How do various forces affect pressure waves in bubbly flows?

This study investigated the weakly nonlinear propagation of pressure waves in compressible, flowing water with spherical microbubbles, considering various forces. Previous theoretical studies on nonlinear pressure waves in bubbly flows did not consider the forces acting on the bubbles, although the validity of ignoring these forces has not been demonstrated. We focused on every possible force such as drag, gravity, buoyancy


I. INTRODUCTION
A pressure wave in a bubbly flow evolves into a shock wave 1-3 or stable wave [sometimes referred to as (acoustic) soliton]; these waves exhibit extremely different properties.A shock wave and an acoustic soliton evolve based on the competition between nonlinearity and dissipation and that between nonlinearity and dispersion, respectively.To predict the evolution waveform, the relative strengths of three properties, i.e., nonlinearity, dissipation, and dispersion, must be clarified.3][14][15][16][17][18][19][20][21][22][23][24][25][26] However, three properties mentioned above are difficult to obtain quantitatively by experiments or numerical analysis alone; nonlinear wave equations based on theoretical analysis 27,28 is an effective method for clarifying these properties.Moreover, the advantage of theoretical analysis is that the factors of wave attenuation can be considered separately and evaluated quantitatively.The weakly nonlinear (i.e., finite but small amplitude) 29 propagation of pressure waves in bubble flows is described by nonlinear wave equations, 30,31 among which the Korteweg-de Vries-Burgers (KdVB) equation [32][33][34][35] for low-frequency long waves is popular.In particular, it is shown that the waveform obtained from the KdVB equation agrees with the experimental outcome. 36As the KdVB equation is expressed as a linear combination of nonlinear, dissipation, and dispersion terms, estimating the functions and values of these three terms leads to the elucidation of the evolution waveform.
2][53][54][55][56][57][58] However, the relationship between the forces acting on bubbles and waves in bubbly flows has not been clarified.Previous theoretical studies [32][33][34][35][36] on nonlinear pressure waves in bubbly flows did not incorporate forces acting on the bubble, although the validity of ignoring these forces has not been demonstrated.This may be because of the preconception that the forces (non-oscillations) do not affect the waves (oscillations).Recently, our previous studies [59][60][61][62][63] introduced the drag force into the KdVB equation and showed that the drag force increased wave dissipation.However, the effects of gravity, buoyancy, and Bjerknes forces on waves have not been discussed.5][66][67][68][69] Therefore, this cannot be ignored when dealing with wave theory.Consequently, this study aimed to elucidate the effects of various forces, such as gravity, buoyancy, and Bjerknes forces on pressure waves in bubbly flows by deriving the KdVB equation.
The remainder of this paper is configured as follows.In Sec.II, we introduce the basic equations of the two-fluid model, including drag, gravity, buoyancy, and Bjerknes forces.In Sec.III, we derive the KdVB equation and show that the gravity and buoyancy forces, such as the drag force, increase the dissipation of the waves, whereas the Bjerknes force increases the nonlinearity, dissipation, and dispersion of the waves.We clarified that the dissipation effect of thermal conduction is the largest, followed by those of acoustic radiation.drag, buoyancy, and gravity, based on numerical analysis.Finally, Sec.IV concludes the paper.Because this study clarifies that the attenuation of waves owing to the forces acting on gas bubbles is quantitatively small, this study is the first demonstration of the validity of ignoring forces for pressure wave propagation in bubbly flows.

II. PROBLEM FORMULATION A. Problem statement
We conduct a theoretical investigation of the weakly nonlinear (i.e., finite but small amplitude) propagation of plane (one-dimensional) progressive pressure waves in flowing compressible water uniformly containing numerous small spherical gas bubbles under various forces, as shown in Fig. 1; i.e., drag, gravity, buoyancy, and Bjerknes (acoustic radiation) forces [see ( 7)-( 10) below] are incorporated.
On the other hand, we shall simply the problem based on the following assumptions: (i) The primary Bjerknes force is accounted for, while the secondary Bjerknes force [70][71][72][73][74] is excluded from consideration; (ii) To simplify the model, direct interactions between bubbles, gas-phase viscosity, Reynolds stress, and phase change and mass transport across the bubble-liquid interface are neglected; (iii) The motion of bubbles is assumed to be spherically symmetric; (iv) Bubbles remain stable, without coalescing, breaking, becoming extinct, or forming anew; (v) The liquid temperature is constant; (vi) In the initial state, both gas and liquid phases flow at constant velocities.

