저 시-비 리- 경 지 2.0 한민

는 아래 조건 르는 경 에 한하여 게

l 저 물 복제, 포, 전송, 전시, 공연 송할 수 습니다.

다 과 같 조건 라야 합니다:

l 하는, 저 물 나 포 경 , 저 물에 적 된 허락조건 명확하게 나타내어야 합니다.

l 저 터 허가를 면 러한 조건들 적 되지 않습니다.

저 에 른 리는 내 에 하여 향 지 않습니다.

것 허락규약(Legal Code) 해하 쉽게 약한 것 니다.

Disclaimer

저 시. 하는 원저 를 시하여야 합니다.

비 리. 하는 저 물 리 목적 할 수 없습니다.

경 지. 하는 저 물 개 , 형 또는 가공할 수 없습니다.

공학박사 학위논문

Splash and cavity formation

by the entry of a projectile

depending its aspect ratio and

surface roughness

입수하는 물체의 종횡비 및 표면 거칠기가

스플래시 및 캐비티 형성에 미치는 영향

2019 년 2 월

서울대학교 대학원

기계항공공학부

김 나 영

Splash and cavity formation

by the entry of a projectile depending

its aspect ratio and surface roughness

입수하는 물체의 종횡비 및 표면 거칠기가

스플래시 및 캐비티 형성에 미치는 영향

지도 교수 박 형 민

이 논문을 공학박사 학위논문으로 제출함

2018 년 `10월

서울대학교 대학원

기계항공공학부

김 나 영

김나영의 공학박사 학위논문을 인준함

2018 년 12 월

위 원 장 (인)

부위원장 (인)

위 원 (인)

위 원 (인)

위 원 (인)

I

Splash and cavity formation

by the entry of a projectile

depending its aspect ratio and surface roughness

Nayoung Kim, Doctor of Philosophy

Mechanical and Aerospace Engineering

Seoul National University, 2018

Advisor: Prof. Hyungmin Park

Abstract

The dynamics of free surface deformation due to an impacting projectile is

a fundamental problem in fluid dynamics. In this study, rounded cylindrical bodies

are investigated as impacting projectiles, while varying their aspect ratio (AR = 1–8)

and surface condition (smooth and rough) of the front side. Two impact speeds are

considered (Uo = 2.5 m/s and 4.2 m/s) and the related non-dimensional numbers are

in the ranges of Fr = 32–90, Re = (5–8.4) × 105, and Ca = 0.0345–0.0579. Jet, splash,

and cavity are generated by the free surface deformation, which highly depends on

the impact condition. When the surface of projectile is smooth, a thin liquid film can

II

rise along the body surface and converge at the pole of the rear side. Subsequently,

a thin jet is produced. As AR increases, the duration for which the liquid film stays

on the side of the body becomes longer, such that the liquid film loses its inertia from

the impact and the jet height is reduced. After generating the thin jet, a thick jet is

produced as the body sinks. There is a flow toward the rear side of body that induces

the free surface to sag. The sagged free surface tends to recover its shape, and the

upward flow becomes a thick jet. The tip of the jet contracts and breakup occurs,

depending on the jet-tip radius and its aspect ratio. The Ohnesorge number, which

represents the ratio of surface tension to viscous force, is Oh ≃ 10−3 for all cases, and

thus the viscosity is dominant in the jet-tip breakup. There is critical jet aspect ratio

for breakup, which is affected by the impact velocity and AR of the body. When the

front part is roughened using sandpaper, the liquid film separates from the surface

and forms a splash crown and cavity. As AR increases, the splash height increases

for Uo = 2.5 m/s and decreases for Uo = 4.2 m/s. The formation time of the dome and

cavity pinch-off depend on AR. The airflow in the wake of body is faster as AR

increases, such that the pressure inside the splash is lowered, which induces the dome

early. Moreover, the cavity pinch-off is delayed as AR increases. This is because the

added mass increases as AR increases, which is the same effect as increasing the

density. Finally, the force coefficient is obtained based on trajectory data, and it

shows that at higher AR, the deceleration is low owing to the increased weight. When

the cavity wraps the body before pinch-off, it reduces the drag force acting on the

body. However, the attached cavity after pinch-off increases the drag force.

Keyword : free-surface impact, water entry, splash, cavity formation, liquid film

Student Number : 2013-20648

III

Contents

Abstract I

Contents III

List of Tables VI

List of Figures VII

Nomenclature XV

1. Introduction 1

2. Experimental methods

2.1 Aspect ratio and surface condition of projectiles 6

2.2 Impact conditions 7

2.3 Measurement technique 8

2.4 Image post-processing 9

3. Water entry of smooth projectiles

3.1 Free surface deformation due to impact of smooth projectile 15

3.2 Water film rise along the free surface

3.2.1 Thin jet formation by liquid film convergence 16

IV

3.2.2 Film rising and falling motion

3.2.2.1 Overall behavior of liquid film motion 18

3.2.2.2 Differential approach for liquid film .

dynamics of AR 8 case 19

3.2.2.3 Control volume analysis for liquid film .

dynamics of AR 8 case 22

3.2.2.4 Comparison of two model equations .

and term contribution 26

3.3 Thick jet formation by flow induced sinking projectile 27

3.4 Jet-tip breakup 28

3.5 Conclusion 30

4. Water entry of rough projectiles

4.1 Free surface deformation due to impact of rough projectile 54

4.2 Formation of a splash

4.2.1 Splash 55

4.2.2 Dome 56

4.3 Formation of a cavity

4.3.1 Overall cavity shapes 57

4.3.2 Cavity volume 58

4.3.2 Mechanism of splash dome and cavity pinch-off 59

4.4 Conclusion 61

V

5. Effect of cavity on the dynamics of a sinking projectile

5.1 Sinking motion of smooth and rough cases 76

5.2 Force coefficient 78

5.3 Tumbling motion of AR 2 79

5.4 Conclusion 80

6. Concluding remarks 89

References 93

VI

List of Tables

Table 2.1 Geometric parameters of projectiles depending on their aspect ratio

(AR). R and L are the radius and length of projectiles, respectively.

·············································································10

Table 2.2 Dimensionless numbers based on two different impact velocities

(Uo): Froude number (Fr = Uo2/gR), Reynolds number (Re =

ρUoR/μ), Weber number (We = ρUo2R/σ), Bond number (Bo =

ρgR2/σ), and capillary number (Ca = μUo/σ). Here, σ, ρ, and μ are

the surface tension, density and dynamic viscosity of the water and

g is the gravitational acceleration. ····································11

Table 3.1 Jet-tip radius (Rtip) when jet reaches the maximum height (Hjet,max),

and corresponding Bond number based on jet-tip radius (Botip),

Ohnesorge number (Oh) and aspect ratio of jet (Hjet,max /(2Rtip)). The

breakup of the jet-tip occurs in the case of AR 1, 2 for Uo = 2.5 m/s

and AR 1 for Uo = 4.2 m/s. ·············································31

Table 4.1 Dome formation time (td), cavity pinch-off time (tp), locations of

cavity pinch-off point (Hp) and frontal center (H), and ratio thereof

(Hp/H) for Uo = 4.2 m/s. ················································62

Table 5.1 Dimensionless time (t*) when front and rear parts are flipped as AR

2 descends with respect of Uo. In this field of view, tumbling occurs

two or three times for the smooth case and once for the rough case.

In the smooth case, the time of the first and second tumbles are all

similar, regardless of Uo. However, in the rough case, the first

tumble is delayed as Uo increases. This induces an increasing form

of drag. ···································································81

VII

List of Figures

Figure 2.1 (a) Projectiles with two different front parts are considered, one

with a smooth surface and the other with a roughened surface.

Each surface has four aspect ratios, AR = 1, 2, 4, and 8. To match

the impact velocity at the free surface, the nose tip of each

projectile is placed at the same height above the free surface. (b)

In this experimental condition, a cavity is not created when the

smooth body enters; however, (c) the roughened body can

generate a cavity. The SEM images of AR 1 with (d) smooth front

surface and (e) rough front surface. (f) The static contact angle

(θc) of the rough surface is larger than that of the smooth surface,

because the three-phase contact line is pinned by the asperity of

the roughness. (g) The static contact angle of front part is also

higher than side part due to its curvature. ·························12

Figure 2.2 (a) Experimental setup of impacting projectile with large water

tank. To adjust the entry angle, a hollow cylinder is used in the

vacuum release system, as shown in the inset image. (b)

Consequently, the angle between the longitudinal axis and the

free surface is obtained as 90°. ·····································13

Figure 2.3 (a) To track the center of the front and rear parts, raw images are

first obtained. (b) Then, binarization and Wiener filter are applied.

(c) Center positions are found using the Hough transform

function. (d) The center of the rear part can be found by selecting

the data located one body length from the front center. The center

of mass is defined as the midpoint between the front and rear

centers. ································································14

Figure 3.1 Overall free surface deformation induced by entering smooth

projectile with different aspect ratios: (a) AR 1, (b) AR 2, (c) AR

4, and (d) AR 8. The impact velocity at the free surface is Uo =

2.5 m/s. The numbers represent t*. ································32

VIII

Figure 3.2 Overall free surface deformation induced by entering smooth

projectile with different aspect ratios: (a) AR 1, (b) AR 2, (c) AR

4, and (d) AR 8. The impact velocity at the free surface is Uo =

4.2 m/s. The numbers represent t*. ································33

Figure 3.3 Formation of jet above the free surface in (a) AR 1, (b) AR 2, (c)

AR 4, and (d) AR 8. In AR 1–2, thin and thick jets are clearly

visible. There is one more jet from the shallow cavity for AR 4

and the jet from quasi-static cavity for AR 8. The impact velocity

at the free surface is Uo = 2.5 m/s. The numbers represent t*.

··········································································34

Figure 3.4 Formation of jet above the free surface in (a) AR 1, (b) AR 2, (c)

AR 4, and (d) AR 8. In AR 1 – 2, the thin jet breaks down into

small droplets as soon as it is generated. There is one more jet

from the shallow cavity for AR 8. The impact velocity at the free

surface is Uo = 4.2 m/s. The numbers represent t*.···············35

Figure 3.5 Thin liquid film is emitted after impacting, and it moves along

the body surface of (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8. In

AR 1–2, the thin liquid film gathers at the pole of the body.

However, the body of AR 4 enters the water before the film

gathers. In AR 8, the liquid film stays only on the side part of the

body. The impact velocity at the free surface is Uo = 2.5 m/s. The

numbers represent t*. ················································36

Figure 3.6 Thin liquid film is emitted after impacting, and it moves along

the body surface of (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8.

The impact velocity at the free surface is Uo = 4.2 m/s. The

movement of the liquid film depending AR is similar to that

shown in figure 3.5, but owing to the increased impact speed,

many water droplets are generated at the end of the liquid film.

The numbers represent t*.···········································37

Figure 3.7 End of liquid film (moving contact line) for (a) Uo = 2.5 m/s and

(b) Uo = 4.2 m/s. ●, AR 1; ○, AR 2; □, AR 4; ◊, AR 8. The

IX

time interval is △t* = 0.05 and the origin is set to the nose of

body first contacting the free surface. The data are averaged 10

times. ··································································38

Figure 3.8 Temporal variation of liquid film height for (a) Uo = 2.5 m/s and

(b) Uo = 4.2 m/s. ●, AR 1; ○, AR 2; □, AR 4; ◊, AR 8. The

data are averaged 10 times and the solid line represents estimated

data from equation 3.5. In the initial rising motion, there is no

difference in AR because they all have same impact speed.

However, when the liquid film meets the rear part of each AR,

the liquid film height falls steeply. The liquid film of AR 8 does

not reach the rear part of the projectile. ····························39

Figure 3.9 Sketch for investigating the mechanism of the liquid film motion.

The liquid film height from the undisturbed free surface is

defined as h(t). The dynamic contact angle (θD) and liquid film

thickness (e) are assumed as constant. The liquid film moves at

a speed of dh/dt(t), and the initial condition dh/dt(0) is given as

positive, which means that the liquid film rises at first. ·········40

Figure 3.10 (a) Tracking frontal center of smooth AR 8 using Hough transform

function. (b) The front part of body descends almost constant

speed. ●, Uo = 2.5 m/s; ○, Uo = 4.2 m/s. The data are averaged

10 times. (c) During t* = 0–8 of the liquid film going up and

down, the changes in the downward velocity are approximately

0.2% for Uo = 2.5 m/s and 1.5% for Uo = 4.2 m/s. ················41

Figure 3.11 Velocity profile of liquid film from equation 3.2. dh/dt is positive

when the liquid film ascends and negative when it descends.

··········································································42

Figure 3.12 Temporal variation of liquid film thickness for (a) Uo = 2.5 m/s

and (b) Uo = 4.2 m/s in AR 8 case (◊). The data are averaged 10

times and the red star symbol represents the time when the liquid

film reaches maximum height. At Uo = 2.5 m/s, the thickness of

liquid film is almost constant initially and increases as the film

X

descends. As the impact speed increases to 4.2 m/s, the thickness

decreases while the liquid film rises and it steeply increases

while liquid film falls. ···············································43

Figure 3.13 (a) To obtain the meniscus curvature of the liquid film in the

smooth AR 8 case, an outline profile is captured and then the

point at which its slope exceeds 0.15 is tracked. (b) The temporal

variation of the liquid film and body profile (solid lines), and

height of the meniscus curvature (●) for Uo = 2.5 m/s. ··········44

Figure 3.14 Temporal variation of meniscus height (●), liquid film height ( ),

and their ratio (●) for Uo = 2.5 m/s in the smooth AR 8 case. The

data are averaged 10 times. There is a maximum limit that the

meniscus curvature can reach.······································45

Figure 3.15 Sketch for the control volume (CV) analysis of liquid film motion.

The rectangular fixed control volume contains both water and air

phases. CS means control surface and the liquid film height from

the undisturbed free surface is defined as h(t). The red arrows

indicate the forces acting on CV, and the blue arrows indicate the

inflow and outflow through CS. ····································46

Figure 3.16 Velocity field of air phase when the smooth AR 8 enters the water.

The air flows upward with the ascending liquid film. In the

opposite site, the air comes down due to the descending projectile.

On the area where they meet (black rectangle), there is radial

flow because downward and upward flows collide. ·············47

Figure 3.17 Velocity field of water phase when smooth AR 8 enters the water.

