Foldy according to which bubbles do interact with the average pressure field but not with each other’s field. Note that the sound pressure is assumed to be harmonic, i.e. pS ~ . The equation relating the liquid pressure to the gas pressure is given by (2.13) also yields the following equation for the equilibrium state: (2.14) To complete the formulation, an equation for the internal pressure is needed. It could be either derived from the enthalpy equation , (2.15) where K is the thermal conductivity or from the polytrophic equation for the expansion of perfect gas: (2.16) γ is the ratio of specific heats. When equation (2.16) is used for the thermal equation of state, it refers to an isothermal process for the value of γ=1 and adiabatic process when γ =1.4. Further γ can have complex values which describe other physical phenomena. A more detailed formulation regarding the thermal phenomena can be found in the work of Prosperetti . Instead of using the polytrophic equation of the gas which does not account for the thermal losses, an ad hoc dissipative term is added to that equation by Prosperetti in order to simulate thermal viscous losses.
An intuitive justification of this approximation can be found in the article by Louisnard . Also, relevant discussions were made in Section 1 on the same fact that real part of the wavenumber is well predicted by linear theory as also reported by experimental studies. Finally, with the use of equations (2.49) and (2.50), the Helmholtz equation for the nonlinear wave propagation can be obtained as; (2.51) In Figure 2.5, the real and imaginary part of the wavenumber for the driving frequency 20 kHz, bubble volume fraction β=0.005 % and the distribution of bubbles with same radius R0=5 µm, is plotted. One can observe that the imaginary part of the wave number increases very quickly for the driving amplitudes over the Blake threshold and it can be as high as the real part for very large driving amplitudes. Figure The real and imaginary part of the wavenumber used in the nonlinear wave propagation.
components of r. Equation (3.2) is used to find the potential while the equations for derivatives p/xj are obtained by differentiating (3.2) with respect to xj, i.e. (3.6)‎ The derivatives of the fundamental solution in the integral kernels of Equation (3.6) can be evaluated analytically as given below: (3.7) and (3.8) Equation (3.2) can be discretized as: (3.9) where Hik and Gik are the matrix coefficients arising from the double layer and the single layer integrals, respectively, and Jbn is the number of fictitious nodes on the boundary. Note that in equation (3.9), pj and qj are the values of the potential and the flux, respectively, at the fictitious nodes on the boundary of local sub-domain, whereas pi is the potential value at the source node. Therefore, the parameters pj and
γ1 and γ2 are the discrete indices referring to the difference of the locations of the two field points considered (in x and y directions, respectively). In the LBIE-RBF, given a constant size of influence domain, a discrete form of the equation (3.30) is written for the interior nodes as; (3.49) with s1,s2 being the normalized (with h) distances of the furthest neighbouring nodal points which remain within the influence domain of (q1,q2) in x and y directions, respectively. Here, is the coefficient of the potential value arising in the discretized form of the equation (3.30) written for the node at (q1,q2). By assuming the harmonic evolution of the acoustic pressure of the nodal values given by (3.48), equation (3.49) becomes; (3.50) For a non-zero distribution of the acoustic pressure, uh, it is evident from equation (3.50) that the numerical wavenumber can be determined, for given values of k and the angle of propagation, by solving; (3.51) Dispersion error: Throughout this thesis the results for dispersion are presented as follows; (3.52)
results are plotted along the line y=150 mm. It can be observed that RBIE agrees with COMSOL better than LBIE in the region x=[-0.10, -0.050] and x=[0.050, 0.10]; whereas LBIE agrees better in the range x=[-0.050, 0.050], and also x=[-0.15, -0.10] and x=[0.10, 0.15]. Figure Calculated pressure for the different solutions along the line y=150 mm for k=60. In Figure 4.23, the results are plotted along the line x=0. It is observed that LBIE shows good agreement with COMSOL; so that the difference is not visible in the given plot. RBIE also shows a good agreement overall. Figure Calculated pressure for the different solutions along the line x=0 for k=60. Similarly, the discontinuity around y=0.2 m represents the thin body and P is assigned as zero at this location. 3D example problem In this sub-section, the acoustic wave propagation around a thin plate immersed in a liquid is investigated for a 3D problem. A solution domain with dimensions 100mm x 100mm x 60mm is considered, 60mm being the height (See Figure 4.24). A thin plate with 50mm x 50mm cross-section is placed at the center of the domain, i.e. at the coordinates (50mm, 50mm, 30mm). Two different cases are considered for the thickness of the thin plate, ht, i.e. ht = 1mm and the null thickness (denoted as ht =0
