Compressible pairwise interaction extended point-particle model for force prediction of shock-particle bed interaction

We propose a pairwise influence framework for the complex unsteady compressible particle-laden flow problem by accounting for the scattered hydrodynamic waves emitting from neighboring particles in a Euler-Lagrange simulation. It has been observed from particle-resolved (PR) simulations of randomly dispersed particle beds under a loading shock that the compressible pseudoturbulence dominates the flow system even after the primary shock has passed, which causes fluctuations observed in the forces experienced by the particle. Moreover, the fact that each particle exists in the vicinity of a random arrangement of other particles modifies the time history of the drag force experienced by each particle during and after the passage of the shock. First, the scattering flow field due to an incoming shock interacting with a single sphere is constructed using an analysis of the flow in the acoustic limit. Then we examine the validity of the compressible Maxey-Riley-Gatignol force model by comparing the force prediction against a PR simulation of two interacting particles for various particle arrangements and incoming shock strength. Subsequently, the neighboring influences are stored as a library of maps that can be used readily in the calculation of the perturbation force. Finally, the pairwise interaction assumption is evaluated by comparing the force predicted with the model with PR simulations of a randomly packed particle bed of 10% volume fraction for both water and air as the fluid medium for an incoming shock Mach number 1.22. With a considerably lower cost for the implementation of the model compared to PR simulations, it is verified that the model is reasonably accurate in pinpointing particles whose peak force is significantly larger or smaller than the mean drag but also to capture the prolonged fluctuations after the initial shock.

Figure 1

The plot of normalized streamwise velocity at the point ${\mathit{x}}_p=(0,0,2D)$ analytically calculated with the acoustic scattering theory: (a) Convergence with increasing frequency range included in the integration. Results are shown for ${\omega }_{\text{tr}}=250$, 500, 1000, and 1500 ($n_{\omega }$ was kept at 1400). (b) Convergence with increasing number of frequency samples used in the integration for $n_{\omega }=1400$, 2100, and 2800 (${\omega }_{\text{tr}}$ was kept at 500). Also shown in (b) is the particle-resolved simulation result for a shock Mach number of $M=1.05$.

Figure 2

A schematic illustrating the concept of scattering time $t_{\text{sc}}$, which captures the delay of the signal received by a point B on the reference particle from the instance when the shock (marked as vertical blue line) first touches the scattering sphere. The sphere radius is 1 in this case.

Figure 3

(a) The velocity signal of the scattered field computed using the acoustic solution Eq. (10) at the points shown in the inset. The time axis has been shifted by the first arrival time $t_{\text{sc}}$, which is a function of the location of the individual points. (b) The corresponding scattered velocity plots computed from a particle-resolved simulation at $M_s=1.05$.

Figure 4

Time of arrival as a function of $l$ to determine the first time of arrival of the scattered field (blue lines) and the comparison with $t_{\text{sc}}$ measured in two-sphere PR simulation (red lines) performed at a shock Mach number of 1.22.

Figure 5

Snapshots of the two-sphere PR simulation pressure field for an inline arrangement where the centers are separated by a distance of $1.25D$. The spheres are marked in white. Incoming shock Mach number is $M_s=1.05$. The unit for the color bar is Pa. The time stamp for each of the figures is (a) 0 s, (b) $1.2\times {10}^{-4}$ s, (c) $2.4\times {10}^{-4}$ s, (d) $3.6\times {10}^{-4}$ s, (e) $4.8\times {10}^{-4}$ s, (f) $6\times {10}^{-4}$ s, (g) $7.2\times {10}^{-4}$ s, and (h) $8.4\times {10}^{-4}$ s. Here $\tau =D/u_s=2.22\times {10}^{-4}$ s.

Figure 6

Snapshots of inviscid PR simulation pressure fields of two side-by-side spheres whose centers are separated by $1.5D$. The spheres are marked in white. Incoming shock mach number is $M_s=1.05$. The unit for the color bar is Pa. The time stamp for each of the figures is (a) 0 s, (b) $1.2\times {10}^{-4}$ s, (c) $2.4\times {10}^{-4}$ s, (d) $3.6\times {10}^{-4}$ s, (e) $4.8\times {10}^{-4}$ s, (f) $6\times {10}^{-4}$ s, (g) $7.2\times {10}^{-4}$ s, and (h) $8.4\times {10}^{-4}$ s.