As the liquid film rises above the free surface, there is an upward

flow below the free surface on the side of body. ··················48

Figure 3.18 Comparison of differential approach (black solid line, equation

3.5) and CV analysis (red solid line, equation 3.20). The symbol

( ) represents the experimental data for (a) Uo = 2.5 m/s and (b)

Uo = 4.2 m/s. ··························································49

XI

Figure 3.19 Term contribution of (a) equation 3.5 (○: first term, : second

term, : third term, : fourth term, : experimental data) and (b)

equation 3.20 (○: first term, : second term : third term, :

fourth term, : fifth term, : experimental data). ················50

Figure 3.20 Velocity filed of water and the process of forming the thick jet on

the free surface at the same time. The numbers in the velocity

filed represent t*. ····················································51

Figure 3.21 Temporal variation of jet height for (a) Uo = 2.5 m/s and (b) Uo =

4.2 m/s. ●, AR 1; ○, AR 2; □, AR 4. The red star symbol represents

the moment when the jet-tip breakup occurs and the data are

averaged 10 times. ···················································52

Figure 3.22 Overall sequence of the free surface deformation by impacting

smooth projectile. (a) When the smooth projectile impacts on the

free surface, (b) thin liquid film is emitted and moves along the

body surface. (c) As the film reaches the pole of the rear part, (d)

a thin jet is generated at the point of film convergence. (e) Below

the free surface, the descending body pushes the surrounding

water out, inducing the flow structure. The upward flow above

the stagnation point becomes a thick jet above the free surface.

(f) The tip of the jet experiences a necking phenomenon

according to Oh and its aspect ratio (Lo). ··························53

Figure 4.1 Overall free surface deformation induced by entering rough

projectile with different aspect ratios: (a) AR 1, (b) AR 2, (c) AR

4, and (d) AR 8. The impact velocity at the free surface is Uo =

2.5 m/s. The numbers represent t*. ································63

Figure 4.2 Overall free surface deformation induced by entering rough

projectile with different aspect ratios: (a) AR 1, (b) AR 2, (c) AR

4, and (d) AR 8. The impact velocity at the free surface is Uo =

4.2 m/s. The numbers represent t*. ································64

XII

Figure 4.3 Thin liquid film is emitted after impacting, and it separates from

the body surface for (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8.

The impact velocity at the free surface is Uo = 2.5 m/s. However,

in AR 1 (a), the liquid film separation does not occur on some

sides of the object. The numbers represent t*. ····················65

Figure 4.4 Thin liquid film is emitted after impacting, and it separates from

the body surface for (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8.

The impact velocity at the free surface is Uo = 4.2 m/s. In contrast

to the low-impact-velocity case, the liquid film separation occurs

on all sides of the object. The numbers represent t*. ·············66

Figure 4.5 Temporal variation of splash height for (a) Uo = 2.5 m/s and (b)

Uo = 4.2 m/s. ●, AR 1; ○, AR 2; □, AR 4; ◊, AR 8. At low impact

speed, the splash height increases as AR increases, but at high

impact speed, the effect is reversed. The data are averaged 10

times. ··································································67

Figure 4.6 Temporal variation of splash width for (a) Uo = 2.5 m/s and (b)

Uo = 4.2 m/s. ●, AR 1; ○, AR 2; □, AR 4; ◊, AR 8. At low impact

speed, the splash width increases as AR increases. However, at

high impact speed, there are no significant differences in the

initial stage, and the splash of AR 8 closes and meets the object

(Ws/D = 1.0). The splash of AR 1 takes the longest time to form

the dome (Ws/D ≃ 0). The data are averaged 10 times. ··········68

Figure 4.7 Cavity dynamics for (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8.

The impact velocity at the free surface is Uo = 2.5 m/s. The cavity

is formed asymmetrically and the descending motion bends to

the side lacking the cavity. The numbers represent t*. ···········69

Figure 4.8 Cavity dynamics for (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8.

The impact velocity at the free surface is Uo = 4.2 m/s. The cavity

is formed symmetrically. In AR 4 & 8, the pinch-off cavity occurs

in the middle of body. The numbers represent t*. ················70

XIII

Figure 4.9 Velocity field of water phase after AR 8 enters the water. (a) As

the projectile impacts the free surface, the surrounding water is

pushed to the radial direction at Uo = 2.5 m/s. (b) As the cavity

shrinks, the water flows toward the point where cavity pinch-off

occurs at Uo = 4.2 m/s. The numbers represent t*. ···············71

Figure 4.10 Characteristic cavity shape behind the projectile with rough

surface. (a) Tooth-like three phase contact line (AR 1, Uo = 4.2

m/s). (b) Close-up view of the rough cavity surface (AR 4, Uo =

4.2 m/s). Oscillating interface of the air cavity attached to the

sidewall of the sinking projectile at Uo = 4.2 m/s in (c) AR 4 and

(d) AR 8. The numbers represent t*. ·······························72

Figure 4.11 Temporal variation of cavity volume for (a) Uo = 2.5 m/s and (b)

Uo = 4.2 m/s. ●, AR 1; ○, AR 2; □, AR 4; ◊, AR 8. Each volume is

assumed to be axisymmetric and subtracted by the volume

occupied by the projectile. The data are averaged 10 times. ····73

Figure 4.12 Velocity field of air around splash. The air enters the splash and

forms the cavity below the free surface. The numbers represent

t*. ······································································74

Figure 4.13 Overall sequence of the free surface deformation by impacting

rough projectile. (a) When the rough projectile impacts on the

free surface, (b) a splash and cavity are formed. Subsequently,

the surrounding air enters the splash and the pressure inside the

splash is lower than the atmospheric pressure (p0) owing to the

airflow. (c) Therefore, the splash is closed to the dome and cavity

begins to contract owing to the hydrostatic pressure of the

surrounding water. The wetted area is always same, even though

AR increases, such that the added mass effect increases as AR

increases. As a result, the cavity pinch-off is delayed with

increasing AR. ························································75

XIV

Figure 5.1 Trajectory of descending projectile for (a) Uo = 2.5 m/s and (b)

Uo = 4.2 m/s. ○, smooth surface; ●, rough surface. The data are

averaged 10 times. ···················································82

Figure 5.2 Velocity of descending projectile for (a) Uo = 2.5 m/s and (b) Uo

= 4.2 m/s. ○, smooth surface; ●, rough surface. The data are

averaged 10 times. ···················································83

Figure 5.3 Acceleration of descending projectile for (a) Uo = 2.5 m/s and

(b) Uo = 4.2 m/s. ○, smooth surface; ●, rough surface. The data

are averaged 10 times. ···············································84

Figure 5.4 Velocity field of water when the body of (a) AR 1 and (b) AR 2

rotate. The numbers represent t*. ··································85

Figure 5.5 Force coefficient of descending projectile for (a) Uo = 2.5 m/s

and (b) Uo = 4.2 m/s. ○, smooth surface; ●, rough surface. The

data are averaged 10 times. ·········································86

Figure 5.6 Trajectory, velocity, acceleration, and force coefficient of

descending AR 2 projectile for (a) Uo = 3.1 m/s and (b) Uo = 3.7

m/s. □, smooth surface; ■, rough surface. The data are averaged

10 times. ······························································87

Figure 5.7 Centers of front and rear parts are tracked using Hough transform

function in the case of AR 2 with smooth surface at Uo = 4.2 m/s.

The red symbol indicates the time when nose and tail of body are

flipped. ································································88

XV

Nomenclature

Roman letters:

A cross-sectional area

AR aspect ratio of projectile (= L/D)

a1, a2 pre-factors for compensation of assumptions

Bo Bond number (Bo = ρgR2/σ)

Botip Bond number based on jet-tip radius (Botip = ρgRtip2/σ)

b1, b2, b3 pre-factors for compensation of assumptions

Ca capillary number (Ca = μUo/σ)

CF force coefficient

CV control volume

CS control surface

D diameter of projectile

DS diameter of splash

Dtank width of water tank

d solid–liquid density ratio (d = ρs/ρ)

e liquid film thickness

Fc capillary force (Fc = 2πRσcosθD)

Fg gravity force (Fg = mfg = ρ2πRehg)

Fr Froude number (Fr = Uo2/gR)

g gravitational acceleration

H distance from frontal center to free surface

XVI

Hjet,max maximum jet height

Hp distance from cavity pinch-off point to free surface

Hs splash height

h liquid film height from undisturbed free surface

h* non-dimensional liquid film height (h*=h/D)

L length of projectile

LC length of control volume

Lo aspect ratio of liquid jet (Hjet,max /(2Rtip))

mf liquid film mass added to film (mf = ρ2πReh)

Oh Ohnesorge number (Oh = μ/(ρRtipσ)0.5

p pressure

p0 atmospheric pressure

R radius of projectile

Rtip jet-tip radius

Ra arithmetical mean deviation of surface

Re Reynolds number (Re = ρUoR/μ)

r coordinate in radial direction

t* dimensionless time (t*= tUo/D)

td time of the dome formation

tp time of the cavity pinch-off

Uo impact velocity at undisturbed free surface

Va volume of air

VC volume of air in cavity

Vd volume of air in dome

XVII

Vw volume of water

We Weber number (We = ρUo2R/σ)

Ws splash width

z coordinate in vertical direction

Greek letters:

σ surface tension of water

θc static contact angle

θD dynamic contact angle

μ dynamic viscosity of water

ρ density of water

ρa density of air

ρs density of solid

1

Chapter 1

Introduction

Free surface deformation due to an impacting projectile is a fundamental

phenomenon in fluid dynamics, and engineering aspects such as ship slamming,

offshore structures, dip coating, ballistics or missiles moving underwater, and diving.

Water entry phenomena have been researched extensively since the pioneering work

of Worthington (1908). When a solid object impacts on the free surface, a thin liquid

film is ejected first, and a splash and cavity are formed depending on whether the

liquid film attaches to the body surface. As the liquid film moves along the body

surface without separation, it converges at the top of body to generate a thin liquid

jet (Kuwabara et al. 1987). On the contrary, when the liquid film is detached from

the surface, the air (or gas) phase enters the void space and a cavity is developed.

Later, the separated liquid film forms a splash crown, which collapses into a dome,

depending on the liquid properties and the rising velocity of the film. Below the free

surface, the cavity shrinks owing to hydrostatic pressure of the surrounding water,

or the splash dome suppresses the development of the cavity.

The liquid film separation is determined by several important parameters

such as the impact condition, liquid properties, body geometry, surface contact angle,

and roughness (May 1951; Duez et al. 2007; Truscott & Techet 2009; Aristoff &

Bush 2009; Zhao et al. 2014). May (1951) founded that the surface condition of a

projectile is an important factor for generating a cavity. Compared to a clean surface,

a much lower impact velocity is required for cavity creation when a spherical surface

is contaminated. Half a century later, Duez et al. (2007) more clearly determined that

2

the separation of the liquid film from a spherical surface occurs according to the

impact velocity (Uo) and surface static contact angle (θc). The critical impact velocity

for cavity creation is high at θc < 90° and in contrast, a relatively low impact velocity

is needed at θc > 90°. Therefore, it is obvious that the wettability of the surface affects

the cavity formation. Zhao et al. (2014) roughened a spherical surface and found that

the critical impact velocity dramatically decreases because the three-phase contact

line is pinned by the asperity of the roughness. In addition to the surface wettability,

the viscosity and speed of the liquid film also determine its separation. Marston &

Thoroddsen (2008) showed that when a highly viscous droplet falls into a liquid pool

of low viscosity, the liquid film follows the drop surface more easily. They also

confirmed that moderate liquid film velocity is required for convergence of the film

at the pole; that is, the liquid film detaches from the surface at high velocity and

cannot reach the end of the body at insufficient velocity. In other words, the liquid

film can move along the spherical surface if the conditions are satisfied, but the body

geometry can also lead to film separation. For example, with a wedge, there is always

liquid film separation at the sharp end (Vincent et al. 2018).

The research focusing on each phenomenon in the water entry problem

has been extensive, and there are many relevant dimensionless parameters such as

the Reynolds number (Re = ρUoR/μ), Froude number (Fr = Uo2/gR), Weber number

(We = ρUo2R/σ), Bond number (Bo = ρgR2/σ), capillary number (Ca = μUo/σ), and

solid–liquid density ratio (d = ρs/ρ). Here, ρs and R are the density and characteristic

length of the body; σ, ρ, and μ are the surface tension, density, and dynamic viscosity

of the liquid; and g is the gravitational acceleration. In the case of the absence of a

cavity, Kuwabara et al. (1987) found that the thin liquid film moves up along the

spherical surface, generating two stages of jet with different widths. The first jet is

thinner than the second jet, and only the sphere radius determines the widths of the

two at Re = 1.5 × 105. Kubota & Mochizuki (2009) also showed that as the impact

3

velocity increases (or We increases), smaller droplets are created at the end of the

liquid film owing to high inertia. On the contrary, in the case of cavity generation,

Thoroddsen et al. (2004) showed experimentally that the radial jet produced by the

separation of the liquid film appears within 1 ms after the first contact with the water

surface and its velocity can be up to 30 times faster than the impact velocity at Re ≃

105. At this moment, Marston et al. (2011) observed that air is entrapped sphere’s

head surface and it shrinks as the sphere descends at We > 1. After the splash is

formed from the separated liquid film, Marston et al. (2016) showed that the surface

tension is important along with the gas pressure in the splash close event for We ≃

103–105 and Re ≃ 104–105. Although the width of the splash increases as the pressure

decreases, the rate of collapse is independent of the surrounding pressure when the

splash collapse begins and it is only affected by the surface tension. At the same time,

under the free surface, the cavity grows and eventually shrinks, and it is classified

according to the closing event. When the necking occurs in the middle of cavity, it is

called a deep seal, whereas a shallow seal and quasi-static seal refer to the cavity

pinch-off occurring near the free surface and near the projectile, respectively. If the

dome comes down to the free surface and blocks the cavity expansion, this is called

a surface seal (Truscott et al. 2014; Aristoff & Bush 2009). Mansoor et al. (2014)

found that the deep seal occurs before the surface seal at 9 < Fr < 32, and the surface

seal is formed before the deep seal at Fr > 32. Aristoff et al. (2010) showed that with

increasing Fr, the cavity pinch-off point height (Hp) and time (tp) increase, and the

ratio of the cavity pinch-off point to the sphere center when pinch-off occurs (Hp/H)

is constant when the deceleration of the sinking sphere is low.

Moreover, several researchers carried out theoretical studies about the

cavity dynamics. Lee et al. (1997) considered that the kinetic energy loss of the

impacting body is equal to the sum of the kinetic energy of the surrounding fluid and

the potential energy due to the cavity expansion against hydrostatic pressure.

4

Duclaux et al. (2007) assumed the cavity shape as cylindrical and predicted its

behavior theoretically by using the Rayleigh–Besant problem and velocity potential.

Subsequent studies by Aristoff & Bush (2009) and Aristoff et al. (2010) extended the

theory of Duclaux et al. (2007) by considering aerodynamic pressure and surface

tension. Truscott et al. (2012) investigated cavity dynamics with the boundary

element method, in the manner of a sphere with a doublet, the cavity with a sheet

and point source, and a ring source at the center of a sphere path.

