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Figure 13. Effect of k(hyo): (a) Evolution of bubble amplitude, h, for cases 1, 3 and (Ninin = 13 k (Ago) = 1.1 x 10~°, 1.1 x 107-4 and 0.011). (b) Evolution of bubble amplitude, h, fc mode-coupling cases 16, 17 and 18 (k(hyo) =4.7 x 10~®, 4.7 x 10-4 and 0.047).

Figure 13 Effect of k(hyo): (a) Evolution of bubble amplitude, h, for cases 1, 3 and (Ninin = 13 k (Ago) = 1.1 x 10~°, 1.1 x 107-4 and 0.011). (b) Evolution of bubble amplitude, h, fc mode-coupling cases 16, 17 and 18 (k(hyo) =4.7 x 10~®, 4.7 x 10-4 and 0.047).

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Ficure 1. Histogram of RT growth constant a from simulations and experiments.
Ficure 1. Histogram of RT growth constant a from simulations and experiments.
cell arrangement is used for the pressure and velocities, with the pressure node located at the cell centre, while the velocities are computed at the cell faces. Explicit timestepping was used in the solution of these equations, with the size of the time step chosen to satisfy a Courant condition. A three-stage fractional step algorithm is used to advance the solution in time. At each step, an advection calculation followed by a Langrangian source term update is performed. In the Lagrangian calculation, intermediate velocities are obtained as where E,N and T are the east, north and top cell faces, and g,=g,=0. The superscript n + 1/2 denotes an intermediate value obtained after the advection update, and the asterisk represents the results of the Lagrangian update. To satisfy continuity, the net mass flux across each cell is set to zero:
cell arrangement is used for the pressure and velocities, with the pressure node located at the cell centre, while the velocities are computed at the cell faces. Explicit timestepping was used in the solution of these equations, with the size of the time step chosen to satisfy a Courant condition. A three-stage fractional step algorithm is used to advance the solution in time. At each step, an advection calculation followed by a Langrangian source term update is performed. In the Lagrangian calculation, intermediate velocities are obtained as where E,N and T are the east, north and top cell faces, and g,=g,=0. The superscript n + 1/2 denotes an intermediate value obtained after the advection update, and the asterisk represents the results of the Lagrangian update. To satisfy continuity, the net mass flux across each cell is set to zero:
viscosity for multi-mode simulations is similar to that of a physical viscosity, as it sets an upper bound the initial conditions, for the fastest growing modes. Smaller wavelengths, present in or generated through nonlinear interactions (mode-coupling), are smeared out by the numerical viscosity. A similar analysis was performed for the nonlinear phase o a Froude number of f single-wavelength bubble evolution. The kA =0.196 case gave ~ 0.6 (obtained by tracking the bubbletip), which is in good agreement with the value of 0.56 for a square periodic lattice of bubbles (Li, Jin & Glimm 1996). Since bubble velocities converged faster in the nonlinear stage, the more stringent resolution criterion of 8 zones/A, obtained from the analysis of the linear stage, was adopted.
viscosity for multi-mode simulations is similar to that of a physical viscosity, as it sets an upper bound the initial conditions, for the fastest growing modes. Smaller wavelengths, present in or generated through nonlinear interactions (mode-coupling), are smeared out by the numerical viscosity. A similar analysis was performed for the nonlinear phase o a Froude number of f single-wavelength bubble evolution. The kA =0.196 case gave ~ 0.6 (obtained by tracking the bubbletip), which is in good agreement with the value of 0.56 for a square periodic lattice of bubbles (Li, Jin & Glimm 1996). Since bubble velocities converged faster in the nonlinear stage, the more stringent resolution criterion of 8 zones/A, obtained from the analysis of the linear stage, was adopted.
FiGureE 3. Initial perturbations for simulation 10 in (a) physical, (c) wavenumber space, and (e) azimuthally averaged in wavenumber space. Initial perturbations for simulation 18 in (b) physical, (d) wavenumber space, and (f) azimuthally averaged in wave number space.
FiGureE 3. Initial perturbations for simulation 10 in (a) physical, (c) wavenumber space, and (e) azimuthally averaged in wavenumber space. Initial perturbations for simulation 18 in (b) physical, (d) wavenumber space, and (f) azimuthally averaged in wave number space.
