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Numerical study of water-confinement geometries for laser propulsion
Optics and Lasers in Engineering 48 (2010) 950–957
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Numerical study of water-confinement geometries for laser propulsion Bing Han, Zhong-Hua Shen, Jian Lu, Xiao-Wu Ni n School of Science, Nanjing University of Science & Technology, Nanjing, Jiangsu 210094, People’s Republic of China
a r t i c l e in f o
a b s t r a c t
Article history: Received 22 October 2009 Received in revised form 28 May 2010 Accepted 31 May 2010 Available online 16 June 2010
The processes of laser propulsion in a water environment are investigated numerically in this paper. Four kinds of propelled surfaces are discussed: a plane, a hemispherical shell, a 901-conical shell and a 301-conical shell. The bubble radius and the velocity of the bubble surface varying with time during the first-expansion of the bubble are investigated. The evolution of the shock wave induced pressure field around the pressure peak on the surface and the evolution of the velocity field of the liquid jet are discussed. The energy that the propelled surface obtains from the first-expansion of the bubble is enhanced by the three kinds of shapes except the plane, among which the 901-conical shell produces the most effective propulsion. During the final collapse of the bubble, the plane propelled surface obtains the most energy, while the 301-conical shell obtains the least. The narrower the propelled surface, the higher the proportion of the energy offered by the plasma-bubble shock wave. But the propulsion efficiency declines as the propelled surface becomes narrower. The momentum coupling coefficient of the hemispherical shell is the highest. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Laser propulsion Water Confinement Bubble formation
1. Introduction Laser propulsion has been investigated in two main aspects, atmospheric breathing [1,2] and ablation under vacuum [3,4]. In order to avoid the main problems that encumber the development of laser propulsion in atmosphere and vacuum, we proposed laser propulsion in a water environment [5] based on the special properties of the laser-induced breakdown in water. When a highintensity laser beam is focused onto the propelled surface in water, plasma of high-temperature and high-pressure will be induced if the power density at the laser focus exceeds the breakdown threshold of water. The plasma expands violently, inducing a quasi-spherical shock wave. In the meantime, the pressure and temperature of the plasma area drop, which finally leads to the formation of steam. A high density steam bubble [6–8] is produced with the initial plasma as the spherical center. The bubble expands and radiates a shock wave. Before finally collapsing, the bubble usually experiences several oscillations. Every time the bubble reaches the minimum volume and starts to rebound, it radiates a quasi-spherical shock wave. As the bubble energy decreases, the energy of the successive shock waves is attenuated [9]. A liquid jet [10] will be induced toward the propelled surface when the bubble collapses. The sources of the propelling force include the plasma shock wave, the bubbleexpanding shock wave and the final collapse impact.
n
Corresponding author. Tel.: +86 25 84315075; fax: + 86 25 84318430. E-mail addresses: [email protected] (B. Han), [email protected] (X.-W. Ni). 0143-8166/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2010.05.013
An increase in the laser pulse energy will increase the intensity of the plasma shock wave, thus increase the propelling force of the shock wave. The increase in the expanding velocity of the bubble surface and the prolongation of the high-speed expansion of the bubble will both increase the propelling force of the bubble-expanding shock wave. There are many factors that can influence the final collapse of the bubble. For example, the ratio between the bubble radius and the distance between the initial bubble center and the propelled surface [11–13], the characteristics of the liquid interface around the bubble [14–17], and the thickness of the plate on which the bubbles are induced [18]. Investigations on laser propulsion in atmosphere show that the propelling effect is influenced by the confined shape of the propelled surface [19]. Different kinds of confined propelled surfaces have been designed, among which the Myrabo-type [2] and the bell-type [4] are the most famous. According to our investigation, variation of the confining shape of the propelled surface influences all three sources of propelling force mentioned at the end of the first paragraph. Therefore, the processes of laser propulsion in a water environment with four kinds of propelled surfaces—a plane, a hemispherical shell, a 901-conical shell and a 301-conical shell—are investigated numerically in this paper. Parabolic shell-shaped surfaces have been widely used in laser propulsion in atmosphere and vacuum. An important purpose is to focus the attenuated laser beam transmitted through a long distance. Thus, plasma can be induced if the power density at the laser focus exceeds the breakdown threshold of air or the working medium. On the one hand, in this paper, the laser system can be put near the propelled surface, thus the excitation beam can be focused on the propelled surface directly. On the other hand, the
B. Han et al. / Optics and Lasers in Engineering 48 (2010) 950–957
parabolic shell and the conical shell are similar when the scale of the propelled surface is mm. Therefore, the hemispherical shell is selected instead of the parabolic shell. In addition, 901-conical and 301-conical shells are compared to investigate the influence of the degree of the confinement on the propulsion. Investigations on the laser-induced bubbles are concentrated on the oscillating properties of the spherical bubble in an infinite liquid environment [20], and the collapsing properties of the quasi-spherical bubble near different interfaces. However, in this paper, the bubbles induced on the four kinds of propelled surfaces are quasi-half, quasi-quarter and quasi-duodecimal spherical cavitation bubbles elongated in the opposite direction of the laser beam, as shown in Fig. 1. The plasma and bubble first-expansion shock waves are emitted almost at the same time [8,21], so they are called the plasma-bubble shock wave in this paper. According to our investigations [5], the majority of the propelling power is from the plasma-bubble shock wave and the final collapse impact. Only less than 1% of the propelling power is from the latter bubble expansion shock waves. Therefore, the characteristics of the forces induced by the plasma-bubble shock wave and the final collapse impact are discussed in this paper. The bubble radius and the velocity of the bubble surface varying with time in the firstexpansion of the bubble are investigated. The evolution of the plasma-bubble shock wave induced pressure field around the pressure peak on the surface and the evolution of the velocity field of the liquid jet are discussed. The characteristics of the propulsion are compared between the plane and the other propelled surfaces, between the hemispherical and the conical shells, and between the two different conical shells.
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2. Numerical simulation The arrangement of laser propulsion in water is shown in Fig. 1. The propelled surface made of copper is in the small dashed cube. The insets show the four kinds of axisymmetric propelled surfaces and the corresponding bubbles. Fig. 2 shows the models used in the numerical simulation, including (a) the plane, (b) the hemispherical shell, (c) the 901conical shell and (d) the 301-conical shell. The unit of the spatial coordinates is mm. z is the rotational symmetric axis of the whole model. Region D is the water environment. Point (0,0) is the laser focus. Due to high intensity of the laser focus, ablation of the metal surface is possible, which will influence the bubble dynamics and contribute to the impulse directly through substance ejection. Because the main object of this paper is the bubble-induced mechanical force, the ablation of the metal is neglected in calculations of this paper. The CFD numerical simulation is based on the conservation laws, including the mass conservation Eq. (1), the momentum conservation Eqs. (2) and (3), and the energy conservation Eq. (4) [22]: @r @ 1@ ðrvÞ ¼ 0 þ ðruÞ þ @z r @r @t
ð1Þ
@ @ 1@ @p ðruÞ þ ðru2 Þ þ ðr ruvÞ þ ¼0 @t @z r @r @z
ð2Þ
@ @ 1@ @p ðrvÞ þ ðruvÞ þ ðr rv2 Þ þ ¼0 @t @z r @r @r
ð3Þ
water environment
Nd:YAG laser
glass cuvette
Inset (a1) quasi-half spherical bubble
Nd: Y AG laser Inset (a2)
velocity direction
Inset (b1) quasi-half spherical bubble
propelled object
Inset (c1) quasi-quarter spherical bubble
Nd: Y AG laser
Nd: Y AG laser
Inset (b2)
Inset (c2)
Inset (d1) quasi-duodecimal spherical bubble
Nd: Y AG laser Inset (d2)
Fig. 1. The arrangement of laser propulsion in water. Insets show the four kinds of axisymmetric propelled surfaces and the corresponding bubbles induced on them. (a1) to (d1) are the plane, the hemispherical shell, the 901-conical shell and the 301-conical shell, respectively. (a2) to (d2) are the quasi-half, quasi-half, quasi-quarter and the quasi-duodecimal spherical bubbles, respectively.
