3. Results and analysis

Before presenting the results from the statistical analysis, we show in figure 1 some characteristic evolution of the acceleration magnitudes, $a$, along tracers and light particles trajectories near strong compression structures (e.g. shocklets) at $M_t=0.51$ and $M_t=1.00$, colour coded in terms of the local fluid compressibility $\theta$. For tracer particles, there is no significant difference between $M_t=0.51$ and 1.00, except for the larger peak of acceleration magnitudes near shocklets at $M_t=1.00$. For bubbles at $M_t=0.51$, the evolution of acceleration magnitudes is similar to that of tracer particles. However, at $M_t=1.00$, bubbles seem to be trapped near the shocklets for a long time (several times $\tau _{\eta }$), and the acceleration magnitudes of particles fluctuate dramatically. For the sake of completeness, the root-mean-square of acceleration magnitudes, $a_{rms}$, is shown in table 2. As one can see, as $M_t$ increases from 0.36 to 1.00, $a_{rms}$ of tracers increases slightly, whereas $a_{rms}$ of bubbles increases dramatically with increasing $M_t$, especially for $M_t > 0.64$. In addition, for light particles the root-mean-square values of the two terms in the right-hand side of (2.6) are also measured (not shown). The root-mean-square of the drag term is comparable to (slightly smaller than) that of the $\beta \,\mathrm {D}\boldsymbol {u}_p/ \mathrm {D}t$ term at all $M_t$.

Figure 1.

The evolution of the acceleration magnitude, $a$, of typical tracers (a,b) and light particles (c,d) at $M_t=0.51$ and $M_t=1.00$. The colourbar indicates the divergence of fluid velocity, $\theta$, at the position of particles. Here $a_{rms}$ and $\theta _{rms}$ represent the root- mean-square values of $a$ and $\theta$, respectively.

Table 2.

Values for $a_{rms}$ of tracers and bubbles at different $M_t$ values.

The p.d.f.s of one component of the particle accelerations, $a_i$, with $i=1, 2, 3$, are illustrated in figure 2. It is apparent that all p.d.f.s have some intense stretched tails, indicating that the accelerations of tracer and bubbles are both intermittent. Unfortunately, the multifractal model used to predict successfully the p.d.f. of acceleration for tracers (Biferale et al. Reference Biferale, Boffetta, Celani, Devenish, Lanotte and Toschi2004) in incompressible flows cannot be translated easily to the case of inertial particles. This is due to the fact that it is not known how to correlate the local fluctuations of the Kolmogorov time with the effects of preferential concentration and filtering induced by the presence of inertia. Furthermore, the compressible effects add another direction that is not fully under control with the multifractal approach. The strength of intermittency can be quantified by the flatness of the acceleration component defined as $F_a = \langle (a_1)^4 \rangle / \langle (a_1)^2 \rangle ^2$, where here by $\langle \cdot \rangle$ we indicate the average among all particles. For tracers, $F_a$ increases steadily with the increase of $M_t$, as seen in figure 3. However, $F_a$ of light particles does not follow this trend. As $M_t$ increases from 0.36 to 1.00, $F_a$ of bubbles first increases, then reduces after $M_t$ exceeds about 0.6. The increase of $F_a$ of tracer particles (also applicable for bubbles at $M_t < 0.6$) is attributed to the fact that increasing compressibility increases the flow intermittency. However, when $M_t > 0.6$, why does $F_a$ of light particles decrease with the increase of $M_t$? Figure 4 shows the marginal cumulated distribution $P(a,\theta _m)=\int _{-\infty }^{\theta _m} P(a,\theta ) \,\mathrm {d}\theta$ of particle acceleration components conditioned to fall in different flow regions. We define CR1, CR2 and CR3 to represent the flow regions where $\theta _m < -\theta _{rms}$, $\theta _m < -2\theta _{rms}$ and $\theta _m < -3\theta _{rms}$ separately. For the sake of clarity, we mark the boundaries of CR1, CR2 and CR3 in figure 5 where the p.d.f.s of the divergence of fluid velocity at different $M_t$ are shown. Figure 4 reflects that the stretched tails of acceleration p.d.f.s result predominantly from the particles in strong compression regions when $M_t > 0.6$. In addition, $F_a$ is impacted heavily by the tails of acceleration p.d.f.s, as shown in the insets of figure 2. Therefore, we argue that the decrease of $F_a$ should be related to the statistical properties of bubbles in compression regions when $M_t>0.6$.

