FINITE-ELEMENT SIMULATIONS OF NONLINEAR FOCUSED ULTRASOUND IN BUBBLY LIQUIDS María Teresa Tejedor Sastre Universidad Rey Juan Carlos, Tulipán s/n 28933, Móstoles, Madrid, Spain email:
[email protected]
Alexandre Leblanc and Antoine Lavie Université d’Artois – LGCgE, Faculté des sciences appliquées, 62400 Béthune, France
Christian Vanhille Universidad Rey Juan Carlos, Tulipán s/n 28933, Móstoles, Madrid, Spain The aim of this work is to study the energy concentration in a bubbly liquid when we excite the system with an ultrasonic source of large amplitude. We use a spherical transducer in an open cylindrical domain. Finite-element simulations are performed in different configurations. In particular, we show results on the symmetry axis by varying some parameters of the problem. The authors acknowledge the support of the Ministry of Economy and Competitiveness of Spain (DPI2012-34613, BES-2013-064399, EEBB-I-15-09737).
1.
Introduction
The propagation of ultrasound of finite amplitude is of interest in many different applications [1]. The analysis of these nonlinear waves requires specific mathematical models, particularly in nonhomogeneous media [2]. Bubbly liquids made of a homogeneous liquid in which small gas bubbles are introduced have been studied intensively in the last years, but the understanding of the interaction between nonlinear oscillating bubbles and nonlinear ultrasonic waves is still a theoretical challenge [3]. Even a small concentration of bubbles in a liquid modifies the acoustic properties of the medium. The liquid becomes strongly dispersive and for some frequency ranges the sound speed, the attenuation, and the nonlinear parameter increase hugely. These changes greatly affect the propagation of acoustic waves [4-6]. In this paper we aim to study the focusing of finite-amplitude ultrasound in a bubbly liquid. This configuration may have some implications for different non-invasive techniques used in diagnostic and therapeutic medicine [7-9]. In this work we perform a new study of focused ultrasound in bubbly liquids based on the finiteelement method (FEM) by considering the bubbles being evenly distributed in the three-dimensional domain excited by a spherical transducer. A system of differential equations that couples acoustic pressure (wave equation) and bubble volume variation (Rayleigh-Plesset equation) is solved [10]. The numerical problem is reduced by considering cylindrical symmetry around the axis of the transducer. Pressure wave patterns are shown and frequency decompositions of time-dependent pressure signals are evaluated. Maximum values along the symmetry axis are calculated after varying the pressure amplitude at the source and the value of bubble density. The position of the focus is found in each case.
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2.
Model
2.1 Physical problem and differential equations We consider a spherical transducer of radius of curvature Rc and diameter D emitting an ultrasonic signal in an open cylindrical domain. The medium is composed of water and air bubbles. We suppose that the bubble distribution is uniform in the liquid, and the bubbles are all spherical and of the same size. The system is assumed to have a symmetry around the axis defined by the main direction of propagation, z. We solve the differential system that couples the pressure p(z, r, t) with the volume variations of bubbles v(z, r, t) = V(z, r, t) - v 0g , where V(z, r, t) is the current volume, v0g = 4pR 30g / 3 is the initial volume of the bubbles, and R 0g is their initial radius. The space coordinates are z and r, t is the time. This interaction is described by the wave equation, Eq. (1), and the Rayleigh Plesset equation, Eq. (2):
¶ 2 p 1 ¶p ¶ 2 p 1 ¶ 2 p ¶2 v + + = -r N , (z, r) Î]0, L zmax [ ´ ]0, D / 2[, t Î]0, t max [ , 0l g ¶r 2 r ¶r ¶z 2 c0l2 ¶t 2 ¶t 2
(1)
æ ¶ 2 v æ ¶v ö2 ö ¶2 v ¶v 2 2 + dw + w v + h p = av + b çç 2v 2 + ç ÷ ÷÷ , (z, r) Î [0, Lzmax ]´[0, D / 2], t Î]0, t max [ . (2) 0g 0g ¶t 2 ¶t è ¶t ø ø è ¶t In these equations, c 0l is the sound speed in the liquid, r0l is the density of the liquid at the equilib2 rium state, and N g is the density of the bubbles. d = 4nl / (w0g R 0g ) is the viscous damping coeffi2 cient, where n l is the cinematic viscosity of the liquid. w0g = 3g g p 0 g / r0l R 0g is the resonance fre-
2 quency of the bubbles, where g g is the specific heat ratio of the gas, p0g = r0g c0g / g g is its atmos-
pheric pressure, r 0g is its density at the equilibrium state, and c 0g is the sound speed in the gas. 2 h = 4pR 0g / r0l , a = (g g + 1)w0g / 2v0g , and b = 1/ (6v 0g ) are constants. The length of the domain is Lzmax and the experiment lasts from t = 0 s to tmax. The initial conditions are:
p(z ¹ 0) = v =
¶p ¶v (z ¹ 0) = = 0, ¶t ¶t
t = 0, (z, r) Î [0, L zmax ]´[0, D / 2] .