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r/mm
r/mm
25
25 r1 = 2
D
r2 = 2
D
z1 = -1
r1
z1 = -1
r1
r1 = 3
z2 = 0
z2 = 0
r2
z3 = 2 z/mm
z/mm -25
z2
z1
-25
25
r/mm
z1
z2
z3
25
r/mm
25
25 r1 = 2
D
z1 = -1
r1 = 0.536
D
z1 = -1 z2 = 0
z2 = 0 z3 = 2
r1
z/mm -25
z1
z2
z3
25
z3 = 2
r1
z/mm -25
z1
z2
z3
25
Fig. 2. The models used in the numerical simulation: (a) the plane, (b) the hemispherical shell, (c) the 901-conical shell and (d) the 301-conical shell.
8 < @I @E @ 1@ ½rvðE þ pÞ ¼ @z þ ½uðEþ pÞ þ : @t @z r @r 0
td o t otp ,ðz,rÞ A O
ð4Þ
otherwise
where t is the time, r is the density, u and v are velocities along axes z and r, respectively, p is the pressure, E is the energy per unit volume, I is the laser intensity, td is the moment when the shock wave appears, tP is the laser pulse width, and O is the area of the laser beam. If one takes e as the specific internal energy, E can be defined as E ¼ reþ 12rðu2 þ v2 Þ
ð5Þ
The pressure field p(z,r,t) of the calculation area can be calculated from Eqs. (1)–(5). The impulse Imi that the propelled surface obtained from t ¼0 to t ¼ti can be calculated by Z ti Z redge ½pðzout ,r,tÞpðzin ,r,tÞ2prdrdt ð6Þ Imi ¼ 0
0
where zout is the coordinate of the outer surface, zin is the coordinate of the inner surface (irradiated by the Nd:YAG laser) and redge is the coordinate of the edge of the surface. Thus the kinetic energy Ekinetic of the propelled surface at t ¼ti can be i calculated by ¼ Ekinetic i
2 Imi 2M
ð7Þ
where M is the mass of the propelled surface. The momentum coupling coefficient Cm of a single laser pulse, expressed in unit of N s/J, can be calculated by Cm ¼
total Im Epulse
Steinberg–Guinan strength model [24] is chosen. Region D is of obvious fluidity, so the Euler algorithm, suitable for the calculation of large deformation, is chosen. The propelled surface is a rigid plane, so the Lagrange algorithm, suitable for the calculation of elastic/plastic finite deformation, is chosen. The VOF model is used. The mass of the propelled surfaces shown in Fig. 2(a), (b), (c) and (d) are 113, 354, 113 and 8 mg, respectively. The parameter that has been chosen is the shape of the propelled surface, not the mass. After the needed surface models are built, the mass of each model is fixed. The wavelength of the laser is 1.06 mm. The energy per pulse is 20 mJ. The laser pulse width is 10 ns. The radius of the laser focus is 50 mm. In the simulation of this manuscript, the detailed interaction between the laser beam and water (e.g., reflection, absorption, scattering) was not calculated. The simulations start from the bubble growth. The oscillations of the bubbles are calculated. Initial conditions of the bubbles are taken from experimental results. In the meantime, the momentum coupling of the propelled surface is calculated. The exact value that determines the dynamic behavior of the laser-induced bubble is the laser energy saved after all the attenuation. In the future when we make the laser propulsion an applicable tool, if we know such saved-energy that is necessary to produce the force we need, we can manage to get it in different conditions and for different lasers.
3. Simulation results and discussion 3.1. The first-expansion of the bubble
ð8Þ
total is the total impulse that the propelled surface where Im obtained, Epulse is the energy of a single laser pulse. Because of the existence of the strong discontinuity, namely the shock wave, the shock equation of state (SEOS) [23] is chosen as the state equation. The material of the propelled surface is copper (Cu), which will be deformed under the impact of the shock wave and the bubble final collapse, namely the cavitation erosion. So the
The newborn bubble expands rapidly and radiates a shock wave. The bubble energy decreases quickly, so the velocity of the bubble surface decreases. After reaching its maximum radius, the bubble starts to constrict. Fig. 3 shows the bubble radius and the velocity of the bubble surface varying with time in the firstexpansion of the bubble on the four kinds of propelled surfaces. The inset of Fig. 3 shows the definition of the bubble radius and the position on the bubble surface where the surface velocity is
2.2 2.0
240 220 200 180 160 140 120 100 80 60 40 20 0
Radius
1.8 1.6 R (mm)
1.4 1.2 1.0 0.8
plane hemisphere
0.6
90°- taper
0.4
30°- taper
Velocity
0.2 0.0 0
5
10
15
20
25 30 35 Time (µs)
40
45
50
V (m/s)
B. Han et al. / Optics and Lasers in Engineering 48 (2010) 950–957
55
Surface velocity read from this point
Bubble radius
Inset
Fig. 3. The bubble radius and the velocity of the bubble surface varying with time in the first-expansion of the bubble on the four kinds of propelled surfaces. Inset shows the definition of the bubble radius and the position on the bubble surface where the surface velocity is monitored.
monitored. In the early stage of the bubbles, during about 0–3 ms, the velocity of the bubble surface reaches 100 m/s, consistent with the experimental results in Refs. [10,15,20]. The same amount of laser energy is transmitted into the water in the laser focus, which means the same amount of plasma can be induced. For example, to get the same volume, plasma on the 901-conical inner peak needs a bigger radius than that on the plane, which further leads to the radius of the bubble induced on the 901conical inner peak is bigger than that on the plane. The plasma on the 901-conical inner peak spreads into a quarter of the whole spatial angle, while the plasma on the plane spreads into half of the whole spatial angle. Thus, compared to the plane, the conical and hemispherical surfaces restrict the bubble to grow in a certain direction, which increases both the expanding velocity and the maximum bubble radius. The narrowest shape (301-conical shell) brings the highest restriction. The restrictive effect of the hemispherical shell is a little lower than that of the 901-conical shell. In addition, the velocity–time curves show that the velocity of the bubble surface of the hemispherical shell is a little higher than that of the plane in the early stage, but the velocity of the hemispherical shell approaches that of the 901-conical shell gradually after about 5 ms. The reason is that the top of the hemispherical shell (around point (0,0)) is flat, similar to the plane. As the bubble becomes larger, the restrictive effect of the hemispherical shell becomes stronger. Fig. 4 shows the evolution of the pressure field, induced by the plasma-bubble shock wave, around the pressure peak on the four kinds of propelled surfaces: (a) the plane, (b) the hemispherical shell, (c) the 901-conical shell and (d) the 301-conical shell. The abscissa, propelled surface (mm), shows the distance from the point (0,0), along the propelled surface irradiated directly by the laser. Fig. 4(e) shows the value of the pressure peak varying with time on the four kinds of propelled surfaces. The pressure peak of the plasma-bubble shock wave reaches 108 Pa. As it
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propagates toward the edge of the propelled surface, the shock wave front broadens and the peak pressure decreases gradually, until it attenuates to a sound wave. Because the top of the hemispherical shell (around point (0,0)) is flat, the peak pressure in the early stage on the hemispherical shell is a little higher than that on the plane and lower than that on the two conical shells. However, the walls of the hemispherical shell approach a cylindrical shape at large radii, which abates the attenuation of the wave, as shown in Fig. 4(b). Thus, the peak pressure on the hemispherical shell becomes the highest after about 0.9 ms, as shown in Fig. 4(e). It is noticed that the pressure peak of the newborn plasma-bubble shock wave is the highest in the 301conical shell, but the attenuation of the shock wave is also the most serious in the 301-conical shell. As shown in Fig. 4(e), the peak pressure is weaker in the 301-conical shell than that in the 901-conical and hemispherical shells after about 0.9 ms. Fig. 5 shows the numerical results of the energy that the four kinds of propelled surfaces obtain from the first-expansion of the bubble. Compared to the plane, the hemispherical and conical surfaces obtain more energy during the first-expansion of the bubble. On the one hand, the hemispherical shell failed to restrict the energy to release in the opposite direction of the propulsion, the positive direction of axis z, in the early stage of the bubble. On the other hand, the shock wave pressure on the propelled surface of the 301-conical shell is perpendicular to the surface, thus the component in the propelling direction is very small. However, the 901-conical shell could both restrict the energy to release in the opposite direction of the propulsion properly, and induce more propelling force to work in the direction of the propulsion. So the energy increase of the 901-conical shell is the highest during the first-expansion of the bubble.
