# Morphological Analysis of a Collapsing Cavitation Bubble near a Solid Wall with Complex Geometry
## 4 Morphological Analysis Method Exerted on the Cavitation Bubble Collapse Model
### Evolution of Bubble Profiles
#### 4.2.1 Complexity of Solid Walls  
The first factor involves the complexity of the solid wall, which can be quantitatively described using the fractal dimension. As shown in Figure 9, there are five solid walls with different fractal dimensions but similar orientations. Figure 10 depicts the morphological metric curves of the Minkowski functionals for \(\lambda=1.5\) and \(D=1.03476\), \(D=1.06289\), \(D=1.08107\), \(D=1.10391\), and \(D=1.13124\).  
Figure 9: Similar solid walls with different fractal dimensions: (**a**) \(D=1.03476\), (**b**) \(D=1.06289\), (**c**) \(D=1.08107\), (**d**) \(D=1.10391\), and (**e**) \(D=1.13124\).  
The three Minkowski measures for the collapsing bubble near a solid wall with a complex geometry depict similar trends regardless of the fractal dimension. It can be obviously observed from Figure 9 that the surface of the solid wall with a higher fractal dimension is rougher than the one of the solid wall with a lower fractal dimension. The maximum scale difference of the solid wall surface is about \(5\) lu in these five solid walls, which is much less than the radius of the bubble \(R_{0}=80\) lu. As a result, it takes more time to reflect the differences in the retarding effect of the solid wall on the collapsing bubble. Therefore, from the beginning to \(t=550\) ts, the variation curves of the fractal dimension and boundary length almost coincide with each other. It is illustrated that the dynamic processes of the collapsing bubbles have very high similarity. In addition, there is a clear distinction between these curves from \(t=550\) ts. This is because the differences in the retarding effect of the solid wall on the collapsing bubble begin to take effect as the time goes on.  
There is no doubt that the retarding effect of the solid wall becomes stronger as the fractal dimension increases. The velocity of the bubble's inward collapse will be different because of the variation in the retarding effect of the solid wall, which directly influences the size of the collapsing bubble at \(t=550\) ts. In Figure 10b,c, as \(D\) increases, the first and second collapses are delayed, and the interval between the first and second collapses evidently increases. These changes are also closely associated with the size of the collapsing bubble. The smaller the bubble, the faster it collapses. Therefore, the retarding effect of the solid wall on the collapsing bubble can be indicated by the occurrence time of the first and second collapses. In conclusion, the retarding effect of the solid wall is proportional to its complexity when the solid walls with complex geometries have similar orientations.  
Figure 11 depicts the jet velocity evolution of the collapsing bubble from \(t=550\) ts to the first collapse. The figure also shows the corresponding boundary length of the collapsing bubble under five different fractal dimensions to clearly interpret the variations in the cavitation bubble jet velocity. The evolution of the jet velocity almost follows a trend, that is, it gradually increases to a peak and then slightly decreases with time. Moreover, the intervals at which the jet velocity reaches its peak are the same. The variation range of the jet velocity decreases as the fractal dimension increases. The trend in the boundary length indicates that an increase in the fractal dimension influences the depression speed of the collapsing bubble and causes the depression degree of the bubble top wall to increase slowly. When the depression degree of the bubble top wall increases with time, the surface tension increases correspondingly, causing the acceleration of the jet velocity to decrease. Therefore, the higher the fractal dimension, the slower the peak jet velocity. Based on the above analysis, an increase in the fractal dimension can also affect the evolution of the jet velocity.  
Figure 10: Morphological analysis of the collapsing bubble under different fractal dimensions: (**a**) fractional area \(F\), (**b**) boundary length \(U\), and (**c**) Euler characteristic \(X\).