Cavitation Onset Caused by Acceleration

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Direct PNAS a is article This interest. of conflict S.L.T., no D.J.D., declare Y.T., authors A.K., The Z.P., and data; paper. analyzed the wrote T.T.T. D.J.D. and and R.H., A.K., Z.P., tools; analytic new uhrcnrbtos ..adTTT eindrsac;ZP,DJD,adT.T.T. contributed and D.J.D., Z.P., research; T.T.T. designed and Y.T. contributions: Author [email protected]. owo orsodnemyb drse.Eal aaaoc.uta.por [email protected] Email: addressed. be may correspondence whom work. To this to equally contributed D.J.D. and A.K., Z.P., hehl twihcvtto ilocr(Ca occur will cavitation which the at predicts validation, threshold number cavitation for alternative the conducted that a were confirming num- than experiments cavitation rather traditional Systematic the acceleration cavi- ber. by sudden prescribed the as a predict velocity by large to the caused suitable of onset more proposed tation derivation The is threshold. alternative number the an dimensionless validate propose and number we cavitation paper this In Significance opeitadsmaiecvtto ne ylreaccelera- large by onset cavitation summarize and predict To experiments similar conducted have researchers past, the In ∼ O c,1 (10 ct .Thomson L. 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Fig. 1. Two cases of cavitation onset introduced by large accelerations in low-speed flows. A bottle filled with water accelerated by the impact of a mallet on the top (A). A test tube filled with silicone oil accelerated by an impact with the ground (B). Both image sets correspond to the impact (first frame), tiny bubble appearance (second frame), bubble expansion (third frame), bubble collapse and cracking (fourth frame), and crack propagation/failure (fifth frame). Although the time between each event is different, the overall behavior is very similar (Movies S1–S4). Relative timing of bubble collapse and fracture incidence suggests that implosion-induced waves are likely responsible for fracture initiation, although further investigation into fracture mechanisms in the case studies presented here would be necessary to confirm this observation (Fig. S1). Relationships between cavitation and structural damage are well-documented elsewhere in biological and man-made systems (27–31).

∂v 1 = − ∇p. [2] To gain physical insight into the interpretation of the quiescent ∂t ρ cavitation number, gravitational acceleration can be introduced Integrating Eq. 2 along the centerline of the liquid from the free and Eq. 4 can be reformulated as surface to the bottom of the column (assuming the depth of the (p − p )/ρgh liquid is h), denoting the magnitude of the vertical component of Ca = r v . [5] ∂v/∂t as a, and solving for the pressure difference in the liquid a/g column yields Gravitational acceleration is not an essential term in the cav- pr − pb = ρah, [3] itation number. However, it is included here to enable a formulation with explicit physical meaning. The numerator is the where pr is the reference pressure at the free surface and pb is the pressure at the bottom of the column. Cavitation is likely to maximum nondimensionalized force that the pressure difference can provide (similar to Eq. 1) and the denominator is occur when pb < pv . Thus, we can establish the nondimensionalized inertial force the liquid experiences p − p Ca = r v [4] under acceleration (in contrast to the ﬂuid momentum of Eq. 1). ρah Thus, once the inertial forces exceed the maximum pressure as an indicator of cavitation onset when the ﬂow undergoes a difference (i.e., Ca < 1), cavitation is likely. However, when violent acceleration. We refer to this expression as the quiescent Ca > 1 the pressure is large enough to balance the vacuum intro- cavitation number. duced by acceleration. Hence, cavitation is not likely.