Furthermore, the generation of a cavity affects the sinking motion (or

hydrodynamic force) of the sphere. Truscott et al. (2012) studied the trajectory and

the force of a descending sphere near the free surface in the cases of both the presence

and absence of a cavity. The force acting on a descending sphere is due to the added

mass, buoyancy, and wake defect when the cavity is not created, and in the case of

cavity generation, the pressure acting on the wetted parts of the frontal side is

important for the sinking motion. Techet & Truscott (2011) showed that when a

cavity forms only on half of the body, lateral motion is induced such that the

trajectory is kinked toward the non-cavity side. Mansoor et al. (2017) confirmed that

when a sphere in the Leidenfrost state descends a long distance while the cavity is

attached after pinch-off, the force coefficient decreases when the cavity is stable, i.e.

in the absence of ripple or unstable wake due to the breakdown of the cavity into

bubbles.

Therefore, the basic shape of the sphere has been extensively investigated

as an impact projectile and is well understood. However, from a more practical point

of view, it is necessary to approach the projectile geometry in terms of engineering

aspects, i.e., a cylindrical body. An object with a larger aspect ratio or an

axisymmetric slender body is closer to the industrial applications and few studies

have investigated its impact phenomena. Here, the aspect ratio is the important

geometry parameter for a cylindrical body and is defined as AR = L/(2R), where L is

5

the body length. Bodily et al. (2014) conducted an experiment with a slender body

(AR = 10), and observed that the entry angle, nose shape, and position of the center

of mass affect the descending trajectory. That is, a projectile moves in the lateral

direction when it has higher entry angle and conical nose, and the center of mass is

far away from the nose at Fr = 56–80 and Re = (3.4–4.0) × 104. Yao et al. (2014)

predicted the cavity shape of a body with AR = 5.7, which has a hemispherical front

and flat rear, and they compared a theoretical analysis with their experimental data

at Re = 1.3 × 105 and Fr = 2600. Gekle et al. (2008) showed that the pinch-off

position of a cylinder (AR = 3.7) cavity is different from a sphere’s pinch-off position

and it is not linear with respect to Fr—there are some discrete jumps. Aristoff &

Bush (2009) observed that when a flat-tail cylinder enters the water, the free surface

is dragged to create the cavity only in the wake of the body. The diameter of a

cylinder’s (AR = 75) cavity does not exceed its body diameter, whereas the diameter

of a sphere’s cavity is longer than the diameter of the sphere.

Considering previous studies, it is necessary to investigate the water entry

problem of a cylindrical body in greater detail, especially focusing on the free surface

deformation and subsequent descending behavior. Therefore, the overall aim of this

dissertation is to investigate closely the dynamics of jets, splashes, and cavities

generated by impacting projectiles with increasing aspect ratio. The impact velocity

is also varied to observe the effect of liquid film motion. To create a cavity, the front

side of the projectile is roughened with sandpaper, i.e., the surface wettability is also

controlled. The dissertation is organized as follows. The detailed experimental

methods and conditions are described in Chapter 2, and the jet and cavity formation

cases are investigated in Chapters 3 and 4, respectively. The descending motion and

force coefficient are investigated in Chapter 5. Finally, in Chapter 6, the results of

the previous chapters are summarized and the conclusion is presented.

6

Chapter 2

Experimental methods

2.1 Aspect ratio and surface condition of projectiles

The projectiles used in the experiment are rounded cylindrical bodies,

including a sphere, the aspect ratios (AR ≡ L/(2R)) of which are 1, 2, 4, and 8 (figure

2.1a). They are made of acrylic and their other characteristic dimensional parameters

such as length and mass are summarized in Table 2.1.

To control the surface wettability, two frontal noses of projectile are

considered, one with a smooth surface and the other with a roughened surface. In the

case of a sphere (AR 1), the cavity is generated only when the surface is roughened

at the same impact velocity (figure 2.1b & c). This means the critical impact velocity

for cavity formation is lowered in the roughened surface case. Thus, the head surface

is roughened with sandpaper (grit number #60) to investigate how the cavity and

splash are affected by the aspect ratio. To measure the surface roughness, SEM

images are captured of the side view of the surface and binarization is applied. The

arithmetic mean deviation is calculated using Matlab, yielding Ra = 12.98 μm for the

rough surface and Ra = 0.26 μm for the smooth surface. All four AR cases have

similar roughness values and the top view of the SEM images of AR 1 are shown in

figure 2.1(d) and (e). Moreover, the static contact angles are measured by a method

based on B-splines (Stalder et al. 2006) and they have no significant differences in

AR, only the differences in the roughness and curvature of surfaces. The static contact

angle of a roughened surface has a higher value than a smooth surface; e.g., the

7

smooth AR 1 has θc ≃ 71° and the rough AR 1 has θc ≃ 108° (figure 2.1f). With a

smooth surface, the static contact angle of the nose (θc ≃ 82°) is higher than the side

of the body (θc ≃ 74°), owing to its curvature (figure 2.1g).

2.2 Impact conditions

As shown in figure 2.2(a), the projectiles fall to a large acrylic tank filled

with static tap water. The ratio of the body diameter to the tank width is D/Dtank =

0.05, and Mansoor et al. (2014) demonstrated that the cavity dynamics are affected

by tank wall at D/Dtank > 0.075. They found that when D/Dtank > 0.075, the cavity is

contracted at several points simultaneously, such that waves (or ripples) are

generated at the cavity wall. Conversely, when D/Dtank < 0.075, the pinch-off time

(tp) and the distance from the free surface to the pinch-off point (Hp) are almost

constant with respect to Fr. Thus, in this setup, the effect of the tank wall can be

ignored because the ratio is smaller than 0.075.

The projectile is held by a vacuum release system, which is positioned

above the center of the tank. The release height can be adjusted to control the impact

speed at the free surface. In order to drop the projectile exactly vertically, a hollow

cylinder with an inner diameter equal to the outer diameter of the object is fitted on

the body when it is held on the vacuum pump (Rocker 300, Rocker), and it is aligned

vertically with the water surface (figure 2.2a). Then, it is removed before the launch

so as not to interfere with the falling object. The vacuum pressure is set to

approximately 10 kPa, because excessive pressure induces rotational motion of the

falling body. Hence, it is confirmed that when the vacuum is gradually removed, the

body falls vertically without tilting. As shown in figure 2.2(b), the angle between the

8

longitudinal axis and free surface is 90°. To obtain the same impact velocity for all

aspect ratios, the front part of each body is placed at the same height; that is, they all

have the same distance from the nose tip to the free surface before freely falling.

Consequently, the two impact speeds considered are Uo = 2.5 m/s and 4.2 m/s, which

are measured from image sequences. Non-dimensional numbers based on the impact

speeds are summarized in Table 2.2. In this Ca range, the cavity is not created with

a smooth sphere. The temperature in the laboratory is 25 °C, and each measurement

is captured 30 min after the previous one to ensure that the water in the tank is static.

2.3 Measurement technique

The image sequences are captured by a high-speed camera (SpeedSense

M310, Dantec Dynamics), which can capture images from 3,200 to 10,000 fps by

controlling the pixel resolution from 1280 × 800 to the 560 × 448. Depending on the

situation, a 105 mm lens (Nikon) or a 50 mm lens (Nikon) is used, and the spatial

and time resolutions are 0.005D and 0.03D/Uo, respectively. The camera position is

adjustable in height, such as above or below the water surface to observe each

phenomenon. A tungsten lamp (Openface 750W, ARRI) is used for illumination.

To obtain the velocity field of the air and water phases, a particle image

velocimetry (PIV) technique is used. A green laser of wavelength 532 nm (RayPower

5000, Dantec dynamics) is used. The two types of particles are used as seeding

particles, which are hollow glass sphere (110P8, Potters Industries Inc.) for the water

phase and oil droplets produced by a fog generator (Safex, Dantec Dynamics) for the

air phase. Their diameters are approximately 10 μm and 1 μm, respectively.

Consequently, one vector has a spatial resolution of approximately 0.025D.

9

2.4 Image post-processing

After capturing the raw images, the binarization and Wiener filter are

applied and then different post-processing methods are used, depending on the data

to be obtained. For the splash height or shape, the outline of the binarized image is

extracted and then averaged more than 10 times.

For tracking the trajectory of descending projectiles, the Hough transform

is applied, which can find a circle using a known radius. In figure 2.3(a) and (b), the

body frontal and rear parts are well visible after binarization, and thus the centers of

each circle can be found. Then, the two centers are averaged, which is considered as

the center of mass. When the circles cannot be distinguished as well because the

cavity obscures them, the probability of the Hough transform function is lowered

and then many circles are found. The data located one body length from the center

of the front part are considered as the center of the rear part. The trajectory of the

descending body of both the smooth and rough surfaces is obtained by tracking the

mass center from the images of the entire time, and the velocity, acceleration, and

force coefficient (CF) are acquired by differentiating the trajectory data. The

trajectory data are splined and differentiated using Matlab, following the method

reported by Truscott et al. (2012).

For the velocity field, the moving body or splash profiles are obtained after

applying binarization to the raw image and the small seeding particles are removed.

In the cross-correlation process, the velocity vectors are calculated not only inside

body but also inside the splash, because the laser sheet can penetrate the liquid film.

Therefore, masking is applied to the cross-correlation data to delete the velocity

vectors on the body or splash where seeding particles do not actually exist. Finally,

spurious vectors are treated with the Gaussian weight kernel method (Agüí 1987;

Landreth 1990).

10

Table 2.1 Geometric parameters of projectiles depending on their aspect ratio (AR). R

and L are the radius and length of the projectiles, respectively.

11

Table 2.2 Dimensionless numbers based on two different impact velocities (Uo): Froude

number (Fr = Uo2/gR), Reynolds number (Re = ρUoR/μ), Weber number (We = ρUo

2R/σ),

Bond number (Bo = ρgR2/σ), and capillary number (Ca = μUo/σ). Here, σ, ρ, and μ are

the surface tension, density, and dynamic viscosity of the water and g is the gravitational

acceleration.

12

Figure 2.1 (a) Projectiles with two different front parts are considered, one with

a smooth surface and the other with a roughened surface. Each surface has four

aspect ratios, AR = 1, 2, 4, and 8. To match the impact velocity at the free surface,

the nose tip of each projectile is placed at the same height above the free surface.

(b) In this experimental condition, a cavity is not created when the smooth body

enters; however, (c) the roughened body can generate a cavity. The SEM images

of AR 1 with (d) smooth front surface and (e) rough front surface. (f) The static

contact angle (θc) of the rough surface is larger than that of the smooth surface,

because the three-phase contact line is pinned by the asperity of the roughness.

(g) The static contact angle of front part is also higher than side part due to its

curvature.

13

Figure 2.2 (a) Experimental setup of impacting projectile with large water tank. To

adjust the entry angle, a hollow cylinder is used in the vacuum release system, as

shown in the inset image. (b) Consequently, the angle between the longitudinal axis

and the free surface is obtained as 90°.

14

Figure 2.3 (a) To track the center of the front and rear parts, raw images are first

obtained. (b) Then, binarization and Wiener filter are applied. (c) Center positions are

found using the Hough transform function. (d) The center of the rear part can be found

by selecting the data located one body length from the front center. The center of mass

is defined as the midpoint between the front and rear centers.

15

Chapter 3

Water entry of smooth projectiles

3.1 Free surface deformation due to impact of smooth projectile

The overall free surface deformations induced by entering objects with

smooth surfaces are shown in figures 3.1 and 3.2, with impact velocities of 2.5 m/s

and 4.2 m/s, respectively. The deformation is highly influenced by the aspect ratio

of the projectile for both impact velocities. In AR 1–4, a jet occurs above the free

surface without generating a cavity. The maximum jet height decreases as AR

increases, because the liquid film loses its inertia more as the body length increases.

However, in AR 8, the cavity is formed only in the wake of the falling object. Aristoff

et al. (2009) observed that when the rear part of a cylinder enters the water, it drags

the free surface and the cavity is generated only in the cylinder wake. This cavity is

called a quasi-static seal, which is different from the deep seal that will be considered

in Chapter 4. Thus, in figure 3.1(d), a long jet is formed as the quasi-static cavity

collapses, and this jet is different in origin from the jets of AR 1–4. For Uo = 4.2 m/s,

the jet from the quasi-static cavity is not long because it descends attached to falling

objects due to the high inertia of the body prior to the collapse (figure 3.2d).

The moment when the jet is created is closely displayed in figures 3.3 and

3.4. The jet above the water surface appears to exhibit a two-step structure, consisting

of thin and thick jets. The two-step jet is clearly visible in AR 1–2 at Uo = 2.5 m/s

and AR 4 has one more jet due to a shallow cavity. In AR 8, the quasi-static seal

creates the jet. At Uo = 4.2 m/s, as Kubota and Mochizuki (2009) mentioned, thin jets

16

break down into small droplets owing to the high momentum of the liquid film, but

the shape of the two-stage jets can still be seen. These thin and thick jets a have

similar shape to granular flow (Marston et al. 2008; Royer et al. 2008) and laser-

induced jet (Chen et al. 2013), for which thick and thin jets were observed, but the

origin is slightly different from those. Kuwabara (1987) mentioned these two jets,

and the reason why these two-stage jets are formed will be investigated in detail.

3.2 Water film rise along the free surface

3.2.1 Thin jet formation by liquid film convergence

The fact that a thin liquid film climbs on the smooth surface of an object

immediately after impacting has been found in many previous studies (Duez et al.

2007; Zhao et al. 2014), and this phenomenon is captured in more detail in figures

3.5 and 3.6 for Uo = 2.5 m/s and 4.2 m/s, respectively. The ascending film gathers at

the pole and forms a thin, long jet over the free surface, and the film gathering can

be seen in AR 1–2. The jet formation by the convergence of liquid film is also

mentioned in previous research (Kuwabara 1987; Marston et al. 2008). In the case

of AR 4 in figure 3.5(c), the liquid film reaches the top of the object; at the same time,

the object sinks below the water surface just before liquid film gathers. As a result,

the thin jet does not clearly occur and a small volume of a cavity is formed below

the water surface (shallow seal), such that one more jet is formed. Although it does

not meet on the top of the object, it almost reaches the pole and still seems to form a

17

crown-like (or finger-like) structure, because the liquid film exists above the free

surface. The body of AR 8 is longer than the limit to which the ejecta sheet can rise,

and thus the liquid film descends due to losing its momentum as the body still enters

the water (figure 3.5d). In other words, as AR increases, the liquid film does not fully

rise up the body and cannot close at the pole, such that there is an empty space or

cavity under the water. Consequently, the jet caused by the cavity comes out. As the

impact velocity increases (figure 3.6), the motion of the liquid film is similar to that

of the low impact velocity, and there are many small droplets at the end of the body

owing to the high inertia of the liquid film.

