Adaptive Mesh Refinement in Filling Simulations Based on Level Set - PowerPoint PPT Presentation

Adaptive Mesh Refinement in Filling Simulations Based on Level Set RICAM Special Semester | Space-Time Methods for PDEs PhD Student: M.Sc. Violeta Karyofylli Advisor: Prof. Marek Behr, Ph.D. This is a joint work with Markus Frings , Loic Wendling and Dr. Stefanie Elgeti at the Chair for Computational Analysis of Technical Systems

Table of Contents Governing Equations Simplex Space-Time Meshes Static Bubble (2D) Rising Bubble (2D) Rising Droplet (3D) Step Cavity Benchmark Case (2D) Coat Hanger Distributer and Rectangular Cavity Benchmark Case (2D) Conclusion 2 of 19 Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

𝜖𝜚 on Ω 𝑗 (𝑢), ∀𝑢 ∈ [0, 𝑈] ∀𝑢 ∈ [0, 𝑈] ∇ ⋅ 𝐯 = 0 ∀𝑢 ∈ [0, 𝑈] Ω 𝑗 (𝑢), in Governing Equations • Two-phase flow – Incompressible, transient, isothermal – Two immiscible Newtonian phases melt phase boundary e i m t s p a c e air Temporal and spatial refinement in the vicinity of moving interfaces. • Navier-Stokes equations: (1) 𝜍 𝑗 (𝜖𝐯 𝜖𝑢 + 𝐯 ⋅ ∇𝐯 − 𝐠) − ∇ ⋅ 𝜏 𝑗 = 0 (2) • Level-Set equation: (3) 𝜖𝑢 + 𝐯 ⋅ ∇𝜚 = 0 3 of 19 Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

Simplex Space-Time Meshes 3d6n 4d8n 3d4n 4d5n (a) Prism-type space-time elements. (b) Simplex-type space-time elements. Comparison of prism- and simplex-type space-time elements. Black nodes correspond to 𝑢 𝑜 and white nodes correspond to 𝑢 𝑜+1 . [1] • Advantages of Simplex Space-Time Meshes – Different temporal refinement in different parts of the domain – Connection of disparate spatial meshes 1 Behr, M. (2008). Simplex space-time meshes in finite element simulations. International Journal for Numerical Methods in Fluids, 57(9), 1421-1434 4 of 19 Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

0.5 Ω 1 1.0 1.0 0.5 0.5 Ω 2 Static Bubble (2D) • Description 2 – A perfectly stationary circular bubble at equilibrium – Laplace-Young law: 𝑞 𝑗𝑜 = 𝑞 𝑝𝑣𝑢 + 𝜏/𝑠 – Surface tension coefficient: 𝜏 = 𝟤 𝗅𝗁/𝗍 𝟥 – Density of both phases: 𝜍 1 = 𝜍 2 – Viscosity of both phases: 𝜈 1 = 𝜈 2 Static bubble in 2D: Computational domain. • Time Discretization • Boundary Conditions – Time slab size: Δ𝑢 = 𝟣.𝟣𝟤 𝗍 – No-slip boundary conditions at all – Total number of time slabs: 𝟤𝟥𝟨 boundaries – A zero reference pressure at one corner 2 Hysing, S. (2006). A new implicit surface tension implementation for interfacial flows. International Journal for Numerical Methods in Fluids, 51(6), 659-672 5 of 19 Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

0.4 𝑄𝑠𝑓𝑡𝑡𝑣𝑠𝑓 2 1 1 0.8 0.6 0 0.2 0 5 𝑦 𝑄𝑠𝑓𝑡𝑡𝑣𝑠𝑓 4 𝑦 5 4 3 2 1 1 0.8 0.6 0.4 0.2 3 Static Bubble (2D) prismatic prismatic simplex simplex (a) ℎ = 1/80 (b) ℎ = 1/160 Pressure cut-lines at 𝑧 = 𝟣.𝟨 after 𝟤𝟥𝟨 time slabs for two different mesh sizes. 6 of 19 Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

0.5 Ω 1 1.0 2.0 0.5 0.5 Ω 2 Rising Bubble (2D) • Description 2 – A bubble rising in a heavier fluid – Surface tension coefficient: 𝜏 = 𝟥𝟧.𝟨 𝗅𝗁/𝗍 𝟥 – Density of both phases: 𝜍 1 = 𝟤𝟣𝟣𝟣 𝗅𝗁/𝗇 𝟦 ; 𝜍 2 = 𝟤𝟣𝟣 𝗅𝗁/𝗇 𝟦 – Viscosity of both phases: Rising bubble in 2D: Computational domain. 𝜈 1 = 𝟤𝟣 𝗅𝗁/𝗇/𝗍; 𝜈 2 = 𝟤 𝗅𝗁/𝗇/𝗍 – Gravity: 𝑔 𝑧 = −𝑕 = −𝟣.𝟬𝟫 𝗇/𝗍 𝟥 • Boundary Conditions • Time Discretization – No-slip boundary conditions at the top and bottom boundary – Time slab size: Δ𝑢 = 𝟣.𝟣𝟤 𝗍 – Slip boundary conditions along the – Total number of time slabs: 𝟦𝟣𝟣 vertical walls – Zero pressure specified at the upper boundary 2 Hysing, S. (2006). A new implicit surface tension implementation for interfacial flows. International Journal for Numerical Methods in Fluids, 51(6), 659-672 7 of 19 Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