3.2. The final collapse of the bubble When the maximum radius of the bubble is mm, as it is for the bubbles discussed in this paper, the duration of the liquid jet is a few 10’s of ms [21], and the radius of the jet is 10 1 mm. That is to say, the process of the liquid jet is instantaneous and highly localized. The evolution details of the liquid jet are investigated numerically in this paper. Fig. 6 shows the velocity distribution of the jet front along the r axis at different moments for the four kinds of propelled surfaces: (a) the plane, (b) the hemispherical shell, (c) the 901-conical shell and (d) the 301conical shell. The numerical simulation starts at the moment when the energy of the propelled surface rises. The starting time of the [recorded time] (ms) marked in Fig. 6 is the same as that of the numerical simulation. The distance between the jet front and the propelled surface at the recorded moment is also marked in Fig. 6. For all four kinds of propelled surfaces, the gradients of the velocity fields around the velocity peaks increase, and the radii of the jets decrease. Except in the 301-conical shell, the velocity of the jet front keeps increasing until it reaches the propelled surface. It is the plane that induces the highest velocity and the widest high-speed range of the jet, when the jet almost impacts the propelled surface. Next is the hemispherical shell, and the 901conical shell. The jet velocity and the high-speed range of the 301conical shell are both the smallest. The velocity of the jet front in the 301-conical shell begins to decrease after about 1.2 ms. The reason is that the area close to the point (0,0) of the 301-conical shell becomes very narrow. Thus the push of the jet compresses the whole bubble seriously, which makes the pressure of the area between the jet front and propelled surface increase dramatically. In other words, the jet front encounters great resistance before it reaches the propelled surface. For other propelled surfaces, the jet penetrates the central section of the bubble, which makes the
B. Han et al. / Optics and Lasers in Engineering 48 (2010) 950–957
300 275 250 225 200 175 150 125 100 75 50 25 0 -25
recorded time (µs)
0.21 0.40 0.61
0.0
Pressure (MPa)
Pressure (MPa)
0.11
300 275 250 225 200 175 150 125 100 75 50 25 0 -25
0.3
0.81
1.00
0.6 0.9 1.2 1.5 Propelled Surface (mm)
1.33
1.8
recorded time (µs) 0.10
0.21
0.40 0.63
0.80
1.00
1.30
1.70
0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 Propelled Surface (mm)
Pressure (MPa)
300 270
plane hemisphere
240
90°- taper
300 275 250 225 200 175 150 125 100 75 50 25 0 -25
recorded time (µs) 0.08
0.18
0.40 0.60
0.0
2.1
Pressure (MPa)
Pressure (MPa)
954
0.4
0.81 1.00 1.21 1.41
1.64
0.8 1.2 1.6 2.0 2.4 Propelled Surface (mm)
2.8
0.08
300 275 250 225 200 175 150 125 100 75 50 25 0 -25
3.2
recorded time (µs)
0.20 0.30 0.50 0.69 1.00 1.13
0.0
0.3
0.6 0.9 1.2 1.5 Propelled Surface (mm)
1.8
2.1
30°- taper
210 180 150 120 90 60 30 0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time (µs) Fig. 4. The evolution of the pressure field, induced by the plasma-bubble shock wave, around the pressure peak on the four kinds of propelled surfaces: (a) the plane, (b) the hemispherical shell, (c) the 901-conical shell and (d) the 301-conical shell. (e) The value of the pressure peak varying with time on the four kinds of propelled surfaces.
bubble become toroidal. Thus the high resistance area between the jet front and propelled surface is avoided. Fig. 7 shows the numerical results of the energy that the four kinds of propelled surfaces obtain from the final collapse impact of the bubble. Except for the 301-conical shell, there are two increase stages of the energy. In the first increase stage, the contracting bubble induces the surrounding liquid flowing toward and impacting the propelled surface, which makes the energy of the propelled surface rise. According to Fig. 6, the starting time of the second increase stage in Fig. 7 is just the moment when the liquid jet reaches the propelled surface. The high-speed and highpressure jet impacts the propelled surface violently, which makes the energy of the propelled surface rise again. The energy that the plane obtains from the first-expansion of the bubble is the least,
but during the final collapse impact the plane obtains the most energy. The reason can be described as follows. During the final collapse impact, it is the contracting surrounding liquid and the jet-transported liquid that offer the energy. The hemispherical and conical surfaces block the contracting of the surrounding liquid. For example, because of the similarity in the structure, the hemispherical and the 901-conical shell obtain a similar amount of energy. But after about 2.2 ms, the energy of the hemispherical shell rises. According to Fig. 4(b), the reason is that the jet induced shock wave propagates to the narrow end of the hemispherical shell at about 2.2 ms. In addition, the first energy increase stage of the 901-conical shell ends earlier than the plane and hemispherical surfaces, and the increase is the least. The reason can be concluded as follows. The prolongation of the bubble in the
B. Han et al. / Optics and Lasers in Engineering 48 (2010) 950–957
expansion is more serious for the 901-conical and 301-conical shells than for the other surfaces. The liquid jet forms earlier and the jet radius is smaller, which can be seen in Fig. 6. Fig. 6(c) shows that the jet forms at about 0.6 ms in the 901-conical shell, but the distance between the jet front and the propelled surface is so big that the jet cannot propel the surface effectively. However, the 301-conical shell is so narrow that the jet can propel the surface earlier, thus there is only one increase stage for the 301-conical shell. Fig. 6(d) shows that the jet velocity and the
200
plane
180
hemisphere 90°- taper
Total Energy (µJ)
160
30°- taper
140 120 100 80 60 40 20 0 0
10
20
30
40 50 60 Time (µs)
70
80
90
100
Fig. 5. Numerical results of the energy that the four kinds of propelled surfaces obtain from the first-expansion of the bubble.
