Shock-induced combustion of aluminum particle clusters investigated with resolved sharp-interface two-dimensional simulations

The vaporization and combustion of clusters of aluminum particles in shocked flows is studied through interface-resolved 2D numerical simulations. These mesoscale simulations elucidate, for the first time, aspects of vaporization and burning in molten aluminum $\left(\mathrm{Al}\right)$ particle clusters that are markedly different from an isolated burning $\mathrm{Al}$ particle. Unsteadiness due to shock-generated baroclinic vorticity (inviscid mechanisms) and interactions between the wakes of molten $\mathrm{Al}$ particles (viscous mechanisms) are found to have significant effects; vortical mixing facilitates kinetically limited combustion of the particles located upstream in the cluster. Whereas, for particles located downstream in the cluster, the interaction with the low-speed, oxygen-lean wake of the upstream particles leads to diffusion-limited combustion. Results show that particles in a cluster have lower rates of vaporization and combustion than isolated particles under the same overall flow conditions. To isolate inviscid and viscous effects, the flame structure and vaporization rate for particles in a cluster are quantified in terms of local flow conditions, i.e., the local Mach number, Reynolds number, location of a particle in the cluster, and volume fraction. The results obtained in this study will be useful in understanding and modeling the mesoscale physics of shock-induced burning of explosively dispersed reactive aluminum particles.

Figure 1

A schematic diagram of computational setup for the calculation of $M_s=1.47$ shock interaction with a cylindrical water particle of 4.8 mm in diameter.

Figure 2

Comparison of current and benchmark results for $M_s=1.47$ shock interaction with a cylindrical water particle of 4.8 mm in diameter. The drag coefficient obtained from the current calculations using three different mesh resolutions $\left(\frac{D}{\mathrm{\Delta }x}=50,100,\mathrm{and}200\right)$ are compared with benchmark numerical results obtained by Terashima and Tryggvason [60] and Igra and Takayama [67] in (a). The numerical Schlierens are compared with the experimental results at time $\frac{t{u}_s}{D}=1.7$ and 3.38 in (b) and (c) respectively. Images in (b) and (c) are reprinted from [59] with permission from Springer.

Figure 3

(a) The temperature rises during the constant volume reaction of the stoichiometric mixture of Al vapor and air, initiated at 2300, 2800, and 3000 K. (b) The evolution of the species mass fraction during constant-volume reaction of Al vapor and air initiated at 2800 K.

Figure 4

(a) The initial conditions for the axisymmetric calculation of the combustion of an aluminum particle of 230 μm in diameter during the interaction with an incoming Mach 2 shock. (b) Temperature contours at 2 μs obtained from the axisymmetric calculation of Al particle combustion under the influence of Mach 2 shock.

Figure 5

Comparison of the ${\dot{m}}^{″}$ of the burning Al particle of 230 $\mu \mathrm{m}$ in diameter during interaction with the Mach 2 shock obtained from the current calculation and the benchmark result [35].

Figure 6

The initial computational setup for the numerical calculations of shock interaction with a reacting Al particle cluster.

Figure 7

$\overline{C_D}$ of the particles in the cluster during interaction with a $M_s=3.5$ shock at $\mathrm{Re}{}_D=1000$, obtained using four different grid resolutions.

Figure 8

A sequence of the temperature contours obtained from the reactive calculation of $M_s=$ 3.5 shock interaction with an Al particle cluster of $\varphi =$ 20%. The diameter of the particles is selected $3.844\mu \mathrm{m}$ to ensure $\mathrm{Re}{}_D=1000$. $T_0=293.54\mathrm{K}$.

Figure 9

The x-t* plots of the temperature extracted from $\frac{y}{L_c}=0.25$ in the computational domain during the combustion of Al particle cluster of $\varphi =20\%$ under influence of an incoming $M_s=3.5$ shock at $\mathrm{Re}{}_D=1000$. Legends: (i) Incoming $M_s=3.5$ shock, (ii) reflected shock, (iii) transmitted shock, and (iv) diffusion flame front (v) deflagration front. $p_0=101325.0\mathrm{Pa}$ and $T_0=293.54\mathrm{K}$.

Figure 10

The contours of (a) $Y_{Al}$ and (b)$Y_{O_2}$ at $t^{*}=1.8$ obtained from the reactive calculation of $M_s=3.5$ shock interaction with a cluster of Al particle $\left(\varphi =20\%,\mathrm{Re}{}_D=1000\right)$.

