our goal is to develop and assess computational modeling techniques to provide

better insight into unsteady turbulent cavitation dynamics, aimed at improving the

handling of the above-mentioned issues in practical engineering problems.

Objective 1: Investigate alternative turbulence modeling strategies in the context of the

Favre-averaged Navier-Stokes approach by doing the following:

Develop a filter-based turbulence model (FBM) as the alternative to the original
Launder-Spalding Model, both employing the k e two-equation closure.

Use the FBM in the cavitation simulations to provide insight into the dynamic
interactions between turbulence and cavitation.

Objective 2: Enhance the cavitation model prediction capability based on the recent

transport equation-based model (TEM) by doing the following:

Further examine the merits of the recent achievements in the TEM, focusing on
the interplay of the two-phase compressibility and turbulence by using correlation
and spectral analysis with the direct evaluation by experimental measurements,
and generalize the speed-of-sound numerical treatment.

Adopt an improved approximation to construct the interfacial velocity by
accounting for phase transformation based on the recently developed interfacial
dynamics-based cavitation model (IDM).














CHAPTER 2
NAVIER-STOKES EQUATIONS AND TURBULENCE MODELS

2.1 Governing Equations

The Navier-Stokes equations in their conservative form governing a Newtonian

fluid without body forces and heat transfer are presented below in the Cartesian

coordinates

ap 0 puj)
+ = 0 (2.1)
at 9x


(pu')+- (puyuj)=- OPK+-- [ + a- 2 0u, (2.2)
at ax 9x, 9x 9x ax 3 Ox,

The viscous stress tensor is given by


S= P + 2 (2.3)
ay Ox x, 3 ax,

In theory, direct numerical simulation (DNS) can be used to solve Equations (2.1)-

(2.3) (Rogallo and Moin 1984). However, in practice the limited computing resources

prevent us from pursuing such endeavors for high-Reynolds-number flows. Since we are

most interested in predicting the mean properties of the turbulent flow, we can first

conduct an averaging procedure to simplify the content of the equations. For time-

dependent flow computations, ensemble averaging is an appropriate conceptual

framework. Specifically, to avoid the additional terms involving the products of

fluctuations between density and other variables in variable density flows in Reynolds









time averaging, Favre-averaging is preferred (Favre 1965) in the following form. Further

details can be obtained from many references (e.g. Shyy et al. 1997, Wilcox 1998)


= ; p = = +0"; pf"= 0 (2.4)
p

Then the Favre-averaged N-S equations become

+ V. (p,) = 0 (2.5)
at

+ V. (puu"i)= -VP + V. (rz -pu") (2.6)


8u, Oui 2 f/ u" B u" 2 c"
r, = P( 2 + --' 3,)+Jp( 2&t+-~- ) (2.7)
adx ax 3 av ax ax 3 a

Note that the viscosity fluctuation is neglected. The nonlinear terms (-pu"u" ),

namely the Reynolds stresses, need additional modeling. The Boussinesq's eddy-

viscosity hypothesis for turbulence closure leads to the following form

j, =- =a t( + )-39, ( + pk) (2.8)
ax 9xI 3 ay

Finally, the Favre-averaged N-S equations reach the following form

S+ V. (p ) = 0 (2.9)
at

P+V. (p ) = VP+V.(, +r+R) (2.10)


R fi7 O 2ai
j, + r, = (U +U,)() (2.11)
cx cx, 3 c-

Compared with the original N-S equations, Favre-averaged N-S equations have the

same apparent structure. The above equations can be cast in generalized curvilinear









coordinates. The procedure for the transformation of these governing equations is well

established (Shyy 1994, Thakur et al. 2002).

2.2 Turbulence Model

There are several types of turbulence models. Commonly the classification can be

presented as: (a) Algebraic (zero-equation) models (Baldwin and Lomax 1978); (b) One-

equation models (Bradshaw et al. 1967); (c) Two-equation model, such as k two-

equation (Launder and Spalding 1974), k o two-equation (Saffman 1970); (d) Second-

moment closure models (Launder et al. 1975). Among the above models, the k-g

turbulence model (Launder and Spalding 1974) has been popular because it is

computationally tractable with deficiencies reasonably well documented (Shyy et al.

1997, Wilcox 1998). In this model, two partial differential equations accounting for the

transport of turbulent kinetic energy k and for dissipation rate s are solved. The following

transport equations follow the concept of Launder and Spalding (1974) and are

commonly adopted as

1 Au"
pk = pu"u; pE = (2.12)


a(pk) a(pi k) a a .)
t+ = P- [( + + ( (2.13)
tj xj x o-k 8j

(pe) d(P ) E E2 A
a+ =C P,- Clp + [ ) -] (2.14)
at Qx- k k x Ua Oxj

where o-k and o-s are the turbulence model constants, Ce; and Cs2 are the turbulence

model parameters that regulate the production and destruction of dissipation of

turbulence kinetic energy, respectively. The turbulent production term P, and the

turbulent viscosity are defined as










p R ; UtP (2p15)


Wall functions are used to address the effect of wall boundaries along with the

k turbulence model. The empirical coefficients originally proposed by Launder and

Spalding (1974), assuming local equilibrium between production and dissipation of

turbulent kinetic energy, as given below

C,, =1.44, C,, =1.92, = 1.3, k = 1.0 (2.16)

In the following, we call it the Launder-Spalding model (LSM).

2.3 Filter-Based RANS Model

Turbulence plays a very important role in flow phenomena, especially since the

Reynolds numbers are high in practical engineering problems. The Reynolds-averaged

Navier-Stokes (RANS) and, for variable density flows, the corresponding Favre-averaged

Navier-Stokes (FANS) models, such as the k-s two-equation closure, have been very

popular in providing good prediction for a wide variety of flows with presently available

computational resources. However, RANS models describe flows in a statistical sense

typically leading to time-averaged pressure and velocity fields. Generally these

approaches are not able to distinguish between quasi-periodic large-scale and turbulent

chaotic small-scale features of the flow field. The representation may lose the unsteady

characteristics when the flow field is governed by both phenomena, even with the help of

the non-equilibrium modifications on the set of empirical constants (Wu et al. 2003a). It

is clear that the statistical turbulence models have difficulties with the complex

phenomena, such as flows past bluff bodies which involve separation and reattachment,

unsteady vortex shedding and bimodal behaviors, high turbulence, large-scale turbulent









structures as well as curved shear layers (Frank and Rodi 1993, Rodi 1997, Shyy et al.

1997).

On the other hand, Large Eddy Simulation (LES) operates with unsteady fields of

physical values. Spatial filtering is applied instead of averaging in time or ensemble, and

turbulent stresses are divided into resolved and modeled parts, such as subgrid-scale

models (SGS) with Smagorinsky's hypothesis (Breuer et al. 2003). Only the large

energy-containing eddies are numerically resolved, accomplished by filtering out the high

frequency component of the flow fields and using the low-pass-filtered form of the N-S

equations to solve for the large-scale component (Kosovic 1997).

In recent years, attempts have been made to remedy the gap between RANS and

LES, called the Hybrid RANS/LES methods (Nakayma and Vengadesan 2001, Breuer et

al. 2003, Nelson and Nichols 2003). Various strategies have been investigated as

summarized below

Ad-hoc models (Kato and Launder 1993, Bosch and Rodi 1996,1998): using a
rotation parameter to replace the original production term.

Filter RANS/LES model (Koutmos and Mavirdis 1997): combining elements
from both LES and standard eddy-viscosity approaches, by comparing the
characteristic length with the mesh size-spatial filter to reconstruct the
viscosity.

Multiple Time-scale (MTS) Method (Hanjalic et al. 1980, Nichols and Nelson
2003): dividing the turbulent energy spectrum as two parts and breaking the
standard k-s equations into two sets of equations.

Detached Eddy Simulation (DES) (Spalart 1997, Roy et al. 2003): keeping the
whole boundary layer (attached eddies) to a RANS model and only the
separated regions (detached eddies) to LES.









2.3.1 Literature Review

Firstly, we will review the recent studies related to the hybrid RANS/LES studies

according to the above categories with a summary in Table 2-1.

Bosch and Rodi (1996) applied the ad-hoc model, which used a rotation term to

reduce the turbulent production, to simulate the vortex shedding past a square cylinder

near a wall. Compared with the standard turbulence model, this modification produced

better unsteady behavior and obtained good agreement with the experimental data.

Following their previous study, Bosch and Rodi (1998) adopted a 2-D ensemble-averaged

unsteady Navier-Stokes equation, with the ad-hoc model to compute the vortex shedding

past a square cylinder. The numerical results agreed well with experimental measurement

and other similar numerical results. They also carried out other versions of turbulence

models in the same configurations for comparisons. They concluded the present

modification was better.

Rodi (1997) simulated turbulent flows over two basic bluff bodies, 2-D square

cylinder and 3-D surface-mounted-cube, using different Reynolds numbers and a variety

of LES and RANS methods. The various calculation results generally agreed with

detailed experimental data. Assessment was given based on performance, cost and the

potential of the various methods based on the comparison with the measurement.

Koutmos and Mavridis (1997) combined LES and the standard k-g models to

formulate the eddy-viscosity by comparing a mesh size A with the characteristic length in

the flow field. Then the turbulent viscosity was constructed in two different ways. The

calculation of unsteady separated flows of square cylinder wake and backward-facing









step re-circulating flows under low- (Re=5000) and high-Reynolds number (Re=37000)

conditions agreed well with the experimental measurements.

Nagano et al. (1997) developed a low Reynolds number multiple time-scale

turbulence model (MTS) which separated the turbulent energy spectrum into two parts,

namely, production and transfer according to the wave-number. Then the eddy-viscosity

was approximated by solving both wall and homogeneous shear flows. The test results

compared well with the DNS and experimental data. They concluded for homogeneous

shear flow the difference between the new model and the standard k-s model from the

estimation of the characteristic time-scales, not from the discrepancy in the eddy

approximation.

Nichols and Nelson (2003) employed different turbulence models, including

RANS, MTS and DES to simulate several unsteady flows. Based on the comparison with

experimental data, they made the assessment of different models, and suggested that the

MTS hybrid RANS/LES model needed more investigations in term of grid and time-step

sensitivities.

Roy et al. (2003) examined Detached Eddy Simulation (DES) and RANS

turbulence modeling in incompressible flow over a square cylinder. They found that the

2-D and 3-D simulations using DES are almost identical, and the results also compared

well with experimental data, while the steady-state RANS significantly over-predicted

the recirculating vortex behind the cylinder.

Table 2-1. Summary of the hybrid RANS/LES studies
Author and year Methods Conclusions
Bosch and Rodi Ensemble-averaged N-S with ad-hoc Capture better unsteady
1996, 1998 modification in turbulence closure shedding than RANS and
agreed well with
experimental data









Table 2-1. Continued
Rodi 1997 RANS with LES as turbulence closure Generally agreed with
experiment measurements
and found 2-D could not
resolve the 3-D effect
Koutmos and Ensemble-averaged N-S with a spatial Resolved good unsteady
Mavridis 1997 filtering form LES features in square cylinder
and backward facing step
under low- and high-
Reynolds numbers
Nagano et al. A low-Reynolds number MTS Agreed with the DNS and
1997 turbulence model in homogeneous shear experiment data
flow
Nichols and RANS, Hybrid RANS/LES, DES Hybrid models is better than
Nelson 2003 RANS with comparable
accuracy with LES
Roy et al. 2003 N-S equation with a DES turbulence Compared well with
closure experimental data

2.3.2 Filter-Based Model Concept (FBM)

In the RANS models developed for steady state flows, the turbulent length scales

predicted by the model extend over a large part of the flow domain. By imposing a filter

on the flow, the turbulent scales smaller than the filter will not be resolved. When the

filter size is smaller than the length scales returned by the RANS models, this will allow

the development of flow structures that are not dissipated by the modeled effective

viscosity. The sub-filter flow may be characterized by transport equations for turbulent

energy, dissipation, and Reynolds stresses. In the present example, we choose to apply

the LSM (Launder and Spalding 1974) as the corresponding RANS model. The filtering

operation will be controlled by the size of the imposed filter A and the size of the RANS

length scale lINs. More detailed can be obtained from Johansen et al. (2004). The brief

concept is given below. We start from the RANS length scale


1/los =c3 ki (2.17)


We may now construct filtered eddy viscosity in the following general form










V C412 C 2 Akc R4NS A-.
vt =C =C -NSf(C3 3) =3 V (C ) (2.18)


where lf is the turbulent length scale survived during the filtering operation. The scaling

function f must impose the filter, and have limiting properties such as


f(C3 ) = 1- exp(-C3 (2.19)
k3/2 k3/2

Here k and E are the non-resolved turbulent energy and dissipation rate separately,

corresponding to unresolved turbulent length scales ,ff


1, = 1- exp(-C3 k ) (2.20)


The proposed model becomes identical to the RANS model in the extremely coarse

filter. In the case of a fine filter, the turbulent length scale is controlled only by the

imposed filter, and the LES type of model is obtained under this condition. The model is

expected to have the following properties

If filter size A is identical to cell size and grid Reynolds numbers are
sufficiently small (Kolmogorov scales) the DNS model is recovered.