The points where the body and the top of the ascending liquid film meet are

plotted in figure 3.7. Here, images from figure 3.5 and 3.6 are binarized and the outer

profile of the body and liquid film is first obtained. Then, the boundary between the

body and the liquid film is defined as the inflection point. At Uo = 4.2 m/s, there are

many droplets around the liquid film, which makes it difficult to track the end of

liquid film; however, their tendency is similar to the case of Uo = 2.5 m/s. The ejecta

sheet of AR 1 and AR 2 converge at the pole of the object to form a thin jet; when the

liquid film reaches r/D ≃ 0.1, their height is z/D ≥ 0.5. However, in AR 4, the object

sinks just before the liquid film convergence on the top; at r/D ≃ 0.1, the root of

finger-like jet is tracked as z/D ≤ 0.3, which means that the projectile is fully

submerged before the liquid film gathers. The liquid film rises and falls vertically

for AR 8, and cannot reach the rear part of body.

18

3.2.2 Film rising and falling motion

3.2.2.1 Overall behavior of liquid film motion

Figure 3.8 shows the non-dimensional liquid film height (h*=h/D) in the z-

direction with respect to non-dimensional time (t*= tUo/D). In all cases, the rising

motion of the liquid film before reaching the maximum height is similar. This is

because, in the initial stage, the liquid film rises owing to the momentum of

impacting and their impact speeds are the same. As the impact velocity increases

(figure 3.8b), it has the same trend as the low-impact-velocity cases, that is the AR

1–2 cases steeply fall with time after the liquid film reaches the rear part, and just

before that moment, it has maximum value. However, in the case of AR 4, it reaches

the highest point at approximately h* = 0.8, and then decreases gradually with as the

object descends. Later, it also steeply falls as it reaches the rear part of body. In AR

8, it has also the maximum height h* = 0.8 and after that, it falls to the free surface

at constant rate. Furthermore, the rate of the liquid film falling motion (∂h*/∂t*)

decreases as AR increases, with values of −0.51 for AR 1, −0.44 for AR 2, −0.21 for

AR 4, and −0.17 for AR 8 at Uo = 2.5 m/s and −0.61 for AR 1, −0.52 for AR 2, −0.20

for AR 4, and −0.10 for AR 8 at Uo = 4.2 m/s. This is because the liquid film loses its

inertia more with increasing AR.

19

3.2.2.2 Differential approach for liquid film dynamics of AR 8 case

In order to investigate the mechanism of liquid film motion when it does

not reach the rear part, the problem is defined as shown in figure 3.9. It is assumed

that the liquid film moves up along the body surface with a constant thickness of e,

which is very thin compared to the body radius (e << R). The undisturbed free

surface is defined as the origin, and the height of the liquid film from the origin is

defined as h, which is a function of time. Moreover, the projectile is considered to

move downward at a constant velocity of Uo as the liquid film goes up and down.

When the frontal nose of the projectile is tracked, as seen in figure 3.10(a), the

downward speed is almost unchanged during t* = 0–8 when the liquid film moves

up and down (figure 3.10b and c). It changes by approximately 0.2% for Uo = 2.5

m/s and 1.5% for Uo = 4.2 m/s. At high impact speed, it loses more inertia than in

the case of low impact speed while entering the water, but the loss is considered

negligible during this early stage. Therefore, the wall surface on which the liquid

film moves has a constant velocity of Uo.

There are generally two approaches to analyze the fluid motion, a

differential approach and control volume (CV) analysis. First, the motion of the

liquid film is analyzed by a differential approach and the net force balance can be

expressed as

(3.1)

where mf is the liquid film mass added to the film as it rises above the water surface,

defined as mf = ρ2πReh. The capillary and gravity forces are defined as Fc =

2πRσcosθD as and Fg = mfg = ρ2πRehg, respectively. The viscous force is calculated

f c v g

d dhm F F F

dt dt

20

using a film flowing on a moving substrate with the boundary condition v(0) = −Uo,

dv/dx(e) = 0, which gives

(3.2)

The flow is considered to be fully developed, and it is set to v(e) = dh/dt. As shown

in figure 3.11, the liquid film goes up when dh/dt has positive value, and it goes

down at dh/dt < 0. The outer surface of the liquid film including the leading edge

moves at the speed of dh/dt. Therefore, the viscous force is obtained from the wall

shear stress.

(3.3)

a1 and a2 are pre-factors to compensate the assumptions of constant film thickness

and fully developed flow. Das et al. (2014) also considered a pre-factor to

compensate the deviation assumption of full development in capillary filling. Then,

(3.1) is differentiated as

(3.4)

Combining (3.4) with the non-dimensional liquid film height (h* = h/D) and time (t*

= tUo/D), it becomes

(3.5)

( ) 2o o

dh x xv x U U

dt e e

1 20

44eo

v w

R UR dhF a h a h

e dt e

22

1 22 2 2

2cos 1 2 1oDUd h dh dh

a a gdt e h e dt e h dt

22

1 22 2 2 2 2

2cos 1 1 2 1oD

o o o

Ud h D dh D dha a g

dt e U h e U dt e U h dt

21

The thickness (e) is assumed constant as 0.8 mm for Uo = 2.5 m/s and 1.1 mm for Uo

= 4.2 m/s. The actual thickness of the liquid film is plotted in figure 3.12; after the

liquid film reaches the maximum height, the film thickness increases gradually. The

change in film thickness increases as Uo increases, as can be seen in figure 3.6; the

instability of the moving liquid film increases owing to its high speed. However, the

camera resolution is 0.005D as mentioned in Chapter 2, which is insufficient to

accurately measure the film thickness, and thus the time-averaged value is used for

equation 3.5. Moreover, as can be seen in figure 3.5(d) and 3.6(d), there is a meniscus

curvature between the free surface and the body, which is ignored in this process. In

figure 3.13(a), an outline profile of the body and liquid film is captured by applying

binarization to the raw image, and then the point at which its slope exceeds 0.15 is

defined as the meniscus height. In figure 3.13(b), as the liquid film rises and falls,

the meniscus height also rises and falls. The temporal variation of the meniscus

height in figure 3.14 shows that the meniscus occupies a small portion in the liquid

film during its rising motion, but it is large as the liquid film falls to the free surface.

However, the meniscus curvature can increase to z/D ≤ 0.3, and thus it is ignored in

this problem for simplicity. Furthermore, the dynamic contact angle (θD) is also

assumed to be constant in time; the equation θD3

= θc3 + 144Ca suggested by Cox

(1986) is used, where Ca is based on the impact velocity, yielding 110° for Uo = 2.5

m/s and 125° for Uo = 4.2 m/s. In addition, the initial conditions h*(0) and dh*/dt*(0)

are determined experimentally. The nonzero value of dh*/dt*(0) means that the

liquid film has inertia; that is, the positive value of dh*/dt*(0) is the driving force to

raise the film. This comes from the impact inertia. Other properties of water such as

its density, surface tension, and viscosity are used and the equation 3.5 is solved

using a fourth-order Runge–Kutta method. The solution is plotted as the solid line in

figure 3.8. The pre-factors are a1 = 15 and a2 = 2.8 for Uo = 2.5 m/s, and a1 = 77 and

22

a2 = 5.6 for Uo = 4.2 m/s. The pre-factors are determined by a least-squares method

to fit the experimental data.

The estimated liquid film height follows the trend of the experimental data;

however, at Uo = 4.2 m/s, the deviation is larger than in the case of Uo = 2.5 m/s. This

is because the thickness (e) is assumed as constant, the velocity inside the film is

considered as fully developed, and the curvature between the free surface and the

body is ignored. As the inertia of the liquid film increases, the liquid film induces

complex flow and the curvature of the three-phase line (moving contact line)

experiences resistance force.

Equation 3.5 contains the surface tension, viscosity, gravity, and liquid film

mass added from the pool. Before the liquid film reaches the maximum height (t* <

1.0), the inertia determines the rising motion against the capillary force, and at t* =

1.0–1.5, they lose inertia, and the surface tension, viscosity, and gravity are balanced.

After that (1.5 < t*), the viscous force and gravity induce falling motion of the liquid

film attached to the body surface.

3.2.2.3 Control volume analysis for liquid film dynamics of AR 8 case

In addition to the previous analysis of the differential approach, the motion

of the liquid film is analyzed by using the CV method. The CV is defined as shown

in figure 3.15. It is rectangular and fixed and contains both air and water phases. The

width is of constant thickness e and the meniscus curvature is ignored as in Section

3.2.2.2. Water enters the CV through the control surface (CS), CS1 and CS5. Air

enters CS2; at this area, the air flows down with the descending projectile. Then, the

liquid film push this downward airflow and induces outward airflows through CS4.

23

When the liquid film descends, the air also comes to CS4, but this is not represented

in figure 3.15. The PIV data in figure 3.16 show that the air flowing in opposite

directions collides to generate an outward airflow.

The Reynolds transport theorem (White 1986) is defined as

(3.6)

where B is an extensive variable, β is dB/dm, which means an intensive variable, V

is the control volume and ṁ is the mass flow rate. Therefore, substitute B = m and β

= 1 into equation 3.6, and then the mass conservation is

(3.7)

where ρ and Vw are density and volume of the water, and ρa, Va are the density and

volume of the air phase, v is the velocity through each CS, and A is the cross-sectional

area. Applying the air and water volumes yields

(3.8)

The air density is low compared to the water density, and v1 is a larger than v2, v4,

and v5. According to the PIV result in figure 3.17, immediately after the object

impacts the free surface, the water flow rises in the vertical direction, such that v5 is

negligible. Moreover, comparing the PIV results, v1 ~ 3Uo, v2 ~ 0.1Uo, v4 ~ 0.1Uo,

and v5 ~ 0. Although the PIV results do not have high accuracy, their orders can be

system i i i iCV

outlets inlets

d dB dV m m

dt dt

4 1 2 5

0system

a a wCV CV

a a w

dm d ddV dV

dt dt dt

Av Av Av Av

4 1 2 5

0a C a a

d dA L h Ah Av Av Av Av

dt dt

24

compared. Therefore, v2 and v4 are also negligible and equation 3.8 is derived as

follows.

(3.9)

Finally, v1 = dh/dt is obtained, which means that the inlet flow of water is same as

the change in the liquid film height. The momentum conservation can be derived by

substituting B = mv and β = v into equation 3.6.

(3.10)

va and vw are the velocity of the air and water phases in the CV, respectively. As with

the above assumptions, ρa, v2, v4, and v5 are negligible. Therefore, it becomes

(3.11)

The water flow in the CV is assumed as an averaged of vw. Equation 3.1 is averaged,

which yields the volume flow rate per unit length.

(3.12)

This yields the following:

(3.13)

1

0d

Ah Avdt

4 1 2 54 1 2 5

CV a a a w wCV CV

a a

d dF v dV v dV

dt dt

v Av v Av v Av v Av

1 11 1

wCV w w

dvd dhF Ahv v Av Av Ah v Av

dt dt dt

0

2

3

e

w o o w

dhq v dx U e U e v e

dt

1

2 2 1 2 1

3 3 3 3 3w o o o o

dh dh dhv U U U b U

dt dt dt

25

Where b1 is a pre-factor to compensate the assumption of constant film thickness and

fully developed flow. v1 = dh/dt from the mass conservation. Combining this and

equation 3.13 into equation 3.11 yields

(3.14)

The forces acting on the CV and CS are the surface tension, pressure, frictional, and

gravitational forces.

(3.15)

Removing va and ρa, this becomes

(3.16)

The third term of the right-hand side is the same as equation 3.3, and the pressures

on CS1 and CS2 are considered to obtain the pressure force. p2 is atmospheric

pressure (p0) and the pressure at the end of the liquid film is p0 + ρ(dh/dt)2/2 = p2 +

ρv12/2, because the outer surface of the liquid film is defined as dh/dt in Section

3.2.2.2. The pressure on CS1 is p1 + ρv12/2 + ρgh, and thus applying the Bernoulli

equation to CS1 and the end of the liquid film yields

(3.17)

(3.18)

2 22

1 1 1 2

2 1 2

3 3 3CV o

dh dh d h dhF b A b AU b Ah A

dt dt dt dt

2 1

0 0

2 cosCV D

w aa

x x

F R p p A

dv dvA h L h Ag

dx dx

2 2

1 12 1

2 2

v vp p gh

2 1p p gh

2 1

0

2 cos wCV D

x

dvF R p p A A hAg

dx

26

Finally, equation 3.14 becomes

(3.19)

The cross-sectional area is A = 2πRe and the pre-factors are b2 ≡ a1/b1, b3 ≡ a2/b1. The

dimensionless form of equation 3.19 using h*=h/D, t*=tUo/D is

(3.20)

The pre-factors (b1, b2, b3) are (30, 28, 3.4) for Uo = 2.5 m/s and (30, 115, 6) for Uo

= 4.2 m/s. The other conditions of e, θD and the initial conditions are same as in

Section 3.2.2.2

3.2.2.4 Comparison of two model equations and term contribution

Two approaches are applied to model the liquid film motion in the smooth

AR 8 case, and these are plotted with experimental data in figure 3.18. The CV

analysis fits the experimental data better than the differential approach. However, at

the impact velocity, neither approach fits the data (figure 3.18b). As mentioned in

Section 3.2.2.2, this is due to the assumption of constant thickness of the liquid film.

As shown in figure 3.6, the liquid film is unstable as Uo increases.

22

2 32 2 2

1 1 1

33cos 1 3 1 3 3 11

2 2 2

o oDU Ud h dh dh g dh

b bdt eb h e dt dt h e b b h dt

2

22 2 2

1

2

3 2 2

1 1

3 cos 1 1 1 1 3

2 2

3 3 3 11

2

D

o o

o

o

d h dh D dhb

dt b e U h h dt e U dt

U D dhb g

e b U b h dt

27

Comparing equation 3.5 and equation 3.20, the terms are similar except that

they have different constants, and equation 3.20 has an additional term of

(dh*/dt*)/h*. The right-hand side of equation 3.5 represents the surface tension,

viscosity, viscosity and gravity, and liquid mass added from pool, respectively. The

first term of the right-hand side of equation 3.20 represents the surface tension; the

second term is the viscosity; the third term is the water flow in the CV; the fourth

term is the viscosity, pressure and gravity; and the fifth term represents the water

flow in the CV and the water flow through CS. Their values with time are plotted in

figure 3.19. The surface tension contributes little in both methods, because the inertia

from the initial conditions is larger in this problem. The additional term of

(dh*/dt*)/h* in the CV analysis makes a large contribution, resulting in the CV

analysis more closely fitting the experimental data.

