An Experimental Study of Worthington Jet Formation After Impact of Solid Spheres Jenna Marie ...

91 Pages·2011·12.11 MB·English

An Experimental Study of Worthington Jet
Formation After Impact of Solid Spheres
by
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
Jenna Marie McKown
JUL 2 9 2011
B.S., Mechanical and Ocean Engineering
Massachusetts Institute of Technology (2010) _.BRARIES
-
Submitted to the Department of Mechanical Engineering ARCHIVES
in partial fulfillment of the requirements for the degree of
Master of Science in Ocean Engineering
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2011
© Massachusetts Institute of Technology 2011. All rights reserved.
A uthor ...........................
_
Dep nt of Mechanic Engineering
May 6, 2011
Certified by....... ...................... .............................
Alexandra H. Techet
Associate Professor of Mechanical and Ocean Engineering
Thesis Supervisor
A ccepted by .............................. .. . ..-. .- ......
of
David Hardt, Professor Mechanical Engineering
Chairman, Department Committee on Graduate Theses

An Experimental Study of Worthington Jet Formation After
Impact of Solid Spheres
by
Jenna Marie McKown
Submitted to the Department of Mechanical Engineering
on May 6, 2011, in partial fulfillment of the
requirements for the degree of
Master of Science in Ocean Engineering
Abstract
The impacts of solid spheres with the free surface have been studied for over one
hundred years. In this thesis, the Worthington jets resulting from the impacts of hy-
drophobic and hydrophilic spheres with the free surface are studied experimentally.
Several impact velocities and three materials of differing mass ratios are used. The
resulting jets are characterized in terms of common non-dimensional physical param-
eters. A single camera is used to image activity both above and below the free surface
simultaneously. The resulting images provide measurements of the height and veloc-
ity of the Worthington jets, as well as information about the breakup of the columns
of fluid into droplets.
In the hydrophilic cases, two separate regimes of jet formation are observed. The
heights of both the first and second jet are recorded with time, and the maximum
heights of each jet are compared to the Froude number at impact. The maximum
height of the second jet is found to scale linearly with We, which is calculated using
the velocity of the jet tip rather than impact velocity. Viscosity is shown to be
negligible while a dependence on gravity is indicated. The time at which the first jet
breaks up is proportional to Froude number, while the time of breakup for the second
jet is nearly constant. Comparisons to energy transfer are presented.
In the hydrophobic cases, the Worthington jet results from the collapse of the
cavity formed below the free surface. Heights of the continuous Worthington jet
are recorded for each time step, and average maximum heights obtained are again
compared to Froude and Weber numbers. The increased variability in these data
hinder clear scaling, and experimental error is calculated from repeated trials of two
representative cases. Unaffected by experimental variation, however, the lifespan
of the Worthington jets is shown to depend linearly on Froude number. Energy
comparisons are explored.
Thesis Supervisor: Alexandra H. Techet
Title: Associate Professor of Mechanical and Ocean Engineering
4
Acknowledgments
This thesis would not have been possible without my advisor, Alexandra Techet,
who has mentored and supported me throughout my time at MIT. Her words of
encouragement and advice kept me focused through this whirlwind of a Masters
program. I am ever grateful for the confidence that she had in me, despite my
intermittent lack of confidence in myself.
Tadd Truscott was instrumental to the completion of this thesis, providing endless
enthusiasm and encouragement, and helping me get as excited about the research as
he is... well, almost. On the really slow days when I didn't much feel like working, it
was Jesse Belden who always seemed to give me the extra drive I needed to keep at
it, mostly because his work ethic would make anyone feel like a slacker. Tadd, Jesse,
and Brenden Epps, my UROP supervisor way back when, all helped introduced me
to this lab and showed me the ropes. Anna Shih was excellent for distracting me
when I most needed it, and sometimes when I didn't. But I'm sorry that all of her
fish died on my watch. Shmu and Jeff were my support team upstairs, giving me
endless support.
My family has been extremely supportive, listening to me whine and complain,
and also putting up with my insane propositions of quitting school and becoming a
boat bum in the British Virgin Islands... I didn't think my mother would steal my
idea and do it first. I would not have been able to stay sane if it weren't for their
constant reminders not to worry so much and to have fun.
This thesis is brought to you by the letter W, and the number 2.
6
Contents
