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An experimental study of shock wave attenuation
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Content
An Experimental Study of Shock Wave Attenuation
by
Hongjoo Jeon
A dissertation presented to the
Faculty of The USC Graduate School
In Partial Fulfillment for the Requirements
for the Degree of
Doctor of Philosophy
in Mechanical Engineering
University of Southern California
Los Angeles, California
August 2017
(Defended June 8, 2017)
© Copyright by Hongjoo Jeon 2017
All Rights Reserved
I
To my family
II
Acknowledgement
I would like to express my gratitude to prof.Veronica Eliasson for giving opportunity,
guidance, support, and great supervision throughout my Ph.D.work. Prof.Eliasson's
enthusiasm spreads to students and colleagues, and makes the shock wave lab as an
exciting place to become a scientist. Joining Eliasson’s lab changed my entire life while
at the University of Southern California.
I want to thank to prof.Iv´ an Bermejo-Moreno and prof.Aiichiro Nakano for being
my dissertation committee. I also greatly appreciate committee members of my quali-
fying exam: prof.James Boedicker, prof.Veronica Eliasson, prof.Iv´ an Bermejo-Moreno,
prof.Larry Redekopp, and prof.Alejandra Uranga, for their valuable advice and time.
Awonderfulresearch grouphashelped metoaccomplishmy workatUSC,especially
Dr.Chuanxi Wang, Dr.Gauri Khanolkar, Dr.Orlando Delpino Gonzeles, Stelios Koum-
lis, Shi Qiu, Nicholas Amen, and Qian Wan. They have always been willing to assist
me in my experiments and to criticize my results and presentations. Especially, thank
you Orlando and Stelios, whom I learnt everything about experimental techniques and
analysis. I was also thanks to Christopher Doughtery, Ryan Miller, Hasmik Geozalian,
and Hang Wei for helping me to finish the experiments.
The funding from the National Science Foundation provided for the construction of
theshocktube,experimental equipment, andresearchassistantship forthreeyears. Iam
grateful to all people working at the USC Viterbi/Dornsife machine shop, especially to
DonaldR.Wiggins. Theyhelpedmetodesignandmanufactureexperimentalinstrument
and always tried to finish by deadlines many times.
Thank my parents for raising me as an independent and hard-working researcher.
Without their love and support that you have given me, I might not be the person
I am today. Finally, I would like to thank my wife, Minji, and two daughters, Irene
and Jeanne, for being by my side with endless amounts of love and understanding. To
Minji, I cannot thank you enough for all your help and encouragement. Your patience,
dedication, and tender-loving care tome and two daughters have been aconstant source
of inspiration for during my M.S.and Ph.D.life at USC.
III
Abstract
The understanding of shock wave attenuation has been, and still is, essential in many
parts of the world. Shock waves appear in daily life (volcanic eruption, lightning, ani-
mals living in the ocean, etc.) and can be beneficial or detrimental depending on the
situation. Therefore, it is important to understand shock wave dynamics, so that we
more efficiently can use shock waves to our benefit (e.g. shock wave lithotripsy) or pro-
tect ourselves from the nonlinear effects of a shock wave. In large-scale applications,
porous materials or liquids have been considered as an economical and practical solu-
tion to provide protection against shocks. In this research, compressible porous foams,
water, a cornstarch and water mixture (a cornstarch suspension), and thin films have
been studied to improve the current understanding of their dynamic response to shock
waves, especially in regards to shock wave reflection, mitigation, and transmission. Ex-
periments were performed using a pressure-driven shock tube. Non-invasive schlieren
visualization along with high-frequency pressure transducers were used to obtain qual-
itative and quantitative data from the experiments. The properties examined during
the current study were then utilized to investigate shock wave reflection configurations,
shock Mach numbers, impulse, and overpressure. At first, experiments were conducted
to understand the difference between the shock wave reflection off water, and cornstarch
suspension surface, and the results were compared with an analytical solution. Shock
wave reflection off the surface of water and cornstarch suspensions were different from
the analytical solution, while the reflection configurations between the water and the
cornstarch suspension were not significantly different. Additionally, shock wave atten-
uation by different thicknesses of water and cornstarch suspension were studied. No
transmitted shock wave was observed through water and cornstarch suspension sheets,
but compression waves induced by the shock-accelerated liquid coalesced into a shock
wave. Furthermore, shock wave attenuation by a thin film was quantitatively studied
with various incident shock Mach numbers and a non-dimensional analysis was per-
formed to generalize the experimental results. Results showed that the thickness of
the film affected the attenuation of shock wave when the film thickness is of similar
order of magnitude compared to shock wave thickness. Finally, shock wave attenuation
IV
using form obstacles was investigated to understand the effect of geometries. No signifi-
cantly different results were found among the five different geometries, nor between the
two types of foam. This dissertation is part of an ongoing effort to understand shock
wave interactions with various materials and geometries, and their dynamic responses
using experimental techniques. The present results can be compared against numerical
simulations for modeling shock wave interactions.
V
Contents
1 Introduction 1
1.1 Motivation and Application . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Experimental Overview 9
2.1 Pressure-Driven Shock Tubes . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Blade Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Pressure Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Visualization Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.1 Direct Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.2 Schlieren Visualization . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Shock Wave Reflection off Liquid Surfaces 20
3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.1 Pixel Intensity Method . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.2 Dimensionless Mach Stem Method . . . . . . . . . . . . . . . . . 33
3.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4.1 Validation of the Experimental Methodology . . . . . . . . . . . . 34
3.4.2 Pixel Intensity Method vs Dimensionless Mach Stem Method . . . 34
3.4.3 Repeatability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4.4 Transition Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 Shock Wave Attenuation by Liquid Sheets 41
4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
VI
4.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 Shock Wave Attenuation by Plastic Sheets and Cotton Wires . . . 46
4.3.2 Shock Wave Attenuation by Liquid Sheets . . . . . . . . . . . . . 48
5 Shock Wave Attenuation by Thin Films 61
5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3.1 Flow Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3.2 Pressure Measurements . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3.3 Analysis and Discussion . . . . . . . . . . . . . . . . . . . . . . . 79
6 Shock Wave Attenuation by Foam Obstacles 81
6.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3 Logarithmic Spiral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.4 Sample Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7 Conclusions and Future Work 99
Appendix A Calculating Local Speed of Sound 103
Appendix B Area Reduction Effects 105
Appendix C Close Cell Foam Results 108
Bibliography 112
VII
List of Figures
1.1 Examples of pressure profiles behind shock waves. . . . . . . . . . . . . . 3
2.1 A schematic description of flow in a shock tube. . . . . . . . . . . . . . . 11
2.2 A schematic description of a blade mechanism device. . . . . . . . . . . . 12
2.3 An overview of the direct visualization setup. . . . . . . . . . . . . . . . 14
2.4 An overview of the schlieren setup used in the current work. . . . . . . . 16
2.5 Schlieren visualization with high temporal and high resolution. . . . . . . 17
2.6 Aschematic diagramoftheexperimental facility andthe dataacquisition
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.1 Schematic illustration of different types of shock wave reflection. . . . . . 21
3.2 Detachment transition angles for a range of Mach numbers. . . . . . . . . 22
3.3 A schematic description illustrating the viscous effect and boundary layer
effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Possible shock wave reflection configurations. . . . . . . . . . . . . . . . . 25
3.5 Schematic description of inclined shock tube. . . . . . . . . . . . . . . . . 27
3.6 Overpressure-time history from the inclined shock tube. . . . . . . . . . . 28
3.7 Schlieren photographs using Phantom V711. . . . . . . . . . . . . . . . . 29
3.8 Schlieren photographs using a Nikon D90 camera. . . . . . . . . . . . . . 30
3.9 A schematic side-view of the test section. . . . . . . . . . . . . . . . . . . 31
3.10 Determination of transition angle. . . . . . . . . . . . . . . . . . . . . . 32
3.11 Pixel intensity images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.12 Schematic illustration of dimensionless Mach stem. . . . . . . . . . . . . 34
3.13 The transition angle versus the incident shock Mach number. . . . . . . . 35
3.14 Dimensionless Mach stem as a function of deflection angle. . . . . . . . . 36
3.15 Dimensionless Mach stem versus deflection angle . . . . . . . . . . . . . . 37
VIII
3.16 The transition angle versus the incident shock Mach number using pixel
intensity and dimensionless Mach stem method. . . . . . . . . . . . . . . 38
3.17 Incident shock Mach number for the entire sets of experiments. . . . . . . 38
3.18 An overall measure of dimensionless Mach stem. . . . . . . . . . . . . . . 39
3.19 The transition angle versus the incident shock Mach number. . . . . . . . 40
4.1 Schematic description of shock tube. . . . . . . . . . . . . . . . . . . . . 44
4.2 Schematic description of detailed cut-view of the test section. . . . . . . . 45
4.3 Schematic description of liquid sheet container. . . . . . . . . . . . . . . 45
4.4 Aseriesofhigh-speedschlieren imagesofshockwave interactionwithjust
the plastic sheets and the cotton wires. . . . . . . . . . . . . . . . . . . . 47
4.5 An x−t diagram depicting the shock wave interaction. . . . . . . . . . . 49
4.6 Comparison of the overpressure and impulse histories. . . . . . . . . . . . 50
4.7 A series of schlieren high-speed images for a 5mm thick water sheet at
M
s
=1.46. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.8 A series of high-speed images for a 10mm thick water sheet at M
s
=1.46. 52
4.9 A series of high-speed images for a 10mm thick cornstarch suspension
sheet at M
s
= 1.46. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.10 Aseriesofhigh-speedimagesfora5mmthickcornstarchsuspensionsheet
at M
s
= 1.46. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.11 Comparison of the overpressure and impulse histories. . . . . . . . . . . . 56
4.12 A schematic x−t diagramof the shock wave interaction with a liquid sheet. 58
4.13 Shock wave reflection and attenuation factors. . . . . . . . . . . . . . . . 60
5.1 A schematic x - t diagram of shock wave interaction with a film. . . . . . 62
5.2 Schematic detailed cutaway view of the test section and the film holder. . 64
5.3 A series of high-speed schlieren images of shock wave interaction with a
12.7µm thick polyester film. . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.4 A series of high-speed schlieren images of shock wave interaction with a
12.7µm thick polyester at M
s
=1.20. . . . . . . . . . . . . . . . . . . . . 67
IX
5.5 A series of high-speed schlieren images of shock wave interaction with a
0.92µm thick polyester film at M
s
=1.36. . . . . . . . . . . . . . . . . . 68
5.6 A series of high-speed schlieren images of shock wave interaction with a
25.4µm thick aluminum film at M
s
= 1.39. . . . . . . . . . . . . . . . . . 69
5.7 Shock wave trajectories of the incident, reflected, and transmitted shock
waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.8 A series of high-speed schlieren images of shock wave interaction with a
25.4µm thick polyester film at M
s
=1.34. . . . . . . . . . . . . . . . . . 71
5.9 An x−t diagram depicting the shock wave interaction. . . . . . . . . . . 72
5.10 Comparison of overpressure traces. . . . . . . . . . . . . . . . . . . . . . 73
5.11 Overpressure versus time for the case of a 0.92µm thick polyester film at
M
s
=1.34. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.12 Overpressure profile at sensor S
3
at M
s
= 1.34. . . . . . . . . . . . . . . 76
5.13 Overpressure profile at sensor S
3
with the 12.7µm thick polyester film. . 76
5.14 Overpressure profile atsensors S
1
andS
3
withthe 50.8µm thick polyester
and aluminum films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.15 Normalized overpressure versus the time. . . . . . . . . . . . . . . . . . . 78
5.16 Normalized overpressure versus the incident shock Mach number. . . . . 80
6.1 Schematic description of the shock tube. . . . . . . . . . . . . . . . . . . 84
6.2 Overpressure plot from the upstream sensor S
1
. . . . . . . . . . . . . . . 85
6.3 Sectional view of shock tube test section. . . . . . . . . . . . . . . . . . . 86
6.4 A schematic description of logarithmic spiral shape. . . . . . . . . . . . . 87
6.5 Scanning electron microscope image. . . . . . . . . . . . . . . . . . . . . 88
6.6 Foam samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.7 Open cell foam, Case NC. . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.8 Open cell foam, Case 1C. . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6.9 Open cell foam, Case 2C. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.10 Open cell foam, Case 3C. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6.11 Open cell foam, Case 4C. . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
X
6.12 Opencellfoamoverpressurerecordingsforallcases: incidentandreflected
shock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.13 Open cell foam overpressure recordings for all cases: transmitted wave. . 94
6.14 Photographs showing translation of the NC open and closed cell foam
sample.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.15 Photographs showing translation and deformation of the 1C open cell
foam sample. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.16 Comparison between open cell and closed cell. . . . . . . . . . . . . . . . 98
A1 Schematic description of determining mixture ratio for the two cases. . . 104
B1 Schematic description ofaconverging shocktubeandcorrespondingpres-
sure diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
B2 Incident shock Mach number as a function of driver pressure. . . . . . . . 107
C1 Closed cell foam, Case NC. . . . . . . . . . . . . . . . . . . . . . . . . . . 108
C2 Closed cell foam, Case 1C. . . . . . . . . . . . . . . . . . . . . . . . . . . 109
C3 Closed cell foam, Case 2C. . . . . . . . . . . . . . . . . . . . . . . . . . . 109
C4 Closed cell foam, Case 3C. . . . . . . . . . . . . . . . . . . . . . . . . . . 109
C5 Closed cell foam, Case 4C. . . . . . . . . . . . . . . . . . . . . . . . . . . 110
C6 Closed cell foam overpressure recordings for all cases: incident and re-
flected shock. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
C7 Closed cell foam overpressure recordings for all cases: transmitted wave. . 111
XI
List of Tables
2.1 Summary of pressure transducers. . . . . . . . . . . . . . . . . . . . . . . 13
3.1 Summary of camera settings. . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 Summary of camera settings . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Reflected and coalesced Mach numbers. . . . . . . . . . . . . . . . . . . . 58
5.1 Sample information. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.1 Camera settings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 Overview of experimental sample configurations. . . . . . . . . . . . . . . 89
A.1 Speed of sound calculation at room temperature. . . . . . . . . . . . . . 104
XII
Chapter 1
Introduction
A shock wave is athin wave front (typically 200nm thick atatmospheric conditions [1]),
which is generated, for example, during a sudden release of energy. When a shock wave
travels through an undisturbed medium, the pressure, density, entropy, and temper-
ature drastically increase behind the shock and this sudden changes can damage its
surroundings, which may include humans and building structures. Shock waves can be
generated by natural reactions on Earth, such as volcanic eruptions, lightning during
a thunderstorm, and animals living in the ocean. From a macroscopic point of view,
volcanic eruptions are created by shock wave interactions between ground water and
magma [2]. In addition, lightning during athunderstorm can produce shock waves when
extremely hot air expands into the cool surrounding air and the shock waves become
sound waves called thunder [3]. Furthermore, even a pistol shrimp stuns its prey with a
loud cracking noise, and a shock wave is generated by a large cavitation bubble collapse
whentheshrimp’s snapperclawsshut [4,5]. Shockwaves arealsousedinawiderangeof
human-made applications, such as extracorporeal shock wave lithotripsy (ESWL) [6,7].
With the advent of new technology, humans have learned to deposit high-density
energy (up to megatons of TNT) by explosives, thereby generating shock waves with
incredible strength. Explosives are often used for mining, excavation, and demolition.
However, there are also negative effects that entail with this technology, which include
explosive weapons that can potentially harm soldiers or civilians. Further, explosions
may occur unintentionally, resulting in industrial accidents that can potentially damage
properties and cost lives. With so much at stake, better understanding of how to reduce
the energy contained in a shock wave is motivated by more than academic curiosity.
Consequently, over the past few decades, many researchers have studied shock wave
dynamics.
1
The understanding of shock wave dynamics can be divided into three fields: shock
wave reflection, shock wave attenuation, andshock wave focusing. Shock wave reflection
is a fundamental aspect of shock wave dynamics. There are important features of shock
wave reflection that are not seen in a linear acoustic or optical phenomena. In the
linear reflection, the angles of incidence and reflection are identical; however, this is
not the case for shock wave reflection. Therefore, the study of shock wave reflection is
often preceded to extend the knowledge into shock wave attenuation and shock wave
focusing. Shock wave focusing is a field of study that seeks to maximize the pressure
and temperature. It is of interest to both military and civilian aims in civil, chemical,
nuclear,andbio-medicalindustries. Shockwaveattenuationisoneofthemostimportant
topics for the safety of humans and properties because shock or blast waves can cause
serious damage [8].
Before introducing possible methods to study shock wave mitigation, the properties
behind the actual shock front should be addressed. Typically, there are two scenarios
that occur depending on the method used to generate the shocks: (1) a shock front
followed by constant properties lasting for some time; and (2) a shock front followed by
an exponential decay in properties (often referred to as a “blast wave”), as illustrated
in Figure 1.1. The first scenario is unrealistic unless the shock wave is confined to
propagate in some direction with a constant cross-sectional area, which is the way most
shock tubes are designed. As shown in Figure 1.1, the overpressure or underpressure
can be defined by the relative pressure induced by shock or blast wave compared to the
ambient pressure. A parameter often used to quantify the potential damage caused by a
shock or blast wave is impulse, which is defined as the integral of the overpressure over
a certain time interval.
The understanding of shock wave mitigation requires knowledge of multidisciplinary
research areas, which include the flow field behind the shock front, the dynamical re-
sponseofmaterials,andstructuresunderextremeloadingconditions. Asdiscussedin[9],
the method of attenuating shock waves can be divided into several categories. Path-
changing methods can be used to attenuate shock waves inside tunnels by using large
scale changes in geometry to force the shock wave to change its direction of propagation
thereby reducing the energy of the shock wave due to friction, heat transfer, diffrac-
2
Constant area shock tubes
Point source explosion
Time
Pressure
Negative impulse
Positive impulse
Figure 1.1 Examples of pressure profiles behind shock waves. For the shock wave produced by
constant area shock tubes, the pressurebehindthe shock remains constant for some time (blue
dashed line). For point source explosions the pressure profile behind the shock front decays
exponentially before returning to the ambient conditions (solid red line).
tion, and multiple shock wave reflections. Igra et al. [10] performed experimental and
numerical studies of shock wave propagation through double-bent ducts with smooth
and rough walls. Results showed that a double-bent duct geometry with a rough wall
was an effective method to attenuate the transmitted shock wave. The authors deduced
that a certain aspect ratio of the width and height of the duct determined the peak
pressure caused by the transmitted shock wave. Obstacle methods partially block the
channel using complex large or small scale geometries to attenuate shock waves. The
energy of the shock wave is dissipated by excessive vorticity and lost due to viscous
fluid interaction. As an example, rigid obstacles with different barrier angles, shapes,
and arrangements have been investigated using various experimental methods [11,12].
Chaudhuriet al. [13]numericallystudiedtheshockwaveattenuationeffectsusingarray-
matrix obstacles placed in both non-staggered and staggered columns. Results showed
that the strength of the transmitted shock waves was dependent on parameters such as
the geometry of the obstacles, obstacle orientation in space, and the relaxation length.
Barrier methods transfer kinetic energy of the shock wave to the potential energy of
a solid or liquid barrier. Many experiments and simulations have been performed in this
3
areausingfoambarriers[14–19],rigidporousmedia[20,21],andsolidbarriers[22]. Many
shock mitigation techniques feature a combination of one or more of these mechanisms.
Among options for a fluid medium, environmentally friendly and readily available
liquids,suchaswater,havebeenconsideredforcost-effectiveshockattenuationmethods.
Theacousticimpedanceofwaterisabout3500timesgreaterthanthatofairatstandard
atmospherecondition. Therefore, onlyasmallfractionofenergycanbetransmittedinto
thewater. Forexample, fourwater-filledballoonscanstopabulletfroma0.44Magnum,
one of the most powerful handguns with a bullet velocity of approximately 440m/s [23].
Similarly, in the scenario of an impact of shock waves, water can potentially replace
conventional concrete barriers to attenuate the strength of shock waves [24].
Thestiffnessofaliquidcanbeincreasedbyaddingdenseparticlestoachievesolid-like
behavior such as shear thickening fluids. Shear thickening is a type of non-Newtonian
fluids in which the liquid’s viscosity increases with shear rate. One common and widely
used shear thickening fluid is created by mixing cornstarch in water, known as a corn-
starch suspension. A cornstarch suspension is a type of discontinuous shear-thickening
fluids because its viscosity dramatically jumps with increasing shear rate. Therefore,
the behavior of cornstarch suspension shows a liquid-like characteristic at a low shear
rate and a solid-like characteristic at a high shear rate. The shear thickening behavior
of the cornstarch suspension can be applied to protective barriers to effectively mitigate
shock waves.
A porous compressible foam is another promising material to attenuate shock waves.
The wave attenuation by porous media is one of the fundamental research topics in-
cluding filtration processes and blast protection. The complicated geometry of porous
compressiblefoamsresultsincomplexdynamicresponseswhensubjectedtoshockwaves.
