Self-Similar Cavitation Onset

• The paper establishes that cavitation onset occurs when a bubble’s radius exceeds its time-dependent unstable equilibrium, extending the classical Blake criterion.
• It employs Buckingham π theorem to derive six non-dimensional groups and reveals universal scaling laws governing transient cavitation behavior across fluids.
• Phase maps and power-law scaling demonstrate that dynamic thresholds exceed static predictions, delivering a predictive framework for engineering applications.

A self-similar description of cavitation onset provides a mathematical and physical framework in which the conditions for nucleation of cavities (bubbles or voids) in a medium can be recast in terms of universal, dimensionless criteria and dynamic response functions that depend on ratios of relevant scales, rather than on the absolute values of system parameters. This approach reveals the unifying scaling laws that govern the threshold for cavitation in a range of transient pressure fields, and demonstrates how the classic quasi-static thresholds (such as the Blake critical tension) emerge as limiting cases. Self-similar descriptions are essential for bridging the gap between static and dynamic regimes, accounting for pulse duration, bubble size, and the pertinent fluid properties, and for predicting cavitation onset across different liquids and timescales.

1. Dynamic Definition of Cavitation Threshold

Traditional criteria for cavitation onset, such as the Blake threshold, rely on the minimum static tension at which surface tension can no longer stabilize a pre-existing bubble, leading to unlimited growth. This quasi-static limit is insufficient in transient or highly dynamic pressure fields, where both the applied tension and the bubble’s response are time-dependent. The paper (Coulombel et al., 19 Oct 2025) establishes a dynamic criterion: cavitation onset occurs at the instant when a bubble’s radius $R(t)$ exceeds the time-dependent unstable equilibrium radius $R_{Ue}(t)$. The latter is obtained by solving the instantaneous balance at the liquid–gas interface,

$$p_{G,0} \left( \frac{R_0}{R^*(t)} \right)^{3\kappa} - \frac{2\sigma}{R^*(t)} - 4\mu \frac{\dot{R}(t)}{R^*(t)} - p_\infty(t) = 0,$$

where $p_{G,0}$ is the initial bubble gas pressure including surface tension, $\kappa$ is the polytropic exponent, $\sigma$ the surface tension, $\mu$ the viscosity, and $p_\infty(t)$ the instantaneous far-field pressure. The bubble transitions from a stable (surface tension dominated) regime to a dynamically unstable state once $R(t) > R_{Ue}(t)$. The critical tension $p_{ng,C}$ is defined as the lowest pressure at which this instability occurs during the imposed transient.

2. Non-Dimensional Parameters and Self-Similarity

Applying the Buckingham $\pi$ theorem to the nine relevant variables (initial radius $R_0$, initial pressure $p_0$, density $\rho$, surface tension $\sigma$, viscosity $\mu$, sound speed $c$, polytropic exponent $\kappa$, pulse duration $\tau$, and tension $p_{ng}$), the authors identify six non-dimensional groups governing cavitation onset. Essential ratios include:

• Inertia vs. viscosity: $t_\mu / t_i$ with $t_i = R_0 \sqrt{\rho/\Delta p}$, $t_\mu = (\rho R_0^2)/\mu$
• Capillarity vs. inertia: $t_\sigma / t_i$ with $t_\sigma = \sqrt{\rho R_0^3}/\sigma$
• Non-dimensional pulse duration: $\tau / t_i$
• Normalized tension: $p_{ng} / p_C$, $-p_{ng}/K$ with $K = \rho c^2$

Once the system is mapped into this reduced parameter space, the onset of cavitation displays self-similar behavior across liquids, scales, and forcing regimes: i.e., solutions to the threshold criterion and bubble response collapse in non-dimensional phase space, independent of the absolute sizes or material parameters.

3. Phase Maps, Power Laws, and Minimal Tension

Detailed phase maps in the $(R_0, \tau)$-plane are constructed, showing contours of critical tension $-p_{ng,C}$ and the corresponding maximal bubble size at cavitation onset. These maps demonstrate that:

• For a given transient, only certain combinations of $R_0$ and $\tau$ permit the onset of cavitation.
• Vertical isobars in the phase map correspond to the situations where the critical tension equals the classical Blake threshold $p_C$; this is realized for sufficiently long pulses (quasi-static limit).
• The minimal tension required for cavitation, $-p_{ng,C}^*$, for each pulse duration $\tau^*$, follows a power-law scaling,

$-p_{ng,C}^* \propto \tau^{-\beta}$

with exponents $\beta$ reported between approximately 1/2 and 2/3 depending on fluid and regime.

4. Blake Threshold as Quasi-Static Lower Bound

The classic Blake threshold emerges as a lower bound in all scenarios. The Blake pressure is given by

$$p_C = \min_R \left( (p_0 + 2\sigma/R_0)(R_0/R)^{3\kappa} - 2\sigma/R \right)$$

with the corresponding critical radius $R_C$ at which the minimum occurs. In the quasi-static limit (long pulses, slow tension changes), the dynamic criterion $R(t) > R_{Ue}(t)$ reduces to the Blake criterion. For shorter, more abrupt pulses, the critical tension $p_{ng,C}$ required for cavitation is always more negative (i.e., a larger magnitude) than $p_C$.

5. Dynamics at the Onset: Transition Regimes

For rapid or high-amplitude tension transients, the response is fundamentally dynamic: surface tension and viscosity can only partially counteract the bubble’s expansion rate. The time dependence of $R_{Ue}(t)$ incorporates not just the instantaneous pressure, but also the inertia and damping due to liquid properties. The dynamic threshold is higher than the quasi-static Blake threshold, with the excess determined by the finite expansion rate and the pulse duration relative to $t_i$. For intermediate pulse durations, the threshold interpolates between the static and extreme dynamic limits, governed by the self-similar scaling laws.

6. Physical Implications and Universality

The self-similar description demonstrates that, regardless of liquid used or the specific pressure history, the conditions for cavitation onset can be recast universally in terms of a small set of dimensionless numbers. Phase maps and derived power-law thresholds provide predictive tools for engineering and experimental design, enabling accurate location of the cavitation boundary across diverse regimes. The framework naturally accommodates liquid-dependent properties (via capillary and viscous timescales) yet predicts that, when cast into the correct similarity variables, the onset of cavitation collapses onto universal curves. Notably, the Blake threshold is valid only as a limit; real, finite rate processes always require larger tension magnitudes for inception.

7. Bridging Cavitation Criteria in Transient Environments

By identifying the dynamic instability criterion ($R(t) > R_{Ue}(t)$) and demonstrating its collapse under self-similar rescaling, this approach unifies disparate prior cavitation criteria (static Blake condition, inertial and viscous thresholds, energy arguments) and provides a rigorous method for extending these concepts to arbitrary transient fields. The regime boundaries, onset curves, and their scaling exponents are all determined by the interplay of non-dimensional pulse duration, initial bubble radius, and material properties. This offers a comprehensive, predictive, and universal account of cavitation onset in both quasi-static and strongly time-dependent environments.