3.3 Thick jet formation by flow induced sinking projectile

As can be seen in figures 3.3 and 3.4, after the thin jet occurs, the thick jet

occurs with a larger width than the thin jet. Thick jets have slightly different shapes,

depending on the aspect ratio. For AR 1, the thick jet is longer than the other cases

and the thick jet of AR 2 has a disk-like shape (figure 3.3b). AR 4 has a short thick

jet and AR 8 does not generate thin and thick jets. To see the mechanism of the thick

jet formation, the PIV technique is used to obtain the velocity field below the free

surface (figure 3.11). As the sphere first impacts on the water surface, the water flows

out in all radial directions in the front part, and subsequently flow toward the back

of the body is induced. In figure 3.3, it is observed that the rupture of the free surface

occurs due to flow in the wake. Moreover, the flow around the AR 1 can be regarded

28

as a doublet, as mentioned by Kubota & Mochizuki (2011). The water flow induces

the stagnation point in the wake of the object, and consequently it causes flow toward

the free surface and as a result, the thick jet is generated. Therefore, flow toward the

rear part of the body and a thick jet are observed in AR 1–4 cases in figure 3.11. In

the case of AR 4, the object is long and there is no upward flow on the side of the

body, but after the tail enters, the flow toward the free surface is formed, which

results in a thick jet. As AR decreases, the radial motion around the front part of the

body influences the flow around the rear part, accelerating it. As the body becomes

longer, the motion of the front part cannot affect the flow around the rear part, and

the upward flow will also be weaker and therefore the thick jet cannot rise very high.

3.4 Jet-tip breakup

These thin and thick jets rise until they reach the peak point and then fall to

the free surface. In figure 3.12, the height of the jet-tip above the free surface (Hjet)

is tracked over time, only in cases in which the thin and thick jet occur. At Uo = 2.5

m/s, the breakup of the jet-tip occurs at t* ≃ 4 for AR 1 and t* ≃ 9.5 for AR 2, and

the jet-tips of the AR 4 cases do not break up. At Uo = 4.2 m/s, the thin jet is broken

into the droplets as soon as it is formed, such that in the early stage of AR 1 in figure

3.12(b), Hjet has a steep slope representing that the thin jet rises and falls quickly, and

subsequently the tip of the thick jet is tracked. In AR 1, the pinch-off of jet-tip occurs

around t* ≃ 25. As can be seen from figure 3.2, the height of the jet increases as the

aspect ratio decreases. This is because the momentum of the impact velocity is

transferred to the jet more quickly as the short body enters the water faster. Thus, the

jet rises more quickly as Uo increases and AR decreases.

29

It is well known that necking phenomena occur to the long jet, which is

called the Rayleigh–Plateau instability (Plateau 1873; Rayleigh 1878). To investigate

the mechanism of jet pinch-off, several parameters are compared in Table 3.1. The

radius of the jet-tip is obtained at the moment when the jet-tip reaches the maximum

height (Hjet,max) in figure 3.12, and the jet-tip is assumed as spherical and the radius

(Rtip) is found using the Hough transform. Based on the tip radius, the Bond number

is calculated (Botip = ρgRtip2/σ), and the AR 1 case at low impact velocity has Botip, <

1, which means that the capillary force is greater than the gravitational force. Thus,

the jet breakup occurs at the initial stage as the jet rises in AR 1 for Uo = 2.5 m/s.

Subsequently, the Ohnesorge number (Oh = μ/(ρRtipσ)0.5) is obtained. This represents

the ratio of the viscosity to surface tension and is an important dimensionless number

in the pinch-off of the jet (Eggers & Villermaux 2008). The aspect ratio of the jet is

defined as Lo = Hjet,max / (2Rtip), because the jet pinch-off occurs after the jet reaches

the maximum height, Hjet,max is considered in Lo. The aspect ratio of the jet (Lo) means

that there is a correlation between the inertia of the jet and the capillary forces. It is

assumed that the tip radius at Hjet,max would be similar to the initial radius in the liquid

filament problem. Notz & Basaran (2004) numerically investigated the critical point

of the breakup of the liquid filament and found that there is the critical Lo at which

jet breakup can occur with respect to Oh, especially Lo = 5.5 ± 0.5 for Oh = 0.001 in

the same range as this experimental condition (Table 3). As mentioned earlier, pinch-

off of jet occurs in the case of AR 1, AR 2 (Uo = 2.5m/s), and AR 1 (Uo = 4.2 m/s).

This is consistent with the results of Notz & Basaran (2004), which is that jet breakup

occurs when Lo is greater than 5.5 at Oh = 0.001. Moreover, these cases are the jet-

tip breakup investigated by Stone et al. (1986). Driessen et al. (2013) suggested that

at Oh < 0.1, the main reason for jet-tip breakup is the viscosity rather than the surface

tension. Therefore, the viscosity is hte main reason for jet-tip breakup. Hence, the

30

higher AR of the body, the lower the rising velocity of the jet with the result that Lo

decreases and Rtip increases, and finally the lower probability of pinch-off.

3.5 Conclusion

When a smooth projectile enters the water, it induces free surface

deformation, as can be seen in figure 3.22. The thin liquid film is emitted as a result

of the impacting body and it can move along the smooth body surface. The liquid

film detaches from the smooth surface when it is very fast, but not under these

experimental conditions. As the film reaches the pole of rear part, a thin jet is

generated at the point of film convergence. At the same time, below the free surface,

the descending body pushes the surrounding water out, inducing the flow structure.

Later, the upward flow above the stagnation point becomes a thick jet above the free

surface. The tip of the jet experiences a necking phenomenon according to Oh, Botip,

and its aspect ratio (Lo). According to Notz & Basaran (2004), the jet breakup occurs

in the condition of Lo > 5.5 for Oh = 0.001. In this condition, Oh ranges 0.0011–

0.0024 and it can be confirmed that jet-tip breakup occurs only in the case of Lo >

5.5. In particular, when Botip < 1, jet-tip breakup occurs while the jet rises. As AR

increases, the liquid film cannot gather at the pole. In this case in which the liquid

film simply moves up and down on the side of the projectile, the liquid film dynamics

are investigated theoretically by a differential method and using CV analysis.

31

Table 3.1 Jet-tip radius (Rtip) when jet reaches the maximum height (Hjet,max), and

corresponding Bond number based on jet-tip radius (Botip), Ohnesorge number (Oh) and

aspect ratio of jet (Hjet,max /(2Rtip)). The breakup of the jet-tip occurs in the case of AR 1,

2 for Uo = 2.5 m/s and AR 1 for Uo = 4.2 m/s.

32

Figure 3.1 Overall free surface deformation induced by entering smooth projectile with

different aspect ratios: (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8. The impact velocity

at the free surface is Uo = 2.5 m/s. The numbers represent t*.

33

Figure 3.2 Overall free surface deformation induced by entering smooth projectile with

different aspect ratios: (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8. The impact velocity

at the free surface is Uo = 4.2 m/s. The numbers represent t*.

34

Figure 3.3 Formation of jet above the free surface in (a) AR 1, (b) AR 2, (c) AR

4, and (d) AR 8. In AR 1–2, thin and thick jets are clearly visible. There is one

more jet from the shallow cavity for AR 4 and the jet from quasi-static cavity for

AR 8. The impact velocity at the free surface is Uo = 2.5 m/s. The numbers

represent t*.

35

Figure 3.4 Formation of jet above the free surface in (a) AR 1, (b) AR 2, (c) AR

4, and (d) AR 8. In AR 1 – 2, the thin jet breaks down into small droplets as soon

as it is generated. There is one more jet from the shallow cavity for AR 8. The

impact velocity at the free surface is Uo = 4.2 m/s. The numbers represent t*.

36

Fig

ure

3.5

Thin

liq

uid

fil

m i

s em

itte

d a

fter

im

pac

ting,

and i

t m

oves

alo

ng t

he

body s

urf

ace

of

(a)

AR

1,

(b)

AR

2,

(c)

AR

4, an

d (

d)

AR

8. In

AR

1–

2, th

e th

in l

iquid

fil

m g

ather

s at

the

pole

of

the

body.

How

ever

, th

e bo

dy o

f A

R 4

en

ters

the

wat

er b

efo

re t

he

film

gath

ers.

In

AR

8, th

e li

quid

fil

m s

tays

only

on t

he

side

par

t of

the

body.

Th

e im

pac

t vel

oci

ty

at t

he

free

surf

ace

is U

o =

2.5

m/s

. T

he

num

ber

s re

pre

sent

t*.

37

Fig

ure

3.6

Thin

liq

uid

fil

m i

s em

itte

d a

fter

im

pac

ting, an

d i

t m

oves

alo

ng t

he

body s

urf

ace

of

(a)

AR

1, (b

) A

R 2

,

(c)

AR

4,

and

(d)

AR

8. T

he

imp

act

vel

oci

ty a

t th

e fr

ee s

urf

ace

is U

o =

4.2

m/s

. T

he

mo

vem

ent

of

the

liq

uid

fil

m

dep

endin

g A

R i

s si

mil

ar t

o t

hat

sh

ow

n i

n f

igure

3.5

, but

ow

ing t

o t

he

incr

ease

d i

mpac

t sp

eed,

man

y w

ater

dro

ple

ts

are

gen

erat

ed a

t th

e en

d o

f th

e li

qu

id f

ilm

. T

he

num

ber

s re

pre

sent

t*.

38

Figure 3.7 End of liquid film (moving contact line) for (a) Uo = 2.5 m/s and (b) Uo

= 4.2 m/s. ●, AR 1; ○, AR 2; □, AR 4; ◊, AR 8. The time interval is △t* = 0.05 and

the origin is set to the nose of body first contacting the free surface. The data are

averaged 10 times.

39

Figure 3.8 Temporal variation of liquid film height for (a) Uo = 2.5 m/s and (b)

Uo = 4.2 m/s. ●, AR 1; ○, AR 2; □, AR 4; ◊, AR 8. The data are averaged 10 times

and the solid line represents estimated data from equation 3.5. In the initial rising

motion, there is no difference in AR because they all have same impact speed.

However, when the liquid film meets the rear part of each AR, the liquid film

height falls steeply. The liquid film of AR 8 does not reach the rear part of the

projectile.

40

Figure 3.9 Sketch for investigating the mechanism of the liquid film motion. The liquid

film height from the undisturbed free surface is defined as h(t). The dynamic contact

angle (θD) and liquid film thickness (e) are assumed as constant. The liquid film moves

at a speed of dh/dt(t), and the initial condition dh/dt(0) is given as positive, which means

that the liquid film rises at first.

41

Figure 3.10 (a) Tracking frontal center of smooth AR 8 using Hough transform

function. (b) The front part of body descends almost constant speed. ●, Uo = 2.5

m/s; ○, Uo = 4.2 m/s. The data are averaged 10 times. (c) During t* = 0–8 of the

liquid film going up and down, the changes in the downward velocity are

approximately 0.2% for Uo = 2.5 m/s and 1.5% for Uo = 4.2 m/s.

42

Figure 3.11 Velocity profile of liquid film from equation 3.2. dh/dt is positive

when the liquid film ascends and negative when it descends.

43

Figure 3.12 Temporal variation of liquid film thickness for (a) Uo = 2.5 m/s and

(b) Uo = 4.2 m/s in AR 8 case (◊). The data are averaged 10 times and the red star

symbol represents the time when the liquid film reaches maximum height. At Uo

= 2.5 m/s, the thickness of liquid film is almost constant initially and increases

as the film descends. As the impact speed increases to 4.2 m/s, the thickness

decreases while the liquid film rises and it steeply increases while liquid film

falls.

44

Figure 3.13 (a) To obtain the meniscus curvature of the liquid film in the smooth

AR 8 case, an outline profile is captured and then the point at which its slope

exceeds 0.15 is tracked. (b) The temporal variation of the liquid film and body

profile (solid lines), and height of the meniscus curvature (●) for Uo = 2.5 m/s.

45

Figure 3.14 Temporal variation of meniscus height (●), liquid film height ( ), and

their ratio (●) for Uo = 2.5 m/s in the smooth AR 8 case. The data are averaged 10

times. There is a maximum limit that the meniscus curvature can reach.

46

Figure 3.15 Sketch for the control volume (CV) analysis of liquid film motion.

The rectangular fixed control volume contains both water and air phases. CS

means control surface and the liquid film height from the undisturbed free surface

is defined as h(t). The red arrows indicate the forces acting on CV, and the blue

arrows indicate the inflow and outflow through CS.

47

Figure 3.16 Velocity field of air phase when the smooth AR 8 enters the water. The

air flows upward with the ascending liquid film. In the opposite site, the air comes

down due to the descending projectile. On the area where they meet (black

rectangle), there is radial flow because downward and upward flows collide.

48

Figure 3.17 Velocity field of water phase when smooth AR 8 enters the water. As

the liquid film rises above the free surface, there is an upward flow below the

free surface on the side of body.

49

Figure 3.18 Comparison of differential approach (black solid line, equation 3.5)

and CV analysis (red solid line, equation 3.20). The symbol ( ) represents the

experimental data for (a) Uo = 2.5 m/s and (b) Uo = 4.2 m/s.

50

Figure 3.19 Term contribution of (a) equation 3.5 (○: first term, : second term,

: third term, : fourth term, : experimental data) and (b) equation 3.18 (○: first

term, : second term : third term, : fourth term, : fifth term, : experimental

data).

51

Figure 3.20 Velocity filed of water and the process of forming the thick jet on the

free surface at the same time. The numbers in the velocity filed represent t*.

52

Figure 3.21 Temporal variation of jet height for (a) Uo = 2.5 m/s and (b) Uo = 4.2

m/s. ●, AR 1; ○, AR 2; □, AR 4. The red star symbol represents the moment when

the jet-tip breakup occurs and the data are averaged 10 times.

53

Figure 3.22 Overall sequence of the free surface deformation by impacting

smooth projectile. (a) When the smooth projectile impacts on the free surface, (b)

thin liquid film is emitted and moves along the body surface. (c) As the film

reaches the pole of the rear part, (d) a thin jet is generated at the point of film

convergence. (e) Below the free surface, the descending body pushes the

surrounding water out, inducing the flow structure. The upward flow above the

stagnation point becomes a thick jet above the free surface. (f) The tip of the jet

experiences a necking phenomenon according to Oh and its aspect ratio (Lo).