(3)
The boundary conditions are:
¶p (r = 0) = 0, ¶r
z Î [0, Lzmax ], t Î]0, t max ] ,
(4)
¶p 1 ¶p (z = L zmax ) = (z = L zmax ), ¶z c0l ¶t
r Î [0, D/ 2], t Î]0, t max ] ,
(5)
¶p 1 ¶p (r = D / 2) = (r = D / 2), ¶r c0 l ¶t
r Î [0, D/ 2] , t Î]0, t max ] .
(6)
The source is:
f(z, r, t) = p0 sin(wf t), r Î [0, D 2] , z = R c - R c2 - r 2 , t Î]0, t max ] ,
(7)
where p 0 is the amplitude and wf = 2pf is the frequency.
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2.2 Numerical model Equations (1) and (2) are solved using the FEM commercial solver COMSOL. Two “Coefficient Form PDE” are thus defined and coupled in an axisymmetric domain (cf. Fig. 1). For the spatial discretization, second order triangular Lagrange elements are used. A better accuracy is obtained than with first order element for similar computational cost. An unstructured mesh is employed with maximum element size h given by:
h=
l , N
(8)
where l is the wavelength and N is the arbitrary number of elements per wavelength. Preliminary studies have set N = 5 as a good compromise between the induced calculation time and accuracy. However, this criterion alone is insufficient to guarantee the stability of the FEM, and therefore the convergence of its solutions. Indeed, the implicit time method used to solve the partial differential equations must lead to minimal dissipation. Thus, backward differentiation formula is avoided in favor of Generalized-α scheme [11]. In order to ensure its stability, a constant error growth rate is achieved using a CFL number and defining the time step Dt as:
Dt =
CFL h , c0l
(9)
where 0 < CFL £ 1 . For our applications where all shape functions are quadratic, the CFL is taken as 0.2.
Figure 1: FEM domain definition.
3.
Results We use the following values: R c = 0.08 m , D = 0.04 m , Lz max = 0.12 m , f = 300 kHz ,
t max = 100T ,
where
R 0g = 4.5 ´10-6 m ,
T =1 f .
c0l = 1500 m s ,
r0l = 1000 kg / m , r0g = 1.29 kg / m , g g = 1.4 , and n l = 1.43 ´ 10 3
3
-6
c0g = 340 m s ,
2
m / s.
3.1 Linear and nonlinear regimes We compare the results obtained for two source amplitudes, p0 = 10 Pa and p0 = 25 kPa . In this section we use a bubble density of Ng = 1´1011 m-3 . Figure 2 shows the maximum absolute values of ICSV23, Athens (Greece), 10-14 July 2016
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acoustic pressure as a function of the distance to the source in both cases (Figs. 2a and 2b). We observe that in both cases the pattern is quite similar. The maximum value is obtained at 53 mm from the source: p = 21.5 Pa in the linear case and p = 56.91 kPa in the nonlinear case, i.e., 215% and 228% of p 0 , respectively. These results suggest that when ultrasound focalizes in a liquid with a homogeneous density of small bubbles, the focus position is independent of the source amplitude. However, proportionally to the source amplitude, the maximum pressure amplitude obtained on the symmetry axis is slightly higher in the nonlinear regime due to the nonlinear effects observed in Fig. 3.
(a)
(b)
Figure 2: Maximum absolute value of acoustic pressure vs. distance to the source. Linear (a) and nonlinear (b) regime.
Figure 3 displays the pressure waveform during one period in the linear (a) and nonlinear (b) regimes at 53 mm from the source. The nonlinearity of the bubbly liquid affects the symmetry of the wave: positive pressure values are higher than absolute values of negative pressure. The broadening of the negative pressure zone is also observed.
