Structure of velocity distributions in shock waves in granular gases with extension to molecular gases

Velocity distributions in normal shock waves obtained in dilute granular flows are studied. These distributions cannot be described by a simple functional shape and are believed to be bimodal. Our results show that these distributions are not strictly bimodal but a trimodal distribution is shown to be sufficient. The usual Mott-Smith bimodal description of these distributions, developed for molecular gases, and based on the coexistence of two subpopulations (a supersonic and a subsonic population) in the shock front, can be modified by adding a third subpopulation. Our experiments show that this additional population results from collisions between the supersonic and subsonic subpopulations. We propose a simple approach incorporating the role of this third intermediate population to model the measured probability distributions and apply it to granular shocks as well as shocks in molecular gases.

Figure 1

Density profile in a normal shock wave with PDFs of particle velocity in the insets, based on the Mott-Smith theory for $\text{Ma}=25$. Experimental data in the inset (B) come from Ref. [6]. Note the presence of a significant number of particles with intermediate velocities not accounted for by the Mott-Smith distribution.

Figure 2

(a) Total volume fraction ${\varphi }_{\text{tot}}$ in the shock front plotted versus the distance $y$ from the obstacle. Note that the particle density is high near the obstacle and decreases as the distance to the obstacle increases. 0 corresponds to the summit of the circular obstacle. Inset: photograph of the shock front induced around a circular obstacle. The yellow rectangle indicates the study area (b), where ${\varphi }_{\text{tot}}$ is measured. (a), (b), $e=2$ mm, ${\varphi }_1^{*}=0.015, V_1=1.4$ m/s, and $\text{Ma}\sim 60$. (b) Photograph of the shock front around a circular obstacle in the study area, shown in the inset of (a). Here the obstacle is at the bottom of the image and is light gray while the particles appear as dark disks with a small bright spot near the middle. Yellow arrows indicate particle velocities. Both photographs [inset in (a) and (b)] are taken using reflection from a broad white light source. (c) Representation of 50 particle paths, with one highlighted in bold, in a normal shock front in a granular flow around a circular obstacle. (d) Histograms of $v$ (black) and $u$ (red), respectively, longitudinal and transverse velocities normalized by the mean velocity of the supersonic stream $V_1$ in three different regions: the initial stream (A), the shock front (B), and near the obstacle (C). Note the presence of intermediate velocity particles in region (B). (c), (d), $e=1.5$ mm, ${\varphi }_1^{*}=0.007, V_1=1.1$ m/s, and $M\sim 30$.

Figure 3

Volume fractions ${\varphi }_1\left(y\right), {\varphi }_3\left(y\right)$, and ${\varphi }_4\left(y\right)$ of supersonic, first-generation, and second-generation intermediate particles, respectively. The open symbols are measurements obtained from direct tracking (see Sec. 3a). The solid lines are from our model where $A_{32}/A_{12}=1.05$ and $A_{42}/A_{12}=1.4$ (see Sec. 3b). The solid symbols are from the fit of the velocity PDFs with four populations (as shown in Fig. 6). The total volume fraction is shown in the inset. ($e=1.5$ mm, ${\varphi }_1^{*}=0.007, V_1=1.1$ m/s, and $M\sim 30$).

Figure 4

PDFs of $v$ obtained just after 1, 2, or 3 collisions along with the PDFs of $u$ and $v$ from which the contribution of the supersonic as well as the histograms of particles after 1 or 2 collisions has been subtracted. Note the isotropy of the velocity once this subtraction is carried out. The PDFs of the subsonic particles will be identified with these latter distributions.

Figure 5

(a) Histograms of $u$ and $v$, normalized by the mean velocity of the supersonic stream $V_1$ in region (B). The different symbols represent the histograms of $v$ for all particles (red), for supersonic particles only (green), $u$ (blue), and $v$ (black) without the contribution of intermediate particles with 1 collision. Note that subtracting the contribution of the supersonic particles and that from intermediate particles render the histograms of $v$ closer to that of $u$ by suppressing an important proportion of intermediate particles. The subtraction of the contribution from intermediate particles to the PDF of $u$ does not change its shape visibly ($e=1.5$ mm, ${\varphi }_1^{*}=0.007, V_1=1.1$ m/s, and $M\sim 30$). While in (a) we have subtracted the contribution of the histogram of the full intermediate particle population (for 1 collision), (b)–(e) show that the histograms of this intermediate population (black symbols) obtained in region (B) are well approximated by the histograms of particle velocities obtained just after a single collision (blue symbols). (b) $e=2$ mm, ${\varphi }_1^{*}=0.022, V_1=1.5$ m/s. (c) $e=2$ mm, ${\varphi }_1^{*}=0.004, V_1=1.5$ m/s. (d) $e=1.5$ mm, ${\varphi }_1^{*}=0.007, V_1=1.1$ m/s. (e) $e=4$ mm, ${\varphi }_1^{*}=0.007, V_1=1.5$ m/s.

Figure 6

Velocity PDFs: experimental data and the model for two different $y$ positions. Green lines are fits with first-generation intermediate particles, and red lines with first- and second-generation intermediate particles. In order to improve visibility, the PDF for $y/\lambda =0.26$ is rescaled by a factor $1/30$. ($e=1.5$ mm, ${\varphi }_1^{*}=0.007, V_1=1.1$ m/s, and $M\sim 30$).

Figure 7

(a) Photographs of the displacement of a rectangular obstacle (with a velocity $V_{\text{obs}}=80$ cm/s) in a vibrated granular gas for ${\varphi }_1^{*}\sim 0.07$ and $\text{Ma}\sim 8$. The photographs are taken using reflection from a broad white light source. The contrast is increased to highlight the steel beads. The photographs show the accumulation of particles in the shock front induced by the displacement of the obstacle, which appears light gray in the left part of the image. The obstacle moves from left to right. In the reference frame of the shock front, the supersonic stream is from right to left. (b) Volume fraction profile ${\varphi }_{\text{tot}}$ with PDFs of particle velocities in the reference frame of the shock front, in the three different regions shown in the insets, in a vibrated granular gas for ${\varphi }_1^{*}\sim 0.07$ and $\text{Ma}\sim 8$. The particle density is large near the obstacle and decreases as the distance to the obstacle increases. Note that this profile is only plotted in the area where the volume fraction does not reach the maximum value near ${\varphi }_{\text{MAX}}$, the random close-packing volume fraction. The photograph shows a zoom of the shock front in the vibrated granular gas.

Figure 8

Experimental PDFs of $v$ for (a) ${\varphi }_1/{\varphi }_1^{*}=0.45$ and (b) ${\varphi }_1/{\varphi }_1^{*}=0.27$ in the vibrated granular gas at $\text{Ma}=6$ with the decomposition into the supersonic particles, the subsonic particles, and the intermediate particles. (c), (d) the same at $\text{Ma}=14$. Note that the subsonic population PDF has a peak at a nonzero velocity given by $V_2$.

Figure 9

(a), (b) Experimental measurements from Ref. [6] for ${\varphi }_1/{\varphi }_1^{*}=0.46$ and ${\varphi }_1/{\varphi }_1^{*}=0.13$ at $M=25$ with the PDFs from Mott-Smith theory and our three-population model. (c), (d) The same from Ref. [5] at $\text{Ma}=7.18$. (e), (f) The same from Ref. [27] at $\text{Ma}=9$. Note that the subsonic population PDF has a peak at a nonzero velocity given by $V_2$.