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Simulations of cavitation from the large vapour structures to the small bubble dynamics Aurélia Vallier June Thesis for the degree of Doctor of Philosophy in Engineering. ISSN ISRN LUTMDN/TMHP-13/1092-SE

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Simulations of cavitation from the large vapour structures to the small bubble dynamics Aurélia Vallier June 2013 . Thesis for the degree of Doctor of Philosophy in Engineering. ISSN ISRN LUTMDN/TMHP-13/1092-SE ISBN (print) ISBN (pdf) c Aurélia Vallier, June 2013 Division of Fluid Mechanics Department of Energy Sciences Faculty of Engineering Lund University Box18 S LUND Sweden Typeset in L A TEX Printed by MediaTryck, Lund, May i Populärvetenskaplig sammanfattning Mycket få människor omkring oss vet innebörden av ordet kavitation, förutom de som såg filmen The Hunt for Red October och kan relatera kavitation till Sean Connery i en ubåt. Kavitation motsvarar bildandet av bubblor, som kan likna kokande vatten i en kastrull. Men den uppstår inte på grund av en hög temperatur utan på grund av ett lågt tryck. Den finns i de flesta tekniska anläggningar som innehåller vätska i rörelse. Problemet med kavitation är dess negativa konsekvenser. Till exempel orsakar den oljud vilket inte är onskvärt för en ubåt. Den kan också leda till förstörelse av ytor, vilket inte är onskvärt i en vattenturbin. Kavitation i vattenturbiner orsakar förändringar och instabilitet i strömningen, och implosion av bubblor. Detta resulterar i en minskning i effektivitet, vibrationer och erosion (skador på ytor). Kavitation kan undvikas om turbinen ställts tillräckligt låg, så att det statiska trycket är tillräckligt högt för att förhindra att vatten övergår till gasform. Men byggkostnaderna för en sådan låg inställning är mycket höga. Därför måste man hitta en kompromiss mellan motstridiga krav på en låg installationskostnad och undvikande av negativa effekter från kavitation. Kavitation är mycket komplex. En stor mängd forskning har gjorts under de senaste 30 åren för att förbättra förståelsen för detta fenomen. För att få mer kunskap om kavitation i vattenturbiner, kan man använda sig av numeriska modeller. Genom att lösa lämpliga ekvationer kan man beskriva hur kavitation börjar och utvecklas. Det finns modeller för varje specifik företeelse. Dock är kavitationsmodellering fortfarande mycket utmanande eftersom fenomenet leder till snabba variationer av strömingsegenskaper och samspelet mellan vatten, ånga och gas. Ångan som bildas vid kavitation kan uppträda i varierande storlek och form, från mikroskopiska sfäriska bubblor, till stora sammanhängande strukturer. Dessutom är strömningen turbulent. Alla dessa egenskaper kräver lämpliga modeller för att exakt förutsäga kaviterande strömningar. I detta arbete utförs beräkningar för att utvärdera resultaten av olika modeller. En ny flerskalig modell utvecklas och används på en kaviterande strömning kring en vingprofil. Den nya modellen omfattar både små sfäriska bubblor, stora icke-sfäriska ånga strukturer och övergången mellan dessa regimer. Det är mycket intressant att ha en modell som kan förutsäga hur dem minsta bubblorna transporteras till regioner med lågt statiskt tryck, där de växer och sen imploderar. Genom att mäta tryckvågen som släpps från bubblan, kan man förutse risken för att närliggande ytor ska skadas. Tack vare den förbättrade modellen, kan man förutse där kavitation orsakar skador. Denna kunskap kan i ett senare skede hjälpa till att utforma geometrier som minskar de negativa effekterna av kavitation. Särskild omsorg kan då tas, så att bubbelimplosionerna sker långt ifrån ytor. Detta skulle minimera skador på ytorna, och därmed minska underhållskostnaderna. ii iii Abstract Very few people around us know the meaning of the word cavitation, except from those who saw the movie The Hunt for Red October and can relate cavitation to Sean Connery in a submarine. Some of them know that it corresponds to the formation of bubbles, due to a pressure drop, and causes erosion and noise. However, cavitation is much more complex. A large amount of research work has been done over the last thirty years in order to improve the understanding of the interactions between the various physical processes involved. The present work aims at gaining more knowledge about cavitation in water turbines. Some of the properties of cavitation at a water turbine runner blade are similar to those at a hydrofoil in a water test tunnel. Therefore, the overall purpose of this work is to improve the numerical models for cavitation inception and development on a hydrofoil. The focus of this thesis lies on numerical methodologies that include the broad range of cavity sizes, using appropriate models for each specific phenomenon. The smallest bubbles, called nuclei, are tracked in the flow with the Discrete Bubble Model, and their dynamics is resolved with the Rayleigh-Plesset equation. This approach can predict how the nuclei are transported