If filter size becomes large, the RANS model is recovered.

The statistical understanding of the advancement of the averaged flow during
one time-step implies that the time step itself should be a part of the model.

The filter should be almost independent to the grid in a specific computation,
though it may vary with different geometries (different characteristic lengths).

2.3.3 Modeling Implementation

The k E two-equation is adopted in similar formula as Eq.(2.13)-(2.14). In

Eq.(2.15), the turbulent viscosity modeled with a filter by the scaling function of

Eq.(2.19) leads to the following









A-c k2
= 0.09.Min[1,c3 cA ]- (2.21)
k3/2 E

with C3 -1, here we choose C3 = 1.0. This choice helped to assure that in near wall

A8g
nodes the scaling function f = Min[l,c3 k- ] will always return f = 1.0. Then the use of


standard wall functions is fully justified as the standard k e model.

In the limit of a very large filter size A, the viscosity becomes v, = 0.09 k2 / e and

the standard k e model is recovered. In the limit of a filter much smaller than the

turbulent scale k3/2 I/, the viscosity model becomes

v, = 0.09 C3Ak1/2 (2.22)

Then the FBM is identical to the one-equation LES models of Schumann (1975) and

Yoshizawa (1993).

2.4 Assessing FBM in Time-Dependent Single-Phase Flow

For the filter-based model (FBM), we first carry out the studies on the single-phase

simulations of vortex shedding past a square cylinder, and compare with experimental

measurements (Lyn and Rodi 1994, Lyn et al. 1995).

The parameters for the computations are given in Table 2-2. The square cylinder,

with a height D, is located at the center of a water channel. Figure 2-1 shows the

geometry of the experimental set-up of Lyn and Rodi (1994) and Lyn et al. (1995), which

is selected to guide the investigation, and the non-dimensionalized coordinates. It consists

of a 2D square cylinder inside a channel. To reduce numerical errors in the vicinity of the

cylinder, we adopt uniform grid spacing within the 4Dx3D domain surrounding the

cylinder. Outside this block, the grid is slightly expanded toward the edges of the







19


computational domain. All variables are non-dimensionalized by the free stream velocity


and the cylinder height. The fluid properties are held unchanged and the Mach number is


zero. At the inlet, the mean velocity U,, is uniform and follows the horizontal direction.


The inlet turbulent intensity is 2%. Based on the definition of the turbulent viscosity


formula adopted in the LSM and by assigning the turbulent Reynolds number to be 200,


we determine the inlet dissipation rate. The flow variables are extrapolated at the outlet.


The wall function (Shyy et al. 1997) is employed for the solid boundary treatment.


Table 2-2. Parameters used in the computation
U.n (m/s) p (kg/m3) p (kg/m.s) Re D (m) k (m2/s2) 2 (mZ/s3)
0.535 1000 1.002x10-3 21357 0.04 1.7174x10 4 2.5x10 6


Solid boundary




D=0.04
020 0.56

6.5D
1.0


4 -
I

3-c


Outlet
Outlet

1 I

Ceterline
0- L
Soihudr 1 2 31 4 1 5 1 6 1


Solid boundary 0 1 2 3 4 5 6
x/D
A B
Figure 2-1. Geometry configuration of square cylinder. A) Computational domain. B)
Non-dimensional coordinates

To investigate the effect of grid resolution on numerical accuracy, we used three


levels of grids including fine, intermediate and coarse grids, which have 25, 20 and 10


nodes on each side of the cylinder respectively. For comparison, the Launder-Spalding


model (LSM) is also carried out on coarse and fine grids. To investigate the sensitivity,


we use four different filter sizes: 0.15D, 0.3D, 0.6D and 0.9D on coarse grid (10


intervals), and 0.15D, 0.25D, 0.6D and 0.9D on fine grid (25 intervals). Unless explicitly


mentioned, A= 0.15D is used as the reference filter size.


Inlet









If (A E) / k3/2 exceeds 1.0, the filter scaling function f Min[1.0, (A )/ k3/2] will

take a value of 1.0. This treatment enables one to apply the wall function of LSM for the

solid wall treatment, which is confirmed by the outcome presented in Figure 2-2.

A pressure reference point was located at position x/D=0.0 and y/D=0.50, which is

the midpoint of the cylinder upper wall. For the fine grid we show predicted pressure

development at the wall reference point in Figure 2-3A. We see a modulated and inexact

periodic signal that qualitatively agrees with the experiments (Lyn and Rodi, 1994). The

FBM produces pressure oscillations corresponding to amplitudes of pressure coefficient

of approximately 2.5, while pressure coefficient amplitude predicted by the LSM varies

from approximately 0.1 to less than 0.05. We also found that the LSM tends to be more

time independent on the finer grid and the oscillations die out eventually. The FFT of the

Cp by filter-based model (Figure 2-3A) shows multiple frequencies with a dominant one

at around 2.15 (Figure 2-3B).












A B
Figure 2-2. Streamline snap-shot on fine grid with A = 0.15D Red shadow indicates
f = Min(.0, 3 ) = 1.0 for recovering the LSM. At other regions the

filter function is employed. A) Zoom in. B) Full view











SLaunder-Spalding 1800
-05 Filter-based
1600
-1 1400
5 11200
C 1000
2 C 800

-2 5 600
400
200

5 10 15 20 1 2 3 4 5
t s AFrequency
A B
Figure 2-3. Pressure behaviors at the reference point on the fine grid. A) Pressure
coefficients Cp time evolutions, LSM and FBM. B) Cp FFT, FBM

Figure 2-4 shows the time-averaged horizontal velocities along the centerline


behind the cylinder. The LSM predicts too long wake lengths 4 behind the cylinder at


about 3.0D for both grids (coarse and fine grids), which are almost identical to the

numerical results by Rodi (1997). Further results of the FBM, using a constant filter size

of A = 0.15D are shown in the same figure. For the intermediate and fine grids, the FBM


results quantitatively agree with the experimental data of Lyn et al. (1995) including the

asymmetric behavior. Even for the coarse grid, the size of the separation zone is well

reproduced. However, the reverse velocity in the wake as well as the velocity defect in

the remaining part of the wake is slightly under-predicted. In the case of the coarse grid

the resolution of the shear layer at the cylinder wall is sub-critical and this affects both

the onset of each shedding cycle as well as the magnitude of the vorticity in the wake.

Flow structures of the solutions at a given time instant on coarse and fine grids, and with

LSM and FBM are highlighted in Figure 2-5. For the FBM, as the grid resolution is

refined, the vortex structure becomes more dispersed and less confined in the wake


region. In contrast, for the LSM, the impact of the turbulent viscosity is dominant, and









the fine grid solution exhibits less fluctuation in time. The streamlines at one time instant

for both models are presented in Figure 2-6. The LSM only catches two separated

pockets at the cylinder upper and lower shoulders, and two wakes behind the cylinder.

The shedding motion is almost gone. The FBM is able to capture the sharp separation in

the shear layer, which agrees well with the experimental observation from Lyn and Rodi

(1994) and Lyn et al. (1995).

From Table 2-3 we see that the predicted Strouhal numbers are about 20% higher

than the St=0.135 from experiment by Lyn and Rodi (1994). By comparing the present

solutions on different grids using the FBM with the same filter size, the variations of the

Strouhal number are less than 4%. Hence, the only significant result is the Strouhal

number being approximately 20% larger. This over-prediction may be due to the 2D

effect, compared with the 3D measurement in the experiment. The large-scale structures

of the flow have a three-dimensional nature and we do not expect to reproduce all

features of this flow correctly in two dimensions. This will be more pressing as the filter

size and grid is reduced and we depart more and more from the LSM. This seems to be

consistent with the results from Yu and Kareem (1997) who needed to use a larger

Smagorinsky coefficient to reproduce the Strouhal number in 2D compared to their full

3D simulations. Another interesting finding is that the LSM required much smaller time

steps, compared with the FBM in order to get stable and convergent solutions using the

PISO algorithm. From Table 2-3 it is seen that the time steps have to be reduced

substantially in order to employ the LSM. Hence, the time consumption with the FBM is

around half smaller in the present study.















08 / /

06 /
/1
'04

02 LSM, coarse
LSM, fine
S- A=0.15D, coarse
0 A ------- A =0.15D, intermediate

-0 2
5 10 15
x/D

Figure 2-4. Time-averaged U-velocity along the horizontal centerline behind the cylinder.
Experimental data are from Lyn and Rodi (1994) and Lyn et al. (1995)


Coarse grid


Fine grid


a


Coar


W-7


Figure 2-5. Snap-shots of velocity contour. Color raster plot of axial velocities (red is
largest, blue lowest). A) FBM: coarse grid, velocity range is -0.31-1.53. B)
FBM: fine grid, velocity range is -0.66-1.88. C) LSM: coarse grid, velocity
range is -0.09 to 1.25. D) LSM: fine grid, velocity range is -0.31 to 1.20


r


[00










__C~

~-"-x~,~ -------
__I~i--:-


Figure 2-6. Streamline snap-shots of two turbulence models on fine grid. A) LSM. B)
FBM

Table 2-3. Comparisons of different turbulence models. /ID is the relative position of
the reattachment, measured from the cylinder center (coarse: 10 intervals,
intermediate: 20 intervals, fine: 25 intervals on each cylinder face)

Model Grid (n, x ny) Filter Size Time Step I/D St
dt/(D/Uin)
0.15 D 0.0134 1.22 0.155
0.30 D 0.0134 1.40 0.151
Coarse (162x92).6D 0.0134 2.12 0.143
0.90 D 0.0134 2.73 0.137
FBM Intermediate (290 x190) 0.15 D 0.0134 1.25 0.163
0.15 D 0.0669 1.23 0.161
0.25D 0.0669 1.25 0.148
Fne(30095) 0.60D 0.0669 1.98 0.140
0.90D 0.0669 2.64 0.134
Coarse (162x92) 0.00268 3.03 0.124
LSM
Fine (300x195) 0.000803 2.80 0.125
Exp. Lyn et
al. (1995)38 01

The predicted velocity profiles are given in Figure 2-7 to Figure 2-10. One thing to

be noted is that the experiments of Lyn and Rodi (1994) and Lyn et al. (1995) were

recorded on a single side, assuming that the ensemble and time averaged flow was

symmetrical across the axial symmetry line. As a result, data were recorded only at one

side of the symmetry line. Hence, we have mirrored the data to enable the visual


I 'i'i \'i i


A1 :~fl









comparison with the current numerical results. By close inspection of the experimental

data, we find that the transversal time averaged velocity on the symmetry line is non-

vanishing. This indicates that the data are not completely symmetrical.