TABLE 2. The nonlinear transition time % for the 17 simulations.
TABLE 2. The nonlinear transition time % for the 17 simulations.
FiGure 4. Evolution of bubble and spike amplitudes (A, and h,), and integral width W for (a) case 10 (Nin =2) and (b) case 1 (Nin = 1).
FiGure 4. Evolution of bubble and spike amplitudes (A, and h,), and integral width W for (a) case 10 (Nin =2) and (b) case 1 (Nin = 1).
image by an autocorrelation procedure described below. The values of a, and ,, and other quantities, are then obtained by averaging for t <t<t,,. Table 2 also the mode-coupling transition times from the calculation. For the simulations initialized with Nmin=1, the transition to pure mode-coupling dominance never occurs. For cases with low-amplitude initial perturbations, the products of mode-coupling may play a greater role in the dynamics owing to their increased relative significance. This is discussed in detail in §5. Thus, it is challenging to completely isolate the contributions of mode-coupling and long-wavelength saturation in such studies. The ratio h,/h, ~ 1.25 observed in figure 4(a) at late time is consistent with experi- mental observations for A ~ 0.5 (Dimonte & Schneider 2000). (Their power-law depen- i ese: Tope Be certo: aol weetien den “ea lt: een] iter since:
image by an autocorrelation procedure described below. The values of a, and ,, and other quantities, are then obtained by averaging for t <t<t,,. Table 2 also the mode-coupling transition times from the calculation. For the simulations initialized with Nmin=1, the transition to pure mode-coupling dominance never occurs. For cases with low-amplitude initial perturbations, the products of mode-coupling may play a greater role in the dynamics owing to their increased relative significance. This is discussed in detail in §5. Thus, it is challenging to completely isolate the contributions of mode-coupling and long-wavelength saturation in such studies. The ratio h,/h, ~ 1.25 observed in figure 4(a) at late time is consistent with experi- mental observations for A ~ 0.5 (Dimonte & Schneider 2000). (Their power-law depen- i ese: Tope Be certo: aol weetien den “ea lt: een] iter since:
There are ~ 120 bubbles at Agt? =2, that coalesce to ~9 large bubbles by the end of the calculation. This corresponds to about log,(120/9) ~ 3.7 generations of bubble evolution. It is evident that even at Agt? =16, the bubbles do not approach the box dimension L = 1cm. Thus, we may assume that the flow is not affected by the bounda- ries throughout the calculation. The radial autospectra Ez,(k) of the bubble front function Z, (x, y) is defined to ensure
There are ~ 120 bubbles at Agt? =2, that coalesce to ~9 large bubbles by the end of the calculation. This corresponds to about log,(120/9) ~ 3.7 generations of bubble evolution. It is evident that even at Agt? =16, the bubbles do not approach the box dimension L = 1cm. Thus, we may assume that the flow is not affected by the bounda- ries throughout the calculation. The radial autospectra Ez,(k) of the bubble front function Z, (x, y) is defined to ensure
FiGureE 7. (a) Test image of radius 20 pixels with a parabolic (b) intensity profile. Jo is the peak intensity value at the centre of the image.
FiGureE 7. (a) Test image of radius 20 pixels with a parabolic (b) intensity profile. Jo is the peak intensity value at the centre of the image.
where the summations are over O< x’, y' <L. The azimuthal average (7), is obtained by averaging n(x, y) rotated through 24 angles. This technique was applied to test images with objects of known diameters, and it was found that the radial location where (7), dropped to a value of 0.3 corresponded to the mean radius of the test objects. Figures 7(a) and 7(b) show a typical test object and the radial profile of intensity (along the dashed line), respectively. A parabolic intensity profile was chosen for the test image, to match the observed profiles of the bubble fronts. The corresponding autocorrelation function n(x, y) and its azimuthal average (7), are shown in figures 8(a) and 8(b), respectively. The threshold value of 0.3 accurately
where the summations are over O< x’, y' <L. The azimuthal average (7), is obtained by averaging n(x, y) rotated through 24 angles. This technique was applied to test images with objects of known diameters, and it was found that the radial location where (7), dropped to a value of 0.3 corresponded to the mean radius of the test objects. Figures 7(a) and 7(b) show a typical test object and the radial profile of intensity (along the dashed line), respectively. A parabolic intensity profile was chosen for the test image, to match the observed profiles of the bubble fronts. The corresponding autocorrelation function n(x, y) and its azimuthal average (7), are shown in figures 8(a) and 8(b), respectively. The threshold value of 0.3 accurately
FiGure 12. Histogram of Froude numbers from the 18 simulations.
