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The Numerical Prediction of Droplet Deformation and Break-Up Using the Godunov Marker-Particle Projection Scheme F. Bierbrauer and T. N. Phillips School of Mathematics, Cardiff University Senghennydd Road, Cardiff, CF24 4AG United Kingdom

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Contents Purpose of the StudyPurpose of the Study Droplet Break-UpDroplet Break-Up Modelling the Droplet Break-Up ProcessModelling the Droplet Break-Up Process The Numerical MethodThe Numerical Method Simulations of Droplet Break-UpSimulations of Droplet Break-Up ConclusionsConclusions ReferencesReferences

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Purpose of the Study Importance in industrial ProcessesImportance in industrial Processes –Ink jet printing –Fuel sprays –Fire quenching –Chemical reaction rates Can the Process be Modelled SuccessfullyCan the Process be Modelled Successfully –Construction of a mathematical model –Should be robust and accurate

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Droplet Break-Up Governed by the initial Weber numberGoverned by the initial Weber number –where a is the density of the ambient fluid, D i the initial diameter of the drop, U r the initial relative velocity between the ambient fluid and the drop and ad is the surface tension Viscous effects play no role when the Ohnesorge numberViscous effects play no role when the Ohnesorge number –Where d is the viscosity of the drop and d the density of the drop

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Droplet Break-Up Modes 1.Vibrational Break-Up: 2.Bag Break-Up: 3.Bag and Stamen Break-Up: 4.Sheet Stripping: 5.Catastrophic Break-Up

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Modelling the Droplet Break- Up Process Must solve a two-phase flow problemMust solve a two-phase flow problem –e.g. a rain droplet falling through air Accurately track fluid interfacesAccurately track fluid interfaces –Interface tracking algorithm Solve the Navier-Stokes EquationsSolve the Navier-Stokes Equations –Often incompressible, includes surface tension forces –Usually involves inflow and outflow boundary conditions

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The Numerical Method The method must deal withThe method must deal with –Discontinuously varying fluid densities and viscosities –Severe stretching and rupture of fluid interfaces Single Domain or One-Field MethodSingle Domain or One-Field Method –Fluid variables are field variables in a single domain –Includes both bulk fluid phases and interface geometry –The interfaces represent a discontinuity in density or viscosity –Interfacial conditions become part of the governing equations –Can define these phases by a phase indicator function C, often the volume fraction

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Grid Volume Fraction Volume fraction information within grid cells C 1 – blue fluid, C 2 – yellow fluid C 1 = 0 C 2 = 1 C 1 = 0 C 2 = 1 C 1 = 0.2 C 2 = 0.8 C 1 = 0.7 C 2 = 0.3 C 1 = 0.95 C 2 = 0.05 C 1 = 1 C 2 = 0 C 1 = 0.7 C 2 = 0.3 C 1 = 0.3 C 2 = 0.7

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Interface Capturing Volume TrackingVolume Tracking –The interface is only implicitly tracked, it is captured –The interface is the contrast created by the difference in phase, e.g. MAC method, Marker-Particle method –Or it can be geometrically re-constructed, e.g. VOF methods SLIC, PLIC

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Density and Viscosity In two-phase flow the density and viscosity fields, using C 1 +C 2 =1 with C=C 1,In two-phase flow the density and viscosity fields, using C 1 +C 2 =1 with C=C 1, are:are: Where the i and i are the constant viscosities and densities within each fluid phaseWhere the i and i are the constant viscosities and densities within each fluid phase We have used a serial mean for the density and a parallel mean for the viscosityWe have used a serial mean for the density and a parallel mean for the viscosity

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Eulerian-Lagrangian Methods Makes use of aspects of both Eulerian and Lagrangian methods Particle-Mesh methods e.g. GMPPS –use an Eulerian fixed grid to store velocity and pressure information –Use Lagrangian particles to keep track of fluid phase and thereby density and viscosity

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Fluid Particle Colour Each fluid phase (m) has a set of marker particles (p) located at position (x p, y p )Each fluid phase (m) has a set of marker particles (p) located at position (x p, y p ) Every marker particle of the m th set is permanently assigned a colour such thatEvery marker particle of the m th set is permanently assigned a colour such that Particles are moved with interpolated grid velocitiesParticles are moved with interpolated grid velocities Particle colours are used to update the volume fraction C through interpolation at each time stepParticle colours are used to update the volume fraction C through interpolation at each time step

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Two-Phase Navier-Stokes Equations

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Non-Dimensionalised Domain

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Boundary Conditions Top:Top: Bottom:Bottom: InflowInflow: OutflowOutflow

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Liquid-Liquid System: Vibrational Break-Up Density Contour Plot: corium droplet in water, We i = 13, U i = 0.816 m/s, d = 9200 kg/m 3, d = 0.000282 Pas, D = 9 mm, = 0.45 N/m d = 9200 kg/m 3, d = 0.000282 Pas, D = 9 mm, = 0.45 N/m w = 996 kg/m 3, w = 0.000854 Pas, 60X30 grid w = 996 kg/m 3, w = 0.000854 Pas, 60X30 grid

