The nearly step reduction in gravity arising in routine drop tower tests leads to numerous interesting large-length-scale capillary flow phenomena. For example, a liquid puddle at equilibrium on a hydrophobic substrate is observed to spontaneously jump from the substrate during such tests. Implementing a modified version of the open-source *Gerris* code, we numerically investigate such a puddle jump phenomenon for a variety of water puddles on flat substrates. We quantify a range of puddle jump characteristics including jump time, jump velocity, and free puddle oscillation modes for an unearthly range of drop volumes between 0.001 ml and 15 ml and substrate contact angles between 60° and 175°. A numerical regime map is constructed identifying no jump, standard jump, bubble ingestion, geyser formation, drop fission, and satellite puddle jump regimes. Favorable agreement is found between the simulations, experiments, simple theoretical models, and scaling laws.

## I. INTRODUCTION

An initially flat liquid puddle will spontaneously jump from a nonwetting substrate when gravity is nullified. Such a fluid phenomenon was first reported in drop tower tests conducted by Kirko *et al.*^{1} for immiscible liquids and by Wollman and Weislogel^{2} for liquids in air. Recently, Attari *et al.*^{3} further investigated this phenomenon with interests relevant to low-gravity capillary fluidics aboard a spacecraft where such large unearthly droplets are commonplace (i.e., fuel tanks, coolant systems, water processing equipment, etc.).

During the puddle jump process, the initially horizontal puddle rolls inward and upward, eventually colliding with itself and detaching vertically from the substrate. The general phenomenon until drop detachment is comparable with the second half of terrestrial drop bounce from hydrophobic substrates.^{4–6} Drop bounce occurs when a small free liquid drop impacts and rebounds from a hydrophobic surface, typically in normal-gravity conditions. The drop bounce phenomenon plays an important role in a variety of engineering applications such as anti-icing,^{7} self-cleansing,^{8} spray coating,^{9} and drag reduction.^{10} de Ruiter *et al.*^{11} report that gravity plays a negligible role at the beginning of serial droplet bounce such that there should be great similarity between puddle jump in microgravity environments and droplet bounce in terrestrial ones.^{2} As one of the diverse outcomes of droplet impact on solid surfaces,^{12,13} drop bounce continues to intrigue researchers to fully characterize the capillary fluid physics present. Previous investigations on drop bounce have revealed that both fluid properties and substrate surface properties have significant effects on the subsequent drop dynamics.

In drop bounce investigations, the drop usually impacts a hydrophobic substrate with prescribed velocity, spreads to maximum extent *D*_{max} by inertia, and then, similarly to puddle jump, recoils with rim accumulation inward and upward eventually self-colliding and detaching from the substrate in the only available direction—upward. Several differences between drop bounce and puddle jump are noteworthy. For example, for drop bounce, for fixed drop volume, the height of the inertial flattened drop is highly variable due to the highly variable impact velocities achievable in such tests. This is not the case for puddle jump where, for fixed puddle volume, the initial flattened height is limited by the constant capillary height; i.e., for large puddles, *H* ≈ (2*σ*(1 − cos *θ*)/*ρg*)^{1/2}.^{14} Thus, the outcomes for puddle jump are primarily limited to drop volume. For drop bounce, the rebound phase is initiated by the maximum unsteady inertial spread of the impacting drop, which for superhydrophobic substrates can maintain *θ* = 180° due to stable air films that prevent adhesion of the drop to the wall.^{11} For puddle jump, the initial condition is the wall-bound “maximum static spread” of the puddle satisfying the static contact angle condition at the contact line. Finally, gravitational effects are truly absent during puddle jump as pursued herein.

Although many terrestrial experimental and numerical studies for drop bounce have been undertaken,^{11,15–18} few have been made to analyze the puddle jump phenomenon in a zero- or micro-gravity environment. Zhang *et al.*^{19} used the level-set method to simulate the drop rebound by reducing gravity without experimental validation or parametric investigation, and Saranin *et al.*^{20} studied puddle jump with limited comparisons to experiments.