It has been shown that the shock wave attenuation effectiveness of the porous media
depends on whether the porous media is fixed or unfixed [17], rigid or flexible [18], and
open or closed [19].
The major mechanism of shock wave attenuation using thin films is mainly achieved
by the acoustic impedance mismatch, the momentum and heat exchanges between the
air and the film. It has been widely investigated that the acoustic impedance is the
dominant parameter in the interaction between shocks and other materials. However,
4
the thickness of the material can possibly affect the degree of attenuation when the
material thickness is of similar order of magnitude compared to the shock thickness. To
the best of this authors’s knowledge, no research has been conducted to understand the
effect of the material thickness on the transmitted shock wave.
The main purpose of this experimental study is to understand shock wave attenua-
tion by different types of media, such as water, cornstarch suspensions, thin films, and
porous materials. To achieve the main purpose of the study, shock wave reflection is ini-
tially investigated to understand the nature of transient shock waves on liquid surfaces.
Utilizingcurrentunderstandingofthereflectionphenomena, thedegreeofattenuationis
quantitatively investigated depending onliquid sheetthicknesses andthetype ofliquids.
Similarly, shock wave attenuation using different types of films and porous foams are
investigated. High-speed direct and schlieren visualization techniques and three types
of shock tubes are mainly used in this experimental study.
1.1 Motivation and Application
Industrial accidents and terror attacks can cause great damage, injuries, and loss of
lives. Therefore, the understanding of the shock wave interaction with different types
of media is important to effectively attenuate the shock and blast waves. The goals of
thisdissertation relyonexperimental investigation usingdifferent typesofmaterialsand
obstacles with various geometries to physically understand the mechanisms of the shock
wave attenuation. Two possible applications with regard to shock wave attenuation are
shown below:
• Underground mines and shelter design. In an explosion in an underground
mine or shelter, the tunnels and ducts work as a wave guide for the shock waves
to propagate in the direction of the tunnels. When the shock wave propagates
throughaduct ortube, overpressure caused by the shock wave results inup tofive
times higher pressure than what can be obtained under static conditions with the
same incident shock wave depending on the length of the tube [25]. Therefore, it
5
is essential to attenuate shock waves through the tunnel systems to protect people
and valuables.
• High-speed train tunnel design. When a high-speed train enters a tunnel
at speeds greater than 250km/h, compression waves are generated ahead of the
train inside the tunnel, which coalesce into a weak shock wave [26]. Due to this
phenomenon, sonic booms can be generated and heard up to 400m away from the
tunnel exit [27]. The loud noise from a sonic boom affects both the passengers
inside the train and residents near the tunnel exits. As a result, researchers are
interested in better understanding sonic booms created by high-speed trains [28–
30].
1.2 Outline and Contributions
The contents of the chapters in this dissertation is described below:
Chapter 2: This chapter presents the basic concept of the governing equations,
relevant theories, experimental techniques, and optical methods on the topic of this
dissertation. The experimental setup is designed to conduct highly repeatable exper-
iments. Three shock tubes are designed depending on the main focus of the study
required. Pressure transducers are mainly used to acquire quantitative data. Three dif-
ferent non-invasive visualization techniques are utilized to obtain qualitative and quan-
titative measurements.
Chapter 3: Pseudo-steady shock wave reflections off the surface of water and corn-
starch suspensions are studied to find the transition angle between regular and irregular
reflections. The experiments are conducted using an inclined shock tube, which can be
rotated between the horizontal and vertical planes, combined with two different visual-
ization methods, schlieren visualization with high temporal and high spatial resolution.
Three shock Mach numbers, M
s
= 1.20, 1.38, and 1.52, are investigated in this study.
To determine the transition angle, shock wave position is detected by measuring pixel
intensity, and the results are validated by an additional method, finding a characteristic
curve by measuring the length of the Mach stem.
6
Chapter 4: Shock wave interactions with a liquid sheet are investigated by im-
pacting planar liquid sheets of varying thicknesses with a planar shock wave. A square
frame is designed to hold a rectangular liquid sheet, with a thickness of 5mm or 10mm,
using plastic membranes and cotton wires to maintain the planar shape and minimize
bulge. A schlieren technique with a high-speed camera is used to visualize the shock
wave interaction with the liquid sheets. High-frequency pressure sensors are used to
measure wave speed, overpressure, and impulse both upstream and downstream of the
liquid sheet.
Chapter 5: Shock wave attenuation due to the presence of thin films in the path of
a planar shock wave is investigated. To understand how the failure mechanism (ductile
or brittle) affects the shock wave attenuation, polyester and aluminum films are used.
The effect of varying thicknesses and types of film materials on shock wave attenuation
was considered based on experimental results. In the current study, thicknesses of 0.92,
12.7, 25.4, and 50.8µm polyester and aluminum films are subjected to planar incident
shock waves with Mach numbers of 1.20, 1.34, 1.39, and 1.46. A high-speed schlieren
visualization together with high-frequency pressure sensors are used to study the shock
wave interaction with the films.
Chapter 6: A shock wave impact study on open and closed cell foam obstacles is
completed to assess attenuation effects with respect to different front face geometries of
the foam obstacles. Five types of geometries are investigated, while keeping the mass of
the foam obstacle constant. The front face, i.e., the side where the incident shock wave
impacts, is cut into the geometries with one, two, three or four convergent shapes, and
the results are compared to a foam block with a flat front face. A shock Mach number
of M
s
= 1.25 is used in all cases.
Chapter 7: The conclusions and future directions ofthis dissertation are presented.
The main contributions of this dissertation are presented below:
Experimentalinvestigationofshockwaveattenuationisessentialtounderstandfluid-
structure and two-phase flow interactions due to the difficulties of performing numerical
simulations dealing with multiphase phenomena such as viscosity and liquid surface
7
tension. To clarify the physics of shock wave attenuation using water, cornstarch sus-
pensions, thin films, and foams, the present study used pressure-driven shock tubes and
non-invasive visualization methods. The main objectives and outcomes of this disserta-
tions are below.
• Experimentalresultsareobtainedforthepseudo-steadystateshockwavereflection
off the surface of water and cornstarch suspension using an inclined shock tube.
This is the first experimental study using schlieren photography with both high
temporalandhighspatialresolutionmethodstofindthetransitionanglesbetween
a regular and an irregular shock wave reflection configurations.
• Shock wave interaction with water and cornstarch suspension sheets are visualized
withhigh-speedschlierenphotographyinpreciselycontrolledpressure-drivenshock
tubeexperiments. Theseresultsshowpromisethatwaterorcornstarchsuspensions
mayhave asignificant effectonattenuatingthestrengthofshock waves, andcould
possibly be incorporated into protective armor and barriers.
• Experimental investigationisperformedtounderstand theshockwave attenuation
by the scale of events on the relative thickness of the films compared to the shock
thickness. Results show that shock wave attenuation is negligible when the film
thickness is the same order of magnitude compared to the shock wave thickness.
• Shock wave attenuation using two types of foam obstacles with five different front
face geometries is studied. This dissertation confirms that geometry of the foam
obstacles in the path of a propagating shock wave does not influence the degree of
shock wave attenuation.
• The future direction of the shock wave research is presented. The experimental
results of this research will be used as a baseline for future studies to compare
with numerical studies for modeling the dynamics of liquid sheet and thin film
interacting with the shock wave.
8
Chapter 2
Experimental Overview
Inthecurrentdissertation,severalexperimentaltechniqueswereusedincludingpressure-
driven shock tubes, optical methods, and pressure transducers. In this chapter, the
principles of the experimental methods are described.
The basic analysis of compressible flow is based on three fundamental equations, the
continuity, the momentum, and the energy equation, as discussed in [31],
∂
∂t
ZZZ
V
ρdV +
ZZ
S
ρV·dS = 0, (2.1)
∂
∂t
ZZZ
V
ρVdV +
ZZ
S
(ρV·dS)V =
ZZZ
V
ρfdV −
ZZ
S
pdS, (2.2)
∂
∂t
ZZZ
V
ρ(e+
V
2
2
)dV +
ZZ
S
ρ(e+
V
2
2
)V·dS = (2.3)
ZZZ
V
˙ qρdV −
ZZ
S
pV·dS+
ZZZ
V
ρ(f·V)dV
where V is a fixed volume, V is the velocity vector V = (u,v,w) in the x,y, and z
directions, ρ is the density, S is the surface area of the volume V, p is the pressure
acting on the surface S, ˙ q is the heat rate added per unit mass, f is the body force per
unit mass, and e is the internal energy. The additional equation required to solve the
above equations is an equation of state for an ideal gas, i.e.
p= ρRT (2.4)
where R is the specific gas constant and T is the temperature. Additional assumptions
9
are required for the above equations since a shock wave is an irreversible process and
more details can be found in Anderson [31].
2.1 Pressure-Driven Shock Tubes
Apressure-driven shock tube isanexperimental instrument oftenused forshock loading
research[16,32–34]inwhichagasatlowpressureandagasathighpressureareseparated
by a diaphragm (see Figure 2.1). The high pressure region is called a driver section and
the low pressure partis called a driven section. The driver section, identified by region4
inFigure2.1(a),canbefilledwithdifferenttypesofgas. Thediaphragmcanberuptured
by the burst pressure itself or by an additional device, which can be used to break the
diaphragm at the desired pressure. A short breakup time of the diaphragm is important
to create a planar shock wave, instead of generating multiple compression waves. When
the diaphragm breaks, a shock wave travels down the driven section and an expansion
fan travels toward the driver section, as shown in Figure 2.1(b). The expansion fan is
reflected back from the end wall of the driver section and propagates into the driven
section. The test time of a shock tube is typically defined by the time interval between
the arrival of an incident shock wave and expansion waves. There are several methods
to increase the test time [35].
The incident shock strength, p
2
/p
1
, can be derived from the Equations (2.1) - (2.3)
as a function of the diaphragm pressure ratio p
4
/p
1
p
4
p
1
=
p
2
p
1
1−
(γ
4
−1)(a
1
/a
4
)(p
2
/p
1
−1)
p
2γ
1
[2γ
1
+(γ
1
+1)(p
2
/p
1
−1)]
−2γ
4
(γ
4
−1)
. (2.5)
Here, γ = C
p
/C
v
is the ratio of the specific heats for constant pressure and constant
volume and a is the speed of sound. The subscripts represent the region in which the
property is valid. The strength ofthe shock wave can be increased by different methods,
such as choosing a light gas, heating the gas in the driver section, or increasing the
cross-sectional area ratio. In the current dissertation, three different types of shock
tubes were designed, an inclined shock tube and horizontal shock tubes, to study shock
wave reflection and interaction with thin films, foam obstacles, water, and cornstarch
10
(a)
(b)
Driver section Driven section
p
p
4
4
1
1
2 3
Shock wave Expansion fan
Membrane
Contact surface
Figure 2.1 A schematic description of flow in a shock tube: (a) Initial pressure distribution;
(b) Pressure distribution after the membrane is broken.
11
R25.40
158
25.4
R19.05 Blade
Figure 2.2 A schematic description of a blade mechanism device. Dimensions in mm.
suspensions. For more detailed information about shock tubes and conditions during
operation see references [31,36–40].
2.2 Blade Mechanism
Insidethedriversection,ablademechanismdevicetohelpbreakthemembraneisplaced,
as shown in Figure 2.2. The device consists of several parts and the blade is loaded by
holding an electromagnet against a permanent magnet with a spring in between with
20VDCpower. Oncethemagnetwaslockedinplace,thedriveranddrivensectionswere
clamped togetherandseparated onlyby aMylar membrane (apolyester film). After the
desired pressure differential was reached, the power source to the blade mechanism was
disconnected, allowingthespringtoreleasetheblade, whichthenpiercedthemembrane.
The blade mechanism inside the driver section can reduce non-ideal transient pressure
effects inside the shock tube, and it can also decrease the time between incident shock
waves and reflected expansion waves reaching the test section [35].
12
Table 2.1 Summary of pressure transducers.
Model number Rise time Resonance frequency Discharge time Housing material
[-] [µs] [kHz] [s] [-]
PCB 113B21 1.0 500 1.0 17-4 Stainless steel
PCB 113B31 1.5 400 0.1 to 1.0 Invar
2.3 Pressure Transducers
Twotypesofpressuretransducers(PCB113B21andPCB113B31)wereusedtomeasure
overpressure and the shock wave speeds, as summarized in Table 2.1. The pressure
transducers were flush mounted to fully measure the high-frequency response of shock
waves. The individual pressure transducer was mounted with a threaded Delrin adapter
on the top of the test section to minimize the influence of disturbances propagating
in the shock tube walls. The pressure signals were collected by a signal conditioner
(PCB 482C) andrecorded by adigital oscilloscope (LeCroy, Wave Surfer 24Xs-A).After
the pressure signals were recorded by the oscilloscope, the values were converted to
overpressure (gauge pressure) by applying the sensitivity of a sensor. Finally, a low pass
Fourier filter was used to remove all frequencies above 100 kHz for data recorded by
PCB113B21, and 80 kHz for data recorded by PCB113B31 to avoid non-linearity of the
signals from the pressure transducers.
To quantify the rise time of the shock loading, the overall rise time can be calculated
based on the sum of the individual rise time as below,
τ =(τ
2
1
+τ
2
2
+···+τ
2
n
)
1/2
(2.6)
where τ isthe overallrise timeandτ
n
isthe risetimes oftheindividual stages. However,
the system rise time usually follows the following rule [41],
τ ≥
2.5
f
n
(2.7)
where f
n
is the resonant frequency of the transducer. On the basis offollowing this rule,
it was confirmed that the current system was capable of measuring pressure rise times
of 6.25µs.
13
Light
source
High-speed
camera
Shock tube
Driver section Driven section Test section
.
Figure 2.3 An overview of the direct visualization setup. The dashed-line represents the light
path.
2.4 Visualization Methods
In this dissertation, non-invasive visualization methods were used to study the shock
wave dynamics. Direct visualization with high-temporal resolution was used to under-
stand the response of various materials when subjected to the shock wave. High-speed
schlieren visualization method was used to qualitatively and quantitatively measure the
incident, reflected, and transmitted shock Mach numbers. The basic principles of the
visualization methods are described in this section.
2.4.1 Direct Visualization
Direct visualization with high temporal resolution was used to understand the breakup
process of water or cornstarch suspension when subjected to a shock wave, as shown
in Figure 2.3. Two point light sources (Cree XLamp, XP-G2 LEDs, Cool white) are
utilized to generate continuous light to illuminate a sample and a high-speed camera
(Phantom V711) was used to record high-speed photographs.
2.4.2 Schlieren Visualization
Theschlieren visualizationtechnique isbasedonthedeflection oflightcausedbydensity
and temperature changes in solids, liquids, and gases. Historically, Robert Hooke (1635-
1703) first observed a thermal air disturbance projected by a concave mirror or lens and
named his technique “the wave of the concave speculum.” August Toepler (1836-1912)
14
was the first scientist to use high-speed imaging and the first principal developer of
the schlieren imaging technique, including adjustable knife-edge cutoff, a lantern light
source, and a telescope for viewing the image directly. Ernst Mach (1838-1916) con-
firmed in 1877, by using spark-illuminated schlieren photography, that finite-strength
nonlinear wave could travel faster than the speed of sound, as showed by Riemann
(1860). For a more detailed historical background of the schlieren visualization method,
see [42]. Figure 2.4 shows a schematic representation of the schlieren visualization tech-
nique. The visualization system contains a z-folded schlieren setup, a light source, anda
camera. Parallel lightbetween the 254mm flat mirrors, originatedfromthe light source,
is achieved by the 254mm spherical mirror. A density change in the test section deflects
the parallelized light and causes a phase difference. A knife edge is located in the focal
point to cut off parts of the light. The knife edge is placed vertically or horizontally to
show the density gradient in the flow normal to the edge. In the present study, the knife
edge is placed vertically such that the schlieren images that usually had a 50% cutoff
satisfied both resolution and sensitivity.
Schlieren visualization with high spatial resolution can increase the image magni-
fication factor to detect smaller features but only one image can be captured per one
experiment. Thehightemporalresolutiontechnique usedinthisstudyusesahigh-speed
camera that provides many image frames up to 1,400,000 fps but it also has drawbacks
such as limited spatial resolution for increasing frame rates. In this study, schlieren
visualization with either high temporal or high spatial resolution is used to compensate
the drawbacks of the two individual visualization techniques. Both methods are set up
to avoid motion blur [43], which can be expressed as
v×Δt≤
p
M
im
, (2.8)
whenv isanobject’sspeed,Δtistheexposuretimeofthecameraorthelightsource,pis
the size of one pixel, and M
im
is the image magnification factor. To obtain quantitative
data from the images, optical distortions resulting from the visualization system are
corrected using the control point tool box in MATLAB [44].
Figure 2.5 shows schlieren visualization with high temporal and high spatial resolu-
tion. Typically, the thickness of the shock wave is of one order of magnitude greater
15
Light source
Spherical mirror
Ø = 254 mm
Flat mirror
Ø = 25 mm
lens Ø = 80 mm
Knife edge
Camera
Shock tube
Driver section
Driven section
Test section
x
y
138 cm
124 cm
307 cm
130 cm
130 cm
Spherical mirror
Ø = 254 mm
Flat mirror
Ø = 254 mm
Flat mirror
Ø = 254 mm
lens Ø = 80 mm
Figure 2.4 An overview of the schlieren setup used in the current work. The dashed-dot line
represents the light path.
than the mean free path of the gas, which at standard atmosphere conditions results in
200nm shock thickness [1]. In practice, the thickness of the dark band representing the
shock wave that is obtained by using schlieren photography can be affected by several
factors such as the strength of the shock wave, a non-parallel incident light angle, and
shadowgraph effects.
Figure 2.6 shows a schematic illustration of the experimental facility and the data
acquisition system. Schlieren visualization with high temporal resolution was achieved
using the z-folded schlieren setup with ahigh-speed camera (Phantom V711). The high-
speed camera was equipped with a 50mm focal length lens (Nikkor 50mm f/1.4) to
achieve higher frame rates by reducing the image resolution. The light source for the
camerawasahigh-powerlight-emittingdiode(CreeXLamp, XP-G2LEDs, Coolwhite).
The exposure time of the high-speed camera was set to 294ns with the continuous LED
outputlightning. Differentframeratesandresolutions wereuseddependingonthemain
focusofthestudyrequired. Signalsfrompressuretransducersinsertedinthetestsection
ofthe shock tube were used totriggerthe high-speed camera. An example ofhigh-speed
16
10 mm
(a) 0 μs (b) 30 μs (c) 60 μs
(d)
Figure 2.5 Schlieren visualization with high temporal (a)–(c) and high resolution (d) showing
shock wave reflection off water surface: (a), (b), and (c) show a series of schlieren photographs
using a Phantom V711 camera at incident shock Mach number M
s
= 1.52 and deflection
angle θ
w
= 47
◦
, scale factor: 0.45mm/pix; (d) a schlieren photograph using a Nikon D90
camera at incident shock Mach number M
s
= 1.38 and deflection angle θ
w
= 45
◦
, scale factor:
0.027mm/pix.
17
Shock tube
Pressure
transducer
Pressure
gauge
Oscilloscope
Signal
conditioner
High pressure
tank
High-speed
camera
Computer
Driver section Driven section Test section
Trigger
Light
source
Trigger
(a)
Shock tube
Pressure
transducer
Pressure
gauge
Oscilloscope
Signal
conditioner
High pressure
tank
DSLR
camera
Computer
Driver section Driven section Test section
Trigger
Light
source
Delay
generator
(b)
Figure 2.6 A schematic diagram of the experimental facility and the data acquisition system:
(a) schlieren visualization with high temporal resolution; (b) schlieren visualization with high
spatial resolution
18
schlieren photographs are shown in Figures 2.5(a)–(c).
Schlieren visualization with high spatial resolution was achieved using a z-folded
schlieren setup with a Digital-Single Lens Reflex camera (Nikon D90 SLR). The DSLR
camera was equipped with a 200mm lens (AF Micro-Nikkor 200mm f/4 IF-ED) to
achieve the lower scale factor. The light source for the camera was a pulsed flash lamp
(Nanolite KL-L Blitzlampe, High-speed Photo-system,±10ns timing accuracy) with an
18ns duration, which was generated by a flash lamp driver (High-speed Photo-system).
The timing of the flash lamp was controlled by a delay generator (BNC Model 575,
± 0.8ns trigger accuracy) operated in single pulse shot mode with different delay times.
The exposure time of the camera was set to 2.5 seconds, manually, in a dark-room
setting and signals from pressure transducers inserted in the test section of the shock
tube were used to trigger the pulsed light source with different delay times. An example
of schlieren visualization with high spatial resolution is shown in Figure 2.5(d).
19
Chapter 3
Shock Wave Reflection off Liquid Surfaces
This work has been accepted for publication in the Journal of Fluid Mechanics under
Numerical and experimental investigation of oblique shock wave reflection off a water
wedge by Wan et al. [45].