Along the vertical centerline (Figure 2-7), we see that the LSM is unable to

correctly reproduce the separation in the shear layer. The shoulders on the velocity

profiles are not captured, presumably due to the high effective viscosity in the incoming

flow. The FBM solution on the coarse grid is similar to the fine grid solution, but only as

a result of poor resolution of the shear layer. The profiles for the FBM match the

experimental data of Lyn and Rodi (1994). The time-averaged transversal velocity in the

wake, one cylinder diameter behind the center of the cylinder, is shown in Figure 2-8B,

where the LSM fails to capture the correct transversal velocity. For both intermediate and

fine grids, the FBM again gives very good results. The time-averaged axial velocity is

shown in Figure 2-8A. Here we see that the FBM solutions become more asymmetrical

by grid refinement. The asymmetry in the time-averaged solutions is clearly seen from

Figure 2-8 to Figure 2-9, and has been confirmed independently by calculations using a

code that employs a staggered grid arrangement. The development of asymmetrical time

averaged solutions seemed to be caused by the initial bias of the solution, which is path-

dependent. Asymmetries in the time-averaged fields were noted by Sohankar et al.

(1999). They reported that the asymmetries became more distinct as the Reynolds

number increased. This is consistent with our findings. As suggested by Sohankar et al.

(1999), the reason could be the two-dimensional geometry that is forced on the flow. In

two dimensions wall attachment may be reinforced due to the imposed two-

dimensionality. This can only be resolved by comparing full 3D computations.
























08 ---_ A=0.15D, coarse
-- A=0.15D, intermediate
3 06 -' A=0.15D, fine
SA Experiment
04

02 I

0-

-02

-04 I
-1 0 1
y/D


Figure 2-7. Time-averaged U-velocity along y at x/D=0.0. Experimental data are from

Lyn and Rodi (1994) and Lyn et al. (1995)


I
; A
8- /




'f i" 1
I
SI I

LSM, coarse
/ / LSM, fine
/ A=0.15D, coarse
) / -.-.-- A=0.15D, intermediate
S A=0.15D,fine
\ AA A Experiment
-2 -1 0 1 2
y/D


LSM, coarse
A LSM, fine
A/ \ A=0.15D, coarse
A \ _._._. A=0.15D, intermediate
S - A=0.15D, fine
S\\ A Experiment










A
-'1^
\ \A



\ A


2 -1 0
y/D


1 2


Figure 2-8. Time-averaged velocities along y at x/D=1.0. Experimental data are from Lyn

and Rodi (1994) and Lyn et al. (1995). A) U-velocity. B) V-velocity


- /


A\

\ I'



'I




\


A

AA-

I


A-


SLSM, coarse
LSM, fine
SA=.15D, coarse
\ A=0.15D, interme
\ A=0.15D,fine
Experiment
, I I I I I


-4 -3 -2 -1 0 1 2 3 4
y/D


Figure 2-9. Time-averaged u-velocity along y direction. Experimental data are from Lyn

and Rodi (1994) and Lyn et al. (1995). A) At x/D=3.0. B) At x/D=8.0









The predicted kinetic energies along the horizontal centerline are compared with

experiments in Figure 2-10. We find that the LSM significantly under-predicts the total

kinetic energy, including the periodic and turbulent parts, mainly due to poor resolution

of the large scale structures in the wake. The FBM results agree well with the experiment,

even in the coarse grid, in terms of the magnitude and the peak position (Figure 2-10A).

The turbulent kinetic energy (the stochastic part only) is under-predicted for both LSM

and FBM, as shown in Figure 2-10B. Compared with the total and turbulent kinetic

energy, the turbulent part dominates for the LSM, while the periodic part is dominant for

the FBM. Since the FBM over-predicts the turbulent kinetic energy, it slightly over-

predicts the periodic kinetic energy at the peak.

Next, we investigate the effect of the filter size. As mentioned above, we present

the results on the coarse grid with four filter sizes 0.15D, 0.3D, 0.6D and 0.9D, and fine

grid with four filter size 0.15D, 0.25D, 0.6D and 0.9D. Figure 2-11 shows the transversal

v-velocity of both coarse and fine grids with different filter sizes. The profiles exhibit a

clear trend toward the LSM when the filter size increases. Table 2-3 further demonstrates

that the computed Strouhal number and the reattachment length 4/ also move toward the

LSM as the filter size increase on the coarse grid. Also, the kinetic energy shows a

similar trend with the various filter sizes on both coarse and fine grid (Figure 2-12).

Overall, the solution with A= 0.15D agrees better with the experiment.

By inspection of the results it appears that the present filter-based calculations give

quite regular solutions. In many calculations of the shedding from a square cylinder,

perturbations of the flow are induced by "numerical noise" that may be caused by a large

number of different phenomena. Examples of such phenomena are unbounded convective











fluxes, reduced numerical order caused by expanded or unstructured grids, and too large


time step sizes. In the filter-based model randomness can be added to the solution by


applying a random force field, similar to what is used in Renormalization-Group analyses


(Smith and Woodruff, 1998). Without inducing randomness to the flow by inlet


conditions or by random forcing we expect that the present model will produce regular


oscillating solutions similar to URANS (unsteady RANS) calculations by laccarino et al.


(2003).


The viscosity contours of LSM and FBM are shown in Figure 2-13. The LSM


predicts a very high viscosity in the upstream of the cylinder, and the distribution is


almost symmetric. The FBM significantly reduces the viscosity around 2-orders of


magnitude. As discussed above, the high effective viscosity in the coming flow can damp


out the unsteadiness behind the cylinder.




LSM, coarse A LSM,coarse
/ LSM, fine A LSM, fine
6 - A=0.15D, coarse A=0.15D, coarse
/ ------- A=0.15D, intermediate A------- A=0.15D, intermediate
1// -. A=0.15D, fine 025 A=0.15D, fine
i A Experiment A A Experiment
4 I 02 A
04

F A / \ -A-n M r
,0'

A A A A A A
02 / A A 01 A A
02 \ A A

"0 05

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7
x/D x/D
A B
Figure 2-10. Mean kinetic energy on different grids. Experimental data are from Lyn and
Rodi (1994) and Lyn et al. (1995). A) Total kinetic energy (mean + turbulent).
B) Turbulent kinetic energy (stochastic)















LSM, fine
A A=0.90D, fine
-V\ -- A=0.60D, fine
IA \ ---A=0.25D, fine
\/ ---A=0.15D, fine
/ \ Experiment

V N






\ M


\\< ,!
I\ l


-2 -1 0
y/D


1 2


A B

Figure 2-11. Comparisons of different filter sizes on time-averaged v-velocity along y at

x/D=1.0. Experimental data are from Lyn and Rodi (1994) and Lyn et al.

(1995). A) Coarse grid. B) Fine grid


LSM, coarse
A=0.90SOD, coarse
A=0.60D, coarse
.----A=0.30D, coarse
A:0?0 .....coa I


-. ....... .. A=0.15D, coar
A /" A Experiment


\..


A/ ... ------ ...- -
/
A / .........


V / -
V/


1 2 3 4 5 6 7
x/D


-

A \

A
I '.


- /
//





I I I I
1 2 3 4
x/D


LSM, fine
A=0.90D, fine
A=0.60D, fine
--- A=0.25D, fine
...- A=0.15D, fine
A Experiment







-A--A-
~'Lr~


03


025


01


LSM, coarse
A=0.SO9D, coarse
.....-- A=0.60D, coarse
---- A=0.30D, coarse
----- A=0.15D, coarse
A Experiment


A


A A


- -


1 2 3 4
x/D


03


025


02


.015


01


5 6 7


5 6 7





- LSM, fine
- A=0.90D, fine
---.--- A=0.60D, fine
---- A=0.25D, fine
-.....- A=0.15D, fine
A Experiment


A A


A A A


A A
A


1 2 3 4
x/D


5 6 7


Figure 2-12. Comparisons of different filter sizes on kinetic energy. Experimental data

are from Lyn and Rodi (1994) and Lyn et al. (1995). A) Coarse grid: total

kinetic energy (periodic + turbulent). B) Coarse grid: turbulent kinetic energy

(stochastic). C) Fine grid: total kinetic energy (periodic + turbulent). D) Fine

grid: turbulent kinetic energy (stochastic)


7"


06






04





021
02


nI


U


)5 -- .


ie


I I


- -
- -
--
- -









S1.81 .g09


0.182 mo.oos








A B

Figure 2-13. Time-averaged viscosity contours of different turbulence models on fine
grid. A) LSM. B) FBM

In summary, a filtered-Navier-Stokes model, originated from the LSM, is applied

to vortex shedding from a square cylinder. The introduction of the filter led to an

effective viscosity that depends on both turbulent quantities and the filter size itself. The

method is capable of working with standard wall-functions, allowing much coarser grids

in the boundary layer compared to common LES methods. Presently the use of wall

functions is justified as y' values for near wall nodes are greater than 20 using the fine

grid (25-node). Based on the discussions above, we have the following conclusions

Both coarse and fine grids reproduce the time averaged experimental results
quantitatively. However, by refining the grid we see improved results for the
velocity profiles. For the investigated filter size of A = 0.15D, the solutions on
both intermediate and fine grids are in agreement with experimental data,
demonstrating that the model can produce better resolutions based on the LSM
by allowing the numerical scheme to simulate the fluid physics at the scales
where numerical resolutions are satisfactory.

The increase of filter size shows that the filter-based model smoothly
approaches the LSM.

The filter-based model is shown to produce improvement over the LSM for all
grids investigated.

The Strouhal number of FBM is generally over-predicted. Further investigation
is needed to investigate whether it can be resolved by full 3D simulations.






31


Generally, the model is expected to yield better results if full 3D solutions are
performed, since the large scale 3D coherent flow structures can be resolved.














CHAPTER 3
CAVITATION MODELS

In this chapter, the single set of the governing equations for the flow field,

including the continuity and momentum equations which were introduced in Chapter2

along with a transport equation for the cavitation, is presented first. Then, the cavitation

models utilized for the present study are provided, along with a review of selected recent

studies. The above equations coupled with turbulence models, including Launder-

Spalding model (LSM) and Filter-based model (FBM), presented in the previous chapter

will complete the whole system of equations for the turbulent cavitating flow

computations.

3.1 Governing Equations

The Favre-averaged Navier-Stokes equations, in their conservative form, are

employed for incompressible flows. The cavitation is governed by a volume fraction

transport equation. The equations are presented below in the Cartesian coordinates

OP. a(Pu, )
+ = 0 (3.1)
at 9x

a(Pmu,) C(Pmu,Uj) op a au, ?uJ 2 al/
+ + [(P+,u)( + 5,)] (3.2)
at dCx 9x 9x O)x cx 3 (x


aa+ (auj) = (ih +i' ) (3.3)
at ox

where p, is the mixture density, u, is the velocity component in Cartesian coordinates, t is

the time, p is the pressure, a, is volume fraction of liquid, ih+ is the condensation rate









and li is the evaporation rate, /t is the turbulent viscosity, and / is the laminar viscosity

of the mixture defined as

/A=/ ,a, +/,(1-a,) (3.4)

with /u is the laminar viscosity of the liquid and /u is the viscosity of the vapor.

The mixture density and turbulent viscosity are respectively defined as below

p, = plo,,+ (l- a,) (3.5)

PmCjik2
=t -= (3.6)


where p, is the density of the liquid and p, is the density of the vapor, C, is the

turbulence model constant and its value is 0.09 generally, k is the turbulence kinetic

energy and E is the turbulence dissipation rate. The relevant details of the different

turbulence closures have been reviewed in detail in Section 2.2-2.4.

3.2 Literature Review of Cavitation Studies

Cavitation can produce negative effects in fluid machinery components and

systems. Details of the existence, extent and effects of cavitation can help to minimize

cavitation effects and optimize the designs. Experiments have been conducted in the past

few decades for different types of fluid machinery devices and components. Ruggeri and

Moor (1969) investigated methods that predicted the performance of pumps under

cavitating condition. Typically, the strategy to predict the Net Positive Suction Head

(NPSH) was developed. Stutz and Reboud (1997, 2000) studied the two-phase flow

structure of unsteady sheet cavitation in a convergent-divergent nozzle. Wang (1999)

used high-speed camera and Laser Light Sheet (LLS) to observe the cavitation in a

hollow-jet valve under different openings. Wang et al (2001) studied broad cavitation









regimes of turbulent cavitating flows, covering from inception to supercavitation, over a

Clark-Y hydrofoil under two different angle-of-attacks.