over a hydrofoil to regions of low static pressure, where they grow and either collapse or contribute to the formation of large-scale vapour cavities. The large non-spherical structures are commonly modelled using the Volume-Of-Fluid method together with a mass transfer model for vaporisation and condensation. This approach predicts the development of the vapour cavity, such as its breakup and the shedding process observed experimentally in the context of cavitating hydrofoils. The present work implements the above-mentioned models in the OpenFOAM C++ library, and performs simulations to assess the performance of the models. A new multiscale model is developed, implemented and used on a cavitating hydrofoil. The multi-scale model includes both the small spherical bubbles, the large non-spherical vapour structures, and the transition between those regimes. iv v List of publications This thesis is based on the following papers. 1. Comparisons of numerical and experimental results for rising air bubbles. Aurélia Vallier and Johan Revstedt. Submitted to Journal of Fluids. 2. Modelling of bubble dynamics related to cavitation. Aurélia Vallier, Johan Revstedt and Håkan Nilsson. Submitted to Computers and Fluids. 3. Mass transfer cavitation model with variable density of nuclei. Aurélia Vallier, Johan Revstedt and Håkan Nilsson. 7 th International Conference on Multiphase Flow, ICMF 2010, Tampa, Florida, Numerical procedure for simulating the break-up of cavitation sheet. Aurélia Vallier, Johan Revstedt and Håkan Nilsson. 4 th International meeting on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Belgrade, Serbia, A new multi-scale approach for modelling cavitation on hydrofoils. Aurélia Vallier, Johan Revstedt and Håkan Nilsson. Submitted to International Journal for Numerical Methods in Fluids. vi vii Acknowledgements The research presented in this thesis was carried out as a part of Swedish Hydropower Center - SVC . SVC has been established by the Swedish Energy Agency, Elforsk and Svenska Kraftnät together with Luleå University of Technology, The Royal Institute of Technology, Chalmers University of technology and Uppsala University. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at LUNARC, center for scientific and technical computing at Lund University. I would like to thank my supervisor Johan Revstedt and my co-supervisor Håkan Nilsson for their guidance and advice during this work. Thanks also to Bengt Sundén for letting me follow my family to Helsinki, to finish this work under the best possible conditions. viii ix Contents 1 Introduction Objectives Achievements Cavitation description Cavitation inception Cavitation effect on efficiency Cavitation development Modelling of multiphase flow The Volume-Of-Fluid (VOF) method Treatment of the advection term Treatment of the surface tension The Discrete Bubble Model (DBM) Bubble equations of motion Influence from the bubbles on the flow Bubble-wall collisions Bubble interactions Cavitation modelling Cavitation models Mass transfer models The model of Sauer and Schnerr [62] The cavitation bubble model Dynamics of a still bubble in an unbounded domain Bubble dynamics in the cavitation bubble model The new multi-scale model Transition from the Eulerian to the Lagrangian frame Algorithm Transition from the Lagrangian to the Eulerian frame Unpublished results Liquid jet breakup Nuclei distribution and its effects on the cavity shape Summary of appended papers 41 x 8 Conclusions and future work 45 xi Nomenclature Roman symbols C drag Drag coefficient [ ] C lift Lift coefficient [ ] c 0 Chord length [m] D Diameter [m] F Force [kg m s 2 ] g Gravitational constant [ms 2 ] m Mass [kg] n Unit normal vector [ ] n nuc Nuclei density [m 3 ] p Pressure [kg m 1 s 2 ] R Radius [m] S Source term [kg m 2 s 2 ]or[s 1 ] t Time [s] t Unit tangential vector [ ] U Velocity [ms 1 ] V Volume [m 3 ] x Position [m] Greek symbols α Liquid volume fraction [ ] η Efficiency [ ] ɛ Coefficient of restitution [ ] κ Curvature [m 1 ] μ Dynamic viscosity [kg m 1 s 1 ] ν Kinematic viscosity [m 2 s 1 ] ρ Density [kg m 3 ] σ st Surface tension coefficient [ ] σ Cavitation number [ ] θ Angle [ ] τ Stress tensor [kg m 1 s 2 ] Subscripts a Acoustic wave B Bubble E Eulerian g Gas xii l L nuc v w Liquid Lagrangian Nuclei Vapour Wall xiii Chapter 1 Introduction Cavitation in water turbines causes flow alterations and instabilities, and collapses of bubbles. This results in a drop in efficiency, vibration, erosion and noise. Cavitation can be avoided if the setting of the turbine (i.e. its location with respect to tailwater elevation) is sufficiently low, yielding a static pressure at the runner that is high enough to prevent evaporation. However, construction costs for such a low setting are very high. Therefore, the setting is a compromise between the conflicting demands of a low cost installation, and a few negative effects from cavitation. A better knowledge of the cavitation phenomenon would help when designing geometries