The 3rd Symposium on Computational Marine Hydrodynamics, Shanghai, China Recent Advancement for High-Fidelity Simulation of Free Surface Flow Cheng LIU chengliu@sjtu.edu.cn State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University
OUTLINE Motivation Numerical Method for Large Density-Ratio Flow • Geometric Method • Algebra Method Surface Tension Model • Continuum Surface Force Model • Sharp Surface Force Model Numerical Simulation for Breaking Wave Conclusion and Ongoing Work High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Motivation Wave Breaking Water Entry Exp. Ocean Sphere Wedge Rojas & Loewen (2010) Vincent L et al, 2018 • Void fraction • Energy dissipation PIV • Bubble/Droplet distribution High Speed Camera Delaware air – sea interaction tank High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Motivation Challenge Bubble Flows generated by Wave Breaking (Fraser, 2017) Physical • Large density ratio • Micro-bubbles/droplets ( 𝟐𝟏 − 𝟐𝟏𝟏𝝂𝒏 ) • Turbulent Flow Numerical Bubble Size Distribution Number of bubbles • Fine mesh resolution Wide-Range of Scales • Robust two-phase model Scale 𝑃(1𝑛𝑛) 𝑃(1𝑛) 𝑃(1𝜈𝑛) High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Motivation Numerical Strategy BAMR Ex. Straight Liquid Jet Spray Fractional Step Method High-Resolution Scheme Block-Structured AMR Two-Phase Treatment Interface Capturing Surface Tension Model Scheme SSF Model CSF Model Curvature Estimation √ √ Coupled LS-VOF HF function (SDF field) √ VOF-PLIC HF function (VOF field) √ THINC/SW HF function (VOF field) High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Numerical Method for Large Density-Ratio Flow Problem Description Numerical Instability comes from • Discretization of pressure gradient • Decoupling of pressure-velocity • Inconsistent advection of Mass/momentum Maybe covered by • Small density-ratio • Large physical viscosity • Surface tension dominated High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Numerical Method for Large Density-Ratio Flow Momentum-Mass Consistent Scheme Corrected by mass flux Solution 1 : Geometric Method +… Single-phase cells Interfacial cells Solution 2 : Algebraic Method Staggered grid Dual grid for VOF Collocated grid High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Numerical Method for Large Density-Ratio Flow Numerical Validation Case (1): Dam Breaking Present Momentum-Mass Consistent Scheme Original Scheme Numerical Validation Case (2): Advection of Droplet Initial volume-fraction field 𝛽 = 0 𝑗𝑜 𝑏𝑡 1 𝑗𝑜 𝑚𝑗𝑟𝑣𝑗𝑒 Initial velocity field 0,0 𝑗𝑜 𝑏𝑡 𝐯 = 0 , 𝑣 𝑧 0 𝑣 𝑦 𝑗𝑜 𝑚𝑗𝑟𝑣𝑗𝑒 High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Numerical Method for Large Density-Ratio Flow Numerical Validation 𝐸 𝑀𝑗𝑟𝑣𝑗𝑒 = 10 6 Case (2): Advection of Droplet 𝐸 𝑏𝑡 Bussmann M et al. (2002) Final l Shape of f Drople let Fuster D, Arrufat T, S. Zaleski (2019) Present Simulation Result High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Numerical Method for Large Density-Ratio Flow Numerical Validation Case (3): Two-Dimensional Jet Flow Present Simulation Experimental Setup Park J , Huh K Y , Li X , et al. Experimental investigation on cellular breakup of a planar liquid sheet from an air-blast nozzle. Physics of Fluids, 2004, 16(3):625. High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Numerical Method for Large Density-Ratio Flow Numerical Validation Case (3): Two-Dimensional Jet Flow Present Simulation From Park J et al. Exp. Raessi M , Pitsch H . Consistent mass and momentum transport for simulating incompressible interfacial flows with large density ratios using the level set method. Computers & Fluids, 2012 High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Surface Tension Model (CSF) Continuum Surface Force (CSF) Model 𝑞 = 𝜏𝐨𝜀 𝑡 = 𝜏𝛼𝐼 𝜗 𝐲 − 𝐲 𝑡 𝐼 𝜗 ≡ 𝑤𝑝𝑚𝑣𝑛𝑓 𝑔𝑠𝑏𝑑𝑢𝑗𝑝𝑜 𝐷 • Add ST force