  In this paper, we numerically study the oscillatory dynamics associated with
the tip vortex cavitation over an elliptical hydrofoil section using our 3D
variational multiphase flow solver at a Reynolds number of $Re=8.95 \times
10^5$ via dynamic subgrid-scale modeling and homogeneous mixture theory. To
begin, we examine the grid resolution requirements and introduce a length scale
that considers both the tip vortex strength and the core radius, which is then
employed to non-dimensionalize the spatial resolution in the tip vortex region
and establish mesh requirements for large eddy simulation of tip vortex
cavitation. We next perform simulations to analyze the dynamical modes of
cavity oscillation at different cavitation numbers and compare them with the
semi-analytical solution. The breathing mode of cavity surface oscillation is
extracted from the results through the definition of an effective radius. The
time-averaged effective radius demonstrates that the cavity experiences a
growth region followed by decay as it progresses away from the tip. Further
examination of the local breathing mode oscillations in these regions indicates
the different behavior of cavity oscillations in the growth and decay regions,
with the oscillations within the former being characterized by lower
frequencies. For representative cavitation numbers $\sigma \in [1.2,2.6]$, we
find that pressure fluctuations exhibit a shift of spectrum towards lower
frequencies as the cavitation number decreases, similar to its influence on
breathing mode oscillations. The results indicate the correlations between the
breathing mode oscillations and the pressure fluctuations, with the
low-frequency and relatively higher-frequency pressure fluctuations being
correlated with the growth and decay regions, respectively.