that reduce the negative effects of cavitation. Special care can be taken, such that the bubble collapse energy is reduced, or that the collapse occurs far away from surfaces. This would minimise the damages on the surfaces, and thus reduce the maintenance costs. Computational Fluid Dynamics (CFD) simulations is an alternative to prototype experiments in order to improve the understanding of how to avoid cavitation problems. CFD has been used, for example, to study the influence of modifications of the shape of the trailing edge, the runner or the curvature of the blade by Göde [20], Zobeiri et al [78], Ingvarsdottir et al. [31] and Mishima [48] respectively. Several numerical cavitation models have recently been introduced in the literature and in general-purpose CFD codes. However, modelling cavitation is still very challenging since it involves the interactions between liquid, vapour and undissolved gas, and moreover rapid temporal and spatial variations of the flow properties. The cavities range in size from microscopic spherical bubbles, to large-scale coherent structures. Furthermore, the flow is turbulent, and highly dynamic and unstable. All these features require appropriate models in order to accurately predict cavitating flow. Many of the cavitation properties found in a water turbine are similar to those at a hydrofoil in a water test tunnel. The numerical models for cavitation inception and development are thus in the present work evaluated for cavitation at a hydrofoil. The focus is on the mechanisms of sheet and cloud cavitation, and the transition between those phenomenon. A sheet cavity is a large attached structure which covers a part of the hydrofoil, while cloud cavitation corresponds to a large number of small bubbles being transported with the flow. The sheet cavity length may oscillate if the rear part is periodically detached from the cavity, and the detached part turns into a cloud of small bubbles. Therefore, the present work has a focus on including the broad range of cavity sizes, using appropriate models for each specific phenomenon. 1 The large-scale cavitation inception, development and break up is frequently modelled using the Volume-Of-Fluid (VOF) method, with a mass transfer model for the vaporisation and condensation [30, 41]. In the mass transfer model developed by Sauer and Schnerr [62], the vaporisation is governed by the number of cavitation nuclei per unit volume in the fluid. The cavitation inception is modelled by a linearized Rayleigh-Plesset equation for the rate of growth of the nuclei. This approach successfully predicts the attached sheet cavity, the re-entrant jet which breaks up the cavity, and the shedding process. The part of the cavity that breaks off is however transported downstream as a large coherent structure rather than a cloud of bubbles. For the small bubbles, a more relevant approach is to use the Discrete Bubble Model (DBM) to track individual bubbles, and a Rayleigh-Plesset equation to resolve the bubble dynamics and model the collapse of individual bubbles. With a four-way coupling method, the interaction between the bubbles is accounted for and the flow is affected by the presence of the bubbles. In the present work, simulations are performed to assess the accuracy of the VOF model, with and without the mass transfer model of Sauer and Schnerr [62]. The mass transfer model is modified to account for a non-uniform distribution of nuclei, and simulations are performed to show the effects on the cavitation development. The DBM approach and the Rayleigh-Plesset equation are implemented, and used to investigate the sensitivity to the model parameters. Finally, a new multi-scale model is described, implemented and used on a cavitating hydrofoil. It models both the small spherical bubbles (using DBM and the Rayleigh-Plesset bubble dynamics model), the large non-spherical vapour structures (using VOF and the Sauer and Schnerr mass transfer model), and the transition between those regimes. 1.1 Objectives The overall purpose of this work is to investigate and improve the numerical models that are suitable for modelling cavitation inception and development on a hydrofoil. The work is performed within the OpenFOAM C++ library [53]. The objectives of this thesis are therefore to evaluate the accuracy of the VOF model implemented in OpenFOAM, to investigate the behaviour of the predicted cavitation on a hydrofoil, using the VOF and Sauer and Schnerr mass transfer model in OpenFOAM, to implement and evaluate an improvement of the Sauer and Schnerr mass transfer model, taking into account a non-uniform nuclei distribution, to implement and investigate the behaviour of the Rayleigh-Plesset equation for bubble dynamics, to implement the DBM model, with four-way coupling and the Rayleigh-Plesset equation for bubble dynamics, and evaluate it under academic and realistic conditions, to implement an interaction between the VOF and DBM methodologies, and evaluate it in academic and realistic conditions. 