The necessity to resolve a very steep pressure gradient att= 0, when an EDC bubble is set up, makes grid convergence studies for axisymmetric calculations an everlasting task. Therefore, in order to show the convergence of the method, the grid study was performed for a bubble in unbounded liquid, which can also be done in spherical symmetry, thus reducing computational cost even at very high resolutions. All investigations were done with adapting the initial bubble energy according to Eq. 2.22: After discretization, the true bubble volume V0 is determined and the values of p0 andRn are adapted accordingly to have the same energy as a bubble of Rinit= 20µm, Rn= 184.1µm. All investigations are carried out with a maximum flow Courant number of 0.2, a maximum Courant number for the interface of 0.08 and maximum acoustic Courant numberof 8 for the whole time domain. The maximum acoustic Courant number is reduced to 1 if the bubble equivalent radius is below 30 % of Rn. The upper size of the time step is∆t= 5·10−8s and the time step size att= 0is set to∆t0 = 10−11s.
Figure B.1 shows the first attempt with a boundary distance of XF = 80Rmax. The spatial resolution is varied in the initial bubble domain for both axisymmetric and spherically symmet-ric calculations with Mesh 3.1.2.a and Mesh 3.1.2.b (version A), respectively. For t = 0 the bubble interface (int.f.) was either set up with a thickness of 3 cells or sharp with approximately zero thickness in theory. Both meshes use the same dimensional parameters. Thecore sizeC of both meshes was set to 80 µm and X is set to1.2Rmax. It is seen that the bubble shows the same collapse time for the same method in each of the applied mesh symmetries. But for very high resolutions a random behaviour seems to occur. The shockwave of the bubble generation travels to the boundary, located at a distance of80Rmax= 80·495µm= 39.6mm, and a reflected part travels back to the bubble within approximately 54.6 µs, assuming a uniform sound speed of 1450 m/s. During this time, the bubble reaches maximum expansion and a state of low pressure
and low interface velocity. In this state it is supposed that it is influencable by impinging waves.
The higher the resolution, the sharper the shockwave is resolved, thus leading to higher influ-ence.
cell size in initial bubble area [µm]
spherical + sharp int.f.
axisymm + sharp int.f.
axisymm + 3 cells int.f.
spherical + 3 cells int.f.
87
cell size in initial bubble area [µm]
spherical + sharp int.f.
axisymm + sharp int.f.
axisymm + 3 cells int.f.
spherical + 3 cells int.f.
Fig. B.1: Time from bubble generation till first minimum volume for different resolutions for an unbounded bubble in both spherical and axial symmetry with a outer boundary distance of80Rmax. For very high resolutions suddenly a non-converging, random behaviour appears, irrespective whether the bubble interface is smeared over 3 cells att = 0or not. The lower curves are the ones with smeared interface. The upper curves use a sharp interface in the beginning. Left and right diagrams are the same but linear or logarithmicx-axis.
In order to show that most probably no other parameter than the boundary distance is capa-ble of enhancing convergence, the global mass correction (GMC) and also zero surface tension (sigma=0) were tested in Fig. B.2, thereby validating that the local mass correction (LMC) pro-duces the same collapse times as the global mass correction. The GMC was tested and validated in Koch et al. (2016).
cell size in initial bubble area [µm]
spherical + sharp int.f.
axisymm + sharp int.f.
spherical + 3 cells int.f.
spherical + GMC + 3 cells int.f.
spherical + GMC + 3 cells int.f. + sigma=0
88
cell size in initial bubble area [µm]
spherical + sharp int.f.
axisymm + sharp int.f.
spherical + 3 cells int.f.
spherical + GMC + 3 cells int.f.
spherical + GMC + 3 cells int.f. + sigma=0
Fig. B.2: Same investigation as Fig. B.1 but including investigations where theglobal mass correction(GMC) was applied to check for better convergence. This, however is not the case. Also, the surface tensionσwas supposed to have an influence, which turned out negligible.
88
cell size in initial bubble area [µm]
spherical + sharp int.f.
spherical + 3 cells int.f.
spherical + 3cells int.f. + BC moved to 100Rmax spherical + mesh 2 spherical + + BC moved to 100Rmax
88
cell size in initial bubble area [µm]
spherical + sharp int.f.
spherical + 3 cells int.f.
spherical + 3cells int.f. + BC moved to 100Rmax spherical + mesh 2 spherical + + BC moved to 100Rmax
Fig. B.3: Same investigation as Fig. B.1 finally having found convergence for setting the outer boundary from80Rmaxto100Rmaxapart from the bubble. Mesh 2refers to
Mesh 3.1.2.b (version B) with parameters: X= 20µm=Rinit, XF=Rmax, XFF= 20Rmax, XFFF= 200Rmax, θ = 1◦.
Finally in Fig. B.3, convergence was found by testing the spherical Mesh 3.1.2.b version B (mesh 2) with a boundary distance of XFFF = 200Rmax. This led to testing Mesh 3.1.2.b version A with a boundary distance of XF= 100Rmax, improving convergence for almost every resolution for the case of an initially sharp interface (red curve). Indeed, it can be seen in comparison with Fig. B.1, that even with a boundary distance of only80Rmaxthe axisymmetric case with a sharp interface att = 0is congruent with the spherical solution that converges. The congruence is best in the resolution interval from 1 µm to 3 µm.
The curves of the equivalent Radius over time for the converged and non-converged series are given in Fig. B.4. It is seen that the rebound radius is underestimated for the resolutions 3 µm and 2 µm. So it can be expected that the solution converges fully from 1.35 µm onward.
The axisymmetric calculation (green curve in Fig. B.1) is in congruence with the converged spherical calculation even though the boundary distance was80Rmax. This is not a contradiction, because the resolutions where non-convergence is observed are higher than the ones that could be tested within reasonable time.
One has to note, too, that the above grid convergence study aims at extremely high precision.
Even the worst non-converging solution of Fig. B.1 has roughly a maximum deviation in twice the collapse time of 4 µs from the value of 92 µs, which is considered the true one. This results in a maximum relative errorr,maxof
r,max= 4µs
92µs = 4.3 %.
This is still below 5.5 % which was the precision considered sufficient in Koch et al. (2016) for similar studies.
The two reference solutions are summarized in Fig. B.5.
0
Fig. B.4: Equivalent radius over time for the converged series (top) of Fig. B.3 and non-converged series (bottom). Both with sharp interface.
91.3 91.4 91.5 91.6 91.7 91.8 91.9 92
0.1 1
2 Tc [µs]
cell size in initial bubble area [µm]
reference spherical. BC at 100Rmax reference axisymm.: polar mesh with BC at 80Rmax
Fig. B.5: Time from bubble generation till first minimum volume for different resolutions for the two reference solutions for the unbounded bubble. Axisymmetric calculation (green) of Fig. B.1 and converged spherical calculation (red) of Fig. B.3. This Figure is the same as Fig. 3.8