3.1 Background
ShockwavereflectionphenomenawerefirstreportedbyMachin1878[46]. Inthe1940’s,
von Neumann [47] speculated that reflected shocks could be classified into two groups
based on their structure, as illustrated in Figures 3.1(a) and (b). The first, regular
reflection (or RR), features an incident and a reflected shock that meet at the surface
of the reflecting body, shown in Figure 3.1(a). The second, known as irregular reflection
(or IR), features an incident shock and a reflected shock as well as a third shock known
as a Mach stem or Mach shock. The three shocks meet at a triple point above the
reflecting surface, shown in Figure 3.1(b). These two configurations are often referred
to as two- and three-shock theory. In pseudo-steady flow, the shock wave configuration
grows with time such that the length of the Mach stem grows gradually while following
the reflecting surface. Hence, RR configuration often transforms into IR configuration.
Therefore, in pseudo-steady flow, the transition angle is defined as the angle for which
the reflection configuration remains RR without transforming into IR.
A theoretical limit, which determines whether the reflection is regular or irregular,
was derived by von Neumann [47] and is known as the detachment criterion, which
assumes the following: (1) perfect gas; (2) two-dimensional inviscid flows; (3) the flow is
pseudo-stationary; (4) when two possible solutions are available (weak or strong shock
waves), the weak shock solution will occur. Based on these assumptions, the transition
20
φ
Incident shock
Reflected shock
θw
(a) RR
Incident shock
Reflected shock
θw
Mach stem
Slipstream
(b) IR
Incident shock
Reflected shock
θw
Mach stem
(c) vNR
Figure 3.1 Schematic illustration of different types of shock waver reflection configurations:
(a) regular shock wave reflection configuration (RR); (b) irregular shock wave reflection con-
figuration (IR); (c) von Neumann reflection, vNR (a type of irregular shock wave reflection)
configuration. φ is the angle of incident shock wave.
angle θ
tr
is determined as a function of three parameters: shock Mach number M
s
,
specific heat ratio γ, and deflection angle θ
w
. The solution to this equation is a fifth-
degree polynomial equation [48]
D
0
+D
1
sin
2
φ+D
2
sin
4
φ+D
3
sin
6
φ+D
4
sin
8
φ+D
5
sin
10
φ=0, (3.1)
where coefficients are given by
D
0
=−16 (3.2)
D
1
= 32M
2
∞
−4M
4
∞
−48M
2
∞
γ−16M
4
∞
γ +16γ
2
−16M
4
∞
γ
2
(3.3)
+16M
2
∞
γ
3
+4M
4
∞
γ
4
D
2
=−16M
4
∞
+4M
6
∞
−M
8
∞
+104M
4
∞
γ +16M
6
∞
γ−4M
8
∞
γ (3.4)
−64M
2
∞
γ
2
−32M
4
∞
γ
2
+8M
6
∞
γ
2
−6M
8
∞
γ
2
−56M
4
∞
γ
3
−16M
6
∞
γ
3
−4M
8
∞
γ
3
−12M
6
∞
γ
4
−M
8
∞
γ
4
D
3
= M
3
∞
−64M
6
∞
γ +4M
8
∞
γ +96M
4
∞
γ
2
+64M
6
∞
γ
2
+14M
8
∞
γ
2
(3.5)
+64M
6
∞
γ
3
+20M
8
∞
γ
3
+9M
8
∞
γ
4
D
4
= 8M
8
∞
γ−64M
6
∞
γ
2
−32M
8
∞
γ
2
−24M
8
∞
γ
3
(3.6)
D
5
= 16M
8
∞
γ
2
. (3.7)
21
1 1.2 1.4 1.6 1.8 2 2.2
25
30
35
40
45
50
55
Detachment criterion
M
s
θ
tr
(deg.)
Regular reflection possible
Irregular reflection possible
Figure 3.2 Detachment transition angles for a range of Mach numbers and a gas with γ = 1.4.
Only one root of equation 3.1 is real and gives the detachment criterion. Figure 3.2
shows detachment criterion for given Mach numbers. Regular reflection is possible in
theregionabovethedetachmentcriterionandirregularreflectionispossibleintheregion
under the detachment criterion.
For a weak incident shock wave and small deflection angles, the reflection pattern,
although being similar to IR, does not have a distinct triple point, a slipstream, or a
straight Mach stem. Instead, the Mach stem asymptotically merges into the incident
shock wave as shown in Figure 3.1(c). Two- and three-shock theory cannot explain this
particularshockreflection, whichiscalledavonNeumannreflection(vNR)andthevNR
is classified into IR. The vNR has prompted many attempts to either prove or disprove
the detachment theory through analytical, experimental, and numerical investigations.
Discrepanciesbetweentheoryandexperimentsarecalled“vonNeumannparadox”[49].
The first paradox was known as the persistence of RR beyond its theoretical limit as
determined by the two-shock theory. This paradox stemmed from von Neumann’s use
of inviscid analysis although the actual flow is viscous, as shown in Figure 3.3(a). Hor-
nung and Taylor [50] successfully explained the persistence of a regular reflection by
including effects of viscosity, and their results were experimentally confirmed by Ben-
22
θ
w
Viscous
flow
Inviscid
flow
(a)
δ
δ
Shock wave Shock wave
Solid surface Liquid surface
(b)
Figure 3.3 (a) A schematic description of illustrating the effect of viscosity on the triple point
path. (b) The difference of the velocity profiles for flow above a solid and a liquid surface. The
boundary layer thickness is denoted by distance δ.
Dor et al. [51]. The second paradox deals with the persistence of IR beyond its theo-
retical limit as determined by the three-shock theory. The presence of expansion waves
behind the reflected shock causes this phenomenon forshocks featuringweak reflections,
such as vNR. This theory was originally suggested by Guderley [52] and experimentally
provedbySkews andAshworth [53]. Hornunget al. [54]suggestedthe“soniccriterion”,
which aided the physical understanding ofdifferent shock wave reflection configurations.
The sonic criterion suggests that in IR, signals generated by the corner of the reflecting
wedge can catch up with the reflecting point, where the Mach stem meets the reflecting
surface, because the flow behind the Mach stem is subsonic. In RR, the sonic signals
cannot catch up with the reflection point. The results from the sonic criterion are very
close to those resulting from the detachment criterion.
In 2007, Ben-Dor [55] systematically classified a wide range of shock reflections over
various surfaces in steady, pseudo-steady, and unsteady flows, including liquid surfaces,
as shown in Figure 3.4. The boundary layer development over liquid surfaces is qual-
itatively different than that of boundary layers on solid surfaces, as the slip condition
generated on the surface of the liquid is different than the no-slip condition of a solid
surface, as shown in Figure 3.3(b). Takayama and Ben-Dor [56] showed that the tran-
23
sition angles over a water surface have a good agreement with the detachment criterion
for 1.47≤ M
s
≤ 2.25 and results disagree for 1≤ M
s
≤ 1.47. The authors speculated
that this behavior was caused by a critical shock Mach number at M
s
= 1.45, which is
the threshold of supersonic and subsonic flow behind the Mach stem. However, apart
from this study [56], shock reflections over liquid surfaces remain largely unexplored.
This may be the result of challenges associated with performing experiments on liquid
boundaries,andperformingnumericalsimulationsfeaturingmultiphasephenomenawith
a deforming free surface.
In addition, no research has yet achieved an understanding of reflection from a non-
Newtonian fluid surface, e.g. shear thickening fluids. Shear thickening fluids have the
combination of fluid-like and solid-like properties depending on the applied shear stress.
Therefore, itwillcertainlybeofgreatvaluetostudyshockwavereflectionoffthesurface
ofcornstarchsuspensions. Recently,thereexistgreatintereststounderstandtheimpact-
activated solidification of shear thickening fluids [57,58]. Earlier work showed that the
transition from fluid-like to solid-like behavior is triggered by a critical shear strain,
rather than a characteristic shear rate [59]. Shock Hugoniot data was experimentally
investigated at different particle velocities in the literature for diverse suspensions [60].
It was shown that there is no difference of the shock Hugoniots between the silicon
carbidesuspensions andpureethylene glycolinthelimitoflowstressloadingconditions.
However, it remains unclear whether this could have the same effect on the shock wave
reflection configuration.
There were many attempts to determine the transition angle with different visual-
ization techniques using either high temporal imaging [48,61–63] or high spatial imag-
ing [64–66]. Because of advances in technology of high temporal imaging systems, many
researchers have been tempted to use a high-speed camera. However, photographs from
high-speed cameras give relatively low resolution in the form of a limited number of
pixels to increase frame rates. Therefore, high temporal imaging can be used to under-
stand succinct details of the entire process, but high spatial imaging will be ultimately
required to give more accurate information [43]. Nevertheless, no studies were found in
the literature to determine the transition angle using both high temporal imaging and
high spatial imaging to complement each other.
24
Types of shock wave reflections
Regular reflection
(RR)
Irregular reflection
(IR)
von Neumann
reflection
(vNR)
Mach
reflection
(MR)
Stationay-Mach
reflection
(StMR)
Direct-Mach
reflection
(DiMR)
Inverse-Mach
reflection
(InMR)
Single-Mach
reflection
(SMR)
Transitional-Mach
reflection
(TMR)
Double-Mach
reflection
(DMR)
Transitioned-regular
reflection
(TRR)
Positive double-Mach
reflection
(DMR+)
Negative double-Mach
reflection
(DMR-)
Terminal double-Mach
reflection
(TDMR)
Pseudo-transitional-Mach
reflection
(PTMR)
Figure 3.4 Possible shock wave reflection configurations. This figure is adapted from “The 13
possible shock wave reflection configurations (p.9),” by Ben-Dor [55].
25
Accordingly, shock wave reflection off the surface of water and cornstarch suspen-
sions, compared to reflections off a solid surface, are investigated using both schlieren
visualizationwithhightemporalandhighspatialresolutiontocompensatetherespective
drawbacks of the two visualization techniques.
3.2 Experimental Setup
For these experiments, the inclined shock tube was used to understand shock wave re-
flectionconfigurationsoverliquids, asshown inFigure3.5. Theshocktubewasdesigned
to be rotated to any angle between 0° and 90° to create wedge-shaped liquid surfaces to
be impinged by a planar shock wave since the liquid surface remained horizontal in the
existence ofgravity. The driver section is 300mm longwith a73mm inner diameter and
the length of the driven section is 650mm, which is approximately 17 times longer than
its side of 38.1mm in order to create a planar shock front [37]. The cross section of the
driver sectionis largerthanthatofthedriven andtestsectiontoproducestrongershock
waves [36]. The area reduction effects are quantitatively investigated in Appendix B.
Between the driver and driven section, moisture-resistant polyester film (12.7, 25.4, and
50.8µm thickness, ±10% tolerance) was used as diaphragm featuring with the driver
pressure of 138, 310, and 586kPa (20, 45, and 85psi, gauge pressure) to create incident
shock Mach numbers, M
s
= 1.20, 1.38 and 1.52, respectively. The driver pressure was
measured by a digital pressure gauge (GE DPI 104, ±0.05%, full-scale accuracy) and
compressed air was used as the driver gas.
The test section has identical square geometry as the driven section to avoid dis-
turbance to the shock front. Three pressure transducers (S1 and S3: PCB 113B21, S2:
PCB 113B31) are used to measure the velocity of the shock wave. The angle of the
shock tube was controlled by the movement of mechanical arms and a threaded rod,
which moved 6.4mm per rotation of the rod, and the angle was measured by a digital
inclinometer (Wixey WR365, ±0.1°). The blade mechanism was located inside of the
driver section to rupture the diaphragm at approximately 90% of the burst pressure, as
shown in Figure 3.5(b). The windows of the test section were made of 12.7mm thick
plexiglass.
26
Driver section Driven section Test section Visible area
Threaded rod
Mechanical arm
300 650 82.6
110 55
S
1
S
2
S
3
64.6
38.1 10
45.5 12.7
73
(a)
Driver section
Driven section
Test section
S
1
S
2
S
3 Pressure transducers:
Windows
Angle inclinometer
Diaphragm
Blade mechanism
(b)
Figure 3.5 Schematic description of inclined shock tube: (a) side and cross view of the inclined
shock tube; (b) detailed cut-view of the inclined shock tube. Dimensions in mm.
27
0 1 2 3 4
Time (ms)
-50
0
50
100
150
200
250
300
350
Overpressure (kPa)
S1
S2
S3
Test time: 700 μs
Figure 3.6 Overpressure-time history from the inclined shock tube at M
s
= 1.38 for pressure
transducers, S
1
, S
2
, and S
3
.
The incident shock Mach number was determined using the elapsed time between
the shock passing the three pressure transducers, as shown in Figure 3.6. The passage
of the shock past each pressure transducer produced an easily identifiable jump in pres-
sure (gauge pressure). By recording the time at which the pressure jumped above a
threshold valueof10kPa, itwas possible to measure the velocity ofthe shock wave with
uncertainties of±3.75m/s in each test.
Schlieren visualization with both high spatial and high temporal resolution are used
and these two methods are summarized in Table 3.1. Figure 3.7 shows regular reflection
offthesurfaceofwateratM
s
=1.52and47°deflectionangleusingschlierenvisualization
with high temporal resolution. Figure 3.8 shows regular and irregular reflection off the
surface of water and cornstarch suspension at M
s
= 1.38 and M
s
= 1.52, respectively,
using schlieren visualization with high spatial resolution.
3.3 Problem Setup
To find the transition angle, experiments were performed by first finding an angle at
which the shock reflection configuration was irregular reflection. To minimize the angle
28
Table 3.1 Summary of camera settings.
Camera Focal length Resolution Scale factor Exposure time Frame rate Time interval
[mm] [pix] [mm/pix] [μs] [fps] [μs]
Phantom V711 50 176× 96 0.45 0.294 200,000 5
Nikon D90 200 4288× 2848 0.027 0.018* 1 N/A
* Obtained by the exposure time of the flash light.
10 mm
(a) 0 μs (b) 10 μs (c) 20 μs
(d) 30 μs (e) 40 μs (f) 50 μs
(g) 60 μs (h) 70 μs (i) 80 μs
Figure 3.7 Schlieren photographs using a Phantom V711 showing shock reflection off a water
surface at M
s
= 1.52 and θ
w
= 47
◦
, resulting in regular reflection. This figure is reproduced
from [45].
29
(a)
1 mm
(b)
(c)
1 mm
(d)
Figure 3.8 Schlieren photographs using a Nikon D90 camera showing (a) regular reflection off
a water surface for M
s
= 1.38 and θ
w
= 45
◦
; (c) irregular reflection off a cornstarch surface for
M
s
= 1.52 andθ
w
= 45
◦
. (b) and (d) are magnified images of the reflecting surface highlighted
by the white rectangle in (a) and (c), respectively. This figure is reproduced from [45].
30
Shock wave
Concave
Convex
Water or
cornstarch suspensions
(a)
Shock wave
Solid wedge
Glass-like
surface
(b)
Figure 3.9 A schematic side-view of the test section: (a) a water or cornstarch suspensions;
(b) a solid wedge. This figure is reproduced from [45].
between the solid and the tangent line tothe liquid surface in the test section due to the
liquidsurfacetension,asshowninFigure3.9(a),ahydrophobiccoating(Rain-X,SOPUS
products)was used onthewindows ofthetestsection. Inordertofindtransitionangles,
the curvature of the liquid was controlled to be slightly convex by filling the test section
with liquid fromthe bottom[56]. Cornstarch suspensions were mixed ataconcentration
of 55 wt.% of cornstarch (Clabber girl) to water [67–69] and the mixture were measure
by a digital scale (Scout pro SP2001, ±0.1g). The dry air in the test section is mixed
withwatervaporinthetestsectionandtheeffectonthespeed ofsoundwasanalytically
investigated in Appendix A. Figure 3.9(b) shows the side-view of the test section with a
solid wedge inserted. The solid wedges were made of polycarbonate sheet (Makrolon
®
GP sheet) with 10 different angles (from 40
◦
to 50
◦
) and the reflecting surface of the
sheet had a surface roughness similar to glass.
Figure 3.10 shows a schematic description of transition angle determined by the
smallest resolvable pixel size. In previous research, the triple point trajectory method
has been used to find the transition angle [56,63,66,70]. In this study, two methods,
pixel intensity and dimensionless Mach stem, are used with schlieren visualization for
both high temporal and high spatial resolution settings. With high temporal resolution,
the existence of the Mach stem and the length of the Mach stem are investigated along
31
Incident shock
Reflected shock
s
θw
(a)
Incident shock
Reflected shock
s
θw
(b)
Figure 3.10 Determination of transition angle. The dotted circle is the smallest resolvable
size of a pixel. Thus, the size of the pixel can change IR to RR: (a) RR with respect to the
resolvable size of the pixel; (b) IR with respect to the resolvable size of the pixel. θ
w
and s
represent deflection angle and the length of the Mach stem, respectively.
the reflecting surface. With high spatial resolution, the existence of the Mach stem is
re-examined to compare the results. The angle of the incident shock wave was increased
until the Mach stem disappears with 1° increment for the pixel intensity method or 5°
for the dimensionless Mach stem method.
3.3.1 Pixel Intensity Method
Figure 3.11 shows high temporal resolution schlieren images and pixel intensities as
a function of pixel locations along the line above the liquid surface. Using the pixel
intensitymethod,theshockwaveconfigurationwasquantitivelydeterminedbydetecting
the existence of the Mach stem. In Figure 3.11(a), two-low pixel intensity locations
are shown corresponding to the location of the incident and reflected shock wave. In
Figure 3.11(b), one low and high pixel intensity locations are detected corresponding to
the Mach stem and slipstream, respectively. In Figures 3.11(a)and (b), decreasing pixel
intensity can be seen behind the reflected shock wave, which was caused by a darker
regionofthe schlieren images. When theshock wave reflects offthesurface ofthe liquid,
large pressure differences exist across the surface between the liquid and the air, and
this pressure differentials create small droplets (or mist) causing the darker region.
32
Pixel location
0 0.2 0.4 0.6 0.8 1
250 50 100 150 200
Pixel intensity
Incident
shock
Reflected
shock
10 mm
(a)
Pixel location
0 0.2 0.4 0.6 0.8 1
250 50 100 150 200
Pixel intensity
Slipstream
Mach stem
10 mm
(b)
Figure 3.11 Pixel intensity images for (a) regular reflection, and (b) irregular reflection. The
pixel values along the white dashed-line are plotted against to normalized location.
3.3.2 Dimensionless Mach Stem Method
Since the results obtained by the pixel intensity method is highly depending on the
spatialresolution,anothermethodwasconsideredtovalidatethereliabilityoftheresults.
Figure 3.12 shows a schematic illustration of dimensionless Mach stem. A length scale
condition in shock wave reflections phenomena was previously suggested, which explains
how a corner-generated signal can affect the characteristic length of the Mach stem [54].
Therefore, for a given value of the incident shock Mach number M
s
and the deflecting
angle θ
w
, the dimensionless Mach stem length s/w is determined since the length of the
Mach stem w gradually increases along the reflecting surface. Then, measured values
of the dimensionless Mach stem s/w are plotted against the deflection angle θ
w
and the
transition angle is obtained by finding the characteristic line.
33
Incident shock
Reflected shock
θw
s
w
Figure 3.12 Schematic illustration of dimensionless Mach stem, s: length of the Mach stem, w:
length of the reflecting surface and θ
w
: deflection angle.
3.4 Results and Discussion
In this section, the experimental results are discussed using two different visualization
setups; schlieren visualization with high temporal or high spatial resolution.
3.4.1 Validation of the Experimental Methodology
Theexperimentalresultsobtainedusingthepixelintensitymethodfromthesolidsurface
were validated against the previous experimental data. Figure 3.13 shows a comparison
of the current experimental study and the previous experimental results [51]. A com-
parison to the previous data was made in terms of the transition angle for hydraulically
smooth surface case. As it can be seen, a good agreement has been found for the tran-
sition angles. Therefore, based on the comparison between the previous and current
results, the methodology in this study is able to ensure confidence in the accuracy for
finding transition angles.
3.4.2 Pixel Intensity Method vs Dimensionless Mach Stem Method
To find the characteristic line in the dimensionless Mach stem method, a weighted non-
linear least-squared method is used with second-order power series equations, then the
line is extrapolated to zero, as shown in Figure 3.14. Different curve fitting equations
wereusedbypreviouswork[66,70]. Inthisstudy,weselectedasecond-orderpower-series
34
1 1.2 1.4 1.6 1.8 2 2.2
25
30
35
40
45
50
55
Detachment criterion
Solid - Previous result
Shock Mach number (M
s
)
Transition angle (deg.)
Solid - Current study
Figure 3.13 The transition angle versus the incident shock Mach number using pixel intensity
compared tothepreviousresults. Uncertainty forpreviousdatapoint isrepresented bymarker
size. This figure is reproduced from [45].
equations because it resulted in the smallest difference between the measured data and
those provided by the model. To provide 95% confidence intervals, both the goodness
of fit and the weighting of individual standard deviations are considered.
Figure 3.15shows the transition angle calculated by different curve fitting equations,
such as a second- and third-orderpolynomial equations and it is clear that allfell within
the range of the confidence intervals.
ResultsfromthepixelintensityandthedimensionlessMachstemmethodsareplotted
inFigure3.16tofindthetransitionanglesfortwodifferentliquidsatthreeincidentshock
Mach numbers. The difference between the two methods is not statistically significant.