Figure 2.

The p.d.f.s, $P(a_{i})$, of one component of the accelerations $a_i$, normalized with its root-mean-square. Because of isotropy, all three components are equivalent. For clarity, here we show only the case with $i=1$. Panel (a) is for tracer particles. Inset: $(a_1)^4P(a_1)$ at $M_t=1.00$ is presented to check the statistical convergence of the fourth-order moments. Panel (b) is for light particles. Inset: $(a_1)^4P(a_1)$ is shown at $M_t=0.64$.

Figure 3.

The flatness factor of particle acceleration components, $F_a$, at changing $M_t$ for tracers (red squares) and light particles (blue diamonds). The error bar is calculated from the scatter among 30 subsamples obtained from 30 time snapshots.

Figure 4.

The cumulated marginal p.d.f.s of particle acceleration components $P(a,\theta _m)$, with $\theta _m=-\theta _{rms}$, $-2\theta _{rms}$ and $-3\theta _{rms}$ for CR1, CR2 and CR3, respectively. The black solid line is the reference p.d.f. obtained by considering the particles on the whole computational domain ($\theta _m=+\infty$). The red dotted, green dashed and blue dash-dotted lines are for CR1, CR2 and CR3, respectively. Panels (a–c) are for tracer particles at $M_t=0.51$, 0.64 and 1.00, respectively; (d–f) are for light particles at $M_t=0.51$, 0.64 and 1.00.

Figure 5.

The p.d.f.s of the divergence of fluid velocity, $\theta$, evaluated in the whole computational domain at $M_t=0.51$, $0.64$ and $1.00$. The three vertical lines separately mark the boundaries of CR1, CR2 and CR3, where $\theta < -\theta _{rms}$, $\theta < -2\theta _{rms}$ and $\theta < -3\theta _{rms}$.

Let us first compare $F_a$ values of light particles in different regions, as shown in figure 6, where: case  1 represents the whole domain; case 2 represents the region with $\theta >-3\theta _{rms}$ (removing CR3 from case 1); case 3 represents the region with $\theta >-2\theta _{rms}$ (removing CR2 from case 1); case 4 represents the region with $\theta >-\theta _{rms}$ (removing CR1 from case 1). We can notice that at $M_t=0.64$, $F_a$ decreases sharply once the light particles in CR3 are removed. In contrast, at $M_t=1.00$, $F_a$ increases steadily as the particles in CR3, CR2 and CR1 are removed. Figure 7 also shows the p.d.f.s of $a_1$ of light particles at cases 1, 2, 3 and 4 when $M_t = 0.64$ and $1.00$. The tails of the p.d.f.s of $a_1$ for $M_t = 0.64$ become much narrower when the light particles in strong compression regions are removed, whereas the tails, at $M_t = 1.00$, become wider since the light particles in CR3, CR2 and CR1 are removed. To explain this difference, in figure 8, we compare the number of bubbles and tracers in CR3. Here, we can see that the number of tracers keeps a quite low level when $M_t \in [0.36, 1.00]$ although it is slightly increasing. On the other hand, the number of bubbles in CR3 increases dramatically when $M_t>0.6$, especially at $M_t=1.00$, accounting for approximately $25\,\%$ of the total particle number in the whole domain. That is, bubbles will accumulate significantly in strong compression regions (near shocklets). Figure 9 shows the distribution of light particles and $\theta$ in a quasi-two-dimensional slice with $M_t = 0.64$ and $1.00$. As one can see, bubbles tend to accumulate in strong compression regions at $M_t = 1.00$, which is different to what happens at $M_t = 0.64$. In order to understand the preferential concentration, it is key to control the divergence of particles’ velocities as derived from (2.6), and assuming that $\beta$ and $\tau _p$ do not vary significantly with space and time, we have