FiGure 12. Histogram of Froude numbers from the 18 simulations.
TABLE 3. Summary of results from the simulations.
TABLE 3. Summary of results from the simulations.
had energy confined to modes 16-32. The values of k(ho,) were 4.7 x 10~°, 4.7 x 10° and 0.047 for cases 16, 17 and 18, respectively (note that for these cases, k(ho, is not constant since a k~? spectrum was not used; the value reported here we computed for k, =21//,). Figure 13(b) shows the time traces of bubble amplitude f¢ the mode-coupling simulations. Although all three cases saturate at different tim« owing to different initial amplitudes, they appear to have the same slope during the self-similar evolution. Inspection of table 3, shows that a, is insensitive to k(ho,) whe the large scales are generated purely by mode coupling. Furthermore, a, takes u a lower-bound and universal value of 0.02—0.03 for the mode-coupling simulatior (within the error bounds reported here). saccaa wen ea = | oy ae - Ba & we are ee a
had energy confined to modes 16-32. The values of k(ho,) were 4.7 x 10~°, 4.7 x 10° and 0.047 for cases 16, 17 and 18, respectively (note that for these cases, k(ho, is not constant since a k~? spectrum was not used; the value reported here we computed for k, =21//,). Figure 13(b) shows the time traces of bubble amplitude f¢ the mode-coupling simulations. Although all three cases saturate at different tim« owing to different initial amplitudes, they appear to have the same slope during the self-similar evolution. Inspection of table 3, shows that a, is insensitive to k(ho,) whe the large scales are generated purely by mode coupling. Furthermore, a, takes u a lower-bound and universal value of 0.02—0.03 for the mode-coupling simulatior (within the error bounds reported here). saccaa wen ea = | oy ae - Ba & we are ee a
Figure 15. Effect of k(hyo): (a) Evolution of bubble wavelength, 2, for cases 1, 3, and € (Nmin = 1; k (Axo) =1.1 x 10%, 1.1 x 10-4, and 0.011) scaled to 2,. (b) Evolution of B, for cases 1, 3, and 6.
Figure 15. Effect of k(hyo): (a) Evolution of bubble wavelength, 2, for cases 1, 3, and € (Nmin = 1; k (Axo) =1.1 x 10%, 1.1 x 10-4, and 0.011) scaled to 2,. (b) Evolution of B, for cases 1, 3, and 6.
FiGurE 16. Comparison of 6, from the model and Navier-Stokes. Open circles show insensitivity of the mode-coupling cases to the initial amplitudes.
FiGurE 16. Comparison of 6, from the model and Navier-Stokes. Open circles show insensitivity of the mode-coupling cases to the initial amplitudes.
to the initial perturbations approaching their nonlinear saturation limit. Note that a, also saturates for k(ho,) = 0.01, in contrast with the model. One reason for the reduced sensitivity of 6, at low amplitudes may be mode- coupling effects. Although these calculations were initialized with k~? spectra, mode- coupling may also be present and contribute to the dynamics at late time. Haan (1991) and Dimonte (2004) suggest adding the mode-coupling contributions in quadrature o the long-wavelength amplitudes already present in the initial conditions (ambient modes). If the amplitudes of the ambient modes are low, the relative significance of the mode-coupling products increases. Consequently, a, and 6, may become less sensitive o k(ho,) at small amplitudes. This is seen in the 6, behaviour and, to a smaller extent, in figure 14 for a». There appears to be a transition point at k(hq) ~ 3 x 10~+, below which mode-coupling desensitizes B,. A similar transition occurs at k(Ao.) ~ 3 x 107+ for the molecular mix fraction, bubble density and the kinetic energy dissipation (§ 5.2). Furthermore, it is possible that mode-coupling has a greater effect on 6, than on ap. For instance, the merger of two bubbles of nearly equal diameter would increase 6, by 2'/3, but the corresponding terminal velocity would be higher by only 2'/°. Of course, in addition to the terminal velocity, a, also depends on the length of time At over which he merger takes place, i.e. the merger rate. (Cheng, Glimm & Sharp (2002) from their analysis of images from the rocket rig experiment, report the distribution of bubble diameters and their positions, and infer a bubble merger rate from applying single- mode dynamics to the bubble envelope.) Three-dimensional simulations initialized with two different wavelengths may clarify the bubble merger process in detail. Note that in the experiments, both mode-coupling and long-wavelength saturation are present simultaneously, and further studies quantifying the relative importance of each are required to resolve these issues.