Figure 7

Normalized drag force experienced by the reference particle as a planar shock passes over it, plotted as a function of time-shifted by the time of shock arrival and normalized by $\tau$. In each case, the PR simulation result is compared against C-MRG prediction for $M_s=1.05$. (a) Force on an isolated particle. (b) Force contribution of only the scattered field due to an upstream neighbor located inline with its center 1.25 $D$ upstream of the center of the reference particle. (c) Scattered flow force due to an inline neighbor located 2.5 $D$ upstream. (d) Drag force due to the scattered field of a neighbor located 1.5 $D$ to the lateral side. In all four cases, the individual components in terms of the undisturbed flow force ${\mathit{F}}_{\text{un}}$ and the inviscid unsteady force ${\mathit{F}}_{\text{iu}}$ are also shown.

Figure 8

Normalized drag force experienced by the reference particle as a planar shock passes over it. In each case, the PR simulation result is compared against C-MRG prediction and the force contribution of only the scattered field is presented. (a) and (b) are for $M_s=1.1$. (c) and (d) are for $M_s=1.22$. (e) and (f) are for $M_s=1.55$. (a), (c), and (e) are for an inline configuration where the two particle centers are separated by $1.25D$. (b), (d), and (f) are for an inline configuration where the two particle centers are separated by $2.5D$.

Figure 9

The normalized lateral force experienced by the reference particle as a planar shock passes over the side-by-side particle pair. In each case, the PR simulation result is compared against C-MRG prediction. (a) $M_s=1.05$ and (b) $M_s=1.55$.

Figure 10

$-S_p{\overline{p\mathit{n}}}^S$ disturbance maps. The inner black half circle denotes the location of the disturbing particle and the region shaded in white denotes locations that the reference particle's center cannot reach. The upper half plots the streamwise component and the lower half plots the transverse component. $D^2{\rho }_0{c}_0^2$ is the normalizing factor and the quantities plotted are noted in the first frame.

Figure 11

${\displaystyle \frac{\partial {\overline{\left(\rho {\mathit{u}}_r\right)}}^S}{\partial t}}$ disturbance maps. The inner black half circle denotes the location of the disturbing particle and the region shaded in white denotes locations that the reference particle's center cannot reach. The upper half plots the streamwise component and the lower half plots the transverse component. ${\rho }_0/D$ is the normalizing factor and the quantities plotted are noted in the first frame.

Figure 12

Unsteady force comparison between PR simulation (red) and prediction using the C-MRG model (blue) for a water shock through the particle bed. The orange curve is the force experienced by that particle without any neighboring disturbances, subject to a pure shock. The green curve is the undisturbed contribution and the cyan curve is the inviscid unsteady contribution. The streamwise location for each of the particle center is p. 11: $1.00D$, p. 57: $4.17D$, p.74: $5.81D$, and p. 145: $11.65D$.

Figure 13

Unsteady force comparison between PR simulation (red) and prediction using the C-MRG model (blue) for an air shock through the particle bed. The orange curve is the force experienced by that particle without any neighboring disturbances, subject to a pure shock. The green curve is the undisturbed contribution and the cyan curve is the inviscid unsteady contribution. The streamwise location of the particle centers are p. 60: $4.42D$ and p. 145: $5.82D$.

Figure 14

Peak force comparisons for the case of (a) water and (b) air shock. Each of the points is colored corresponding to the quarter in which the center is located. Colors red, blue, green, and purple correspond to upstream, left-of-center, right-of-center, and downstream sections of the bed. Several particles are highlighted in each figure and the force curves experienced by these particles are shown in Figs. 12 and 13. The mean peak drag has been subtracted: $C_{D,\text{peak}}$ denotes the peak value for each particle and the overline $\overline{\left(\right)}$ denotes the average value.

Figure 15

The time delay between when the drag force in PR simulation reaches the maximum and that predicted by the model. The horizontal axis is the depth in $z$ where the particles sit and the vertical axis is the time delay observed. The blue circles denote results for the water shock case and the red crosses are for the air shock case.