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Fig. 6.28 shows the effects of the load duration, Az, on the pit depth and width when Pp and Ar are kept constant (Py = 1 GPa and Ar = 100 um). The figure clearly shows that both pit depth and pit radius increase nonlinearly with the duration of the applied load, Az, up to about At = 2 us in this case. For the considered values of Py and Ar, for Al 7075, and At > 2 us, the permanent deformation becomes independent of the duration of application of the load. This implies that the dynamic effects are reduced with increasing At and the results become equivalent to quasi-static loading when the loading duration exceeds a certain period of time. Fig. 6.27. Depth and width of the pit generated by the collapse of a bubble under a synthetic load of the form in Eq. (6.15) for different maximum pressure magnitude, Po. The duration and radial extent were maintained constant: At = 0.1 ps and Ar = 250 um. 5 mm thick Al 7075 plate. — Figure 6 28 shows the effects of the load duration, Az, on the pit depth and width when Pp and Ar are kept constant (Py = 1 GPa and Ar = 100 um). The figure clearly shows that both pit depth and pit radius increase nonlinearly with the duration of the applied load, Az, up to about At = 2 us in this case. For the considered values of Py and Ar, for Al 7075, and At > 2 us, the permanent deformation becomes independent of the duration of application of the load. This implies that the dynamic effects are reduced with increasing At and the results become equivalent to quasi-static loading when the loading duration exceeds a certain period of time. Fig. 6.27. Depth and width of the pit generated by the collapse of a bubble under a synthetic load of the form in Eq. (6.15) for different maximum pressure magnitude, Po. The duration and radial extent were maintained constant: At = 0.1 ps and Ar = 250 um. 5 mm thick Al 7075 plate.

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where p is the liquid density, p is the pressure, u, v, and w are the velocity components in the x, y, z directions respectively, g is the acceleration of gravity, and e=e+0.5(u?+v’+w”) is the total energy with e being the internal energy. Eq. (6.8) is applied to solve gas, water, or a mixture of both, with an equation of state, which relates pressure, density, and energy for each phase. Se ews: oo oe pag ' i é ian where p 1s the liquid density, p is the pressure, u, v, and w are the velocity components in the x, y, z directions respectively, g is the acceleration of gravity, and e,=e+0.5(u’+v’+w”) is the total energy with e being the internal energy. Eq. (6.8) is applied to solve gas, water, or a mixture of both, with an equation of state, which relates pressure, density, and energy for each phase. The code applies a high order Godunov-type method [16] to accurately capture the location of discontinuities such as shock waves and contact surfaces. A mixed cell algorithm [8] is applied to enable the code to simulate multi-phase and multi-material flows. GEMINI can be coupled with a structural code to predict the response of nearby structures, and can be linked to 3DYNAFS-BEM*®, as described in Section 6.2.1.c, to model various stages of bubble collapse accurately and efficiently [3-7, 11, 12]. GEMINI has been validated against large scale experiments [8-10, 17, 18]. GEMINI is a compressible Euler equation solver developed by the Naval Surface Warfare Center, Indian Head division, based on a finite difference scheme. The code solves continuity and momentum equations for a compressible inviscid liquid in Cartesian coordinates, which can be written in the following format:

In order to take advantage of the strengths of both the BEM and compressible Euler methods, a hybrid method was developed to address underwater explosion bubble problems and is used here for cavitation erosion bubble dynamics problems. Underwater explosion bubble studies can provide good visualization and measurement since the maximum bubble sizes are in the range of meters rather than millimeters, and time scales are in seconds rather than microseconds. The large spatial and temporal scales enable one to measure the jet speed and the pressures generated at the boundary with high accuracy both in space and time [26-28]. Fig. 6.1. Schematic showing the compressible-incompressible hybrid modeling procedure: link from GEMINI to 3DYNAFS-BEM® during bubble growth and shock propagation and from 3DYNAFS-BEM® to GEMINI following jet impact and shock emission.