1 Introduction
1.1 Review of Literature...........
1.1.1 Studies of Impacts.......
1.1.2 Experimental Variations
.
1.1.3 Cavity Dynamics........
1.1.4 Jet M odels . . . . . . . . . . . .
1.1.5 Jet Instabilities and Breakup
1.2 Outline of Thesis . . . . . . . . . . . .
2 Experimental Methods
2.1 Apparatus . . . . . . . . . . . . . . . .
2.2 Test parameters . . . . . . . . . . . . .
2.3 Data Processing . . . . . . . . . . . . .
2.3.1 Calibration . . . . . . . . . . .
2.3.2 Height Measurements . . . . . .
2.3.3 Cavity Measurements . . . . . .
3 Results
3.1 Hydrophilic Cases . . . . . . . . . . . .
3.1.1 Two Regimes of Jet Formation
3.1.2 Froude Number and Mass Ratio Comparisons
3.2 Hydrophobic Cases . . . . . . . . . . .
3.2.1 Surface Closure . . . . . . . . .
7
3.2.2 Froude Number and Mass Ratio Comparisons . . . . . . . . .
4 Discussion 53
4.1 Hydrophilic Cases . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.1 Worthington Jet Heights . . . . . ...... . . . . . 56
4.1.2 Bond and Ohnesorge Numbers . . 61
4.1.3 Time to Breakup . . . . . . . . . 61
4.1.4 Energy Transfer . . . . . . . . . . 63
4.2 Hydrophobic Cases . . . . . . . . . . . . 66
4.2.1 Worthington Jet Heights . . . . . 69
4.2.2 Experimental Error Quantification 74
4.2.3 Rayleigh-Plateau Instability . . . 76
4.2.4 Jet Lifespan . . . . . . . . . . . . 80
4.2.5 Energy Transfer . . . . . . . . . . 81
5 Conclusions 85
5.1 Hydrophilic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2 Hydrophobic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3 Future Wo rk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
List of Figures
1-1 Time series of Worthington jet. . . . . . . . . . . . . . . . . . . . . . 24
2-1 Apparatus schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-2 Example frames from single camera . . . . . . . . . . . . . . . . . . .
2-3 Time evolution of jet height. . . . . . . . . . . . . . . . . . . . . . . .
3-1 Initial formation of hydrophilic Worthington jet . . . . . . . . . . . .
3-2 Comparison of hydrophilic jets at Fr = 5.32. . . . . . . . . . . . . . .
3-3 Comparison of hydrophilic acrylic spheres with different impact velocities.
3-4 Maximum heights of the first hydrophilic jet . . . . . . . . 39
3-5 Maximum heights of the second hydrophilic jet . . . . . . . . . . . . . 41
3-6 Acrylic hydrophobic series with Fr = 4.86. . . . . . . . . . . . . . . . 42
3-7 Cavities of acrylic hydrophobic spheres at different Fr . . . . . . . . 44
3-8 Comparison of depth of pinch-off for different Fr . . . . . . . . . . . 45
3-9 Surface closure of hydrophobic impact cavity. . . . . . . . . . . . . . 46
3-10 Hydrophobic jets with surface closure . . . . . . . . . . . . . . . . . . 47
3-11 Hydrophobic jets at one Fr . . . . . . . . . . . . . . . . . . . . . . . 48
3-12 Acrylic hydrophobic jets at various Fr . . . . . . . . . . . . . . . . . 50
3-13 Hydrophilic and hydrophobic jets with same Fr . . . . . . . . . . . . 52
4-1 Stages of Worthington jet formation for hydrophilic cases
4-2 Maximum heights of first Worthington jets vs. Froude number . .
4-3 Weber number vs. Froude number for 1st jet in hydrophilic cases
4-4 Weber number vs. Fr2 for 2nd jet in hydrophilic cases . . . . . .
4-5 Maximum heights of second hydrophilic Worthington jets vs. Fr2 . 59
4-6 Maximum heights of second hydrophilic Worthington jets vs. We . 60
4-7 Times to hydrophilic jet breakup . . . . . . . . . . . . . . . . . . . . 62
4-8 Non-dimensional time to breakup for first and second Worthington jets
vs. W eber number . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4-9 Hydrophilic sphere velocity after impact . . . . . . . . . . . . . . . . 64
4-10 Energy transfered from hydrophilic spheres to fluid . . . . . . . . . . 65
4-11 Stages of Worthington jet formation for hydrophobic cases . . . . . . 67
4-12 Depth of pinch-off scaling with Froude number and mass ratio . . . . 68
4-13 Acrylic Hydrophobic Non-dimensional Jet Heights . . . . . . . . . . . 69
4-14 Ceramic Hydrophobic Non-dimensional Jet Heights . . . . . . . . . . 71
4-15 Steel Hydrophobic Non-dimensional Jet Heights . . . . . . . . . . . . 72
4-16 Weber number vs. Froude number for the hydrophobic cases . . . . . 73
4-17 Mean maximum hydrophobic jet heights vs. Weber number . . . . . . 74
4-18 Average maximum heights of acrylic and ceramic hydrophobic Wor-
thington jets, 30 cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4-19 Local velocity vs. non-dimensional radius of the jet tip . . . . . . . . 77
4-20 Local velocity vs. wavenumber of the perturbation . . . . . . . . . . . 78
4-21 Non-dimensional maximum jet height vs. Rayleigh-Plateau breakup
tim e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4-22 Non-dimensional time to jet fall vs. Fr . . . . . . . . . . . . . . . . . 81
4-23 Non-dimensional cavity volume vs. Fr . . . . . . . . . . . . . . . . . . 82
4-24 Non-dimensional average maximum height vs. potential energy of the
cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83