Puddle jump from hydrophobic substrates is investigated numerically herein. The experimental data published by Wollman and Weislogel^{2} and Attari *et al.*^{3} are used to help validate the model. The experiments are simple: establish a horizontal equilibrium puddle configuration on a hydrophobic substrate in a terrestrial environment and then observe the jump phenomenon in the microgravity environment routinely provided via a drop tower test facility. The drop tower experiments were conducted using distilled water (surface tension *σ* = 0.0719 N/m, density *ρ* = 1000 kg/m^{3}, and dynamic viscosity *μ* = 0.001 kg/m s). Initially, a Cassie state with apparent contact angle *θ* ≈ 150° was achieved by placing a water droplet of prescribed volume *V* on a microstructured hydrophobic substrate. The approximate duration of the drop tower free-fall is 2.1 s. Attari *et al.*^{3} report several capillary fluidic response characteristics from the experiments such as critical jump volume, jump time, jump velocity, and variations in the drop profiles as functions of puddle volume.

In this work, the numerical simulations are carried out using the open-source computational fluid dynamics code *Gerris*,^{21,22} which has been adapted and used in our group for a variety of microscopic flows.^{23–27} The 2-dimensional axisymmetric model is employed. We first briefly introduce the numerical methods of solution for initial 1 − *g*_{o} equilibrium configuration and low-g dynamic puddle jump, which we then validate by comparisons to experiments. The numerical results of puddles of various volume and contact angle are then analyzed—including cases outside those pursued experimentally. A numerical regime map is constructed presenting the variety of interfacial phenomena observed.

## II. NUMERICAL METHOD

Computational multiphase flows have advanced significantly in recent decades. The variety of numerical methods for simulating multiphase flows includes the level set,^{28} front tracking,^{29} lattice-Boltzmann,^{30} and volume of fluid (VOF) methods.^{31} The VOF method is perhaps the most popular for drop dynamics modeling due to good mass conservation, efficient robust treatment of large topological changes in the interface, and facility of integration with surface tension models. In this study, we adopt the modified open-source code *Gerris*^{21,22} which solves multiphase flows using the Cartesian grid based finite volume method with the adaptive mesh refinement (AMR) and the VOF method. The AMR technique allows for high-fidelity simulations at low computational costs because the fine mesh can be focused in regions where it is needed most, as shown in Fig. 1. A brief overview of the numerical methods used in *Gerris* is presented here. Interested readers may find more details in Refs. 21 and 22.

*Gerris* utilizes the octree-based AMR method to solve the incompressible, variable-density, Navier-Stokes (NS) equations and the VOF method for the phase transport equation,

where ∇ is the gradient operator, ** u** is the velocity vector,

*t*is the time,

*ρ*is the density,

*p*is the pressure,

*μ*is the dynamic viscosity, and

**is the strain rate tensor given by $Dij=(\u2202ui/\u2202xj+\u2202uj/\u2202xi)$/2. The term**

*D**σκδ*

_{S}

**represents the body force specified by the surface tension, where**

*n**σ*is the surface tension,

*κ*is the local free surface curvature,

*δ*

_{S}is the Dirac distribution function signifying the interfacial concentration of surface tension, and

**is the unit normal vector of the interface.**

*n**T*is the volume fraction function varying from 0 to 1 depending on the location. For cells in the gas and liquid phases,

*T*= 0 and 1, respectively. For cells containing the liquid-gas interface,

*T*is between 0 and 1. Density

*ρ*and viscosity

*µ*both depend on the volume fraction

*T*and are expressed as

*ρ*=

*Tρ*

_{l}+ (1 −

*T*)

*ρ*

_{g}and

*µ*=

*Tµ*

_{l}+ (1 −

*T*)

*µ*

_{g}, respectively. The subscript

*l*represents the liquid phase and

*g*the gas phase.

