Far-Field Noise Induced by Bubble near Free Surface

The motion of a bubble near the free surface is solved by the boundary element method based on the linear wave equation, and the influence of fluid compressibility on bubble dynamics is analyzed. Based on the solution of the bubble motion, the far-field radiation noise induced by the bubble is calculated using Kirchhoff moving boundary integral equation, and the influence of free surface on far-field noise is researched. As the results, the oscillation amplitude of the bubble is weakened in compressible fluid compared with that in incompressible fluid, and the free surface amplifies the effect of fluid compressibility. When the distance between the bubble and an observer is much larger than that between the bubble and free surface, the sharp wave trough of the sound pressure at the observer occurs. With the increment of the distance between the bubble and free surface, the time of the wave trough appearing is delayed and the value of the wave trough increase. When the distance between the observer and the bubble is reduced, the sharp wave trough at the observer disappears.


Introduction
The perturbation induced by a bubble in fluid will propagate to far field as the form of shock wave or noise. When there are boundaries such as wall or free surface near the bubble, its motion will be affected obviously. For example, the bubble near the free surface will generate high speed jet opposite to the free surface at the contraction stage, while a spike occurs at the free surface . And the perturbation propagation will also be affected by the boundaries. Both of the free surface and wall will reflect the perturbation, and change the characteristics of far-field noise under water. Analyzing the influence of the free surface on the propagation of noise induced by the bubble could be the references for the research about cavitation noise in the wake flow of ship or wavelet properties at far field of air gun array used in seismic exploration and so on. Now, the boundary element method (Wang et al., 1996;Robinson et al., 2001) and some CFD methods (Shyue, 2006;Hu et al., 2009) are used to solve the interaction between the bubble and free surface. Boundary element method usually uses the incompressible fluid assumption, and the fluid characteristics are described as Laplace equation. However, as the severe motion, the Mach number of the bubble surface will be O(10 -2 -10 -1 ), and then the compressibility of fluid affects the bubble motion obviously. Therefore, in the boundary element method it should not assume that the fluid is incompressible, and the perturbation can propagate to far field at a finite speed. Compared with incompressible fluid, the amplitude of bubble oscillation in compressible free fluid field is weak, and the velocity of the jet tip decreases .
Far-field noise induced by the bubble motion is usually solved by boundary element method. But the motion and noise should be solved respectively, and the motion of bubble surface will be the boundary condition of the noise calculation. Qi (1999), Qi and Lu (2001) used the classical boundary element method to calculate single or multiple bubbles near different boundaries. And a virtual surface was set to keep a certain distance from the bubble and encircling the bubble. Then the far-field noise is calculated based on this virtual surface using the fix boundary element method. Jamaluddin et al. (2011) adopted the FLM (Free Lagrange Method) to solve the motion of the bubble subjected to shock wave, and also arranged the fix virtual surface to obtain the far-field noise. Ye et al. (2013Ye et al. ( , 2015 utilized a new form of boundary element method  considering the weak compressibility of fluid to solve the dynamics of bubbles, and presented a method using the moving boundary integral equation to obtain the far-field noise. This method can calculate the far-field noise directly, without setting the virtual surface. In this paper, the dynamics of the bubble near the free surface is solved by the boundary element method considering weak compressibility of fluid. Compared with the results obtained by the classical boundary element method (Wang et al., 1996;Li et al., 2016;Zhang and Liu, 2015), the influence of compressibility on the bubble motion near the free surface is analyzed. Based on the motion of the bubble and the free surface, the Kirchhoff moving boundary integral equation is used to obtain the far-field noise induced by the bubble and the free surface, and the influence of the free surface on noise is researched.