However, the nominal values obtained by the dimensionless Mach stem method have
a tendency to show slightly higher transition angles compared to the pixel intensity
method because the transition angles determined by the pixel intensity method can
be more limited by the spatial resolution. Therefore, the transition angles determined
by the pixel intensity method is possibly underestimated. The pixel intensity method
results inasmaller rangeofuncertainties,±0.5
◦
, asthe experiment was performedwith
1
◦
increments. The dimensionless Mach stem method shows a minimum uncertainty of
35
10 20 30 40 50
θ
w
(deg.)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
s/w
Weighted fit
95% Confidence limits
Detachment criterion
Data
(a) Water, M
s
= 1.20
10 20 30 40 50
θ
w
(deg.)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
s/w
Weighted fit
95% Confidence limits
Detachment criterion
Data
(b) Cornstarch suspensions, M
s
= 1.19
10 20 30 40 50
θ
w
(deg.)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
s/w
Weighted fit
95% Confidence limits
Detachment criterion
Data
(c) Water, M
s
= 1.38
10 20 30 40
θ
w
(deg.)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
s/w
Weighted fit
95% Confidence Limits
Detachment criterion
Data
50
(d) Cornstarch suspensions, M
s
= 1.38
10 20 30 40 50
θ
w
(deg.)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
s/w
Weighted fit
95% Confidence limits
Detachment criterion
Data
(e) Water, M
s
= 1.52
10 20 30 40 50
θ
w
(deg.)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
s/w
Weighted fit
95% Confidence Limits
Detachment criterion
Data
(f) Cornstarch suspensions, M
s
= 1.52
Figure 3.14 Dimensionless Mach stem s/w as a function of deflection angle θ
w
. Averaged data
was fitted usinga second-order power-series least squares method for each shock Mach number
and liquid case. Water is shown in (a), (c), and (e) and cornstarch suspension is shown in (b),
(d), and (f). The asterisk represents the detachment criterion.
36
2nd-order power-series
Data
10 20 30 40 50
θ
w
(deg.)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
s/w
95% Confidence limits
Detachment criterion
2nd-order polynomial
3rd-order polynomial
Figure3.15DimensionlessMachstems/w versusdeflectionangleθ
w
withdifferentcurvefitting
methods: Water, M
s
= 1.38.
0.9
◦
and a maximum of 2.4
◦
, which are generated by the 95 % confidence intervals.
3.4.3 Repeatability
Various factors can affect the repeatability of the incident shock waves, such as tem-
perature variations and driver pressure variations; however, the main feature causing
repeatability is breaking of the diaphragm. The uncertainty of measurement of the inci-
dent shock wave is calculated based on the error propagation method [71]. Figure 3.17
shows maximum and minimum range, interquartile range, and median values of each
of the three cases. The actual incident shock Mach number variations are 1.20 ± 0.03
(2.5%), 1.38 ± 0.02 (1.4%), and 1.52 ± 0.03 (2.0%) and the standard deviation in the
three sets are 0.02 (1.7%), 0.01 (0.7%), and 0.01 (0.6%), respectively. Figure 3.18 shows
an overall repeatability of the dimensionless Mach stem method.
3.4.4 Transition Angle
Figure3.19showsthetransitionangleasafunctionofincidentshockMachnumberalong
with the detachment criterion and previous data [56]. The transition angles for solid
37
Pixel intensity
Dimensionless Mach stem
1.1 1.2 1.3 1.4 1.5 1.6
38
40
42
44
46
48
Shock Mach number (M
s
)
Transition angle (deg.)
(a)
1.1 1.2 1.3 1.4 1.5 1.6
38
40
42
44
46
48
Transition angle (deg.)
Shock Mach number (M
s
)
Pixel intensity
Dimensionless Mach stem
(b)
Figure 3.16 The transition angle versus the incident shock Mach number using pixel intensity
and dimensionless Mach stem method: (a) water; (b) cornstarch suspensions.
Cases
1.2
1.3
1.4
1.5
Shock Mach number (M
s
)
Cornstarch suspensions
Water
M
s
= 1.20 M
s
= 1.38 M
s
= 1.52
Figure 3.17 Incident shock Mach number for the entire sets of experiments. Upper, lower,
and square boundary represent maximum, minimum, and interquartile range of incident shock
Mach number, respectively. The center line inside of the square indicates median values.
38
1 2 3 4 5
Test No.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
s/w
Figure 3.18 An overall measure of dimensionless Mach stem s/w at M
s
= 1.38 and θ
w
=
30degree; five independent experiments.
surface were measured 1.5
◦
lower than the detachment criterion. The primary reason of
the persistence of RR below the detachment criterion is likely caused by the boundary
condition over a solid because the detachment criterion was based on an inviscid-solid
boundary condition as illustrated in Figure 3.3(a).
The observed transition angle of the shock off the solid wedge using the pixel inten-
sity method is 2 ± 0.5
◦
higher than that of the results from the water or cornstarch
suspensions. This is because the flow velocity behind the reflected shock wave over
liquids becomes slower due to the dispersed droplets compared to solid boundary sur-
face, as shown in Figure 3.8. Moreover, the difference is primarily caused by the energy
transmission intowater orcornstarch suspensions, which doesnotexist inthe solidcase.
There is a slight difference between the previous and the current data from shock
Mach number M
s
= 1.38 and higher. It is not yet clear what is the cause for the
discrepancy withthe results obtainedby Takayama andBen-Dor et al. [56], butneither
error bars or spatial resolution were presented in the previous work so a fair comparison
is hardtomake. The criticalvalue ofM
s
=1.45which was presented asthemain reason
to the change in transition behavior is not observed in the current study.
The difference inthetransitionanglebetween water andcornstarch suspension using
39
1 1.2 1.4 1.6 1.8 2 2.2
25
30
35
40
45
50
55
Detachment criterion
Water - Pixel intensity
Cornstarch - Pixel intensity
Water - Previous result
Water - Dimensionless Mach stem
Cornstarch - Dimensionless Mach stem
Shock Mach number (M
s
)
Transition angle (deg.)
Solid - Pixel intensity
Figure 3.19 The transition angle versus the incident shock Mach number.
the two methods is not statistically significant even though the viscosity and density
betweenthetwoliquidsaredifferent. Thisbehaviorhasagoodagreementtotheprevious
study in terms of the shock Hugoniot data in low stress loading condition [60].
40
Chapter 4
Shock Wave Attenuation by Liquid Sheets
This work was published in Experiments in Fluids under Shock wave interactions with
liquid sheets by Jeon and Eliasson [72].
4.1 Background
In response to the need for a cost-effective method to attenuate shocks in large-scale
applications, there has been interest in using water, which is readily available and envi-
ronmentally friendly. By introducing liquid droplets, the momentum ofthe shocked flow
can be transferred to the droplets. One of the first attempts to mitigate shock waves
using water invoked the process of water droplet atomization [32,73,74]. The process
of water deformation and fragmentation by aerodynamic forces is known as secondary
atomization [75] and is divided into two stages. During the first stage, each droplet
experiences deformation. If the interfacial tension and viscous forces are smaller than
the outside forces, the droplets enter the second stage. In the second stage, known as a
fragmentation process, the larger-deformed droplets are broken into smaller ones. Dur-
ing the fragmentation process, the energy of the shock waves can be used to increase
the exchange surface by droplet evaporation between gas and droplets. This process can
also decrease the pressure and gas velocity of the shocked gas flow, due to momentum
exchange [32].
Another method employed to mitigate the effects of blast or shock waves involves
the use of bulk water. A large body of water can be used to replace conventional con-
crete barriers or protective gears and armors. Previous investigations mostly focused
on blast wave mitigation with bulk water and showed a significantly reduced amount of
shocked-gas pressure and impulse from explosions [73]. Meekunnasombat et al. [76] in-
41
vestigated shock wave accelerated liquid layers using a vertical shock tube. The authors
of the studies used X-ray radiography to measure the volume fraction of fragmented-
water induced by the passage of a shock wave and measured end-wall overpressure to
understand the shock mitigation effect. They concluded that multiple water layers for
a given volume of water can be used to decrease the end-wall loading. Chen et al. [77]
showed that the blast overpressure can be reduced by 97% using body armorcontaining
water-filled plastic tubings compared with cases without any protection. However, the
results are not clear in terms of how much attenuation was achieved due to the wa-
ter itself. Son et al. [78] studied blast mitigation using unconfined 3mm thick water
sheets with an explosively driven shock tube. They revealed a significant degree of blast
mitigation, by as much as 82% of the peak pressure and 75% of the impulse using a
water sheet. However, the drawback was that the results could not provide a detailed
explanation of the pressure-time history due to a limited visualization area.
Addingdenseparticlesisonewaytoincreasethestiffnessofliquid,inordertoachieve
solid-like behavior. Shear thickening is a type of non-Newtonian behavior in which the
liquid’s viscosity increases with shear rate. One common and widely available shear
thickening fluid is created by mixing cornstarch in water. A cornstarch suspension is
oftenreferredtoasadiscontinuousshear-thickeningfluidbecauseitsviscositydrastically
jumps with increasing shear rate [79]. Possible applications of discontinuous shear-
thickening fluids include shock absorptive gears, such as flexible protective armors [57],
which can dissipate vibrations without losing stiffness. These shear-thickening fluids
can also be used for skis and tennis rackets [80]. Recently, there has been an interest
from the physics community in shear-thickening materials for soft matter systems using
the concept of dynamic jamming [79]. To understand the impact resistance and solid-
like behavior of shear-thickening fluids at high-strain loading rates, Kolsky bar (split
Hopkinson pressure bar) experiments were performed on a silica-based suspension to
measurepressurepulsemitigation. Resultsshowed thatthetransmittedpulseamplitude
and width was much weaker and longer, respectively, compared to cases using ethylene
glycol [81]. Petel et al. [82] investigated ballistic resistance in different particle-based
shear-thickening fluids and showed that the loss of ballistic resistance was caused by the
material strength of the particles in suspensions. Furthermore, aluminum rod impact
42
experiments were conducted to understand how a cornstarch suspension absorbs the
impact energy of a mechanical rod impact [58]. The authors showed that the growth of
adynamic jammingregion generated by the impact is the main mechanism thatabsorbs
theimpactenergyofthemechanicalrod. Rocheet al. [83]investigateddynamicfracture
of a cornstarch suspension impacted by a cylindrical metal rod and showed a fluid-to-
solid transition. They quantified the velocity of the crack tip and the number of cracks
andthenestimatedtheshearmodulusofthefracturedmaterial. However, itisunknown
if conclusions drawn from the physical impact of a rod can be extended to shock wave
absorption.
In this study, shock wave interactions with planar water and cornstarch sheets are
investigated experimentally. The goals are to investigate if a transmitted shock wave
through liquid sheets can be observed; obtain detailed information of the pressure-time
history during and after the shock wave impacts onto the liquid sheet; and finally to
quantify the shock wave attenuation effect caused by the liquid sheets.
4.2 Experimental Setup
Figure 4.1 shows a conventional horizontal shock tube, which was used to investigate
shock wave interaction with liquid sheets. The driver section is 96.8mm in diameter
with a circular cross-section, while the driven and test section have 63.5mm sides with
square cross-sections. To create a planar shock front, the length of the driven section
is about 30 times longer than the side of the driven section [37]. The driver pressure
was measured by a digital pressure gauge (GE DPI 104, ±0.05%, full-scale accuracy)
and compressed air was used as the driver gas. The blade mechanism inside of the
driver section ruptures the diaphragm, as discussed in Section 2.2. Between the driver
and driven sections, plastic diaphragms (moisture-resistant polyester film, 50.8µm and
102µmthickness,±10%tolerance)areusedfeaturingwiththedriverpressureof310kPa
and 586kPa (45 and 85psi) to create incident shock Mach numbers M
s
= 1.34 and
1.46, respectively. The driven and test sections contained air at atmospheric pressure
(101kPa) and room temperature (297K). Four pressure transducers (S
1
and S
4
: PCB
113B21, S
2
and S
3
: PCB 113B31, flush mounted to the shock tube wall) were installed
43
Driver section Driven section Test section
648 1860
1260
350 200 200 200 200
96.8 63.5
S
1
S
2
S
3
S
4
Visible area
Figure 4.1 Schematic description of shock tube: (a) side view of the shock tube. Dimensions in
mm. S
1
and S
2
will be referred to as the upstream transducers and S
3
and S
4
will be referred
as the downstream transducers.
to measure overpressure and the velocity of the shock wave.
The shock Mach numbers could be reproduced within±0.75% with a measurement
uncertainty less than±0.82% in each experiment, including a±0.1% machining toler-
ance, a±0.79% uncertainty of measuring pressure rise time, and a±0.37% uncertainty
of temperature measurement. The uncertainty was calculated using error propagation
method. Previous published studies also show the same order of the uncertainties in
Mach number measurements of±0.1% and±0.17% [62,63].
Figure4.2showsdetailedschematicviewofthetestsection. Thetestsectionconsists
of three separated parts with constant inner cross-section: upstream, downstream and a
liquid sheet container. The sides of the test section were made of transparent windows
(impact-resistant polycarbonate, 12.7mm thickness), which were machined by a water-
jet cutter to minimize stress concentration at the edges.
To find a method of suspending liquids in the test section, different alternatives were
considered in the design process. One of the first concepts was a sheet from free falling
water sheet, as previously shown in [78]. After preliminary investigations, it was seen
that this method prevented precise pressure measurements caused by leakage from gaps
between the liquid supply system and the test section. In addition, the thickness of the
liquids was not constant and buildup of liquid on the floor of the test section was an
issue. Instead, a novel liquid sheet container was designed (inspired by [76]) to provide
shock-liquid interactions with a planar 5 or 10mm thickness, as shown in Figure 4.3. To
suspend liquids inside of the liquid container, plastic sheets (12.5µm, Saran wrap) and
44
S
1
S
2
S
3
S
4
Shock
wave
Pressure
transducers
Windows
Liquid sheet
container
Figure 4.2 Schematic description of detailed cut-view of the test section. Dimensions in mm.
S
1
and S
2
will be referred to as the upstream transducers and S
3
and S
4
will be referred as
the downstream transducers.
Cap
12.7 mm
Support wires
O-ring
Support membrane
Figure 4.3 Schematic description of liquid sheet container. One layer of support membrane
and 4 cotton wires were used on each side to maintain the shape and minimize bulge.
45
Table 4.1 Summary of camera settings. This table is reproduced from [72].
Visualization Resolution Scale factor Exposure time Frame rate Time interval
[pix] [µm/pix] [µs] [fps] [µs]
Direct 720× 264 312 10 35,015 28.6
Schlieren 272× 160 384 0.294 96,045 11.4
cotton wires (5µm thickness, 12.7mm distance) were used to maintain a near constant
sheet thickness and to minimize bulge. The cornstarch suspension was prepared as the
same procedure introduced in Section 3.3. The sides of the liquid sheet containers have
O-rings to prevent leakage when joined to the upstream and the downstream sections of
the shock tube.
To visualize the shock wave interaction with liquid sheets, both schlieren and direct
visualization with high temporal resolution were used as described in Section 2.4. See
more details in Table 4.1.
4.3 Results and Discussion
To clarify shock wave interactions with just the plastic sheets and the cotton wires
of the liquid sheet container, both high-speed schlieren visualization and overpressure-
time history were collected. Then, shock wave interactions with water and cornstarch
suspensions were investigated and analyzed.
4.3.1 Shock Wave Attenuation by Plastic Sheets and Cotton Wires
Figure 4.4 shows a series of high-speed schlieren images of shock wave interactions with
the plastic sheets and cotton wires, but without liquid in between the plastic sheets.
Stress concentrations caused by the bolted joints were visible as dark regions between
the square frame and windows. The transmitted and reflected shock waves are clearly
seen in Figure 4.4(c). The plastic sheets instantaneously break upon contact with the
shock wave. Two reflected shock waves are generated by the front and back membranes,
respectively, as shown in Figure 4.4(d). In Figures 4.4(d)–(f), the second reflected shock
46
Incident
shock
Liquid
container
10mm
(a) 0µs (b) 57µs
Reflected
shock
Transmitted
shock
(c) 114µs
Plastic
sheet
(d) 171µs (e) 228µs (f) 285µs
Figure 4.4 A series of high-speed schlieren images of shock wave interaction with just the
plastic sheets and the cotton wires on incident shock Mach number of M
s
= 1.46. The white-
dashed lines denote the position of the empty liquid sheet container. This figure is reproduced
from [72].
wave catches up with the first and they coalesce into a single shock wave due to the
different flow condition behind the first reflected shock wave. The transmitted and
reflected shock waves are slightly curved, because of the slightly uneven surface of the
plastic sheets.
Figures 4.5(a) and (b) show x−t diagrams of the paths of the incident, reflected,
compression and expansion waves in the just a shock tube and a shock tube equipped
with two plastic sheets and cotton wires, but without water between the plastic sheets.
In Figure 4.5(a), when the membrane between the driver and driven sections bursts
at the desired pressure, an incident shock wave travels toward the driven section that
is followed by a contact surface. At the same time, the expansion wave travels in the
opposite direction along the driver section and is reflected by the end of the driver
section. In Figure 4.5(b), the incident shock wave interacts with the polyester films in
the test section and yields the transmitted and reflected shock wave. The compression
47
and expansion waves are generated when the plastic sheets start to move downstream.
The time, t
0
, is the arrival time of the incident shock wave and t
1
is the arrival of the
expansion fan at S
1
and t
3
at S
3
.
In Figure 4.6, pressure and impulse histories measured from the sensors along the
test section are plotted to understand the effect of the plastic sheets and cotton wires at
two shock Mach numbers, M
s
= 1.34 and 1.46. Each experiment was performed at least
three times under the same conditions, and only averaged pressure impulses are shown
here. Figures 4.6(a) and (b) shows sharp jumps occurring between time t = 2 and 3ms,
which are generated by the reflected shock wave. Impulse, I, can be calculated as
I =
Z
t
1,3
t
0
(p−p
0
)dt, (4.1)
where t is the time, p is the overpressure, and p
0
is the atmospheric pressure. In the
present study, the time, t
1
, is chosen to be the arrival of the expansion fan at S
1
and t
3
at S
3
to determine the impulse. The time, t
0
, can be defined as the time of the arrival
of the incident shock wave. where t is the time, p is the overpressure, and p
0
is the
atmospheric pressure. In Figures 4.6(c) and (d), the impulse shows a small deviation
of less than 5% caused by the reflected wave from the plastic sheets and cotton wires.
Mach numbers measured by the upstream and downstream sensors show a consistent
wave speed with the plastic wrap and cotton wires. Therefore, shock wave attenuation
by the plastic sheets and cotton wires is negligible (<5%), which is in agreement with
previous results [76].
4.3.2 Shock Wave Attenuation by Liquid Sheets
Figure 4.7 shows a sequence of schlieren high-speed photographs for a 5mm thick water
sheet impacted by M
s
= 1.46. It can be seen that no transmitted wave is observed
through the water sheet. The multiple circular arcs seen behind the reflected shock
wave were generated by the cotton wires.
Direct high-speed images, shown in Figures 4.8 and 4.9, were obtained to observe
the breakup processes of the two types of liquid sheets when impacted by a planar
shock wave. The time instant t = 0ms corresponds to the time when the shock wave
48
S
1
S
2
S
3
S
4
Position (x)
Time (t)
t
0
t
3
Incident shock wave
Contact surface
Expansion wave
t
1
(a)
S
1
S
2
S
3
S
4
Position (x)
Time (t)
t
0
t
3
Incident shock wave
Plastic sheet
Reflected shock wave
Contact surface
Compression
wave
Expansion wave
Trasmitted
shock wave
t
1
(b)
Figure 4.5 An x−t diagram depicting the shock wave interaction for (a) just the shock tube,
and (b) a shock tube containing the two plastic sheets and cotton wires but without water in
between the plastic sheets. This figure is reproduced from [72].
49
0 1 2 3
Time (ms)
0
50
100
150
200
Overpressure (kPa)
-0.01 0 0.01 0.02
Time (ms)
0
100
200
Overpressure (kPa)
Plastic sheet & cotton wires (S
1
)
Plastic sheet & cotton wires (S
3
) Empty (S
3
)
Empty (S
1
)
(a)
0 1 2 3
Time (ms)
0
50
100
150
200
Overpressure (kPa)
-0.01 0 0.01 0.02
Time (ms)
0
100
200
Overpressure (kPa)
Plastic sheet & cotton wires (S
1
)
Plastic sheet & cotton wires (S
3
) Empty (S
3
)
Empty (S
1
)
(b)
0 0.5 1 1.5 2 2.5 3 3.5
Time (ms)
0
100
200
300
400
500
600
Impulse (kPa · ms)
Plastic sheet & cotton wires (S
1
)
Plastic sheet & cotton wires (S
3
) Empty (S
3
)
Empty (S
1
)
(c)
0 0.5 1 1.5 2 2.5 3 3.5
Time (ms)
0
100
200
300
400
500
600
Impulse (kPa · ms)
Plastic sheet & cotton wires (S
1
)
Plastic sheet & cotton wires (S
3
) Empty (S
3
)
Empty (S
1
)
(d)
Figure 4.6 Comparison of the overpressure and impulse histories recorded by the sensors S
1
andS
3
for two shock Mach numbers of M
s
= 1.34 and 1.46 during experiments with an empty
test section containing only the two plastic sheets and cotton wires: (a) and (b) overpressure
versus time; (c) and (d) impulse versus time. The inserts in (a) and (b) show overpressure
versus time from -0.01ms to 0.02ms. This figure is reproduced from [72].