The governing equations are discretized using a time-staggered finite-volume formulation on the collated Cartesian grid (primitive variables located at the cell center) as

where the superscript is the time-step number and ∆*t* is the increment in time. The transient terms are treated with the second order Crank-Nicholson scheme. By using the projection method, Eq. (4) can be converted to

where the superscript * denotes interim or provisional quantities. The advection term is estimated using the second order Godunov method.^{21,22} With provisional velocity *u*^{*}, the velocity at time *m* + 1 can be obtained using the correction equation

Taking the divergence of Eq. (6) and using the continuity equation [Eq. (2)], we obtain the Poisson equation

The right-hand side of Eq. (7) is calculated at every control volume by the finite-volume approximation. The pressure calculated in Eq. (7) is used to obtain *u*^{m+1} using Eq. (6).

The calculation of the surface tension force term $(\sigma \kappa \delta sn)m+12$ is significant for solving capillary flows. The height-function (HF) method becomes unstable when the interface curvature radius becomes comparable with mesh size. To deal with this under-resolved interface issue, Popinet^{21} proposed a hierarchy of consistent approximations that greatly improves the accuracy in estimating the surface curvature with the Eulerian mesh. *Gerris* also employs averaging of face-centered pressure gradients to obtain the cell-centered pressure gradient. These features help to accurately formulate the capillary driven flows in *Gerris*. The Courant-Friedrichs-Lewy (CFL) condition and shortest capillary wave control the time step ∆*t* in the numerical analysis by

where *C*_{f} is the CFL number (set to 0.5 in this study), *x* is the cell size, and *u*_{i} is the *i*-direction face- or cell-centered velocity. A no-slip boundary condition with a constant contact angle is applied on the solid substrate.

*Gerris* reads in a user-specified parameter file to initialize the computational data including the grid, fluid properties, mesh adaptation strategies, initial and boundary conditions, etc. The code solves the NS equations in an iterative manner using the numerical method discussed above. In each time step, Eq. (5) is first solved for provisional velocity using the velocity of the previous time step. The grid is then adapted according to the user defined AMR method such as the interface topology, variables (e.g., pressure, velocity, temperature) and their gradients, or functions. Next, Eq. (7) is solved for the cell-centered pressure field with the octree-based multigrid scheme.^{22} Finally, the velocity of the current step is obtained by updating provisional velocity with the pressure gradient using Eq. (6). The code is implemented entirely in language C using the object-oriented programming method, which enables relatively easy and fast modification for new applications. The details about the programming aspect of *Gerris* (e.g., flowchart, functions, subroutines, object hierarchy, etc.) can be found in Refs. 21, 22, and 32.

## III. MODEL VALIDATION

We carry out our numerical simulations of the puddle jump process by first establishing the equilibrium puddle configuration on a hydrophobic substrate under normal gravity conditions. We then allow the puddle to jump after setting gravity to zero. Each simulation assumes axisymmetry and room temperature water in air. The equilibrium puddle configuration is solved by settling an initially spherical drop in a gravitational field. Once the “flattened” equilibrium puddle is formed with velocity components in the *r*- and *z*-directions being zero, gravity is nullified. In the absence of gravity, capillary forces dominate the flow phenomena. During the process, the “puddle” forms a “drop,” which separates from the hydrophobic substrate with constant jump velocity.

All computations are performed using eight 2.0 GHz processors with 16 GB RAM and 200 GB hard drive. Grid-independence is carried out for the case of a 2 ml puddle on a flat substrate with *θ* = 150°. The maximum refinement levels 9 and 11 are demonstrated here with a 4 cm box size resulting in finest mesh sizes (= *box size*/2^{level}) of 78 *µ*m and 19.5 *µ*m, respectively. Puddle centroid displacement and total velocity are computed for the two meshes in Fig. 2. A comparison of dynamic profiles is provided in Fig. 2(c). The level 11 computation requires 247 h compared to 7 h for level 9. Level 9 is thus selected for most computations. In special cases where puddle volumes are less than 1 ml, a maximum refinement level of 10 is used. As the puddle volume increases, the contact length of the initial static equilibrium puddle increases. The box size of the computational domain must increase in such instances. Table I lists more details for the simulations.

