Direction of the microjet produced by the collapse of a cavitation bubble located in a corner of a wall and a free surface

PHYSICAL REVIEW FLUIDS 6, 083601 (2021) Direction of the microjet produced by the collapse of a cavitation bubble located in a corner of a wall and a free surface Akihito Kiyama Institute of Global Innovation Research, Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan and Department of Mechanical and Aerospace Engineering, Utah State University, Logan, Utah 84322, USA Takaaki Shimazaki Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan José Manuel Gordillo Área de Mecánica de Fluidos, Departamento de Ingeniería Aeroespacial y Mecánica de Fluidos, Universidad de Sevilla, Avenida de los Descubrimientos s/n 41092, Sevilla, Spain Yoshiyuki Tagawa * Department of Mechanical Systems Engineering, Tokyo University of Agriculture and Technology, Tokyo 184-8588, Japan (Received 11 November 2020; accepted 8 July 2021; published 5 August 2021) In this paper, we present a simplified theoretical model based on the method of images that predicts the direction of the microjet produced after the implosion of a cavitation bubble created in a corner of a free interface and rigid wall. Our theoretical predictions have been verified by means of a thorough experimental study in which the distances of the pulsed-laser cavitation bubble to the wall and the free surface are varied in a systematic manner. In addition, we extend the predictions to arbitrary values of the corner angle, π/(2n) with na natural number. The present analytical solution might be a hint to a practical design for preventing cavitation-induced damage. DOI: 10.1103/PhysRevFluids.6.083601 I. INTRODUCTION Cavitation bubbles can form a microjet directed towards a rigid wall when the collapse takes place near the solid surface [1]. The impingement of the microjet on the solid boundary is one of the sources of cavitation-induced damage (e.g., Ref. [2]). A number of contributions have been devoted to predict and analyze the direction of the high-speed microjet with different geometries or properties, such as a rigid flat surface [3], a rigid curved surface [4], an ice surface [5], an elastic surface [6], a composite surface [7] and a free surface [8], and they all reveal that the jet direction is largely affected by the rigidity and the geometry of the flow boundaries. Kucera and Blake [9] have studied the collapse of a cavitation bubble placed at the corner of two solid substrates forming an angle of either α=π/2orα=π/4. Brujan et al. [10] experimentally *tagaw[email protected] 2469-990X/2021/6(8)/083601(11) 083601-1 ©2021 American Physical Society KIYAMA, SHIMAZAKI, GORDILLO, AND TAGAWA FIG. 1. High-speed images of a collapsing bubble with a maximum diameter Din the vicinity of a solid surface and of an air bubble (a) and that in the absence of the air bubble (b). The two cases shown have similar values of the ratios w/Dand h/D, with wand hdenoting, respectively, the distances of the center of the cavitation bubble to the wall and the free surface. showed the direction of the microjet at the corner of two solid plates (α=π/2) is proportional to the ratio of distances from the plates to the bubble. Tagawa and Peters [11] employed the method of images for solving the case α=π/n, where nis a natural number, and obtained analytical solutions for the jet direction that were confirmed experimentally. The method of images has been also adopted to analyze the motion of a bubble parallelly sandwiched by both a free surface and solid bottom [12]. Recently, Wang et al. [13] conducted numerical simulations that show a fair agreement with the published data for α=π/2[10] and their experiments for α=π/4. The microjet direction inside the confined geometries [14,15] and that near the slotlike structured surface [16]havealso been studied. In applications, it is desirable to prevent the impingement of the jet over the solid since, in this way, the lifetime of different types of fluid machinery, such as ship propellers, can be substantially increased [17]. An approach to achieve this is the installation of another surface with a different impedance. Luo et al. [18] placed an air bubble right next to the cavitation bubble and found that the