2 1.2 Achievements An overview of the main achievements of this work is given below. The Volume-Of-Fluid (VOF) method is assessed under non-cavitating conditions. Simulations are performed for deformable air bubbles in a water channel and for the breakup of a liquid jet. The results obtained with OpenFOAM are compared to experimental data and numerical results from an in-house code. The Discrete Bubble Model (DBM) approach is implemented and used under non-cavitating conditions to study the nuclei distribution at a hydrofoil. The non-uniform nuclei distribution is included in the mass transfer model of Sauer and Schnerr [62] in order to highlight its effects on the cavity shape and behaviour. The Rayleigh-Plesset equation is implemented to model the bubble dynamics. It is used to study the influence of the various model parameters on the results. The Discrete Bubble Model and the Rayleigh-Plesset equation are coupled into a cavitation bubble model. This model is used under cavitating condition in the case of nuclei travelling above a rectangular cylinder. The results include the trajectory of the bubbles, the evolution of their radius and the pressure wave emitted at collapse. A method is developed to couple the VOF model and the Discrete Bubble model under non-cavitating condition. It is based on the detection of the small bubbles and their conversion from the VOF description to the DBM approach. This model is implemented and used for modelling the breakup of large vapour bubbles by a liquid jet and the formation of small Lagrangian bubbles. A multi-scale cavitation model is developed and implemented. It uses the VOF model with the mass transfer model of Sauer and Schnerr [62] to predict the large vapour structures, the cavitation bubble model (the Discrete Bubble Model with the Rayleigh-Plesset equation) to predict the small spherical bubbles, the model for the detection of the small bubbles and the conversion from the VOF description to the DBM approach, a model for the conversion from the DBM approach to the VOF description. This model is used in the case of a cavitating hydrofoil. 3 Chapter 2 Cavitation description Cavitation involves many complex phenomena. Here some of them are discussed, such as the initiation (inception), the effect on efficiency, and the development which includes the forms commonly taken, the transition from sheet and cloud cavitation, and the dynamics of the smallest bubbles. 2.1 Cavitation inception Cavitation inception can be described as the transition of a liquid into vapour due to a local reduction in static pressure. It requires the presence of nuclei in the fluid. Nuclei are small bubbles containing gas and vapour with diameters in the range of 10 3 to 10 1 mm. They are present in most technical systems where liquids are transported. If a nucleus enters a zone of low static pressure its radius grows. This may yield cavitation, which can take different forms, depending on the flow conditions. During the process, only nuclei over a certain size are stimulated into growth. Indeed the behaviour of a single nucleus influences the flow, and therefore it also influences the behaviour and the stability of the nuclei nearby. Masato [42] studied the interaction between nuclei and concluded that when the largest nuclei affects the flow, the smaller ones do not grow anymore or do not even become unstable. Other authors, such as Arora et al. [2], Brennen [8] and Morch [49] point out the nuclei content and size as crucial factors for cavitation inception. Apart from the free stream nuclei mentioned above, nuclei can also exist on the surface. They are small attached cavities which develop on the roughness of the surface. Therefore the type of surface influences cavitation inception and development. Another important parameter for cavitation inception is the turbulence because turbulence mixing may enhance the presence of nuclei in the turbulent boundary layer, and affect the form of the cavitation at inception. Franc [16] considered three different configurations to show that the type of the boundary layer affects the influence of the nuclei content and distribution on cavitation inception. In the case described in Figure 2.1(a), the laminar boundary layer separates from the wall and instabilities develop in the shear layer downstream separation. In that case, cavitation inception occurs in the core of the vortices, where the nuclei are trapped. In the case described in Figure 2.1(b), the separation of the laminar boundary layer is followed by the transition to turbulence and the reattachment of 4 the turbulent boundary layer to the wall. In that case, cavitation inception occurs in this small recirculation region, where the nuclei are trapped. In the case of an attached turbulent boundary layer, cavitation inception occurs as isolated bubbles, originating from the nuclei attached to the wall or transported by the fluid. (a) (b) Figure 2.1: (a) The nuclei

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