approximation of the surface by using higher order basis functions (see the literature on h-p approximation in Finite and Boundary Elements). The radial functions discussed in Tong (1997) can give spectrally accurate approximation to smooth surfaces which aids computational efficiency and can be necessary for representation of the modes of bubble oscillation.

The occurrence of numerical instabilities in other methods seems exaggerated in the present paper. The breakdown of the computation reported in Blake et al. (1995) in due to the jet tip reaching the opposite bubble surface and the computation goes as far as equivalent axisymmetric cases. Some discussion of the causes of instability is given in Baker & Nachbin (1998).

References

Baker, G. and Nachbin, A. 1998 Stable methods for vortex sheet motion in the presence of surface tension. SIAM J. Sci. Comput. 19, 1737–1786.

Blake, J.R., Tomita, Y., and Tong, R.P. 1998 The art, craft and science of modelling jet impact in a collapsing cavitation bubble. Appl. Sci. Res. 58, 77–90.


Tong, R.P. 1997 A new approach to modelling an unsteady free surface in boundary integral methods with application to bubble-structure interactions. Math. Comput. Sim. 44, 415–426.


Zinchenko, A.Z., Rother, M.A., and Davis, R.H. 1997 A novel boundary-integral algorithm for viscous interaction of deformable drops. Phys. Fluids 9, 1493–1511.

AUTHORS’ REPLY

The free surface breaks into a water spike covered by a spray for an underwater explosion near it. The phenomena cannot be fully simuated by BIM, but it is the first step toward more accurate modeling. For very small bubbles, the surface tension may help to maintain the smoothness of the jet. However, the smooth jet may occur for cases whereby the surface tension is small and negligible. Some examples can be found in the experiments of Blake & Gibson (1981).

A high-order local-surface-fitting is suitable for a surface when the surface-fitting behavior matches with the surface. It is hard to provide a high-order local-surface-fitting that is good for an arbitrary surface. It is hard to compare the interpolation scheme on how accurate they are for an arbitrary shape. Besides, the high order panel methods are known to be more susceptible to numerical instability. Therefore, the authors prefer global linear interpolation coupled with the weighted averaging approach.

The jets were not smooth at the end of collapse phase, and sometimes broke in the middle stage of jetting in the previous simulations (Blake et al. 1995, 1997). In the present work, the smooth jets were simulated nearly until the jet impacts upon the opposite bubble surface. The very sharp jets were simulated in the present work that had not been simulated in the previous papers.

Reference

Blake & Gibson 1981 J. Fluid Mech. 111, 124–140.



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