(3.1)\begin{equation} \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{v}_p \approx \theta + \tau_p(1-\beta)\left(Q-\frac{\mathrm{D}\theta}{\mathrm{D}t}\right). \end{equation}

Here, $Q=\varOmega _{ij}\varOmega _{ij}-S_{ij}S_{ij}$ is the second-order invariant of fluid velocity gradients, where $\varOmega _{ij}$ and $S_{ij}$ are the rate-of-rotation tensor and rate-of-strain tensor, respectively, and $\mathrm {D}\theta / \mathrm {D}t = \partial \theta / \partial t + \boldsymbol {u}\boldsymbol {\cdot } \boldsymbol {\nabla } \theta$ is the material derivative of $\theta$. From (3.1) one can see that light particles ($\beta >1$) tend to accumulate in the regions $\theta <0$, $Q>0$ and $\mathrm {D}\theta / \mathrm {D}t<0$. From the above expression, it is clear that the prediction on preferential concentration for the case of inertial particles in compressible flows is much more complicated than in the incompressible case, being connected to cross-correlations among compressibility of the carrying flow, inertia parameters and local flow topology. In order to achieve a systematic understanding of such a complex phenomenon, one would need refined scanning at changing of $\tau _p$ and $\beta$ parameters too, which is outside the scope of this paper.

Figure 6.

$F_a$ of light particles in different regions: case 1, the whole domain; case 2, the regions with $\theta > -3\theta _{rms}$ (removing CR3 from case 1); case 3, the regions with $\theta > -2\theta _{rms}$ (removing CR2 from case 1); case 4, the regions with $\theta > -\theta _{rms}$ (removing CR1 from case 1).

Figure 7.

The p.d.f.s of $a_1$ of light particles in different regions when $M_t=0.64$ (a) and $M_t=1.00$ (b).

Figure 8.

The ratio of the particle number in CR3 to the total particle number in the whole domain.

Figure 9.

The distribution of light particles and $\theta$ in a two-dimensional slice when $M_t=0.64$ (a) and $M_t=1.00$ (b). Here, all particles lie in a quasi-two-dimensional slice with a width in the third dimension of the order of the Kolmogorov length scale. X and Y represent the x-coordinate and y-coordinate of slices.

According to figure 4, one can understand that the particles with extremely large accelerations come predominantly from the strong compression regions (such as CR3) at high $M_t$. Then, if the number of particles in CR3 is small (like tracers), the probability of the particles with large accelerations is still small. Therefore, a slight increase of particle number in CR3 will make a contribution to the enhancement of the acceleration intermittency. Hence $F_a$ of tracers increases with the increase of $M_t$. On the other hand, if the number of particles in CR3 is large, as for the case of bubbles at $M_t = 0.77 - 1.00$, there will be a large portion of particles with large accelerations, which can no longer be regarded as rare events. Then the increase of particle number in CR3 will no longer contribute to the increase of acceleration intermittency. As a result, the preferential concentration of bubbles in strong compression regions (e.g. CR3) results in the decrease of $F_a$ when $M_t > 0.6$. In order to further validate this mechanism, the flatness factor of acceleration components of fluid elements at the position of bubbles, $F_a^u$, is measured, shown in figure 10 where $F_a^u$ varies with the same trend as $F_a$ of light particles. Therefore, we confirm that the decrease of $F_a$ of light particles at $M_t > 0.6$ is attributed to the preferential concentration of light particles in strong compression regions.

Figure 10.

The flatness factor of acceleration components of fluid elements at the position of light particles, $F_a^u$, versus $M_t$.