high-speed range of the 301-conical shell are both the smallest, and they decrease before reaching the propelled surface. Therefore, the 301-conical shell obtains the least energy, as shown in Fig. 7. Except for the 301-conical shell, the velocity of the jet front (radius about 20 mm) reaches 1000 m/s when the jet arrives at the propelled surface. The jet induced pressure on the propelled surface reaches 109 Pa, which induces the shock wave, namely the jet induced shock wave. The propagation and the attenuation of the jet induced shock wave are similar to that of the plasmabubble shock wave on the propelled surface shown in Fig. 4. Furthermore, after the jet penetrates the bubble to make it become toroidal, the vortex circling the ring shaped bubble will be induced and it will transport the liquid behind the jet onto the propelled surface, which is consistent with the experimental and numerical results in Refs. [25,26]. The vortex offers a continual propelling force (lasting about several ms). Fig. 8 shows the sketches of (a) the beginning of the liquid jet and (b) the vortex after the jet on the hemispherical shell. The liquid jet and the vortex on other propelled surfaces except the 301-conical shell are similar to Fig. 8. For the 301-conical shell, the jet compresses the whole bubble. When the jet arrives at the surface, it is reflected back, which blocks the liquid behind the jet. Thus, the 301-conical shell obtains the least energy during the final collapse impact of the bubble. The propelled surface is accelerated by every impingement from the shock wave or the jet. These propelling forces vary with time and disappear after some time. So the kinetic energy of the propelled surface also varies with time. The characteristic values representing the propelling effect listed in Table 1, including the kinetic energy that the propelled surface obtains during the
0
0
-100
-100
[0.20]
-200
[0.40] [0.60] [0.80]
-400
[1.00]
-500
{0.58}
-600
{0.28}
-700
[1.30]
-800
{0.17}
[0.60]
-300
{0.54}
{0.49}
{0.40}
[1.20]
[0.20] [0.40]
-200
{0.60}
V (m/s)
V (m/s)
-300
{0.52}
{0.47}
{0.42} [0.80]
-400
{0.32}
-500 [0.82]
-600 [recorded time] (µs) {distance to point (0,0)} (mm)
{0.27}
-700
[recorded time] (µs) {distance to point (0,0) }(mm)
[0.86]
-800
-900 -1000
{0.25}
-900 0.0
0.1 0.2 0.3 0.4 Distance to Axes z (mm)
0.0
0.5
-100 [0.60]
-300
{0.60}
-700
0.7
[0.20] {0.99} [0.40]
V (m/s)
{0.49} [0.80] {0.35}
-150
[0.60] [0.80]
-200
[recorded time] (µs) {distance to point (0,0) }(mm)
-350
{0.70}
{0.63} [1.60]
-300
{0.19}
{0.97}
{0.92}
[1.20] {0.86} [1.40]
-250
[1.00]
-800 -900 0.0
0.6
-100
{0.57}
-500 -600
0.2 0.3 0.4 0.5 Distance to Axes z (mm)
-50
[0.20] [0.40]
-200 -400
0.1
0
0
V (m/s)
955
{0.55} [2.20]
[recorded time] (µs) {distance to point (0,0)} (mm)
{0.35}
0.1
0.2 0.3 0.4 0.5 0.6 Distance to Axes z (mm)
0.7
-400 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 Distance to Axes z (mm)
Fig. 6. Velocity distribution of the jet front along r axis at different moment for the four kinds of propelled surfaces: (a) the plane, (b) the hemispherical shell, (c) the 901conical shell and (d) the 301-conical shell.
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2000
plane hemisphere
1800
90°- taper
1600
30°- taper
Total Energy (µJ)
1400 1200 1000 800 600 400 200 0 0.0
0.5
1.0
1.5 Time (µs)
2.0
2.5
3.0
Fig. 7. Numerical results of the energy that the four kinds of propelled surfaces obtain from the final collapse impact.
Jet
Vortex
Fig. 8. Sketches of (a) the beginning of the liquid jet and (b) the vortex after the jet on the hemispherical shell.
Table 1 Several characteristic values represent the propelling effect of the four kinds of propelled surfaces during the first-expansion and the final collapse of the bubble. energy Propelled surface shape Ekinetic (mJ) P1energy (%) Pcollapse (%) Ef (%) Cm (N s/J)
Unconfined plane Hemispherical shell 901-conical shell 301-conical shell
2.0 103 9.5 102 8.9 102 2.6 102
3.6 10.3 21.8 40.2
96.4 89.7 78.2 59.8
9.8 4.8 4.5 1.3
3.3 10 2 4.1 10 2 2.2 10 2 3.2 10 3
first-expansion and the final collapse of the bubble (Ekinetic), the proportion of the energy offered by the first-expansion of the bubble (P1energy ), the proportion of the energy offered by the final energy collapse impact (Pcollapse ), the propulsion efficiency (Ef) and the momentum coupling coefficient (Cm), are calculated from the maximum kinetic energy that the propelled surface could reach under the action of each propelling force. It is noticed that the narrower the propelled surface, the higher the P1energy . But Ef declines as the propelled surface becomes narrower. Cm of the hemispherical shell is the highest.
4. Conclusions Compared to the plane, the hemispherical and conical surfaces restrict the bubble to grow in a certain direction, which increases the expansion velocity of the bubble, the maximum bubble radius and the amplitude of the plasma-bubble shock wave. Thus the energy that the propelled surface obtains during the firstexpansion of the bubble is increased. The expansion velocity of the bubble and the plasma-bubble shock wave of the
hemispherical shell are close to that of the plane in the early stage of the bubble, then approach that of the 901-conical shell gradually, because of the narrowing at the end of the hemispherical shell. For the 301-conical shell, the expansion velocity of the bubble, the plasma-bubble shock wave and the maximum bubble radius are all the highest. But the attenuation of the shock wave is also the most serious in the 301-conical shell, and the propelling force is not concentrated in the direction of the propulsion properly. However, the 901-conical shell could both restrict the energy to release in the opposite direction of the propulsion properly, and induce more propelling force to work in the direction of the propulsion. So the 901-conical shell obtains the highest energy during the first-expansion of the bubble. During the final collapse impact, the contracting surrounding liquid and the jet-transported liquid offer the propelling energy. But except the plane, the other three kinds of surfaces block the contracting of the surrounding liquid. Thus the plane obtains the most energy during the final collapse impact. Except for the 301conical shell, the vortex is induced and supports a continual propelling force to the propelled surface after the jet impact. The jet impact on the 301-conical shell is the weakest, and the vortex is not induced. Thus, the 301-conical shell obtains the least energy during the final collapse impact. In addition, the narrower the propelled surface, the higher the proportion of the energy offered by the first-expansion of the bubble. But the propulsion efficiency declines as the propelled surface becomes narrower. The momentum coupling coefficient of the hemispherical shell is the highest.
References [1] Yabe T, Phipps C, Yamaguchi M. Microairplane propelled by laser driven exotic target. Appl Phys Lett 2002;80:43l8–20. [2] Myrabo LN. World record flights of beam-riding rocket lightcraft: demonstration of ‘‘disruptive’’ propulsion technology. In: 37th AIAA/ASME/SAE/ASEE joint propulsion conference, AIAA 2001-3798. [3] Ogata Y, Yabe T, Ookubo T. Numerical and experimental investigation of laser propulsion. Appl Phys A 2004;79:829–31. [4] Bohn WL, Schall WO. Laser propulsion activities in Germany. In: Pakhomov AV, editor. Beamed energy propulsion: first international symposium on beamed energy propulsion, CP664. American Institute of Physics, 2003. p. 79–91. [5] Han B, Shen ZH, Lu J, Ni XW. Laser propulsion for transport in water environment. Mod Phys Lett B 2010;24(7):641–8. [6] Vogel A, Busch S, Parlitz U. Shock wave emission and cavitation bubble generation by picosecond and nanosecond optical breakdown in water. J Acoust Soc Am 1996;100(1):148–65. [7] Petkovˇsek R, Mozˇina J, Mocˇnik G. Optodynamic characterization of shock waves after laser-induced breakdown in water. Opt Express 2005;13: 4107–12. [8] Petkovsek R, Gregorcic P, Mozina J. A beam deflection probe as a method for optodynamic measurements of cavitation bubble oscillations. Meas Sci Technol 2007;18:2972–8. [9] Zhao R, Xu RQ, Shen ZH, Lu J, Ni XW. Experimental investigation of the collapse of laser-generated cavitation bubbles near a solid boundary. Opt Laser Technol 2007;39:968–72. [10] Philipp A, Lauterborn W. Cavitation erosion by single-laser produced bubbles. J Fluid Mech 1998;361:75–116. [11] Blake JR, Tomita Y, Tong RP. The art, craft and science of modelling jet impact in a collapsing cavitation bubble. Appl Sci Res 1998;58:77–90. [12] Tong RP, Schiffers WP, Shaw SJ, Blake JR, Emmony DC. The role of ‘splashing’ in the collapse of a laser-generated cavity near a rigid boundary. J Fluid Mech 1999;380:339–61. [13] Vogel A, Lauterborn W, Timm R. Optical and acoustic investigations of the dynamics of laser-produced cavitation bubbles near a solid boundary. J Fluid Mech 1989;206:299–338. [14] Rattray M. Perturbation effects on cavitation bubble dynamics. PhD thesis, California Institute of Technology, Pasadena, CA, 1951. [15] Brujan EA, Nahen K, Schmidt P, Vogel A. Dynamics of laser-induced cavitation bubbles near an elastic boundary. J Fluid Mech 2001;433:251–81. [16] Kodama T, Tomita Y. Cavitation bubble behavior and bubble-shock wave interaction near a gelatin surface as a study of in vivo bubble dynamics. Appl Phys B 2000;70:139–49. [17] Gregorcˇicˇ P, Petkovˇsek R, Mozˇina J. Investigation of a cavitation bubble between a rigid boundary and a free surface. J Appl Phys 2007;102: 094904.