Fig. 6.2. Pressure and velocity field evolution during non-spherical bubble collapse near a wall predicted by 3DYNAFS-BEM® for a bubble with an initial radius of 10 um and a gas pressure of 1 atm subjected to a pressure drop to -1 atm for 1 ms and then a steep pressure rise to 50 atm (¢ = 0). The bubble center is initially at a standoff 0.65 mm away from the wall. The maximum bubble radius is Rnax = 1.3 mm at t= 0. The upper left contour is at bubble maximum volume (¢ = 20.6 us). The other three contours are at t= 21.1 ps (upper right), t= 21.6 ps (lower left), and ¢ = 21.8 1s (lower right). Bubble collapse close to a rigid boundary can be highly asymmetrical and results in the formation of a fast re-entrant jet towards the boundary. High pressure loading on the boundary will develop from the impact of the jet on the boundary after it crosses the bubble. The intensity of the impact depends mainly on the relative distance between the bubble and the rigid boundary and on the “strength” of the collapse. Fig. 6.2 illustrates the bubble dynamics near a rigid boundary using an axisymmetric version of 3DYNAFS-BEM®.

Fig. 6.3. Crosscuts showing bubble contour shapes during bubble growth (left) and collapse (right). Results obtained from 3DYNAFS- BEM® simulation. Note that a water layer remains between the bubble and the wall as observed experimentally with large scale bubbles. The bubble has an initial radius of Ro = 10 pm and an initial gas pressure of 2.6 x 10’ Pa and is initially at a standoff distance of 75 um from a rigid wall, while the ambient pressure is 1 atm.

Fig. 6.4. Comparison between experimental measurements and numerical computations of the pressures due to bubble collapse on a rigid wall [7]. The bubble was generated by 10 g of TNT at a depth of 50 m on an instrumented circular plate. The generated bubble had a maximum radius of 0.18 m and the standoff from the plate was 0.14 m. The re-entrant jet speed was as high as 240 m/s. The axisymmetric version of 3DYNAFS-BEM® was used to conduct the numerical simulations and reproduced well the re-entrant jet impact pressure peak at t ~ 15.6 ms and to a lesser extent the bubble ring collapse pressure at about 16.7 ms. (1000 psi = 6.9 MPa).

Fig. 6.5. 3DYNAFS-BEM® solution (left) where the solution was handed back to GEMINI and the first time step from GEMINI (right). The bubble has an initial radius of Ro = 10 ym and an initial gas pressure of 2.6 x 10’ Pa charged at a standoff distance of 75 um from the rigid wall with an ambient pressure of 1 atm. The Link procedure continues the simulations starting from the 3DYNAFS-BEM® solution until a few computation steps before the jet impacts the opposite side of the bubble. The left side of Fig. 6.5 shows the 3DYNAFS-BEM® gridded bubble shape used to proceed with the compressible flow solution. The right side of Fig. 6.5 shows pressure contours from the first time step of GEMINI right after the solution was passed from 3DYNAFS-BEM®.

Fig. 6.6. Pressure contours at different instances of time due to jet impact and ring collapse obtained from GEMINI proceeding from the 3DYNAFS-BEM¢ solution. The bubble had an initial radius of Ro = 10 pm and an initial gas pressure of 2.6 x 107 Pa charged at a standoff distance of 75 um from the rigid wall with an ambient pressure of | atm.

Fig. 6.7. Pressure due to bubble jetting and ring collapse monitored at two different radial locations at the wall, y= 0 and r = 60 um. The bubble had an initial radius of Ro = 10 pm and an initial gas pressure of 2.6 x 107 Pa at a standoff distance of 75 um from the rigid wall while the ambient pressure was | atm. Pressure peaks A are due to the water hammer like pressure generated by the re-entrant jet impact, while pressure peaks B are due to ring bubble collapse. Fig. 6.6 shows the pressure contours and the bubble dynamics from GEMINI at three subsequent times. At t = 3.63 us, the jet impacts the wall generating a localized high pressure impact region on the axis leaving around it a bubble ring (white contours). At t =3.85 ts, the bubble ring has completed its collapse (the white contours reach the minimum) resulting in a ring like high pressure region on the wall around r= 50 um. The high pressure or shock waves travel towards the axis resulting in a high pressure region near the axis, as shown at 3.89 us in the last contours where the bubble ring is rebounding (white contours increasing area again).

