Influential parameters on scattering cross section

In this section, the influence of several parameters of the dual-frequency

approach, e.g., the acoustic pressure amplitude, the energy allocation

between two frequencies (N) and the ratio of the two frequencies ( f₂ f₁), on the scattering cross section is numerically investigated.

Figure 3.7 illustrates the scattering cross section curves under

dual-frequency excitation with different acoustic pressure amplitudes (P_e). As with the single-frequency approach, the main resonances, the

they lean over towards smaller bubble radii at higher acoustic pressure

amplitudes. Furthermore, the amplitude of the combination resonances

[e.g., resonances marked as (1,1) and (-1,1) in Figure 3.7] increase

significantly with the increase of P_e. The width of the resonances is also growing with the increase of P_e, leading to the merging of the resonances. For instance, in the case with P Pe 0 0.4, resonance

1 0, 2       merges into resonance (1,0) while resonance (0,1) merges with the resonance (2,0) as

well.

Figure 3.7 Acoustical scattering cross section versus equilibrium bubble

radius under dual-frequency acoustic excitation with P P_{e 0} 0.1 (dash-dotted line), 0.2 (dash line), 0.3 (dotted line) and 0.4 (solid line)

respectively. f₁180 kHz. f₂ 320 kHz. N P_A2 P_A1 1. The two numbers with brackets above the peaks stand for (a,b).

The energy allocation between the two component waves, indicated byN, can also affect the scattering cross section. Figure 3.8 shows the

predictions of the scattering cross section curves under single-frequency

and dual-frequency acoustic excitation with different N. For N <1, as N

decreases, the peaks of the main resonance (0,1) become much lower and

narrower, while those of main resonance (1,0) almost remain the same as

those under single-frequency excitation with frequency f₁. ForN>1, with the increasing N, the peaks of main resonance (1,0) become much lower and narrower, while those of main resonance (0,1) almost remain the same

as those under single-frequency excitation with the frequency f₂ . Furthermore, N influences the values of the scattering cross section near subharmonics in the same way, i.e., the more energy allocated to one of

the component frequencies, the lower the peaks of main resonance and

subharmonics of the other component frequency, andvice versa. However, for combination resonances [e.g., (1,1) and (-1,1)], the maximum values of

scattering cross section are obtained in the case with N 1. The influence ofNon harmonics is much more complicated. Nevertheless, for N<1, the values of scattering cross section near the harmonics [e.g., (0,2), (2,0),

(3,0)] decrease with decreasing N while they do not differ much when 1

N  . So generally speaking, N 1 could generate a relatively high value of scattering cross section within the whole range of R₀.

Figure 3.8 The predictions of acoustical scattering cross section under

single-frequency [marked as “ f₁” (dash line) and “ f₂” (dotted line)] and dual-frequency [marked as “ f₁ f₂” with differentN] acoustic excitation.

1 180

f  kHz. f₂ 320 kHz. N P_A2 P_A1  0.2, 0.5, 1, 2, 5 respectively. The two numbers with brackets above the peaks stand for

The influence of the ratio of frequencies is indicated in Figure 3.9, in

which f₁ is 100 kHz and f₂ is 160 kHz, 200 kHz and 300 kHz respectively. Obviously, f₂ f₁ would influence the positions of the resonances. Particularly, the “distance” between the two main resonances

increases with the increase of f₂ f₁. Meanwhile, the values of scattering cross section in the region between the two resonances decreases with the

increase of f₂ f₁. With increasing f₂ f₁ but fixing f₁, the values of scattering cross section increase in the region below the main resonance of

2

f , decrease in the region between two main resonances, and nearly keep constant above the main resonance of f₁.

Figure 3.9 Acoustical scattering cross section versus equilibrium bubble

radius under dual-frequency acoustic excitation with f₁ 100 kHz and 2 160

f  kHz (dashed line), 200 kHz (solid line), 300 kHz (dotted line) respectively. P P_{e 0} 0.1. NP_A2 P_A1 1. The two numbers with bracket above the peaks indicated the main resonances.

3.6 Summary

In this chapter, both the analytical and numerical solutions of acoustical

scattering cross section of gas bubbles under dual-frequency excitation are

obtained. It is assumed that the bubble oscillates spherically symmetrically

in a Newtonian fluid (i.e., water). Only radial motion of the bubble is

considered. And vapour pressure in the bubble is omitted. The predictions

of the values of scattering cross section by the analytical method agree

well with the numerical simulations at low acoustic pressure amplitudes

for both single-frequency excitation and dual-frequency excitation.

However, the validity of the analytical solution will be violated when the

acoustic pressure amplitude increases because many nonlinear features

(e.g., bending phenomenon, harmonics, subharmonics, and ultraharmonics)

will be present.

The nonlinear characteristics of the scattering cross section curve under

dual-frequency excitation are investigated numerically. Generally, the

scattering cross section curves show typical nonlinear features such as

harmonics, subharmonics and ultraharmonics. Furthermore, the

dual-frequency approach displays more resonances termed as

“combination resonances”. Compared to the single-frequency approach,

section significantly within a much broader range of bubble sizes due to

the generation of more resonances.

The influence of several essential parameters in the dual-frequency system

on the prediction of scattering cross section has been discussed. With the

increase of the pressure amplitude, all amplitudes of resonances increase

and the peaks of resonances lean over towards smaller bubble radii. The

energy allocated between the two frequencies (i.e.,N) and the ratio of the two component frequencies (i.e., f₂ f₁) can also affect the scattering cross section. The driving frequency affects the positions of the resonances.

According to these results, for an effective enhancement of scattering

cross section, the energy allocated to the two component frequencies

should be almost equal and the ratio of two frequencies should be

Chapter 4 The Secondary

Bjerknes Force under

Dual-Frequency Excitation

In this chapter, the characteristics of the secondary Bjerknes force under

dual-frequency excitation are investigated. Parts of this chapter have been

submitted as a journal paper (Zhang and Li, 2015, under review)

The chapter is organized as follows: in Sec. 4.1, the basic equations are

introduced and the corresponding analytical solution is obtained; in Sec.

4.2, the results calculated from the analytical and numerical solutions are

compared, together with the discussions of the valid region of the

analytical solution; in Sec. 4.3, the basic features of the secondary

Bjerknes force under dual-frequency excitation are investigated based on

the numerical simulations; in Sec. 4.4, the influence of pressure amplitude

is discussed. In this chapter, it is assumed that bubbles oscillate spherically

symmetrically in a Newtonian fluid (i.e., water). Only radial motion of

bubbles is considered. For solving the equations of bubble motion, the

translational motion of bubbles is not taken into consideration. And vapour