Three-Dimensional Simulations of Free Surface Flows Senior Thesis Author: Mingru Li Research Supervisor: Frederic Gibou Graduate Student Mentor: Raphael Egan Department of Mechanical Engineering College of Creative Studies University of California Santa Barbara Contents 1 Introduction 3 1.1 WhatAreFreeSurfaceFlow?.................................... 3 1.2 TheModel .............................................. 4 2 Computational Techniques 6 2.1 LevelSetMethods .......................................... 6 2.1.1 IntroductiontoLevelSetMethod. 6 2.1.2 TheEvolvingLevelSet ................................... 8 2.2 Grids&Discretization........................................ 9 2.2.1 EfficientStructuredGrids .................................. 9 2.2.2 DealingwithTreeStructure. 10 2.2.3 TreesinParallelComputing................................. 10 2.2.4 Finite Di↵erenceDiscretization ............................... 13 2.2.5 ReinitializationSchemes................................... 18 2.3 ProjectionMethod .......................................... 20 2.3.1 WhatisProjectionMethod? ................................ 20 2.3.2 MarkerandCell ....................................... 21 2.3.3 Solving from the past–Semi-Lagrangian Method . 22 2.3.4 Finite Volume Method on Voronoi Meshes . 23 2.3.5 Discretization of the Projection Step . 26 3 Free Surface Flow 30 3.1 TwoModels.............................................. 30 3.1.1 Sinusoidal Wave in 2D . 30 3.1.2 OscillatingCylinder/Ball .................................. 31 3.2 Boundary Condition Report . 32 3.2.1 Boundary Condition: Pressure . 32 3.2.2 Boundary Condition: Stress . 33 3.3 LiquidVolumeReport ........................................ 37 3.3.1 LevelofRefinement ..................................... 38 3.3.2 Frequency........................................... 38 3.3.3 SurfaceTension........................................ 39 4 Summary 41 Bibliography 43 1 Acknowledgment Writing this senior thesis would be impossible without the support from these people. IwanttofirstthankmyresearchadvisorProfessorFredericGibouforintroducingmeto this area. I met him because of a small visualization project, and I’m still so glad that he allows me to further study computational science in his lab. He also supports me with lots of lab resources, which is essential to the research I present in this thesis. I’ll forever remember his insight about this subject, and his email ends with energetic ”Cheers”. Also, I want to thank my graduate student mentor Raphael Egan. He guided me through my learning process. Every time I cannot figure out a concept in the paper or a bug in the code, he kindly gives great illustration, which builds up my ability to a higher level. His familiarity about the computational fluid dynamics always surprises me, along with his lightening fast proofread skill. At last, I really appreciate all the people in the lab for creating a friendly environment to learn and to research. I’m really glad that I got this chance to grow here. 2 Chapter 1 Introduction 1.1 What Are Free Surface Flow? Free surface flow, also known as open channel flow, is one of the fundamental subjects to various branches in computational fluid dynamics (CFD). It describes a situation where the fluid surface is subject to zero tangential stress. This model can be applied to many fluid- gas interface problems. It can also be extended to approximate a multi-phase system. For example, Zappa’s researches about the microwave breaking process involves studying the sea-air interface[20]. He introduces that the flux of the air into the fluid is a↵ected by the slope of the fluid surface and the di↵erence in concentration between air and the fluid. Also for combustion problem, the formation of the droplets and sprays which subsequently burns originates from the interfacial instability[17]. Or even in simpler cases, the collision between two travelling waves, or two water droplets also have the essence of the free surface flow in them. Because of its wide applications, free surface flow has attracted lots of research e↵ort, both theoretical and computational. For example, Meniko↵developed the theory evolution in 2D with periodic boundary conditions[10]. He analyzed both the stable and unstable modes for a flow containing two kinds of fluid. However, due to the complexity of fluid problem, it’s usually impossible to gain a analytic solution under realistic assumptions. The compu- tational approaches is the mainstream in studying this subject. However, even though the computational technology has been developed for a long time, the fluid simulation is still expensive and delicate, which is embodied by various aspects. First it’s still not realizable to run a complex simulation, like multi-phase systems with distinct properties and dynamics, without the power of clusters. Especially in 3D, the number of nodes and equations increases tremendously compared to 2D simulation, which requires advanced parallel CPU processing and large memory storage. On the other hand, because the governing equation is intrinsi- cally non-linear, for di↵erent scenarios, we need to apply di↵erent methods to let the whole simulation converge. Without the proper treatment even to one small section, the simulation tends to either diverge or give nonphysical results. This still poses various obstacles to our research to this relative simple fluid problem. 