A Novel Methodology for Simulating Contact-Line Behavior in Capillary-Driven Flows Thesis by Gerry Della Rocca In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2014 (Defended May 20, 2014) ii (cid:13)c 2014 Gerry Della Rocca All Rights Reserved iii For my parents. iv Acknowledgments An investment in knowledge pays the best interest. — Benjamin Franklin I would first like to thank my doctorate advisers. My official adviser, Professor G. Blanquart, had a critical guiding role in this work despite (or perhaps because of) all our friction and email storms. I really grew as an individual and as a researcher working with him. While most people are limited to a single adviser, I had the privilege of two additional unofficial advisers. I worked with Professor S. M. Troian for approximately a year in the middle of my doctorate and while none of that research appears in the present work, my experience with her was invaluable. It may have arguably been the turning point in my graduate school career. Professor T. Colonius has been a great counselor; I could always count on blunt, honest, and fair criticism even if I didn’t want to hear it. GraduateschoolisundoubtedlyastressfulexperienceandIwouldn’thavemadeitthroughwith- out many people, too many to name. First year at Caltech was mentally and emotionally taxing; I am indebted to the classmates with whom I went through it. In later years, my lab mates (Sid, Jason, Phares, Yuan) and pseudo-lab mates (Vedran, Jomela) were always willing to empathize, givefeedback, andsharealaugh. Finally, Ihavebeenveryfortunatetohavehadanawesomegroup of friends, e.g., Tom / Caroline / Dex / Max / Archie, Nick / Andreas, Jay / Anu, Ding / Bilin, Francesco, Alex, Nick, Dylan. I would like to single out Tom and Serena (Tomerena) who were the first people I met at Caltech and were steady, insistent proponents of a work-life balance. v Myelderbrother,Joe,hashadadynamicimpactonmylife. Icannotclaimtohavebeenagood youngerbrother; Icertainlydidn’tlookuptohimgrowingupandsayingaloathedhimwouldn’tbe far from the truth. I pushed myself to beat him in every way. This competition created the person I am today. Surprisingly, I realized a few years ago that, regardless of our differences, I’m really proud to call him my brother. These are words I never thought I’d write. Lastly,IcouldnothavegottenwhereIamwithoutmyparents. Myparents(VinandPam)have always emphasized education’s importance. I learned at an early age that, while they would never buy me video games, they would buy me any book I wanted. Additionally, they sacrificed much to get me through an expensive high school and college. I hope I can be worthy of this exceptional investment and support. This material is based upon work supported by the National Science Foundation Graduate Re- search Fellowship under Grant No. DGE-1144469. vi Abstract Despite the wide swath of applications where multiphase fluid contact lines exist, there is still no consensus on an accurate and general simulation methodology. Most prior numerical work has im- posedoneofthemanydynamiccontact-angletheoriesatsolidwalls. Suchapproachesareinherently limited by the theory accuracy. In fact, when inertial effects are important, the contact angle may behistorydependentand,thus,anysinglemathematicalfunctionisinappropriate. Giventheselim- itations, the present work has two primary goals: 1) create a numerical framework that allows the contactangletoevolvenaturallywithappropriatecontact-linephysicsand2)developequationsand numerical methods such that contact-line simulations may be performed on coarse computational meshes. Fluid flows affected by contact lines are dominated by capillary stresses and require accurate curvature calculations. The level set method was chosen to track the fluid interfaces because it is easy to calculate interface curvature accurately. Unfortunately, the level set reinitialization suffers fromanill-posedmathematicalproblematcontactlines: a“blindspot”exists. Standardtechniques to handle this deficiency are shown to introduce parasitic velocity currents that artificially deform freely floating (non-prescribed) contact angles. As an alternative, a new relaxation equation reini- tialization is proposed to remove these spurious velocity currents and its concept is further explored with level-set extension velocities. vii To capture contact-line physics, two classical boundary conditions, the Navier-slip velocity boundary condition and a fixed contact angle, are implemented in direct numerical simulations (DNS). DNS are found to converge only if the slip length λ is well resolved by the computational mesh. Unfortunately, since λ is often very small compared to fluid structures, these simulations are not computationally feasible for large systems. To address the second goal, a new methodology is proposed which relies on the volumetric-filtered Navier-Stokes equations. Two unclosed terms, an average curvature κ¯ and a viscous shear VS, are proposed to represent the missing microscale physics on a coarse mesh. All of these components are then combined into a single framework and tested for a water droplet impacting a partially-wetting substrate. Very good agreement is found for the evolution of thecontactdiameterintimebetweentheexperimentalmeasurementsandthenumericalsimulation. Such comparison would not be possible with prior methods, since the Reynolds number Re and capillary number Ca are large. Furthermore, the experimentally approximated slip length ratio (cid:15) is well outside of the range currently achievable by DNS. This framework is a promising first step towards simulating complex physics in capillary-dominated flows at a reasonable computational expense. viii Contents Acknowledgments iv Abstract vi Contents viii List of Figures xiii List of Tables xxi Nomenclature xxii 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Prior contact-angle models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Molecular kinetic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Hydrodynamic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 Non-unique contact angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Numerical methods for multiphase flow and contact lines. . . . . . . . . . . . . . . . 7 1.3.1 Atomistic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2 Continuum methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.2.1 Front-tracking methods . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2.2 Front-capturing methods . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 ix 1.5 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2 Numerical methods 18 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Fluid mechanics numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.1 Implementation of the Navier-Stokes equations . . . . . . . . . . . . . . . . . 20 2.3.2 Multiphase flow treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3.3 Numerical stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Level set formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.1 Level set advection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.4.2 Reinitialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.4.3 Narrow band methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.4.4 Curvature calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 An improved method for level set reinitialization at a contact line 1 31 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Blind spot methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2.2 Transport scheme errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2.3 WENO stencils with wall ghost values . . . . . . . . . . . . . . . . . . . . . . 38 3.2.3.1 Zero Neumann boundary condition . . . . . . . . . . . . . . . . . . 38 3.2.3.2 Extrapolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.3.3 Ghost interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.2.4 Offset finite-difference stencils . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.3 Relaxation equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.4 Method extensions and limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 x 3.5 Example 1: sliding droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5.2 Evolution of a 2D gravity-driven droplet on a wall . . . . . . . . . . . . . . . 52 3.5.3 Evolution of a 3D gravity-driven droplet on a Wall . . . . . . . . . . . . . . . 53 3.6 Example 2: wedge of fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.6.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.6.2 Evolution of a fluid wedge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4 Extension velocities and angle propagation 58 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 Construction of extension velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2.1 Implementation outside of the blind spot . . . . . . . . . . . . . . . . . . . . 61 4.2.2 Implementation in the blind spot . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.2.2.1 Fast Marching Method (FMM) in the blind spot . . . . . . . . . . . 64 4.2.2.2 Propagation of isocontour angle θ˜ . . . . . . . . . . . . . . . . . . . 65 4.3 Spurious currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3.2 Velocity error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 Example: Pinned droplet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.4.1 Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.4.3 Solution h¯ behavior for different inputs, Bo and x¯ . . . . . . . . . . . . . . 72 W 0 4.4.4 Simulation comparison to the exact solution . . . . . . . . . . . . . . . . . . . 72 4.5 Extensions and limitations of blind spot extension velocities . . . . . . . . . . . . . . 73 4.6 Reinitialization with angle propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76