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Liquid-Liquid System: Bag Break-Up Density Contour Plot: corium droplet in water, We i = 26, U i = 1.155 m/s, d = 9200 kg/m3, d = 0.000282 Pas, D = 9 mm, = 0.45 N/m d = 9200 kg/m3, d = 0.000282 Pas, D = 9 mm, = 0.45 N/m w = 996 kg/m3, w = 0.000854 Pas, 60X30 grid w = 996 kg/m3, w = 0.000854 Pas, 60X30 grid

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Liquid-Liquid System: Sheet Stripping Break-Up Density Contour Plot: corium droplet in water, We i = 177, U i = 2.98 m/s, d = 9200 kg/m 3, d = 0.000282 Pas, D = 9 mm, = 0.45 N/m w = 996 kg/m 3, w = 0.000854 Pas, 60X30 grid d = 9200 kg/m 3, d = 0.000282 Pas, D = 9 mm, = 0.45 N/m w = 996 kg/m 3, w = 0.000854 Pas, 60X30 grid

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Conclusions and Future Work Currently:Currently: –Course grids, simple outflow conditions, simple Poisson solver Given the current limitations the GMPPS is able to qualitatively model some aspects of the break-up processGiven the current limitations the GMPPS is able to qualitatively model some aspects of the break-up process Refine the grids, implement better outflow conditions and increase the accuracy of the Poisson solver for variable density coefficientsRefine the grids, implement better outflow conditions and increase the accuracy of the Poisson solver for variable density coefficients Move to 3D simulationsMove to 3D simulations

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References Bierbrauer, F., Zhu, S.-P., A Numerical Model for Multiphase Flow Based on the GMPPS Formulation, Part I: Kinematics, to appear in Computers and Fluids, 2007.Bierbrauer, F., Zhu, S.-P., A Numerical Model for Multiphase Flow Based on the GMPPS Formulation, Part I: Kinematics, to appear in Computers and Fluids, 2007. Bierbrauer, F., Zhu, S.-P., A Numerical Model for Multiphase Flow Based on the GMPPS Formulation, Part II: Dynamics, submitted Computers and Fluids, 2007Bierbrauer, F., Zhu, S.-P., A Numerical Model for Multiphase Flow Based on the GMPPS Formulation, Part II: Dynamics, submitted Computers and Fluids, 2007. Rider, W.J., Kothe, D.B., Mosso, S.J., Cerutti, J.H., Hochstein, J.I., Accurate Solution Algorithms for Incompressible Multiphase Flows, in AIAA Paper No. 95-0699, 33 rd Aerospace Sciences Meeting, Reno, NV, 1995Rider, W.J., Kothe, D.B., Mosso, S.J., Cerutti, J.H., Hochstein, J.I., Accurate Solution Algorithms for Incompressible Multiphase Flows, in AIAA Paper No. 95-0699, 33 rd Aerospace Sciences Meeting, Reno, NV, 1995 Puckett, E.G., Almgren, A., Bell, J.B., Marcus, D.L., Rider, W.J., A High- Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows, J. Comput. Phys., 130 (1997), 269-282Puckett, E.G., Almgren, A., Bell, J.B., Marcus, D.L., Rider, W.J., A High- Order Projection Method for Tracking Fluid Interfaces in Variable Density Incompressible Flows, J. Comput. Phys., 130 (1997), 269-282 Almgren, A.S., Bell, J.B., Szymczak, W.G., A Numerical Method for the Incompressible Navier-Stokes Equations Based on an Approximate Projection, SIAM J Sci Comput, 17 (1996), 358-369.Almgren, A.S., Bell, J.B., Szymczak, W.G., A Numerical Method for the Incompressible Navier-Stokes Equations Based on an Approximate Projection, SIAM J Sci Comput, 17 (1996), 358-369. Brackbill, J.U., Kothe, D.B., Zemach, A, A Continuum Model for Modeling Surface Tension, J. Comput. Phys., 100 (1992), 335-354.Brackbill, J.U., Kothe, D.B., Zemach, A, A Continuum Model for Modeling Surface Tension, J. Comput. Phys., 100 (1992), 335-354. Nomura, K., Koshizuka, S., Oka, Y., Obata, H., Numerical Analysis of Droplet Breakup Behavior Using Particle Method, J. Nucl. Sci. Technol, 38 (2001), 1057-1064.Nomura, K., Koshizuka, S., Oka, Y., Obata, H., Numerical Analysis of Droplet Breakup Behavior Using Particle Method, J. Nucl. Sci. Technol, 38 (2001), 1057-1064. Zhu, J., The Second-Order Projection Method for the Backward-Facing Step Flow, J. Comput. Phys., 117 (1995), 318-331.Zhu, J., The Second-Order Projection Method for the Backward-Facing Step Flow, J. Comput. Phys., 117 (1995), 318-331.

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