54

Chapter 4

Water entry of rough projectiles

4.1 Free surface deformation due to impact of rough projectile

As a roughened projectile enters the water, the free surface deformation

occurs, as shown in figures 4.1 and 4.2. At Uo = 4.2 m/s in figure 4.2, as Truscott,

Epps & Belden (2014) mentioned, the ejecta develops into a splash (or crown) after

impact, and the crown collapses to a dome. Under the free surface, the cavity

develops and contracts. The necking (or pinch-off) phenomenon occurs and two jets

are created, one of which is emitted to the free surface and the other is ejected to the

descending body. At Uo = 2.5 m/s (figure 4.1), the cavity and the splash are

asymmetric, and thus the splash height and cavity volume are lower than in the high-

impact-speed case. In other words, the axisymmetry and repeatability are lower than

those of high impact velocity, in which case the cavity and splash are always created

uniformly.

The splash formation process is shown in figure 4.3 for Uo = 2.5 m/s and in

figure 4.4 for Uo = 4.2 m/s. As soon as the rough surface of the body impacts on the

free surface, in contrast to the smooth surface discussed in Chapter 3, the splash is

produced because the liquid film cannot move along the rough surface (figure 4.4).

However, at low impact velocity, the speed of the liquid film is also low and therefore

the liquid film can rise up a portion of the body while filling the roughness gaps. As

shown in figure 4.3(a), the liquid film rises up on some part of the body surface,

whereas on the other side it does not. Zhao, Chen & Wang (2014) considered a

55

roughened sphere and suggested that no splash is formed as the speed of the liquid

film is low enough to fill the gaps of the roughness. As mentioned in Chapter 2, the

body nose is roughened using sandpaper and it has a random roughness structure.

When the three-phase contact line moves at low velocity, its advancing motion may

be disturbed due to asperity of roughness and some parts may not be disturbed. For

this reason, in figure 4.1(a) and (b), the splash and cavity are formed asymmetrically.

However, at Uo = 2.5 m/s, the deceleration of the body decreases with increasing AR

and the liquid film velocity also increases, such that the liquid film separates from

more parts of body.

4.2 Formation of a splash

4.2.1 Splash

Figure 4.5 shows the result of tracking the splash height (Hs) with respect

to time. At Uo = 2.5 m/s, the splash is not formed axisymmetrically and this

asymmetry decreases with increasing AR (figure 4.1). Although AR 1–8 have same

the speed just above the free surface, the impact force acting on each body of AR will

be different during the entry, such that the inertia of the liquid film increases as AR

increases. As a result, the separation of the liquid film occurs more at high AR, and

thus the weight of the projectile rather than its shape may contribute to the behavior

of the splash. As a result, the descending speed seems to vary depending on AR, such

that the deceleration is reduced as AR increases, and thus AR 1 has the lowest HS.

At high impact speed, the splash and cavity shapes are always generated

axisymmetrically in all AR cases, and thus it is also affected by the shape and length

56

of the body. In the initial stage (t* = 0–5) of figure 4.5(b), there is no HS difference

according to AR. This is because the liquid film separates from all parts of the body

surface without filling the roughness gaps, and the splash rises because the

momentum of the impact and their impact velocity is all the same. After that, the

splash collapses to a dome, and thus Hs decreases at t* = 5–7. Below the free surface,

the cavity begins to shrink and consequently, Hs increases after cavity pinch-off at t*

= 7–15. After cavity pinch-off, the Worthing jet is emitted and air inside the cavity

also comes out (figure 4.2), such that for AR 1–4, Hs rises. They increase to the same

height at t* ≃ 12.5, which means that the amounts of air in the cavity and dome are

the same. To support this, at t* = 5, the sum of the dome volume (Vd/D3) and cavity

volume (VC/D3) is calculated and they are similar, in the range 11.6–12.0. Finally,

the dome decreases at t* > 15. For AR 8, Hs does not rise as much as in the other

cases, because the cavity is less developed owing to the longer body and the dome is

formed in the middle of the body.

4.2.2 Dome

Figure 4.6 displays the diameter (DS) of the crown (or splash). At Uo = 2.5

m/s, as AR increases, the diameter of the dome becomes larger because the inertia of

the liquid increases as AR increases. Therefore, the AR 1 case has the lowest DS.

However, their speed is insufficient to form the dome, and thus they all have values

of Ws/D > 0.5, which means that the dome is not formed. At Uo = 4.2 m/s, the

diameter of the dome does not differ with AR at the early stages, but it differs later.

The splash of AR 8 meets the body before forming the dome (Ws/D = 1.0), and it can

57

be seen at t* = 5.0 in figure 4.2(d). The dome of AR 2 and 4 occur next (Ws/D ≃ 0.1),

and the splash of AR 1 forms the dome last.

4.3 Formation of a cavity

4.3.1 Overall cavity shapes

As mentioned earlier, as a cavity develops, it collapses owing to the

hydrostatic pressure of the surrounding water and experiences the necking

phenomenon (so-called pinch-off). Figure 4.7 and 4.8 show the pinch-off process of

a cavity at Uo = 2.5 m/s and 4.2 m/s, respectively. The cavity is formed

asymmetrically at Uo = 2.5 m/s (figure 4.7) because of the low inertia of liquid film.

Consequently, the sinking motion of projectile is kinked. Truscott & Techet (2009)

found that as a cavity is formed on only half of the sphere, the trajectory of the object

is bent to the side where the cavity is not formed. It is also observed in figures 4.7(a)

and (b), that the projectiles of AR 1 and AR 2 have curved trajectories to the left,

where the cavity is lacked. At Uo = 4.2 m/s, the pinch-off of the cavity occurs in the

middle of the body in AR 4 (figure 4.8c) and AR 8 (figure 4.8d) cases. In particular,

for AR 8 in figure 4.2(d), after pinch-off, the Worthington jet forms an annular shape,

because the pinch-off occurs in the middle of body. Immediately after the object

impacts the free surface, the water flows in the radial direction because the cavity

expands in the radial direction (figure 4.9a). However, as the cavity develops and

contracts later, the water flows toward the cavity, and it comes to the pinch-off point

(figure 4.9b).

58

On the surface of cavity, it is found that there are longitudinal lines (or

grooves) and ripples rather than being smooth. The longitudinal grooves are

extended from the tooth-like three-phase contact line on the body surface, where the

applied surface roughness ends (figure 4.10a), whereas the irregular protrusions on

the cavity surface are induced by the water droplets falling from the collapse of the

splash (or dome) over the free surface (figure 4.10b). Mansoor et al. (2014)

investigated this phenomenon in detail. They artificially removed the dome above

the free surface and confirmed that the cavity wall is smooth all the time. Because

the dome occurs later than in other cases (figure 4.6b), the case of AR 1 has a

smoother cavity wall than the others (figure 4.8). In addition, for the cases of AR 2–

8, the cavity attached to the object after the breakup experiences a ripple

phenomenon (figure 4.10c & d), and this instability is thought to be related to the

interaction between the long sidewall of the projectile and the jet flowing downward

after the pinch-off (Grumstrup et al. 2007).

4.3.2 Cavity volume

In figure 4.11, the volume of the cavity (VC) is obtained with respect to time.

The volume of the cavity excludes the volume occupied by the body and it is

assumed as axisymmetric. In the cases of AR 4 and AR 8 at low impact speed (figure

4.11a), the cavity volume is small and suddenly increases at t* > 4 for AR 4 and t* >

7.5 for AR 8. After the cavity starts to contract, it seems to expand because the contact

line of cavity sticks to the rear part and consequently it descends with the falling

body, such that it seems to increase suddenly. This can also be observed in figure

4.7(c) at t* = 4.0–4.6 and figure 4.7(d) at t* = 7.3–8.3. It is the same reason why in

59

the smooth AR 8 case at Uo = 2.5 m/s, the cavity is formed only at the rear part and

it pulls down the free surface to create the cavity (figure 3.1d). On the contrary,

because the body of AR 1 and AR 2 occupy a small portion of the inside of the cavity

and the pinch-off point is located above the body, the area of the cavity volume is

large from the beginning. After it expands to the maximum volume, it contracts

owing to the hydrostatic pressure of the surrounding water. The cavity of AR 1 starts

to shrink earliest (figure 4.11a).

At high impact velocity (figure 4.11b), the cavity volumes of all AR are

similar until t* < 4. In the dome formation process at t* = 4–7 in figure 4.5b, the air

inside the splash comes down to the cavity. Consequently, the cavity volume

increases at t* ≥ 4 (figure 4.11b). Moreover, the cavity volume of AR 1 shrinks first

because the dome closure occurs latest. This means that the hydrostatic pressure is

dominant over the incoming airflow due to forming dome. In AR 8, the splash closure

and cavity pinch-off occur on the side of the body, such they occur earlier than other

cases. As a result, the cavity is less developed and small in AR 8.

4.3.2 Mechanism of splash dome and cavity pinch-off

At high impact velocity (Uo = 4.2 m/s), the splash closure and cavity pinch-

off are well distinguished, and thus only this impact velocity is considered in this

section. The dome formation time (td*) and cavity pinch-off (tp*), pinch-off location

from the free surface (Hp*), distance from the frontal center to the free surface at the

moment of pinch-off (H*), and ratio thereof (Hp/H) are summarized in Table 4.1. The

dome and cavity pinch-off occurs in the side of AR 8 body, such that td and tp are the

lowest. Therefore, considering only the case of AR 1–4 in which the body does not

60

disturb their dynamics, the dome of AR 1 is formed earliest and the cavity pinch-off

of AR 1 occurs latest.

In the development of the cavity and splash, airflow into the cavity is

important and figure 4.12 shows the air velocity field above the free surface as AR 1

enters the water. It can be seen that the surrounding air comes into the crown, which

causes the cavity expansion below the free surface. For the larger AR, the

deceleration of the body before and after impact is low, and thus it is expected that

the speed of the airflow in the wake of the object will be faster. The spatially averaged

air velocity at −0.35 ≤ r/D ≤ 0.35 just above the splash, the non-dimensional impact

velocity (va/Uo) is founded to be 0.14 for AR 1, 0.20 for AR 2, and 0.27 for AR 4 at

the same t*. Increased air velocity induces the low pressure owing to the Bernoulli

equations (Marston et al. 2016); therefore, as AR increases, the dome is formed more

quickly (Table 4) because the pressure difference between atmospheric pressure and

the internal cavity pressure increases.

The pinch-off of the cavity has been a major concern of previous studies.

In particular, Aristoff et al. (2010) observed the cavity pinch-off phenomenon by

changing the density of the sphere. They mentioned that in the case of a low-density

sphere, the descending speed is decelerated and the cavity development is delayed

compared to a higher-density case at the same location. The radial velocity of the

surrounding water and rate of cavity expansion decrease and the cavity is less

expanded, such that the cross-sectional area of the cavity decreases. As a result, the

cavity pinch-off occurs faster and shorter, that is, tp and Hp decrease. In addition, the

location of the body from the free surface (H) decreases owing to deceleration,

resulting in the dimensionless pinch-off location (Hp/H) increasing.

In the roughened cylindrical body, only the front part is exposed to the water,

and thus the behavior is similar to the spherical case. With the larger AR, the added

mass increases, which is the same effect as increasing the density of the sphere.

61

Therefore, as can be seen in the Table 4, at AR 1–4 except for AR 8, as AR increases,

the cavity pinch-off is delayed (tp increases) and Hp also increases. In addition, the

speed of the body increases, such that H increases and consequently, Hp/H decreases.

This is consistent with the results of the higher-density case in Aristoff et al. (2010).

4.4 Conclusion

When a projectile with a roughened surface impacts on the water surface,

the roughened surface repels the water and a cavity is generated. The overall

sequence of the free surface deformation by an impacting rough projectile is as

shown in figure 4.13. When the rough projectile impacts on the free surface, the

splash and cavity are formed. Subsequently, the surrounding air enters the splash and

the pressure inside the splash is lower than the atmospheric pressure (p0), owing to

the airflow. Therefore, the splash is closed to the dome and cavity begins to contract

owing to the hydrostatic pressure of the surrounding water. The wetted area is always

the same, even though AR increases, such that the added mass effect increases as AR

increases. As a result, the cavity pinch-off is delayed with increasing AR.

62

Table 4.1 Dome formation time (td), cavity pinch-off time (tp), locations of cavity pinch-

off point (Hp) and frontal center (H), and ratio thereof (Hp/H) for Uo = 4.2 m/s.

63

Figure 4.1 Overall free surface deformation induced by entering rough projectile with

different aspect ratios: (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8. The impact velocity

at the free surface is Uo = 2.5 m/s. The numbers represent t*.

64

Figure 4.2 Overall free surface deformation induced by entering rough projectile with

different aspect ratios: (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8. The impact velocity

at the free surface is Uo = 4.2 m/s. The numbers represent t*.

65

Figure 4.3 Thin liquid film is emitted after impacting, and it separates from the body

surface for (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8. The impact velocity at the free

surface is Uo = 2.5 m/s. However, in AR 1 (a), the liquid film separation does not occur

on some sides of the object. The numbers represent t*.

66

Figure 4.4 Thin liquid film is emitted after impacting, and it separates from the body

surface for (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8. The impact velocity at the free surface

is Uo = 4.2 m/s. In contrast to the low-impact-velocity case, the liquid film separation occurs

on all sides of the object. The numbers represent t*.

67

Figure 4.5 Temporal variation of splash height for (a) Uo = 2.5 m/s and (b) Uo =

4.2 m/s. ●, AR 1; ○, AR 2; □, AR 4; ◊, AR 8. At low impact speed, the splash

height increases as AR increases, but at high impact speed, the effect is reversed.

The data are averaged 10 times.

68

Figure 4.6 Temporal variation of splash width for (a) Uo = 2.5 m/s and (b) Uo =

4.2 m/s. ●, AR 1; ○, AR 2; □, AR 4; ◊, AR 8. At low impact speed, the splash width

increases as AR increases. However, at high impact speed, there are no significant

differences in the initial stage, and the splash of AR 8 closes and meets the object

(Ws/D = 1.0). The splash of AR 1 takes the longest time to form the dome (Ws/D

≃ 0). The data are averaged 10 times.

69

Figure 4.7 Cavity dynamics for (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8. The

impact velocity at the free surface is Uo = 2.5 m/s. The cavity is formed

asymmetrically and the descending motion bends to the side lacking the cavity.

The numbers represent t*.

70

Figure 4.8 Cavity dynamics for (a) AR 1, (b) AR 2, (c) AR 4, and (d) AR 8. The

impact velocity at the free surface is Uo = 4.2 m/s. The cavity is formed

symmetrically. In AR 4 & 8, the pinch-off cavity occurs in the middle of body.

The numbers represent t*.

71

Figure 4.9 Velocity field of water phase after AR 8 enters the water. (a) As the

projectile impacts the free surface, the surrounding water is pushed to the radial

direction at Uo = 2.5 m/s. (b) As the cavity shrinks, the water flows toward the

point where cavity pinch-off occurs at Uo = 4.2 m/s. The numbers represent t*.