. | Box size . | Mesh refinement . | Finest grid . |
---|---|---|---|

Volume (ml) . | (cm) . | level . | size (cm) . |

<1 | 1 | 10 | 0.0009 |

1–5 | 4 | 9 | 0.0078 |

7, 10, 15 | 6 | 9 | 0.0117 |

. | Box size . | Mesh refinement . | Finest grid . |
---|---|---|---|

Volume (ml) . | (cm) . | level . | size (cm) . |

<1 | 1 | 10 | 0.0009 |

1–5 | 4 | 9 | 0.0078 |

7, 10, 15 | 6 | 9 | 0.0117 |

The initial equilibrium shape of the puddle in the presence of gravity plays an important role in the puddle jump in the absence of gravity. We first compare the computational equilibrium shape with the experiments. The dimensions compared are the maximum puddle length *L*_{max}, maximum height *H*_{max}, and contact length *D*_{max}, as shown in Fig. 3(a) and listed in Table II for *θ* = 150°. The equilibrium shapes of puddles of different volume are plotted in Fig. 4. If the initial Bond number Bo_{i} (=*ρgV*^{2/3}/*σ*) ≫ 1 (e.g., Bo_{i} = 13.6 for *V* = 1 ml), the puddle is nearly flat with height *H* nearly constant everywhere except near the contact line. Thus, the equilibrium shape of large puddles may be approximated as a cylindrical disk, as shown in Fig. 3(b). In the study of Attari *et al.*,^{3} the height *H* of the disk-shaped puddle is estimated by *H* ≈ (2*σ*(1 − cos *θ*)/*ρg*)^{1/2},^{14} with *θ* ≡ 180°, and diameter *D* = 2(*V*/*πH*)^{1/2}. For varying puddle volume, we compare *H* and *D* to numerical values of *H*_{max} and *D*_{max} in Fig. 5. It is clear from Fig. 5(b) that the dependence of contact length on the volume follows the power law as indicated by the analytical solution, which is in good agreement with the result obtained from numerical integration of the equilibrium equation of sessile droplets balanced between capillary and gravitational forces.^{20} The cylindrical disk approximation is reasonable for large puddles of *V* > 1 ml. The jump velocity can then be estimated from a simple energy analysis in which the high initial surface energy is converted to reduced surface energy and non-zero kinetic energy with ignorance of viscous dissipation.

Volume V (ml)
. | H_{max} (cm)
. | D_{max} (cm)
. | L_{max} (cm)
. | ||||
---|---|---|---|---|---|---|---|

Num. . | Expt. . | Num. . | Expt. . | Num. . | Expt. . | Num. . | Expt. . |

0.03 | 0.032 ± 0.001 | 0.296 | 0.31 ± 0.01 | 0.26 | 0.25 ± 0.10 | 0.42 | 0.42 ± 0.01 |

2 | 1.97 ± 0.03 | 0.542 | 0.55 ± 0.01 | 2.23 | 2.1 ± 0.2 | 2.41 | 2.3 ± 0.2 |

5 | 5.1 ± 0.2 | 0.544 | 0.54 ± 0.01 | 3.53 | 3.5 ± 0.6 | 3.61 | 3.7 ± 0.6 |

10 | 10.03 ± 0.03 | 0.532 | 0.54 ± 0.02 | 5.01 | 5.2 ± 0.7 | 5.16 | 5.4 ± 0.7. |

Volume V (ml)
. | H_{max} (cm)
. | D_{max} (cm)
. | L_{max} (cm)
. | ||||
---|---|---|---|---|---|---|---|

Num. . | Expt. . | Num. . | Expt. . | Num. . | Expt. . | Num. . | Expt. . |

0.03 | 0.032 ± 0.001 | 0.296 | 0.31 ± 0.01 | 0.26 | 0.25 ± 0.10 | 0.42 | 0.42 ± 0.01 |

