Predicting laser‐induced cavitation near a solid substrate

The asymmetric collapse of cavitation bubbles near a solid substrate generates large wall shear stresses, the precise magnitude of which is still not known with certainty. By comparing numerical simulations and experiments of a laser‐induced cavitation bubble near a solid substrate, we demonstrate that an accurate measurement of the pressure pulse emitted during bubble inception and of the maximum bubble radius allow a unique initialisation of the simulation. This allows an accurate reproduction of the asymmetric collapse, with reliable predictions of the shear stress and pressure generated at the substrate.


Introduction
A cavitation bubble near a solid substrate collapses asymmetrically and produces a fast liquid jet, which in turn generates large shear stresses on the substrate that can exceed 100 kPa [1]. Such jetting cavitation bubbles may cause detrimental material erosion but are also utilised successfully, for instance, in laser ablation technologies and in ultrasonic cleaning. Laser-induced cavitation offers a reproducible and controllable means of generating such cavitation bubbles individually in predefined locations, through optical breakdown of the liquid and the subsequent formation of a gas cavity. While experiments and Rayleigh-Plesset models have provided a very good understanding of the related bubble dynamics, especially the shear stresses generated on the substrate and nearby objects are still not known in detail. To this end, accurate numerical simulations can provide valuable insight, however, previous experiments [2] and numerical simulations [1] produced contradictory results. Typically, the maximum radius of the cavitation bubble is the only reference value used in simulations when reproducing a given experiment [1,3]. In this article, we show that a measurement of the pressure pulse emitted during bubble inception together with the maximum bubble radius provide a unique set of reference values to guide the selection of the initial conditions for accurate and reliable simulations of the asymmetric collapse of laser-induced cavitation bubbles.

Methods
In the experiments, the cavitation bubble is created inside the test section at a distance h 0 above the substrate by a focused laser beam using a Q-switched Nd:YAG laser (New Wave Research, wavelength 532 nm, pulse duration 6 ns, laser beam diameter 2.75 mm), as described in more detail in [4]. Fig. 1a shows high-speed images of the considered bubble, with maximum radius R max = 638 µm and stand-off distance γ = h 0 /R max = 0.4. The generated pressure is recorded by a fiber optic probe hydrophone (690 ONDA, 150 MHz bandwidth) positioned off-centre at a distance of d i = 950 µm to the bubble inception site, ≈ 80 µm above the substrate, to avoid damage to the hydrophone and distortion of the hydrophone signal.
The simulations are conducted using a fully-coupled pressure-based algorithm for interfacial flows [5], which is based on a second-order finite-volume discretisation [6]. The bubble is initialised with radius R 0 and gas pressure p 0 at distance h 0 above the substrate, as illustrated in Fig. 1b. The ambient pressure is p ∞ = 10 5 Pa. Air is assumed to be an ideal gas with polytropic exponent κ = 1.4, density ρ air,0 = 1.2 kg m −3 at p ∞ , and viscosity µ air = 1.82 × 10 −5 Pa s. Water is modelled by the Noble-Abel-stiffened-gas model with the properties proposed by Le Métayer and Saurel [7] and a viscosity of µ water = 10 −3 Pa s. Surface tension and gravity are neglected. Following previous work [1], the computational domain is resolved with a rectilinear mesh with mesh spacing ∆x = 2 µm and a gradual mesh refinement near the wall, with a minimum mesh spacing of ∆x min = 50 nm. The time-step is adaptively chosen to satisfy a Courant number of Co = ∆t|u|/∆x < 0.7.
Acoustic signal of a bubble collapsing close to a solid wall, = 0.4, R b = 638 µm; b) clos c) signal produced by the first collapse of the bubble; d) signal due to the second collap of the signal shown in figures a-d. The length of the bar in the first image from the left is m.
70 Bar. After reaching its minimum volume, the now toroidal bubble expands and ⇠ 200 µs; however, the acoustic trace emitted was perhaps too weak to be detected ustic signal can be seen from figure 2d at the time the bubble collapses for the sec e 3a, the acoustic trace recorded for a larger is shown; the trace is similar t ng peaks can be seen corresponding to the bubble inception and the first collapse tly shorter third peak is observed, this peak corresponds to the second collapse of t WHM at inception are 51 ns and 87 ns, respectively, as can be seen in 3b, the rise but the width of the pulse time is ⇡ 20% larger; as mentioned in the supplementa ation in both the rise time and the pulse width was observed, we believe this is du omplex interaction of the reflected and refracted waves at the surface of the fiber on the other hand, the acoustic trace rises slower than at inception, here t r = 197 , figure 2c, two peaks of approximately the same magnitude can be observed, in fig etected. Also, in this figure the pressure slowly decays and does not recover. At race displays a rather complex pattern with two acoustic pressure peaks at t = 27 , we can correlate the appearance of the pressure peaks in figures 3a-d with the in    Fig. 1c, is in very good agreement with the experimental measurements, regarding the maximum bubble radius, R max = 638 µm, and the amplitude of the pressure pulse emitted at bubble inception, p i,1 = 6.67 MPa, see Fig. 2c. The amplitude of the pressure pulse emitted during bubble collapse, p i,2 , predicted by the simulation is also in very good agreement with the experimental measurements, considering the tolerance of the hydrophone measurements is approximately 0.5 MPa. The peak shear stress, τ w,2 = 68.3 kPa, and the peak pressure amplitude, p w,2 = 7.64 MPa, associated with the collapse of the bubble, which are also shown in Fig. 2c, are similar to the values found in previous numerical studies [1,3]. While the peak shear stress, τ w,2 , is a consequence of the fast liquid jet developed during the bubble collapse, see Fig. 1c, impinging on the substrate, the peak pressure amplitude, p w,2 , is the result of the shock wave emitted by the bubble when it reaches its minimum volume.

Conclusions
Conducting a direct comparison of experimental measurements and numerical simulations, we introduced a unique combination of reference values (p i,1 , R max ) to select the initial conditions (p 0 , R 0 ) of the simulations. Selecting the initial conditions (p 0 , R 0 ) to match both reference values (p i,1 , R max ) measured in the experiment reproduces the dynamic behaviour of the collapsing bubble accurately and provides reliable predictions of the shear stress and pressure generated at the substrate, which we find to be in line with previous numerical studies [1,3] for the considered representative bubble.