Besides experimentation, using CFD simulation to analyze cavitation phenomena

has become convenient and popular with the development of computer hardware and

software. A number of cavitation models have been developed. These studies can be put

into two categories, namely interface tracking methods and homogeneous equilibrium

flow models. A review of the representative studies is presented by Wang et al. (2001).

Taking the advantage of homogeneous equilibrium flow theory, the mixture concept is

introduced. And a unique set of mass and momentum equations along with turbulence

and cavitation models is solved in the whole flow field. Within the homogeneous

equilibrium flow theory, two approaches can model the cavitation dynamics. The first

one is the arbitrary barotropic equation model, which suggests that the relationship

between density and pressure is p = f(p), and the second one is the transport equation-

based model (TEM). Barotopic equations were proposed by Delannoy and Kueny (1990).

They assumed that density was a continuous function of pressure where both pure phases

were incompressible, and the phase change could be fitted by a sine curve. They could

not correctly produce the unsteady behaviors in the venturi simulation. Arbitrary

barotropic equation models (density is only a function of pressure) do not have the

potential to capture baroclinic vorticity production because the baroclinic term of the

vorticity transport equation yields zero by definition (Senocak 2002, Senocak and Shyy

2004a). In addition to agreement with the experimental study of Gopalan and Katz

(2000), the above two references have demonstrated computationally that the baroclinic

vorticity generation is important in the closure region. In TEM, a transport equation for









either mass or volume fraction, with appropriate source terms to regulate the mass

transfer between vapor and liquid phases, is adopted. The apparent advantage of this

model comes from the convective character of the equation, which allows modeling of

the impact of inertial forces on cavities like elongation, detachment and drift of cavity

bubbles, especially in complex 3-D interface situations (Wang et al. 2001). Different

modeling concepts embodying qualitatively similar source terms with alternate numerical

techniques have been proposed by various researchers (Singhal et al. 1997, Merkle et al.

1998, Kunz et al. 2000, Ahuja et al. 2001, Senocak and Shyy 2003, 2004a,b).

Numerically, Singhal et al. (1997, 2002) and Senocak and Shyy (2002a,b, 2003, 2004a,b)

utilized pressure-based algorithms, while Merkle et al. (1998) and Kunz et al. (2000)

employed the artificial compressibility method. In addition, Vaidyanathan et al. (2003)

performed a sensitivity analysis on a transport equation-based cavitation model to

optimize the coefficients of its source terms. More recently, attempts have been carried

out on numerical simulation of cavitating flows in turbomachines, such as pumps and

inducers. Medvitz et al. (2002) utilized the pre-conditioned two-phase N-S equations to

analyze the performance of cavitating flow in a centrifugal pump. Hosangadi et al. (2004)

simulated the cavitating flow in the full geometry of a 3-blade Simplex inducer. Couties-

Delgosha et al. (2005) presented a 3D model for cavitating flow through a 4-blade

inducer adopting only one blade-to-blade passage. However, a complete robust and

accurate CFD framework in this field is still a longstanding work.

3.2.1 Cavitation Compressibility Studies

Since the flow fields are rich in complexity at the cavity interface region, as pointed

out in chapter 1, compressibility effect is one of the major issues in cavitation studies.









Nishiyama (1977) developed a linearized subsonic theory for supercavitating

hydrofoils to clarify the compressibility correction of Mach number effect in pure water.

The 2-D & 3-D steady characteristics of super-cavitating hydrofoils in subsonic flow

were compared to those in incompressible flow. The essential differences, including the

compressibility effect between vapor and liquid and the co-relation between cavitation

and Mach number, were shown in detail.

Saurel and Cocchi (1999) focused on cavitation in the wake of a high-velocity

underwater projectile. They presented a physical model based on the Euler equations in

terms of two-phase mixture properties. The mathematical closure was achieved by

providing state equations for the possible thermodynamic states: compressible liquid,

two-phase mixture and pure vapor. The model was then solved using a hybrid

computational scheme to accurately maintain the property profiles across the

discontinuities. The results demonstrated a reasonable agreement compared with the

known analytical solutions.

Saurel and Lemetayer (2001) proposed a compressible multiphase unconditionally

hyperbolic model to deal with a wide range of application: interfaces between

compressible materials, shock wave in condensed multiphase mixtures, homogeneous

two-phase flows and cavitation in liquids. The model did not require a mixture equation

of state and was able to provide themodynamic variables for each phase. The results and

validations with analytical solutions were provided.

Kunz et al. (2000), followed by Venkateswaran et al. (2002), developed a

preconditioned time-marching algorithm for the computation of multiphase mixture flows

based on carrying out perturbation expansions of the underlying time-dependent system.









The method was efficient and accurate in both incompressible and compressible flow

regimes. However, it was not a fully compressible formulation for the flow fields. It

could not be accurate if the bulk flows were supersonic, which indicated that

compressibility should be considered even in pure liquid phase.

Senocak and Shyy (2004a,b) used the pressure-based method to simulate the

cavitating flows through convergent-divergent nozzles. They presented two different

treatments of speed-of-sound (SoS), including SoS-1 and SoS-2, to build up the

relationship between the pressure and density for steady and unsteady cavitating flows.

The comparison with the experimental data and other simulation results displayed that

SoS-2 was more likely better in predicting the unsteady characteristics.

Table 3-1. Overview of the compressible cavitation studies
Author and year Methods (Analytic/Numerical) Conclusions(Analytic/Numerical)
Nishiyama Proposed linearized theory linearized relationships between
1977 velocity, pressure and sound speed
Saurel and A physical model based on the Reasonable agreed with analytic
Cocchi 1999 Euler equations, with state solutions and could reliably deal
equations with strong shock wave and
complex EOS
Saurel and An unconditional hyperbolic model, Compared with analytic solutions
Lemetayer 2001 with accurate treatment for non- And clouded it could deal with a
conservative form to ensure mass wide range of application, such as
conservation shock wave, homogeneous two-
phase flow and cavitation.
Kunz et al. A preconditioned time-marching Agreed with experiment data
2000, CFD method for isothermal under good precondition, and
Venkateswaran multiphase mixture flows, suggested the compressible
et al. 2002 associated artificial compressibility treatment could improve the
dynamics description than
incompressible computations
Senocak and Two numerical models for of Agreed well with experimental
Shyy 2004a,b speed-of-sound in steady and data and other numerical
unsteady cavitating flows simulations, and demonstrated the
capability in unsteady
computation.









3.2.2 Cavitation Studies on Fluid Machinery Components and Systems

Ruggeri and Moor (1969) investigated similarity methods to predict the

performance of pumps under cavitating condition for different temperatures, fluids and

operating conditions. Typically, the strategy to predict the NPSH was developed based on

two sets of available test data for each pump at the concerned range of operating

conditions. Pumps performance under various flow conditions such as discharge

coefficient, impellor frequency as assessed for different fluids, such as water, LH2 and

butane.

Wang (1999) studied the cavitating phenomena in a hollow-jet valve under

different openings and cavitation conditions, using high speed camera and LLS. Further

more, he also studied the induced vibration mechanism, cavitation damage characteristics

and ventilation effect in the valve.

Athavale and Singhal (2001) presented a homogeneous two-phase approach with a

transport equation for vapor, and the reduced Rayleigh-Plesset equations for bubble

dynamics based on local pressure and velocity. Compared with the experimental

measurements, they obtained reasonable predictions of cavitating flows in two typical

rocket turbo-pump elements: inducer and centrifugal impeller.

Lee et al. (2001a,b) analyzed the cavitation of the pump inducer sequentially. They

found that the forward-swept blade demonstrated more resistance to vortex cavitation

than the conventional one. They also observed that the cavity length of surface cavitation

at various conditions was closely related with the cavitating number and event duration,

but inconsistent with the mode predicted by linear cascade analysis in the cross-flow

plane at far off-design points.









Medvitz et al. (2002) used the homogeneous two-phase RANS equations to analyze

the performance of 7-blade centrifugal pumps under cavitating conditions. By using the

quasi-3D analysis, the numerical results were found to be qualitatively comparable with

the experimental measurements across a wide range of flow coefficients and cavitation

numbers, including off-design flow, blade cavitation and breakdown.

Duttweiler and Brennen (2002) experimentally investigated a previously

unrecognized instability on a cavitating propeller in a water tunnel. The cavitation on

blades and in the tip vortices was explored through visual observation. The cyclic

behavior of the attached blade cavities had strong similarities to that of partial cavity

oscillation on single hydrofoils. Furthermore, the reduced frequency of the instability was

consistent with the partial cavity instability on a single hydrofoil.

Friedrichs and Kosyna (2002) described an experimental investigation of two

similar centrifugal pump impellers at low specific speed. The high-speed-film displayed

rotating cavitation over a wide range of part loaded operating points, and illustrated the

development of this instability mechanism, which was mainly driven by an iteration of

the cavity closure region and the subsequent blades.

Hosangadi et al. (2004) numerically simulated the performance characteristics of an

inducer using an N-S methodology coupled with a two-equation turbulence model. The

simulations were performed at a fixed flow rate with different Net suction Specific

Speeds (NSS). The numerical results showed the head loss was related to the extent of

cavitation blockage. The breakdown NSS and the head loss prediction agreed well with

experimental data. The insights provided a sequence of traveling and alternate cavitation

phenomena in blade passages.










Coutier-Delgosha et al. (2005) presented a 3D model for a cavitating flow in 2-D

venture and in a 4-blade inducer with the comparison to experimental data and visuals.

They assumed symmetry in the inducer and only considered one blade-to-blade channel

in the inducer. The quasi-static results showed a consistent agreement with the

experiment, but did not catch the performance breakdown.

Table 3-2. Overview of cavitation on pumps, propellers, inducers and turbine blades
Author and year Problem study (Experimental) Findings(Experimental)
Methods (Numerical) Conclusions(Numerical)
Ruggeri and Similarity to predict the pumps Provide strategies to predict the
Moore 1969 performance under cavitation pump NPSH
conditions
Wang 1999 High-speed camera and LLS in a Cavitation, induced vibration,
hollow-jet valve damage characteristics and the
ventilated cavitation effect
Athavale and A homogeneous two-phase Plausibly agreed the experiment,
Singhal 2001 approach, with Rayleigh-Plesset with robustness and stability in
equations. numerical convergence behaviors
Lee et al. 2001a,b Tested the inducer performance, Forward-swept blade was more
inlet pressure signals and event resistive to vortex cavitation than
characteristics the conventional one
Medvitz et al. Homogeneous two-phase RANS Obtained qualitative performance
2002 model trends with experimental data in
off-design flow, blade cavitation
and breakdown
Duttweiler and Surge instability on a cavitation Cyclic oscillation of the attached
Brennen 2002 propeller in water tunnel blade cavities
Friedrichs and High-speed camera in a Rotating cavitation was mainly
Kosyna 2002 centrifugal pump impeller driven by an interaction of cavity
cavitation and the leading edge of the
following blades.
Hosangadi et al. N-S equations with an Well agreed with the experimental
2004 compressible multiphase model data in terms of the head loss and
breakdown NSS, and pointed out
the performance loss was strongly
correlated with cavitation
blockage.
Coutier-Delgosha 3-d code FINETURBOTM, with a Obtained good agreement in 2-D
et al. 2005 time-marching algorithm and low- venturi computation, And
speed preconditioner for low qualitatively agreed with the
Mach number flows experimental measurements of
efficiency and cavity visuals in 3-D
inducer.









3.3 Transport Equation-based Cavitation Model (TEM)

Cavitation process is governed by thermodynamics and kinetics of the phase

change dynamics in the system. These issues are modeled with the aid of a transport

equation with source terms regulating the evaporation and condensation of the phases. In

the present study, two different cavitation models, with similar source terms for Eq.(3.3),

are presented:

Heuristic model (HM). The liquid volume fraction is chosen as the dependent

variable in the transport equation (Kunz et al. 2000). The evaporation term mi is a

function of pressure whereas the condensation term m' is a function of the volume

fraction

-Cdes va, Min(0, p p)
p( (pU2m /2)t,

=h. Cproda O (1 _a) (3.7)
pit

where Cd,, =9.0 x 10 and CPd = 3.0 x104 are empirical constant values, to is the ratio of

the characteristic length scale to the reference velocity scale (L /U).