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[18] Sasoh A, Watanabe K, Sano Y, Mukai N. Behavior of bubbles induced by the interaction of a laser pulse with a metal plate in water. Appl Phys A 2005;80:1497–500. [19] Sinko JE, Dhote NB, Lassiter JS, Gregory DA. Conical nozzles for pulsed laser propulsion. In: Claude R, editor. High-power laser ablation VII, Phipps, Proceedings of the SPIE 2008; 7005, 70052Q. [20] Petkovˇsek R, Gregorcˇicˇ P. A laser probe measurement of cavitation bubble dynamics improved by shock wave detection and compared to shadow photography. J Appl Phys 2007;102:044909. [21] Gregorcic P, Petkovsek R, Mozina J, Mocnik G. Measurements of cavitation bubble dynamics based on a beam-deflection probe. Appl Phys A 2008;93:901–5.
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[22] Yang YN, Zhao N, Ni XW. Reflection effects of spherical shock wave. Mod Phys Lett B 2005;19:1451–4. [23] Kohn BJ. Compilation of Hugoniot equations of state. Air Force Weapons Laboratory, Report AFWL-TR-69-38, 1969. [24] Steinberg DJ, Cochran SG, Guinan MW. A constitutive model for metals applicable at high-strain rate. J Appl Phys 1980;51:1498–504. [25] Zwaan E, Gac SL, Tsuji K, Ohl CD. Controlled cavitation in microfluidic systems. Phys Rev Lett 2007;98:254501. [26] Brujan EA, Keen GS, Vogel A, Blake JR. The final stage of the collapse of a cavitation bubble close to a rigid boundary. Phys Fluids 2002;14: 85–92.
Contents lists available at ScienceDirect
Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng
Numerical study of water-confinement geometries for laser propulsion Bing Han, Zhong-Hua Shen, Jian Lu, Xiao-Wu Ni n School of Science, Nanjing University of Science & Technology, Nanjing, Jiangsu 210094, People’s Republic of China
a r t i c l e in f o
a b s t r a c t
Article history: Received 22 October 2009 Received in revised form 28 May 2010 Accepted 31 May 2010 Available online 16 June 2010
The processes of laser propulsion in a water environment are investigated numerically in this paper. Four kinds of propelled surfaces are discussed: a plane, a hemispherical shell, a 901-conical shell and a 301-conical shell. The bubble radius and the velocity of the bubble surface varying with time during the first-expansion of the bubble are investigated. The evolution of the shock wave induced pressure field around the pressure peak on the surface and the evolution of the velocity field of the liquid jet are discussed. The energy that the propelled surface obtains from the first-expansion of the bubble is enhanced by the three kinds of shapes except the plane, among which the 901-conical shell produces the most effective propulsion. During the final collapse of the bubble, the plane propelled surface obtains the most energy, while the 301-conical shell obtains the least. The narrower the propelled surface, the higher the proportion of the energy offered by the plasma-bubble shock wave. But the propulsion efficiency declines as the propelled surface becomes narrower. The momentum coupling coefficient of the hemispherical shell is the highest. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Laser propulsion Water Confinement Bubble formation
1. Introduction Laser propulsion has been investigated in two main aspects, atmospheric breathing [1,2] and ablation under vacuum [3,4]. In order to avoid the main problems that encumber the development of laser propulsion in atmosphere and vacuum, we proposed laser propulsion in a water environment [5] based on the special properties of the laser-induced breakdown in water. When a highintensity laser beam is focused onto the propelled surface in water, plasma of high-temperature and high-pressure will be induced if the power density at the laser focus exceeds the breakdown threshold of water. The plasma expands violently, inducing a quasi-spherical shock wave. In the meantime, the pressure and temperature of the plasma area drop, which finally leads to the formation of steam. A high density steam bubble [6–8] is produced with the initial plasma as the spherical center. The bubble expands and radiates a shock wave. Before finally collapsing, the bubble usually experiences several oscillations. Every time the bubble reaches the minimum volume and starts to rebound, it radiates a quasi-spherical shock wave. As the bubble energy decreases, the energy of the successive shock waves is attenuated [9]. A liquid jet [10] will be induced toward the propelled surface when the bubble collapses. The sources of the propelling force include the plasma shock wave, the bubbleexpanding shock wave and the final collapse impact.
n
Corresponding author. Tel.: +86 25 84315075; fax: + 86 25 84318430. E-mail addresses: [email protected] (B. Han), [email protected] (X.-W. Ni). 0143-8166/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2010.05.013
An increase in the laser pulse energy will increase the intensity of the plasma shock wave, thus increase the propelling force of the shock wave. The increase in the expanding velocity of the bubble surface and the prolongation of the high-speed expansion of the bubble will both increase the propelling force of the bubble-expanding shock wave. There are many factors that can influence the final collapse of the bubble. For example, the ratio between the bubble radius and the distance between the initial bubble center and the propelled surface [11–13], the characteristics of the liquid interface around the bubble [14–17], and the thickness of the plate on which the bubbles are induced [18]. Investigations on laser propulsion in atmosphere show that the propelling effect is influenced by the confined shape of the propelled surface [19]. Different kinds of confined propelled surfaces have been designed, among which the Myrabo-type [2] and the bell-type [4] are the most famous. According to our investigation, variation of the confining shape of the propelled surface influences all three sources of propelling force mentioned at the end of the first paragraph. Therefore, the processes of laser propulsion in a water environment with four kinds of propelled surfaces—a plane, a hemispherical shell, a 901-conical shell and a 301-conical shell—are investigated numerically in this paper. Parabolic shell-shaped surfaces have been widely used in laser propulsion in atmosphere and vacuum. An important purpose is to focus the attenuated laser beam transmitted through a long distance. Thus, plasma can be induced if the power density at the laser focus exceeds the breakdown threshold of air or the working medium. On the one hand, in this paper, the laser system can be put near the propelled surface, thus the excitation beam can be focused on the propelled surface directly. On the other hand, the
B. Han et al. / Optics and Lasers in Engineering 48 (2010) 950–957
parabolic shell and the conical shell are similar when the scale of the propelled surface is mm. Therefore, the hemispherical shell is selected instead of the parabolic shell. In addition, 901-conical and 301-conical shells are compared to investigate the influence of the degree of the confinement on the propulsion. Investigations on the laser-induced bubbles are concentrated on the oscillating properties of the spherical bubble in an infinite liquid environment [20], and the collapsing properties of the quasi-spherical bubble near different interfaces. However, in this paper, the bubbles induced on the four kinds of propelled surfaces are quasi-half, quasi-quarter and quasi-duodecimal spherical cavitation bubbles elongated in the opposite direction of the laser beam, as shown in Fig. 1. The plasma and bubble first-expansion shock waves are emitted almost at the same time [8,21], so they are called the plasma-bubble shock wave in this paper. According to our investigations [5], the majority of the propelling power is from the plasma-bubble shock wave and the final collapse impact. Only less than 1% of the propelling power is from the latter bubble expansion shock waves. Therefore, the characteristics of the forces induced by the plasma-bubble shock wave and the final collapse impact are discussed in this paper. The bubble radius and the velocity of the bubble surface varying with time in the firstexpansion of the bubble are investigated. The evolution of the plasma-bubble shock wave induced pressure field around the pressure peak on the surface and the evolution of the velocity field of the liquid jet are discussed. The characteristics of the propulsion are compared between the plane and the other propelled surfaces, between the hemispherical and the conical shells, and between the two different conical shells.