It is to be noted that for such small cavitation bubbles, the distinction between the two events is extremely difficult experimentally using conventional transducers. For spark generated bubbles or underwater explosion bubbles however, these differences in the signal can be clearly measured [25-28]. The difference between the measured signals on-axis and off-axis, further illustrated in Fig. 6.8, could explain why discussions are still on-going about whether cavitation damage is from shock waves or from jets as time histories of the pressure appear different depending on the precise location of the gauge. Fig. 6.8. Wall pressures recorded during the collapse and rebound of a spark generated bubble located at 1.8 cm from a vertical wall. The equivalent maximum bubble radius was 2.4 cm and the static pressure at the bubble center was 8,590 Pa for the left figure and 7,900 Pa for the right figure. In the left figure the PCB 101A04 piezoelectric transducer was located at the bubble re-entrant jet impact location. In the right figure the transducer was 0.5 cm off-center. These tests were conducted in the DYNAFLOW spark test facility.

For all values of the driving pressure, a strong re-entrant jet develops and advances to hit the rigid wall with a large impact velocity. As illustrated in Fig. 6.9, the jet velocity increases with the square root of thi driving pressure, Peoiapse. This is consistent with previous studies of bubble collapse dynamics [44]. In the present case, the re-entrant jet velocity can be approximated by: Fig. 6.9. Relationship between the re-entrant jet velocity and the pressure driving the bubble collapse. The red diamond symbols are jet impact velocities computed by the 3DYNAFS-BEM® for different ambient pressures, and the black curve is a linear fit. Maximum bubble radius 2 mm. Distance of bubble center to wall 1.5 mm (normalized standoff distance of 0.75).

Fig. 6.10. Relationship between the peak impact pressure and the jet velocity. The jet velocities are computed by the 3DYNAFS-BEM® a the time of jet impact. The red symbols are from computed maximum peak impact pressures by the hybrid approach (3DYNAFS-BEM® and GEMINI) (such as in Fig. 6.11). Maximum bubble radius 2 mm, distance from bubble center to wall 1.5 mm (normalized standoff distance of 0.75).

Fig. 6.11. Pressures computed by the hybrid approach (3DYNAFS-BEM® and GEMINI) at a nearby rigid wall from the collapse of a bubble under different bubble collapse driving pressures. Initial maximum bubble radius, Rina = 2 mm. Initial standoff distance, X¥ = 1.5 mm. The details of the impact pressure versus time at the center point on the wall are shown in Fig. 6.11 fora set of collapse driving pressures for the same bubble collapse conditions as in the previous two figures. All pressure histories have been shifted to the same starting point to show times from the moment of jet impact at the wall. As previously discussed in Fig. 6.7, all of the signals but the one corresponding to P suapse = 4 atm Clearly exhibit at least two pressure peaks following the impact: the first corresponds to re- entrant jet tip liquid impact at the wall, and the second (and sometimes third) is generated by the bubble ring collapse (and sometimes second collapse after rebound) and the focusing of the resulting shock wave y

Fig. 6.12. Maximum pressures versus radial distance generated at a nearby rigid wall from the center of a bubble collapsing under different driving pressures. Initial maximum bubble radius, Rmax = 2 mm. Initial standoff distance, X= 1.5 mm. As the driving pressure increases, T,,,,,,,., decreases and the spacing between the two peaks also decreases. On the other hand, as the jet speed increases, the resulting jet impact pressure and the bubble ring collapse pressure increase. The above observations provide a scaling of the level of the impulsive pressure generated on the wall at bubble collapse, P,,,,,,,..aS a function of the ambient pressure driving the bubble collapse, P.,,,,,,.. 5 combining Eqs. (6.10) and (6.12):

