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Direct Numerical Simulation of Fragmentation of Droplets

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Direct Numerical Simulation of Fragmentation of Droplets Direct Numerical Simulation of Fragmentation of Droplets by Maziyar Jalaal B.Sc., University of Tabriz, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIRMENTS FOR THE DEEGREE OF Master of Applied Science in The College of Graduate Studies (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) May 2012 © Maziyar Jalaal, 2012 Abstract The work described in the present thesis is related to a series of projects that I worked on toward the better understanding of fragmentation phenomena. In the past decades, the science of fragmentation has attracted many attentions within the researchers due to its wide range of applications. However, because of the complexity of the subject, even its basic concepts need more investigations. This thesis starts with an introduction to fragmentation of droplets using experimental or numerical approaches. It is discussed that the current mathematical and experimental tools are not able to describe all the details. Thus, high performance numerical simulations are the best alternatives to study the breakup of droplets. The introduction is followed by a discussion on the numerical method and the ranges of the non-dimensional groups. It is described that an adaptive, volume of fluid (VOF) method based on octree meshing is used, providing a notable reduction of computational cost. The rest of the thesis basically discusses the obtained results using direct numerical simulations. Two main geometries are investigated: falling droplets and droplets in a stream. For the case of falling droplets, three simulations with different Eötvös numbers are performed. For the case of droplets in a stream, two-dimensional and three-dimensional simulations are performed for a range of Weber number. The results are compared with the available mathematical theories and it is shown that the analysis presented here precisely demonstrates the mechanism of the bag breakup of falling droplets and instability growth over the droplets in an external high-speed flow. The outcomes can significantly assist the development of the secondary atomization and turbulent two-phase flows modelling. ii Table of Contents Abstract ............................................................................................................................................. ii Table of Contents ............................................................................................................................. iii List of Tables ..................................................................................................................................... v List of Figures .................................................................................................................................. vi Acknowledgements ......................................................................................................................... xii Dedication ...................................................................................................................................... xiv 1. Introduction ................................................................................................................................... 1 1.1 Fragmentation of Droplets ................................................................................................. 1 1.2 Motivation .......................................................................................................................... 7 1.3 Outline of This Thesis ........................................................................................................ 8 2. Methodology & Non-Dimensional Groups ................................................................................... 9 2.1 Geometries ......................................................................................................................... 9 2.2 Governing Equations ........................................................................................................ 11 2.3 Flow Solver (Gerris) ....................................................................................................... 12 2.3.1 Temporal Discretisation ................................................................................... 13 2.3.2 Spatial Discretisation ....................................................................................... 14 2.4 Non-Dimensional Numbers ............................................................................................. 17 3. Fragmentation of Falling Droplets .............................................................................................. 21 3.1 General Description of The Fragmentation Procedure ..................................................... 21 3.2 Drop Deformation and Bag Formation ............................................................................ 24 3.3 Bag Breakup ..................................................................................................................... 30 3.4 Number of Fragments ...................................................................................................... 46 4. Statistical Characteristics of Fragments ...................................................................................... 51 4.1 An Overview on Droplet Deformation and Breakup ....................................................... 51 4.2 Fragment Number Variation ............................................................................................ 52 4.3 Fragment Size and Velocity Distribution ......................................................................... 55 4.4 Variation of Non-Dimensional Groups ............................................................................ 65 5. Deformation of Droplet in a Stream ............................................................................................. 71 5.1 An Overview on Droplet Dynamics in a Stream .............................................................. 71 iii 5.2 Two-Dimensional Results ................................................................................................ 73 5.2.1 Grid Tests ......................................................................................................... 73 5.2.2 Deformation Mechanism .................................................................................. 76 5.2.3 Effect of Weber Number .................................................................................. 78 5.2.4 Comparison with Theories (Shear Instabilities) ................................................ 82 5.2.4.1 Combination of Potential Flow and Very Thin Vorticity Layer ...... 82 5.2.4.2 Non-Zero Vorticity Layer ................................................................ 85 5.3 Three-Dimensional Results .............................................................................................. 87 5.3.1 Deformation Mechanism .................................................................................. 88 5.3.2 Comparison with Theories (Transverse Azimuthal Modulation) ..................... 92 6. Conclusion ................................................................................................................................... 97 6.1 Future Work ..................................................................................................................... 98 Bibliography .................................................................................................................................. 100 iv List of Tables 2.1. Non-dimensional numbers studied for the case of falling droplets .......................................... 18 2.2. Dimensionless groups for the case of a droplet in a stream ..................................................... 19 3.1. Values of Reynolds and Weber numbers ................................................................................. 28 4.1. Parameter in log-normal PDF modelling ................................................................................. 60 5.1. Grid properties at 0.87 ....................................................................................................... 74 5.2. Comparison of the present direct numerical simulations and equation 5.6 .............................. 85 5.3. Comparison of the present direct numerical simulations and equations 5.7 and 5.10 .............. 87 5.4. Comparison of the present direct numerical simulations and equation 5.14 ............................ 95 5.5. Comparison of the present direct numerical simulations and equations 5.15 and 5.16 ............ 95 v List of Figures 1.1. a) Droplet in an external flow. b) Droplet falling in a constant gravity (acceleration). ................. 1 1.2. The algorithm of droplet disintegration and fragments distribution. ............................................. 2 1.3. Main modes of droplet breakup. .................................................................................................... 3 1.4. Breakup modes in terms of Weber number. ................................................................................... 5 2.1. Schematic sketch of the initial condition of simulations for falling droplets. ............................... 10 2.2. Droplet in a stream. Flow direction is from left to right. ............................................................. 11 2.3. Quad-tree grid adaption from level 0 (dotted box) to level 4 (green boxes). ............................... 14 2.4. Grids generated for a case of falling droplet ( 0.761 for Eo 216 , Ohd 0.05, Ohc 0.05
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