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2. Numerical simulation The arrangement of laser propulsion in water is shown in Fig. 1. The propelled surface made of copper is in the small dashed cube. The insets show the four kinds of axisymmetric propelled surfaces and the corresponding bubbles. Fig. 2 shows the models used in the numerical simulation, including (a) the plane, (b) the hemispherical shell, (c) the 901conical shell and (d) the 301-conical shell. The unit of the spatial coordinates is mm. z is the rotational symmetric axis of the whole model. Region D is the water environment. Point (0,0) is the laser focus. Due to high intensity of the laser focus, ablation of the metal surface is possible, which will influence the bubble dynamics and contribute to the impulse directly through substance ejection. Because the main object of this paper is the bubble-induced mechanical force, the ablation of the metal is neglected in calculations of this paper. The CFD numerical simulation is based on the conservation laws, including the mass conservation Eq. (1), the momentum conservation Eqs. (2) and (3), and the energy conservation Eq. (4) [22]: @r @ 1@ ðrvÞ ¼ 0 þ ðruÞ þ @z r @r @t
ð1Þ
@ @ 1@ @p ðruÞ þ ðru2 Þ þ ðr ruvÞ þ ¼0 @t @z r @r @z
ð2Þ
@ @ 1@ @p ðrvÞ þ ðruvÞ þ ðr rv2 Þ þ ¼0 @t @z r @r @r
ð3Þ
water environment
Nd:YAG laser
glass cuvette
Inset (a1) quasi-half spherical bubble
Nd: Y AG laser Inset (a2)
velocity direction
Inset (b1) quasi-half spherical bubble
propelled object
Inset (c1) quasi-quarter spherical bubble
Nd: Y AG laser
Nd: Y AG laser
Inset (b2)
Inset (c2)
Inset (d1) quasi-duodecimal spherical bubble
Nd: Y AG laser Inset (d2)
Fig. 1. The arrangement of laser propulsion in water. Insets show the four kinds of axisymmetric propelled surfaces and the corresponding bubbles induced on them. (a1) to (d1) are the plane, the hemispherical shell, the 901-conical shell and the 301-conical shell, respectively. (a2) to (d2) are the quasi-half, quasi-half, quasi-quarter and the quasi-duodecimal spherical bubbles, respectively.
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r/mm
r/mm
25
25 r1 = 2
D
r2 = 2
D
z1 = -1
r1
z1 = -1
r1
r1 = 3
z2 = 0
z2 = 0
r2
z3 = 2 z/mm
z/mm -25
z2
z1
-25
25
r/mm
z1
z2
z3
25
r/mm
25
25 r1 = 2
D
z1 = -1
r1 = 0.536
D
z1 = -1 z2 = 0
z2 = 0 z3 = 2
r1
z/mm -25
z1
z2
z3
25
z3 = 2
r1
z/mm -25
z1
z2
z3
25
Fig. 2. The models used in the numerical simulation: (a) the plane, (b) the hemispherical shell, (c) the 901-conical shell and (d) the 301-conical shell.
8 < @I @E @ 1@ ½rvðE þ pÞ ¼ @z þ ½uðEþ pÞ þ : @t @z r @r 0
td o t otp ,ðz,rÞ A O
ð4Þ
otherwise
where t is the time, r is the density, u and v are velocities along axes z and r, respectively, p is the pressure, E is the energy per unit volume, I is the laser intensity, td is the moment when the shock wave appears, tP is the laser pulse width, and O is the area of the laser beam. If one takes e as the specific internal energy, E can be defined as E ¼ reþ 12rðu2 þ v2 Þ
ð5Þ
The pressure field p(z,r,t) of the calculation area can be calculated from Eqs. (1)–(5). The impulse Imi that the propelled surface obtained from t ¼0 to t ¼ti can be calculated by Z ti Z redge ½pðzout ,r,tÞpðzin ,r,tÞ2prdrdt ð6Þ Imi ¼ 0
0
where zout is the coordinate of the outer surface, zin is the coordinate of the inner surface (irradiated by the Nd:YAG laser) and redge is the coordinate of the edge of the surface. Thus the kinetic energy Ekinetic of the propelled surface at t ¼ti can be i calculated by ¼ Ekinetic i
2 Imi 2M
ð7Þ
where M is the mass of the propelled surface. The momentum coupling coefficient Cm of a single laser pulse, expressed in unit of N s/J, can be calculated by Cm ¼
total Im Epulse
Steinberg–Guinan strength model [24] is chosen. Region D is of obvious fluidity, so the Euler algorithm, suitable for the calculation of large deformation, is chosen. The propelled surface is a rigid plane, so the Lagrange algorithm, suitable for the calculation of elastic/plastic finite deformation, is chosen. The VOF model is used. The mass of the propelled surfaces shown in Fig. 2(a), (b), (c) and (d) are 113, 354, 113 and 8 mg, respectively. The parameter that has been chosen is the shape of the propelled surface, not the mass. After the needed surface models are built, the mass of each model is fixed. The wavelength of the laser is 1.06 mm. The energy per pulse is 20 mJ. The laser pulse width is 10 ns. The radius of the laser focus is 50 mm. In the simulation of this manuscript, the detailed interaction between the laser beam and water (e.g., reflection, absorption, scattering) was not calculated. The simulations start from the bubble growth. The oscillations of the bubbles are calculated. Initial conditions of the bubbles are taken from experimental results. In the meantime, the momentum coupling of the propelled surface is calculated. The exact value that determines the dynamic behavior of the laser-induced bubble is the laser energy saved after all the attenuation. In the future when we make the laser propulsion an applicable tool, if we know such saved-energy that is necessary to produce the force we need, we can manage to get it in different conditions and for different lasers.
3. Simulation results and discussion 3.1. The first-expansion of the bubble
ð8Þ
total is the total impulse that the propelled surface where Im obtained, Epulse is the energy of a single laser pulse. Because of the existence of the strong discontinuity, namely the shock wave, the shock equation of state (SEOS) [23] is chosen as the state equation. The material of the propelled surface is copper (Cu), which will be deformed under the impact of the shock wave and the bubble final collapse, namely the cavitation erosion. So the
The newborn bubble expands rapidly and radiates a shock wave. The bubble energy decreases quickly, so the velocity of the bubble surface decreases. After reaching its maximum radius, the bubble starts to constrict. Fig. 3 shows the bubble radius and the velocity of the bubble surface varying with time in the firstexpansion of the bubble on the four kinds of propelled surfaces. The inset of Fig. 3 shows the definition of the bubble radius and the position on the bubble surface where the surface velocity is
2.2 2.0
240 220 200 180 160 140 120 100 80 60 40 20 0
Radius
1.8 1.6 R (mm)
1.4 1.2 1.0 0.8
plane hemisphere
0.6
90°- taper
0.4
30°- taper
Velocity
0.2 0.0 0
5
10
15
20
25 30 35 Time (µs)
40
45
50
V (m/s)
B. Han et al. / Optics and Lasers in Engineering 48 (2010) 950–957
55
Surface velocity read from this point
Bubble radius
Inset
Fig. 3. The bubble radius and the velocity of the bubble surface varying with time in the first-expansion of the bubble on the four kinds of propelled surfaces. Inset shows the definition of the bubble radius and the position on the bubble surface where the surface velocity is monitored.
