3.1. Vorticity Field
A comparison of the vorticity field
(Equation (
19)) based on both boundary conditions is presented in
Figure 4. Note that only the plane-normal component is unequal to zero. In the interior phase, the general shape of the solution is the same in both cases. Quantitative differences are determined by
, i.e., the radius at which the near-field boundary condition is imposed. The vorticity jump at the interface is given by Equation (
22), and the phase in which the maximum vorticity magnitude appears, is eventually determined by the dynamic viscosity ratio
. In the bubble case shown here, the vorticity peak can be found in the gaseous phase.
In the exterior phase, the solution is not even qualitatively the same. In both cases and in both phases, the vorticity is zero for
as well as
and it peaks at
. However, the vorticity steadily decays to zero for
with far-field boundary conditions. According to Equation (
21), the zero-vorticity iso-contour in the exterior phase (black line in
Figure 4) is given by
, which is smaller than
for near-field boundary conditions. Note that
for far-field boundary conditions in contrast. It might be questioned whether the imposed near-field boundary condition (Equation (
25)) is an adequate choice. The idea of a constant parallel flow for
implies vanishing velocity gradients, which is not satisfied at
. The differences in the vorticity field are also reflected in the topology classification discussed subsequently.
3.2. Invariants and Flow Topologies
As introduced in
Section 2.1, the result of the flow topology classification with far-field boundary conditions is shown in
Figure 5,
Figure 6 and
Figure 7 for different dynamic viscosity ratios. In addition, the corresponding invariant fields (Equation (
4)) are plotted since these also contain information on the strength of the underlying fields. First, it can be observed that the focal topologies (S1 and S4), according to their vortical nature, prevail in the high vorticity regions where
. Secondly, the topology behavior at the phase interface is continuous in the neutral case (
), whereas a discontinuous behavior can be observed for the bubble case (
) and the droplet case (
). Accordingly, the second invariant
Q and the third invariant
R exhibit a continuous or discontinuous behavior depending on the case.
Following the flow pathlines (which coincide with the streamlines for steady flows, as depicted in
Figure 2), the sequence of the adopted flow topologies (nodal-to-focal-to-nodal transition) is generally the same in all cases. Regarding the distribution of nodal topologies, mainly S2 can be found in the upstream and S3 in the downstream part of the domain, respectively. However, a close inspection of the topology fields reveals a narrow band of intermediate nodal topologies (switching between S2 and S3) in both phases. It may be questioned whether this behavior is a consequence of the particular stream function ansatz (Equation (
6)) or whether it is strictly necessary due to physical reasons. The existence of theses intermediate topologies cannot be explained by numerical inaccuracies. In fact, it can be shown analytically that the S2–S3 transition in the interior phase is specified by the constraint
:
which is satisfied at
and
, respectively. In contrast, the S1–S2 transition is specified by the constraint
:
which is satisfied at
For symmetry reasons, the S3–S4 transition occurs at
. Remarkably, the topology borders in the interior phase are straight lines, which depend on neither the radial coordinate
r nor the dynamic viscosity ratio
. The constant topology volume fractions can thus be calculated to be
for nodal topologies (S2 + S3), and consequently 80% for focal topologies (S1 + S4). Exactly the same values were theoretically predicted by Hasslberger et al. [
11], but with a very different approach.
In the exterior phase, the S2–S3 transition is specified by the constraint
:
which is satisfied at
and
, respectively. The S1–S2 transition is specified by the constraint
:
which is satisfied at
Again, for symmetry reasons, the S3–S4 transition occurs at
. Equations (
34) and (
36) reveal that the topology borders in the exterior phase are generally curved and furthermore depend on the viscosity ratio. To determine the extent of the focal region in the exterior phase
(a finite value only for near-field boundary conditions),
, representing the horizontal line through the origin, can be inserted in Equation (
36), which yields a fifth-order equation that can be solved numerically:
Independent of the phase, the transition between focal topologies S1 and S4 occurs at .
The topology and invariant fields with near-field boundary conditions are shown in
Figure 8,
Figure 9 and
Figure 10 for
. To facilitate a direct comparison with far-field boundary conditions, the same range of dynamic viscosity ratios is investigated as in
Figure 5,
Figure 6 and
Figure 7. In addition, in this case, the topology and invariant fields are continuous at the phase interface only if
. The topology distribution in the interior phase is identical for both types of boundary condition. The main difference concerns the shape of the focal region in the exterior phase. The focal region extends to the end of the domain (and probably to infinity) with far-field boundary conditions, whereas it is bounded to a tire-like zone around the bubble or droplet with near-field boundary conditions.
The functional relationships describing the topology borders are already given by Equations (
29)–(
36). It is worth noting that the topology distribution in the interior phase is identical for both boundary conditions since the stream function constants do not appear in Equations (
29) and (
31).
To check the universality of the findings, very small (
) and very large (
) dynamic viscosity ratios were tested in addition to the water–air bubble (
) and droplet case (
). The topology fields are depicted in
Figure 11 for far-field boundary conditions and in
Figure 12 for near-field boundary conditions. It appears that the behavior for very small and very large dynamic viscosity ratios is not significantly different to the bubble and droplet case. For very small ratios and with far-field boundary conditions, the topology borders seem to approach straight lines even in the exterior phase. For very large ratios, the focal region in the exterior phase seems to completely enclose the interior phase for both boundary conditions. In general, differences in the topology field due to the boundary conditions increase with increasing distance from the bubble or droplet.
Finally, the influence of
, i.e., the radius where the near-field boundary condition is imposed, on the topology field was investigated in
Figure 13. The behavior corresponding to
of the previously discussed cases was compared to an even smaller value of
and a much larger value of
. As can be expected, the general shape of the solution is the same for a constant dynamic viscosity ratio (
here). It is clear that the solution based on near-field boundary conditions approaches the solution based on far-field boundary conditions for very large values of
, i.e., when
.
3.3. Phase-Space Projection
To obtain the scatter plots of the second invariant
Q and the third invariant
R, the solution was discretized by steps of
and
in
r- and
-direction, respectively. The full range of data points is contained in
Figure 14 and a magnified view close to the origin of the phase-space diagram is presented in
Figure 15. The topology borders
,
and
, as introduced in
Section 2.1, are included as well. A high density of data points seems to occur near the origin of the phase-space diagram. It is interesting to note that direct transitions between all four topologies are possible via the origin. Furthermore, strong alignment of the points with the topology borders
and
can be found in the region representing nodal topologies S2 and S3. Even for highly irregular turbulent bubble flows, the same observation was made by Hasslberger et al. [
10] by means of three-dimensional direct numerical simulations. The distribution of points is generally similar for both phases and both boundary conditions. It can be observed that the population in the phase-space diagram is also symmetric with respect to
, i.e., the volume fractions of nodal topologies S2 and S3 as well as focal topologies S1 and S4 are identical. However, the disparity of extreme values of
Q and
R is clearly different. In the bubble case shown here, the disparity of extreme values is larger in the interior, i.e., gaseous, phase. The disparity of extreme values is generally larger with near-field boundary conditions.