2 | 1.97 ± 0.03 | 0.542 | 0.55 ± 0.01 | 2.23 | 2.1 ± 0.2 | 2.41 | 2.3 ± 0.2 |

5 | 5.1 ± 0.2 | 0.544 | 0.54 ± 0.01 | 3.53 | 3.5 ± 0.6 | 3.61 | 3.7 ± 0.6 |

10 | 10.03 ± 0.03 | 0.532 | 0.54 ± 0.02 | 5.01 | 5.2 ± 0.7 | 5.16 | 5.4 ± 0.7. |

Experimental and computed dynamic puddle jump profiles for *V* = 2 ml and *θ* = 150° are shown in Fig. 6. The agreement is favorable. The initial puddle shape is largely flat at *t* = 0 s (Bo_{i} ≈ 27 ≫ 1). For *t* > 0 s, we quantitatively observe characteristic rim formation, roll-up, and jump with subsequent oscillations during flight. The predicted jump time and velocity are 0.12 s and 11.0 cm/s, respectively. The experimentally measured jump time and velocity are 0.113 s ± 0.013 s and 11.83 cm/s ± 0.40 cm/s, respectively. Minor observational differences between the experimental and numerical results are largely due to the fact that the numerical profiles employ a 2-dimensional sectional view compared to the frontal 3-dimensional view of the experiments. Such benchmarks provide measured confidence as we explore parametric puddle jump ranges beyond those pursued experimentally by Attari *et al.*

## IV. RESULTS AND DISCUSSION

### A. Regime map

A wide range of puddle volumes (0.001 ml ≤ *V* ≤ 15 ml) and contact angles (60° ≤ *θ* ≤ 175°) are employed in the simulations. Figure 7 plots the evolution of the puddle profile for *V* = 2 ml, 5 ml, 10 ml, and 15 ml with *θ* = 135°. The results are qualitatively identical to drop bounce investigations. When the growing rim eventually collides with itself at the puddle axis, the radial motion is rectified into the vertical motion by internal flows as the puddle detaches from the substrate. For large puddles, air is trapped at the moment of rim collision, referred to as bubble ingestion. This phenomenon is clearly seen for *V* = 10 ml and 15 ml with *θ* = 135° in Figs. 7(c) and 7(d). The collision of the rim with sufficiently high inertia can lead to the rapid vertical ejection of satellite droplets, or a geyser that quickly breaks up into an array of satellite droplets, as also seen in Figs. 7(c) and 7(d). The events of bubble ingestion and geyser formation typically occur at the same time. To better understand both events, the evolution of the velocity and pressure field during the jump process for a 10 ml puddle with *θ* = 135° is plotted in Fig. 8. As the rim grows and rolls up toward the puddle center due to capillary force, an air cavity is gradually formed, which can be clearly seen in Fig. 8 (0.05 s → 0.09 s). The upper portion of the air cavity retracts faster than the lower portion. As a result, an air bubble is entrapped at the moment when the rim is colliding with itself radially at the speed of 1 m/s at the top, as shown in Fig. 8 (0.93 s). Simultaneously, an upward moving jet with a velocity of 3.74 m/s is caused by the inertial focusing mechanism.^{33} As the jet stretches upward, the tip of the jet grows into a blob until it pinches-off two satellite droplets one after another, as shown in Fig. 8. In fact, such jet or geyser formation induced by collapsing surface singularities has been seen in a variety of capillary flow phenomena, such as droplet impact on hydrophobic solid surfaces^{34} or liquid pool,^{35} bubble bursting,^{36} and oscillating droplets.^{37}

After the events of bubble ingestion and geyser formation, we further observe a variety of drop fissions due to large unstable undulations in the puddle. From the numerical results, we note that the strong roll-up motion occurs for *V* ≥ 5 ml and that bubble ingestion and geyser formation occur when *V* ≥ 7 ml. Attari *et al.*^{3} reported that geyser formation was only observed for puddle volumes *V* ≥ 10 ml. Though our numerical results largely confirm the results of Attari *et al.*, we note again that, for large puddles, Attari *et al.* employed slighted dished substrates to enable the practical symmetric balancing of the puddles in their drop tower experiments. Flat substrates could not be balanced adequately to prevent the large and often superhydrophobic puddles from rolling off the substrate while hanging in the tower prior to the tests. We performed exploratory computations of puddle jumps from slightly dished substrates to find that jump macro-phenomena remain fairly well-predicted such as jump time and velocity but that the finer details of higher harmonic waves, ingestions, ejections, and thus, dissipation are more sensitive to the substrate profile. We will not further address the dished substrate case herein.