monitored. In the early stage of the bubbles, during about 0–3 ms, the velocity of the bubble surface reaches 100 m/s, consistent with the experimental results in Refs. [10,15,20]. The same amount of laser energy is transmitted into the water in the laser focus, which means the same amount of plasma can be induced. For example, to get the same volume, plasma on the 901-conical inner peak needs a bigger radius than that on the plane, which further leads to the radius of the bubble induced on the 901conical inner peak is bigger than that on the plane. The plasma on the 901-conical inner peak spreads into a quarter of the whole spatial angle, while the plasma on the plane spreads into half of the whole spatial angle. Thus, compared to the plane, the conical and hemispherical surfaces restrict the bubble to grow in a certain direction, which increases both the expanding velocity and the maximum bubble radius. The narrowest shape (301-conical shell) brings the highest restriction. The restrictive effect of the hemispherical shell is a little lower than that of the 901-conical shell. In addition, the velocity–time curves show that the velocity of the bubble surface of the hemispherical shell is a little higher than that of the plane in the early stage, but the velocity of the hemispherical shell approaches that of the 901-conical shell gradually after about 5 ms. The reason is that the top of the hemispherical shell (around point (0,0)) is flat, similar to the plane. As the bubble becomes larger, the restrictive effect of the hemispherical shell becomes stronger. Fig. 4 shows the evolution of the pressure field, induced by the plasma-bubble shock wave, around the pressure peak on the four kinds of propelled surfaces: (a) the plane, (b) the hemispherical shell, (c) the 901-conical shell and (d) the 301-conical shell. The abscissa, propelled surface (mm), shows the distance from the point (0,0), along the propelled surface irradiated directly by the laser. Fig. 4(e) shows the value of the pressure peak varying with time on the four kinds of propelled surfaces. The pressure peak of the plasma-bubble shock wave reaches 108 Pa. As it
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propagates toward the edge of the propelled surface, the shock wave front broadens and the peak pressure decreases gradually, until it attenuates to a sound wave. Because the top of the hemispherical shell (around point (0,0)) is flat, the peak pressure in the early stage on the hemispherical shell is a little higher than that on the plane and lower than that on the two conical shells. However, the walls of the hemispherical shell approach a cylindrical shape at large radii, which abates the attenuation of the wave, as shown in Fig. 4(b). Thus, the peak pressure on the hemispherical shell becomes the highest after about 0.9 ms, as shown in Fig. 4(e). It is noticed that the pressure peak of the newborn plasma-bubble shock wave is the highest in the 301conical shell, but the attenuation of the shock wave is also the most serious in the 301-conical shell. As shown in Fig. 4(e), the peak pressure is weaker in the 301-conical shell than that in the 901-conical and hemispherical shells after about 0.9 ms. Fig. 5 shows the numerical results of the energy that the four kinds of propelled surfaces obtain from the first-expansion of the bubble. Compared to the plane, the hemispherical and conical surfaces obtain more energy during the first-expansion of the bubble. On the one hand, the hemispherical shell failed to restrict the energy to release in the opposite direction of the propulsion, the positive direction of axis z, in the early stage of the bubble. On the other hand, the shock wave pressure on the propelled surface of the 301-conical shell is perpendicular to the surface, thus the component in the propelling direction is very small. However, the 901-conical shell could both restrict the energy to release in the opposite direction of the propulsion properly, and induce more propelling force to work in the direction of the propulsion. So the energy increase of the 901-conical shell is the highest during the first-expansion of the bubble.
3.2. The final collapse of the bubble When the maximum radius of the bubble is mm, as it is for the bubbles discussed in this paper, the duration of the liquid jet is a few 10’s of ms [21], and the radius of the jet is 10 1 mm. That is to say, the process of the liquid jet is instantaneous and highly localized. The evolution details of the liquid jet are investigated numerically in this paper. Fig. 6 shows the velocity distribution of the jet front along the r axis at different moments for the four kinds of propelled surfaces: (a) the plane, (b) the hemispherical shell, (c) the 901-conical shell and (d) the 301conical shell. The numerical simulation starts at the moment when the energy of the propelled surface rises. The starting time of the [recorded time] (ms) marked in Fig. 6 is the same as that of the numerical simulation. The distance between the jet front and the propelled surface at the recorded moment is also marked in Fig. 6. For all four kinds of propelled surfaces, the gradients of the velocity fields around the velocity peaks increase, and the radii of the jets decrease. Except in the 301-conical shell, the velocity of the jet front keeps increasing until it reaches the propelled surface. It is the plane that induces the highest velocity and the widest high-speed range of the jet, when the jet almost impacts the propelled surface. Next is the hemispherical shell, and the 901conical shell. The jet velocity and the high-speed range of the 301conical shell are both the smallest. The velocity of the jet front in the 301-conical shell begins to decrease after about 1.2 ms. The reason is that the area close to the point (0,0) of the 301-conical shell becomes very narrow. Thus the push of the jet compresses the whole bubble seriously, which makes the pressure of the area between the jet front and propelled surface increase dramatically. In other words, the jet front encounters great resistance before it reaches the propelled surface. For other propelled surfaces, the jet penetrates the central section of the bubble, which makes the
B. Han et al. / Optics and Lasers in Engineering 48 (2010) 950–957
300 275 250 225 200 175 150 125 100 75 50 25 0 -25
recorded time (µs)
0.21 0.40 0.61
0.0
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Pressure (MPa)
0.11
300 275 250 225 200 175 150 125 100 75 50 25 0 -25
0.3
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1.00
0.6 0.9 1.2 1.5 Propelled Surface (mm)
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recorded time (µs) 0.10
0.21
0.40 0.63
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0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 Propelled Surface (mm)
Pressure (MPa)
300 270
plane hemisphere
240
90°- taper
300 275 250 225 200 175 150 125 100 75 50 25 0 -25
recorded time (µs) 0.08
0.18
0.40 0.60
0.0
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Pressure (MPa)
Pressure (MPa)
954
0.4
0.81 1.00 1.21 1.41
1.64
0.8 1.2 1.6 2.0 2.4 Propelled Surface (mm)
2.8
0.08
300 275 250 225 200 175 150 125 100 75 50 25 0 -25
3.2
recorded time (µs)
0.20 0.30 0.50 0.69 1.00 1.13
0.0
0.3
0.6 0.9 1.2 1.5 Propelled Surface (mm)
1.8
2.1
30°- taper
210 180 150 120 90 60 30 0 0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Time (µs) Fig. 4. The evolution of the pressure field, induced by the plasma-bubble shock wave, around the pressure peak on the four kinds of propelled surfaces: (a) the plane, (b) the hemispherical shell, (c) the 901-conical shell and (d) the 301-conical shell. (e) The value of the pressure peak varying with time on the four kinds of propelled surfaces.
bubble become toroidal. Thus the high resistance area between the jet front and propelled surface is avoided. Fig. 7 shows the numerical results of the energy that the four kinds of propelled surfaces obtain from the final collapse impact of the bubble. Except for the 301-conical shell, there are two increase stages of the energy. In the first increase stage, the contracting bubble induces the surrounding liquid flowing toward and impacting the propelled surface, which makes the energy of the propelled surface rise. According to Fig. 6, the starting time of the second increase stage in Fig. 7 is just the moment when the liquid jet reaches the propelled surface. The high-speed and highpressure jet impacts the propelled surface violently, which makes the energy of the propelled surface rise again. The energy that the plane obtains from the first-expansion of the bubble is the least,
but during the final collapse impact the plane obtains the most energy. The reason can be described as follows. During the final collapse impact, it is the contracting surrounding liquid and the jet-transported liquid that offer the energy. The hemispherical and conical surfaces block the contracting of the surrounding liquid. For example, because of the similarity in the structure, the hemispherical and the 901-conical shell obtain a similar amount of energy. But after about 2.2 ms, the energy of the hemispherical shell rises. According to Fig. 4(b), the reason is that the jet induced shock wave propagates to the narrow end of the hemispherical shell at about 2.2 ms. In addition, the first energy increase stage of the 901-conical shell ends earlier than the plane and hemispherical surfaces, and the increase is the least. The reason can be concluded as follows. The prolongation of the bubble in the
B. Han et al. / Optics and Lasers in Engineering 48 (2010) 950–957
expansion is more serious for the 901-conical and 301-conical shells than for the other surfaces. The liquid jet forms earlier and the jet radius is smaller, which can be seen in Fig. 6. Fig. 6(c) shows that the jet forms at about 0.6 ms in the 901-conical shell, but the distance between the jet front and the propelled surface is so big that the jet cannot propel the surface effectively. However, the 301-conical shell is so narrow that the jet can propel the surface earlier, thus there is only one increase stage for the 301-conical shell. Fig. 6(d) shows that the jet velocity and the
200
plane
180
hemisphere 90°- taper
Total Energy (µJ)
160
30°- taper
140 120 100 80 60 40 20 0 0
10
20
30
40 50 60 Time (µs)
70
80
90
100
Fig. 5. Numerical results of the energy that the four kinds of propelled surfaces obtain from the first-expansion of the bubble.
