Discussion of the vortex results

The simulations in the previous section have been re-made obeying thebest practice guidelines for solution convergence (Crit. 3.2 mentioned in Sec. 3.1.4). Special care was taken to reduce the amount of cells while putting the outer boundary from 80R_max to a distance of 100R_max. The time interval of the reduction of Rn, which was coinciding with the time of minimum bubble volume in the previous section (90µs to105µs), is now put well before the time of minimum volume, namely to60µs to75µs. The basis for the mesh is the axisymmetric, polar Mesh 3.1.2.a that is tested in Sec. B.1.3 and was used in the previous section, too. In the previ-ous section the resolution was 2 µm, which is now reduced to 1 µm. The resolution is taken as the name tag for the two kinds of respective parameter sets, which compare as given in Tab. 4.3.

Xi Xii X XF grading factorgf refinement at wall initial data

2 µm calc. 80µm 1.2·√

Table 4.3: Setup and initial data of the two approaches to calculate the vortices of a collapsing bubble close to a solid boundary.

The termrefinement at wallrefers to whether the cell size was halfed in a rectangular region at the wall (x ∈ [0; 0.7875D_init], y ∈ [−D_init; −0.875D_init], with y = −D_init at the solid bound-ary). This was done in order to better resolve wall shear flows. It turned out, however, that parasitic currents occur, when the bubble interface is in that region. This is the case at times of maximum expansion for D^∗ . 0.4, as well as during rebound in all the studied values of D^∗. Therefore, in this dataset, the data point D^∗ = 0.2is omitted due to spurious currents. For

D^∗ = 0.4and 0.42 the simulations were repeated without the refinement at the wall.

Cylinderandsphereininitial datain Tab. 4.3 refer to the the shape and algorithm the bubble initial data is set up. The 2 µm-approach was set up with algorithm 1 of Tab. 3.1, while the 1 µm approach makes use of algorithm 2.

The time stepping procedure is the same as in the previous Sec. 4.1.1.

It is left to evaluate the influence of the choice of time interval for the reduction ofR_n, before going on with the 1 µm calculations. A small convergence study with the same simulation as in the previous Sec. 4.1.1, including the boundary distance at80R_max, but in unbounded liquid was carried out with bothR_n-reduction intervals (60µs to75µs and90µs to105µs). The resulting values for2Tcare given in Fig. 4.10 compared to the reference solutions of Fig. 3.8.

90.2

cell size in initial bubble area [µm]

converged spherical calc.

axisymm. polar mesh spherical, adiabatic pressure adjusting axisymm. cyl. bubble approach, R_n reduce at 60µs axisymm. cyl. bubble approach, R_n reduce at 90µs

Fig. 4.10: Same diagram as Fig. 4.2, but including the distinguishing of theR_n-reduction interval times in the cylinder bubble approach.

It can be seen that more gas content inside the bubble prolongs the collapse time a bit and eventually pushes it more to the convergence value. Since the axisymmetric calculations are too time consuming for cell sizes lower than 1 µm, no statement about solution convergence can be done here, but it can be deduced that the time interval of R_n-reduction does not play a major role.

Figures 4.11 to 4.14 show the results for the 1 µm calculation. Some more data points were added to the transition region from free vortex to wall vortex. Figure 4.15 then shows the vortex direction data points plotted with the data of Fig. 4.1. A qualitative comparison of the main differences can be summarized like the following:

stationary vortex vortexpureness free vortex appar-ent speed

2 µm, Sec. 4.1.1 atD^∗ = 0.4 mainly one major stays in sight 1 µm, Sec. 4.1.2 atD^∗ = 0.42 sometimes twofold out of sight

Table 4.4: Summarizing qualitatively the differences between the 2 µm and 1 µm calculations.

Without wall refinement:

Without wall refinement:

From here including wall refinement:

D^∗ = 0.8: 1000 µs n.a.

Fig. 4.11:Free and stationary vorticesgenerated by theD^∗ ∈[0.4, 0.42, 0.6, 0.8]bubbles, simulated with the 1 µm-approach for two very late time instants: 550 µs (left) and 1 ms (right). In this approach, theStationary vortexis more generated atD^∗ = 0.42 than atD^∗ = 0.4. TheD^∗ = 0.8simulation crashed before reaching the limiting time.

Transition from free vortex to wall vortex, without wall refinement:

Fig. 4.12:Free vortices and the transition to wall vortexgenerated by the D^∗ ∈[1.0, 1.2, 1.29, 1.30]bubbles.

Fig. 4.13:Wall vorticesgenerated by theD^∗ ∈[1.32, 1.33]bubbles and atransition vortexat D^∗ = 1.31.

From here again including wall refinement:

Fig. 4.14:Wall vorticesgenerated by theD^∗ ∈[1.4, 1.6, 1.8]bubbles.

The evaluation of the vortex direction is not as clear with the 1 µm calculation as with the 2 µm calculation. Therefore, a floating point criterion is introduced:

v_d=











1 clear free vortex

0.5 stationary vortex till 1000 µs 0 clear wall vortex

else gradual variation between the three

(4.4)

The cases where it is not fully clear are the bubbles atD^∗ ∈[0.4, 1.29, 1.30]. Their values have been decided to be:

v_d(D^∗ = 0.4) = 0.75, v_d(D^∗ = 1.29) = 0.4, v_d(D^∗ = 1.3) = 0.1 (4.5)

At this point it is also discussable how to convert D^∗ to γd. As stated in the introduction of Sec. 4.1, the authors determined the distance dof the bubble from the solid boundary with the first available recorded image from the camera and the maximum radius is taken from the maximum bubblevolume. From the numerical calculations the maximum volume is determined precisely and the distance from the solid boundary is taken either at 4 µs or at 10 µs to stress the influence of the time instant here. The values are fit with the function

γ_d =a(D^∗)^b +c. (4.6)

The results are plotted in Fig. 4.15. It is seen that the two different conversion rules mainly influence the data points where D^∗ ≤ 1. However, the main result that the vortex direction changes with D^∗ ≈ 1.29 remains untouched. Furthermore the following conclusions can be consolidated:

1. The vortex phenomenon is well captured by the solver, thus long-term dynamics can be resolved as well.

2. The vortex phenomenon does not necessitate a full 3D calculation, hence the main, global aspect of the dynamics of the phenomenon is axisymmetric.

0

Fig. 4.15: Left:Non-linearfit for convertingD^∗ →γ_dwith rules γ_d=d(t= 4µs)/R_max(t=t(V_max))(top) and

γ_d=d(t= 10µs)/R_max(t=t(V_max))(bottom). Data for vortex experiments from (Reuter et al. 2017a). Depending on theD^∗(γ_d)-rule, the data points for low normalized distances shift inγ_d-direction.