Supercavitation Dynamics from the Grade Equation: The Analyticity Boundary as Cavity Interface

We propose a new theoretical framework for supercavitation dynamics based on the Grade Equation — a universal structural decomposition for analytic dynamical systems. The central insight is that the cavity boundary in supercavitating flow is precisely the surface where the local analyticity radius \(\rho(\mathbf{x})\) of the velocity field vanishes: the liquid phase is analytic (\(\rho > 0\)), the vapor phase is analytic (\(\rho > 0\)), but the interface between them is a singularity surface (\(\rho = 0\)) across which the grade hierarchy reorganizes. This identification transforms three open problems in supercavitation — closure dynamics, cavity stability, and cavitation noise statistics — into questions about the geometry and statistics of the \(\rho = 0\) level set. We derive three testable predictions: (i) the re-entrant jet velocity at cavity closure scales as \(U_{\text{jet}} \sim U_\infty \cdot \sigma^{-1/2}\) where \(\sigma\) is the cavitation number, following from grade-2 momentum conservation across the interface; (ii) the cavity oscillation frequency satisfies a grade-balanced dispersion relation \(\omega_n \sim (n \pi / L_c) \cdot U_\infty \cdot (1 + \sigma)^{1/2}\) where \(L_c\) is the cavity length; and (iii) the peak pressure distribution during bubble collapse follows a log-Poisson law analogous to turbulence intermittency, with the codimension parameter determined by the cavity geometry (codimension-1 for sheet cavitation, codimension-2 for tip vortex cavitation). The framework unifies partial cavitation, supercavitation, and cloud cavitation within a single analytical structure, and all predictions are testable against existing experimental and DNS data.