Interfacial dynamics-based cavitation model (IDM). The interfacial dynamics-

based cavitation model (IDM) was developed by Senocak and Shyy (2004a,b). A

hypothetical interface is assumed to lie in the liquid-vapor mixture region (Figure 3-1).

By applying the mass and normal momentum conservation equations at the interface, and

normalizing with a characteristic time scale t, = L /U, the evaporation mi and


condensation tih are given by









p laMin(0, p- )
P (Vv, Vn)2(p1 )t (3.8)
S (1- a)Max(O, p )
(V"" VT"n) 2 (P1 P )t

with vapor phase normal velocity, the interface normal, and the interfacial velocity


V=u-n, n= -
Va'1

1-f pA (3.9)
,n v,n, f =-0.9
1 PL


wherefis found by computational satisfaction and a value of -0.9 is used. For steady

state, we have V,,, = 0.0. This approach will be called original IDM.









Vn V





Figure 3-1. Sketch of a cavity in homogeneous flow

Regarding the above original IDM, the cavity interfacial velocity was linked to the

local fluid velocity in the time-dependent computations. Such an approach lacks

generality because the interfacial velocity is supposed to be a function of the phase

change process. Fundamentally, the interfacial velocity can possibly be estimated more

accurately based on the moving boundary computational techniques (Shyy et al. 2004).










Here, we estimate the interfacial velocity via an approximate procedure by accounting for

the phase transformation process in each computational cell. By integrating Eq. (3.8)

through the control volume, we have ihi AV and ir AV. The net interface velocity (the

interface velocity relative to the local flow field) becomes

Vf"A = AM
In
n A abs(i AV)-abs(i AV) (3.10)
'" A A

where A is the interface area between vapor and liquid phases.

Practically, the control volume face area, Areacy, is projected to the interface

normal direction, which can be obtained by taking the gradient of the volume fraction,

denoted as S, as shown in Figure 3-2 for the 2-D situation

Va -
= VctL = fly]
n=- --=n+nJ

= Area (3.11)






Liquid
l,n

fLt Interface

Areav
Vapor


Figure 3-2. Interface vector sketch in a CV

Substituting S into Eq. (3.10) leads to


vne A abs(i AV) -abs(AV) (3.12)
V (3S.2)
I'" S S









The interface velocity includes two parts: flow field local velocity V"LO1 C V= V and

net velocity vnt. Then we can have the following derivation
( -V 2= [ -(et + VLoco ,, (V,) (3.13)


Finally, the source terms assume the format, which we call modified IDM

SPLLMin(0,p pv)
Pv(vt )2(PL Pv)t.
(3.14)
(1- aL)Max(0, p p )
(Vne )2 (PL PV)tm


Recalled that for steady-state condition, we have the relationships of

I, = In + VLco" = 0 and v" = -_VL, = -,,. In this case, the modified approach is

identical to the original IDM.














CHAPTER 4
NUMERICAL METHODS

The governing equations, presented in the previous chapters, are discretized using a

finite volume approach in the present study. In this approach, the flow domain is divided

into control volume cells and the governing equations are integrated over each control

volume. The main advantage of the finite volume method is that the conservation laws

are satisfied locally for each control volume. A non-staggered grid system is defined at

the center of the cell. The decoupling of velocity-pressure can be handled by the

momentum interpolation method, originally proposed by Rhie and Chow (1983). While,

the original momentum interpolation had some problems, such as under-relaxation factor-

dependent, time step size-dependent, and even checkerboard pressure field, which can be

solved using linear interpolation of the two neighbor nodes in the cell-face velocity

evaluation. A detailed review can be obtained in Yu et al. (2002).

In cavitating flow computation, the conventional computational algorithm of

single-phase incompressible flow meets severe convergence and stability problems. The

situation is improved by using either density-based method or pressure-based method.

Both have been successful to compute turbulent cavitating flows in different

configurations with comparable accuracy (Wang et al. 2001, Senocak and Shyy 2004a).

However, generally the density-based method needs pre-conditioning or artificial density

for flows which are largely incompressible (Merkle et al. 1998, Kunz et al. 2000, Ahuja

et al. 2001). Hence we choose the pressure-based method (Shyy 1994, Senocak and Shyy

2002a). To take the advantage of non-iteration in the time-dependent computations, we









use the Pressure-Implicit Splitting Operator (PISO) algorithm other than the Semi-

Implicit Method for Pressure-Linked (SIMPLE) method (Patankar 1980). The original

PISO method was introduced by Issa (1985) and was modified later to be suitable for

large density variation (Thakur and Wright 2002, Thakur et al. 2004). Senocak and Shyy

(2002a) further extend it to cavitating flows by addressing the large density jump

between phases.

4.1 Pressure-Based Algorithm

The pressure-based algorithm for steady-state computation follows the SIMPLE

algorithm (Patankar 1980). The momentum equations can be discretized as

AIp = Z Aii -Vp(VdP), +b (4.1)

where A", A' are the coefficients of the cell center and neighboring nodes from

convection and diffusion terms, and Vp, b" are the cell volume and source term

separately. Vd is the discrete form of the gradient operator. When there is no source

term, the above equation turns into

p = H(ib)-Dp(VdP)p (4.2)

with

VP [ /A; 0
Dp= 0 Vp/A' 0

P P

The pressure-correction equation in the pressure-based method has been revised to

achieve successful solutions for highly compressible flows (Shyy 1994, Senocak and

Shyy 2002a). Generally, the mixture density at the phases interface region has high









variations. Here we will illustrate some key computational issues by focusing on the flux

terms in the continuity equation, which can be decomposed as

pu = (p + p')(i* + i')= p*u* + p*u' + p'u' + p' (4.3)

where the starred variables represent the predicted values and primed variables represent

the correction terms. And it leads to

pp = CP' (4.4)

-Vd .(pDVdP')+Vd .(Cp*P')= -Vd .(p**)+Vd .(CpPDVdP') (4.5)

The relative importance of the first and second terms in Eq. (4.3) depends on the

local Mach number (Shyy 1994). For low Mach number flows, only the first term

prevails; for high Mach number flows, the second term dominates. The fourth term is a

nonlinear second-order term and can be either neglected or included in the source term

for stability in early iterations. In the present algorithm, the following relation between

density and pressure correction is taken

pp =C(1-a )pP (4.6)

where C is an arbitrary constant and it does not affect the final converged solution. The

further details of the model can be obtained from Senocak and Shyy (2002a, 2004a) and

Senocak (2002).

4.2 Pressure Implicit Splitting of Operators (PISO) Algorithm for Unsteady
Computations

In the SIMPLE-type of the pressure-based methods (Patankar 1980, Shyy 1994),

the equations are solved successively by employing iterations. In cavitating flow

computations, the typical relaxation factors used in the iterative solution process are

smaller than the ones used in single-phase flows, and hence smaller time steps are needed









to study the cavitation dynamics. Issa (1985) developed the PISO method for the solution

of unsteady flows. The splitting of pressure and velocity makes the solution procedure

sequential in time domain and enables the accuracy at each time step. It also eliminates

the need for severe under-relaxation as in SIMPLE type algorithm. Bressloff (2001)

extended the PISO method for high-speed flows by adopting the pressure-density

coupling procedure in all-speed SIMPLE type of methods. Oliveira and Issa (2001)

followed the previous PISO work to combine the temperature equation to simulate

buoyancy-driven flows. Thakur and Wright (2002) and Thakur et al. (2004) developed

approaches using curvilinear coordinates with suitability to all speeds. Senocak and Shyy

(2002a) further extended this PISO algorithm to enhance the coupling of cavitation and

turbulence models and to handle the large density ratio associated with cavitation.

The PISO algorithm contains predictor and correction steps. In the predictor step

the discretized momentum equations are solved implicitly using the old time pressure to

obtain an intermediate velocity field. A backward Euler scheme is used for the

discretization of the time derivative term


iu = H[Ii], -D,(VdP"n1), + u)- (4.7)
6t

The intermediate velocity field does not satisfy continuity and needs to be corrected

using the continuity equation as a constraint. In the first corrector step, a new velocity

field, u** and pressure field P* are expected. The discretized momentum equation at this

step is written as


i* = H[ i], -DP(VdP*), + (4.8)
P9


Subtracting Eq. (4.7) from Eq. (4.8), we have









ui = -Dp(VdP'), (4.9)

If the density field depends on the pressure field, such as in high Mach number

flows or in cavitating flows, the density is corrected using the pressure-based method

p = p1 + P, P, = CP' (4.10)

The discretized continuity equation written for the new velocity field and density

field becomes

n 1
P* -P"
V, +A[p*u** -iS ] = 0 (4.11)
St

Combining Eq. (4.9), (4.10) and (4.11), a first pressure-correction step equation is obtain

below


p +A[p" 'D(VdP').-S] ],+A[CP'U ] = -A[p 'U ] (4.12)
9t

To satisfy the mass conservation, the second corrector step is conducted to seek a

new velocity field, '*** and pressure field P**


iu = H[**] -D(Vd**)p + P (4.13)


Subtracting Eq. (4.9) from Eq. (4.13) leads to the correction term

S= ii + H[i** -u ] -D (VdP")p (4.14)

The corrected density field leads to

p =pp + p = p + p' +p (4.15)

The second pressure-correction step can be derived from Eq.(4.14), Eq. (4.15) and

the mass continuity equation to reach the below format










+ p A[p D(VdP").iSCf] + A[CP"U** ] = -A[p*U ]
9t (4.16)
A[p*H(ti** i*) ). Sf ]p

Then, by solving the above predictor and correction steps coupled with the

cavitation model, which was formulated in Chapter 3, and turbulence closures given in

Chapter 2, the solution procedure for turbulent cavitating flow computations is

accomplished.

4.3 Speed-of-Sound (SoS) Numerical Modeling

As mentioned in chapter 1, the harmonic speed-of-sound in the two-phase mixture

is significantly attenuated. Therefore the multiphase flow fields are characterized by

widely different flow regimes, such as incompressible in pure liquid phase, low Mach

compressible in the pure vapor phase, and transonic or supersonic in the mixture.

Consequently, an accurate evaluation of speed-of-sound is necessary and important.

From Eq. (4.6) and the definition of speed-of-sound, the relation between C, and

the speed-of-sound is

ap 1
C, = () = (4.17)
OP c2

In high-speed flows, the exact form of the speed-of-sound can be computed easily

from the equation of state. However, in cavitating flows, computation of the speed-of-

sound is difficult. Each transport equation-based cavitation model defines a different

speed-of-sound as a result of a more complex functional relationship. In the literature,

there have been theoretical studies on defining the speed-of-sound in multiphase flows

(Wallis 1969). One-dimensional assumption and certain limitations are typical in these

studies. These definitions do not necessarily represent the actual speed-of-sound imposed









by the cavitation models of interest in this study. They can only be an approximation. On

the other hand, the fundamental definition of speed-of-sound as given in Eq. (4.17) could

be useful, and the path to compute the partial derivative is known. From these arguments,

it is clear that the computation of the speed-of-sound in cavitating flows is an open

question.

Due to the lack of a dependable equation of state for multiphase mixtures modeling

sound propagation, Senocak (2002) and Senocak and Shyy (2003) present two numerical

forms of speed-of-sound, called SoS-1 and SoS-2, and showed that the SoS-2 was more

likely to produce the correct unsteady behavior in unsteady simulations. In the present

study, these two different definitions for the speed-of-sound are further investigated in

the pressure-density coupling scheme. The SoS-1 is the previous pressure-density

coupling scheme with an order of 1 constant coefficient C as

ap 1
SoS-1: C, = (), = -C(1- a) (4.18)


The SoS-2 is based on an approximation made to the fundamental definition of


speed-of-sound. It is assumed that the path to compute the partial derivative is the
OP

mean flow direction (4) other than the isentropic direction (s), because the details of

thermodynamic properties are not known and the entropy can not be directly computed.