direction of bubble movement upon its collapse was altered. Indeed, our preliminary observation [Fig. 1(a)] shows that the jet produced by the implosion of a laser-induced cavitation bubble generated in the vicinity of an air bubble is directed away from it. More recently, Avila et al. [19] demonstrated that a solid wall covered with micron-sized gas bubbles can alter the jet direction and help prevent the collision of the jet. However, to our knowledge, the theoretical prediction of the microjet direction in a corner of both free and rigid surfaces has not yet been provided. Here, we provide a theoretical prediction of the microjet direction in a corner of a wall and free surface as sketched in Fig. 2, which contains the major physical ingredients behind the control of the direction of the microjet by means of the addition of air bubbles near the collapsing vapor bubble. Making use of ideas in a previous contribution [11], we employ the potential flow approximation and theoretically derive the direction of the jet as a function of the bubble location (Sec. II). To make the derivation as simple as possible and without loss of generality, this model is particularized for the simplest case where the corner angle between a wall and free surface α=π/2(=90◦). Our analytical predictions are compared with the results of a systematic experimental study in which a laser-induced cavitation bubble is generated at well-defined distances from the wall and the free surface (Secs. III and IV). In addition, we derive a generalized model for the jet angle βfor corner angles α=π/(2n) with na natural number (Sec. V). 083601-2 DIRECTION OF THE MICROJET PRODUCED BY THE … FIG. 2. (a) The schematic of the system on x-ycoordinates marked in gray. Green markers ()andred markers () respectively represent the sources (s1and s2) and sinks (s0and s3). The velocity vectors affecting the bubble induced from mirrored images are denoted as ur1,ur2,andur3. (b) The illustration of the microjet direction β. The horizontal (ux) and vertical (uy) components of the velocity vector acting on the bubble ujare denoted. II. THEORY A flow field induced by a spherically collapsing bubble is obtained by considering a sink [20] [Fig. 2(a),s0at (−w,−h)]. A solid wall and free surface are respectively modeled by placing the sink [Fig. 2(a),s3at (w,−h)] and source [Fig. 2(a),s1at (−w,h)] [21]. This sink-source system is appropriate since the bubble expansion does not significantly deform the boundaries in our experiments. Noteworthy, another source s2must be placed at (w,h) to satisfy the boundary conditions as Tagawa and Peters [11] found the microjet direction β[Fig. 2(b)] could not be modeled by simply considering the superposition of only two mirror images. We now derive the microjet direction β. The radial speed urat a certain distance Rfrom the sink/source is generally described as ur=Q/(4πR2), where Qis the flow rate. The magnitudes of the velocity vectors at the bubble location ur1,ur2, and ur3originated from respectively the sink/source s1,s2, and s3are ur1=Q 16πh2,ur2=Q 16π(h2+w2),ur3=Q 16πw2.(1) The microjet direction βcan be calculated as β=arctan ux uy=arctan 1−(1 +χ2)−3/2 χ−2+χ(1 +χ2)−3/2,(2) where uxand uyare the magnitudes of both the horizontal and the vertical components of the flow velocity vector as a function of ur1,ur2,ur3and the dimensionless bubble location parameter χ= h/wexpressed as ux=ur3−ur2 w √h2+w2=Q 16πw21−1 (1 +χ2)3/2,(3) uy=ur1+ur2 h √h2+w2=Q 16πw21 χ2+χ (1 +χ2)3/2.(4) Equation (2) indicates that the direction of the microjet only depends on the dimensionless bubble location χ. For extreme conditions, the model predicts limχ→∞ β=90◦and limχ→0β=0◦.We 083601-3 KIYAMA, SHIMAZAKI, GORDILLO, AND TAGAWA FIG. 3. (a) Experimental setup (not to scale). An aluminum block is used as a wall. The edge of the aluminum block reaches the free surface. A high-speed camera films the laser-induced cavitation bubble by using the backlight method. (b) An example image of a bubble with the illustration of the angle β. (c) Parameter space, where h/D=0andw/D=0 respectively indicate a free surface and wall. Circles show the data taken from the series of experiments with diameters Dup to 3.6 mm, where the overall bubble shape was mostly