1 Introduction

Cavitation is a major source of erosion, for instance, of ship propellers, pumps and water turbines. In such systems, low pressure regions (pockets) exist where the water pressure suddenly becomes very low, almost a vacuum range. These growing pockets, i.e. the “cavitation bubbles” propagate to high pressure regions, where they collapse spontaneously [1].

To investigate the cavitation phenomena experimentally, different methods of bubble generation are possible. One of them uses a powerful short laser pulse that is focused into a liquid generating an optical breakdown. During the successive plasma recombination, a fast growing and nearly spherical bubble arises. When the radius has reached its maximum, the bubble contracts continuously to a minimum and finally collapses with one or more rebounds. The onset and collapse of the bubble are usually combined with shock wave emission (laser-generated bubble and shock waves, LGBS). The life time of a cavitation bubble depends on the initial laser energy into the liquid breakdown and bubble formation usually in the order of a few microseconds up to several milliseconds. In the case of LGBS, referring to the shock waves, the time is much shorter, i.e. of the order of 10 ns to a few microseconds, respectively. Consequently, a corresponding temporal resolution although it is not easy to obtain would be required in LGBS experiments.

For experimental investigations on cavitation bubble dynamics, in general, there are two different methods, i.e., high-speed shadow photography with 104–108 frames/s [2, 3] and probe beam deflection (PBD) [4, 5]. With increasing distances from the plasma, these authors were able to obtain information on bubble dimension and wall velocities. A further method limited to very small bubbles is based on Mie scattering [6].

The present work reports on measurements of an alternative, novel and inexpensive optical method to enable a time-resolved insight into the formation and collapse of cavitation bubbles. Different energies, temperatures, and pressures have been studied in liquids with different viscosity (glycerine (propane-1,2,3-triol) and distilled water).

The method provides also information on resulting shock waves from the initial state until the end of the rebound process. Under the assumption of a nearly spherical bubble, the method allows continuous measurements with temporal and spatial resolution of a complete cycle of bubble dynamics with a temporal resolution better than 20 ns depending on the sample rates of photodiode and oscilloscope. Contrary to Englert et al. [7], who studied luminescence of laser-induced bubbles by using a photomultiplier, a linear photodiode and a Gaussian diagnostic laser beam permit the calculation of the bubble radius in the present work.

2 Method of bubble diameter measurement

The principle of the method relies on laser beam deflection from a bubble that is centered on the axis of a HeNe laser beam of diameter r 0 (“diagnostic beam”, see Fig. 1). At a distance L behind the bubble, an aperture of radius r a discriminates most of the deflected light. The undeflected part of the beam is collected and focused onto a photodiode. With respect to the signal in absence of a bubble, the measured signal is reduced by a polarizer and allows the bubble diameter to be deduced with temporal resolution. The temporal resolution is only limited by the readout system consisting of a fast photodiode and a fast oscilloscope. Laser beam deflection at the bubble could be simply estimated from geometrical optics (for details see Appendix A). Briefly, the parallel diagnostic beam hits the bubble of radius r b , which is generated within a liquid by a further laser system (“driving pulse”) on the beam axis. The bubble is described as a homogeneous sphere of gas (or steam) with a refraction index (n b ) smaller than that of the ambient liquid (n a ).

Fig. 1
figure 1

Principle of measurement

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Parallel rays of the diagnostic beam which propagate at y>y 3 (Fig. 9) from the optical axis are refracted by the bubble with a total deflection angle, ϵ, that depends on y coordinate and r b . When y exceeds a certain value y>r b ⋅n a /n b , total reflection occurs. In both cases, this results in divergent rays behind the bubble, which are mostly blocked by the aperture located far behind the bubble. Rays propagating y≥r b are not affected and may still pass through the aperture where they will be detected.

To estimate the resulting “intensity” J of the diagnostic beam behind the aperture as a function of r b , first an estimate of J in the absence of the bubble is made by integration of the diagnostic beam profile over the aperture area (Gaussian profile with 1/e2 diameter r 0, and intensity on axis I 0; see Appendix A):

$$ J_0=\frac{\pi r^{2}_{0}I_0}{2} \bigl[1-e^{-2 (\frac{r_a}{r_0} )^2} \bigr] $$
(1)

If now, a bubble with a relative radius ζ=r b /r a is generated on axis of the diagnostic beam, ray deflection occurs. To deduce the bubble diameter, even for expanded bubbles, the following requirement must always be fulfilled: r 0>r a >r b .