72

Figure 4.10 Characteristic cavity shape behind the projectile with rough surface.

(a) Tooth-like three phase contact line (AR 1, Uo = 4.2 m/s). (b) Close-up view of

the rough cavity surface (AR 4, Uo = 4.2 m/s). Oscillating interface of the air

cavity attached to the sidewall of the sinking projectile at Uo = 4.2 m/s in (c) AR

4 and (d) AR 8. The numbers represent t*.

73

Figure 4.11 Temporal variation of cavity volume for (a) Uo = 2.5 m/s and (b) Uo

= 4.2 m/s. ●, AR 1; ○, AR 2; □, AR 4; ◊, AR 8. Each volume is assumed to be

axisymmetric and subtracted by the volume occupied by the projectile. The data

are averaged 10 times.

74

Fig

ure

4.1

2 V

eloci

ty f

ield

of

air

around

sp

lash

. T

he

air

ente

rs t

he

spla

sh a

nd

fo

rms

the

cavit

y b

elow

the

free

surf

ace.

The

num

ber

s re

pre

sent

t*.

75

Figure 4.13 Overall sequence of the free surface deformation by impacting rough

projectile. (a) When the rough projectile impacts on the free surface, (b) a splash

and cavity are formed. Subsequently, the surrounding air enters the splash and

the pressure inside the splash is lower than the atmospheric pressure (p0) owing

to the airflow. (c) Therefore, the splash is closed to the dome and cavity begins

to contract owing to the hydrostatic pressure of the surrounding water. The wetted

area is always same, even though AR increases, such that the added mass effect

increases as AR increases. As a result, the cavity pinch-off is delayed with

increasing AR.

76

Chapter 5

Effect of cavity on

the dynamics of a sinking projectile

5.1 Sinking motion of smooth and rough cases

The cavity formation affects the motion of a sinking projectile and many

previous studies have confirmed this using spheres (Truscott et al. 2012; Shepard et

al. 2014; Mansoor et al. 2017; Vakarelski et al. 2017). In this chapter, the cavity

formation of a cylindrical body is investigated in detail, with the trajectory, velocity,

acceleration, and force coefficient. After tracking the mass center as described in

Chapter 2, it is splined with a quintic function and differentiated by the method of

Truscott et al. (2012). Figure 5.1 shows the trajectory of the projectiles and in AR 1,

there is no difference between smooth and rough surfaces in the early stage at t* <

10. After cavity pinch-off, a jet is emitted toward the descending body, and thus the

roughened sphere sinks faster than the smooth case at t* > 10, and it also has higher

velocity at t* = 10–30 in figure 5.2. The sphere decelerates as it descends, owing to

buoyancy against inertia, and AR 1 reaches terminal velocity in this field of view

(figure 5.2). However, there is a region where the velocity increases before reaching

terminal velocity in the smooth AR 1 case. As explained by Truscott et al. (2012),

there is the vortex pair in the body wake, and vortex shedding transfers momentum

to the body. In other words, as the sphere descends, it changes its direction of motion

for a short period of time. In figure 5.4, the vortex structure around the wake affects

77

body of AR 1, and consequently the velocity of AR 1 increases at t* = 20–40 in figure

5.2. As a result, the velocity of smooth AR 1 decreases, then accelerates slightly and

again decelerates. In figure 5.3, smooth AR 1 has a negative value at t* = 20–40,

which indicates downward acceleration. However, in the rough case, the vortex

formation is suppressed by to the cavity (Truscott et al. 2012), such that the

acceleration region is not clearly observed compared to the smooth case. Finally,

both cases reach terminal velocity, and they have zero acceleration.

The bodies of AR 2 and AR 4 have no significant difference in the z-

directional motion for both the smooth and rough surfaces (figure 5.1), and the

velocity of these cases gradually decreases, but it does not reach terminal velocity in

this field of view (figure 5.2). In the case of AR 2 at low impact velocity, the velocity

of the rough body is faster at first (t* < 14) because the cavity reduces the drag force.

However, the velocities of the smooth and rough surfaces become similar at t* > 14

after cavity pinch-off. In figure 5.4(b), there is also a vortex structure in the wake of

smooth AR 2 while it rotates. In this field of view, the body of AR 2 tumbles twice

for the smooth case, and once for the rough case, which is suppressed, by the cavity.

On the contrary, at high impact velocity, the speed is the same at first and then the

rough body decelerates more (t* > 9). In the early stage, there is no difference

between the smooth and rough surfaces because of the high inertia, but later, it is

suggested that a large cavity attached to the body affects the velocity as a form drag

increases. Moreover, at high impact speed, they tumble three times for the smooth

case and once for the rough case.

In AR 4, they have the same velocity at first (t* < 12) in figure 5.2, and later

the smooth projectile decelerates more for both impact speeds. This is because the

smooth body is about to rotate at the end of the field of view, and thus it decelerates

rapidly. Both the smooth and rough projectiles of AR 8 fall at almost constant speed,

78

which can be seen in figures 5.2 and 5.3, where the velocity is almost constant and

the deceleration is almost zero for both Uo.

5.2 Force coefficient

Finally, the force coefficient is obtained by the equation (Mansoor et al.

2017)

(4.1)

where m is mass of the projectile as shown in Table 2.1. It can be seen that as AR

increases, CF decreases (figure 5.5). The rough case of AR 1 has a smaller CF at t* <

14 for Uo = 2.5 m/s and t* < 17 for Uo = 4.2 m/s. After cavity pinch-off, CF decreases

in the rough cases because of the jet heading toward the body. The smooth case of

AR 1 has a negative value at t* = 22–40 for Uo = 2.5 m/s and at t* = 25 – 45 for Uo

= 4.2 m/s. This is because, as mentioned earlier, the projectile is accelerated owing

to the vortices in the wake of AR 1 (Truscott et al. 2012). After that, CF increases

again and becomes almost zero, which means it reaches the terminal velocity. For

the rough case of AR 1, wake-induced instability is suppressed by the cavity, such

that CF saturates faster than in the smooth case. In AR 2, CF of the roughened body

is larger than the smooth case. This is because the cavity attached to the body after

pinch-off increases the pressure drag. For AR 4, the smooth case is about to rotate at

the end of the field of view, but it does not tumble perfectly. Therefore, it has a higher

value than the roughened object whose rotation is delayed owing to the cavity. In

this field of view, CF of AR 8 is close to zero for both the rough and smooth cases.

2 221 1

2 8

F

F t ma tC

Av t D v t

79

5.3 Tumbling motion of AR 2

As mentioned earlier, the projectile of AR 2 tumbles as it descends, which

means the leading and trailing edges are flipped. Moreover, the trend of velocity of

AR 2 with smooth and rough surfaces is reversed as Uo increases in figure 5.2. To

investigate this more closely, an additional experiment was carried out by changing

the impact speed from 3.1 m/s to 3.7 m/s.

At Uo = 2.5 m/s, there is no significant difference in smooth and rough

surfaces in figure 5.1(a). However, in figure 5.6 and 5.1(b), the trajectory differences

of smooth and rough surfaces increase as Uo increases. With respect to the velocity

in figure 5.2(a), as mentioned earlier, the roughened AR 2 is faster than the smooth

at first and they become the same later. Because Uo is above 3.1 m/s in figure 5.6,

they are same in the early stage and the smooth projectile decelerates later. Moreover,

the velocity difference of smooth and rough surfaces increases as Uo increases.

Table 5.1 represents the dimensionless time (t*) when the front and rear

parts are flipped as AR 2 descends with respect of Uo. The time at which tumbling

takes place is defined as the point where the frontal center overlaps with the rear

center, as shown in figure 5.7. The tumble occurs two or three times for the smooth

case and once for the rough case in this field of view. In the smooth case, the time of

the first tumble and second tumble are all similar, regardless of Uo. However, in the

rough case, the first tumble is delayed as Uo increases. This induces an increasing

form of drag. That is, as Uo increases, the time to first tumbling also increases, and

thus an unstable cavity attached to the body induces a form drag until the first tumble.

Some of the unstable cavity is detached during the first tumbling.

80

5.4 Conclusion

Interestingly, the downward velocity of AR 1 with smooth surface increases

as it descends owing to vortex shedding in the wake. In the rough case, the vortex is

suppressed by the cavity. In the early stage, the smooth AR 1 decelerates more but

slightly accelerates, such that the smooth and rough case of AR 1 descend with

similar velocities. In the AR 2 case, it is more complicated because it tumbles, which

needs to be investigated further and more closely. The velocity of the roughened AR

2 is faster at first for Uo = 2.5 m/s, but in the range of Uo = 3.1–4.2 m/s, the velocity

of the smooth and rough cases are almost the same at first and the roughened case

decelerates more later. This is related to the tumbling motion. Some of the unstable

cavity attached to the body is separated during the first tumbling. As Uo increases,

the difference in velocity of the smooth and rough cases also increases. In the AR 4

case, the smooth projectile is about to tumble at the end of tank. Moreover, in AR 8,

they do not decelerate in this experimental setup.

81

Table 5.1 Dimensionless time (t*) when front and rear parts are flipped as AR 2 descends

with respect of Uo. In this field of view, tumbling occurs two or three times for the

smooth case and once for the rough case. In the smooth case, the time of the first and

second tumbles are all similar, regardless of Uo. However, in the rough case, the first

tumble is delayed as Uo increases. This induces an increasing form of drag.

82

Fig

ure

5.1

Tra

ject

ory

of

des

cen

din

g p

roje

ctil

e fo

r (a

) U

o =

2.5

m/s

and (

b)

Uo

= 4

.2 m

/s. ○

, sm

oo

th s

urf

ace;

●, ro

ugh

surf

ace.

The

dat

a ar

e av

erag

ed 1

0 t

imes

.

83

Fig

ure

5.2

Vel

oci

ty o

f d

esce

nd

ing p

roje

ctil

e fo

r (a

) U

o =

2.5

m/s

and (

b)

Uo

= 4

.2 m

/s.

○,

smoo

th s

urf

ace;

●,

rough s

urf

ace.

Th

e d

ata

are

aver

aged

10 t

imes

.

84

Fig

ure

5.3

Acc

eler

atio

n o

f d

esce

nd

ing p

roje

ctil

e fo

r (a

) U

o =

2.5

m/s

and (

b)

Uo

= 4

.2 m

/s.

○,

smo

oth

surf

ace;

●,

rough s

urf

ace.

Th

e d

ata

are

aver

aged

10 t

imes

.

85

Fig

ure

5.4

Vel

oci

ty f

ield

of

wat

er w

hen

the

body o

f (a

) A

R 1

and (

b)

AR

2 r

ota

te. T

he

num

ber

s re

pre

sent

t*.

86

Fig

ure

5.5

Fo

rce

coef

fici

ent

of

des

cen

din

g p

roje

ctil

e fo

r (a

) U

o =

2.5

m/s

and (

b)

Uo

= 4

.2 m

/s. ○

, sm

oo

th s

urf

ace;

●,

rough s

urf

ace.

The

dat

a ar

e av

erag

ed 1

0 t

imes

.

87

Fig

ure

5.6

Tra

ject

ory

, vel

oci

ty, ac

cele

rati

on,

and f

orc

e co

effi

cien

t of

des

cendin

g A

R 2

pro

ject

ile

for

(a)

Uo =

3.1

m/s

and (

b)

Uo =

3.7

m/s

. □

, sm

oo

th s

urf

ace;

■,

rough s

urf

ace.

The

dat

a ar

e av

erag

ed 1

0 t

imes

.

88

Figure 5.7 Centers of front and rear parts are tracked using Hough transform

function in the case of AR 2 with smooth surface at Uo = 4.2 m/s. The red symbol

indicates the time when nose and tail of body are flipped.

89

Chapter 6

Concluding remarks

In the present dissertation, the water entry of a rounded cylindrical body

was investigated experimentally, focusing on the free-surface deformation related to

the liquid film dynamics. We systematically varied the aspect ratio, surface condition

(smooth and rough), and impact velocity, and the resulting free-surface deformation

was optically measured using a high-speed camera, which was quantified with a

series of image processing techniques. We found that the moving liquid film (or

three-phase contact line) behavior determines the cavity generation; the liquid film

attached to the body induces no cavity formation, whereas the early separation of

liquid film generates a cavity. To control the liquid film separation, the front part was

roughened using sandpaper, such that it had a random roughness structure. In

summary, the following five issues were investigated and their mechanisms were

suggested.

1. Free surface deformation into thin and thick jets

2. Breakup of jet-tip

3. Analytical prediction of liquid film rise and fall

4. Free surface deformation into splash and cavity

5. Splash and cavity closure mechanism

6. Effect of cavity dynamics on the drag on sinking body

90

In the smooth surface case, the liquid film was emitted immediately after

impact, and consequently moved and gathered at the pole of the body, resulting in

forming a thin jet at the pole. However, as AR increased, the liquid film lost its

momentum and the converging inertia was weaker; the jet height decreased. A thin

jet was produced at the pole before the body sank, and a thick jet was produced after

the sinking. As the body sank, the flow structure toward rear part was generated and

this flow caused the rupture of the free surface. Consequently a stagnation point was

induced, and above this stagnation point, the upward flow was created to make a

thick jet above the free surface.

Later, for some of the conditions among the tested cases, the jet-tip was

broken down. The pinch-off of jet occurred only in AR 1, 2 for Uo = 2.5 m/s and AR

1 for Uo = 4.2 m/s. The necking of jet-tip was due to viscosity rather than capillary

force (i.e., it was not due to the well-known Rayleigh–Plesset instability) and Oh is

calculated based on the tip radius, which is the ratio of the viscous time scale to the

capillary time scale.

As AR increased to 8, the liquid film rose and lost its momentum while the

body still entered, such that the liquid film could not reach the rear part. It only

underwent vertical motion. Surprisingly, in both Uo, the maximum heights to which

liquid film could rise were similar to each other. Moreover, in AR 4, although they

reached at the rear part, the maximum height was h* ~ 0.8, as in the AR 8 case.

Therefore, the force balance for the liquid motion was composed of viscosity,

capillary force, and gravity, and more importantly, the initial condition of dh*/dt*(0)

was imposed as a positive value that comes from the impact velocity (transfer of

energy). As a result, the initial motion was mainly due to inertia against capillary,

and then they were balanced for some period and the viscosity and gravity later

caused the liquid film to fall with the entering body.

91

When the front part was roughened, the movement of a liquid film was

prevented (early separation of liquid film) and this was more frequent in the case of

larger inertia of the liquid film. As a result, they formed a splash above the free

surface, and a cavity below the free surface. Subsequently, the cavity and splash were

closed when they had enough inertia. After cavity pinch-off, a jet was emitted, which

was called a Worthington jet. As AR increased, the dome and splash tended to form

on the side of body, such that the Worthington jet had an annular shape.