Fig. 6.13. Bubble shape variations during collapse of a 100 tm radius bubble under a collapse driving pressure of 34 atm at three standoffs _X, initial distance from the bubble center to the wall: X/Rmax = 1.05 (left), X/Rmax = 1.25 (middle), and X/Rinax = 2 (right). Rmax =100 jum is the initial maximum bubble size. The bubble has an initial gas pressure of | atm. Fig. 6.14 shows the pressures due to bubble collapse at the center of the nearby rigid boundary. The left and right figures present computations for a bubble with initial maximum radii of 10 um and 500 pm, respectively. The bubbles are initially spherical and in equilibrium with the surrounding liquid at a pressure of 1 atm. Three standoff distances are considered as in Fig. 6.13: X/Rimax = 1.05, 1.25 and 2.0. Unlike previous cases in Fig. 6.13 where the initial bubble radius was 100 pm, it is here 10 and 500 um. The reason for the choice of two very different radii is to highlight the fact that the amplitude of the pressure generated by the bubble collapse for the same driving condition is practically independent of the initial bubble radius. This is very well highlighted in Fig. 6.14, where, for the three standoff distances, here is practically no difference in the amplitude of the pressures between the 10 um (left) and the 500 um (right) cases. The time scale on the other hand, follows the Trayieign scaling (Eq. (6.13)). The first peaks in Fig. 6.14 (~ 0.18 us and ~ 9 us on the left and right figure respectively) occur at re-entrant jet impact. The timing of the re-entrant jet impact is seen to scale with Rinax and is 50 times larger for the 500 um bubble (~ 9.5 us) than for the 10 um bubble (~ 0.19 us). The same applies for the time delay between he jet impact and the bubble ring collapse (~ 0.2 us vs. ~ 0.004 ps).

Fig. 6.14. Comparisons for Rnax = 10 pm (left) and Rnax = 500 pm (right) of the pressure on the wall at the bubble axis due to re-entrant jet impact and bubble ring collapse for different normalized standoffs: X/Rmax = 1.05 (red), 1.25 (blue), and 2.0 (green). The bubbles have an initial gas pressure of 1 atm and are subjected to an ambient pressure of 34 atm. Notice the very close correspondence of the amplitudes between the two cases indicating very small sensitivity of the pressure amplitude to the bubble initial radius, and a ratio of 50 between the times indicating linear relationship between the bubble radius and the time scale.

Fig. 6.15. Comparison of the time history of the normalized bubble equivalent radius for initial bubble maximum radii of 10, 50, 100, 250 and 500 pm at X/Rmax = 1.05 (blue line) and X/Rnax = 2.00 (red line). All the bubbles have an initial gas pressure of 1 atm and are subjected to an ambient pressure of 34.5 atm. The bubble equivalent radius is normalized by Rmax while the time scale is normalized by the Rayleigh time. the dimensional jet velocity curves versus normalized times are insensitive to the bubble size especially for the smaller standoff distances, and are mostly controlled by the collapse driving pressure and the normalized distance from the wall. Near the wall, asymmetry is predominant and bubble size does not show an important effect. Farther from the wall, surface tension and initial gas pressure, which introduce nonlinear effects not captured by the Rayleigh scaling, have a non-negligible effect on the jet velocity and are responsible for the differences between the curves.

Fig. 6.16. Comparison of the time history of the re-entrant jet velocity for initial maximum bubble radii of 10, 50, 100, 250 and 500 um at X/Rmax = 1.05 (solid lines) and X/Rmax = 2 (dashed lines). All the bubbles have an initial gas pressure of 1 atm and are subjected to a sudden pressure jump to 34 atm driving the bubble collapse.

Fig. 6.17. Time history of the normalized pressure on the wall at the axis (r = 0) for different bubble sizes at two normalized standoff distances: X/Rmax = 1.05 (left) and 1.25 (right). All the bubbles have an initial gas pressure of 1 atm and are subjected to a sudden pressure rise of 34.5 atm driving the bubble collapse. Fig. 6.17 compares the normalized time histories of the wall pressures for X/Rinax = 1.05 and 1.25 for various bubble sizes. The collapse of all pressure loading curves for a given standoff is remarkable. This highlights that, independent of the bubble size, the pressures generated at the wall (see Eq. (6.14)) scale very well with the water hammer pressure due to the re-entrant jet velocity, which depends only on the driving pressure (see Eq. (6.10)). The time also scales very well with the Rayleigh time (see Eq. (6.13)). The normalized peak pressure for X/Rinax = 1.05 is of the order of 0.5pcV,,,, while for X/Rmax = 1.25 it decreases to about 0.25pc/V,,,.

Fig. 6.18. Variations of the pressure at the wall with time and with the radial direction for two standoffs X/Rmax = 1.05 (left) and 2.00 (right). The bubble has an initial radius Rmax = 10 um and is in balance with an ambient pressure of | atm. It is then subjected to a sudden pressure rise to 34 atm.