50
10 mm
Incident
shock
Water sheet
(a) 0.740ms (b) 0.792ms
Reflected
wave
(c) 0.844ms
(d) 0.896ms (e) 0.948ms (f) 1.00ms
Figure 4.7 A series of schlieren high-speed images for a 5mm thick water sheet at M
s
= 1.46.
The incident shock wave is reflected by the water sheet. The multiple circular arcs behind the
reflected shock wave are generated by the cotton wires.
arrives at sensor S
1
. Figure 4.8 shows a series of high-speed images for the 10 mm thick
water sheet at M
s
= 1.46. In Figure 4.8(b), the water sheet, after the incident shock
wave had been reflected, was deformed due to the high-pressure at the upstream sec-
tion. The deformation was associated with the upstream-propagating expansion waves
and the downstream-propagating compression waves. The top and bottom portions of
the sheet were broken first and the water droplets were fragmented immediately after
the deformation by the shock wave as shown in Figures 4.8(c) and (d). The larger
fragmented-water volume fraction indicates more energy dissipation of the shocked-gas
flow. The fragmented water droplets arrived at sensor S
3
after approximately 3.3ms.
Figure 4.9 shows the 10mm thick cornstarch sheet breakup process at the incident
shock wave Mach number M
s
=1.46. In Figure 4.9(b), the evenly-deformed surface was
caused by shock-activated solidification ofcornstarch suspensions. The shock-driven im-
pact onto the surface ofa cornstarch suspension created a compression ofthe cornstarch
particles and resulted in a rapidly growing jammed region similar to that observed in
51
Water sheet
S
3
20 mm
(a) 0.572ms (b) 1.43ms
(c) 2.29ms (d) 3.14ms
(e) 4.00ms (f) 4.86ms
Figure 4.8 A series of high-speed images for a 10mm thick water sheet at M
s
= 1.46. S
3
is
located in the right-top side of the test section. This figure is reproduced from [72].
52
Cornstarch sheet
20 mm
S
3
(a) 0.575ms (b) 1.43ms
Cotton
Wires
(c) 2.29ms
Water
droplets
(d) 3.15ms
(e) 4.00ms (f) 4.86ms
Figure 4.9 A series of high-speed images for a 10mm thick cornstarch suspension sheet at
M
s
= 1.46. This figure is reproduced from [72].
53
the mechanical impacts by [58]. After the sheet underwent the deformation process,
the cornstarch suspension showed piston-like motion or a solid-like behavior contrary
to the water sheet, as illustrated in Figures 4.9(c) and (d). The breakup time of the
cornstarch sheet, compared to the water sheet, was longer because of the increased vis-
cosity between cornstarch particles in the suspension. After the cornstarch sheet was
broken, fragmented water droplets were immediately squeezed out from the cornstarch
suspension, as shown in Figure 4.9(d). Theoretical models to understand the shock
wave attenuation by the cornstarch sheet are not available at this point, but it can
be qualitatively explained by shear thickening in colloidal dispersions. Since multiple
colloidal cornstarch particles approach each other by the high pressure caused by the
shock wave, the hydrodynamic pressure between the particles extracts water from the
cornstarch suspension [84]. The particle concentration became denser by water being
extracted from the suspension, so the transmitted energy was dissipated by a higher
viscosity. Brittle fracture was observed on the surface of the cornstarch suspension to
dissipate the impact energy. In Figure 4.9(f), the larger size cornstarch particles were
broken up into smaller sizes, but the particle sizes were larger than those of the water
droplets under the same experiment conditions.
A similar breakup behavior was observed for the 5mm thick cornstarch sheet im-
pactedbyanincidentshockwavewithMachnumberM
s
= 1.46,asshowninFigure4.10.
The breakup time of the 5mm thick cornstarch sheet was reduced and the size of the
fragmented particles was smaller than compared to the 10mm thick cornstarch sheet.
Figure 4.11 shows the overpressure and impulse measured by S
1
and S
3
at incident
shock Mach numbers, M
s
= 1.34 and 1.46, during experiments with 5mm and 10mm
thick water and cornstarch sheets. In Figure 4.11(a), the first pressure jump at t = 0
is generated by the arrival of incident shock wave. The same amplitude and width of
the first jump at M
s
= 1.34 and 1.46 represent the reproducibility of the repeated at
least three experiments each case. The second pressure jump is created by the reflected
shock wave and the arrival time is different because of higher reflected Mach numbers
caused by higher incident shock Mach numbers. The arrival time and overpressure
of the reflected shock wave, for an incident shock Mach number of t = 2.10ms are
M
s
= 1.34. For an incident shock Mach number M
s
= 1.48, the arrival time is t =
54
Cornstarch sheet
20 mm
(a) 0.575ms
Cotton
Wires
(b) 1.43ms
Water
droplets
(c) 2.29ms (d) 3.15ms
(e) 4.00ms (f) 4.86ms
Figure 4.10 A series of high-speed images for a 5mm thick cornstarch suspension sheet at
M
s
= 1.46.
55
0 0.5 1 1.5 2 2.5
t (ms)
0
50
100
150
200
250
300
350
400
Overpressure (kPa)
5mm water
10mm water
5mm cornstarch
10mm cornstarch
Line thickness
M
s
= 1.46
M
s
= 1.34
Line type
Incident
shock
Reflected
shock
(a) S
1
1 1.5 2 2.5 3 3.5
Time (ms)
0
20
40
60
80
100
Overpressure (kPa)
Compression
wave
Liquid
impact
Coalesed shock
wave
(b) S
3
0 0.5 1 1.5 2 2.5
t (ms)
0
100
200
300
400
500
600
Impulse (kPa · ms)
(c) S
1
1 1.5 2 2.5 3 3.5
t (ms)
0
20
40
60
80
100
120
Impulse (kpa · ms)
(d) S
3
Figure 4.11 Comparison of the overpressure and impulse histories measured at S
1
and S
3
during experiments with 5 mm and 10 mm thick water or cornstarch suspension sheets for
incident Mach shock numbers of M
s
= 1.34 and 1.46: (a) overpressure versus time at S
1
; (b)
overpressure versus time at S
3
; (c) impulse versus time at S
1
; (d) impulse versus time at S
3
.
This figure is reproduced from [72].
56
2.05ms and overpressure is 358kPa (5mm thick water sheet) and 378kPa (10mm thick
water sheet). The variation in reflected overpressure is caused by an early breakup time
of the 5mm liquid sheets for both water and cornstarch suspensions.
In Figure 4.11(b), the peak overpressure measured at the downstream sensor S
3
is much lower than that recorded at the upstream sensor S
1
since nearly all of the
incident shock wave is reflected by the liquid sheets in agreement with previous results
by [85]. Oscillatory traces of the pressure measurements (frequency≈ 3kHz) are caused
by vibrations of the side windows (natural frequency ≈ 3.5kHz), which occur in every
experiment no matter if the test section is empty or equipped with a liquid sheet. In the
case of the 5mm thick liquid sheets at a higher Mach number of M
s
= 1.46, a higher
peak overpressure is obtained. Two sharp jumps, denoted compression wave and liquid
impact in Figure 4.11(b), are observed for incident shock Mach number M
s
= 1.46,
with an exception for the case of the 10mm thick cornstarch sheet. As previously
discussed by Henderson et al. [85] and Sakurai [86], only a small fraction of the energy
(3.5% at M
s
= 3.5) from the incident shock wave is transmitted into the liquids due
to the impedance mismatch. Consequently, the first pressure jump is caused by the
arrival of compression waves, which are generated by the breakup of the liquid sheet.
Further downstream, the compression waves coalesce into a shock wave that propagates
faster than the shocked liquid layer. A similar behavior was also seen in the study
by meekunnasombat et al. [76]. The second pressure jump is generated by the arrival
of water droplets or cornstarch suspension particles. Water sheets show higher peak
overpressure than cornstarch sheets forthe same incident shock Mach number and sheet
thickness. In Figure 4.11(c), no discrepancy of the total impulse can be seen at M
s
=
1.34,butasmalldiscrepancy (6%)isshownatM
s
=1.46aftert=2ms,whichiscaused
by the different amplitude of the reflected shock waves. In Figure 4.11(d), the overall
trend shows a similar result in the overpressure history.
Based on the pressure profiles, a schematic description of x - t diagram of the paths
ofthe incident, reflected, expansion, compression waves alongthe shock tubeis depicted
in Figure 4.12.
Table 4.2 shows reflected and coalesced Mach numbers, M
r
and M
c
, calculated by
the upstream and downstream sensors using a speed of sound value calculated based
57
S
1
S
2
S
3
S
4
Position (x)
Time (t)
t
0
t
3
Incident shock wave
Liquid sheet
Reflected shock wave
Contact surface
Compression wave
Expansion wave
Coalesced
shock wave
t
1
Figure 4.12 A schematic x−t diagram of the shock wave interaction with a liquid sheet. This
figure is reproduced from [72].
Table 4.2 Reflected and coalesced Mach numbers (M
r
and M
c
). M
r
is calculated based on the
shock arrival time between S
1
and S
2
and M
c
is calculated between sensors S
3
and S
4
. This
table is reproduced from [72].
Type
M
s
= 1.34 M
s
= 1.46
M
r
M
c
M
r
M
c
5 mm water 1.26 1.03±0.06 1.35 1.22±0.02
5 mm cornstarch 1.27 0.844 ±0.06 1.35 1.03
10 mm water 1.27 1.00±0.02 1.35 1.00±0.08
10 mm cornstarch 1.27 [ - ] 1.36 [ - ]
on analytical values at room temperature. The reflected and coalesced Mach numbers
increase with the higher incident shock Mach number. The coalesced Mach numbers M
c
are near or under sonic velocity, except for the 5 mm water sheet featuring M
s
= 1.46.
The coalesced Mach number resulting from the case with the 10mm cornstarch sheet
is not available since the overpressure is significantly lower and cannot be distinguished
from the noise level of the sensor.
Based on the pressure and impulse traces, several observations can be made: (a)
no transmitted shock wave is detected in this study; instead, compression waves are
generated by the accelerated liquid layer in the downstream section and compression
58
waves later coalesce into a shock wave; (b) a thinner liquid sheet results in a higher
side-wall peak overpressure (up to 200% higher) and impulse (up to 15% higher), and
it results in a stronger coalesced shock wave, especially for the case of a 5mm water
sheet at M
s
= 1.46 in which M
c
= 1.22± 0.02; (c) a cornstarch sheet shows a lower
peak overpressure (up to 35% lower) and impulse (up to 12% lower) when water and
cornstarchsuspensionofthesamethicknessarecompared,whilethestrengthofcoalesced
wave results in similar speed (near sonic velocity); and (d) a cornstarch sheet shows a
delayedbreak-uptimecomparedtowatersheetsbecauseliquid-to-solidtransitionoccurs
with the impact of the shock wave.
On the basis of the impulse profile, shock wave reflection factors K
R
and a shock
wave attenuation factor K
A
are calculated for combinations of liquid sheets for incident
shock Mach numbers M
s
= 1.34 and 1.46, as shown in Figure 4.13. The shock wave
reflection and attenuation factors, K
R
and K
A
, are defined as
K
R
=
I
1
I
0
, (4.2)
K
A
=
I
3
I
0
, (4.3)
where I
1
is the impulse recorded at sensor S
1
with a liquid sheet (water or cornstarch),
I
3
is the impulse recorded at sensor S
3
with a liquid sheet, and I
0
is the impulse for
a square frame without liquid between the plastic sheets. Therefore, the shock wave
reflection and attenuation factors represent the value of the impulse with a liquid sheet
normalized by the impulse without the liquid.
As shown in Figure 4.13(a), the shock wave reflection factor obtained for an incident
shockMachnumberofM
s
= 1.34isnotstatisticallydifferentfromcasetocase. However,
there is a difference in the results for the higher shock Mach number M
s
= 1.46. This
difference was mainlycaused byavariationofbreakup timeinstants oftheliquid sheets.
The shock wave attenuation factor for 10mm water and 5mm cornstarch sheets at
M
s
= 1.34 and 1.46 is not significantly different, as shown in Figure 4.13(b). However,
thecornstarchsheetshowslowerattenuationfactor(higherattenuationeffect)compared
to the same thickness of the water sheet. The shock wave attenuation factor was mainly
influencedbythecompressionwaves(duetotheshock-acceleratedliquidlayer)thatlater
59
1.34 1.46
M
s
1.3
1.35
1.4
1.45
1.5
1.55
K
R
5mm water
5mm cornstarch
10mm water
10mm cornstarch
(a) Shock wave reflection factor
1.34 1.46
M
s
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
K
A
5mm water
5mm cornstarch
10mm water
10mm cornstarch
(b) Shock wave attenuation factor
Figure 4.13 Shock wave reflection and attenuation factors, K
R
and K
A
, for 5mm and 10mm
thick water and cornstarch sheets for incident shock Mach numbersM
s
= 1.34 and 1.46. Error
bars represent standard deviations.
coalesced into a shock wave. One can note that the shock wave attenuation was mainly
achieved by bulk viscosity of the cornstarch suspension caused by rapid compression
regions [87].
60
Chapter 5
Shock Wave Attenuation by Thin Films
5.1 Background
When a planar shock wave interacts with a deformable thin film, the major mechanism
of shock wave attenuation using thin films is mainly achieved by the acoustic impedance
mismatch between the shock and the film (the acoustic impedances Z is defined as the
product of density ρ and sound speed c). In Figure 5.1, the incident shock wave I
impacts the film and a reflected shock R
1
and a transmitted shock T
1
are generated by
the impedance mismatch in air-film-air interface (zoom in region A). Since the acoustic
impedance of the film is several orders of magnitude higher than that of air, most of the
incident energy is reflected and atransmitted wave propagates tothe right. Consecutive
reflectionsofwavestrappedwithinthefilmgeneratethesecondreflectedwaveR
2
andan
finite number of reflections will occur in the film before the film propagates to the right.
When the film thickness is similar to the shock wave thickness, successive reflections
by the interface within the film occur in a short time depending on the thickness and
the acoustic impedance of the film, then compression waves coalesce into a shock wave.
Later, the film is accelerated in the same direction as the incident shock wave and the
initial motion of the film generates compression and expansions waves. Compression
waves propagate into the region ahead of the film and then coalesce into a shock wave.
While there has been a few experimental and numerical work focused on shock wave
attenuation using thin materials, these investigations were conducted with materials
whose thickness was at least three orders magnitude greater than the shock thickness
(200nm at standard temperature and pressure [1]).
The interaction between a shock wave and a film was originally investigated by
Meyer [88]. An approximate theoretical solution was derived in one-dimensional motion
61
x
t
Transmitted shock
Incident shock
Reflected shock
Compression
wave
Expansion
wave
Film
I
R
1
T
1
T
2
T
3
R
2
R
3
Air Air Film
A
Zoom in region A
Figure 5.1 A schematic x - t diagram of shock wave interaction with a film.
withoutconsideringspecificentropychangesthroughthetransmittedandreflectedshock
waves. The results were compared to an experimental investigation by using 254, 432,
and762µmthickcelluloseacetate,whichwerefreetomoveunderimpact,inashocktube
at the incident shock Mach number of 1.5. Results showed a good agreement (within
approximately 7.0%) between the analytical and experimental methods. It also has
been shown that the transmitted shock wave and the film trajectories were independent
of the mass of the film.
Nakagawa et al. investigated the shock wave attenuation using surgical materials to
protect brain tissue from shock waves [89]. Underwater shock waves were generated
by a Ho:YAG laser with a maximum overpressure of 5MPa(50 bar) and shadowgraph
visualization together with pressure measurements by ultrasonic polyvinylidene flouride
(PVDF) needle hydrophones were used to obtain the experimental results. Results
showed that surgical cotton, 0.7mm thick polytetrafluoroethylene (PTFE) membrane
(Gore-Tex membrane), and surgical gauze reduced the shock wave overpressure up to
50%, 96%, and 20%, respectively. Unfortunately, the thickness of surgical cotton and
surgical gauze was not presented. The authors of the studies explained that the 0.7mm
thick Gore-Tex membrane showed higher degree of attenuation because of its highly
closed porous structure caused by the three-dimensionally oriented thin fibers.
Saito et al. used analytical, numerical, and experimental methods to understand un-
derwater shock wave attenuation by a 0.3mm thick Gore-Tex membrane for protecting
soft tissue [90]. Experiments were performed by using a Nd:YAG laser, double expo-
sure holographic interferometry, and PVDF needle hydrophones. Results showed that
62
a Gore-Tex layer can decrease overpressure over 90%, 99.9%, and 99.8% by the ex-
perimental, analytical and numerical methods, respectively. The numerical result was
obtainedby0.7mmthickGore-Texmembrane, whichwasdifferentfromtheexperiments
due to numerical simulation difficulties. The thickness was neglected in the analytical
derivation. The authors concluded that shock wave attenuation was mainly achieved by
the impedance mismatch, i.e. the air-trapped in a complex Gore-Tex structure.
Murray et al. investigated the influence of the diaphragm mass on the transmitted
shock wave through thin films [91]. Different mass of polyvinylidene chloride (PVC)
and polyethylene diaphragms were used to measure the velocity and strength of the
transmittedshockinshocktubeexperiments. Resultsshowedthatthetransmittedshock
characteristics were dependent on the diaphragm material properties and experimental
results did not agree with the analytical solution suggested by Meyer [88].
Despite these investigations, there is no research performed to understand the shock
wave attenuation by the scale of events on the relative thickness of the materials com-
pared to the shock thickness. The main focus of the present study is to determine if
the thickness of thin films could affect the strength of the transmitted shock wave when
the material thickness is of same or two orders of magnitude greater than the shock
wave thickness. Here, polyester and aluminum film were chosen, with the intent to also
investigate how the failure mechanism (ductile or brittle) influences the shock wave at-
tenuation. A high-speed schlieren imaging technique with simultaneous measurements
from high-frequency pressure sensors were used to understand the resulting interaction
of the shock wave and the film.
5.2 Experimental Setup
The experiments were performed with a conventional pressure driven shock tube, as in-
troducedinSection4.2. Betweenthedriveranddrivensections,fourdifferentthicknesses
of plastic diaphragms (moisture-resistant polyester film, 25.4, 50.8, 76.2 and 102µm
thickness, ±10% tolerance) were used. The driver pressure was measured by a digital
pressure gauge (GE DPI 104,±0.05%, full-scale accuracy) and compressed air was used
as the driver gas. The driver pressures were set to 137, 310, 427 and 586kPa (20, 45, 62,
63
S
1
S
2
S
3
S
4
Film holder
Upstream
section
Downstream
section
Pressure
transducers
Windows
Shock
wave
Film
O-ring
Bolted joint
Bolted joint
Figure 5.2 Schematic detailed cutaway view of the test section and the film holder. Four
pressure transducers are mounted with 100mm distance.
and 85psi, gauge pressure) to create incident shock Mach numbers of 1.20, 1.34, 1.39,
and 1.46, respectively.
Figure 5.2 presents a schematic view of the test section and the film holder of the
shock tube. The test section has three separate parts: an upstream section, a film
holder, and a downstream section. The film holder was designed in the desired planar
shape to hold different thicknesses of a single layer of polyester or aluminum films using
o-rings and bolted joints to prevent gas leakage. The polyester or aluminum film was
held in the downstream side. The sides of the upstream and downstream sections in the
test section were equipped with optically clear polycarbonate sheets which each have
a thickness of 12.7mm and a length of 200mm. The shock Mach number M
s
and the
overpressure P are obtained by four pressure transducers installed in the upstream and
downstream sections(S
1
andS
4
: PCB113B21,S
2
andS
3
: PCB113B31,flushmounted)
with 100mm distance (±0.1%).
A high-speed schlieren visualization was set up to record the shock wave interaction
with the films. A high-framerateof84054fps was used tovisualize thetest section with
64
Table 5.1 Sample information. Normalized thickness is defined as the ratio of the material
thickness relative to the shock wave thickness (200nm at standard temperature and pres-
sure [1]).
Sample
Thickness Normalized thickness Tolerance Density Sound speed Impedance
Reference
h (µm) (Order of magnitude) (%) (kg/m
3
) (m/s) (kg/m
2
s)
Polyester
0.92 0–1
±10 1360 2.3 × 10
3
3.1 × 10
6
[92]
12.7 2
25.4 2
50.8 2
Aluminum
25.4 2
±10 2780 6.4 × 10
3
1.8 × 10
7
[93]
50.8 2
an exposure time of 0.294µs and an interframe rate of 11.9µs. The spatial resolution of
304× 184 pixels was used to visualize shock wave interaction with the film in between
the locations of pressure sensors S
2
and S
3
, and 352× 184 pixels was used to visualize
between the film holder and sensor S
3
. The scale factor of the current setup was 370
µm/pix.