It is also worth explaining why $F_a$ of light particles increases dramatically at $M_t\in [0.5,0.6]$. One understands that the particles in CR3 (near shocklets) have large accelerations in general. Figure 8 shows that the number of tracers in CR3 always keeps a quite low level, and it slightly increases as $M_t$ increases from 0.36 to 1.00, resulting in a significant increase of the intermittency of acceleration. Therefore, $F_a$ of tracers increases steadily with increasing $M_t$. The number of bubbles is also low when $M_t < 0.6$, but it increases faster than the number of tracers at $M_t\in [0.5,0.6]$, so that the number of light particles in CR3 at $M_t=0.64$ has been up to that of tracers at $M_t = 1.00$, leading to a fast increase of $F_a$.

3.1. Longitudinal acceleration at different $M_t$

The longitudinal acceleration, $a_L=(\boldsymbol {a}\boldsymbol {\cdot }\boldsymbol {v}_p)/ |\boldsymbol {v}_p|$, is the projection of acceleration ($\boldsymbol {a}$) to the direction of particle velocity ($\boldsymbol {v}_p$), which determines how fast the magnitude of velocity varies. Figure 11 presents the p.d.f.s of $a_L$ of particles at $M_t = 0.36 - 1.00$. One can see that for tracers, the p.d.f.s of $\boldsymbol {a}_L$ are skewed towards negative values. However, for bubbles, the p.d.f.s of $\boldsymbol {a}_L$ skew only moderately to the negative value at low $M_t$ (e.g. $M_t=0.51$), and this trend vanishes as the p.d.f.s become almost symmetric at high $M_t$ (e.g. $M_t=1.00$). To quantify this difference, we measure the skewness factor of $\boldsymbol {a}_L$, $S_a$, as shown in figure 12. For tracers, $S_a$ becomes more negative as $M_t$ increases from 0.36 to 1.0. As is well known, shocklets will appear in compression regions with the increase of $M_t$. When tracer particles pass through the shocklets from upstream to downstream, they will be decelerated significantly owing to the negative pressure gradient (Anderson Reference Anderson2010). As $M_t$ increases, the shocklets become stronger (the deceleration is also more remarkable). Therefore, the p.d.f. of $\boldsymbol {a}_L$ is more skewed to negative values and $S_a$ becomes more negative for tracers when $M_t$ increases.

Figure 11.

The p.d.f.s of longitudinal accelerations for (a) tracer particles, (b) light particles.

Figure 12.

The skewness factor of particle longitudinal accelerations, $S_a$, varies with $M_t$.

As shown in figure 12, for light particles, $S_a$ also decreases with $M_t$ when $M_t$ is small, similar to the situation for tracers. However, $S_a$ gradually goes back to zero after $M_t > 0.6$. A similar phenomenon was suggested by Yang et al. (Reference Yang, Wang, Shi, Xiao, He and Chen2014), who discovered that $S_a$ of light particles in compression regions is close to 0 at $M_t \approx 1.03$, and there are many light particles whose velocities have an obtuse angle with local pressure gradient. To understand the above difference between tracers and bubbles, we show in figure 13 the p.d.f. of $\cos \alpha$ in CR1 (compression region with $\theta <-\theta _{rms}$), where $\alpha$ is the angle between the velocity of particles and the local pressure gradient. One can see that most tracers in CR1 have angle $\alpha <90^{\circ }$ ($\cos \alpha >0$) at all $M_t$. However, for light particles, the number of particles with $\alpha >90^{\circ }$ ($\cos \alpha <0$) increases steadily as $M_t$ increases from 0.6 to 1.0 approximately. Then we measure how many particles have angle $\alpha < 90^{\circ }$ before entering CR1, denoted as $N_{CR1}^{in}$, and once inside CR1, denoted as $N_{CR1}$. Figure 14 shows that for tracer particles, $N_{CR1}$ is always close to $N_{CR1}^{in}$ at different $M_t$. That is, tracers rarely change their direction in CR1. However, for light particles, $N_{CR1}$ is noticeably smaller than $N_{CR1}^{in}$ at high $M_t$, and this deviation becomes more significant as $M_t$ increases. That is, as $M_t$ increases, more bubbles reverse their direction of velocity from $\alpha < 90^{\circ }$ to $\alpha > 90^{\circ }$ in CR1, so that more light particles that were decelerated turn to be accelerated. This is the main reason for the p.d.f. of $\boldsymbol {a}_L$ being more symmetric when $M_t>0.6$. Also, $N_{CR1}$ of light particles decreases moderately as $M_t$ increases from 0.6 to 1.0, which implies that at high $M_t$, more light particles have positive longitudinal accelerations when they are entering CR1.

