Fluid Mechanics Research Laboratory Vibration Induced Droplet Ejection Ashley James Department of Aerospace Engineering and Mechanics University of Minnesota Marc K. Smith George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Supported by NASA Microgravity Research Division and Hoechst Celanese Corp.

Fluid Mechanics Research Laboratory Outline Problem definition Project overview Transducer-drop interaction Numerical simulations Conclusions and future work

Fluid Mechanics Research Laboratory Vertical vibration induces the formation of capillary waves on the free surface. When the forcing amplitude is large enough secondary droplets are ejected from the wave crests. Ejection Schematic

Fluid Mechanics Research Laboratory Literature Faraday (1831) - wave formation due to vibration Benjamin & Ursell (1954) - stability analysis Sorokin (1957) - vibration induced droplet ejection Woods & Lin (1995) - stability on an incline, ejection Lundgren & Mansour (1988) - vibration of an unattached drop Wilkes & Basaran (1997,1999) - vibration of an attached drop Goodridge et al. (1996, 1997) - vibration induced droplet ejection

Fluid Mechanics Research Laboratory Applications Fuel atomization and injection for engine combustors Thermal management and control Electronic cooling Mixing processes Material processing Encapsulation Emulsification

Fluid Mechanics Research Laboratory Heat Transfer Cell for high power electronic cooling (100 W/cm 2 ) Printed Circuit Board Integrated Circuit Condensation Surface Fins Resonance Atomizer

Fluid Mechanics Research Laboratory Low Frequency Forcing Axisymmetric motion Single drop ejected from center 0 to 100 Hz Driver is a rigid piston Experiments performed to determine ejection behavior Focus of simulations Photographs courtesy of Kai Range

Fluid Mechanics Research Laboratory High Frequency Forcing Chaotic motion Multiple droplet ejection across drop surface ~ 1 kHz Driver is a flexible diaphragm Coupling between driver and ejection dynamics Experimental investigation of spray characteristics unforcedejectionatomization

Fluid Mechanics Research Laboratory Close-up of High Frequency Ejection A crater forms on the drop surface. As the crater collapses an upward jet is created. One or more secondary droplets are ejected from the end of the jet. crater Photographs courtesy of Bojan Vukasinovic

Fluid Mechanics Research Laboratory Transducer-Drop Interaction Model

Fluid Mechanics Research Laboratory Amplitude Response Unloaded Transducer 0.16 V 1.85 V 4.06 V

Fluid Mechanics Research Laboratory Effect of Drop Size on Response 0  L 100  L 200  L Driving Voltage: 0.74 V

Fluid Mechanics Research Laboratory Response of System to f = 0.99 Forcing 5.91 V 6.20 V 6.50 V a f

Fluid Mechanics Research Laboratory Response of System to f = 1.04 Forcing 5.91 V 6.20 V 6.50 V 6.79 V f a

Fluid Mechanics Research Laboratory Comparison of Model to Experiment 5.91 V 6.20 V 6.50 V 6.79 V f a Model Experiment

Fluid Mechanics Research Laboratory Response Behavior 0  L 100  L 200  L f < f r f > f r

Fluid Mechanics Research Laboratory Computational Method Transient, axisymmetric, incompressible governing equations. Forcing is an oscillating body force in inertial reference frame. Finite volume discretization on a uniform, staggered grid. Explicit projection method for Navier Stokes solver. Incomplete-Cholesky conjugate gradient method for solution of pressure-Poisson equation.

Fluid Mechanics Research Laboratory Volume of Fluid Method The position of the interface is tracked via a volume fraction, F. The evolution of the volume fraction is governed by a convection equation. The interface is approximated by a straight line in each cell. To prevent false smearing of the interface the volume fraction flux is computed from the straight line approximation.

Fluid Mechanics Research Laboratory Continuum Surface Force The surface tension forces are incorporated as a source term in the momentum equation. Surface cells and interior cells are treated the same. The source term is nonzero only near the interface. The surface tension is distributed over a small region near the computed interface. The curvature is calculated directly from the volume fraction.

Fluid Mechanics Research Laboratory Continuity: Radial momentum: Vertical momentum: Volume fraction: Governing Equations

Fluid Mechanics Research Laboratory Verification Translation of a fluid region. Exact solution of Poisson equation. Poiseuille flow. Transient Couette flow in an annular region. Stability of a drop in equilibrium.

Fluid Mechanics Research Laboratory Parameters Range Viscous effects Forcing amplitude 0 - 5Forcing frequency 0 - 5Gravity effects Ejection Simulations

Fluid Mechanics Research Laboratory Initial and Boundary Conditions Symmetry line Outlet No-slip walls 80 cells 30 cells

Fluid Mechanics Research Laboratory Video Cases Re = 475Re = 10Re = 10Re = 10Re = 10 A = 8.7A = 18A = 20A = 25A = 30  = 1.2  = 1  = 1  = 1  = 1 Bo = 1.3Bo = 0Bo = 0Bo = 0Bo = 0

Fluid Mechanics Research Laboratory Comparison of Simulation and Experiment Re = 475, A = 8.7,  = 1.2, Bo = 1.3 Scale: 1 cm Forcing stepped on Forcing slowly ramped up

Fluid Mechanics Research Laboratory Ejection Simulation - Case 2 Re = 10, A = 18,  = 1, Bo = 0 t = 2.8t = 3t = 3.2t = 3.4t = 3.6t = 3.8t = 4

Fluid Mechanics Research Laboratory Ejection Simulation - Case 3 Re = 10, A = 20,  = 1, Bo = 0 t = 1.8t = 2t = 2.2t = 2.4t = 2.6t = 2.8t = 3

Fluid Mechanics Research Laboratory Ejection Simulation - Case 4 Re = 10, A = 25,  = 1, Bo = 0 t = 0.6t = 0.8t = 1t = 1.2t = 1.4t = 1.6t = 1.8

Fluid Mechanics Research Laboratory Ejection Simulation - Case 5 Re = 10, A = 30,  = 1, Bo = 0 t = 0.8t = 1t = 1.2t = 1.4t = 1.6t = 1.8

Fluid Mechanics Research Laboratory Effect of Forcing Amplitude on Ejection Bo = 0, Re = 10,  = 1

Fluid Mechanics Research Laboratory Effect of Bond Number on Ejection Re = 10, A = 25,  = 1

Fluid Mechanics Research Laboratory Effect of Reynolds Number on Ejection Bo = 0, A = 25,  = 1

Fluid Mechanics Research Laboratory Effect of Forcing Frequency on Ejection Re = 10, Bo = 0, A = 25

Fluid Mechanics Research Laboratory Ejection Threshold Ejection No ejection Simulations Range et al. Goodridge et al. low viscosity Goodridge et al. high viscosity

Fluid Mechanics Research Laboratory Conclusions Although the forcing frequency has a dramatic effect on the response, ejection may occur when a crater collapses to form a spike in both the low and high frequency regimes. The bursting behavior is explained by the coupling of the diaphragm vibration with the changing drop mass. The single degree-of-freedom model with linear droplet ejection is sufficient to describe the system dynamics. Low-frequency ejection is promoted by increasing A, decreasing Bo, increasing Re, or decreasing . The simulated drop behavior and the ejection threshold compare well with experiments.

Fluid Mechanics Research Laboratory Future Work Extend simulations to three dimensions. Improve computational methodology. Investigate the formation of satellite drops. Determine effect of contact line condition. Simulate the vibration of a liquid layer. Improve understanding of high-frequency atomization. Design systems involving high-frequency spray formation.