3 CHAPTER1.INTRODUCTION 1.2.THEMODEL Fig. 1.1: Example of tracking the free surface using structured (left) and unstructured (right) grids 1.2 The Model There are many methods that are applicable in solving the free surface flow and they have their own advantages and disadvantages. In most scenarios, a grid (structured or unstruc- tured) is present (Fig. 1.1), with di↵erent discretization methods to the governing equations. Other methods, like moving particle semi-implicit method[7], does not require a grid in the whole computational domain. The set of governing equations can be usually Navier-Stokes equations solving the velocity of fluid @ (⇢u)+ (⇢u u)= p + σ + f (1.1) @t r· ⌦ r r· or its equivalent expression about the vorticity of the incompressible fluid. @! +(u ) ! =(! )u + ⌫ 2!. (1.2) @t ·r ·r r Here σ is the stress tensor corresponding to the fluid, and f is the external body forces. ! = u is the vorticity of the velocity field.⌫ is the kinematic viscosity. Inr⇥ this research, our interest focuses on the water-air interface of water with horizontal periodic boundary condition. We use adaptive Cartesion grids with adaptive time-stepping to simulate the evolution of the free surface.With such characteristic, we can efficiently use the computational power to the region where the details need to be examined more carefully. In such region, we refine the meshes, which means that we reduce the size of the mesh to increase the accuracy of the simulation. Adaptive time step control also has similar advantages. It allows us to gain more details about the evolution from tn to tn+1 if the motion is complicated at tn. Also, the combination of this two features stabilizes the simulation by satisfying criterion like Courant–Friedrichs–Lewy (CFL) condition given by ∆t u C . (1.3) | |max · ∆x max where ∆t and ∆x are the time and spatial separation. C is the Courant number. This condition ensures satisfactory resolution of advection. 4 CHAPTER1.INTRODUCTION 1.2.THEMODEL We set up the problem using the Navier-Stokes solver built in CASL library, which is developed from P4EST and PETSC. The CASL library is able to setup non-uniform struc- tured grids, called octree (or quatree in 2D). The computational domain is first tessellated with macro-cells called the ”root cells”. Each cell can be divided into 4 (or 8 in 3D) children cells, and the operation can be recursively repeated onto chosen children cells. Illustration are provided in Fig. 1.21 for 2D and 3D cases with one single root cell, minimum level of refinement equal to 1 and maximum level of refinement equal to 4(in 2D) and 3(in 3D). Comparing to uniform grid shown in Fig. 1.1, the number of cells (leaves in the tree) is greatly reduced. Along with the benefits also come the challenges. We need to apply new spatial discretization to solve the governing equations. Also when we march in time, the grid needs to be changed according to the current situation. Therefore, there are a couple key techniques: level set methods, projection methods, etc, which will be introduced in chapter 2. Along with Eq. 1.1 to be solved come boundary conditions. The boundary conditions to be applied onto the irregular free surface determine the quality of the model and its physical relevance. Usually, to maintain the accuracy and stability of the solver, people usually set homogeneous Dirichlet and Neumann boundary conditions for the variables. Specifically, at the free surface Γ, + (p p−) =0, (1.4) − |Γ µ( u nˆ) = 0, (1.5) r · |Γ + where p and p− are the pressure above and below the free surface. These two values come from the interpolation given by the data stored at certain location. These two boundary conditions indicate that across the interface, the pressure jump is zero. However, by Young- Laplace law, we know that this shouldn’t be the case due to the surface tension. Similarly, the gradient of velocity field is related to the stress tensor of the fluid, which can be treated in a more physical way. The old version of the free surface solver implements the boundary condition in these two ways. After the modification, I’ll present the di↵erence brought by setting more realistic boundary conditions in Chapter 3.