2 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1702502114 Pan et al. Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 h tfns ftecnanradannudmdu a loivsiae ssonin on shown based as green) investigated in also (shaded was none medium and nonfluid red) a and in container (shaded the formation of cavity stiffness between the separation theoretical represent Lines detection. (p pressures 3. Fig. 2. Fig. .Tesaldfeec rvldbetween traveled difference small The (B). ground in glass the shown a with uses evidence experiment collision (experimental TUAT by acceleration The relative imparted mallet. the is rubber a emphasize impact by to the imparted lines where is dashed bottom, impact as rounded the where a (A), with facilities tube USU/BYU test at done Experiment (g). acceleration gravitational a tal. et Pan in region in (red-squared region varied (blue-squared was varied pressure was reference type the fluid where region the in points iust aiaeEq. validate tech- to measurement and niques setups fol- different the with experiments conducted lowing independently groups separate in Two found be Methods. can experiments the about details More Setup Experimental ,rfrnepesr (p pressure reference (h), height column liquid mean column: fluid the of diagram body Free hs iga o h aiainostb ceeaini the in acceleration by onset cavitation the for diagram Phase r smre ntelgn.Oe akr eoecvtto eeto n le akr eoeasneo cavitation of absence denote markers filled and detection cavitation denote markers Open legend. the in marked as (h) depths fluid and ), 4 or h AB 5. B A g t accelerometer = No CavitationNo t 0 A). mallet p p b r aeil and Materials t = Cavitation t 1 p p {(p b r r − ee ihamxmmmaual ceeaino 1,000 of acceleration measurable maximum accelero- an a and fitted pressure was with internal tube control meter to cavitation tap The pressure 20). a built with (1, tube acrylic cavitation transparent cylindrical from a used (USU/BYU) University p v h ru rmUa tt nvriyadBihmYoung Brigham and University State Utah from group The )/ρgh p p p p p p p Ca = r r r r r r r h =101.3kPa, .Coeu iw(C view Close-up A). = 19.0kPa, = 85.7kPa, = 86.9kPa, = 101.3kPa, = 101.3kPa, = 101.3kPa, C is S3 Figs. 1 i.S2 Fig. , a/g g t = } t and 0 and p p D), ( diameters container types, fluid various for (A) plane D D D D D D D b r = 55.0 mm, Water, USU/BYU Water, =55.0mm, USU/BYU Water, =55.0mm, USU/BYU Water, =55.0mm, TUAT Silicone =27.2mm, oil, TUAT Silicone =14.2mm, oil, TUAT Silicone =8.0mm, oil, = 14.2 mm, Ethanol, TUAT =14.2mm, Ethanol, TUAT oi S4). Movie S4 rigid floor r and ,pesr ttebto ftecnanr(p container the of bottom the at pressure ), fclasddt onsi h einweethe where region the in points data collapsed of ) fcnetae data concentrated of (B) view Close-up S2. Table t = t 1 p p b r NSEryEdition Early PNAS Ca = nEq. in 1 t 0 and Changing 5. t 1 sshown is | b ,and ), f5 of 3 g