to the initial perturbations approaching their nonlinear saturation limit. Note that a, also saturates for k(ho,) = 0.01, in contrast with the model. One reason for the reduced sensitivity of 6, at low amplitudes may be mode- coupling effects. Although these calculations were initialized with k~? spectra, mode- coupling may also be present and contribute to the dynamics at late time. Haan (1991) and Dimonte (2004) suggest adding the mode-coupling contributions in quadrature o the long-wavelength amplitudes already present in the initial conditions (ambient modes). If the amplitudes of the ambient modes are low, the relative significance of the mode-coupling products increases. Consequently, a, and 6, may become less sensitive o k(ho,) at small amplitudes. This is seen in the 6, behaviour and, to a smaller extent, in figure 14 for a». There appears to be a transition point at k(hq) ~ 3 x 10~+, below which mode-coupling desensitizes B,. A similar transition occurs at k(Ao.) ~ 3 x 107+ for the molecular mix fraction, bubble density and the kinetic energy dissipation (§ 5.2). Furthermore, it is possible that mode-coupling has a greater effect on 6, than on ap. For instance, the merger of two bubbles of nearly equal diameter would increase 6, by 2'/3, but the corresponding terminal velocity would be higher by only 2'/°. Of course, in addition to the terminal velocity, a, also depends on the length of time At over which he merger takes place, i.e. the merger rate. (Cheng, Glimm & Sharp (2002) from their analysis of images from the rocket rig experiment, report the distribution of bubble diameters and their positions, and infer a bubble merger rate from applying single- mode dynamics to the bubble envelope.) Three-dimensional simulations initialized with two different wavelengths may clarify the bubble merger process in detail. Note that in the experiments, both mode-coupling and long-wavelength saturation are present simultaneously, and further studies quantifying the relative importance of each are required to resolve these issues.
Figure 18. Effect of k(hyo): (a) Evolution of global molecular mix fraction © for cases 1, 3 and 6 (Minin =1; k(Ayo) = 1.1 x 10°, 1.1 x 10-* and 0.011). (b) Evolution of K.E./AP.E. for cases 1, 3 and 6. (k(ho,.) = 0.011) the bubblefronts were extracted at Agt? ~ 6. From a visual inspection of figure 17, both images show leading bubbles of roughly the same diameter (and hence the same f, since the amplitudes are the same) consistent with the results of the autocorrelation analysis. We conclude that while it is possible that 6, is truly insensitive to k(ho,) in disagreement with the model, the sensitivity of this parameter may also be muted owing to other factors such as mode-coupling.
Figure 18. Effect of k(hyo): (a) Evolution of global molecular mix fraction © for cases 1, 3 and 6 (Minin =1; k(Ayo) = 1.1 x 10°, 1.1 x 10-* and 0.011). (b) Evolution of K.E./AP.E. for cases 1, 3 and 6. (k(ho,.) = 0.011) the bubblefronts were extracted at Agt? ~ 6. From a visual inspection of figure 17, both images show leading bubbles of roughly the same diameter (and hence the same f, since the amplitudes are the same) consistent with the results of the autocorrelation analysis. We conclude that while it is possible that 6, is truly insensitive to k(ho,) in disagreement with the model, the sensitivity of this parameter may also be muted owing to other factors such as mode-coupling.
FiGureE 19. Global molecular mix fraction © vs. k{hyo).
FiGureE 19. Global molecular mix fraction © vs. k{hyo).