B. Basic equations
To introduce various forces into the interfacial momentum transport, we apply the conservation equations of mass and momentum for the gas and liquid phases based on a two-fluid model 75,76 as follows: where t Ã is the time, x Ã is the space coordinate normal to the wavefront, a is the void fraction (0 < a < 1), l Ã is the viscosity, q Ã is the density, u Ã is the velocity, p Ã is the pressure, and P Ã is the liquid pressure averaged over the bubble-liquid interface. 76The superscript Ã denotes a dimensional quantity, and the subscripts G and L denote the volumeaveraged variables in the gas and liquid phases, respectively.As interfacial momentum transport terms, the following model of virtual mass force 50 is introduced: where b 1 , b 2 , and b 3 are constants that can be set as 1/2 for a spherical bubble.The Lagrange derivatives D G =Dt Ã and D L =Dt Ã are defined as follows: Furthermore, we introduce a model for the drag force term F Ã dr for spherical bubbles, 59 where R Ã is a representative bubble radius and C D is the drag coefficient for a single spherical bubble.We also introduce the gravity, buoyancy, and (primary) Bjerknes forces, 48,49 where g Ã is the acceleration of gravity and B is a constant.Note that gravity acting on each phase is considered.We will focus on the presence of B to verify the effect of the Bjerknes force, i.e., B ¼ 0 and B ¼ 1 correspond to the cases of without and with the Bjerknes force, respectively.9][80] This approach integrates the dynamics of bubble oscillation and translation to comprehensively describe their motion: FIG. 1. Schematic illustration of one-dimensional propagation of pressure waves in bubbly flows; drag, gravity, buoyancy, and acoustic radiation (primary Bjerknes) forces acting on each bubble.The direction of these forces is an example.
where c Ã L0 is the initial speed of sound in pure water.In this study, the energy equation 81 for thermal conduction at the bubble-liquid interface is introduced to account for the thermal effects within the bubble: where T Ã G is the gas temperature, j is the ratio of specific heats, r Ã is the radial distance from the center of the bubble, and k Ã G is the thermal conductivity of the gas inside the bubble.Prosperetti 81 did not use a temperature-gradient model.However, certain models for the temperature-gradient as the first term on the right-hand side of (12) were proposed.This study uses the model proposed by Sugiyama et al., 82

@T
where Re and Im denote the real and imaginary parts, respectively.The physical quantities in the initial state are denoted by the subscript 0 and are constants.Certain symbols are defined as follows: 82 where x Ã B is the eigenfrequency of a single bubble, c e is the effective polytropic exponent, l Ã e0 is the initial effective viscosity, r Ã is the surface tension, i denotes an imaginary unit, and C N , a N , and e L Ã p are complex numbers.
To close the set of (1)-( 4), (11), and ( 12), the equation of state for an ideal gas, the Tait equation of state for liquid, the mass conservation law of gas inside the bubbles, and the balance of normal stresses across the bubble-liquid interface, are introduced as follows: where n is a material constant (e.g., n ¼ 7.15 for water).

