Géza Pattantyús-Ábrahám Doctoral School of Mechanical Engineering 

Department of Hydrodynamic Systems

Supervisor: Dr. Hegedűs Ferenc

Experimental and theoretical investigation of micro-chaos

Introducing the research area

The irradiation of liquids with ultrasound (sonochemistry) is a new method in chemistry with great potential. The physical background is a special case of cavitation, the so called acoustic cavitation. As a result of ultrasound irradiation, thousands of micron sized radially pulsating bubbles are formed in the liquid. The collapse of the radially pulsating bubbles can be so strong that the temperature inside the bubbles can reach even several thousand degrees of Kelvin [1], which is beneficial for chemical reactions (sonochemistry [2]). In this way, the efficiency of chemical processes can be significantly increased that may revolutionize conventional technologies. An innovative breakthrough can lead to the extension of the chemical industry that currently accounts for an 8.6% share of the industrial production. This can contribute to the compensation for the overweight of the automotive industry in Hungary (30%). The major challenge of the research is to scale up the production for industrial applications. Due to the specifics of this field of science (complicated physics and large number of parameters), it is difficult to carry out measurements that lead to well established and adequate results; thus, the basis of my research method is to exploit the capabilities of High Performance Computing.

 

Brief introduction of the research place

The present research was carried out at the Department of Hydrodynamic Systems of the Faculty of Mechanical Engineering BUTE supervised by Dr. Ferenc Hegedűs. The activity of the research group covers a wide range of sonochemical applications via exploiting the computing capacity of high performance graphics cards. The research group emphasizes the importance of international cooperation; at present, there is active cooperation with renowned German universities. Furthermore, we emphasize the importance of educating young people, and work jointly with several BSc and MSc students.

 

History and context of the research

The analysis of the dynamics of cavitation bubbles dates back to 1917. The Rayleigh-Plesset equation [1], named after Lord Rayleigh and Milton Plesset, is still commonly used in cavitation research today. At this point, a bubble containing an ideal gas-vapour mixture placed in an infinite domain of liquid domain, which is considered as a nonlinear oscillator with one degree of freedom, where the state of the system is described by the time-dependent bubble radius. Ultrasound generates a periodic pressure fluctuation in the liquid domain causing the bubble to oscillate. Above an ultrasonic irradiation of adequate intensity (proportional to the square of the pressure amplitude), the bubble expands several times larger than its initial size, and then suddenly collapses with high wall velocity. The wall velocity can approach to the sound velocity of the liquid. At this point, the consideration of liquid compressibility of the liquid domain is necessary. Therefore, during my research, the Keller-Miksis equation was applied [3] that takes into account the compressibility of the liquid up to the first order.

 

In the cavitation bubble cloud generated by ultrasound, the physics of bubbles is rather complex. A bubble moves in space during its lifetime affected by the pressure gradient (Bjerknes-force [4]), grow via rectified diffusion, or dissolve into the liquid [5], it may interact with other bubbles, or possibly disintegrate into smaller bubbles. Due to the complex processes and the different physical time scales, the investigation of a cavitation cloud is rather complex, thus we follow the so-called bottom-up approach; that is, single bubbles forming the acoustic cavitation cloud were examined first. This is a good initial approach because a bubble is relatively far from the other bubbles in most of its lifetime.

The research goals, open questions

The aim of the research is to increase the efficiency of sonochemical applications by numerical simulations. Sonochemistry and the closely related bubble dynamics are quite active research areas with many studies are conducted annually. With a few exceptions, these studies are carried out using water as a media. Comprehensive parameter study for viscous liquids is missing from the literature. On one hand, high viscosity results in high damping, which reduces the intensity of collapse strength, and can therefore be one of the limitations of applications. On the other hand, it can be seen from the differential equation describing the bubble surface dynamics that the viscosity helps maintain the spherical shape of the bubble. This is essential to maintain strong, focused and long-lasting collapse-like oscillations. Shape unstable bubbles are more easily disintegrated into smaller bubbles, which are more difficult to excite due to the surface tension that is inversely proportional to the bubble radius. The main purpose of the thesis is to propose an optimal viscous damping that satisfies both of the two aforementioned criteria.