Figure 9 presents the evolution of the puddle profile of *V* = 7 ml for *θ* = 90°, 120°, and 165°. The contact angle has an obvious influence on the puddle jump process. For a 7 ml puddle, for *θ* < 90°, the puddle does not jump. For *θ* = 90° also, the puddle does not jump, but a portion of the puddle is ejected or fissioned at a low velocity of *U* = 1.44 cm/s, with the rest of the puddle remaining wall-bound. We call this the satellite jump regime since a portion of the puddle does not jump. There is some difficulty establishing such outcomes in practice, but satellite jumps have been observed at large scale in drop tower experiments. For *θ* = 120° and 165°, the droplet fission occurs after the puddle detaches from the substrate, as shown in Figs. 9(b) and 9(c). The jump velocities for *θ* = 120° and 165° are 8.4 cm/s and 12.4 cm/s, respectively. For a given volume, the jump velocity increases as the contact angle increases, as will be discussed shortly.

A numerical regime map is provided in Fig. 10. Six regimes are identified: (i) the puddle jump limit, where the puddle does not leave the surface because the initial condition Bo_{i} ≫ 1 is not satisfied or the contact angle *θ* is low,^{3,19} (ii) standard single drop puddle jump, (iii) satellite jump with a portion of the puddle remaining wall-bound, (iv) drop fission, (v) bubble ingestion, and (vi) geyser formation. According to the experimental results,^{3} for *θ* = 148°, the limiting puddle volume below which no puddle jump occurs is 0.03 ml, which compares to the unity Bond number scale *V* ∼ (*σ*/*ρg*)^{3/2} = 0.02 ml and is also included in Fig. 9. Sufficiently below this volume, the initial “puddle” is a capillary dominated sessile drop of height less than the capillary height *H*. Our numerical prediction for the “no jump” limit for *θ* = 150° is between 0.015 ml and 0.02 ml. As observed from the map, the complexity of the interfacial phenomena increases with increasing *V*. Perhaps most striking about the map is the linearity of the “no jump limit” which, despite dramatic changes in the drop geometry over the 4-decade variation in puddle volume, correlates well with *V* ≈ 5000·e^{−5θ}, with *θ* in radians, *V* in ml, and goodness of fit 0.99.

### B. Jump time, $\tau c$

The puddle jump time is the period from the initial state to the time when the puddle detaches from the surface. As discussed earlier, this time is analogous to the contact time or the inertia-capillary time *τ*_{c} ∼ (*ρR*^{3}/*σ*)^{1/2}, where *R* is the drop radius in terrestrial drop bounce investigations.^{4,30,31} For drop bounce, the contact time *τ*_{c} is a measure of the energy conversion efficiency. A fit coefficient is often employed such that

Richard *et al.*^{38} found *c* = 2.6 ± 0.1 for their substrates, Gilet *et al.*^{6} found *c* = 1.86 for theirs, Bro *et al.*^{39} found *c* = 0.56 for pancake bouncing, and Bird *et al.*^{4} proposed *c* = 2.2 to achieve the shortest contact time from the lowest-order oscillation period, as found by Lord Rayleigh^{40} where *c* = *π*/2^{1/2} for free drop oscillation. All of these studies are relevant to the phenomenon of drop bounce where the impinging drop spreads to the maximum elongation and then retracts and detaches from the hydrophobic substrate. For puddle jumping, the first stage of the drop elongation is absent and the contact time, at first guess, is likely closer to half that for drop bounce experiments, i.e., *c* ≈ 1.3 from Richard *et al.*^{38} The success of this approach is addressed by Attari *et al.*,^{3} who suggest *c* ≈ 2.04 such that *τ*_{c} ≈ (*ρV*/*σ*)^{1/2} for puddle jump from their *θ* ≈ 150° substrates. Numerical jump times for *θ* = 135°, 150°, and 165° are shown in Fig. 11. The data of Richard *et al.* (assumed *θ* ≈ 180°^{11}) and Attari *et al.* (*θ* ≈ 150°) are included for comparisons in the applicable range of drop/puddle volumes. The *θ*-dependent numerical predictions fall between *c* = 1.3 and *c* = 2.04 with larger contact angles leading to shorter jump times as well as better agreement with the former.