This definition is referred to as SoS-2 in the rest of the study and given below


SoS-2: C = ( = Ap ( 1 (4.19)
O QP A c2 AP \ -










The partial derivative is computed based on central differencing of the neighboring

nodes. The absolute value function is introduced to make sure a positive value is

computed.

Since the above approximation in SoS-2 is based on the mean flow direction, to

generalize it, an averaged form is adopted. It will be referred as SoS-2A, taking the

format as


C coffu- P+1,j,k P,1i,j,k + coff Pi,j+1,k P,j-1,k
C =coffu -- +coffrv -- -
P,+1,j,k P,-1,],k P,,j+1,k P,,j-1,k
SoS-2A: (4.20)
SPl ,j,k+l Pl,j,k-1
+coffw -- -
coff ,j,k+ P,j,k- 1

with the velocity weight coefficients


coffu = U-jk O ,coff = k
f ,k + jk + Wk U ,k + ,,,k + W,,,k

C &Wlj,k
coffw = -- ,k + wV,,k
U^Jk + V,,k + Wl,,,k














CHAPTER 5
ASSESSING TIME-DEPENDENT TURBULENT CAVITATION MODELS

5.1 Cavitating Flow through a Hollow-Jet Valve

There are serious implications on the safe and sound operation in flow-control

valve cavitation phenomena. A limited number of experimental studies have been

published on this topic, such as those by Oba et al. (1985) and Tani et al. (1991a,b). In

addition, Wang (1999) used high-speed cameras and Laser Light Sheet (LLS) to observe

the cavitation behavior in a hollow-jet valve under various cavitation conditions and for

different valve openings. However, to the best of our knowledge, to date, no

comprehensive numerical study has been done in this respect. Furthermore, complex

geometries and inaccessible regions of occurrence restrain experimental investigations in

cavitation. Hence, we investigate the capability of transport equation-based cavitation

models to predict incipient level cavitation.

As documented in Wang (1999), Figure 5-1 shows the geometry and the main

configurations of the valve. A key component is the needle, used to control the flow rate

by moving to different location in the x-direction. A cylindrical seal supports the needle.

There are six struts supporting the cylinder in the pipe center, which are called splitters.

The gear is used to control the needle position moving through the x-axis.

Figure 5-2 illustrates the computational domain in selected planes according to the

geometry. A multi-block structured curvilinear grid is adopted to facilitate the

computation. Figure 5-2A shows the configuration on the X-Y plane, and Figure 5-2B

from the Y-Z plan. Figure 5-2C shows the plane's location, and Figure 5-2D shows the









boundary conditions in the computations. In the present study, the splitter thickness is

neglected and its shape is considered to be rectangular. The Reynolds number is 5x 105

and the cavitation number is 0.9, with the density ratio between the liquid phase and

vapor phase p, l/P being 1000 in the water, and the valve opening is 33%.

Two investigations are conducted here using the original interfacial dynamics-

based cavitation model (IDM) (Section 3.3), using the Launder-Spalding model (LSM)

(Section 2.2) as turbulence closure. We study steady and unsteady computations. In

steady-state, we adopt SoS-2. In time-dependent simulations, we further examine the two

different SoS impact, including SoS-1 and SoS-2 (Section 4.3), which have been firstly

investigated by Senocak and Shyy (2004b) in cavitating flows. For the time-dependent

computation, the steady single-phase turbulent flow, without considering cavitation, is

computed and then the solution is adopted as the initial condition for the unsteady

cavitating turbulent flow. The results and discussion are presented in the following.


Figure 5-1. Valve geometry. (1) Splitter. (2) Cylinder. (3) Plunger. (4) Needle. (5) Needle
seal overlay. (6) Seal seat inlay. (7) Stroke. (8) Ventilation duct. (9) Gear










bZ




Upper boundary

I :net Splitter outlet

Lower boundary
alve
Pipe center

A B
Y

Lz

No-slip
BSplitterary c
planes tead No-slip s lioutlet
Inlet / No-slip
No-slip
Middle Slis-
p la n e S lipD

C D
Figure 5-2. Computational domains and boundary conditions. A) Computational domain
in X-Y plane. B) Computational domain in Y-Z plane. C) Planes location. D)
Boundary conditions

5.1.1 Steady and Unsteady Turbulent Cavitating Flows

Figure 5-3 shows the density distributions for the steady situation. The cavity is

located at the valve tip. The comparison shows that the cavities on the splitter plane and

the middle plane are slightly different in size. Figure 5-4 shows how the cavity shape and

location vary with time.

For time-dependent computations, the cavity fluctuates quasi-periodically. At non-

dimensional time t* = t/t =0.4, the cavity is the biggest with smallest density, see Figure

5-4A. As time passes, the cavity size reduces to the smallest in Figure 5-4(B, C) at time

t*=0.6 and t*=0.8, respectively. After one cycle it reaches the maximum again (Figure 5-

4D at time t*=1.2s). Both steady and unsteady results exhibit the cavity at the needle tip










(Figure 5-3 and Figure 5-4). The steady case corresponds to a single instantaneous result

in the time-dependent solution (Figure 5-3B and Figure 5-4C). The time-dependent

results are qualitatively consistent with the experiment in size and shape (Figure 5-4).

Unfortunately, there is insufficient experimental information to ascertain the time-

dependent characteristics in detail.

The experiment observed that the cavities incept, grow, then detach from the needle

tip and transport to the downstream periodically, which is clearly shown in Figure 5-4E.

However, as already discussed in Senocak (2002), with the current combination of

turbulence and cavitation models, the detachment of the cavity is not captured, possibly

due to the representation of the turbulence via a scalar eddy viscosity. Henceforth, the

issue is under further investigation in following by means of adopting different

turbulence models.



5 0.99
4 0.98 5 0.99
3 0.96 4 0.98
2 0.9 3 0.96
1 0.8 2 0.9
1 0.8
5 5
4 4


Needle tip Needle tip





A B
Figure 5-3. Density contour lines of the steady state solution (The mixture of vapor and
liquid inside the outer line forms the cavity), original IDM with SoS-1, LSM.
A) At splitter plane. B) At middle plane








57




5 099 5 099
4 098 4 098
3 096 3 096
2 09 2 09
1 098 1 08

5 ~-
4 4




Needle Needle





A B


F5 099 5 099
4 098 4 098
3 096 3 096
2 09 2 09
1 08 1 08




3/ 3



Needle Needle





C D














Valve tip








Figure 5-4. Middle section density contours at different time instants (The mixture of
vapor and liquid inside the outer line forms the cavity), original IDM with
SoS-2, LSM. A) Time t*=0.4. B) Time t*=0.6. C) Time t*=0.8. D) Time
t*=1.2. E) Experimental observations from Wang (1999): cavity at needle tip
and cavitation aspects around the needle











To further demonstrate the cavity-induced quasi-periodic characteristics of the flow


field, Figure 5-5 highlights the time evolutions at selected locations. Figure 5-5A shows


the locations of the points selected. The pressure is at the middle points, and the density is


near the bottom boundary, just one point away from it. From Figure 5-5B, at the places


away from the cavity, A, B and D, the density is constant since only the liquid phase


exists there, and is quasi-periodic inside the cavity at Point C. On the other hand pressure


oscillates in the whole domain except on the inlet plane, where the flow condition is fixed


(Figure 5-5C).




C D


A B,



A





095 7 -
SLocation A 6 -
09 Location B 5 Location A
SLocation C 4 Location B
G Location D V Location C
085
C o 3 Location D
2 -
08 2
08

0 75

07 --2
0 1 2 3 4 0 1 2 3 4
SB t C

Figure 5-5. Time evolutions at different locations, original IDM with SoS-2, LSM. A)
Samples locations. B) Density. C) Pressure

Figure 5-6 presents the flow structure on the middle plane. For the steady case, the


flow fields of both single phase (without invoking the cavitation model in the course of


computation) (Figure 5-6A) and cavitating flows are almost identical (Figure 5-6B). It







59


indicates that in the present case, the detailed cavitation dynamics does not exhibit

substantial influence on the overall flow pattern. For the time-dependent case, at different

time instants, t*=0.4 and t*=0.8 (Figure 5-6C), the flow field around the needle remains

largely the same. The comparison between the steady and unsteady case indicates that

there is not very much difference in the flow pattern. As expected, there are two

recirculating zones: one behind the needle and the other one downstream around the

splitter region, which is located at about x=3.4-4.2. Compared with the experimental

observation in Figure 5-6D (the cavitating flow structures behind the needle, and in the

splitter region), the present results are in general agreement.





1 0 -




0- 0-


3 35 4 45 5 3 35 4 45 5 55









0 0 -


3 35 4 45 5 3 35 4 45 5
X X
C
Figure 5-6. Projected 2-D streamlines at middle plane and experimental observation. A)
Steady single-phase flow, LSM. B) cavitating flow, original IDM with SoS-2,
LSM (Right). B) Unsteady cavitating flow at time t*=0.4 (Left) and t*=0.8
(Right). D) Flow pattern from Wang (1999), behind valve about x=3.0-3.4
(Left) and at splitter region about x=3.4-4.2 (Right)









Flow -















Figure 5-6. Continued


5.1.2 Impact of Speed-of-sound Modeling

We further examine the behavior of the different handling of speed-of-sound

(Chapter 4), that is, SoS-1 and SoS-2.

Figure 5-7 shows the variation in cavity shapes and location for LSM at maximum

and minimum fraction instants. Compare the half period between maximum and

minimum instant, the cavity oscillation for SoS-1 is distinguished by a significantly

higher frequency and moderately higher amplitude than for SoS-2 (Figure 5-7A, B).

These observations, as also discussed later, are a precursor to the significant impact

produced by the speed-of-sound definition on the flow time scales. The qualitative shape

and location of the cavity for both SoS-1 and SoS-2, however, are similar to the

experimental observations (Figure 5-6D). Senocak and Shyy (2003,2004b) pointed out

that SoS-2 performed better than SoS-1 in simulating the oscillatory behavior of the

cavitating flow in the convergent-divergent nozzles, because SoS-2 successfully

produced a quasi-steady solution at high cavitation number and a periodic solution at low

cavitation number. In Figure 5-8, SoS-2 and SoS-1 behave similarly in this computed








61



case, and they both are nearly constant when away from the pure phase. These


qualitatively agree with the theoretical analysis (Figure 1-2). Also, this helps to explain


why the cavity shapes of the two different SoS models are qualitatively close. One thing


needs to be pointed out here that the volume fraction is very close to the pure liquid


phase. Hence the linear approximation SoS-1 can have similar behavior as SoS-2.


5 099
4 098
3 096
2 09
1 08


5 099
4 098
3 096
2 09
1 08



-5


4


5 099
4 098
3 096
2 09
1 08



// ^

5 099
1 4 098
S3 096
2 09
1 1 08




^^ ^4 6


Figure 5-7. Middle section density contours of different SoS at different instants (the
mixture of vapor and liquid inside the outer line forms the cavity), original
IDM, LSM. A) SoS-2, maximum fraction at t*=0.32 (Left) and minimum
fraction at t*= 0.77 (Right). B) SoS-1, maximum fraction at t*=5.92 (Left)
and minimum fraction at t*=5.86 (Right)


Lack of substantial experimental data prevents us from making more direct


comparisons at this stage. Various experimental studies have shown that, besides


oscillating, the cavities incept, grow, detach from the needle tip and get periodically


3//









transported downstream (Figure 5-4E). The periodic detachment and inception of cavities

is, however, difficult to capture through currently known turbulence and SoS models

(Wang et al. 2001, Senocak and Shyy 2003, Coutier-Delgosha et al. 2003, Wu et al. 2003

a). Consequently, fundamental understanding on the above modeling aspects, which is

discussed in the following sections, is imperative to encounter the above limitation. The

following discussions serve to probe the sensitivity of the various modeling concepts,

guided by our qualitative, but incomplete, insight into the fluid physics.