 

The parameter studies were carried out using glycerine whose viscosity is three orders of magnitude larger than that of water at room temperature. Moreover, the viscosity changes by two orders of magnitude by the alteration of the temperature. Thus, the degree of viscous damping can be manipulated via the ambient temperature. Note that the damping rate can also be influenced by other means, e.g. by using glycerine-water mixtures.

 

According to the ultrasonic excitation, the two primary control parameters are the pressure amplitude and the frequency of the excitation (besides the liquid temperature). During the research, I conducted a comprehensive parameter study in the pressure amplitude-frequency-temperature parameter space to seek for strong collapse-like, but also shape stable bubble oscillations, thereby supporting the development of effective operation strategies.

Methods

A common way to investigate the spherical stability of a bubble is to apply a small initial perturbation to the surface of the bubble. The growth or decay of this perturbation implies the shape unstable or stable bubble oscillations, respectively. The perturbed bubble shape can be written as

                                               r(t,Θ,Φ)=R(t)+a_nY_n(Θ,Φ),                                                (1)

where R(t) is the instantaneous mean bubble radius, a_n is the amplitude of surface perturbation, Y_n(Θ,Φ) is the spherical harmonics of order n. The parameter study is limited to rotationally symmetrical shapes, since the external perturbation usually causes an initial deformation from a particular direction (proximity of wall or other bubble). Two examples of such distorted bubble oscillations are shown in Fig. 1, where a bubble deformed by the third mode (to the left), and a bubble deformed by the fourth mode (to the right) can be seen as a function of time.

 

Figure 1: Non-spherical bubble shapes deformed by the third and fourth surface modes (the perturbations are artificially magnified for better visualisation).

 

During the computations, the Keller-Miksis equation was used to model the volumetric oscillation, which is a second-order nonlinear ordinary differential equation. It is worth mentioning that this model is validated by measurements in a wide range of parameters and describes well the dynamics of the bubbles. To examine the shape-stability, ordinary differential equations must be coupled to the Keller-Miksis equation [6] to describe the dynamics of the surface waves for each mode.

Results

To explore the shape-stable bubble oscillations, the ambient temperature was varied by 5 °C between 20–70 °C, and bi-parametric parameter studies were carried out on the excitation frequency-pressure amplitude parameter space. The stability maps obtained from the computation are shown in Figure 2. In the figure, the frequency is normalized with the eigenfrequency of the bubble. The resolution of the stability maps is p_A×ω_R=500×1151, which requires approximately 576 thousand simulations to create a stability diagram in case of a single initial condition. With such a high computational demand, it is essential to exploit advanced computing techniques and the high computing capacity of GPUs (graphics cards). The white domains in the figure represent the shape-stable bubble oscillations, while the colour coded domains shows the most unstable surface oscillations, i.e. the mode number that exhibits the highest growth rate, up to mode number six. The figures also show an iso-line representing the theoretical limit of the minimum collapse strength required for sonochemical applications (solid black curve).

Figure 2: Stability maps on the pressure amplitude-excitation frequency parameter space with increasing the ambient temperature.

 

According to the figure, these iso-lines are rapidly shifted towards the lower amplitudes as the initial temperature increases; that is, smaller intensity (pressure amplitude) is required for active cavitation (inducing chemical reactions). This threshold does not change significantly over an ambient temperature of 45 °C, i.e. the pressure amplitude required to reach the active cavitation threshold at a given frequency cannot be reduced further by raising the temperature. In addition, at higher temperatures, lower viscosity results in the loss of shape-stability. The results show that in a temperature range of approximately 45–50 °C, spherical stability and active cavitation can be achieved simultaneously. Thus, an optimum viscous damping can be found between these two temperature values, where a compromise between collapse strength and shape-stability can be obtained.