“ where the integral is performed over the ent ire computational domain. The ratio K.E./AP.E. is plotted for cases 1, 3 and 6 in figure 18(b). The fraction of energy jissipated is given by 1 — K.E./AP.E., and ap 50% for these simulations. This is in good proaches a value slightly greater than agreement with the experiments of Ramaprabhu & Andrews (2004), who report a value of D/AP.E.=49 % from their simultaneous measurements of density and ve ocity fields. Figure 20(a) is a plot of K.E./AP.E. from all the simulations, and shows a slight increase with k(/o;,). Dimonte ot al. (2004) argue that the ratio K.E./AP.E. bears a strong correlation with a, for the leading bubbles. In the limit of small densi y differences, they obtain
“ where the integral is performed over the ent ire computational domain. The ratio K.E./AP.E. is plotted for cases 1, 3 and 6 in figure 18(b). The fraction of energy jissipated is given by 1 — K.E./AP.E., and ap 50% for these simulations. This is in good proaches a value slightly greater than agreement with the experiments of Ramaprabhu & Andrews (2004), who report a value of D/AP.E.=49 % from their simultaneous measurements of density and ve ocity fields. Figure 20(a) is a plot of K.E./AP.E. from all the simulations, and shows a slight increase with k(/o;,). Dimonte ot al. (2004) argue that the ratio K.E./AP.E. bears a strong correlation with a, for the leading bubbles. In the limit of small densi y differences, they obtain
Figure 21. Variation of bubble density p, with k (hyo).
Figure 21. Variation of bubble density p, with k (hyo).
peaks at mode numbers 1, 2 and 3, respectively, (thus reducing the mode-coupling contribution), and all had the same amplitude k(ho,) =0.004. Figures 23(a) and 23(b) are plots of a, and #, for various values of Nnin. Although both parameters vary slightly with the minimum mode number, there appears to be no clear trend. As found previously for the initial amplitude and mode-coupling cases, the global molecular mix fraction © and the kinetic energy dissipation showed little dependence on the initial spectral content. WU ce Bees Bcc ct © wee BE eee a ee eee es SB) eee eo ee lO Mo So ee OS ee ow ce ff ff sy
peaks at mode numbers 1, 2 and 3, respectively, (thus reducing the mode-coupling contribution), and all had the same amplitude k(ho,) =0.004. Figures 23(a) and 23(b) are plots of a, and #, for various values of Nnin. Although both parameters vary slightly with the minimum mode number, there appears to be no clear trend. As found previously for the initial amplitude and mode-coupling cases, the global molecular mix fraction © and the kinetic energy dissipation showed little dependence on the initial spectral content. WU ce Bees Bcc ct © wee BE eee a ee eee es SB) eee eo ee lO Mo So ee OS ee ow ce ff ff sy
FiGure 23. Effect of longest wavelength imposed (Nj) on (a) a» and (b) Bp.
FiGure 23. Effect of longest wavelength imposed (Nj) on (a) a» and (b) Bp.
where (e), denotes azimuthal averaging. As before, this definition ensures that Jo. Ep. (k) dk = (f?)—(fi)?. Figure 24(a) is an image of fi(x, y) at z=0 from simulation 18 at Agt? =22, where the large scales are generated purely through mode-coupling. The corresponding azimuthally averaged spectra for the mode-coupling cases 16, 17 and 18 are shown in figure 24(b). As described earlier, (e), was obtained by azimuthally averaging over 30 angles. The solid line indicates the —5/3 slope and shows the presence of a short inertial range with k~°/?. The Reynolds number at Agt? =22 may be estimated as
where (e), denotes azimuthal averaging. As before, this definition ensures that Jo. Ep. (k) dk = (f?)—(fi)?. Figure 24(a) is an image of fi(x, y) at z=0 from simulation 18 at Agt? =22, where the large scales are generated purely through mode-coupling. The corresponding azimuthally averaged spectra for the mode-coupling cases 16, 17 and 18 are shown in figure 24(b). As described earlier, (e), was obtained by azimuthally averaging over 30 angles. The solid line indicates the —5/3 slope and shows the presence of a short inertial range with k~°/?. The Reynolds number at Agt? =22 may be estimated as
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