Effect of Mach number and volume fraction in air-shock interacting with a bed of randomly distributed spherical particles

Shock-particle interaction is a fundamental problem in many engineering applications, with the dynamics being heavily influenced by the incident-shock Mach number and the particle volume fraction. In this paper we present fully resolved inviscid simulations of an incident-shock wave traveling through a bed of randomly distributed spherical particles. We vary the strength of the incident shock along with the particle volume fraction in order to study the complex wave interaction during shock-particle interaction. In this study we are interested in the early-time behavior during which the particles do not move and hence in our simulations all the particles are fixed in space. We compute the streamwise average of flow field quantities to generate the $x\text{−}t$ contour plots to study the unsteady oscillations inside the particle bed. We observe that the transmitted shock slows down under certain conditions and it is partly due to tortuosity and partly due to weakening caused by energy dissipation. We also present the force histories of the streamwise drag and lift forces for all the particles. The random distribution of particles leads to high variability in the drag force experienced by the particles. We compute the mean peak drag force as a function of the streamwise location to study the mean behavior of the transmitted shock. Based on our findings, we propose simple modifications to improve the current point-particle models used in Euler-Lagrange simulations of shock interacting with a bed of particles.

Figure 1

Simulation setup at time zero.

Figure 2

Histogram plot of ${\varphi }_{fl{u}_i}$ along with the normal fit for (a) ${\varphi }_1=2.5\%$, (b) ${\varphi }_1=10\%$, and (c) ${\varphi }_1=20\%$.

Figure 3

Isosurface plot of the nondimensional velocity magnitude for $M_s=3$ and for (a) ${\varphi }_1=2.5\%$ and (b) ${\varphi }_1=20\%$.

Figure 4

Contour plot of nondimensional pressure at $t/\tau =12$ along the $x\text{−}z$ plane at $y=0$ for ${\varphi }_1=2.5\%,10\%$, and $20\%$ and (a)–(c) $M_s=1.22$, (d)–(f) $M_s=1.66$, and (g)–(i) $M_s=3$, respectively.

Figure 5

Zoomed-in contour plot of the Mach number along the $x\text{−}y$ plane at $z=0$ and $t/\tau =7$ for (a) $M_s=1.22$, (c) $M_s=1.66$, and (e) $M_s=3$ and contour plot of the nondimensional vorticity magnitude along the $x\text{−}z$ plane at $y=0$ and $t/\tau =12$ for (b) $M_s=1.22$, (d) $M_s=1.66$, and (f) $M_s=1.66$, with ${\varphi }_1=20\%$.

Figure 6

The $x\text{−}t$ contour plot of the nondimensional cross-sectional averaged pressure $\langle P\rangle /P_{ps}$ for ${\varphi }_1=2.5\%,10\%$, and $20\%$ and (a)–(c) $M_s=1.22$, (d)–(f) $M_s=1.66$, and (g)–(i) $M_s=3$, respectively.

Figure 7

The $x\text{−}t$ contour plot of the nondimensional cross-sectional averaged streamwise velocity $\langle u\rangle /u_{ps}$ for ${\varphi }_1=2.5\%,10\%$, and $20\%$ and (a)–(c) $M_s=1.22$, (d)–(f) $M_s=1.66$, and (g)–(i) $M_s=3$, respectively.

Figure 8

The $x\text{−}t$ plot of the transmitted shock for (a) $M_s=1.22$, (b) $M_s=1.66$, and (c) $M_s=3$.

Figure 9

Plot of tortuosity as a function of particle volume fraction for $M_s=1.22$ (black curve), $M_s=1.66$ (red curve), and $M_s=3$ (blue curve).

Figure 10

Plot of the normalized nondimensional drag force $C_D$ as a function of nondimensional shifted time $(t-t_{\mathrm{arrival}})/\tau$ for ${\varphi }_1=2.5\%,10\%$, and $20\%$ and (a)–(c) $M_s=1.22$, (d)–(f) $M_s=1.66$, and (g)–(i) $M_s=3$, respectively.

Figure 11

Plot of the normalized drag coefficient $C_D/C_{D,\mathrm{peak}}$ as a function of logarithmic shifted nondimensional time $log[(t-t_{\mathrm{arrival}})/\tau ]$ for six particles in the random bed for $M_s=3$ and ${\varphi }_1=20\%$.

Figure 12

Plot of the peak drag coefficient $C_{D,\mathrm{peak}}$ along with the exponential curve fit for ${\varphi }_1=2.5\%,{\varphi }_1=10\%,{\varphi }_1=20\%$, and (a) $M_s=1.22$, (b) $M_s=1.66$, and (c) $M_s=3$.

Figure 13

Plot of the normal distribution fit for the fluctuating peak drag coefficient for (a) $M_s=1.22$ and ${\varphi }_1=2.5\%$, (b) $\varphi =2.5\%$, (c) $\varphi =10\%$, and (d) $\varphi =20\%$.

Figure 14

Plot of the nondimensional $y$ lift force $C_{D,y}$ as a function of the nondimensional shifted time $(t-t_{\mathrm{arrival}})/\tau$ for ${\varphi }_1=2.5\%,10\%$, and $20\%$ and (a)–(c) $M_s=1.22$, (d)–(f) $M_s=1.66$, and (g)–(i) $M_s=3$, respectively.