spherical. Data taken from the series of experiments with the larger diameter Dare marked by diamonds. Most of those diamonds are located at smaller h/Dor w/Dthan the circles, where the bubble often leads to its nonspherical growth or a noticeable deformation of the free surface. note that in contrast with the case of two rigid walls, where β=45◦for χ=1[11], in the case at hand the deflection angle is β=45◦for χ≃1.38 by virtue of Eq. (2). III. EXPERIMENTAL SETUP Figure 3(a) shows the experimental setup we used. A pulsed laser beam (Nd:YAG laser Nano S PIV, Litron Lasers Ltd., UK, wavelength 532 nm, pulse duration 6 ns) passes through an objective 083601-4 DIRECTION OF THE MICROJET PRODUCED BY THE … lens [MPLN series, Olympus Co., magnification 20 times, numerical aperture (NA) value 0.25] and illuminates a point inside ultrapurified water (produced by Milli-Q) inside an acrylic container (130 mm ×165 mm ×127 mm). A cavitation bubble emerges and expands. A microjet then forms when the bubble collapses near the surface. For details of the laser and optical components, see Ref. [22]. A high-speed camera (FASTCAM SA-X, Photron Co.) is synchronized with the laser pulse through a delay generator (model 575 Pulse/Delay Generator, BMC). The typical resolutions of recording were set at 20 000–100 000 fps and 0.1–0.2mm/pixel. We controlled the maximum bubble diameter D, and the bubble location (χ=h/w), and measured the direction of the microjet β[see Fig. 3(b)]. Experiments are repeated five times under each condition. We first varied the location of a cavitation bubble (hand w). The bubble diameters are up to D=3.6 mm. The laser light was introduced vertically from the free surface. We observed the intermediate jet [open circles in Fig. 3(c),D=2.98 ±0.46 mm] and the weak jet [solid circles in Fig. 3(c),D=1.39 ±0.98 mm] based on a classification [23]. Our experiments reveal that bubbles collapse preserving the spherical symmetry until the very end, when a jet is issued. We only analyzed the experimental data corresponding to the intermediate-jet regime because the jet is stronger, causes more severe damage, and it is much easier to identify experimentally. The weak jet was observed when the bubble is located far enough from the surfaces, or the bubble diameter is small enough [Fig. 3(c)], as expected. We then performed two series of experiments with bubbles of diameters D⩽4.7 mm, where the laser light was introduced horizontally through the container wall. We first varied both hand wfrom 2 to 10 mm (corresponding χ=h/wranges from 0.3 to 3.5) at larger cavitation diameters D=4.35 ±0.19 mm. We then placed a cavitation bubble at the locations h∼w∼2 mm and h∼ w∼3mm(χ∼0.9 for both conditions), and varied the cavitation diameter Dfrom 2.0 to 4.7 mm. We note that a relatively large diameter D, or the relatively small distance to one surface (i.e., the small hor w), can introduce nonspherical bubble growth as well as a complex shape of the jet (see Sec. IV). In our experiments, nonspherical bubble growth, a noticeable deformation of the free surface, or the onset of cavities at the surfaces could be observed when h/D⩽1.5orw/D⩽ 1.5. Rigorously speaking, our model does not target these phenomena. Thus, the model does not necessarily predict the direction of the jet in such conditions. However, it is interesting to discuss these regimes especially from a practical point of view. We therefore included these experimental data and marked them with diamonds in Fig. 5. IV. RESULTS The typical behaviors of the smaller cavitation bubble near the surfaces are shown in Fig. 4.The microjet caused by the collapse of a bubble located near the wall (χ≃4.27) is directed towards the solid substrate, with β≃81◦. In contrast, the bubble situated near the free surface (χ≃0.23) produces a nearly vertical jet, with β≃1◦. When the bubble is in a corner of the rigid wall and free surface, with χ≃1.5, the deflection angle is β≃45◦. Figure 5summarizes the experimental data and compares the measured jet angle βvs χ=h/w with the values predicted by Eq. (2), finding a good agreement between observations and predictions. The solid line denotes the prediction [Eq. (2)], while the circles and diamonds represent experimental results (Fig. 