Normal impact force of Rayleigh jets

The surface-normal impingement of a liquid jet emanating from a nozzle is of use in numerous technological applications (e.g., surface cleaning, waterjet cutting, and needle-free injection). Upon impact, the (Rayleigh) jet is characterized as either a steady-state (time-invariant) jet, a wavy jet, or a droplet train. The present study experimentally investigates the force imparted by these three distinctly different processes for identical nozzle flow conditions (i.e., velocity and jet diameter). Force models are developed based on Rayleigh jet theory, and they are shown to compare well with the measurements. The force induced by the wavy jet is sinusoidal, oscillating about the steady-state force, while the droplet train force is discrete with periods of zero induced force. Due to conservation of momentum, the peak force experienced by the droplet train is significantly higher than that of the steady stream. This identifies the droplet train as the dominant process to obtain higher peak forces than the continuous stream counterparts during impact with a surface. This can then be used as a means to increase the performance of applications such as waterjet cutting and surface cleaning.

Figure 1

Schematic of the normal impingement of Rayleigh jets: (a) steady-state jet, (b) wavy jet, and (c) droplet train.

Figure 2

Rayleigh's dispersion relation identifying the dependence of wave growth rate $\omega$ on the normalized wave number $k{R}_0$. The fastest growing wave (Rayleigh mode) occurs for $k{R}_0=0.697$, labeled with a green star. The wave number at impact for our experiments is $k{R}_0=0.634$, and it is represented with a purple diamond.

Figure 3

Force induced by a droplet train according to the Rayleigh mode (solid blue), and force induced by a steady-state jet (dashed green), shown for three cycles. The area under each profile (impulse) is approximately equal (i.e., both scenarios exchange the same cycle-averaged momentum with the surface). Due to its discrete nature, the droplet train experiences a greater peak force than the steady-state jet.

Figure 4

Schematic of the experimental setup. A Rayleigh jet is formed and impacts normal to a piezoelectric force sensor. A speaker stimulates the free surface to form a wavy jet/droplet train.

Figure 5

High-speed images of the normal impingement of Rayleigh jets: (a) steady-state jet, (b) wavy jet, and (c) droplet train. The progression of images is from left to right in increments of 370 $\mu \mathrm{s}$ (a), (b) and 593 $\mu \mathrm{s}$ (c). The time in which these impacts occur corresponds to the shaded green regions in Fig. 6.

Figure 6

Force induced by Rayleigh jets: (a) steady-state jet, (b) wavy jet, and (c) droplet train. Arrows indicate model predictions (orange/green), e.g., $F_a$, $F_b$, and $F_c$ from Eqs. (10), (12), and (16), respectively. The (dashed black) lines indicate integration bounds.

Figure 7

Wavy jet free-surface profile. (a) The free-surface along the axial direction $z$, at $t=10$ ms (red line), with an envelope representing the time-averaged amplitude (dashed blue line). (b) The time-averaged amplitude of $z$ locations near the plate (blue dot), with the line of best fit (green line). (c) The free surface with respect to time at locations $z=104$ mm (purple line) and $z=91$ mm (red line).

Figure 8

Force induced by the droplet train (blue-raw), and (cyan-filtered) for primary droplet impact numbers 2–6 (a) and 10–14 (b). The force predicted by Eq. (16) is shown in orange. For successive impacts (b), the presence of a liquid pool increases the peak force and alters the force profile toward symmetric conditions about the peak force.

Figure 9

Normalized force of the steady-state jet (green) and droplet train (blue). Both jets have identical nozzle flow conditions, yet peak force induced by the droplet train is over three times greater. Both profiles exhibit approximately the same impulse, as required by momentum conservation.