The implications of speed-of-sound (SoS) definition are quantified by a correlation

study (Table 5-1 and Table 5-2) and spectral analysis (Figure 5-10) on a series of

pressure-density time history at various nodes in cavitation vicinity. The Pearson's

correlation (r) between pressure and density is calculated as

N
Y[(X, -)(Y -)]
r= I ; X =P, Y= p (5.1)
V x,- (X-rY)2


Table 5-1 and Table 5-2 indicate that within similar variation limits of liquid

volume fraction, SoS-1 has a broader range of pressure-density correlation coefficients

than SoS-2. Furthermore, SoS-2 consistently exhibits much stronger pressure-density

coefficients than SoS-1. The distinct effects produced by the speed-of-sound definition

are further corroborated by the dramatic time scale differences observed from the time

history and FFT plots of pressure and density (Figure 5-9 and Figure 5-10A, B, C). SoS-1

plots clearly indicate dominance of a single high-frequency compressibility effect unlike

SoS-2, which are characterized by a wider bandwidth for pressure and density, and a

lower dominant frequency. It is worth noting here that although the plots in Figure 5-9(B,







63


C) are plotted against a smaller time range, we have used a convincingly long time

history for our analysis.


70

60


0 40
0
U,


0

09 095
liquid fraction


Liquid fraction


Figure 5-8. Two different SoS in hollow-jet valve flow, original IDM, LSM

Table 5-1. Time-averaged liquid volume fraction v/s pressure-density correlation at
multiple points inside the cavity, original IDM with SoS-1, LSM
Time averaged liquid volume Pearson's correlation (r) between
fraction (al) pressure and density (SoS-1)
0.862 -0.674
0.882 -0.09
0.936 0.093
0.970 -0.045
0.978 -0.077
0.983 -0.096
0.986 -0.103
0.988 -0.12
Mean(r)= -0.13; SD(r)= 0.9

Table 5-2. Time-averaged liquid volume fraction v/s pressure-density correlation at
multiple points inside the cavity, original IDM with SoS-2, LSM
Time averaged liquid volume Pearson's correlation (r) between
fraction (al) pressure and density (SoS-2)
0.858 -0.733
0.868 -0.106
0.934 -0.106
0.968 -0.46
0.977 -0.47
0.982 -0.412
0.985 -0.412
0.987 -0.412
Mean(r)= -0.42; SD(r) = 0.1


























SLocation A
Location B
Location C
SLocation D



.- E


B
Figure 5-9. Pressure time evolutions of different
locations. B)SoS-1. C) SoS-2


Location A
Location B
Location C
Location D


-2

0 1 2
tS

SoS, original IDM, LSM. A) Samples


I I I IA
Figure 5-10. Time evolution and spectrum of pressure and density of two SoS definitions
at a point at the cavitation vicinity, original IDM, LSM. A) Pressure (Left) and
density (Right) time history. B) Spectral analysis on pressure (Left) and
density (Right, SoS-1. C) Spectral analysis on pressure (Left) and density
(Right), SoS-2





























s "
B








I I I3 2 25I

C


Figure 5-10. Continued

5.2 Turbulent Cavitating Flow through a Convergent-Divergent Nozzle

To study the performance of the filter-based model (FBM, Section 2.3), and the

modified IDM (Section 3.3) in time-dependent computations, the turbulent cavitating

flow through a convergent-divergent nozzle is investigated.

Stutz and Reboud (1997, 2000) have studied the unsteady cavitation formed in a

convergent-divergent nozzle that has a convergent part angle of 180 and a divergent part


angle of 80. The experimental Reynolds number is Re = 106 3 3 x 106 based on the

reference velocity and cavity length. Cavitation formed in this nozzle is described as

"unsteady and vapor cloud shedding". Senocak and Shyy (2003, 2004b) conducted the

numerical simulations using IDM coupled with the LSM. They found that under

cavitation number c = 1.98, the cavity length L,, matched the experiments (Stutz and


I


$a 0 I 2D 5
rhap-y


- I I I I I





ILm q. mm U l I" ir m d m, J .. b









Reboud 1997, 2000), which did not specify the cavitation number. Therefore, to be

consistent, we use c = 1.98 in the present study.

First, we apply the original IDM with SoS-2 for the simulations to assess the FBM

performance.

To test the grid and filter size dependency, two grids are adopted including a coarse

grid with A = 0.5Lo and a fine grid with A = 0.25L av. The time-averaged eddy viscosity

contours of two grids are given in Figure 5-11. The FBM gives effective viscosities for

both grids about half an order of magnitude lower than that produced by the LSM. The

differences of the time-averaged eddy viscosity are noticeable between the two grids. The

largest viscosity on the fine grid is lower and its location shifts to the upstream, compared

with Figure 5-11 (A, B) left and right columns correspondingly. However, the time-

averaged vapor volume fraction does not show significant difference (Figure 5-12A, B).

The time-averaged u-velocity profiles have the similar trend.

Hereafter, all the results are based on the fine grid with A = 0.25Lv Figure 5-13

shows the time-averaged velocity and vapor volume fraction profiles within the cavity at

four different sections, using two different turbulence models. The boundaries of the

cavitating region, from experiments, LSM and FBM, are also included. Although the

numerical results of the cavitating boundaries are about 5% higher than that from the

experiments, the computations capture the main cavity body and the overall trends are

agreeable (Figure 5-13B). The velocity profiles qualitatively agree with the experimental

data, especially in the core of the reverse flow (Figure 5-13A). Comparing the LSM and

FBM, both the time-averaged velocity and volume fraction show marginal difference.

However, the instantaneous velocity and volume fraction profiles display significant











differences (Figure 5-14). The current results show an auto-oscillating cavity. Here only


the maximal- and minimal-volume fraction time instants are presented. The maximal-


volume fraction of the FBM is bigger than that of the LSM. On the contrary, the minimal-


volume fraction is much smaller by comparing Figure 5-14A and Figure 5-14B. The


reverse velocity profiles show the same trend. The increase in fluctuations is due to the


FBM reducing the eddy viscosity (Figure 5-11). The observation reveals that the time-


averaging process can be misleading and not suitable as the only indicator for


performance evaluation, which is similar to the previous study by Wu et al. (2003b).


I 4111E 11 1005

U U-- 6-04 E04

005 005
-0 41 0 4904







-15 LSM o i,. FBM
-05 0 5 1 15 -05 0 05 1 15
x (m) x(m)
S25617E-04 7 2 4 56E-04








-- 4 3541E 044 1E-04








2 4191E04191E









LSM FBM
-0 15 -0 15 -










-05 0 D5 1 15 0- 5 o 05 1 15
x(m) x(m)
B

Figure 5-11. Time-averaged eddy viscosity contours of different grids, original IDM with
SoS-2, = 1.98. A) Coarse grid with A= 0.50LZav. B) Fine grid with

A= 0.25L
A = 0.25L,













Vapor volume fraction, coarse grid
Vapor volume fraction, fine grid
Vapor volume fraction, Exp. data
Boundary of cavitating flow, coarse grid
Boundary of cavitating flow, fine grid

25%
25%


25%


25(%)
25%


Vapor volume fraction, coarse grid, A=0.5L_
Vapor volume fraction, fine grid,A=0.25L_
15 Vapor volume fraction, Exp. data
Boundary of cavitating flow, coarse grid
Boundary of cavitating flow, fine grid
25% 25%
25%
10

25%





5 n


0 20 40 60 80 0 20 40 60 80
X (mm) X (mm)
A B
Figure 5-12. Time-averaged vapor volume fraction comparisons of different grids,
original IDM with SoS-2, a = 1.98. The vertical scale is the distance from the
wall. Experimental data are from Stutz and Reboud (1997&2000). A) LSM.
B) FBM

Velocity, Standard Vapor volume fraction, Standard
Velocity, filter-based Vapor volume fraction, filter-based
15 A Velocity, Exp. data 15 A Vapor volume faction, Exp. data
Boundary of cavitating flow, Exp. Boundary of cavitating flow, Exp.
Boundary of cavitating flow, Standard Boundary of avtating flow, Standard
Boundary of cavitating flow, Filter-based U (m/s) Boundary of cavitating flow, Filter-based (%)
10 10 25% 25%
10 25%
10 A 10- A

25% 5
E 7 A E


X (mm) A X (m)
A B
Figure 5-13. Time-averaged comparisons of different turbulence models on fine grid with
A = 0.25L original IDM with SoS-2, a = 1.98. The vertical scale is the

distance from the wall. Experimental data are from Stutz and Reboud
(1997&2000). A) U-velocity. B) Vapor volume fraction


An illustration that the FBM can induce noticeably stronger flow oscillations is


presented in Figure 5-15. It is observed that time-averaged pressure contours and


streamlines are similar at the cavity zone, although a recirculation region produced by the


FBM exists at downstream near the lower wall (Figure 5-15A). However, the


I


//


ii








69



instantaneous pressure contours and velocity field produced by the two turbulence


models are distinctly different. The FBM yields wavy flow structures, which induce the


auto-oscillation of the flow fields in Figure 5-15(B, C). The experimental measurements


(Figure 5-14) are confined in the throat region, which have missed the large scale


unsteadiness of the flow field.


-- Velocity, minimal instant
Velocity, maximal instant
15 Boundary of cavitating flow, Minimal
Boundary of cavitating flow, Maximal

10
10 L


/j 'I


10


S 10 20 30 40 50
X(mm)


60 70 80


-- Velocity, minimal instant
Velocity, maximal instant
5 Boundary of cavitating flow, Minimal
S Boundary of cavitating flow, Maximal


10




/


10








* I II


u (m/s)
10

//


- Vapor volume fraction, minimal instant
Vapor volume fraction, maximal instant
Boundary of cavitating flow, Minimal
Boundary of cavitating flow, Maximal (y)
25%
25%


- Vapor volume fraction, minimal instant
Vapor volume fraction, maximal instant
- Boundary of cavitating flow, Minimal
Boundary of cavitating flow, Maximal ^ (o)
25%


0 10 20 30 40 50 60 70 80 _0 10 20 30 40 50 60 70 80
X (mm) X (mm)

Figure 5-14. Instantaneous profiles on fine grid with A = 0.25Lc, original IDM with

SoS-2, a = 1.98. The vertical scale is the distance from the wall. A) LSM: u-
velocity and vapor volume fraction. B) FBM: u-velocity and vapor volume
fraction


I I


P
j












Pressure contours: color
S-0.842


0.1 -


LSM


0 05
x (m)


Pressure contours: color
-0.842











LSM


05
x(m)


Pressure contours: color
-0.842


Pressure contours: color
*-0.842


LSM


1-1.26


05
S(m)


Figure 5-15. Pressure contours and streamlines comparison of two turbulence models on
fine grid, original IDM with SoS-2, cr = 1.98. A) Time-averaged. B) At
maximal-volume fraction instant. C) At minimal-volume fraction instant









Obviously, both turbulence and cavitation models directly affect the cavity

dynamics such as the shedding pattern and frequency. Hence in this section, we further

investigate the implication of the cavitation model in the context of the interfacial

velocity estimated, as discussed in Section 3.3 which is called the modified interfacial

dynamics-based cavitation model. The speed-of-sound (SoS) is based on the SoS-2A,

formulated in Section 4.3. The turbulence closure is employed the original Launder-

Spalding model (LSM) described in Section 2.2. The detailed results are presented below.

Figure 5-16 gives the time history of the pressure at a reference point near the inlet.

The pressure of the original IDM, which adopts an empirical factor to assign the

interfacial velocity based on the local fluid velocity in Section 3.3 Eq. (3.9), exhibits

much smaller variation than that computed with the modified IDM. As given in table 5-3,

although both IDMs under-predict the Strouhal number, the dynamically adjusted

interfacial velocity embodied by the modified IDM yields better agreement compared

with the experimental measurement.