Figure 13.

The p.d.f.s of $\cos \alpha$ in CR1. Here, $\alpha$ is the angle between the velocity of particles and the local pressure gradient: (a) tracer particles; (b) light particles.

Figure 14.

The ratios of $N_{{CR1}}^{in}$ (red squares) and $N_{{CR1}}$ (blue diamonds) to the total number of particles in CR1. The empty symbols are for tracers, and the filled symbols are for light particles. Here, $N_{{CR1}}^{in}$ denotes the number of particles with $\alpha <90^{\circ }$ when entering CR1, and $N_{{CR1}}$ denotes the number of particles with an angle of $\alpha <90^{\circ }$ in CR1.

Why do many light particles reverse their velocity direction from $\alpha < 90^{\circ }$ to $\alpha > 90^{\circ }$ in CR1, but few tracers do? As one can see from (2.6), the effect of local fluid acceleration on light particles is amplified by $\beta$ ($\beta \approx 2.94$ in our simulations). Thus it is easier to decelerate light particles by the negative pressure gradient near shocklets, compared with tracers. Assuming that a tracer and a light particle enter the shocklets from upstream at the same time, both will be decelerated under the negative pressure gradient. In general, the tracer particle can pass through the shocklets successfully. However, the velocity of the light particle could decrease to zero, or even have its direction reversed since $\beta$ amplifies the effect of negative pressure gradient. For example, the comparison between $N_{{CR1}}^{in}$ and $N_{{CR1}}$ in figure 14 reflects quantitatively that at $M_t=1.00$, $51\,\%$ of light particles in CR1 reverse their velocity direction from $\alpha < 90^{\circ }$ to $\alpha > 90^{\circ }$, but few tracer particles do.

3.2. Autocorrelation function of acceleration components at different $M_t$

Figure 15 shows the autocorrelation function of the first component of accelerations $R_1(\tau )=\langle a_1(t)\,a_1(t+\tau ) \rangle / \langle a_1(t)^2 \rangle$ (because of isotropy, $R_2$ and $R_3$ would be equal to $R_1$). One can notice that in the small time scale, the $R_1$ values for tracers and bubbles both decrease faster for increasing $M_t$. However, for tracers, the zero-crossing time, $\tau _0$, barely varies with $M_t$, and it is approximately close to $2.3\tau _{\eta }$, which is similar to the results in incompressible turbulence (Yeung Reference Yeung1997; Mordant et al. Reference Mordant, Crawford and Bodenschatz2004; Volk et al. Reference Volk, Calzavarini, Verhille, Lohse, Mordant, Pinton and Toschi2008a). For bubbles, $\tau _0$ is also approximately equal to $2.3\tau _{\eta }$ when $M_t<0.6$. In fact, it is slightly smaller than the value for tracer particles, as shown in figure 16. Volk et al. (Reference Volk, Calzavarini, Verhille, Lohse, Mordant, Pinton and Toschi2008a) found that $\tau _0$ of light particles in incompressible homogeneous isotropic turbulence is significantly smaller than the value for tracers, which is slightly different from our results. This might be attributed to the fact that ${St}=1.64$ in the work by Volk et al. (Reference Volk, Calzavarini, Verhille, Lohse, Mordant, Pinton and Toschi2008a), whereas ${St}\approx 0.2$ in our simulations. Also shown in figure 16 is that when $M_t>0.6$, $\tau _0$ of light particles will gradually decrease with the increase of $M_t$, unlike the value for tracer particles.

Figure 15.

The autocorrelation function, $R_1(\tau )$, of $a_1$ at different $M_t$, for (a) tracers, (b) light particles.