C. Analysis on multiple scales
Using the method of multiple scales, 31 we introduce four scales as extended independent variables.This approach is based on the assumption of a finite but small nondimensional wave amplitude, denoted as ð(1Þ: where the nondimensional independent variables are defined by t ¼ t Ã =T Ã and x ¼ x Ã =L Ã ; T Ã is a typical period and L Ã is a typical wavelength.Here, the subscripts 0 and 1 correspond to the near and far fields, 31 e.g., t 0 is the nondimensional time for the near field.Note that the difference between the constant is denoted by subscript 0 and near field by 0. The dependent variables are then nondimensionalized and expanded in the power series of , as follows: where U Ã ð L Ã =T Ã Þ is the typical propagation speed, and the initial nondimensional pressures p G0 and p L0 are defined as Further, the ratio of the initial densities of the gas and the liquid phases is sufficiently small.
By using nondimensional ratios based on , the low-frequency long wave is described by x where V, D, and X are the constants of O(1).Equations ( 32)- (34) correspond to the present acoustic properties of bubbly flows; i.e., the speed of sound within these flows is significantly lower compared to that in pure water, the initial bubble radius is markedly smaller than the typical wavelength observed in such environments, and the incident frequency of waves within bubbly flows is substantially lower than the eigenfrequency of individual bubbles.We determine the sizes of the nondimensional numbers for the thermal effect: 61,63 The nondimensionalization of the acceleration owing to gravity g Ã is where g is a constant of O(1).The nondimensionalization of the liquid viscosity l Ã L is defined by where l L is a constant of O(1).The drag coefficient C D is defined by where A is a constant (e.g., A ¼ 16), and C D depends on the Reynolds number Re (C D ¼ A=Re). 83I.RESULTS A. Linear propagation at near field By substituting ( 24)-( 38) into (1)-( 4), (11), and ( 12), we obtain a set of linear equations using ( 20)-( 23) from the leading-order approximation: Although gravitational and buoyancy forces do not appear here, the effect of the Bjerknes force (F Ã bje ¼ ÀBa@p Ã L =@x Ã ) is described by the last term on the left side of ( 41) and (42).
Combining ( 39)-( 44) results in a linear wave equation for the first-order variation in the bubble radius R 1 , where v p is the phase velocity expressed as The linear Lagrange derivative D=Dt 0 is defined as For simplicity, the initial velocities of both phases are assumed to be the same (u G0 ¼ u L0 u 0 ).However, the perturbations of the velocities are not the same (u G1 6 ¼ u L1 ).Setting v p ¼ 1 yields the explicit form of U Ã as Value of the typical propagation speed U Ã is increased by considering the Bjerknes force as shown in Table I.By focusing on the right-running wave (i.e., by introducing the moving coordinates u 0 ¼ x 0 À v p t 0 ), a 1 , u G1 , u L1 , p L1 , and T G1 are expressed in terms of R 1 .

B. Nonlinear propagation at far field
As in the case of OðÞ, the following set of inhomogeneous equations for Oð 2 Þ is derived: where the inhomogeneous terms K i (1 i 6) are explicitly presented in the Appendix.Consequently, ( 51)-( 56) are combined into a single inhomogeneous equation, where f ¼ f ðt 1 ;x 1 ;u 0 Þ is the first order perturbation of the nondimensional bubble radius R 1 and K is given by Based on the solvability condition for (57), K ¼ 0 is required. 31From ( 24), the original independent variables x and t are restored Finally, we obtain the KdVB equation 2. The effect of Bjerknes forces on each coefficient: (a) Absolute value of nonlinearity jP 1 j, (b) dissipation owing to liquid compressibility jP 2 j (see also Table II), (c) dispersion P 3 (see also Table III), and (d) dissipation P 4 as a function of a 0 for the case of R Ã 0 ¼ 500 lm, using a variable transform where the constant coefficients are expressed as where P 0 is the advection coefficient, P 1 is the nonlinear coefficient, P 2 and P 4 are the dissipation coefficients, and P 3 is the dispersion coefficient.Furthermore, P 2 , P 4buo , P 4gr , P 4dr , and P 4th are the dissipation coefficients owing to acoustic radiation (i.e., liquid compressibility), buoyancy, gravity, drag, and thermal conduction, respectively.3. Comparison of the dissipation coefficients: (a) The red and black curves represent the coefficient owing to thermal conduction and the sum of drag, buoyancy, and gravity; (b) the black, green, and blue curves represent the coefficient owing to drag, buoyancy, and gravity, respectively (see also Table IV).The condition is the same as that in Fig. 2.
In this way, the present theory could divide the total attenuation of waves into independent attenuation components due to various forces.

