Sliding droplets in a laminar or turbulent boundary layer

In this study we report an experimental investigation of droplet sliding under the influence of a laminar or turbulent airflow for water and glycerin droplets. The onset of sliding is described thanks to a critical Weber number, based on the mean airflow velocity impacting the droplet, depending upon the contact angle hysteresis and a drag coefficient (that also depends on the Reynolds number). A fairly good agreement is observed with our experiments and various data from the literature. The transitions between the various droplet shapes observed during sliding (oval, corner, and rivulet) are characterized in a phase diagram built on the droplet capillary and Bond numbers.

Figure 1

Critical Weber number We as a function of the hysteresis at the contact line $cos\left({\theta }_r\right)-cos\left({\theta }_a\right)$ as reported in literature for water droplets. The symbols represent different droplet sizes with respect to the boundary layer: open symbols for $H/\delta <5$ and solid symbols for $H/\delta >5$. ($•$) Milne and Amirfazli [10]; ($\blacksquare$) Roisman et al. [14]; ($◂$) Seiler et al. [15]; ($▸$) Barwari et al. [16]; ($\diamond$) Hooshanginejad and Lee [22]; ($▴$) White and Schmucker [19]; and ($▿$) Zhang [23].

Figure 2

Experimental setup: (a) the wind tunnel showing the bottom view, (b) bottom view setup, (c) side and bottom view setup, and (d) examples of droplet photos taken by the cameras from side and bottom view.

Figure 3

Measurement of flow profiles for (a) laminar flow and (b) turbulent flow.

Figure 4

Side and bottom views of a glycerol droplet submitted to a laminar shear flow on a glass surface, $V_d=40\mu \text{L}-U_{\infty }=22$ m/s.

Figure 5

Side and bottom views of a shed water droplet submitted to a laminar shear flow on a glass surface, $V_d=40\mu \text{L}-U_{\infty }=20$ m/s.

Figure 6

Cartography of shape and behavior for water droplets: (a) laminar flow, (b) turbulent flow, ($\times$) oval immobile, ($+$) oval oscillation and immobile, ($\square$) oval oscillation and moving, ($\circ$) oval moving, ($▵$) corner, ($▹$) rivulet, and ($\diamond$) long rivulet.

Figure 7

Cartography of shape and behavior for glycerol droplets (laminar flow only): ($\circ$) oval moving, ($▹$) rivulet, ($▿$) rivulet wave, and ($⬠$) rivulet breakup.

Figure 8

Evolution of the critical shear rate ${(\partial U/\partial z)}_c$ as a function of Eq. (1): (a) water droplet in laminar flow, (b) glycerol droplet in laminar flow, and (c) water droplet in turbulent flow.

Figure 9

Evolution of the critical Weber number, ${\text{We}}_c$, for water and glycerol as a function of $8[cos\left({\theta }_r\right)-cos\left({\theta }_a\right)]/\left(\pi C_d\right)$: ($\circ$) water droplets and ($\square$) glycerol droplets. Droplets immersed in a laminar boundary layer are represented in open symbols and in solid symbols for a turbulent boundary layer.

Figure 10

The distribution of critical Weber numbers determined using the shear stress ${(\partial U/\partial z)}_c$ as defined in the text for both our data and the literature on water droplets in laminar flow, function of $8[cos\left({\theta }_r\right)-cos\left({\theta }_a\right)]/\left[\pi C_d\right]$. ($\circ$) Water droplet – our data, ($•$) Milne and Amirfazli (2009), ($\blacksquare$) Roisman et al. (2015), ($\diamond$) Hooshanginejad and Lee (2017), ($▴$) White and Schmucker (2021), and ($▿$) Zhang (2021). The symbols represent different droplet sizes with respect to the boundary layer: open symbols for $H/\delta <5$ and solid symbols for $H/\delta >5.$

Figure 11

Dimensionless length $\overline{L}$ of the droplet function of dimensionless time $\overline{t}$ to show the condition of transition between oval and corner for two different velocities $U_{\infty }$ = 12 and 20 m/s.

Figure 12

Advancing and receding contact angles as a function of the capillary number: (a) water droplet, (b) glycerol droplet, ($▵$) advancing contact angle ${\theta }_a$, and ($▿$) receding contact angle ${\theta }_r$.

Figure 13

Time evolution of the droplet's length as a function of capillary number: (a) water droplet and (b) glycerol droplet.

Figure 14

Capillary number Ca function of Bond number $Bo$. Droplets immersed in a laminar boundary layer are represented in open symbols and in solid symbols for a turbulent boundary layer: (a) water droplet and (b) glycerol droplet.