If one assumes that all deflected rays are blocked by the aperture (this is the case for L→∞), the detected intensity is reduced to

$$ J(\zeta)=J_0 \biggl[1-\frac{q-e^{-2 (\frac{r_a}{r_0}\zeta )^2}}{1-e^{-2 (\frac{r_a}{r_0} )^2}} \biggr] $$
(2)

(q is a correction which here is identical to 1). Due to the temporal evolution of the bubble size, ζ and hence J(ζ) depend on time t. When J is focused onto a diode with a linear response, from Eqs. (1) and (2) the bubble diameter could be deduced as a function of t (still q≡ 1):

$$ r_b\bigl(u(t)\bigr)=r_0\sqrt{{-\frac{1}{2}}\ln \bigl[q- \bigl(1-u(t) \bigr) \bigl[1-e^{-2 (\frac{r_a}{r_0} )^2} \bigr] \bigr]} $$
(3)

where u(t)=J(ζ(t))/J 0, i.e. the ratio of the photodiode signal with bubble at time t to the signal in absence of the bubble.

If the deflected rays are not totally blocked, the correction q in Eqs. (2) and (3) is no more identical to one, but becomes a function of ζ(u(t))=r b (u(t))/r a :

$$ q(\zeta)=e^{-2 [\frac{r_a}{2L (\frac{n_a}{n_b}-1 )}\zeta ]^2} $$
(4)

Hence, Eq. (3) becomes more complicate and has to be solved by iteration (the initial value of ζ and r b , respectively, could be estimated from Eq. (3) (with q=1); furthermore, for r a ≪L Eq. (4) may be linearized). However, as may be seen from following sections, a correction is not always necessary. As an example, within the present the work, r a =2 mm, L≥500 mm, n a =1,33, n b ≈1 (water and steam, respectively) and thus, for ζ≤1 the deviation of q from 1 is smaller than 7×10−5!

3 Experimental setup

Cavitation bubbles have been induced by two different laser systems into distilled water and glycerine, respectively (driving pulse in Fig. 1). The liquid has been kept in a quartz glass cuvette (volume=3×3×3 cm3) at a temperature T=25 °C and a pressure p=1 bar. Alternatively a stainless steel container (80 mm inner diameter and 200 mm height) is used to allow the temperature to be controlled in the range of 20 ∘C<T<100 °C and the pressure to be kept between 0.2 bar<p<1 bar (absolute values).

Two different laser systems have been used for inducing cavitation bubbles. A femtosecond laser was used for the experiment with different laser energies in water (laser fluence of max. 80 J/cm2, pulse duration (FWHM) Δτ=150 fs, wavelength λ=775 nm, CPA 2110 Ti:Sapphire, Clark-MXR, Inc.). A Q-switched Nd:YAG laser (laser fluence of 0.2 MJ/cm2, Δτ=6 ns, λ=532 nm, Solo III PIV 15, New Wave Research) was used for measuring glycerine and water, respectively, at different temperatures and pressures. The experimental setup using the container is shown in Fig. 2. Measurements with the femtosecond laser have been carried out by using the quartz glass cuvette.

Fig. 2
figure 2

Schematic sketch of the experimental setup

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An expanded, collimated diagnostic beam of a HeNe laser (λ=633 nm, D-7517, Polytec) in TEM00 mode crosses the bubble driving beam perpendicularly and is then detected by means of a PIN photodiode (BPW 34, Osram). The cavitation bubble generation is centered exactly in the middle of the diagnostic beam proved by a beam profiler (LBP-1, Newport). Figure 3 demonstrates the dip located in the maximum of the Gaussian beam.

Fig. 3
figure 3

Image in false color of the HeNe laser beam profile to verify the alignment of the cavitation bubble in the middle of the Gaussian beam

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The diagnostic beam (HeNe laser) is focused to the PIN diode, covered with a 633 nm interference filter (Schott, 10 nm band width) to discriminate environmental light. An oscilloscope (TDS7154B, Tektronix) is used to read out the voltage U(t). A polarizer allows the intensity of the diagnostic beam to be adjusted. Part of the beam is blocked by the cavitation bubble. Thus, the temporal evolution of the diode voltage U(t) depends on the temporal evolution of the bubble volume V(t). Then, under the assumption of a spherical bubble geometry, the radius r b (t) of the bubble can be calculated as described earlier (Sect. 2). The assumption of the spherical bubble geometry is verified by a separate series of CCD images (SensiCam, PCO, fitted on a microscope) during the measurements (see Fig. 4, Table 1). This is only valid within the first bubble before the collapse. During the first rebound, the spherical condition for the shape of the shadow area (two-dimensional) is not fulfilled as shown in Fig. 4 (Δt=190 µs). However, the measurement from the image of the non-spherical area leads to an “equivalent-spherical” area of nearly the same bubble dimension estimated by the novel measurement method presented in this paper (Table 1).

Fig. 4
figure 4

Comparison of bubble radii measured in water by the introduced method (r b , open triangles, black line as an example of bubble evolution) and CCD images (r pho, solid squares) at various time delays Δt after plasma ignition; Nd:YAG laser

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Table 1 Radii of bubble size deduced from CCD images (r pho) and comparison to r b deduced from present method at time delays Δt after plasma ignition
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4 Results and discussion

The aperture of the experimental setup (Fig. 1) acts like a ‘schlieren device’, if strong gradients occur. It is a very sensitive possibility to detect initial and collapse shock waves as well as the time-depending bubble formation which can be seen in Fig. 5.