Two different types offilms, polyester and aluminum, were used in the current study
and sample information is summarized in Table 5.1. A 0.92µm thick polyester film was
provided from DuPont Teijin Films and the other films were purchased from McMaster-
Carr. Here, the normalized thickness was defined as the ratio of the material thickness
relative to the shock wave thickness. The acoustic impedance of the aluminum film was
of one order of magnitude greater than that of the polyester film.
5.3 Results
Results are presented using flow visualization from high-speed images and overpressure
profiles obtained through high-frequency pressure sensors.
5.3.1 Flow Visualization
The high-speed schlieren images in Figure 5.3 show the shock wave interactions with a
12.7µm thick polyester film at the incident shock Mach number M
s
= 1.39 in between
65
Incident
shock
10 mm
Film holder
(a) t = 528µs (b) t = 540µs
Tranmitted
wave
Polyester
film
(c) t = 552µs
Reflected
wave
(d) t = 564µs (e) t = 576µs (f) t = 588µs
Figure 5.3 A series of high-speed schlieren images of shock wave interaction with a 12.7µm
thick polyester film initially located at the right edge of the film holder. The incident shock
Mach number is M
s
= 1.39. The resolution is 304×184 pixels.
the locations of pressure sensors S
2
and S
3
. The experimental visualizations emphasize
the observation of the reflected and transmitted shock waves when the incident shock
wave interacts with the film. The two-dimensional configuration of the test section
allows for observations of the breakup of the film to help us with the understanding of
the overpressure traces. The time t =0 corresponds to the arrival of the incident shock
wave at sensor S
1
.
In Figure 5.3(a), the incident shock wave moves toward the film holder. When the
incident shock wave impacts the polyester film, the reflected and transmitted waves
are generated by impedance mismatch between the air and the film, as shown in Fig-
ures 5.3(c)and (d). Following the transmitted wave, the polyester film expands radially,
resulting in downstream propagating compression waves and upstream propagating ex-
pansion waves. The polyester film starts to expand slowly and later bursts from the
center at the pressure threshold after severe ductile deformation, as illustrated in Fig-
ure 5.3(e). Subsequently, the deformation of the polyester film is attributed to the
66
Incident
shock
10 mm
Film holder
(a) t = 417µs (b) t = 477µs
Tranmitted
wave
(c) t = 536µs
Reflected
wave
Polyester
film
(d) t = 596µs (e) t = 655µs (f) t = 715µs
Figure5.4Aseriesofhigh-speedschlierenimagesofshockwaveinteractionwitha12.7µmthick
polyester at the incident shock Mach number of M
s
= 1.20. The resolution is corresponding
to 304×184 pixels.
momentum transfer from the shocked gas flow to the film.
Figure 5.4 shows high-speed images of shock wave interaction with a 12.7µm thick
polyester film impacted by the incident shock Mach number M
s
= 1.20. The polyester
film at the incident shock Mach number of M
s
=1.20 was slowly deformed compared to
Figure 5.3.
Figure 5.5 shows a sequence of high-speed images of shock wave interaction with
a 0.92µm thick aluminum film at the incident shock Mach number M
s
= 1.36. In
Figures 5.5(d)–(f), the accelerated polyester film is evenly deformed due to the planar
incident shock wave, and the film is fragmented throughout the test section. Comparing
Figures 5.3–5.5, it can be seen that the breakup process of the polyester films differs
with respect to the strength of the incident shock wave and the thickness of the film.
Figure 5.6 shows high-speed schlieren photographs of shock wave interaction with a
25.4µm thick aluminum film at the incident shock Mach number M
s
=1.39. The shock
waveinteractionwiththealuminumfilmshowsnosignificantdifferenceexceptthebrittle
67
Incident
shock
10 mm
Film holder
(a) t = 382µs (b) t = 424µs
Reflected
wave
Tranmitted
wave
(c) t = 467µs
Polyester
film
(d) t = 510µs (e) t = 553µs (f) t = 598µs
Figure 5.5 A series of high-speed schlieren images of shock wave interaction with a 0.92µm
thick polyester film at the incident shock Mach number of M
s
= 1.36. The resolution is
corresponding to 304×184 pixels.
68
Incident
shock
10 mm
Film holder
(a) t = 528µs (b) t = 540µs
Tranmitted
wave
(c) t = 552µs
Reflected
wave
Aluminum
film
(d) t = 564µs (e) t = 576µs (f) t = 588µs
Figure 5.6 A series of high-speed schlieren images of shock wave interaction with a 25.4µm
thick aluminum film at the incident shock Mach number of M
s
= 1.39. The resolution is
corresponding to 304×184 pixels.
69
0 20 40 60 80 100 120
x (mm)
300
350
400
450
500
550
600
650
700
t (µs)
Incident shock
Reflected shock
Transmitted shock
Measured from pressure transducers
Film
holder
Figure 5.7 Shock wave trajectories of the incident, reflected, and transmitted shock waves for
the case of 12.7µm thick polyester film at the incident shock Mach number of M
s
= 1.34.
Results are obtained from high-speed schlieren photographs (markers) and measured shock
wave velocities from pressure sensors (solid line). The location x = 0 corresponds to the
left-end of high-speed images. Uncertainty for each data point is represented inside marker.
breakup process of the film, as illustrated in Figures 5.6(e) and (f). Compared to the
polyester film, the aluminum sheet breaks up in multiple pieces with less deformation.
Figure 5.7 displays the shock wave trajectories of the incident, reflected, and trans-
mitted shock waves at the incident shock Mach number of 1.34 with 12.7µm thick
polyester film using high-speed images. The location of the shock waves in the film
holder cannot be observed. The shock wave locations are compared to the shock wave
velocity calculated by the arrival time between pressure sensors of S
1
and S
2
for the the
incident and reflected waves, and S
3
and S
4
for the transmitted wave. Comparison be-
tween theresultsfromthehigh-speedimagesandthevaluesofthevelocity measurement
from the pressure sensors show a good agreement (within 2%). Figure 5.7 confirms that
shock wave acceleration ordeceleration during the interaction with thefilm is negligible.
Thehigh-speedschlieren imagesinFigure5.8showtheshockwaveinteractionwitha
25.4µmthickpolyester film attheincident shock Machnumber M
s
=1.34. The window
of visualization is located in between the film holder and S
3
to observe the downstream
of the test section. The stress concentrations caused by the bolted joints on the top
and the bottom of the test section leave visible marks showing up as dark regions. The
70
10 mm
Film holder
S
3
(a) t = 328µs
Transmitted
shock wave
(b) t = 387µs
Polyester film
(c) t = 467µs
(d) t = 506µs
1
st
major
compression wave
(e) t = 566µs
2
nd
major
compression wave
(f) t = 625µs
Figure 5.8 A series of high-speed schlieren images of shock wave interaction with a 25.4µm
thick polyester filmattheincidentshockMach numberofM
s
= 1.34. A resolution of 352×184
pixels was used.
location of the pressure sensor S
3
is illustrated in Figure 5.8(a). When the polyester
film experiences the maximum expansion caused by the large upstream pressure, the
membrane behaves a piston-like motion, resulting in the first major compression, as
shown in Figure 5.8(e). After the polyester film bursts, the compressed air contained in
the upstream region starts to propagate downstream of the test section and generates
the second major compression wave, as shown in Figure 5.8(f). However, the interaction
between theplanarshock wave andthefilminduces complex expansion andcompression
waves and those also depend on the strength of the incident shock wave and the film
thickness.
5.3.2 Pressure Measurements
Figure5.9showsaschematicx-tdiagramoftheshockwaveinteractionwithandwithout
a film for the entire shock tube. An incident shock wave is generated by the membrane
breakupbetweenthedriveranddrivensection. Acontactsurface,whichseparatesdriven
and driver gases, follows the incident shock wave with a lower velocity. Meanwhile,
71
S
1
S
2
S
3
S
4
x
t
Incident shock wave
Film
Reflected shock wave
Contact surface
Compression
wave
Expansion wave
Trasmitted
shock wave
t
Incident shock wave
Contact surface
Expansion wave
Position (x)
Test without film
Test with film
Shock wave
by film breakup
x
Figure 5.9 A schematic x - t diagram of the shock wave interaction without (top) and with
(bottom) a film.
expansion waves travel toward the driver section and the waves are reflected by the end
wallofthedriversection. Whentheincidentshockwaveinteractswiththefilm,reflected
and transmitted shock waves are generated due to the impedance mismatch between
the air and the film. During the expansion stage of the film, downstream propagating
compression waves andupstream propagatingexpansion waves aregeneratedby thefilm
motion. Later, when the film breaks up after a threshold pressure, another shock wave
is generated ahead of the film.
Figure 5.10 represents the overpressure traces measured on the top-wall at the inci-
dent shock Mach number of M
s
= 1.34 during experiments with and without a 25.4µm
thick polyester film. Here, three different experiments were performed but only the av-
eragedvaluesareplotted. InFigures5.10(a)and(b),theanalyticalvaluewascalculated
based on the incident shock Mach numbers measured between sensors S
1
and S
2
and
the Rankine-Hugoniot shock relation. The experimental results show a good agreement
with the analytical solution.
In Figure 5.10(a), the incident shock wave shows nearly constant overpressure until
72
t (ms)
0
50
100
150
200
P (kpa)
S
1
S
2
S
3
S
4
0 1 2 3 4
Analytical
-0.01 0 0.01 0.02
t (ms)
0
100
200
P (kPa)
(a)
t (ms)
0
50
100
150
200
P (kpa)
0 1 2 3 4
Reflected
shock wave
Transmitted
shock wave
P
t
0.6 0.8 1.2
t (ms)
0
40
80
120
P (kpa)
1.0
1
2
3
P
i
S
1
S
2
S
3
S
4
Analytical
(b)
Figure 5.10 Comparison of overpressure traces obtained (a) without and (b) with a 25.4µm
thick polyester film at the incident shock Mach number of M
s
= 1.34. The inserts in (a) and
(b) show overpressure versus magnified time.
the arrival of the expansion fans about 2.5ms. The insert in Figure 5.10(a) represents a
timezoomintheregionbetween−0.01and0.02ms toshow theresponse onthepressure
sensor at S
1
. It can be seen that the system rise time of the current setup was measured
about 6.25µs. In the current study, the shock Mach numbers were reproduced within
±1.03% with a measurement uncertainty less than±0.82% by considering the system
rise time.
In Figure 5.10(b), the incident shock wave was reflected by the polyester film, re-
sulting in the the second peak at sensors S
1
and S
2
. The overpressure generated by
the reflected shock wave is 37% lower than that induced by the shock wave reflection
off of a rigid wall (P = 254kPa). The incident shock wave is transmitted through the
polyester film, showing the overpressure rises at sensors S
3
and S
4
. After the polyester
film is completely broken, the overpressure in the upstream and downstream sections
reaches about 94kPa, which is the same overpressure without the polyester film, as
shown in Figure 5.10(a). At sensors S
3
and S
4
, more complicated overpressure profiles
are shown in Figure 5.10(b). The insert in Figure 5.10(b) represents a time zoom in the
region between 0.6 and 1.2ms at sensors S
3
and S
4
. The first rapid overpressure rise is
73
t (ms)
0
50
100
150
P (kpa)
0 1 2 3 4
S
1
S
2
S
3
S
4
P
t
(a)
0.6 0.7 0.8 0.9 1
t (ms)
0
50
100
150
P (kpa)
S
3
S
4
(b)
Figure 5.11 Overpressure versus time for the case of a 0.92µm thick polyester film at the
incident shock Mach number of M
s
= 1.34 for (a) time from 0 to 4.0ms, and (b) time from
0.6 to 1.0ms.
caused by a transmitted shock wave. It is apparent that the overpressure caused by the
transmitted shock wave is significantly lower than that created by the incident shock
wave. Thisisbecausethemomentumoftheshockedflowwaspartiallytransferredtothe
film. The peak overpressure induced by the incident shock wave P
I
and the transmitted
shock wave P
T
is the main parameters to quantify shock wave attenuation by the film,
as previously discussed [12]. The first overpressure rise is followed by two additional
overpressure rises. The second rise is produced by the first major compression wave
caused by the acceleration of the polyester film as visualized in Figure 5.8(e). The third
rise corresponds to the second major compression wave, which is generated by the fully-
opened film, as shown in Figure 5.8(f). The downstream overpressure traces recorded
at sensors S
3
and S
4
show a higher amplitude of oscillatory trends compared to those at
sensors S
1
and S
2
, which is mainly caused by the vibration of the side windows in the
test section, as explained in Section 4.3.2.
Figure 5.11 shows overpressure as a function of time for the case of a 0.92µm thick
polyester film at the incident shock Mach number M
s
= 1.34. In Figure 5.11(a), no
reflectedshockwaveisrecordedatsensorsS
1
andS
2
. InFigure5.11(b),theoverpressure
74
induced by the transmitted shock wave at sensors S
3
and S
4
shows nearly the same
amplitude compared to the incident shock wave measured at sensors S
1
and S
2
(< 5.0%
difference). ThedifferencebetweentheincidentandthetransmittedshockMachnumber
is less than 2.5% based on the arrival time between S
1
and S
2
for the incident shock
and S
3
and S
4
for the transmitted shock. This result shows a good agreement to the
previous results [76].
Figure 5.12 presents the overpressure as a function of the time recorded by sensor
S
3
for the case of different polyester film thicknesses of h = 12.7, 25.4, and 50.8µm at
the incident shock Mach number of 1.34. Here, we present only the overpressure traces
at sensor S
3
since there are no different traces recorded at sensor S
4
. Figure 5.12(a)
shows that different thickness of the polyester film causes slightly different transmitted
overpressure trends. In Figure 5.12(b), the shock wave interaction with a 12.7µm thick
polyester film at the incident shock Mach number of 1.34 clearly shows three steps of
overpressure jumps, as discussed in Figure 5.10(b), but this trend cannot be observed
in other cases. This is because different film thicknesses requires different threshold
overpressure for the breakup. A 50.8µm thick polyester film shows 19% lower peak
overpressure and 5.2% lower transmitted shock Mach number compared to a 12.7µm
thick polyester film.
Another set of experiments were conducted in which the film thickness was kept
constant for four different incident shock Mach numbers and the same polyester film
thickness. Figure 5.13 shows the overpressure versus the time measured at sensor S
3
for
the case of a 12.7µm thick polyester film with different incident shock Mach numbers of
M
s
=1.2,1.34,1.39,and1.46. Figure5.13(a)showsthatastrongerincidentshockMach
number results in a higher transmitted overpressure (up to 125%) and an earlier arrival
time of the transmitted shock wave (up to 16%). When the incident shock wave Mach
numberisM
s
=1.20,thetransmittedshockwaveisrelativelyweakanddirectlyfollowed
by expansion waves, as shown in Figure 5.13(b). This is because the flow propagating
though the polyester film loses velocity due to the existence of polyester film, resulting
in overpressure decrease from 0.8ms. The overpressure decrease ends after the breakup
of the polyester film and gradually approaches constant pressure.
The measured overpressure between two different materials, polyester and aluminum
75
t (ms)
0
50
100
150
P (kpa)
h = 12.7µm
h = 25.4µm
h = 50.8µm
0 1 2 3 4
(a)
0.6 0.7 0.8 0.9 1
t (ms)
0
20
40
60
80
100
P (kpa)
h = 12.7µm
h = 25.4µm
h = 50.8µm
(b)
Figure 5.12 Overpressure profile measured at sensor S
3
at the incident shock Mach number of
M
s
= 1.34 with different polyester film thicknesses for (a) time from 0 to 4.0ms, and (b) time
from 0.6 to 1.0ms.
t (ms)
0
50
100
150
200
250
300
P (kpa)
M
s
= 1.20
M
s
= 1.34
M
s
= 1.39
M
s
= 1.46
1 2 3 4 0
(a)
0.6 0.7 0.8 0.9 1
t (ms)
0
50
100
150
200
P (kpa)
M
s
= 1.20
M
s
= 1.34
M
s
= 1.39
M
s
= 1.46
(b)
Figure5.13 OverpressureprofileatsensorS
3
withthe12.7µmthick polyesterfilmanddifferent
incident shock Mach numbers for (a) time from 0 to 4.0ms, and (b) time from 0.6 to 1.0ms
for (a) time from 0 to 4.0ms, and (b) time from 0.6 to 1.0ms.
76
t (ms)
0
50
100
150
200
250
300
P (kPa)
S
1
_Poly
S
3
_Poly
S
1
_Al
S
3
_Al
0 1 2 3 4
(a)
0.6 0.7 0.8 0.9 1
0
20
40
60
80
100
P (kPa)
t (ms)
S
3
_Poly
S
3
_Al
(b)
Figure 5.14 Overpressure profile at sensors S
1
and S
3
with the 50.8µm thick polyester (Poly)
and aluminum (Al) films at the incident shock Mach numbers of 1.34 for (a) time from 0 to
4.0ms, and (b) time from 0.6 to 1.0ms.
films, with the same incident shock Mach number and thickness is shown in Figure 5.14.
Although, acoustic impedance of the materials is different by one order of magnitude,
overpressure profiles show the similar overall trend, as shown in Figures 5.14(a) and (b).
ThetransmittedoverpressureandshockMachnumbersweredifferentby5.1%and3.7%
between two types of films, respectively. One likely reason for the similar overpressure
profile between two materials is that the incident shock wave was already transmitted
throughthematerialsbeforethedeformationoccurs. Thus, thematerialthickness seems
todominatetheshockwave attenuationbyovercomingdifferentacousticimpedanceand
deformation process.
Figure 5.15 shows normalized overpressure versus the time elapsed from the passage
of the transmitted shock wave at S
3
for the case of polyester films. The inserts in
Figure5.15showmoredetailedoverpressuretracescausedbythetransmittedwaveinthe
range between 0.6to 1.0ms. Here, normalized overpressure represents the overpressure
induced by the transmitted shock wave P
T
relative to the overpressure induced by the
incident shock wave P
I
. In Figure5.15,stronger incident shock waves cause the stronger
transmitted shock waves, resulting in the earlier arrival of transmitted waves and the
77
t (ms)
0
0.5
1
1.5
2
M
s
= 1.20
M
s
= 1.34
M
s
= 1.39
M
s
= 1.46
0.6 0.7 0.8 0.9 1
t (ms)
0
0.5
1
1.5
P
t
/P
i
1 2 3 4 0
P
T
/ P
I
(a) h = 12.7µm
t (ms)
0
0.5
1
1.5
2
P
T
/ P
I
0.6 0.7 0.8 0.9 1
t (ms)
0
0.5
1
1.5
P
t
/P
i
1 2 3 4 0
M
s
= 1.20
M
s
= 1.34
M
s
= 1.39
M
s
= 1.46
(b) h = 25.4µm
t (ms)
0
0.5
1
1.5
2
1 2 3 4 0
0.6 0.7 0.8 0.9 1
t (ms)
0
0.5
1
1.5
P
t
/P
i
P
T
/ P
I
M
s
= 1.20
M
s
= 1.34
M
s
= 1.39
M
s
= 1.46
(c) h = 50.8µm
Figure 5.15 Ratio of the overpressure induced by the transmitted shock wave relative to the
overpressure induced by the incident shock wave P
T
/P
I
versus the time at sensor S
3
with
different thicknesses of polyester films. The inserts show the overpressure from 0.6to 1.0ms.
78
higher amplitude of transmitted overpressure. The normalized overpressure of M
s
=
1.20 shows different trends compared to incident shock Mach numbers 1.34, 1.39, and
1.46. This is because the higher incident shock Mach numbers breaks the film earlier
than the lower Mach number. It can be seen that normalized overpressure caused by
the transmitted shock wave is not significantly affected by the different incident shock
Mach numbers (up to 11.6%). The experimental results demonstrate that by using
different thickness of the films, while keeping the incident shock Mach number constant,
the normalized overpressure caused by the transmitted shock wave is changed (up to
36%).
5.3.3 Analysis and Discussion
Figure 5.16 compares the normalized overpressure obtained with a shock wave with
incident shock Mach numbers 1.20, 1.34, 1.39, and 1.46 at sensor S
3
during experiments
with 12.7, 25.4, and 50.8µm thick polyester films and 25.4 and 50.8µm thick aluminum
films to understand the influence of the different incident shock Mach numbers and the
film thicknesses. In Figure 5.16(a), it can be seen that the shock wave attenuation
by a 0.92µm thick polyester film is nearly negligible. The thicknesses of 12.7, 25.4, and
50.8µmpolyesterfilmsshowdifferentnormalizedoverpressureatthesameincidentshock
Machnumber. ByincreasingtheincidentshockMachnumbers, normalizedoverpressure
shows a slightly decreasing trend in the range of M
s
= 1.20 and 1.46. The difference
in normalized overpressure caused by different thicknesses of polyester films is up to
125%. In Figure 5.16(b), normalized overpressure of a 25.4µm thick aluminum film at
the incident shock Mach number of 1.46 shows higher values compared to other cases
and this iscaused by the early breakup ofthe aluminum film. Inthis case, the measured
overpressure was not caused by the transmitted shock wave, but generated by the shock
wave from the breakup of the aluminum film. Comparing the results between polyester
and aluminum films, the difference in normalized overpressure of two materials with the
same thickness is within 9.2% except the case of the early breakup of aluminum film in
the current study.