When the body sank, the surrounding air went into the splash and cavity.

As AR increased, the air velocity in the wake increased, such that the pressure

decreased owing to the Bernoulli equation (Marston et al. 2016). Therefore, as AR

increased, the splash dome was formed earlier, except for AR 8. In AR 8, the splash

met the long body before forming dome. In addition, below the free surface, the front

part was the only wetted area, which induced an increasing added mass effect.

Aristoff et al. (2010) showed that when the density of sphere increased, the cavity

pinch-off was delayed. Therefore, cavity pinch-off was delayed as AR increased.

The calculation of the force coefficient (CF) acting on each projectile

showed that AR 1 with a smooth surface had a rotational motion due to vortex

shedding in the wake. This induced the negative CF. However, the rough case (with

cavity) suppressed the vortex structure (Truscott et al. 2012) and the rotational

motion was delayed. When body of AR 2 sank, it tumbled and the smooth case

rotated more than the rough case, again due to the cavity. This tumble induced the

lower CF. AR 4 with the smooth case was about to rotate at the end of field of view,

which induced a sudden deceleration, such that the CF of the smooth AR 4 was larger

than the rough case. AR 8 had an almost zero CF in this field of view, and they did

not decelerate.

As a final remark, the entry of rounded cylindrical bodies with different

aspect ratios and surface conditions was investigated. We suggested a detailed

92

mechanism for different features such as jets, splashes, and cavities. We think that

the results of this study could further improve the previous water entry problem by

using a more practical body.

93

References

Agüí, J. C. & Jimenez, J. 1987 On the performance of particle tracking. J. Fluid

Mech. 185, 447–468.

Aristoff, J. M. & Bush, J. W. M. 2009 Water entry of small hydrophobic spheres. J.

Fluid Mech. 619, 45–78.

Aristoff, J. M., Truscott, T., Techet, A. H. & Bush, J. W. M. 2010 The water entry of

decelerating spheres. Phys. Fluids 22, 032102.

Benkreira, H. & Khan, M. I. 2008 Air entrainment in dip coating under reduced air

pressures. Chem. Eng. Sci. 63, 448–459.

Bodily, K. G., Carlson, S. J. & Truscott, T. T. 2014 The water entry of slender

axisymmetric bodies. Phys. Fluids 26, 072108.

Chen, R. C., Yu, Y. T., Su, K. W., Chen, J. F. & Chen, Y. F. 2013 Exploration of water

jet generated by Q-switched laser induced water breakdown with different depths

beneath a flat free surface. Opt. Express 21, 445–453.

Clanet, C., Hersen, F. & Bocquet, L. 2004 Secrets of successful stone-skipping.

Nature 427, 29.

Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. J. Fluid

Mech. 168, 169–194.

Das, S., Chanda, S., Eijkel, J. C. T., Tas, N. R., Chakraborty, S. & Mitra, S. K. 2014

Filling of charged cylindrical capillaries. Phy. Rev. E 90, 043011.

Driessen, T., Jeurissen, R., Wijshoff, H., Toschi, F. & Lohse, D. 2013 Stability of

viscous long liquid filaments. Phy. Fluids 25, 062109.

Duclaux, V., Caille, F., Duez, C., Ybert, C., Bocquet, L. & Clanet, C. 2007 Dynamics

of transient cavities. J. Fluid Mech. 591, 1–19.

94

Duez, C., Ybert, C., Clanet, C. & Bocquet, L. 2007 Making a splash with water

repellency. Nat. Phys. 3, 180–183.

Dussan, E. B. V. 1979 On the spreading of liquids on solid surfaces: static and

dynamic contact lines. Annu. Rev. Fluid Mech. 11, 371–400.

Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.

Ern, P., Risso, F., Fabre, D. & Magnaudet, J. 2012 Wake-induced oscillatory paths

of bodies freely rising or falling in fluids. Annu. Rev. Fluid Mech. 44, 97–121.

Gekle, S., van der Bos, A., Bergmann, R., van der Meer, D. & Lohse, D. 2008

Noncontinuous Froude number scaling for the closure depth of a cylindrical

cavity. Phys. Rev. Lett. 100, 084502.

Grumstrup, T., Keller, J. B. & Belmonte, A. 2007 Cavity ripples observed during the

impact of solid objects into liquids. Phys. Rev. Lett. 99, 114502.

Harrison, S. M., Cohen, R. C., Cleary, P. W., Barris, S. & Rose, G. 2016 A coupled

biomechanical-smoothed particle hydrodynamics model for predicting the

loading on the body during elite platform diving. Appl. Math. Model. 40, 3812–

3831.

Kubiak, K. J., Wilson, M. C. T., Mathia, T. G. & Carval, P. 2011 Wettability versus

roughness of engineering surfaces. Wear 271, 523–528.

Kubota, Y. & Mochizuki, O. 2009 Splash formation by a spherical body plunging

into water. J. Visual. 12, 339–346.

Kubota, Y. & Mochizuki, O. 2011 Influence of head shape of solid body plunging

into water on splash formation. J. Visual. 14, 111–119.

Kuwabara, G., Tanba, H. & Kono, K. 1987 Splash produced by a smooth sphere or

circular cylinder striking a liquid surface. J. Phys. Soc. Jpn 56, 2733–2743.

Landreth, C. C. & Adrian, R. J. 1990 Impingement of a low Reynolds number

turbulent circular jet onto a flat plate at normal incidence. Exp. Fluids 9, 74–84.

95

Latka, A., Strandburg-Peshkin, A., Driscoll, M. M., Stevens, C. S. & Nagel, S.

R. 2012 Creation of prompt and thin-sheet splashing by varying surface

roughness or increasing air pressure. Phys. Rev. Lett. 109, 054501.

Lee, M., Longoria, R. G. & Wilson, D. E. 1997 Cavity dynamics in high-speed water

entry. Phys. Fluids 9, 540–550.

Mansoor, M. M., Marston, J. O., Vakarelski, I. U. & Thoroddsen, S. T. 2014 Water

entry without surface seal: extended cavity formation. J. Fluid Mech. 743, 295–

326.

Mansoor, M. M., Vakarelski, I. U., Marston, J. O., Truscott, T. T. & Thoroddsen, S.

T. 2017 Stable-streamlined and helical cavities following the

impact of Leidenfrost spheres. J. Fluid Mech. 823, 716–754.

Marston, J. O., Seville, J. P. K., Cheun, Y.-V., Ingram, A., Decent, S. P. & Simmons,

M. J. H. 2008 Effect of packing fraction on granular jetting from solid sphere

entry into aerated and fluidized beds. Phys. Fluids 20, 023301.

Marston, J. O. & Thoroddsen, S. T. 2008 Apex jets from impacting drops. J. Fluid

Mech. 614, 293–302.

Marston, J. O., Truscott, T. T., Speirs, N. B., Mansoor, M. M. & Thoroddsen, S. T.

2016 Crown sealing and buckling instability during water entry of spheres. J.

Fluid Mech. 794, 506–529.

Marston, J. O., Vakarelski, I. U. & Thoroddsen, S. T. 2011 Bubble entrapment during

sphere impact onto quiescent liquid surfaces. J. Fluid Mech. 680, 660–670.

May, A. 1951 Effect of surface condition of a sphere on its water-entry cavity. J.

Appl. Phys. 22, 1219–1222.

May, A. 1952 Vertical entry of missiles into water. J. Appl. Phys. 23, 1362–1372.

Notz, P. K. & Basaran, O. A. 2004 Dynamics and breakup of a contracting liquid

filament. J. Fluid Mech. 512, 223–256.

96

Otsu, N. 1979 A threshold selection method from gray-level histograms. IEEE Trans.

Syst. Man. Cybern. 9, 62–66.

Plateau, J. 1873 Experimental and Theoretical Steady State of Liquids Subjected to

Nothing but Molecular Forces. Paris: Gauthiers-Villars.

Rayleigh, F. 1878 On the instability of jets. Proc. Lond. Math. Soc. 5, s1–10.

Rosellini, L. Hersen, F., Clanet, C. & Bocquet, L. 2005 Skipping stones. J. Fluid

Mech. 543, 137–146.

Royer, J. R., Corwin, E. I., Conyers, B., Flior, A., Rivers, M. L., Eng, P. J. & Jaeger,

H. M. 2008 Birth and growth of a granular jet. Phys. Rev. E 78, 011305.

Shepard, T., Abraham, J., Schwalbach, D., Kane, S., Siglin, D. & Harrington, T.

2014 Velocity and density effect on impact force during water entry of sphere. J.

Geophys. Remote Sens. 3, 129.

Stalder, A. F., Kulik, G., Sage, D., Barbieri, L., & Hoffmann, P. 2006 A snake-based

approach to accurate determination of both contact points and contact angles.

Colloid. Surface. A, 286(1-3), 92-103.

Stone, H. A., Bentley, B. J. & Leal, L. G. 1986 An experimental study of transient

effects in the breakup of viscous drops. J. Fluid Mech. 173, 131–158.

Sun, H. & Faltinsen, O. M. 2006 Water impact of horizontal circular cylinders and

cylindrical shells. Appl. Ocean Res. 28, 299–311.

Techet, A. H., & Truscott, T. T. 2011 Water entry of spinning hydrophobic and

hydrophilic spheres. J. Fluid. Struct. 27(5-6), 716-726.

Thoroddsen, S. T., Etoh, T. G., Takehara, K. & Takano, Y. 2004 Impact jetting by

a solid sphere. J. Fluid Mech. 499, 139–148.

Truscott, T. T., Epps, B. P. & Belden, J. 2014 Water entry of projectiles. Annu. Rev.

Fluid Mech. 46, 355–378.

Truscott, T. T., Epps, B. P. & Techet, A. H. 2012 Unsteady forces on spheres

during free-surface water entry. J. Fluid Mech. 704, 173–210.

97

Truscott, T. T., & Techet, A. H. 2009 A spin on cavity formation during water entry

of hydrophobic and hydrophilic spheres. Phys. Fluid, 21(12), 121703.

Vakarelski, I. U., Klaseboer, E., Jetly, A., Mansoor, M. M., Aguirre-Pablo, A. A.,

Chan, D. Y. C. & Thoroddsen, S. T. 2017 Self-determined shapes and velocities

of giant near-zero drag gas cavities. Sci. Adv. 3, e1701558.

Vincent, L., Xiao, T., Yohann, D., Jung, S., & Kanso, E. 2018 Dynamics of water

entry. J. Fluid Mech. 846, 508-535.

Worthington, A. M. 1908 A Study of Splashes. Green: Longmans.

White, F. M. 1986 Fluid mechanics. McGraw-hill.

Yao, E., Wang, H., Pan, L., Wang, X. & Woding, R. 2014 Vertical water-entry of

bullet- shaped projectiles. J. Appl. Math. Phys. 2, 323–334.

Zhao, M. H., Chen, X. P. & Wang, Q. 2014 Wetting failure of hydrophilic surfaces

promoted by surface roughness. Sci. Rep. 4, 5376.

Zhao, R. & Faltinsen, O. 1993 Water entry of two-dimensional bodies. J. Fluid Mech.

246, 593–612.

98

국문 초록

물체가 입수 할 때 자유 표면과의 충돌로 인해 발생하는 현상은

유체 역학에 있어서 기초가 되는 분야 중에 하나이다. 본 논문에서는 양

끝단이 반구 형태를 가진 긴 원통형 물체를 가지고 실험을 진행

하였으며, 물체의 종횡비(AR = 1 - 8)와 표면 상태(부드럽거나 거친)를

변화시키면서 실험을 진행하였다. 물체가 수면에 충돌하는 순간의

속도(Uo = 2.5 m/s 및 4.2 m/s)를 변화시켰으며, 이로 인한 무차원 수는

Fr = 32 - 90, Re = 5 × 105 - 8.4 × 105 및 Ca = 0.0345 –

0.0579이다. 입수 조건에 따라 자유 표면의 변형이 각각 다르게

나타나는데, 예를 들어, 제트 혹은 스플래시 및 공동이 생성된다. 물체의

표면이 매끄러우면, 입수 하는 순간 얇은 액체 필름이 입수 하는 물체

표면을 따라 올라가고 위쪽 끝에서 모여 얇은 제트가 생성된다. AR이

증가함에 따라, 액체 필름이 물체 측면에서 머무르는 시간이 길어져

액체 필름은 충격으로 인한 관성을 더 많이 잃게 되고 이로 인한 제트가

도달할 수 있는 최고 높이 또한 감소한다. 얇은 제트를 생성 한 후에는

바로 이어서 비교적 두꺼운 제트가 생성된다. 두꺼운 제트는 물체가

입수하면에 따라 자유 표면을 끌어 당겨, 이러한 운동의 반작용으로

수면위로 튀어 올라가는 흐름에 의해 발생한다. 즉, 처진 자유 표면은

그 모양을 회복하는 경향이 있으며, 상방 흐름은 두꺼운 제트가 된다.

제트의 반지름과 종횡비에 따라 제트 끝 부분이 수축하여 액적으로

절단되는 현상이 발생한다. 이는 표면 장력과 점성력의 비를 나타내는

Ohnesorge수를 보게 되면 본 실험은 Oh ≃ 10-3의 범위 있다. 이는

99

제트 끝 부분이 끊어지는 현상이 점성 때문이라는 것을 의미한다. 충돌

속도와 몸체의 종횡비에 따라서 제트 끊어짐 현상이 발생하는 기준 제트

종횡비가 있다. 반면에, 물체의 앞 반구를 사포로 거칠기를 주면 충돌

직후 액체 필름이 표면에서 분리되어 스플래시 크라운과 캐비티를

형성한다. AR이 증가함에 따라 Uo = 2.5 m/s에서는 스플래시 높이가

증가하고 Uo = 4.2 m/s에서는 높이가 감소한다. 돔과 캐비티의 핀치-

오프 시간은 AR에 따라 다른데, AR의 증가에 따라 물체 후류가 더

빨라져 스플래시 내부의 압력이 낮아지고 돔이 일찍 형성된다. 또한,

캐비티 핀치-오프는 AR이 증가함에 따라 지연되는데, 이는 AR이

증가함에 따라 질량이 증가하기 때문이며 밀도를 높이는 것과 같은

효과이다. 마지막으로, 물체가 가라앉는 궤적 데이터를 기반으로 힘

계수를 얻었는데 높은 AR에서는 무게 때문에 감속이 작다. 캐비티가

핀치-오프 전에 물체를 감싸면 물체에 작용하는 저항이 감소하나,

반대로 핀치-오프 후에 부착 된 캐비티는 저항을 증가시킨다.

주요어 : 자유 표면, 입수, 제트, 스플래시, 캐비티, 핀치-오프, 액체막

학 번 : 2013-20648