Basic theory and method for bubble motion
The fluid characteristics are described using the linear wave equation, and the sound speed c in fluid is assumed to be constant. By using the method presented by Zhang et al. (2013), the boundary integral equation considering compressibility is:

‫,א‬ Θ, andℜ
where, is the velocity potential induced by the bubble and free surface, I is the unit matrix, Π is the local curvature matrix of the bubble surface. Operators are: ds, h respectively, and s is the bubble surface, is the solid angle, l pq is the distance between the field point p and source point q. The dynamics boundary conditions of the bubble surface and free surface are: where, D/Dt is the material derivative; γ=1.25 is the gas specific heat; V 0 is the initial volume of the bubble; ε=P 0 /ΔP is the intensity parameter, P 0 being the initial inner pressure of the bubble; ΔP=P ∞ -P c , P ∞ being the pressure at infinity, and P c being the saturated vapor pressure; δ=ρgR m is the buoyance parameter, R m being the maximum radius of the bubble; β=α/(R m ΔP) is the surface tension parameter with α=0.0728 N/m; λ is the distance between the free surface and the bubble initial center. The kinematic boundary condition of bubble and free surface is: where l is the position vector of the bubble surface and free surface. Fig. 1 shows the numerical model of the bubble motion near the free surface, L f is the calculation range of the free surface in the boundary element method, R 0 is the initial radius of the bubble. The virtual face is added since the boundary element method needs closed surface. Here, the discretion and integral equation calculation of the free surface is the same as that of the bubble, but the velocity potential and velocity on the virtual surface are set to be zeroforced. The calculation range of the free surface should be large enough but still satisfy the assumption of later stage  to reduce the influence of boundary. In order to consider the contribution of the infinite free surface to the bubble motion and noise, the far-field free surface can be equivalent as dipole (Pearson et al., 2004), and the height, velocity potential and velocity at the free surface are attenuated as the distance's cube (Pearson et al., 2004): where r is the coordinate on the free surface out of the calculation range, r max is the maximum coordinate inside the calculation range; f represents the physical quantities on the free surface out of the calculation range, and f max is the value of f at r max .

Calculation method for far-field noise induced by bubble motion
The Kirchhoff moving boundary integral equation (Farassat, 1988) is: where t is the field time, and τ is the source time; Φ t(y) = ; Mach number M=v/c, v being the velocity of the moving boundary; normal Mach number M n =v·n/c=M·n; radiation Mach number , and is the τ * τ * unit radiation radius vector; θ is the angle between the radius vector and outer normal vector of moving boundary; "pro" means the time of noise propagation from points on the moving boundary to observer should be considered. Sound pressure at field point p at time t is induced by perturbation of the source point at time , and is the solution of Eq. (7): Notice that, the displacement and velocity of the bubble surface are the same with those of fluid clinging the bubble. If we treat the bubble surface as a deformable moving boundary, then . To apply Eq. (6) to the bubble surface directly, the ele-ment discretization and mapping are used Ye et al. (2013Ye et al. ( , 2015. The bubble surface is discretized into triangular elements, and every triangular element is mapped to isosceles right triangle coordinate system ξ-η. The Jacobi matrix J is brought in by coordinate transformation. The coordinate system ξ-η is independent of time, but Jacobi matrix J is related to time, and can consider the deformation of moving boundary. If the perturbation induced by Node q' on the bubble surface at time t* propagates to the observer at time t, then we define Node q' as the perturbation node of the observer at time t*. During the bubble's movement, the velocity potential at the observer induced by one of its perturbation nodes can be obtained by Eq. (6): where are directional derivatives of normal, tangential and radiation, respectively; T an is the unit tangential vector of the bubble surface; ∑ ele represents the sum of perturbation from elements around the perturbation node to the observer. The time of the physical quantities at the right side of the equation is source time, while at the left side of the equation is observer time. The total velocity potential at time t at the observer is: where ∑ pernode represents the sum of perturbation induced by all perturbation nodes of the observer at time t. The distances between different perturbation nodes and the observer at the same time are different, and the distance between a certain perturbation node and the observer at different time is also different. Thus the perturbation at the observer at a certain observer time is superimposed by the perturbation induced by different perturbation nodes at different source time. In order to obtain the history of the sound pressure at the observer, the perturbation propagating to the observer required to be interpolated as time and superimposed at the same observer time. Thus, the motion of the bubble is solved with the boundary integral equation, and the velocity potential at the observer can be calculated by Eqs. (8) and (9), finally the sound pressure at the observer is obtained by the derivative of the velocity potential with respect to observer time:

Validation
To record the maximum radius during the bubble motion, the convergence of L f is proved as shown in Fig. 2. The parameters of the bubble are R 0 =0.13, ε=143.6, δ=0.12, and λ=0.8. When L f >6, the result is convergent.
Compared with the experimental results of Zhang et al. (2011), it can be seen that the method presented in this paper can simulate the motion of the bubble near the free surface as well. The initial distance between the bubble center and the free surface is 0.948 cm, and the height of spike at the moment jet impact is 2.10 cm. The detail conditions about this experiment can be found in Zhang et al. (2011).
The far-field sound pressure induced by single bubble in free field is calculated to prove the noise calculation method presented in this paper. The parameters for this case are R 0 =0.35, ε=10, and the distance between the observer and the initial bubble center is 15. The bubble motion is solved using Eq. (1) to consider the compressibility and boundary integral formula of Laplace equation (Wang et al., 1996) without considering the compressibility, respectively. And the sound pressure at far field is calculated using Eqs. (8), (9) and (10). The comparison shown in Fig. 4 indicates that the result of the method presented in this paper (solid line) is matched well with the analytical result obtained by the Fig. 2 noise of monopole (Ross, 1976): where R is the radius of the bubble obtained from the bubble motion equation in compressible fluid presented by Lezzi and Prosperetti (1987); r m is the distance between the observer and the bubble; (r m -R)/c represents the retarded time; '.' means the derivative of R with respect to time. As shown in Fig. 4, the peak value of the sound pressure with the incompressible assumption (dash line) is not decayed in time at a certain observer.

Influence of compressibility on bubble motion
During the bubble expansion, the center of the free surface will move upward, and generate the spike. The fluid will be attracted quickly to the narrow area between the bubble and free surface, and form a local high pressure. Thus the bubble will be "repelled" by the free surface during its contraction and form the high speed jet back to the free surface, as shown in Fig. 5. Since the energy at the near field will propagate to far field in compressible fluid, the local high pressure between the bubble and free surface decreases. And the height of the spike in compressible fluid is smaller than that in incompressible fluid. Notice that, at the moment of the jet impact, for the bubble just once oscillation, the shapes of the bubble in compressible and incom-pressible fluid are similar. Fig. 6 shows the influence of compressibility on the maximum radius with different distances between the initial bubble center and the free surface, and ΔR is the maximum radius difference of the bubble in compressible and incompressible fluid. After considering the compressibility, the maximum radius of the bubble decreases. With the decrease of the distance between the initial bubble center and the free surface, ΔR increases, i.e. the free surface enhances the influence of compressibility on the bubble motion.