3). Overall, most of the data collapse into the solid line, which is independent of the bubble diameters tested herein. For a majority of data points, the difference between the measured angle and that estimated by the model fell within the range ±10◦. Our model, which is a function of only the bubble location parameter χ=h/w, predicts the general trend of the microjet direction βthat occurs upon the collapse of a cavitation bubble affected by both the free surface and the wall. Interestingly, the data marked by the diamonds, which is often accompanied by nonspherical bubble growth or a noticeable free-surface deformation, largely lies within the range of ±10◦as well. We further discuss the influence of cavitation diameter D. Figure 6summarizes the cavitation behavior with different h(2 mm ⩽h⩽10 mm at fixed w=8 mm) and w(2 mm ⩽w⩽10 mm 083601-5 KIYAMA, SHIMAZAKI, GORDILLO, AND TAGAWA FIG. 4. High-speed images of collapsing cavitation bubbles. (a) The microjet goes towards the solid wall (β≃81◦)whereχ≃4.27. (b) β≃1◦for χ≃0.23. (c) β≃45◦for χ≃1.51. The solid wall is on the righthand side of the bubble. at fixed h=8 mm) values. The averaged diameter Din this series of experiments was D= 4.35 ±0.19 mm and thus significantly larger than that presented in Fig. 4. The relatively small distance from the boundaries, either h/Dor w/D, leads to nonspherical bubble growth [Figs. 6(a)– 6(d)] as well as vertical jet formation above the free surface [Fig. 6(a)], free-surface deformation [Fig. 6(b)], and cavity inception at the wall surface [Fig. 6(c)]. Interestingly, we did not observe much of a discrepancy between the experimental data and model even in these cases (see Fig. 5). We conjecture that the situations of cavitation bubble dynamics in these cases were similar to that near the single boundary as the distance to one surface, either h/Dor w/D, gets smaller [e.g., Figs. 6(a) and 6(c)]. It is probably the reason why the model describes the data even though the bubble causes the phenomena, which the model does not take into account. FIG. 5. A comparison with the experimental data (circles, diamonds, and stars) and the model [black line, Eq. (2)] as a function of χ. The larger markers (circles and diamonds) show the mean value calculated by the multiple runs disclosed by the small gray markers. The error bars represent the standard deviations estimated from multiple runs if possible. The solid and open stars represent the values estimated from Fig. 1. 083601-6 DIRECTION OF THE MICROJET PRODUCED BY THE … FIG. 6. A comparison of the cavitation behavior at different conditions (a)–(d). The approximate bubble locations were h=2.1 mm, w=8.0 mm, and D=3.9 mm for (a), h=4.1 mm, w=8.0 mm, and D= 4.4 mm for (b), h=7.9 mm, w=2.2 mm, and D=4.5 mm for (c), and h=7.9 mm, w=6.0 mm, and D=4.3 mm for (d). Orange lines on the last frame of the images show the approximate angle of the microjet predicted by the model. The solid wall is on the left-hand side of the bubble. Figure 7shows the behavior of a cavitation bubble at the corner (χ=0.85 ±0.06 for h∼w∼ 2 mm and χ=0.92 ±0.02 for h∼w∼3 mm) with varying diameters (2.0 mm ⩽D⩽4.7 mm). A relatively small bubble at some distances from the boundaries [Fig. 7(a),h/D∼w/D∼1.5] shows a smooth jet that the angle βmatches with the model prediction. The cavitation bubble maintains its shape as mostly spherical, as expected from the results shown in Fig. 6.Neither the deformation of the free surface nor the bubble onset at the boundaries was noticeable. Increasing the bubble diameter D[Fig. 7(b)], and further decreasing the distances hand w [Fig. 7(c)], introduce the nonspherical cavitation bubble as well as the complex jet as discussed in Fig. 6. The deformation of the free surface, and bubble inception at the wall and corner [Figs. 7(b) and 7(c)], were noticeable. Especially in Fig. 7(c), the emerged cavities at the boundaries seemed to merge into the laser-induced cavitation bubble. However, we again note that the difference between the measured and predicted angles remains at a similar level to other results (see Fig. 5). Figure 5also shows the results from Fig. 1using star symbols. The bubble shown in Fig. 1(b), located at χ≃2.9, produces a jet with β≃73◦, a value which agrees well with the prediction in Eq. (2) (see an open star in Fig. 5). To define χfor the data shown in Fig. 1(a), we took the distance between the cavitation bubble center and the surface of the air bubble, λ, as the characteristic length, instead of h, where we find χ=λ/w≃0.52. Since the jet direction in Fig. 1(a) seems to be almost vertical, we consider β≃0◦(see a solid star in Fig. 5). The fair agreement suggests that our model could even be applied to the situation where the jet direction is modified by an air bubble instead of by the free surface. 