Head-on collisions of dense granular jets

We study head-on impacts of equal-speed but unequal-width incompressible jets in two dimensions with a focus on dense granular jets. We use discrete particle simulations to show that head-on impact of granular jets produces a quasi-steady-state where a fraction of the excess incident momentum from the larger jet is captured by an impact center that drifts steadily over time. By varying the dissipation in our discrete particle simulations and through additional analogous continuum jet impacts of different rheologies, we show that this central drift speed is remarkably dependent primarily on the total dissipation rate to the power $1.5$, and largely independent of the dissipation mechanism. We finish by presenting a simple control volume analysis that qualitatively captures the emergence of the drift speed but not the scaling.

Figure 1

Two-dimensional head-on impact of two jets produces a steady-state flow with collimated ejecta and a steadily drifting flow configuration. (a) In two dimensions, A jet of width $2R$, shown in blue, is incident against a jet of with $2kR$, shown in red. Both jets are initially at $81\%$ packing fraction and traveling toward each other at the same speed $U_0$. Here, $R=100R_{\mathrm{G}}$, where $R_{\mathrm{G}}$ is the grain width, and $k=0.465$. (b) Snapshot at $t=2R/U_0$ after the two jets directly collide. Here we indicate the coordinates $x$ and $y$ whose origin is located at the point of initial contact. (c) Snapshot at $t=10R/U_0$, where the central saddle point, shown with a dot, is drifting toward the smaller oncoming jet with drift speed $U_d$. Here, $U_d=0.052U_0$. Additionally, the impact produces 2 collimated ejecta streams with widths $2R_{\mathrm{out}}$ and ejecta velocities with magnitude $U_{\mathrm{out}}$ exiting at an angle ${\mathrm{\Psi }}_0$ from the horizontal. Note ${\mathrm{\Psi }}_0$ is not quite the same as the spatial angle formed between the ejecta jet and the horizontal due to the fact that the ejecta streams are drifting alongside the saddle point.

Figure 2

Following a short transient, the central saddle point drifts at a relatively steady speed $U_d$. Normalized (a) saddle position and (b) corresponding drift speed versus time for collision between two jets at $k=0.465$ (red) and $k=0.25$ (blue). The impact initially occurs at $x_{\mathrm{saddle}}=0$. Following a short transient of duration $\sim 4R/U_0$, the saddle point translates linearly in time. The direct measurement of the drift speed obtained by differentiating the saddle position. The drift speed starts high in the transient state and then fluctuates about the mean drift speed after steady drift has been reached. The drift speeds above are $U_d/U_0=0.052$ for $k=0.465$ and $U_d/U_0=0.100$ for $k=0.25$.

Figure 3

The entire flow configuration arising from the head-on collision of unequal jets drifts steadily in time. (a) As viewed in the fixed laboratory reference frame, we show horizontal velocity $u_x/U_0$ versus position $x/R$ along the line running through the centers of the jets defined by $y=0$. Velocities are shown at a variety of times, ranging from very soon after impact (blue) to long after impact (red). The velocity fields at these different times does not coincide when viewed from the laboratory frame. (b) Horizontal velocity profile $\overline{x}/R$ along the center line as viewed in the comoving reference frame which is translating steadily at the drift speed $U_d$ along with the drifting saddle. All of the profiles from (a) are included, where all of the later ($t>2R/U_0$) snapshots directly overlap in the comoving frame. Hence, excluding the profiles obtained in the initial transient, the flow configuration translates at a constant drift speed $U_d$, remaining preserved as it translates.

Figure 4

As the jets approach equal widths, the drift speed continuously vanishes. (a) In the laboratory frame, we show the saddle point drift speed $U_d/U_0$ versus the ratio of the incident jet widths $k$ while keeping $R=100R_{\mathrm{G}}$ and the restitution and friction coefficients $0.9$ and $0.2$, respectively in the discrete particle simulation (red points). We furthermore include the outcome from our control volume analysis for comparison, where we see the analysis perform quite well despite its simplicity. We see that $U_d$ decreases monotonically as $k$ increases, and eventually $U_d$ vanishes when the jets are equal widths at $k=1$. (b) Ejecta velocity in the laboratory (red) and comoving (blue) frames. Note that the ejecta velocity appears relatively constant for all $k$, especially in the drifting frame. For the grains used here, that constant is approximately $B=0.73$. (c) To complete the characterization of the flow configuration resulting from the collision we include the ejecta angle in both the laboratory and comoving frames.

Figure 5

Momentum balance incorporates momentum absorption into the drifting saddle. Here we show the momentum absorbed by the drifting saddle $P_{\mathrm{saddle}}$ as a fraction of the net incident momentum $P_{\mathrm{in}}$ as we vary the jet-width-ratio $k$. We see that the drifting saddle absorbs a considerable fraction of the net incident momentum. In the discrete particle jets used here, at least $15\%$ of the net incident momentum is captured by the drifting saddle.

Figure 6

Drift speed reaches a fraction of its maximum based only on how much of the available incident energy is dissipated. This fraction is achieved entirely independently of all jet-width-ratios and dissipation mechanisms. Shown in (a) is the drift speed $U_d/U_0$ versus the normalized energy dissipation (from Eqs. (18) and (15)) for the discrete particle simulations and the continuum fluids. The discrete particle simulations (DPS) with coefficient of restitution (COR) 0.9 (black, triangles), 0.1 (dark grey, squares), and 0.05 (light gray, circles) are shown for $k=0.465$. The continuum impacts are shown with $k=0.314$ ($\times +$), $k=0.465$ (+), and $k=0.688$ (×) for the shear thickening fluid (ST, magenta), the frictional fluid (FF, red), and the Newtonian (Liq, blue) cases. At any given $k$, all of these cases produce similar drift speeds at a given energy dissipation rate. For comparison, we additionally include our control volume analysis (green, dashed) at $k=0.465$, which qualitatively captures the increasing drift speed with increasing dissipation, but is completely unable to capture other features, most notably the concavity. (b) Drift speed $U_d$ normalized by the maximum drift ${\overline{U}}_d^{\max }$ versus the energy dissipation rate ${\overline{E}}_{\mathrm{diss}}$ normalized by its maximum ${\overline{E}}_{\mathrm{diss}}^{\max }$. Firstly, we find a dramatic collapse of the data, indicating that when the drift speed reaches a given fraction of its maximum dependent only on the dissipation rate and independent of the details of the highly varying impact materials and jet-widths-ratios. Secondly, we find that this fraction is robustly dependent on the normalized dissipation rate to the power $1.5$.

Figure 7

Dissipation rate versus (a) friction coefficient $\mu$ and (b) inverse Reynolds number for frictional fluid and Newtonian impacts, respectively. These continuum simulations are at $k=0.465$. The green dashed lines are linear fits to the low dissipation data points. These lines highlight that at low dissipation, the dissipation increases linearly with the dissipation coefficients ($\mu$ and $\eta$), corroborating the intuition that when the coefficients are small, the overall flow field remains relatively undisturbed. At higher coefficients, the energy dissipation rate begins to saturate to its maximum.