To sum up, shock wave attenuation is negligible if the film thickness is of the same
79
1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
M
s
0
0.2
0.4
0.6
0.8
1
1.2
P
T
/ P
I
h_Poly = 12.7µm
h_Poly = 25.4µm
h_Poly = 50.8µm
h_Poly = 0.92µm
(a) Polyester film
1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
M
s
0
0.2
0.4
0.6
0.8
1
1.2
P
T
/ P
I
µm
µm
h_Al = 25.4
h_Al = 50.8
Early breakup
(b) Aluminum film
Figure 5.16 Ratio of the overpressure induced by the transmitted shock wave relative to the
overpressure induced by the incident shock Mach numbers, normalized overpressure, versus
the incident shock Mach numbers. Error bars represent the standard deviation of a data set.
order of magnitude compared to the shock wave thickness for polyester film at M
s
=
1.20, 1.34, 1.39, and 1.46. This is because compression waves caused by an infinite
number of reflections within the film layer coalesced into a shock wave in a short time
with a negligible amount of energy dissipation. When the film thickness is of two orders
of magnitude relative to the shock wave thickness, successive reflections within the film
occurs relatively longer time with decrease in overpressure intensities. On the basis of
order of magnitude considerations, the difference of one order of magnitude of the film
thickness affects the shock wave attenuation more than that of the acoustic impedance.
Althoughitispossiblethatthisresultismorelikelyaffectedbythecurrentexperimental
setup inwhich theshock wave interactiontakes placebetween theairandthefilms. The
acoustic impedance of the air (4.2×10
2
kg/m
2
s) is several orders of magnitude lower
compared tothatofthefilms andthere is oneorder ofmagnitude difference between the
polyester and aluminum films. However, the findings regarding the different strength
of the transmitted shock wave and its dependence on thickness of the materials can be
importantaspects, whichhave notyetquantitatively beenreportedinpreviousresearch.
80
Chapter 6
Shock Wave Attenuation by Foam Obstacles
This work was published in Aerospace under Shock Wave Attenuation Using Foam Ob-
stacles: Does Geometry Matter? by Jeon et al. [19].
6.1 Background
Porous structures featuring pores several orders of magnitude smaller than the diameter
of the duct have demonstrated agreat deal ofpromise. A review ofthe shock wave liter-
ature reveals many studies investigating various methods of mitigation. Some methods
that have proven effective for this purpose include the introduction of bends inside a
duct [94,95], the addition of rigid or semi-rigid structures in the path of the shock wave
and baffles, orifices, perforated plates with holes or slits, grids and obstacles of varying
geometries placed in different patterns [13,22,96–102]. The most common examples
of such structures are rigid or flexible granular filters [33,103,104] or foams made of
solid materials or liquids [14,15,17,105–109]. A three-part article series with reviews on
the current standing on aqueous foams and experimental observations is presented by
Britan et al. [14,15,105].
Polymer foams, such as polystyrene (one variety of which is sold under the brand
nameStyrofoam)andpolyurethanearewidelyavailableandpossespropertieswellsuited
for use in shock wave experiments. Responses to shock wave loading are dependent on
several attributes of the foam, such as the cell structure and foam density, as well as
environmentalattributes,suchastheambienttemperatureandhumidity. Theproperties
ofthesefoamshavebeenquantifiedforarangeofconditionsusingexperimentalmethods,
such as drop weights, impact sleds, split Hopkinson pressure bars and shock tubes [110–
113]. The results of these investigations have shown that polymeric foams subjected
81
to high dynamic loads respond first with linear-elastic behavior for small strains. This
involves linear bending of the cell walls. Next, the foams displayed elastic buckling
and plastic yielding as the strain increases. Finally, the cell walls are crushed, and the
collapsed cell matrix starts to deform.
Despite these investigations, a comprehensive understanding of foam properties and
how they are coupled to peak pressure and impulse reduction under shock loading is
still lacking. Results from shock tube experiments show that foams can be successfully
used to mitigate shock waves when placed in front of a surface that they are supposed
to protect. One of the more recent experimental works, and perhaps one of the most
extensive to date, is that by Seitz and Skews [109]. In this study, seven different types
of foams (with varying foam density, permeability, tortuosity and cell properties) were
impacted by shock waves at different strength. The work resulted in a large dataset
valuable for the computational simulation community.
In addition to experimental investigations of shock wave attenuation with foam,
numerical studies of this method have also been performed. In the work of Ball and
East [114], pressure histories ahead of the exit duct of a shock tube were examined us-
ing Lagrangian CFD simulations, for cases in which the duct was obstructed by foam
of varying macroscopic geometries. These included semi-infinite and finite length foam
sheets and cylindrical foam caps covering the exit duct. There was no significant dif-
ference in the attenuation properties of the various geometries. The authors concluded
that varying the density of the foams, and therefore, the impedance mismatch of the
foam/air interface, had a larger effect on attenuation of blast waves than varying the
geometry of the foam or the dimensions of each type of geometry. In 2013, the work
of Ram and Sadot [115] resulted in a constitutive model to predict the pressure history
at a shock tube wall behind a foam obstacle. The authors used a single experiment to
quantify thefoambehaviorandwere thenabletopredict itsattenuationproperties fora
wide range of conditions. Ram and Sadot concluded that the pressure history recorded
at the wall depended on the incident shock Mach number, the stand-off distance to the
wall behind the foam obstacle, the length of the foam obstacle and a lump coefficient
that accounts for foam properties. The model was also applied to previous foam at-
tenuation experiments by other research groups (e.g., [101]), and with success, the new
82
modelcouldpredicttheresultsaccurately. RamandSadotconcludedthatthegeometric
shape of the foam obstacle did not influence the pressure history results at the rear wall
after experimenting with triangular and cylindrical cross-sectional grids.
In this work, an obstacle made of polymeric foam was inserted in a shock tube to
block the path of an incident shock wave. Five differently-shaped obstacles made from
two types of foam were used in this study. The geometries consisted of blocks with one,
two, three or four convergent shapes, as well as a rectangular block with a flat front
face. The types of foam used include Styrofoam brand polystyrene and polyurethane.
Data were acquired in the form of pressure traces collected ahead of and behind the
obstacles, as well as high-speed schlieren visualization of the shock waves encountering
the obstacles. This yielded useful insight into how geometrical properties of the front
face of the foam influence the attenuation capabilities. Of particular interest is how the
large-scale curvatures of the front face of the foam samples affect the flow.
6.2 Experimental Setup
Figure 6.1 shows an horizontal shock tube, which was used to understand shock wave
mitigation through foam obstacles. The total length of the shock tube setup is 4.7 m
with a 0.65 m-long driver section followed by a 3.30 m-long driven section. The inner
diameters of the driver and the driven sections are 91.4 and 72.8 mm, respectively, thus
leading to a reduction in cross-section area of 37%. The larger size of the driver section
produces stronger shocks, and it can also delay the time it takes the reflected expansion
wave to reach the driven section [36,116]. Compressed air was used to fill the driver
section while the driven section was left open to atmospheric conditions. The driver
pressure was measured by a digital pressure gauge (GE DPI 104, ±0.05%, full-scale
accuracy). At the end of the driven section is a transition section, 95 mm long, that
transforms the cross-section area from circular to square, while simultaneously also re-
ducing the cross-section area by 38%. A cross-section areareduction increases the shock
Machnumber. Previousresearchhasshownthatadistanceof5–40diametersissufficient
to produce a planar shock wave front downstream of the membrane location [40], and
hence, it is even shorter for a transition section where a shock wave is already formed.
83
Driver section Driven section
Mylar diaphragm
Test section
ID 72.8
648 3277 685
50.8
Transition area
279
ID 91.4
0
S
2
S
1
S
152.5 152.5
95
Figure 6.1 Schematic description of the shock tube. Both the driver and driven sections have
circular cross-sections, and the test section has a rectangular cross-section. Three pressure
sensors denotedS
0
, S
1
andS
2
areused to measurepressurebeforeandafter thefoam obstacle.
Dimensions in mm; not to scale. This figure is reproduced from [19].
The test section, 685 mm long, is located downstream of the transition section. It is
equipped with two sets of windows, one outer polycarbonate pair of windows, 6.4 mm
(1/4”) thick, and an inner acrylic pair, 3.2 mm (1/8”) thick. The inner windows are
replaceable to avoid the accumulation of scratches, as samples are pushed through the
test section due to the pressure increase of the reflection of the incident shock wave.
Pressure wasmeasuredbyusingpiezoelectricpressure transducers (PCB113B21and
113B31, flush mounted) placed at 152.5, 305 and 584 mm from the beginning of the test
section; see Figure 6.1. Sensors S
1
and S
2
will be referred to as the upstream and the
downstream sensor, respectively, through the rest of this paper.
Figure 6.2 shows a pressure plot obtained from the upstream sensor S
1
mounted in
an empty test section. This pressure trace shows a Δt = 2.3 ms time interval between
theincident shock andthehead ofthereflected expansion wave. The pressure peak near
0.4msinFigure6.2,indicated bythearrow, iscaused byweak reflections ofaremovable
panel inside the test section; see the detailed view in Figure 6.3. This removable panel
is designed to change the samples without causing any deformation of the samples.
However, when the samples are inserted into the test section, this panel is covered by
the sample, so that this pressure peak does not occur.
Schlieren visualization technique is used in the current study, as shown in Sec-
tion 2.4.2. The camera settings for the various cases are summarized in Table 6.1.
84
Time (ms)
#10
-3
-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4
) a P k ( Overpressure
-20
0
20
40
60
80
100
∆t
Figure 6.2 Overpressure plot from the upstream sensor S
1
with no obstacle in the test section.
Time interval Δt = 2.3 ms represents the time between the shock wave and the head of the
reflected expansion wave. The arrow points at a reflection caused by a removable panel of the
shock tube. This figure is reproduced from [19].
Table 6.1 Camera settings. Note: the same settings were used for all open cell foam
schlieren photographs. This table is reproduced from [19].
Case
Exposure Time Frame Rate Resolution
[μs] [Frames/s] [mm/pix]
Open cell 0.38 86,075 0.387
NC (closed cell) 0.94 210,526 0.414
1C (closed cell) 0.38 150,110 0.414
2C (closed cell) 0.38 210,526 0.414
3C (closed cell) 0.38 210,526 0.414
4C (closed cell) 0.38 210,526 0.414
85
Sensor S
Sensor S
Sensor S
Sample
Double windows
Transition section
Removable panel
0
1
2
Safety hole
Figure 6.3 Sectional view of shock tube test section with transition section, sensor and sample
placements. This figure is reproduced from [19].
6.3 Logarithmic Spiral
In previous work, a logarithmic spiral has shown the ability to minimize shock wave
reflections offthe surrounding boundary, andeffectively focus an incident shock wave on
its focal point in air [117–119] and water [120,121]. A logarithmic spiral for an incident
shock Mach number M
s
is described in Figure 6.4. Using polar coordinates, an equation
representing a logarithmic spiral can be expressed as [122]:
r =
L
cosχ
e
χ−θ
tanχ
(6.1)
The characteristic angle χ is determined by the following relations [123]
tanχ =
M
2
s
−1
λ(M
s
)M
2
s
1
2
, (6.2)
86
Ms
R
O L
r
χ
θ
Figure6.4Aschematicdescriptionoflogarithmicspiralshape. Oistheoriginofthelogarithmic
spiral, L represents the first characteristic length of the spiral, and χ is the characteristic
angle. The size of the logarithmic spiral can be controlled by the characteristic length L.
Here, R =L/cosχ, where r is the distance between the point on the logarithmic spiral curve
and the origin, and θ is the angle formed by r and the centerline. This figure is reproduced
from [19].
where
μ
2
=
(γ−1)M
2
s
+2
2γM
2
s
−(γ−1)
, (6.3)
λ(M
s
)=
1+
2
γ +1
1−μ
2
μ
1+2μ+
1
M
s
. (6.4)
Here, M
s
is the shock Mach number, and γ is the ratio of the specific heat. Based on
these two values, the characteristic angle, χ, is obtained through equations (6.2)-(6.4).
Wan andEliasson [102]showed logarithmic spiral obstacles in two-dimensional ducts
aremoreefficient toattenuateboththetransmitted andreflected shock waves compared
to an arrangement of squares, cylinders, and triangles placed in staggered and non-
staggered subsequent columns. In addition, they found a higher ability of attenuating
the overpressure of the transmitted and reflected shock waves especially at the design
Mach number.
87
Figure 6.5 Scanning electron microscope image of the open-celled structure of polyurethane
foam. This figure is reproduced from [19].
6.4 Sample Preparation
Two different types of foams were used: (1) a closed cell polystyrene foam (ASTM C578
Type IV) with a density of 24.8 kg/m
3
; and (2) an open cell polyurethane foam with a
density of28.83kg/m
3
(alsoknown as Aquazone). The densities were obtained from the
manufacturer data. Figure 6.5 shows a micrograph of the polyurethane foam taken at
40× magnification using a JEOL-JSM 7001 scanning electron microscope (JEOL Ltd,
Japan). The top surface of the foam sample was sputtered with platinum coating, and
the edges were painted with colloidal graphite to make it conductive. The micrograph
shows the foam’s open cell structure, as evidenced by the pores in each cell leading to
other cells. Each cell opening is roughly 0.4 mm wide.
ThefivedifferentfoamsamplesareshowninFigure6.6. Eachsamplewasconstructed
usingafoamcoresandwichedbetweentwo2.54mm-thickplywoodsheets. Eachplywood
sheet was cut with a laser, and the plywood grains were oriented in the direction of the
incident shock wave. The front face of the foam sample, i.e., the side facing the incident
shock wave, was cut into different convergent shapes using a hot wire, and the samples
were cut in the same directions from the original foam sheets. Small variations in the
front surface occurred due to varying hot wire temperatures and cutting speeds. The
88
Figure 6.6 Foam samples: (a) NC; (b) 1C; (c) 2C; (d) 3C; (e) 4C. This figure is reproduced
from [19].
Table 6.2 Overview of experimental sample configurations. This table is reproduced from [19].
Case
# of LS L d
[–] [mm] [mm]
NC – – 47
1C 1 87 20
2C 2 46 33
3C 3 29 37
4C 4 19 39
plywood was used to prevent the foam from deforming as it was inserted into the test
section of the shock tube.
In these experiments, the logarithmic spiral shape was chosen for an incident shock
Mach number of M
s
= 1.2, and depending on the case, the characteristic length was
varied to accommodate one, two, three or four curves. To keep the total mass of the
foam samples constant, the distance from the focal point of the logarithmic spiral curve
to the rear end of the sample, d, defined in Figure 6.6(b), was different forall cases. The
mass of the Styrofoam samples (without the wood panels) was 3.9± 0.2 g, and for the
polyurethane samples, it was 3.7± 0.1 g. A summary of sample configurations is shown
in Table 6.2.
The samples were designed to fill the entire test section with a slight interference
fit to not leave any open air gaps for the shock to propagate through, following the
experiments presented in [106]. No visible deformation was found in the samples after
insertion in the test section. A sketch of a sample inserted in the test section is shown
89
in Figure 6.3 together with the transition section.
6.5 Results and Discussion
To begin, the shock wave speed was measured using three pressure sensors inserted
in the test section. A shock Mach number of M
s
= 1.25± 0.01 was measured for
the experimental conditions used in all cases. Based on the pressure ratio between the
driverandthedrivensection, ananalyticalshockMachnumberof1.16canbecalculated
assuming no losses, which is smaller than the measured Mach number. The reasons are
the two cross-section area reductions; the first one between the driver and the driven
section and the second one between the driven section and the test section. Through
repeated experiments, the shock wave speed was shown to decrease no more than 1.8%
between the first and the last sensor placed 279 mm apart. This decrease is caused by
two open holes in the test section, placed ahead of the foam sample for safety reasons
(see the location in Figure 6.3).
Figures 6.7–6.11 show schlieren photographs for the open cell foam experiments. In
these plots, the incident and reflected shock wave fronts are annotated with blue and
red arrows, respectively. Case NC is shown in Figure 6.7. The incident shock is close
to planar with only a two-pixel difference (0.387 mm/pix) between the middle section
and the top and bottom sections. This is also true for the reflected shock, which has a
slight convex shape with only a two-pixel deviation from a straight line. Additionally,
reflections from the upper and lower sides are visible in the photographs.
The deflection of the inner pair of acrylic windows of the test section was estimated
andturnedouttoresultinaminimalcross-sectionareachange. Usinganapproximation
of a distributed pressure loading of a simple beam, the bending of the inner acrylic
window of the test section has been estimated using the maximum pressure behind
the reflected shock. Using these methods, the maximum window deflection under the
reflectedwavepressurewascalculatedas1.37mm(0.054”). Atrapezoidalapproximation
was then made to calculate the new cross-section area given this deflection. Under the
maximum pressure from the reflected wave, the change of the cross-sectional area was
found to be approximately 3.6%. This effect could result in a convex reflected shock
90
(a) 0 μs (b) 34.86 μs (c) 69.72 μs
(d) 104.58μs (e) 139.44μs (f) 174.30μs
Figure 6.7 Open cell foam, Case NC. This figure is reproduced from [19].
(a) 0 μs (b) 81.34 μs (c) 162.68 μs
(d) 244.02μs (e) 325.36μs (f) 406.70μs
Figure 6.8 Open cell foam, Case 1C. This figure is reproduced from [19].
wave.
However, itisofimportancetonotethatsince allsamplesweretestedusingthesame
experimental setup and approach, a comparison of results is valid because all samples
encountered the same testing conditions.
The black arrowsmarkingdarkregionsinFigure6.7show locationswhere screws are
drilled into the outer windows of the shock tube. Recall that the flow is only in contact
with the inner windows, so while minute deformation of the outer windows affects the
image captured by the camera, it does not affect the flow.
Pressure traces for the incident and reflected waves are shown in Figure 6.12 for the
five different configurations. The transmitted pressure wave through the different foam
91
(a) 0 μs (b) 46.48 μs (c) 92.96 μs
(d) 139.44μs (e) 185.92μs (f) 232.40 μs
Figure 6.9 Open cell foam, Case 2C. This figure is reproduced from [19].
(a) 0 μs (b) 34.86 μs (c) 69.72 μs
(d) 104.58μs (e) 139.44μs (f) 174.30μs
Figure 6.10 Open cell foam, Case 3C. This figure is reproduced from [19].
(a) 0 μs (b) 46.48 μs (c) 92.96 μs
(d) 139.44μs (e) 185.92μs (f) 232.40μs
Figure 6.11 Open cell foam, Case 4C. This figure is reproduced from [19].
92
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
(a) NC
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
(b) 1C
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
(c) 2C
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
(d) 3C
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
(e) 4C
Figure 6.12 Open cell foam overpressure recordings for all cases: incident and reflected shock.
This figure is reproduced from [19].
specimens is shown in Figure 6.13. In these pressure plots, raw temporal profiles of the
pressure traces obtained from four or five repeated experiments are shown in gray, and
the average is represented by the solid black line. The scatter in the experimental data
is too large to conclude if there is a significant difference between the different cases.
In Figure 6.12(a), the blue dotted line represents the approximate time when the foam
sample starts to translate, and thus, the pressure upstream of the sample is reduced.
Schlieren photographs of the case with a single logarithmic spiral, Case 1C, are
shown in Figure 6.8. The front edges of the foam start to deform, clearly visible in
Figures 6.8(c)–(f). The foam is pulled along with the flow behind the shock wave.
Schlieren photographs of the case with two logarithmic spirals, Case 2C, is shown
in Figure 6.9. A similar behavior is observed as in the 1C case: the upper and lower
edges of the foam are pulled inwards by the flow behind the shock wave. The center
93
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
(a) NC
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
(b) 1C
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
(c) 2C
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
(d) 3C
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
(e) 4C
Figure 6.13 Opencell foam overpressure recordings for all cases: transmitted wave. This figure
is reproduced from [19].
94
piece remains close to its original shape initially and generates a cylindrical shock wave
centered on the tip of the center piece. At later times, Figures 6.9(e)–(f), the front of
the center piece is compressed and becomes blunt.
Schlieren photographsofthecasewiththreelogarithmicspirals, Case3C,isshownin
Figure 6.10. The upper and lower edges, and the center sections where the logarithmic
spirals meet cause cylindrical reflected shock waves.
Schlieren photographs of the case with four logarithmic spirals, Case 4C, is shown in
Figure 6.11. As in the previous cases, cylindrical reflected shock waves are generated.
The reflected cylindrical shock waves coalesce into a shock front that becomes planar
sooner than the previous cases; see Figures 6.11(c) and (d). The reflection from the
shock wave that entered the logarithmic spiral curves (red arrows) also coalesces, as it
exits the foam sample; see Figure 6.11(e).