083601-7 KIYAMA, SHIMAZAKI, GORDILLO, AND TAGAWA FIG. 7. A comparison of the cavitation behavior at the corner, where the diameter Dranges from 2.0 to 4.7 mm (a)–(c). The approximate bubble locations were h/D∼w/D∼1.5 for (a), h/D∼w/D∼0.6 for (b), and h/D∼w/D∼0.5 for (c). Orange lines on the last frame of the images show the approximate angle of the microjet predicted by the model. The solid wall is on the left-hand side of the bubble. V. AN EXTENSION TO VARIOUS αVALUES The approach of Tagawa and Peters [11] allows us to generalize the model for various α [=π/(2n)]. As mentioned in the literature [11], the magnitude of the velocity vector  usat the bubble located at  rs0=leiθs0induced by, for example, a sink, located at  rs=leiθsis expressed as | us|=Q/(4πl2)=Q/{8πl2[1 −cos (θs0−θs)]}, where lis the distance between the sink and the center of the system O. The unit vector for the sink with respect to the bubble is expressed as (eiθs−eiθs0)/{√2[1 −cos (θs0−θs)]}. The general expression of the velocity vector can be obtained by multiplying those relationships as  us=Q π√128l2 eiθs−eiθs0 [1 −cos (θs0−θs)]3/2.(5) The system for n=1 considered here [Fig. 8(a)] corresponds to that shown in Fig. 2. By applying the same strategy, we may place the sinks/sources for n=2 [i.e., α=π/4(=45◦)] as shown in Fig. 8(b), or even higher nvalues [α=π/(2n)]. The vector for the microjet  ujis given by summing 083601-8 DIRECTION OF THE MICROJET PRODUCED BY THE … FIG. 8. The schematic of the system on the polar coordinates. Green markers () and red markers () respectively represent the sources and sinks, where s0represents the cavitation bubble. The corner opening angle αis π/2(=90◦) in (a) and π/4(=45◦) in (b). The sinks/sources in (a) are referred to as s1,s2,ands3 for comparison purposes to Fig. 2. The angle of the image sinks/sources is denoted as a function of both the corner opening angle αand the bubble location θs0. up all  usbelonging to the image sinks/sources, as  uj C=eiθs0−ei(2π−θs0) [1 −cos (2θs0)]3/2 + 2n−1  k=1eiθs0−ei(2kα−θs0) [1 −cos (2θs0−2kα)]3/2−eiθs0−ei(2kα+θs0) [1 −cos (2kα)]3/2(−1)k. (6) The first term represents the influence of the induced flow by a source located at θs=2π−θs0, while the latter terms indicate those by the sinks/sources (n−1 for each) located at θs=2kα±θs0. Note that we assumed that all sinks/sources are located at the same distance lfrom the center of the system O. The natural number krepresents the index from 1 to 2n−1. The prefactor Cin the equation above is expressed as C=Q/(π√128l2). The jet direction is thus obtained as β=arctan −Re(  uj) Im(  uj),(7) without using the prefactor C. Figure 8shows the normalized jet angle β/α as a function of the location of the bubble θs0/α predicted by Eq. (7). We note that, for n=1 [i.e., α=π/2(=90◦)], the prediction matches with Eq. (2) (not shown). The well-defined nondimensional axes reveal the fact that increasing nvalues (i.e., the finer corner angle α) induces a slight shift of the curve as seen in the inset. However, even though we increase nsignificantly, the influence of the first term in Eq. (7) remains in effect and the curve does not cross the point β/α =θs0/α =0.5. It perhaps belongs to the same reason as to why Eq. (2) predicts β=45◦not at χ=1butχ≃1.38, meaning that this is a unique feature of such a system made by two different boundaries; the free surface has a relatively dominant effect on the jet direction βwhen compared to the solid wall. Though the situation for α<π/2isabit difficult to test experimentally and thus speculative, our extension of the previous work [11] to our system might deepen our understanding of the jet formation upon the cavity collapse at the complex geometry. 083601-9