Neighbor-induced unsteady force in the interaction of a cylindrical shock wave with an annular particle cloud

Multiphase volume-fraction-dependent quasisteady force models have recently been developed. These models account for the mean force on a particle in the presence of neighboring particles. Additionally, in shock-particle interaction, there is an unsteady effect to the perturbation flow induced by the neighbors and their arrival to a particle. Namely, there is an unsteady component to the volume fraction effect. The present work advances a simple model to capture this time-dependent perturbative influence of neighbors in the context of a shock propagating through a cloud of particles. This model makes use of existing quasisteady force correlations that account for volume fraction within the framework of the compressible Maxey-Riley-Gatignol equation. This new unsteady neighbor force is applied via a time history kernel. The problem is examined in the context of a shock propagating outward in cylindrical geometry. It is observed that for the radii of curvatures examined, the model prediction is accurate in recovering the time history of force and time-integrated impulse on the particles. By comparing with a classical model, we also highlight the importance of volume fraction correction in accurately extracting the long-time force and the importance of unsteady contribution in accurately predicting the peak shock-induced force.

Figure 1

Schematic of the simulation setup in cylindrical geometry. The dashed lines enclose the simulation domain, and the solid lines encircle the particle cloud.

Figure 2

Ensemble-averaged Mach number, $\langle M\rangle$, plotted as a function of scaled radius, $(r-R_0)/L$, at varying radii of curvature, $R_0/L$, at several times. ${\tau }_L=L/u_s$.

Figure 3

Ensemble-averaged pressure, $\langle p\rangle$, plotted as a function of scaled radius, $(r-R_0)/L$, at varying radii of curvature, $R_0/L$, at several times. ${\tau }_L=L/u_s$. The reference pressure, $p_{\mathrm{ref}}$, is the initial postshock pressure.

Figure 4

Ensemble-averaged density, $\langle \rho \rangle$, plotted as a function of scaled radius, $(r-R_0)/L$, at varying radii of curvature, $R_0/L$, at several times. ${\tau }_L=L/u_s$. The reference density, ${\rho }_{\mathrm{ref}}$, is the initial postshock density.

Figure 5

Average (spatial, not ensemble) pressure plotted as a function of radius at several times for $R_0/L=1.0$. The ensemble average of these make up some the results in Fig. 3. The only difference between cases is the particle distribution, which has the same volume fraction but a different microstructure. The initial condition has been omitted from this figure since the solutions are an exact match at that point. See Fig. 3 for the initial condition. The reference pressure, $p_{\mathrm{ref}}$, is the initial postshock pressure. ${\tau }_L=L/u_s$.

Figure 6

Slice of Mach number for $R_0/L=2.0$, plotted at $t/{\tau }_L$ values of (a) 0.20, (b) 0.96, (c) 1.51, (d) 2.00, (e) 2.49, and (f) 3.02.

Figure 7

C-MRG comparison to PR data without time delay (a) with volume fraction effects and (b) without volume fraction effects: $\mathrm{\Phi }(M,\text{Re},{\alpha }_p=0)$. $R_0/L=2.0$. Every fifth bin is shown here to make the plot easier to read.

Figure 8

C-MRG comparison of peak drag to PR data without time delay (a) with volume fraction effects and (b) without volume fraction effects: $\mathrm{\Phi }(M,\text{Re},{\alpha }_p=0)$. $R_0/L=2.0$. Note that while the C-MRG model includes all three force contributions (undisturbed flow, quasisteady, and inviscid unsteady), only the form of the quasisteady force changes between (a) and (b). The orange shading represents the 95% confidence interval for the PR data.

Figure 9

C-MRG drag prediction compared to average PR data with the QS volume fraction delay kernel for $R_0/L=2.0$ with (a) ${\beta }_{nu}=1/3$, (b) ${\beta }_{nu}=2/3$, and (c) ${\beta }_{nu}=1$. Notice that for ${\beta }_{nu}$ values too small, the volume fraction effects are applied too early, and for ${\beta }_{nu}$ values too large, the volume fraction effects are applied too late.

Figure 10

C-MRG drag prediction comparison to PR for early to intermediate times for (a) $R_0/L=1.0$, (b) $R_0/L=1.25$, (c) $R_0/L=1.5$, (d) $R_0/L=2.0$, and (e) $R_0/L=\infty$. Every fifth bin is shown for clarity.

Figure 11

Peak drag for the C-MRG prediction compared to the PR data plotted with its mean for (a) $R_0/L=1.0$, (b) $R_0/L=1.25$, (c) $R_0/L=1.5$, (d) $R_0/L=2.0$, and (e) $R_0/L=\infty$. The bounds surrounding the mean PR values indicate the 95% confidence interval. (f) A comparison of the mean peaks from the other plots to highlight the effect of curvature.

Figure 12

Long-time drag comparison for the C-MRG prediction compared to the PR data plotted with its mean for (a) $R_0/L=1.0$, (b) $R_0/L=1.25$, (c) $R_0/L=1.5$, (d) $R_0/L=2.0$, and (e) $R_0/L=\infty$. Every fifth bin is shown for clarity.

Figure 13

Long-time impulse at $t/\tau =100$ for the C-MRG prediction compared to the PR data plotted with its mean for (a) $R_0/L=1.0$, (b) $R_0/L=1.25$, (c) $R_0/L=1.5$, (d) $R_0/L=2.0$, and (e) $R_0/L=\infty$. The bounds surrounding the mean PR values indicate the 95% confidence interval.

Figure 14

C-MRG prediction components compared to the PR data plotted with its mean for (a) $R_0/L=1.0$, (b) $R_0/L=1.25$, (c) $R_0/L=1.5$, (d) $R_0/L=2.0$, and (e) $R_0/L=\infty$. Every fifth bin is shown for clarity.

Figure 15

Drag model used by Jacobs et al. [53] for shock-particle interaction compared to the present particle-resolved data for $R_0/L=2$. The plots are comparisons of (a) time history of drag, (b) peak drag as a function of position, and (c) long-time impulse ($t/\tau =100$) as a function of position.