Properties of noise induced by the bubble near the free surface
The properties of noise induced by the bubble near the free surface are discussed here. Fig. 7 shows the history of the far-field sound pressure at the observer. The parameters of the bubble are R 0 =0.15, ε=95.94, and λ=0.9. The observer is arranged at the bottom of the bubble, and the coordinate of the observer is (0, -50.9), i.e. the distance between the observer and the initial bubble center is 50, which is much larger than the size of the bubble. The dash line in Fig. 7 represents the sound pressure induced by the bubble in the free field with the same parameters. When the motion amplitude of the free surface is small, according to the linear free surface theory, it can assume that there is a virtual bubble existing at the other side of the free surface, and the motion phase of this virtual bubble is opposite to the real    YE Xi et al. China Ocean Eng., 2018, Vol. 32, No. 1, P. 26-31 29 bubble. Now the virtual bubble and the real bubble form a "dipole" with different source and sink strengths, while the single bubble in the free field can be treated as monopole.
That is to say, the characteristics of the far-field radiation noise of the bubble near the free surface are different from those of the bubble in the free field. It is known that the influence of monopole on a certain point p in the free field is decayed as 1/r p , r p being the distance between the bubble and Point p. So for the observer, with the linear free surface theory, the influence ratio of real bubble and virtual bubble is: where l is the distance between the real bubble and the virtual bubble. We can find that when r m l, ε≈1. For this case the observer is far from the source, and the perturbation at the observer induced by the bubble is similar to that at the free surface, i.e. the influence of the free surface will be obvious. When l>r m , ε>2. For this case the influence of the bubble on the observer is larger than that of the free surface, and with the increment of ε, the characteristics of the sound pressure at the observer tend to those in free field. As shown in Fig. 7, with the initial high speed expansion of the bubble, the sound pressure at the observer forms a peak at first. With the influence of the free surface, the expansion speed of the bubble increases. Thus the perturbation induced by the bubble at this motion stage is enlarged, and the peak value of the sound pressure at the observer also increases compared with that in free field. Later on, the perturbation induced by the free surface propagates to the observer, which is anti-phased to the perturbation induced by the bubble. This anti-phased perturbation accelerates the sound pressure tending to negative, so the sound pressure is decayed to the wave trough more quickly than that in the free field and forms a sharp wave trough. On the one hand, the existence of the free surface shortens the bubble oscillation period; on the other hand, it also limits the maximum volume during the bubble expansion. Thus, the bubble near the free surface begins to contract earlier than that in the free field. At this moment, the bubble becomes non-spherical obvious, the high speed jet is formed opposite to the free surface, and the volume is reduced fast. Then, the sound pressure at the observer begins to tend to positive. At the jet stage, the distance between the bubble and free surface is just small, and the free surface will limit the motion of the bubble. So the bubble volume is still large when the jet is formed, and has not enough contraction leading to less perturbation. Thus the induced sound pressure at the observer is smaller than that in the free field. Fig. 8 shows the sound pressure with different distances between the bubble and free surface. In this figure, λ=0.9, 2.1, 4, and the parameters of the bubble are unchanged. The observer is arranged at the bottom of the bubble on the symmetry axis, and r m is always 50. With the increment of λ, the propagating time of perturbation induced by the free surface to the observer increases, i.e. the anti-phased sound pressure is later to affect the observer. Thus, the appearance of the wave trough will be put off, and after superimposing with the perturbation induced by the bubble, the value of the wave trough increases. At the jet stage, with the increment of λ, the limit from the free surface to the motion of the bubble is reduced. Thus, the volume of the bubble at this stage decreases, and the bubble can contract enough leading to more perturbation, increasing the sound pressure at the observer. Fig. 9 shows the sound pressure with different r m (λ=0.9). With the reduction of r m , the influence of the free   surface on the observer increases. But the influence ratio of the free surface and the bubble on the observer is reduced. Thus, the sound pressure at the observer tends to that in the free field, and the sharp wave trough caused by the free surface disappeares gradually.

Conclusions
The boundary integral equation considering the compressibility of fluid is used to solve the motion of the bubble near the free surface. Based on the bubble motion solution, the moving boundary Kirchhoff integral equation is utilized to calculate the sound pressure at far field, and the influence of the free surface on the noise is researched. The main conclusions are as follows.
(1) With the compressibility of fluid, the amplitude of the bubble oscillation is reduced, and the free surface amplifies the influence of compressibility on the bubble motion.
(2) The influence of the free surface and bubble on the far-field noise at the observer depends on the distance between the observer, bubble and free surface. If the distance between the observer and bubble is much larger than that between the bubble and free surface, the influence of the bubble and free surface on the sound pressure at the observer is similar, and the sharp wave trough appears. If the distance between the observer and bubble approximates that between the bubble and free surface, the sound pressure at the observer is affected by the bubble motion primarily, and the sharp wave trough disappears.