Numerical values of *τ*_{c} are plotted in the semi-log form in Fig. 12(a) as a function of 1 − cos *θ* revealing nearly linear behavior. For the range of contact angles computed, we find that numerical contact angle effects may be collapsed via

A useful fit is found for *c* = 1.85 and *n* = 3/2 such that Eq. (10) is accurate to ±11% for all cases computed. The degree of collapse is fair as demonstrated in Fig. 12(b), where *τ*^{*} ≡ *τ*_{c} (1 − cos *θ*)^{n}/*c*(*ρV*/*σ*)^{1/2} ≈ 1.0 ± 0.11 is plotted against *θ*.

### C. Jump velocity, *U*

A simple analytical model for jump velocity *U* was proposed based on the conversion of the puddle surface energy to jumped drop kinetic energy neglecting dissipation during the process.^{3} In the model, as sketched in Fig. 3, a large puddle is assumed as a cylindrical disc with height *H*, which becomes a single non-oscillating spherical drop of radius *R* = (3*V*/4π)^{1/3} after detaching from the substrate. The result provides the estimation

which expectedly over-predicts the jump velocity when compared to experimental and present computational results. Somewhat accounting for dissipation, setting *b* = 0.72, Eq. (11) is presented in Fig. 13 along with numerical predictions for *U* as functions of *V* and *θ*. The numerical results verify the limited region of applicability of Eq. (11) consistent with the assumptions of large enough puddles to be flat 3*H*^{3}/*V* < 1, yet not too large to avoid large distortions of the puddle after jump. The onset of the flat puddle assumption is identified in the figure as a shaded region for the contact angle range computed numerically.

Using Eq. (11), for *θ* ≥ 110°, we determine the drop jump Reynolds number

below which single standard puddles jump and above which geysers, fissions, etc. are observed, as seen in Fig. 9. For drop volume of diameter *D*, the Weber number criterion We_{D} ≡ *ρU*^{2}*D*/*σ* < 3.3 is similar to (12), but less quantitative. The numerical results show increased jump velocity with increased contact angle across the full range of puddle volumes. The numerical results also identify the region of jump velocity leveling (*V* ≥ 4 ml). The data of Attari *et al.* for puddle jumps with *θ* = 150° from slightly dished substrates are added to the figure, providing a qualitative agreement with the overall trends of the numerical results.

We note that, in the large puddle limit 3*H*^{3}/*V* ≪ 1, Eq. (11) suggests that for such puddles, *U* varies with *θ* in a similar manner regardless of puddle volume, namely, *U* ∼ (1 − cos *θ*)^{1/2}. Thus, we plot the numerically computed jump velocity *U* against (1 − cos *θ*)^{1/2} for puddles of varying volume in Fig. 14(a). An overall linear trend is observed for the full range of *V* investigated, 0.002 ml ≤ *V* ≤ 15 ml. We again observe how *U* increases with *V* until approximately *V* ≈ 5 ml, beyond which the data are nearly coincident as discussed in connection with Fig. 12. When the puddle becomes sufficiently large, the more complex and energetic interfacial phenomena discussed in Sec. IV A appear. These phenomena result in higher viscous dissipation rates for the kinetic energy, and higher end-state surface energies due to more than one unconnected varying velocity droplets and satellites produced. Thus, increases in volume above *V* ≈ 5 ml produce nearly balancing increases in driving surface energy with increases in dissipation modes and end-state surface and kinetic energy states such that *U* ≠ fcn(*V*) for 5 ml ≤ *V* ≤ 15 ml. A collapse of the numerical *U* data is found for *V* ≥ 5 ml and *V* < 5 ml such that