Selected time evolutions of the cavity shape and the associated flow structures are

presented in Figure 5-17 with the available experimental observation. The original IDM

demonstrates an attached periodic cavity, while the modified IDM produces cavity break-

up and detachment, in manners qualitatively consistent with the experimental

observations. The cavity break-up and collapse of the modified IDM also helps to explain

the Cp behavior shown Figure 5-16. Furthermore, because of the shedding of the cavity,

the modified IDM better predicts the vapor volume fraction profiles than the original

IDM, with less difference in the velocity profiles between these two cavitation models

(Figure 5-18).











Table 5-3. Comparisons of Strouhal number of original and modified IDM with SoS-2A
Original IDM, SoS-2A Modified IDM, SoS-2A Experiment
0.07 0.13 0.30


- Modified IDM
Original IDM


HP
y


1 2 3 4 5


Figure 5-16. Pressure evolutions of original
reference point


and modified IDM with SoS-2A at a


Figure 5-17. Cavity shape and recirculation zone during cycling of original and modified
IDM. The experimental observation is adopted from Stutz and Rebound
(1997, 2000)


.1,i
-
































Figure 5-17. Continued


X (mm)


X (mm)


A B
Figure 5-18. Time-averaged volume fraction and velocity comparisons of original and
modified IDM. The vertical scale is the distance from the wall. Experimental
data are from Stutz and Reboud (1997, 2000). A) Vapor volume faction. B) u-
velocity

5.3 Turbulent Cavitating Flow over a Clark-Y Hydrofoil

The capability of the modified IDM (Section 3.3) and turbulence models are further

investigated in unsteady cavitating flows over a Clark-Y hydrofoil, assessed by

experimental data from Wang et al. (2001). To facilitate the performance evaluation, the









generalized treatments of the SoS-2, including SoS-2, SoS-2A described in Section 4.3,

are coupled.

For the computational set-up (Figure 5-19A), the computational domain and

boundary conditions are given according to the experimental set-up. The Clark-Y

hydrofoil is located at the water tunnel center. Two angles-of-attack (AoA) considered

are 5 and 8. The hydrofoil chord length is c and the hydrofoil leading edge is 3c away

from the inlet. The two important parameters are the Reynolds number and the cavitation

number, which is based on inlet pressure P. and vapor pressure P1 with inlet velocity

U.


Re Uc 7 x 105 (5.2)
V

P -P
S= (5.3)
pU /2

The filter size in the present study is chosen to be larger than the largest grid scale

employed in the computation, and is set to be A = 0.08c.

Computations have been done for two AoA and several cavitation numbers.

Specifically, for AoA=5 under four flow regimes: no-cavitation (o- = 2.02), inception

(o- = 1.12), sheet cavitation (o = 0.92), and cloud cavitation (o = 0.55), and AoA=8

under three cavitation numbers: no-cavitation (o = 2.50), sheet cavitation (o = 1.40),

cloud cavitation (o-= 0.80). All cases above are at the same Reynolds number

Re =7x105.

Two different turbulence models, LSM and FBM, have been employed to help

probe the characteristics of cavitation and turbulence modeling interactions, coupled with









different SoS numerical models. The outlet pressure is fixed and adjustment of the vapor

pressure is needed to be consistent with the prescribed cavitation number in each case.

Grid sensitivity analysis. To investigate the grid dependency, two grids are

adopted in the computation: coarse grid and fine grid. The grid blocks and numbers of the

coarse grid are shown in Figure 5-19B. The fine grid has 60% more number of nodes

than that of the coarse grid in the vertical direction while maintaining the same

distribution in the horizontal direction. Two different cavitation numbers, o = 2.02 and

a = 0.55, both at AoA=5 have been investigated. Overall, the solutions on both grids are

in good agreement. The time-averaged u-velocity and v-velocity profiles with o = 2.02

and a = 0.55 using the LSM are shown in Figure 5-20. The results based on the FBM are

of similar nature and will not be repeated.

Hereafter, to reduce the cost of time-dependent computations, we use the coarse

grid in the computations.

Visualization of cavity and flow field. First, we focus on the LSM results to

analyze the performance of the IDM. Figure 5-21 and Figure 5-22 show the time-

averaged flow structure and cavity shape under varied cavitation numbers. With no

cavitation, the flow field is attached without separation for both AoA=5 and AoA=8

(Figure 5-21A left column and Figure 5-22A left column). This is consistent with the

experimental observation. When cavitation appears, the density will change by a factor of

1000 between liquid and vapor phases. Consequently, there is a drastic reduction in the

amount of mass inside the cavity, and a contraction of the fluid flow behind the cavity.

With the reduction of the cavitation number, the cavity and recirculation zone become

bigger. At cloud cavitation regime, the cavity experiences shedding, causing multiple










recirculating flows (Figure 5-21D left column and Figure 5-22C left column). Compared

to the experimental data for both AoAs (Table 5-4 and Table 5-5), the predicted cavity

sizes demonstrate qualitatively consistent rending, albeit generally over-predicted. For

incipient cavitation, the experiment observed the recurring formation of hair-pin type

cavitating vortex structures, which are not attached to the solid surface (Wang et al.

2001). This type of flow structure is not captured in the computation. The time-averaged

flow structures associated with sheet and cloud cavitation, on the other hand, seem to be

reasonably captured computationally. Furthermore, the time-averaged outcome of

employing both LSM and FBM seems compatible.


No-Slip


e i- Hydrofoil: No-Slip
_E 0
10c

No-Slip A

Coarse Grid

Block 2:70x50 Block 4:70x50 Block 1: 80x50






Block 1: 70x50 /Block 3: 70x50 Block 5: 80x50



B
Figure 5-19. Clark-Y geometry sketch and Grid blocks. A) Geometry configuration and
boundary conditions, c is the hydrofoil chord. B) Grid blocks and coarse grid
numbers











77





A Experiment
Fine grid
O Coarse grid


04 I




o 02 -





4 1.0
1. 0_ulu -


02 04 06 08 1 12
X/C


A Experiment
-- Fine grid
Coarse grid
04






02





S1..
0-1..


-1.0

002 04 06 08 1
XIl A


A Experiment
Fine, LSM
4 Coarse, LSM


04 -


A A A
A Aa






S 1.01 u/U


1.0
0< Uv/U

1.0
02 04 06 08 1 12
X/c


A Experiment
a- Fine, LSM
04 Coarse, LSM




A A

02 -I *A A ,






S.VU A




0 02 04 06 08 1

XC B

Figure 5-20. Grid sensitivity of the time-averaged u- and v-velocity, LSM, AoA=5.

Experimental data are from Wang et al. (2001). A) No-cavitation. B) Cloud

cavitation c = 0.55









Table 5-4. Time-averaged cavity leading and trailing positions of different turbulence
models (Taking aL = 0.95 as cavity boundary), modified IDM with SoS-2A,
AOA=5 degrees
Position Inception = 1.12 Sheet cavitation a = 0.92 Cloud cavitation a = 0.55
LSM FBM Exp. LSM FBM Exp. LSM FBM Exp.
Leading 0.13 0.13 0.13 0.13 0.23 0.16 0.17 0.22
Trailing 0.62 0.57 0.64 0.64 0.68 1.01 1.04 0.90

Table 5-5. Time-averaged cavity leading and trailing positions of different turbulence
models (Taking aL = 0.95 as cavity boundary), modified IDM with SoS-2A,
AoA=8 degrees.
Position Sheet cavitation a = 1.40 Cloud cavitation a = 0.80
LSM FBM Exp. LSM FBM Exp.
Leading 0.094 0.098 0.15 0.12 0.13 0.15
Trailing 0.53 0.52 0.56 1.07 1.10 0.84


Figure 5-21. Time-averaged volume fraction contours and streamlines of different
turbulence models, AoA=5. A) No-cavitation. B) Incipient cavitation. C)
Sheet Cavitation c = 0.92. D) Cloud cavitation o = 0.55



















B





C
Figure 5-22. Time-averaged volume fraction contours and streamlines of different
turbulence models, AoA=8. A) No-cavitation. B) Sheet cavitation a = 1.40.
C) Cloud cavitation a = 0.80

The temporal evolution of the computed and experimentally observed flow

structures with cloud cavitation, under two AoAs, are shown in Figure 5-23 and Figure 5-

24. Figure 5-23A left column shows the time sequences of flow structures predicted by

LSM at AoA=5. The corresponding flow predicted by the FBM is shown in the right

column. The experimental visual image is shown in Figure 5-23B. The flow structures at

AoA=8 are shown in Figure 5-24. Both the computations and the experiment indicate

that as the AoA increases, the cavity exhibits a more pronounced recurrence of the size

variation. The FBM predicts stronger time-dependency than the LSM. We will further

investigate this aspect in the following discussion. To help elucidate the main features of

the cavity dynamics from both numerical and experimental studies, we show in Figure 5-

25 three stages of the cavity sizes, for cloud cavitation c = 0.55 at AoA=80. The figure

demonstrates that the numerical simulation is capable of capturing the initiation of the

cavity, growth toward trailing edge, and subsequent shedding, in accordance with the

qualitative features observed experimentally.







80






t=0.00




t=3.36 ms




t= 10.1ms




t=11.8 ms




t=15.1 ms




t=20.2 ms A




t=0.0 t=2.2 ms





t=4.4 ms t=6.7 ms






t=8.9 ms t=ll.1 ms B
Figure 5-23. Time evolutions of cloud cavitation c = 0.55, AoA=5. A) Numerical
results of different turbulence models. B) Side views of the experimental
visuals from Wang et al. (2001)
























































Figure 5-24. Time evolutions of cloud cavitation c = 0.80, AoA=8. A) Numerical
results of different turbulence models. B) Side views of the experiment visuals
from Wang et al. (2001)


LSM. t=24.5




































































t=22.2 ms
Figure 5-24. Continued


t=)U ms


Flow























Figure 5-25 Cavity stage comparisons, cloud cavitation c = 0.80, AoA=8: FBM (Left)
and Experiment (Right)0. Experimental visuals are from Wang et al. (2001).
A) Early stage: cavity formation. B) Second stage: cavity growth toward
trailing edge. C) Third stage: cavity break-up and shedding

Velocity profiles and lift/drag coefficients. The mean horizontal u- velocity and

vertical v-velocity of the flow field are illustrated in Figure 5-26 (AoA=5) to Figure 5-27

(AoA=8). The time-averaged velocity profiles are documented at 6 chordwise locations,

0%, 20%, 40%, 60%, 80%, and 100% of the leading edge, under different cavitation

numbers. With no cavitation, the numerical results agree well with the experiment, and

the results of the two turbulence models are virtually identical (Figure 5-26A and Figure

5-27A). With the cavitation number decreasing, the differences between prediction and

measurement become more substantial, especially at the cavity closure region. In fact,

recirculation breaks up the cavity and tilts the rear portion of the cavity more upward,

which makes the cavity thicker in comparison to the experiment. With the reduction in

cavitation number, it becomes more distinguished in Figure 5-23 and Figure 5-24. The

cavity tilt-up results in over-shooting the u-velocity at the closure region (0.8c and 1.0c

sections) (Figure 5-26 and Figure 5-27). Overall, considering the difficulties in

experimental measurement (Wang et al. 2001) the agreement is reasonable.


















A Experiment
-- Filter-based
Launder-Spalding


A Experiment
- Filter-based
--- Launder-Spalding


A Expenment
-- Filter-based
Launder-Spalding
----- FBM, cavity boundary
- LSM, cavty boundary


A Experiment
Filter-based
Launder-Spalding
FBM, cavity boundary
LSM, cavity boundary


"xC B


Figure 5-26. Time-averaged u- and v-velocities of two turbulence models, AoA=5.

Experimental data are from Wang et al. (2001). A) no-cavitation. B) Inception

a = 1.12. C) Sheet cavitation a = 0.92. D) Cloud cavitation c = 0.55


04





02


02


12




















A Expenment
---Filter-based
SLaunder-Spalding
-...-- FBM, cavity boundary
LSM, cavity boundary


A Experiment
-- Filter-based
Launder-Spalding
FBM, cavity boundary
LSM, cavity boundary


Figure 5-26. Continued


04





02


04





02


12