(a)
(b)
Figure 3: Pressure waveform in the linear (a) and nonlinear (b) regimes at z=53 mm.
Figure 4 represents the frequency decomposition of the pressure signal in the linear (a) and nonlinear (b) regimes at 53 mm from the source. As a result of the nonlinearity provoked by the presence of the bubbles in the liquid, a second harmonic component appears.
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(a)
(b)
Figure 4: Frequency decomposition of the pressure signal in the linear (a) and nonlinear (b) regimes at 53 mm from the source.
3.2 Influence of bubble density In this section we study the influence of the bubble density on the focusing of the wave. The pressure source amplitude p0 = 15 kPa is used. Figure 5 shows the maximum pressure amplitude obtained for different bubble densities. This maximum value decreases when the bubble density is raised. The position of this maximum value is closer to the source when the bubble density is higher. For Ng = 1.5 ´1010 m-3 , Ng = 1´1011 m-3 , and
Ng = 2.5 ´1011 m-3 , this position is, respectively, 77 mm, 55 mm, and 30 mm from the source.
Figure 5: Maximum pressure amplitude vs. bubble density.
The generation of second harmonic of pressure is weakened when the bubble density is higher, as shown in Fig. 6. The second harmonic amplitude is 16.17%, 6.43%, and 3.87% for Ng = 1.5 ´1010 m-3 , Ng = 1´1011 m-3 , and Ng = 2.5 ´1011 m-3 , respectively. This is probably due to the important population of bubbles that produces a huge attenuation from the transducer up to pre-focal area, i.e., before the formation of the intense focus.
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(a)
(b)
(c) Figure 6: Frequency decomposition of the pressure signal for (a) Ng = 1.5 ´1010 m-3 , (b) Ng = 1´1011 m-3 and, (c) Ng = 2.5 ´1011 m-3 .
REFERENCES 1 Gallego-Juárez, J. A and Graff, K. F. Eds., Power ultrasonics: applications of high-intensity ultrasound, Woodhead Publishing Series in Electronic and Optical materials, vol. 66, Elsevier, Amsterdam (2015). 2 Naugolnykh, K. and Ostrovsky, L. Nonlinear Wave Processes in Acoustics, Cambridge University Press, New York (1998). 3 Tudela, I., Sáez, V., Esclapez, V., Diez-Garcia, M. I., Bonete, P. and González-García, J. Simulation of the spatial distribution of the acoustic pressure in sonochemical reactors with numerical methods: A review, Ultrasonics. Sonochemistry, 21, 909-919, (2014). 4 Grieser, F., Choi, F. P. K., Enomoto, N., Harada, H., Okitsu, K. and Yasui, K. Eds., Sonochemistry and the Acoustic Bubble, Elsevier, Amsterdam (2015). 5 Hamilton, M. F. and Blackstock, D. T. Eds., Nonlinear Acoustics, Academic Press, San Diego (1998). 6 Young, F. R. Cavitation, McGraw-Hill, London (1989). 7 Lo, A. H., Kripfgans, O. D., Carson, P. L. and Fowlkes, J. B. Spatial control of gas bubbles and their effects on acoustic fields, Ultrasound In Medicine and Biology, 32 (1), 95-106, (2006). 8 Hosseini, S. H. R., Zheng, X. and Vaezy, S. Effects of gas pockets on high-intensity focused ultrasound field, IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control, 58 (6), 1203–1210, (2011). 6
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9 Kajiyama, K., Yoshinaka, K., Takagi and S., Matsumoto, Y. Micro-bubble enhanced HIFU, Physics Procedia, 3, 305-314, (2010). 10 Tejedor Sastre, M. T., Leblanc, A., Lavie, A. and Vanhille, C. Propagación no lineal de ultrasonidos en líquidos con burbujas: simulaciones mediante un modelo por elementos finitos, Proceedings of the 46º Congreso Español de Acústica, Encuentro Ibérico de Acústica, European Symposium on Virtual Acoustics and Ambisonics (TecniAcústica 2015) Valencia, Spain, 21-23 October, 1569– 1574, (2015). 11 Chung, J. and Hulbert, G. M. A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α, Journal of applied mechanics, 60 (2), 371-375, (1993).
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