effect: 𝑗𝑔 𝐷 𝑗,𝑘,𝑙 − 𝐷 𝑗−1,𝑘,𝑙 > 𝜗, 𝐷 𝑗,𝑘,𝑙 − 𝐷 𝑗−1,𝑘,𝑙 1 𝑣 𝑗−1 2,𝑘,𝑙 = 𝑣 𝑗−1 2,𝑘,𝑙 − ∆𝑢𝜏 𝑗−1 ∙ , 1 ∆𝑦 2,𝑘,𝑙 2 𝜍 𝑗,𝑘,𝑙 + 𝜍 𝑗−1,𝑘,𝑙 averaged from 𝑗−1,𝑘,𝑙 and 𝑗,𝑘,𝑙 • Solve pressure Eq. to recover pressure jump Smoothed High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Surface Tension Model (CSF) Continuum Surface Force Model (CSF) Improved Height function method for curvature estimation Local curvature Local curvature True from Eq. (2) from Eq. (1) Whether 𝐷 𝑡,𝑢,𝑛 is monotonic in 𝑛 = [𝑙 − 3, 𝑙 + 3] Approximate Identify the False quadratic interface inactive cells Disperse curvature values along anti/normal direction 𝑛=𝑙+3 𝐼 𝑡,𝑢 = 𝐷 𝑡,𝑢,𝑛 𝑛=𝑙−3 2 + 𝐼 𝑧𝑧 𝐼 𝑦 2 − 2𝐼 𝑦𝑧 𝐼 𝑦 𝐼 𝑧 𝑗,𝑘,𝑙 = 𝐼 𝑦𝑦 + 𝐼 𝑧𝑧 + 𝐼 𝑦𝑦 𝐼 𝑧 Eq. (1): 2 1.5 2 + 𝐼 𝑧 1 + 𝐼 𝑦 MLS interface-fit 𝐺 𝐲 = 𝑑 1 𝑦 2 + 𝑑 2 𝑧 2 + 𝑑 3 𝑦𝑧 + 𝑑 4 𝑦 + 𝑑 5 𝑧 + 𝑑 6 2 + 1 + 𝑑 2 𝑑 4 2 + 1 − 𝑑 3 𝑑 4 𝑑 5 = 2 𝑑 1 𝑑 5 Eq. (2): 1.5 2 + 𝑑 4 2 + 1 𝑑 5 𝜖 𝐼 = 0 𝑗𝑔 𝑑𝑓𝑚𝑚 𝑗,𝑘,𝑙 ∈ 𝕁 ∪ 𝕄 , 𝜖𝜐 + 𝐼 𝐨 ∙ 𝛼 = 0 1 𝑗𝑔 𝑑𝑓𝑚𝑚 𝑗,𝑘,𝑙 ∈ 𝐻, High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Surface Tension Model (CSF) Example : Curvature estimation of a circular interface Interfacial cells 𝑴 𝟑 and 𝑴 ∞ error △ : smoothed VOF [28], ◇ : re-distancing function based on VOF [28], □ : height function (HF) method of S.J. Cummins et al., 2005, ● : present HF, – – – : second order, – ∙ – ∙ – : first order. High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Surface Tension Model (CSF) Example : Curvature estimation of a circular interface Interfacial/neighbor cells 𝑴 𝟑 and 𝑴 ∞ error Fig. Curvature estimation errors ( L ∞ ) of interfacial cells and extrapolation cells (as shown in Fig. 3) with ( ● ) or without ( ○ ) curvature populating, – – – : second order, – ∙ – ∙ – : first order. High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Surface Tension Model (CSF) Example : Capillary Breakup of Liquid Jet Rep Repeat eat th the e th theor eoret etica ical study by L. l study by L. Pa Para ramet eters ers Raylei Ra yleigh(1 gh(1892 892)* )* and and C. . Weber Weber(19 (1931 31)** )** • Domain 𝑦, 𝑧, 𝑨 ∈ [−1,1] . Three cases with Laplace number: 2000, • Water/air density & viscosity ratio: 238.34, 23.834. 𝜍 𝑚 /𝜍 = 1000.0/1.2 , 𝜈 𝑚 /𝜈 = 1.0 × 10 −3 /1.8 × 10 −5 • Initial free surface profile: cylindrical interface with sinusoidal perturbation, 𝑠 𝑦 = 𝑠 0 1 + 𝜁 sin 𝑙𝑦 , 𝑠 0 = 0.2 , 𝜁 = 0.02 and 𝑙 = 𝜌 . • Periodical b. c. for pressure and velocity 𝜓 𝑛𝑏𝑦 = 𝑠 𝑛𝑏𝑦 − 𝑠 0 , 𝜓 𝑛𝑗𝑜 = 𝑠 0 − 𝑠 𝑛𝑗𝑜 𝜁𝑠 0 𝜁𝑠 0 𝑛𝑏𝑦 , 𝑠 𝑛𝑗𝑜 : 最大，最小半径 𝑠 *Rayleigh L. XVI. On the instability of a cylinder of viscous liquid under capillary force. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science , 1892, 34(207): 145-154. **Weber C. Zum zerfall eines flüssigkeitsstrahles. ZAMM‐Journal of Applied Mathematics and Mechanics/ Zeitschrift für Angewandte Mathematik und Mechanik , 1931, 11(2): 136-154. High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Surface Tension Model (CSF) Example : Capillary Breakup of Liquid Jet (a) (b) Fig. Comparison of relative deformations of the interface. (a). Maximum radius variation 𝜓 𝑛𝑏𝑦 , (b). Minimum radius variation 𝜓 𝑛𝑗𝑜 . – – – : L. Rayleigh [45] ( 𝑀𝑏 = 2000 ), ∙∙∙∙∙: C. Weber [46] ( 𝑀𝑏 = 238.34 ), ∙–∙–∙: C. Weber [46] ( 𝑀𝑏 = 23.834 ), ○ : present BAMR ( 𝑀𝑏 = 2000 ), △ : present BAMR ( 𝑀𝑏 = 238.34 ), □ : present BAMR ( 𝑀𝑏 = 23.834 ). High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
Surface Tension Model (CSF) Example : Droplet Collisions Rep Repeat eat th the e exper experim iment ental s al stu tudy dy by by Qian and Law, 1 Qian and Law, 1997 997 * Four collision mode. Two equal-sized droplets ) 𝜍 𝑚 𝐸(2𝑉 0 𝜇 = 𝑐 𝐸 , 𝑆𝑓 = , 𝜈 𝑚 𝜃 = 𝑥 𝑋𝑓 = 𝜍 𝑚 𝐸 2𝑉 0 2 , 𝐸 𝜏 Table. Fluid properties for present droplet collisions. High-Fidelity Numerical Simulation for Free Surface Flow —— Cheng LIU
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