 

Expected impact and further research

The purpose of my further research is to apply the computations to acoustic cavitation bubble clouds, which requires solving a large number of coupled Keller-Miksis equations. This would allow a more accurate modelling of a sonochemical reactor. One of the main questions is: how these techniques used to control a spherical stability of a single bubble (alteration of the damping rate) could be applied to bubble clouds. More precisely, can a coherent and synchronised collapse (thus, higher cavitation activity) of bubbles be induced by controlling the viscous damping? Developing such a model is a rather complex task due to the size of the problem; hence this is a field that is still actively researched. Considering that our research group has been actively using high performance GPU programming techniques – which are superior in this field at present – for more than two years, our research group can be the first to achieve significant progress in the computation of bubble clouds.

 

Publications, references, links

List of corresponding own publications

S1. K. Klapcsik, F. Hegedűs F. Study of non-spherical bubble oscillations under acoustic irradiation in viscous liquid. Ultrason. Sonochem. 54:256–273, 2018. IF:6.012

S2. K. Klapcsik, R. Varga, F. Hegedűs. Bi-parametric topology of subharmonics of an asymmetric bubble oscillator at high dissipation rate: The exoskeleton, its internal structure and the missing fine substructure, Nonlinear Dyn., 94:2373–2389, 2018. IF: 4.339

S3. K. Klapcsik, F. Hegedűs. The effect of high viscosity on the evolution of the bifurcation set of a periodically excited gas bubble, Chaos Solitons Fract., 104:198–208, 2017. IF: 2.213

S4. F. Hegedűs, K. Klapcsik. The effect of high viscosity on the collapse-like chaotic and regular periodic oscillations of a harmonically excited gas bubble, Ultrason. Sonochem., 27:153-164, 2015. IF: 3.816

S5. K. Klapcsik és F. Hegedűs. Harmonikusan gerjesztett gázbuborék nemlineáris dinamikai vizsgálata nagy viszkozitású folyadékban. in OGÉT 2014. XXII. Nemzetközi Gépészeti Találkozó:189–185, Nagyszeben, 2014.

S6. K. Klapcsik és F. Hegedűs. Two-Parameter Bifurcation Analysis for the Seeking of High Amplitude Oscillation of a Periodically Driven Gas Bubble in Glycerine. in Proceedings of Conference on Modelling Fluid Flow (CMFF’15): 16th event of the International Conference Series on Fluid Flow Technologies. Paper 116. 8p., Budapest, 2015.

S7. F. Hegedűs, R. Varga és K. Klapcsik. Bifurcation Structure of a Periodically Driven Bubble Oscillator Near Blake’s Critical Threshold. in Proceedings of Conference on Modelling Fluid Flow (CMFF’15): 16th event of the International Conference Series on Fluid Flow Technologies. Paper 115. 8p., Budapest, 2015.

S8. R. Varga, K. Klapcsik és F. Hegedűs. Towards Physical Modelling of the Utilization of Ultrasound in Modern Medical Therapeutic Applications. in First European Biomedical Engineering Conference for Young Investigators (ENCY 2015):114–117, Budapest, 2015.

S9. K. Klapcsik és F. Hegedűs. Modelling of non-spherical bubble shape oscillations in viscous liquid. in Proceedings of the 5th International Scientific Conference on Advances in Mechanical Engineering (ISCAME 2017):261–267, Debrecen, 2017.

 

Table of links

http://www.hds.bme.hu/research/BubbleDynamics/index.html

https://www.gpuode.com/

 

List of references

[1] C. E. Brennen. Cavitation and Bubble Dynamics. Oxford University Press, New York, 1995.

[2] K. Yasui. Acoustic Cavitation and Bubble Dynamics. Springer International Publishing, Cham, 2018.

[3] J. B. Keller, M. Miksis. Bubble oscillations of large amplitude. J. Acoust. Soc. Am., 68(2):628–633, 1980.

[4] I. Akhatov, R. Mettin, C. D. Ohl, U. Parlitz, és W. Lauterborn. Bjerknes force threshold for stable single bubble sonoluminescence. Phys. Rev. E, 55:3747–3750, 1997.

[5] M. M. Fyrillas, A. J. Szeri. Dissolution or growth of soluble spherical oscillating bubbles. J. Fluid Mech., 277:381–407, 1994.

[6] J. Holzfuss. Surface-wave instabilities, period doubling, and an approximate universal boundary of bubble stability at the upper threshold of sonoluminescence. Phys. Rev. E, 77(6):066309, 2008.