Effectiveness of the dispersed-phase continuum model for investigating the airflow in the print gap of inkjet printers

To investigate the effectiveness of the dispersed-phase continuum (DPC) approximation to model the airflow in the print gap of inkjet printers, three-dimensional simulations using the DPC model were compared against those using the classic particle-in-cell (P-in-C) approach. The DPC approximation, due to the separation of time scales, models the dispersed phase with a momentum source that depends on a predefined temporally averaged particle number density field. The results demonstrated that the steady DPC model correlated well to the time-averaged P-in-C solution when the former's formulation accounted for the droplet deceleration. The steady DPC model requires less than 0.1% of the computational resources used by the transient P-in-C approach to compute the mean flow field. Further analyses indicated that the DPC model captured a supercritical pitchfork bifurcation, where the airflow shifted from a steady spanwise uniform regime to a standing wave regime at a critical number density. The P-in-C model computed a smooth continuous transition that characterizes an excited supercritical pitchfork bifurcation. An excellent correlation between the models was observed at print densities above the transition point, but at low number densities a certain level of discrepancy was observed. This was a result of the pseudoturbulence or spottiness induced by the local and instantaneous motion of droplets that excited the standing wave solution even at low number densities. The results, thus, demonstrated that the DPC model is effective at estimating the mean flow field and approximating the bifurcation diagram, while being simultaneously more computationally efficient than the P-in-C model.

Figure 1

Schematic of the physics in the numerical model where the origin of the system is located at the center of the injection face.

Figure 2

Example print of a 10% green noise gray image where the nonuniform air flow in the print zone gives rise to optical density variations. The magnified region shows the optical density variations are caused by misplaced satellites which are much smaller than the main droplets. Image reproduced from Mallinson et al. [12].

Figure 3

On the left: flow field snapshots adapted from Mallinson et al. [12]; on the right side: prints of flat gray on an A4 page provided by Memjet with print speed of 0.41 m/s and green noise of 10%; (a2) $b/H=1/2.2$, (b2) $b/H=1/3.2$, and (c2) $b/H=1/4$.

Figure 4

Flow chart exhibiting the solution algorithm for the P-in-C (a) and DPC (b) models.

Figure 5

Comparison of velocity profiles captured by meshes with cell sizes of $0.0167H,0.025H$, and $0.0375H$ in simulations using the DPC (a) and P-in-C (b) models.

Figure 6

Comparison of velocity profiles in steady-state and transient simulations with time step of $\mathrm{\Delta }\tilde{t}=0.25,0.1$, and 0.05.

Figure 7

Vector field overlayed on $\tilde{Q}$-criterion contour comparing the DPC and P-in-C models at a vertical plane located at $x/H=0.9375$ for a case with $N=0.267\times {10}^{-3}$, $V_p/W=0.027$, $b/H=1/3$, and $\chi =0$; (a) DPC model that disregards the droplet deceleration, (b) DPC that accounts for droplet deceleration, (c) P-in-C snapshot at $\tilde{t}=5\times {10}^4$, and (d) P-in-C time averaged. A, B, and C are, respectively, the upstream and downstream vortices and the vortex spottiness. Paper moving from left to right.

Figure 8

Point particles colored by their velocity magnitude (left) and horizontal velocity (right) overlayed on the flow field velocity magnitude contour indicating the droplets' velocity variation as they transit across the print gap. Paper moving from left to right.

Figure 9

Velocity magnitude measured by a longitudinal transect located $x/H=0.9375$ and $z/H=-0.5$ comparing the DPC with and without droplet velocity correction (dark red and blue lines, respectively) and P-in-C (green dots) for a case with $N=0.267\times {10}^{-3}$, $V_p/W=0.027$, $b/H=1/3$, $\chi =0$, $R=2978$, and ${\text{Re}}_d=16$.

Figure 10

Top: $x$-velocity signal measured by a probe located at $x/H=0.9375$, $y/H=0$, and $z/H=-0.5$; bottom: DFT of the velocity signal showing the frequency behavior of the spottiness. Printing conditions: $N=0.267\times {10}^{-3}$, $V_p/W=0.027$, $b/H=1/3$, $\chi =0$, $R=2978$, and ${\text{Re}}_d=16$.

Figure 11

On the left side: top view of an $xy$ plane located at $z/H=-0.5$ showing isosurface of $\tilde{Q}$ criterion overlayed on a plane located at $z/H=-0.5$ and colored by ${\tilde{u}}_x$ comparing, while on the right side: ${\tilde{\omega }}_x$ contour at a vertical plane located at $x/H=0.9375$; both set of plots comparing (a) spanwise-uniform flow field regime with $N=0.267\times {10}^{-3}$, (b) standing wave regime for a case with $N=0.267\times {10}^{-3}$ and $\lambda /H=1.667$, and (c) standing wave regime for a case with $N=0.267\times {10}^{-3}$ and $\lambda /H=2.667$. Printing conditions: $V_p/W=0.027$, $b/H=1/3$, $\chi =0R=2978$, and ${\text{Re}}_d=16$. Paper moving from top to bottom.

Figure 12

Bifurcation diagram showing the transition from uniform flow to standing wave regime for DPC solutions with wavelength $\lambda /H=2.143$ and 1.875 at $b/H=1/3$, $V_p/W=0.027$, $\chi =0$, $R=2978$, and ${\text{Re}}_d=16$.

Figure 13

Isosurface of $\tilde{Q}$ criterion overlayed on a plane located at $z/H=-0.5$ and colored by ${\tilde{u}}_x$ comparing the DPC ($\lambda /H=1.875$) and P-in-C for a case with $b/H=1/3$, $V_p/W=0.027$, $N=0.355\times {10}^{-3}$, $\chi =0$, $R=2978$, and ${\text{Re}}_d=16$. (a) DPC solution, (b) P-in-C time average, and (c) snapshot at $\tilde{t}={10}^5$. Paper moving from left to right.

Figure 14

Lateral velocity profile taken at a spanwise transect located at $y/H=0$ and $z/H=-0.5$ for a case with $b/H=1/3$, $V_p/W=0.027$, $N=0.355\times {10}^{-3}$, $\chi =0$, $R=2978$, and ${\text{Re}}_d=16$ comparing the DPC ($\lambda /H=1.875$) and P-in-C models.

Figure 15

Bifurcation diagram at $b/H=1/3$, $V_p/W=0.027$, $\chi =0$, $R=2978$, and ${\text{Re}}_d=16$ comparing the DPC ($\lambda /H=1.875$) and P-in-C models.

Figure 16

Isosurface of $\tilde{Q}$ criterion overlayed on a plane located at $z/H=-0.5$ and colored by ${\tilde{u}}_x$ for a case with $b/H=1/3$, $V_p/W=0.027$, $N=0.267\times {10}^{-3}$, $\chi =0$, $R=2978$, and $R{e}_d=16$ comparing (a) the DPC with $\lambda /H=1.875$, (b) P-in-C time averaged, and (c) P-in-C snapshot at $\tilde{t}=5\times {10}^4$. Paper moving from right to left.