Figure 16.

$\tau _0$ varies with $M_t$ for tracers (red squares) and light particles (blue diamonds).

Why is the trend of the changes of $\tau _0$ between tracer and light particles different from each other at high $M_t$? Yang et al. (Reference Yang, Wang, Shi, Xiao, He and Chen2013) found that the autocorrelation function of acceleration of tracers near shocklets decreases much faster than that near vortices in CHIT. In addition, we use the persistence time $\tau _c$ to denote the period needed for the particles to pass across the strong compression regions (CR3), as shown in figure 17(a). It is found that the average of $\tau _c$ among tracers, $\langle \tau _c \rangle$, is smaller than $\tau _{\eta }$, and it decreases slightly with the increase of $M_t$ ($\langle \tau _c \rangle \approx 0.2\tau _{\eta }$ at $M_t = 1.00$), as shown in figure 17(b). For light particles, $\langle \tau _c \rangle$ is similar to that for tracer particles when $M_t < 0.6$. However, as $M_t$ increases further, $\langle \tau _c \rangle$ increases dramatically, up to around 4$\tau _{\eta }$ at high $M_t$. Then one can argue that $R_1(\tau )$ of tracers is barely influenced by shocklets since the time that tracers stay in strong compression regions is quite short. Therefore, $\tau _0\approx 2.34\tau _{\eta }$ of tracers is independent of $M_t$ and close to the value of the small-scale eddy turnover time. For light particles, at low $M_t$ (e.g. $M_t=0.51$), $\langle \tau _c \rangle$ is similar to that for tracers so that $\tau _0$ of bubbles is quite close to that of tracers. However, as $M_t$ increases, $\langle \tau _c \rangle$ of light particles increases gradually because more bubbles will be trapped in strong compression regions (near shocklets) for several $\tau _{\eta }$, as shown in figure 18(b), where the p.d.f.s of $\tau _c$ of light particles present two peaks when $M_t > 0.6$, which is different to what happens for tracers; see figure 18(a). The position of the first peak ($\tau _c / \tau _{\eta }=0.2 - 0.3$) is similar to that of tracers. However, the second peak occurs in the range of $\tau _c / \tau _{\eta }=3.0 - 4.0$, indicating that many bubbles are trapped by shocklets, hence the influence on $R_1(\tau )$ becomes remarkable, and $\tau _0$ of light particles decreases with increasing $M_t$. In order to confirm the above conclusion further, we have measured $R_1(\tau )$ for light particles when the events of $\tau _c > 1.0$ are removed at $M_t=1.00$, as shown in figure 19. As expected, $R_1(\tau )$ for light particles decreases much more slowly when the events of $\tau _c > 1.0$ are removed, and $\tau _0$ increases by approximately five times. Therefore, we confirm the observation that the decrease of $\tau _0$ with increasing $M_t$ is attributed to the larger number of light particles that are trapped near shocklets for a long time.

Figure 17.

(a) The history curve of $\theta$ along one trajectory of bubble at $M_t=1.00$. Here, $\tau _c$ denotes the time over which particles pass through CR3. (b) The average value of characteristic time of particles staying in CR3, $\langle \tau _c \rangle$, at different $M_t$.

Figure 18.

The p.d.f.s of the time ($\tau _c$) that particles stay in strong compression regions (CR3) at different $M_t$, for (a) tracers, (b) light particles.

Figure 19.

For light particles, $R_1(\tau )$ at $M_t = 1.00$ after the events of $\tau _c > 1.0$ are removed (blue squares) compared with the unconditioned case (red circles).

It is also worth noting the oscillation of $R_1(\tau )$ for light particles at high $M_t$, as shown in figure 15(b). From the preceding analysis, bubbles could be trapped in vortices and near shocklets for comparable time scales. We believe that the strong oscillatory behaviour is the combined effect of both vortices and shocklets, as confirmed by plotting the conditioned autocorrelation function in figure 19, where the oscillation disappears when the events of $\tau _c > 1.0$ have been removed.