Figure 11

(a) $C_D$, (b) $D_{\mathrm{eff}}$, and (c) $\mathrm{Sh}$ of four selected particles in the cluster of 20% volume fraction during the interaction with the $M_s=3.5$ shock. The $C_D$, Sh, and $D_{\mathrm{eff}}$ of the individual particles in the cluster are compared with the $\overline{C_D}\text{and}\overline{\mathrm{Sh}}$ of the whole cluster and an isolated particle.

Figure 12

The comparison of the Favre averaged (a) pressure $\left(\tilde{p}/p_0\right)$, (b) temperature $\left(\tilde{T}/T_0\right)$, and (c) axial velocity $\left(\tilde{u}/u_{us}\right)$ of the gas phase within the particle clusters with $\varphi =10\%,20\%,\mathrm{and}30\%$ during the interaction with $M_s=3.5$ shock. The initial diameter of the particles is 3.844 $\mu \mathrm{m}$ and the corresponding $\mathrm{Re}{}_D\mathrm{is}1000$. $p_0=101325.0\mathrm{Pa}$, and $T_0=293.54\mathrm{K}$.

Figure 13

The temperature contours during the $M_s=3.5$ shock interaction with cluster of particles of volume-fractions (a) $\varphi =10\%$, (b) $\varphi =20\%$, and (c) $\varphi =30\%$ at $t^{*}=1.73$. The initial diameters of the particles are $3.844\mu \mathrm{m}$. $T_0=293.54\mathrm{K}$.

Figure 14

$\overline{\mathrm{Sh}}$ of the particles in the clusters are compared with the $\mathrm{Sh}$ of isolated particles in (a), (b), and (c) for different flow conditions characterized by $M_s,\mathrm{Re}{}_D,\text{and}\varphi$. (a) shows the comparison of $\overline{\mathrm{Sh}}$ for $\varphi =10\%$, 20%, and 30% with an isolated particle at $M_s=3.5$ and $\mathrm{Re}{}_D=1000.$ (b) shows how $\overline{\mathrm{Sh}}$ in a cluster of particles vary with $M_s$. The $\overline{\mathrm{Sh}}$ in a particle cluster for $M_s=1.5,2.5$, and 3.5 are compared with an isolated particle sujected to the respective $M_s$. $\mathrm{Re}{}_D=1000$ and $\varphi =20\%$ in these calculations. (c) shows the effect of $\mathrm{Re}{}_D$ on the $\overline{\mathrm{Sh}}$ of the particles. $\mathrm{Re}{}_D=100,1000,$ and 2000 are selected for this comparison. $M_s=3.5$ and $\varphi =20\%$ in these calualtions. In (b) and (c), solid lines show the $\overline{\mathrm{Sh}}$ of the particle cluster and the dashed lines of the same color show the $\mathrm{Sh}$ of an isolated particle for the respective $M_s$ and $\mathrm{Re}{}_D$.

Figure 15

The temperature contours during the (a) $M_s=1.5$, (b) $M_s=2.5$, and (c) $M_s=3.5$ shock interaction with cluster of particles of volume fraction $\varphi =20\%$ at $t^{*}=1.73$. $\mathrm{Re}{}_D=1000$ for all three cases. $T_0=293.54\mathrm{K}$.

Figure 16

The temperature contours during $M_s=3.5$ shock interaction with particles clusters of $\varphi =20\%$ and (a) $\mathrm{Re}{}_D=100$, (b) $\mathrm{Re}{}_D=1000$, and (c) $\mathrm{Re}{}_D=2000$ at $t^{*}=1.73$. $T_0=293.54\mathrm{K}$.

Figure 17

The scaled burn time $\left(t_b^{\prime }\right)$ of the $\mathrm{Al}$ particles in the cluster obtained from the current calculations is plotted against the corresponding initial particle diameter $\left(D\right)$. Current results are compared with the empirical correlation between ${t_b}^{\prime }$ and $D$ obtained by Beckstead [7] from experimental data. The black line shows the empirical correlations obtained from experiments in [7]. The symbols show the burn time of $\mathrm{Al}$ particle clusters in shocked flows obtained from the current calculations. The red line is a linear least-square fit to the ${log}_{10}{t_b}^{\prime }\mathrm{versus}{log}_{10}D$ .