ENGINEERING (Figs. 2A and 1A). The experiment was performed with inter- C (velocity) with an inertial one in Ca (acceleration) in cases nal reference pressures (pr ) of 85.7, and 19.0 kPa. A high-speed of acceleration-induced cavitation. Multiple sets of independent camera was used to image the cavitation tube to detect the pres- experiments were conducted by two separate groups to test the ence of cavitation bubbles. The cavitation tube was filled with theory. Results show that cavitation occurs for Ca < 1 (Eq. 3), distilled water to various depths (h) and the tube was acceler- consistent with the theoretical prediction, establishing that the ated by striking the top with a rubber mallet. alternative cavitation number Ca is a reasonable criterion for the The group from Tokyo University of Agriculture and Tech- onset of acceleration-induced cavitation. The alternative cavita- nology (TUAT) used glass test tubes filled with silicone oil or tion number can potentially be used as a criterion for brain injury ethanol of varying depth (h) impacting vertically on a flat rigid caused by impact-induced cavitation (5, 6), prediction of water surface after free fall (Figs. 1B and 2B). The acceleration and hammer (4), and potentially applied to the development of safety appearance of cavitation bubbles was estimated with calibrated devices (e.g., helmet design). high-speed video. Materials and Methods Results and Discussion USU/BYU Experiments. An acrylic tube (55.0 mm inner diameter) is partially filled (h = 1−200 mm) with distilled water (ρ = 1.0 × 103 kg/m3, Fig. 3 reports the experimental results from the two groups; 3 oranges and reds are from the USU/BYU group and blues and pv = 2.3 × 10 Pa, and ν = 1 cSt). Acceleration is introduced by striking the greens are from the TUAT group. The vertical axis and the hori- top of the tube with a mallet. The pressure in the tube is controlled by a vacuum. An accelerometer is connected to the bottom of the tube. The tube zontal axis show the numerator and the denominator of Eq. 5, is illuminated by a backlight and images are recorded by a high-speed cam- respectively. Open markers represent cavitation detection and era [Photron APX, SA-3, or Phantom V-1610, 3,000 to 100,000 frames per s closed markers represent no cavitation. The black solid line is (fps) and 0.1 mm per pixel]. Note that a very short paper focusing primarily plotted according to Ca = 1. Most of the open markers are dis- on the artistic nature of a similar set of photographs rather than the scien- tributed in the region of Ca < 1 (shaded green in Fig. 3). How- tific findings was recently published (1) and an early derivation of Ca was ever, closed markers dominate the region where Ca > 1 (shaded given in ref. 20. orange in Fig. 3). These results indicate that the proposed cavitation number explains the trend for a wide range of accelerations TUAT Experiments. A glass test tube (inner diameter 8.0–27.2 mm, tube thickness ≈1 mm) is partially filled (h = 5−120 mm) with silicone oil and water column depths. 3 3 2 (ρ = 1.0 × 10 kg/m , pv = 7.0 × 10 Pa, and ν = 10 cSt) or ethanol The influence of the reference pressure pr is shown in Fig. 2 3 3 (ρ = 8.0 × 10 kg/m and pv = 6.0 × 10 Pa, and ν = 1 cSt). The tube is held 3B and the influence of diameter and fluid type in Fig. 3C as above the floor (22–135 mm) by an electromagnet until it is released. The zoomed-in regions of Fig. 3A. In contrast to the more difficult to tube is illuminated by a backlight and images are recorded by a high-speed apply or ambiguous result of the conventional cavitation number camera (Photron SA-X, up to 90,000 fps and 0.1 mm per pixel). The basic C 1 and the values proposed by numerical simulation Ca 1 concept of the experimental setup is similar to previous research (3, 24). (23), our data suggest that the onset of cavitation is located in a Acceleration, a, is determined by the high-speed camera images, measuring region near Ca < 1, regardless of the reference pressure, inner the impact speed (sum of the downward speed of the tube just before the diameter of the tube, liquid column height, and fluid type. impact and the upward speed just after the impact) and dividing the impact speed by the duration of the collision of the tube bottom with the floor (32). Summary Note that results appearing in this paper mirror our previous report (3) that focused on the motion of the gas–liquid interface rather than the onset of In this paper we introduced an alternative formulation of the cavitation. cavitation number that captures the cavitation onset caused by acceleration. In practice, the conventional cavitation number C ACKNOWLEDGMENTS. A.K. and Y.T. thank Y. Noguchi, C. Kurihara, 1 K. Hirose, Y. Watanabe, M. Maeshima, Y. Mukai, H. Kudo, K. Hayasaka, and (Eq. ) can inaccurately predict that no cavitation will occur R. Yukisada for their help during experiments. This work was supported when cavitation may, in fact, be occurring in suddenly acceler- by Japan Society for the Promotion of Science KAKENHI Grants 26709007, ated flows. For example, the impact of a fluid-filled bottle with 16J08521, and 17H01246. This paper was partially written while A.K. was the ground can cause cavitation (Fig. 1) although the velocities visiting the Department of Mechanical and Aerospace Engineering, Utah C State University. A.K. acknowledges financial support from Tokyo Univer- are too small to predict using . Thus, we propose an alterna- sity of Agriculture and Technology. T.T.T. pursued the study first at Brigham tive cavitation number Ca (Eq. 4) by applying the equation of Young University and finalized these ideas and helped write the paper while motion, resulting in a replacement of the momentum term in at Utah State University.

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ENGINEERING