high-speed range of the 301-conical shell are both the smallest, and they decrease before reaching the propelled surface. Therefore, the 301-conical shell obtains the least energy, as shown in Fig. 7. Except for the 301-conical shell, the velocity of the jet front (radius about 20 mm) reaches 1000 m/s when the jet arrives at the propelled surface. The jet induced pressure on the propelled surface reaches 109 Pa, which induces the shock wave, namely the jet induced shock wave. The propagation and the attenuation of the jet induced shock wave are similar to that of the plasmabubble shock wave on the propelled surface shown in Fig. 4. Furthermore, after the jet penetrates the bubble to make it become toroidal, the vortex circling the ring shaped bubble will be induced and it will transport the liquid behind the jet onto the propelled surface, which is consistent with the experimental and numerical results in Refs. [25,26]. The vortex offers a continual propelling force (lasting about several ms). Fig. 8 shows the sketches of (a) the beginning of the liquid jet and (b) the vortex after the jet on the hemispherical shell. The liquid jet and the vortex on other propelled surfaces except the 301-conical shell are similar to Fig. 8. For the 301-conical shell, the jet compresses the whole bubble. When the jet arrives at the surface, it is reflected back, which blocks the liquid behind the jet. Thus, the 301-conical shell obtains the least energy during the final collapse impact of the bubble. The propelled surface is accelerated by every impingement from the shock wave or the jet. These propelling forces vary with time and disappear after some time. So the kinetic energy of the propelled surface also varies with time. The characteristic values representing the propelling effect listed in Table 1, including the kinetic energy that the propelled surface obtains during the
0
0
-100
-100
[0.20]
-200
[0.40] [0.60] [0.80]
-400
[1.00]
-500
{0.58}
-600
{0.28}
-700
[1.30]
-800
{0.17}
[0.60]
-300
{0.54}
{0.49}
{0.40}
[1.20]
[0.20] [0.40]
-200
{0.60}
V (m/s)
V (m/s)
-300
{0.52}
{0.47}
{0.42} [0.80]
-400
{0.32}
-500 [0.82]
-600 [recorded time] (µs) {distance to point (0,0)} (mm)
{0.27}
-700
[recorded time] (µs) {distance to point (0,0) }(mm)
[0.86]
-800
-900 -1000
{0.25}
-900 0.0
0.1 0.2 0.3 0.4 Distance to Axes z (mm)
0.0
0.5
-100 [0.60]
-300
{0.60}
-700
0.7
[0.20] {0.99} [0.40]
V (m/s)
{0.49} [0.80] {0.35}
-150
[0.60] [0.80]
-200
[recorded time] (µs) {distance to point (0,0) }(mm)
-350
{0.70}
{0.63} [1.60]
-300
{0.19}
{0.97}
{0.92}
[1.20] {0.86} [1.40]
-250
[1.00]
-800 -900 0.0
0.6
-100
{0.57}
-500 -600
0.2 0.3 0.4 0.5 Distance to Axes z (mm)
-50
[0.20] [0.40]
-200 -400
0.1
0
0
V (m/s)
955
{0.55} [2.20]
[recorded time] (µs) {distance to point (0,0)} (mm)
{0.35}
0.1
0.2 0.3 0.4 0.5 0.6 Distance to Axes z (mm)
0.7
-400 0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32 Distance to Axes z (mm)
Fig. 6. Velocity distribution of the jet front along r axis at different moment for the four kinds of propelled surfaces: (a) the plane, (b) the hemispherical shell, (c) the 901conical shell and (d) the 301-conical shell.
956
B. Han et al. / Optics and Lasers in Engineering 48 (2010) 950–957
2000
plane hemisphere
1800
90°- taper
1600
30°- taper
Total Energy (µJ)
1400 1200 1000 800 600 400 200 0 0.0
0.5
1.0
1.5 Time (µs)
2.0
2.5
3.0
Fig. 7. Numerical results of the energy that the four kinds of propelled surfaces obtain from the final collapse impact.
Jet
Vortex
Fig. 8. Sketches of (a) the beginning of the liquid jet and (b) the vortex after the jet on the hemispherical shell.
Table 1 Several characteristic values represent the propelling effect of the four kinds of propelled surfaces during the first-expansion and the final collapse of the bubble. energy Propelled surface shape Ekinetic (mJ) P1energy (%) Pcollapse (%) Ef (%) Cm (N s/J)
Unconfined plane Hemispherical shell 901-conical shell 301-conical shell
2.0 103 9.5 102 8.9 102 2.6 102
3.6 10.3 21.8 40.2
96.4 89.7 78.2 59.8
9.8 4.8 4.5 1.3
3.3 10 2 4.1 10 2 2.2 10 2 3.2 10 3
first-expansion and the final collapse of the bubble (Ekinetic), the proportion of the energy offered by the first-expansion of the bubble (P1energy ), the proportion of the energy offered by the final energy collapse impact (Pcollapse ), the propulsion efficiency (Ef) and the momentum coupling coefficient (Cm), are calculated from the maximum kinetic energy that the propelled surface could reach under the action of each propelling force. It is noticed that the narrower the propelled surface, the higher the P1energy . But Ef declines as the propelled surface becomes narrower. Cm of the hemispherical shell is the highest.
4. Conclusions Compared to the plane, the hemispherical and conical surfaces restrict the bubble to grow in a certain direction, which increases the expansion velocity of the bubble, the maximum bubble radius and the amplitude of the plasma-bubble shock wave. Thus the energy that the propelled surface obtains during the firstexpansion of the bubble is increased. The expansion velocity of the bubble and the plasma-bubble shock wave of the
hemispherical shell are close to that of the plane in the early stage of the bubble, then approach that of the 901-conical shell gradually, because of the narrowing at the end of the hemispherical shell. For the 301-conical shell, the expansion velocity of the bubble, the plasma-bubble shock wave and the maximum bubble radius are all the highest. But the attenuation of the shock wave is also the most serious in the 301-conical shell, and the propelling force is not concentrated in the direction of the propulsion properly. However, the 901-conical shell could both restrict the energy to release in the opposite direction of the propulsion properly, and induce more propelling force to work in the direction of the propulsion. So the 901-conical shell obtains the highest energy during the first-expansion of the bubble. During the final collapse impact, the contracting surrounding liquid and the jet-transported liquid offer the propelling energy. But except the plane, the other three kinds of surfaces block the contracting of the surrounding liquid. Thus the plane obtains the most energy during the final collapse impact. Except for the 301conical shell, the vortex is induced and supports a continual propelling force to the propelled surface after the jet impact. The jet impact on the 301-conical shell is the weakest, and the vortex is not induced. Thus, the 301-conical shell obtains the least energy during the final collapse impact. In addition, the narrower the propelled surface, the higher the proportion of the energy offered by the first-expansion of the bubble. But the propulsion efficiency declines as the propelled surface becomes narrower. The momentum coupling coefficient of the hemispherical shell is the highest.
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