Fig. 5
figure 5

Bubble time-distribution in water induced by a femtosecond laser: (a) for different laser energies; (b) inset for indicating the bubble initiation (A) and the first collapse (B)

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Figure 5a shows the bubble radius r b (t) for different laser energies induced by a femtosecond laser. The advantage of this method can be clearly seen in Fig. 5b. Beside the radius distribution r b (t), also the onset and collapse of the bubble are detectable (insert A and B). A temporal resolution of 20 ns and a spatial resolution of 3 µm have been obtained, which depend only on the sample rate of the oscilloscope and read out of the diode. Measurement accuracies of U(ζ) and r 0 lead to an error of r b of ±5 %. Comparison of r pho determined from shadow images of the bubble and measured radius r b can be realized with an error of ±10 % (see Fig. 3).

In Fig. 5a it can be seen that the collapse radius r c decreases with increasing laser energy E, indicated by a dashed line. However, the bubble structure at the initiation and immediately before the collapse is not exactly spherical but more or less parabolic. As a consequence, the used measuring method delivers the correct bubble start and collapse times, respectively, but the ’bubble radius r b ’ is an ’equivalent radius’ to the real shadow area in the measuring plane of the bubble.

In Fig. 6, the bubble radius r b has been measured in glycerine as a function of time with a constant temperature T 0=30 °C and different initial liquid pressures p 0=1 bar (black) and p 0=0.43 bar (red). With decreasing initial liquid pressure p 0, the bubble dimension increases as well as the collapse time t c (time difference between maximum bubble radius and collapse). The measurements in glycerine depected in Fig. 6, do not show collapse shocks either for p 0=1 bar or for p 0=0.43 bar.

Fig. 6
figure 6

(a) Bubble radius in glycerine as a function of time at constant temperature T 0=30 °C and various initial liquid pressures p 0 (\(E_{\mathrm{(Nd:YAG)}}\approx10\) mJ, r 0=1.836 mm); (b) time-enlarged scale of the first bubble

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The temperature dependence of the bubble radius is shown in Fig. 7. The initial pressure remains p 0=1 bar=const. with T 0=30 °C (black curve) and T 0=66 °C (red curve). Similar to Fig. 7, also both the bubble dimension r b and the collapse time t c increase with increasing temperature T 0 of the liquid. In contrast to Fig. 6b, now a collapse shock wave is visible due to a reduced viscosity at higher temperature (Fig. 7b, red curve).

Fig. 7
figure 7

Bubble radius in glycerine as a function of time for constant pressure p 0=1 bar and different initial liquid temperatures T 0 (Nd:YAG, r 0=1.863 mm)

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Figure 8 shows the bubble wall velocities v b as a function of the bubble radius r b during the shrinking-process from the maximum bubble radius r max to the minimum bubble radius r c (collapse radius, Fig. 8a). Low Nd:YAG laser energy (7 mJ) has been used to achieve a nearly spherical bubble geometry. The results are compared to those obtained by Rayleigh’s equation [8]:

$$ v_b=\frac{2p_\infty}{3\rho_1} \biggl[ \biggl(\frac{r_{\max}}{r_b} \biggr)^3 -1 \biggr]^\alpha; \quad \alpha=0.5 $$
(5)

where p ∞ is the pressure in infinite liquid, ρ 1 is the liquid density, and r max is the maximum bubble radius. The first phase of the bubble shrink process shows excellent agreement, but the last phase (before collapse) is different. Experimental results for the complete bubble shrinking process could be found in the investigation of Peel et al. [5], where an exponent of α=0.17 was found (Fig. 8b). The discrepancy of α is due to the fact that Peel et al. [5] applied the higher laser energy in their investigation (i.e. 140 mJ) as compared to the present work (7 mJ).

Fig. 8
figure 8

Measurements of wall velocities and comparison to Rayleigh’s equation in water (\(E_{\mathrm{(Nd:YAG)}}\approx7\) mJ)

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A detailed investigation is beyond the scope of the present work and will be presented elsewhere [9].

5 Conclusion

A novel and powerful optical method to diagnose cavitation bubbles in liquid is presented. It allows the experimental analysis of the dynamics of laser-driven cavitation bubbles. The main advantages of bubble measurements is the high temporal and spatial resolution from the onset to the collapse of a single cavitation bubble as well as the detection of the combined shock waves generated by a single shot of a high power laser. Values of 20 ns and 3 µm, respectively, are limited by the read out system consisting of a fast photodiode and a fast oscilloscope. The error of r b is ±5 %.

Although limited to spherical bubbles, CCD images during the measurements do confirm that the present method is well applicable. Especially during onset and collapse, deviation from spherical geometry was observed. However, even in that case, the bubble radius determined with the present method could be considered to be an equivalent radius.

The advantage of the novel method is its easiness to assemble, whereas the acquisition of CCD images and also the setup of the illumination are much more complicated. Furthermore, the presence of shock waves and temporal evolution of the cavitation bubble is indicated immediately.