than in 2D problems. An orthogonal frequency dependent basis may be written as follows , (4.17) ϕ is the polar angle and θ is the azimuthal angle. In Figure 4.25, the wave profile obtained by using LBIE for k=5 and ht=0 is given. In the subsequent figures, the wave profile on the same surface is plotted along the line x=[25 mm, 75 mm], y=50 mm. Figure Wave profile on the top surface of the thin for k=5 and ht=0. In Figure 4.26, the pressure wave profile along the line x= [25 mm, 75 mm], y=50 mm is plotted for the null thickness case. It can be observed that the LBIE methods produce very similar results. The COMSOL result indicates slightly higher wave amplitude. Figure Pressure wave profile along the line (x=[25 mm, 75 mm], y=50 mm, z=30mm) for k=5 and ht=0. In Figure 4.27, the pressure wave profile along the line x= [25 mm, 75 mm], y=50 mm is plotted for 1mm thick plate. It can be observed that the LBIE methods
Figure Displacement due to transverse vibrations along the transducer in the z –direction. In Figure 5.12, the wave propagation on the y-z is investigated. The radius distribution range is 1-10 µm, whereas the bubble volume fraction is increased. Similar conclusions to the one made regarding the results in Figure 5.8 can be made. The wavelength decreases with increasing the bubble volume fraction and the attenuation gets higher. The cases β=0.005% and β=0.01% produce similar peak amplitude values; though the pressure profiles are significantly different. Figure Wave distribution for the y-z plane (p∞=101kPa). Effect of bubble volume fraction (R=1-10 µm). In Figure 5.13, the results are presented for the case of bubble radius range 5-50 µm. High damping is observed for larger values of bubble volume fraction. For example, the pressure amplitude decreases down to O(p∞) for β=0.02%. In comparison with Figure 5.12, one can notice that the amount of dissipation is higher for this distribution of bubbles. Figure Wave distribution for the y-z plane (p∞=101kPa). Effect of bubble volume fraction (R=5-50 µm).
where n is the number of influencing nodes (including the source node for which the equation is written) and p is the potential, i.e. ps=p(xs, ys). The coefficient αi’s are evaluated from the boundary integral equation (3.31) and RBF interpolation, which is explained in Section 3.3. Note that the right hand side of (A.1) is zero since the equation is written for a node entirely within the domain, i.e. there is no acoustic sourcing term (unlike equation (3.34)). In the following, we will refer to the harmonic evolution theorem for dispersion error measurement in order to interpret the potential values, pi, (i=1,2,…,n), in (A.1) in terms of the potential at the source node, ps. According to harmonic evolution theorem, the potential at any point i can be written in terms of the potential at point s; (A.2) though the coefficients τi will be left un-evaluated for now (the explicit expressions are given in equation (3.48)). Using the relation (A.2), equation (A.1) can be as; (A.3) Equation (A.3) has the following interpretation. For an acoustic problem within the domain Ω, the rows of the system matrix for the nodes lying entirely within the domain are identical. Further, for a non-trivial solution of the problem, i.e. non-zero values of ps, the following equation is being solved numerically; (A.4) where the coefficients αi and τi are functions of the wavenumber. Therefore, one can solve an equation of type (A.4) by using root-finding algorithms, for one source node only, in order to find out the numerical wavenumber. Hence, the discrepancy between the numerical wavenumber and the exact wavenumber can be determined in order to
A.3 Numerical Results As detailed in Section A.2, RBIE has a different formulation which yields different dispersion errors. In Figure A.2, the dispersion error for the RBIE with the second order polynomial approximation (RBIE-m=6) is shown for the propagation angle . It can be observed that an increase of the radius of influence domain leads to more accurate results in the range . For kh=2, the error still remains less than 10-2 which indicates lower dispersion in comparison to LBIE for this particular case (See Figure 4.12b). In the range , rinfl=4h is preferable to the other two cases. Figure A.2: Dispersion error vs. the non-dimensionalized exact wavenumber for the propagation angle . In Figure A.3, the results obtained for RBIE-m=6, RBIEand RBIE-RBF- for kh=1, are shown. Recalling Figure 4.12a, RBIE perform at the same order of accuracy with the LBIE methods, when rinfl=3h or rinfl=4h, however RBIE produces less accurate results by one order of magnitude when rinfl=2h. RBIE-is the most accurate of the RBIE methods in general with the exceptions obviously occurring when the propagation angle is very different from. Figure A.3: Dispersion over the propagation angle for the non-dimensional wavenumber; (a) rinfl=2h, (b) rinfl=3h and (c) rinfl=4h.
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