C. Discussion
The relationship between the Bjerknes force and coefficients is shown in Fig. 2 and Tables II and III.The absolute values of nonlinear, dissipation, and dispersion coefficients increased owing to the Bjerknes force.In particular, the effect of the Bjerknes force on P 4 is significant.A comparison of the dissipation coefficients P 4 is shown in Fig. 3 and Table IV.The dissipation effect of thermal conduction was the largest, followed by those of drag, buoyancy, and gravity.It should be noted that, as this result depends on the temperature-gradient model, 82 the influence of thermal conduction on the waves may be overestimated.
The dissipation term owing to acoustic radiation (i.e., P 2 @ 2 f =@n 2 ) has a different mechanism from that owing to drag, gravity, buoyancy, and thermal conduction (i.e., P 4 f ) with respect to the unknown variable.We then conducted a numerical analysis using a spectral method based on the split-step Fourier method used in previous studies 59,63,84 to compare each dissipation effect.Our previous study 63 considered three dissipation effects and indicated that the dissipation effect of the thermal conduction was the largest, followed by those of acoustic radiation and drag force obtained from the numerical analysis.Figure 4 illustrates the temporal evolution of the numerical solutions to the KdVB equation (60).The black, blue, and red curves represent waveforms with only acoustic radiation, with drag, gravity, and buoyancy forces and with only thermal conduction, respectively.The initial waveform of the solution is assumed to be a cosine wave.The dissipation effect of thermal conduction was the largest, followed by those of acoustic radiation, drag, buoyancy, and gravity.This order was effective in the range from R Ã 0 ¼ 50 lm to 1 mm.As shown in Table IV, the larger the initial bubble radius R Ã 0 the smaller the damping effect of drag P 4dr .At R Ã 0 ¼ 5 mm, the damping effect of drag was smaller than that of gravity.At R Ã 0 ¼ 10 lm, the damping effect of drag was larger than that of acoustic radiation.

IV. CONCLUSIONS
Many previous theoretical studies (e.g., Refs.32-36) have not clarified the relationship between the forces acting on bubbles and waves in bubbly flows.Although the validity of ignoring forces acting on the bubble has not been demonstrated, previous theoretical studies on nonlinear pressure waves in bubbly flows did not incorporate these forces.In this study, we theoretically examined the weakly nonlinear propagation of plane (one-dimensional) pressure progressive waves in water flows uniformly containing many spherical bubbles, particularly focusing on the effects of gravity, buoyancy, and (primary) Bjerknes forces acting on bubbles.Using the singular perturbation method (multiple scales analysis), the KdVB equation describing weakly

FIG. 4.
Example of the numerical solution of (60) for a 0 ¼ 0:001 and R Ã 0 ¼ 500 lm.Horizontal axis is the nondimensional space coordinate n.The vertical axis is the first order perturbation of the nondimensional bubble radius f.The period is (a) 0, (b) 20, (c) 100, (d) 200, and (e) 300.The black, blue, and red curves represent the waveforms with only the acoustic radiation, with the drag, gravity, and buoyancy force, and with only the thermal conduction, respectively.The condition is the same as that in Fig. 2. Other conditions are grid steps 1024, the duration of the numerical integration of time 0.001, and the size of computational domain 8p.
nonlinear propagation of long waves with a low frequency was derived.The following findings were obtained: (i) The Bjerknes force acting on the bubbles contributed to the nonlinearity, dissipation, and dispersion of waves and increased the three effects.(ii) The drag, gravity, and buoyancy forces acting on the bubbles contributed to dissipation and increased the value of the dissipation coefficients.(iii) In the range from R Ã 0 ¼ 50 lm to 1 mm, the dissipation effect decreased in the order: thermal conduction, acoustic radiation, drag, buoyancy, and gravity.
This study revealed that the attenuation of waves owing to the forces acting on gas bubbles is quantitatively small.This is the first study to demonstrate the validity of ignoring forces for pressure wave propagation in bubbly flows.
A lift could not be introduced here because this study considered only the one-dimensional case.The effect of forces such as lift [44][45][46] and buoyancy force on waves will be investigated in future research in the framework of multidimensional problem.
Note that s 1 , s 2 , and s 3 change due to the effects of the Bjerknes force, whereas s 4 and s 5 do not change.

TABLE III .
Detailed value of the dispersion coefficient P 3 in Fig.2.

TABLE IV .
Detailed value of comparison of drag, buoyancy, and gravity in Fig.3.