A finite element mesh for a circular plate with 1 m radius and 5 mm thickness is generated using the axisymmetric setup il ustrated in Fig. 6.19. The size of the plate (i.e. computational domain) was selected to be sufficiently large relative to the micron size bubbles, in order to minimize numerical reflections from the edges of the plate. The mesh elements near the axis of the plate are 5 um by 5 um. This fine element size was maintained up to 1.5 mm radius from the axis and down to 0.6 mm distance from the surface. From there on, the mesh size was gradually increased both in the outward and in the thickness direction. The mesh s retch factor was 10% up to a radius of 10 cm, and then increased to 30% beyond 10 cm. This resulted in 377 elements in the radial direction and 258 elements in the thickness direction. Fig. 6.19. Finite element axisymmetric grid used in DYNA3D to study elasto-plastic material response to loads due to collapsing cavitation bubbles. The cells in the dark region near 7 = 0 are 5 um by 5 um. The mesh size was stretched by a factor of 10% in the magenta region and 30% in the green region. (Note: r and z do not have the same scale.)

Table 6.1. Material properties used for DYNA3D computations. These were obtained from available information online [49].

Fig. 6.20. Stress-strain curves utilized in DYNA3D to model elasto-plastic behavior of the materials studied. These were obtained from information available online [49].

Fig. 6.21. Time history of pressure acting on an Al 7075 plate with 5 mm thickness and resulting pit depth for a collapse driving pressure of 100 atm. Initial maximum bubble radius, Rnax= 2.0 mm. Initial bubble pressure, latm. Initial standoff distance from bubble center, X = 1.5 mm. us the bubble ring collapse loading vanishes and the surface starts returning as much as possible y»wards the initial state. However, since there was significant deviation from the elastic behavior during 1e loading, a permanent deformation results and the center point in the plate remains depressed by about 0 um. One can also note successive oscillations at this center point as the stress/strain waves propagate nd reflect back and forth at the rigidly held bottom and at the plate-water interface. The above dynamics lustrates the importance of the thickness of a plastic layer on the resulting deformations. No element 1ilure or erosion was observed during this computation as the highest strain observed were 0.025, well elow the failure strain limit of 0.11 for Al 7075.

Fig. 6.22. Profile of the permanent deformation (pit depth versus r) on a 5 mm thick Al 7075 plate for different collapse driving pressures. Initial maximum bubble radius, Rac = 2.0 mm, and initial standoff distance, X= 1.5 mm.

Fig. 6.23. Permanent deformation (maximum depth at the center of a 5 mm thick Al 7075 plate) for different collapse driving pressures. Initial maximum bubble radius, Rmax = 2.0 mm. Initial standoff distance, X¥ = 1.5 mm.

Fig. 6.24. Pit radius and pit depth for different initial maximum bubble radius, Rmax, for bubble collapse on a 5 mm thick Al 7075 plate with the same collapse driving pressure of 10 MPa. Normalized initial standoff distance for all cases is 0.75. The pit depth on the other hand, appears to be more strongly dependent on Ra, and increases as Rinax increases. For Rmnax< 500 um the pit depth is below the computational resolution for a collapse driving pressure of 10 MPa and a normalized standoff of 0.75.

Fig. 6.25. Relationship between pit depth, pit radius, and the jet velocity. Permanent material deformation for different collapse driving pressures on a 5 mm thick Al 7075 plate. Initial maximum bubble radius, Rinax = 2.0 mm. Initial standoff distance, X = 1.5 mm.

Fig. 6.26. Relationship between the pit shape ratio and the jet velocity. Permanent material deformation of a 5 mm thick Al 7075 plate for different collapse driving pressures. Initial maximum bubble radius, Rmax = 2.0 mm. Initial standoff distance, X= 1.5 mm.

Fig. 6.28. Depth and width of the pit generated by the collapse of a bubble under a synthetic load of the form in Eq. (6.15) for different pressure durations, At. The pressure amplitude and lateral extent were maintained constant: Po =1 GPa and Ar =100 pm. 5 mm thick Al 7075 plate.

Fig. 6.29. Depth and width of the pit generated by the collapse of a bubble under a synthetic load of the form in Eq. (6.15) for different pressure lateral extent, Ar. The pressure amplitude and duration were maintained constant: Po =1 GPa and At = 0. 1s. Computations on a 5 mm thick Al 7075 plate.