The transmitted wave is not a shock wave, but rather a weak compression wave due
to downstream translation of the open cell foam block occurring at about 450 μs.
Both the open cell and closed cell foam samples show similar trends in the down-
stream pressure data, rising at approximately 1 ms. In these experiments, the closed
cell and open cell samples begin to translate downstream in the shock tube atabout 350
and 450 μs, respectively. A comparison between open and closed cell foam samples is
shown in Figure 6.14.
The displacement of the samples creates a compression wave that propagates down-
stream, thus reducing the pressure upstream of the sample. An example of the trans-
lation and deformation of the 1C open cell foam sample is shown in Figure 6.15. The
foam core moves faster downstream than the plywood sidewalls; see Figure 6.15(c).
To compare the five different cases for both the open and closed foams, averaged
pressure profiles from each case are plotted in Figure 6.16. Detailed information about
the closed foams is shown in Appendix C. The top row shows the incident and reflected
pressures. Dotted lines correspond to the analytical pressure for incident shock waves
corresponding to the experimentally measured Mach number M
s
= 1.25± 0.01. The
bottom row shows the averaged pressure profiles for the downstream sensor S
2
. As a
result of the downstream translation of the samples shown in Figures 6.14 and 6.15,
the pressure behind the incident and reflected shocks decreased for all cases; see Fig-
95
Closed cell
Open cell
10 mm
1 2
(a)
Closed cell
Open cell
10 mm
(b)
Closed cell
Open cell
10 mm
(c)
Figure 6.14 Photographs showingtranslation of theNC open andclosed cell foam sample. The
dashed Lines 1 and 2 represent the original locations of the front and the rear end of the foam
block, respectively. (a) t = 0 μs, both samples; (b) t = 502 μs (closed foam), t = 500 μs
(open foam); (c) t = 702 μs (closed foam), t = 710 μs (open foam). This figure is reproduced
from [19].
Reflected shock
10 mm
1 2
(a) t = 453 μs
Deformation of
open cell foam
10 mm
(b) t =1000 μs
Plywood
Open cell
foam
10 mm
(c) t =1348 μs
Figure6.15Photographsshowingtranslation anddeformationofthe1Copencell foamsample.
The dashed Lines 1 and 2 represent the original locations of the tip of the logarithmic spiral
and the end of the foam block, respectively. This figure is reproduced from [19].
96
ures 6.16(a) and (b). The pressure of the closed cell samples decreased sooner than that
of the open cell samples, because the closed cell samples experienced earlier translation.
A small contribution of the pressure drop is due to the safety hole upstream of the sam-
ple; see theholelocationinFigure6.3. Apartfromthat,thereareonlyminordifferences
between the various cases, or between the open and closed cell foam. Shock attenuation
by the foams examined in this study seems to be nearly independent of geometry and
porosity; however, the acoustic impedance of the closed cell foam is likely lower than
that of the open cell foam, and yet, reflected shock strengths are nearly alike. A time
lag between the time of arrival of the reflected wave for different geometries is observed.
This is due to the varying lengths of the logarithmic spirals resulting in different travel
times for the shock waves.
An artificial negative pressure reading was observed before the incident shock wave
inFigure6.16(a)andalsoforthecompression wave inFigure6.16(d). Thesource ofthis
negativepressurewasdeterminedtobetheresultofasmallvariationintheconstruction
of the threaded Delrin adapter used to connect the transducers to the shock tube test
section. Additional tests showed that the reading of the positive pressure was otherwise
accurate and comparable with the results of the other Delrin adapters. The artificial
negative pressure is also apparent in Figures 6.12(b) and (e).
97
Time (ms)
0 1 2
Pressure (kPa)
0
40
80
120
160
1C
2C
3C
4C
NC
(a) Open Foam
Time (ms)
0 1 2
Pressure (kPa)
0
40
80
120
160
1C
2C
3C
4C
NC
(b) Closed Foam
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
1C
2C
3C
4C
NC
(c) Open Foam
0 1 2
0
40
80
120
160
Time (ms)
Pressure (kPa)
1C
2C
3C
4C
NC
(d) Closed Foam
Figure 6.16 Overpressure profile between open cell and closed cell. Top row: (a) and (b) show
the incident and reflected shock wave. Thedashedlines show analytical solutions for the range
of Mach numbers measured in the experiment. Bottom row: (a) transmitted compression
wave; (d) “piston like” driven compression wave. This figure is reproduced from [19].
98
Chapter 7
Conclusions and Future Work
The present study studied shock wave attenuation by water, cornstarch suspensions,
films, and foam obstacles. High-frequency pressure transducers and non-invasive vi-
sualization techniques were used to obtain quantitative and qualitative measurements.
Shock tubes were designed to allow pressure measurements and visualization techniques
in order to obtain reproducible experiments.
The transition angle between RR and IR was experimentally investigated for solid,
water, andcornstarchsuspensions usingthreedifferentshockMachnumbers, M
s
=1.20,
1.38, and 1.5 in a pseudo-steady flow. The current study used two schlieren visualiza-
tion methods to determine the shock position and Mach stem length: (1) high temporal
dimensionless Mach stem method and (2) high spatial resolution pixel intensity mea-
surements method. Results showed that the transition angles using both methods were
not significantly different, but the pixel intensity method resulted in lower values of un-
certainty. The transition angle for solid surface showed 1.5
◦
lower than the detachment
criterion. This is not surprising since the detachment criterion assumes inviscid flow.
Moreover, the transition angle of water or cornstarch suspensions was 4.5
◦
lower than
thedetachmentcriterion. Thedifferenceisprimarilyduetotheenergyabsorptionbythe
dispersed droplets and the energy transmission into liquids, which do not exist in solid
surface. The transition angle of water and cornstarch suspensions was not significantly
different within the range of uncertainties.
The attenuation of shock waves by a planar liquid sheet was investigated experimen-
tally at two incident shock Mach numbers, M
s
= 1.34 and 1.46. A square frame was
developed to create a 5mm or a 10mm thick liquid sheet filled with water or cornstarch
suspensions. Plastic sheets and cotton wires were used to hold liquids inside the square
frame. A negligible amount of shock wave attenuation effect was found by the plastic
99
sheets and cotton wires of the square frame. Based on the pressure profiles, the flow
visualization, and theattenuation factor, thefollowing conclusions canbe drawn: (a)no
transmitted shock wave could be observed through a liquid sheet of water or cornstarch
suspensions; (b)compressionwaves weregeneratedbytheliquidsheetbreakupandlater
transition into a coalesced shock wave; and (c) a cornstarch sheet showed shock-driven
solidification, which resulted in a lower peak pressure and impulse compared to a water
sheet. These results showed that water or cornstarch suspensions had asignificant effect
on attenuating the strength of shock waves, and could possibly be incorporated into
protective armors and barriers.
Shock tube experiments were conducted to study shock wave attenuation by thin
films at four different incident shock Mach numbers of M
s
= 1.20, 1.34, 1.39. Different
thicknesses of h = 0.92, 12.7, 25.4, and 50.8µm were chosen to study the shock wave
attenuation by the scale of events on the relative thickness of the materials compared to
theshockthickness. Tounderstandhowthefailuremechanism (ductileorbrittle)affects
the shock wave attenuation, polyester and aluminum films were used. Results showed
that shock wave attenuation was dominated by the film thickness (< 125% difference)
rather than the acoustic impedance (< 9.2% difference) when the normalized thickness
(the film thickness compared to the shock wave thickness) was within two orders of
magnitude. When the normalized thickness was of the same order of magnitude, shock
wave attenuation by the film was negligible. The failure mechanism of the materials did
not affect the strength of the transmitted shock wave relative to the that of the incident
shock wave.
Shock wave attenuation by open and closed cell foam obstacles was studied to un-
derstand the attenuation effect depending on different front face geometries of the foam
obstacles. The pressure magnitudes of the incident shock wave, M
s
= 1.25±0.1, were
obtained by high-speed image processing and were close to the analytical results. The
reflected shock waves were not compared to the analytical results, since the open and
closedcellsampleswerenotfixedinsidetheshocktubeandmovedaftertheimpact. The
experiments were repeated four to five times for each case, but no significant difference
was observed within the results for all the five different geometry cases, except the time
lag. The geometry of the obstacles did not influence the degree of attenuation. The
100
type of foams explored in this study seemed to influence the reflected shock wave. The
rounding of the pressure pulse of the reflected shock wave in the case of the open cell
foam could be attributed to a longer time for shock coalescence, also noted in [124]. It
is noted that the strength of the reflected wave was considerably weaker than that of
closed cell foam. In the case of the closed cell, there was only “piston like” driven com-
pression wave and edge effects. However, the well-known transmitted compression wave
phenomena [17,106]were found in the open cell caused by its porosity and permeability,
even though it was attenuated by the complex geometry of the open cell foam. The
strength of the transmitted compression waves will increase corresponding to a increase
in incident shock wave speed.
A few recommendations for the future work on this topic are listed as follows:
• Further investigations will be needed to quantify the difference in the breakup
processes of a liquid sheet of water to that of cornstarch suspensions. To better
understand the details of the dispersal characteristics of liquid droplet clouds dur-
ing and after interactions with shock waves, a new experimental approach needs
to be undertaken. Future experiments can include a liquid sheet curtain generator
to create a liquid sheet without the plastic sheets and cotton wires since these
can generate unnecessary boundary conditions that are complex and unnecessary.
To analyze the results, scaling studies are required to extend the knowledge from
well-controlled small scale experiments to large scale applications. For example,
particle size, particle number, and the characteristic time for the liquid sheet ex-
pansion process can be quantified using images acquired by a high-speed camera.
Finally,theresultscouldbecomparedwiththedynamicsofdensesolidcloudparti-
cles subjected to shock waves to find different scaling laws [34,125]. Additionally,
the results can be used to compare against numerical simulations for modeling
shock-liquid interaction induced flows.
• More research will be required to complete an analytical and numerical solution
of the interactions between a shock wave and a thin film. The analytical solu-
tion can be derived using the one-dimensional interface of an air-film-air while
101
considering the thickness of the film. More specifically, the transmitted pressure
throughair-film-airinterface canbe calculated based ontheacoustic impedance of
each material and the film thickness. In addition, more experiments are required
using diverse materials in a wide range of incident shock Mach numbers to fully
understand the interactions between the shock wave and the film.
102
Appendix A
Calculating Local Speed of Sound
The speed ofsoundinthis study is different fromthatofdry air(approximately 346m/s
at T = 25
◦
C) because the dry air is mixed with water vapor in the test section. The
mixture ratio can be calculated based on the weight percentage of water or cornstarch
suspension, asshowninFigureA1. Themixtureofthespecificheatratio,γ
mix
,isdefined
as
γ
mix
=
(C
p
)
mix
(C
v
)
mix
, (A.1)
where C
p
and C
v
are the specific heat capacities at constant pressure and constant
volume. The specific heat capacities were calculated as [56]
(C
p
)
mix
=
αρ
a
(C
p
)
a
+(1−α)ρ
v
(C
p
)
v
αρ
a
+(1−α)ρ
v
, (A.2)
and
(C
v
)
mix
=
αρ
a
(C
v
)
a
+(1−α)ρ
v
(C
v
)
v
αρ
a
+(1−α)ρ
v
, (A.3)
where α is the mixture ratio and the subscripts v and a indicate water vapor and air,
respectively. Equations (A.1), (A.2), and (A.3) yield
γ
mix
=
ρ
a
γ
a
ρ
a
+δρ
v
+
ρ
v
γ
v
1/δρ
a
+ρ
v
, (A.4)
where δ =(C
v
)
v
/(C
v
)
a
, γ
a
=(C
p
)
a
/(C
v
)
a
, and γ
v
= (C
p
)
v
/(C
v
)
v
. Similarly, the mixture
of the specific gas constant is
R
mix
= (C
p
)
mix
−(C
v
)
mix
, (A.5)
which can be expressed using Eqs. (A.1) and (A.2) as
R
mix
=
αρ
a
R
a
+(1−α)ρ
v
R
v
αρ
a
+(1−α)ρ
v
. (A.6)
103
Air Water Cornstarch
(a)
(b)
Figure A1 Schematic description of determining mixture ratio for the two cases. The number
of circles represent different mixture ratios that are approximately calculated based on the
weight percentage of the liquid wedge: (a) water to air ratio 1:1; (b) cornstarch with water to
air ratio 1:3.
Table A.1 Speed of sound calculation at room temperature.
Content Mixture ratio (α) Specific heat ratio (γ) Gas constant (R) Speed of sound (a)
[ - ] [ - ] [J/kgK] [m/s]
Air 1 1.40 287 346
Water vapor 0 1.33 462 428
Air and water vapor 0.5 1.40 291 348
Air and cornstarch suspensions 0.666 1.40 289 347
In this study, the properties of the mixed gases are taken as constant. The mixture
ratio is approximately calculated based on the weight percentage, as illustrated in Fig-
ure A1. A summary of calculating the local speed of sound is shown in Table A.1. Note
that despite the assumption of a mixture ratio, there is no significant difference in the
properties of the mixed gases.
104
Appendix B
Area Reduction Effects
AspresentedinSection3.2,thecrosssectionofthedriveroftheshocktubeislargerthan
that of the driven section. In this study, the ratio of the driver to driven cross section
is A
4
/A
1
= 2.90. To quantify the enhancement of the shock properties, the analytical
model suggested by Alpher and White [36] is used, as shown in Figure B1. This model
assumes the area reduction between the driver and driven section as a convergent nozzle
to increase the Mach number of the flow. When a general case of convergent nozzle flow
is considered, the pressure ratio can be expressed as
p
4
p
1
=
p
4
p
3a
p
3a
p
3b
′
p
3b
′
p
3b
p
3b
p
3
p
3
p
2
p
2
p
1
(B.1)
where p is the pressure in different flow regions. This pressure ratio can be expressed as
p
4
p
1
=
p
2
/p
1
g
1−
u
2
a
1
a
1
a
4
γ
4
+1
2
g
−(γ
4
−1)
2γ
4
−2γ
4
γ
4
−1
, (B.2)
where a is sound velocity, u is the particle velocity, γ is specific heat ratio, and g is an
equivalence factor defined as [116]
g =
2+(γ
4
−1)M
2
3a
2+(γ
4
−1)M
2
3b
1
2
2+(γ
4
−1)M
2
3b
2+(γ
4
−1)M
2
3a
2γ
4
(γ
4
−1)
. (B.3)
The relationship between the area reduction ratio and the shock Mach number, the
area-Mach number relation, can be written as
A
4
A
1
=
M
3b
M
3a
2+(γ
4
−1)M
2
3a
2+(γ
4
−1)M
2
3b
(γ
4
+1)
2(γ
4
−1)
. (B.4)
The shock Mach number in region 3, M
3
, can be written as
M
3
=
a
1
u
2
a
4
a
1
g
(γ
4
−1)
2γ
4
−
γ
4
−1
2
−1
. (B.5)
105
Expansion fan 1
Expansion fan 2
Contact surface
Shock wave
Driver section Diaphragm Driven section
p
4
3a
3
′
b
3
b
3 2 1
p
4
p
3a
p
3b
′
p
3 p
2
p
1
section
Figure B1 Schematic description of a converging shock tube and corresponding pressure dia-
gram.
There are two possible solutions for the area-Mach number relation, one in the subsonic
regime, and one in the supersonic regime. Therefore, the condition of the flow can
be changed by the throat condition. When the flow is in the subsonic domain, the
correlation is expressed as M
3
=M
3b
and when the flow is in the supersonic domain, the
correlation is M
3b
′ = 1. Then, the above equations allow the calculation of the shock
Mach number withanareareduction ratioasp
4
/p
1
=f(A
4
/A
1
, γ). In FigureB2, shock
Mach numbers are presented for a range of driver pressure with different area reduction
ratio. It shows that higher Mach number can be achieved with lower driver pressure
for an equivalent shock tube with higher reduction ratio. The measured Mach number
data shows moderate agreement to analytical values at M
s
= 1.38 and M
s
= 1.52. The
discrepancy between the analytical line and measured data at M
s
= 1.20 is possible
caused by the deformation of the polyester diaphragm.
106
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
10
2
10
3
Shock Mach number (M
s
)
Driver pressure (kPa)
A
4
/ A
1
= 1
A
4
/ A
1
= 2.90
Data
(a) Water
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7
10
2
10
3
Shock Mach number (M
s
)
Driver pressure (kPa)
A
4
/ A
1
= 1
A
4
/ A
1
= 2.90
Data
(b) Cornstarch suspensions
Figure B2 Incident shock Mach number as a function of driver pressure (gauge pressure) for
(a) water and (b) cornstarch suspensions.
107
Appendix C
Close Cell Foam Results
Schlieren photographs of the close cell foam cases are shown in Figures C1–C5. The
schlieren photographs show no significant difference compared to the open cell foam
results. Again, the blue and red arrows indicate the incident and reflected shock waves.
The yellow arrows indicate the shock wave that propagates in between the test section
windows, which is not interacting with the foam sample. The closed cell foam block
starts to move at 350 μs, which is 100 μs earlier than the open cell foam samples.
The pressure plots for the incident and reflected shock waves for all five cases are
shown in Figure C6. Figure C7 shows the data from the transmitted compression wave.
The pressure profiles in gray represent the individual raw data from 4–5repeated exper-
iments per case, and the black line shows the average. As can be seen, the experiments
are repeatable. The transmitted wave is not a shock wave, but rather a weak compres-
sion wave due to downstream translation of the open cell foam block occurring at about
450 μs.
(a) 23.75 μs (b) 47.50μs (c) 71.25 μs (d) 95.00 μs (e) 118.75 μs
Figure C1 Closed cell foam, Case NC. This figure is reproduced from [19].
108
(a) 66.60 μs (b) 133.20 μs (c) 199.80 μs
(d) 266.40μs (e) 333.00μs
Figure C2 Closed cell foam, Case 1C. This figure is reproduced from [19].
(a) 28.50 μs (b) 57.00 μs (c) 85.50μs (d) 114 μs (e) 142.50 μs
Figure C3 Closed cell foam, Case 2C. This figure is reproduced from [19].
fig:Sequence3CC
(a) 23.75 μs (b) 47.50 μs (c) 71.25μs (d) 95.00μs (e) 118.75 μs
Figure C4 Closed cell foam, Case 3C. This figure is reproduced from [19].
109
(a) 23.75 μs (b) 47.50μs (c) 71.25 μs (d) 95.00 μs (e) 118.75 μs
Figure C5 Closed cell foam, Case 4C. This figure is reproduced from [19].
(a) NC (b) 1C (c) 2C
(d) 3C (e) 4C
Figure C6 Closed cell foam overpressure recordings for all cases: incident and reflected shock.
This figure is reproduced from [19].
110
(a) NC (b) 1C (c) 2C
(d) 3C (e) 4C
Figure C7 Closed cell foam overpressure recordings for all cases: transmitted wave. This figure
is reproduced from [19].
111
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Abstract (if available)
Abstract
The understanding of shock wave attenuation has been, and still is, essential in many parts of the world. Shock waves appear in daily life (volcanic eruption, lightning, animals living in the ocean, etc.) and can be beneficial or detrimental depending on the situation. Therefore, it is important to understand shock wave dynamics, so that we more efficiently can use shock waves to our benefit (e.g. shock wave lithotripsy) or protect ourselves from the nonlinear effects of a shock wave. In large-scale applications, porous materials or liquids have been considered as an economical and practical solution to provide protection against shocks. In this research, compressible porous foams, water, a cornstarch and water mixture (a cornstarch suspension), and thin films have been studied to improve the current understanding of their dynamic response to shock waves, especially in regards to shock wave reflection, mitigation, and transmission. Experiments were performed using a pressure-driven shock tube. Non-invasive schlieren visualization along with high-frequency pressure transducers were used to obtain qualitative and quantitative data from the experiments. The properties examined during the current study were then utilized to investigate shock wave reflection configurations, shock Mach numbers, impulse, and overpressure. At first, experiments were conducted to understand the difference between the shock wave reflection off water, and cornstarch suspension surface, and the results were compared with an analytical solution. Shock wave reflection off the surface of water and cornstarch suspensions were different from the analytical solution, while the reflection configurations between the water and the cornstarch suspension were not significantly different. Additionally, shock wave attenuation by different thicknesses of water and cornstarch suspension were studied. No transmitted shock wave was observed through water and cornstarch suspension sheets, but compression waves induced by the shock-accelerated liquid coalesced into a shock wave. Furthermore, shock wave attenuation by a thin film was quantitatively studied with various incident shock Mach numbers and a non-dimensional analysis was performed to generalize the experimental results. Results showed that the thickness of the film affected the attenuation of shock wave when the film thickness is of similar order of magnitude compared to shock wave thickness. Finally, shock wave attenuation using form obstacles was investigated to understand the effect of geometries. No significantly different results were found among the five different geometries, nor between the two types of foam. This dissertation is part of an ongoing effort to understand shock wave interactions with various materials and geometries, and their dynamic responses using experimental techniques. The present results can be compared against numerical simulations for modeling shock wave interactions.
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Jeon, Hongjoo
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An experimental study of shock wave attenuation
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