Rising Bubble (2D) (a) 𝑢 = 𝟤.𝟣 𝗍 (b) 𝑢 = 𝟥.𝟣 𝗍 (c) 𝑢 = 𝟦.𝟣 𝗍 Bubble position at various time instances, obtained using a prism-type space-time discretization and a simplex-type space-time discretization. Light grey color corresponds to the prismatic space-time discretization (left half of the bubble) and dark grey color corresponds to simplex-based space-time discretization (right half of the bubble). 8 of 19 Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

1.3 𝑧 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 0.1 𝑦 Rising Bubble (2D) prismatic simplex TP2D Comparison of the bubble at 𝑢 = 𝟦.𝟣 𝗍 with reference data published by [3]. 3 Hysing, S. R., Turek, S., Kuzmin, D., Parolini, N., Burman, E., Ganesan, S., & Tobiska, L. (2009). Quantitative benchmark computations of two-dimensional bubble dynamics. International Journal for Numerical Methods in Fluids, 60(11), 1259-1288 9 of 19 Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

2.4 1.1 2.8 0.95 2.5 1 1.05 1.1 𝑢 𝑧 − 𝑑𝑓𝑜𝑢𝑓𝑠 𝑝𝑔 𝑛𝑏𝑡𝑡 𝑧 − 𝑑𝑓𝑜𝑢𝑓𝑠 𝑝𝑔 𝑛𝑏𝑡𝑡 𝑢 1.2 1 3 0.9 0.8 0.7 0.6 0.5 3 2.5 2 1.5 1 0.5 0 2.9 Rising Bubble (2D) • Comparison between the results of in-house solver and reference data 3 prismatic prismatic simplex simplex TP2D TP2D 2.6 2.7 (a) Over the whole simulation time. (b) Between the time instances 𝑢 1 = 𝟥.𝟧 𝗍 and 𝑢 2 = 𝟦.𝟣 𝗍 . The position of the center of mass X 𝑑 in 𝑧 -direction of the rising bubble in 2D. 3 Hysing, S. R., Turek, S., Kuzmin, D., Parolini, N., Burman, E., Ganesan, S., & Tobiska, L. (2009). Quantitative benchmark computations of two-dimensional bubble dynamics. International Journal for Numerical Methods in Fluids, 60(11), 1259-1288 10 of 19 Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

0.5 Ω 1 1.0 2.0 0.5 0.5 0.5 Ω 2 Rising Droplet (3D) • Description 4 – A droplet rising in a heavier fluid – Surface tension coefficient: 𝜏 = 𝟥𝟧.𝟨 𝗅𝗁/𝗍 𝟥 – Density of both phases: 𝜍 1 = 𝟤𝟣𝟣𝟣 𝗅𝗁/𝗇 𝟦 ; 𝜍 2 = 𝟤𝟣𝟣 𝗅𝗁/𝗇 𝟦 – Viscosity of both phases: 𝜈 1 = 𝟤𝟣 𝗅𝗁/𝗇/𝗍; 𝜈 2 = 𝟤 𝗅𝗁/𝗇/𝗍 – Gravity: 𝑔 𝑧 = −𝑕 = −𝟣.𝟬𝟫 𝗇/𝗍 𝟥 Rising droplet in 3D: Computational domain. • Time Discretization • Boundary Conditions – Time slab size: Δ𝑢 = 𝟣.𝟣𝟤 𝗍 – No-slip boundary conditions at all the – Total number of time slabs: 𝟦𝟣𝟣 boundaries – Zero pressure specified at the upper boundary 4 Adelsberger, J., Esser, P., Griebel, M., Groß, S., Klitz, M., & Rüttgers, A. (2014, March). 3D incompressible two-phase flow benchmark computations for rising droplets. In Proceedings of the 11th World Congress on Computational Mechanics (WCCM XI), Barcelona, Spain, 2014 11 of 19 Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016

Rising Droplet (3D) (a) 𝑢 = 𝟤.𝟣 𝗍 (b) 𝑢 = 𝟥.𝟣 𝗍 (c) 𝑢 = 𝟦.𝟣 𝗍 Droplet position at various time instances, obtained using a prism-type space-time discretization and a simplex-type space-time discretization. Light grey color corresponds to the prismatic space-time discretization (left half of the droplet) and dark grey color corresponds to simplex-based space-time discretization (right half of the droplet). 12 of 19 Prof. Marek Behr, Ph.D. | M.Sc. Violeta Karyofylli Chair for Computational Analysis of Technical Systems | RWTH Aachen University RICAM Special Semester | Space-Time Methods for PDEs | 08.11.2016