Fig. 6.30. Normal stress component on the first plate element below the interface plate-water at the bubble axis. The stresses are the material response to an impulsive loading due to a bubble collapse over the plate. The bubble had an initial radius of 400 um and was located above the plate at a distance of 400 um to the bubble center. The collapsing bubble is driven by pressure rise from 0.1 MPa to 5 MPa. Fig. 6.30 and Fig. 6.31 present t values of Peottapse: 5 MPa and 10 he norma MPa. Bo stresses o,,, on the material during h figures show the influence of fluid he bubble collapse for two -structure interaction on the results. When the response of the material, its deformation, and its feedback to the flow field are included the normal stresses are noticeab four materials feel practically the same no rigid body (Fig. 6.30). For the higher exci as the material gets softer, with Al 1100-H19 (Fig. 6.31). This e y reduced as compared to the rigid wall case. rmal stresses, which are about 20% he peak s resses almost 50% lower than those ffect has been observed experimentally in [50] For Peottapse = 5 MPa, all ower than those felt by the ation, Peottapse = 10 MPa, normal stresses show further decrease felt by the rigid body for through water jet impacting on an instrumented deformable plate; the measured pressures were significantly lower than when impacting a rigid plate.

Fig. 6.31. Normal stress component on the first plate element below the interface plate-water at the bubble axis. The bubble had an initial radius of 400 um and was located above the plate at a distance of 400 yum to the bubble center. The collapsing bubble is driven by pressure rise from 0.1 MPa to 10 MPa.

Fig. 6.32. Radial stress component on the first element below the free surface of the plate at the bubble axis. The bubble had an initial radius of 400 um and was located above the plate at a distance of 400 um to the bubble center. The collapsing bubble is driven by pressure rise from 0.1MPa to 1OMPa.

Fig. 6.33. Equivalent stress on the first element below the free surface of the plate at the bubble axis. The bubble had an initial radius of 400 um and was located above the plate at a distance of 400 um to the bubble center. The collapsing bubble is driven by pressure rise from 0.1 MPa to 10 MPa. compared to the material yield stress. Fig. 6.33 shows the equivalent stresses computes in the first element at the surface of each material for the same computations as in Fig. 6.31 and 6.32. One should note the sharp change in slope when the stresses approach the yield stress, partially explaining the stress drop. From that point on, the material follows the plastic portion of the stress-strain curve, which has a much milder slope, thus resulting in a significant change of the slopes of stress versus time and a significant reduction in the value of the maximum equivalent stress relative to the imposed impulsive pressure.

Fig. 6.34 shows the vertical displacement versus time for the four materials. It is this quasi-final deformation that is remaining in the pits after exposure of the materials to the impulsive loads due to the bubble collapse. The much larger maximum deformation in each of the curves of Fig. 6.34 is not recorded by the material and is unknown to the observer of the pit characteristics. The lengthy computations in Fig. 6.34 were stopped at 3 us, a relatively very long time after the loads due to the bubble collapse have died out (see Fig. 6.31 and Fig. 6.32). However, computations for a longer duration are needed until no change in the depth is observed; nevertheless the conclusions drawn here will not change significantly but towards slightly smaller permanent deformations. Fig. 6.34. Normal displacement of a node located on the surface of the plate at the axis, indicating the evolution of pit depth. The bubble had an initial radius of 400 um and was located above the plate at a distance of 400 um to the bubble center. The collapsing bubble is driven by a pressure rise from 0.1 MPa to 10 MPa.

Fig. 6.35. Idealized plastic zone diagram due to spherical indentation If we use an analysis similar to that done in [51, 52], we can deduce the relationship between the maximum strain, ¢,,,., and the mean strain, ¢ Fig. 6.35 shows an idealized sketch of the plastic zone max ? mean * of the cavitation pit, which is similar to Fig. 3.18.

Table 6.2. Impulsive pressure loads, F,,,,...., imposed in the computation, resulting maximum equivalent stresses in the material and stress values deduced from the computed pits resulting from permanent deformation and Eq. (6.21). Pit diameter was computed using a cutoff depth of 1 um. Bold values correspond to maximum equivalent stresses exceeding yield stress.