## V. CONCLUSION AND FUTURE WORK

The spontaneous puddle jump of water from hydrophobic substrates due to a step reduction in the acceleration level common to drop tower test facilities is studied numerically with a focus on understanding the effects of puddle volume and contact angle on the jump dynamics. The simulations are performed using the open-source code *Gerris*. Our simple axisymmetric model requires computation of the puddle equilibrium in the presence of gravity and then the puddle jump process in the absence of gravity. The model is first validated with experimental data. We then carry out simulations for puddles of varying volumes 0.001 ml ≤ *V* ≤ 15 ml and contact angles 60° ≤ *θ* ≤ 175°, beyond those cases studied experimentally.^{3}

Unlike historical drop bounce investigations, a typical puddle jump begins from a gravitationally flattened wall-bound puddle of fixed capillary height that satisfies the static contact angle condition at the contact line. Upon step-reduction of the gravity level common in drop tower tests, a rim forms along the edge of the puddle, which recoils toward the puddle axis. This growing rim eventually collides with itself, and if ample inertia is available, the puddle detaches from the substrate. For large puddles, the self-collision of the rim can ingest bubbles, form geysers, and eject jets leading to a cascade of satellite droplets. Drop fission may also occur while the puddle remains in contact with the substrate or after leaving the substrate. Similar to drop bounce phenomena, we find that puddle volume has a strong effect on rim formation, recoil, bubble ingestion, geyser formation, and the manner of drop fission. However, contact angle plays a significant role in puddle jump phenomena, whereas for drop bounce, dynamic non-wetting gas layers essentially assure *θ* ≈ 180°. A numerical regime map is constructed to show the influence of the puddle volume and contact angle on these phenomena. The dissipative effects of contact angle hysteresis are not included in the model, and we expect a degree of over-predictions in regime behavior. For a given contact angle, the larger the puddle, the wider the variety of regimes observed during the puddle jump process. We compute a minimum puddle jump volume below which the puddle cannot jump due to insufficient initial surface energy. This puddle jump limit is a strong function of contact angle and obeys an approximate exponential form *V* ∼ 5000·e^{−5θ} in ml. This correlation also provides an estimate of the minimum contact angle below which the puddle cannot jump.

Our numerical results further demonstrate how puddle volume and contact angle play an important role for the jump time *τ*_{c} and jump velocity *U*. As found by others, the relationship between the jump time *τ*_{c} and the puddle volume can be expressed by *τ*_{c} = *c*(*ρR*^{3}/*σ*)^{1/2} with 1.3 ≤ *c* ≤ 2.02, with the lower limit providing improved agreement as the contact angle *θ* approaches 180°. Our numerical results provide a correlation of jump time *τ*_{c} with *θ* such that *τ*_{c} ≈ 1.85·(*ρV*/*σ*)^{1/2}/(1 − cos *θ*)^{3/2} ± 11%. Our simulations also show that, for given *θ*, puddle jump velocity *U* increases with increases in *V*, but only until *V* ∼ 4 ml, after which it remains roughly constant for *V* > 5 ml due to the competition between surface energy and viscous dissipation. For relatively large puddles, the jump velocity predicted from the simulations is in fair agreement with a highly simplified analytical solution based on the energy analysis over a narrow puddle volume range 0.4 ml < *V* < 4 ml. However, the analytical solution reveals that the dependence of the jump velocity on contact angle can be expressed by *U* ∝ (1 − cos *θ*)^{1/2}, which is in good agreement with the numerical results and useful for presenting and collapsing the data.

These numerical results are of practical value in identifying the quantitative influence of puddle volume, contact angle, and fluid properties on the puddle jump process. Because the process is so simple, puddle jump can serve as a natural, passive, no-moving-parts method to eject enormous drops for further drop dynamics investigations in the unique parametric space arising in low-gravity environments routinely encountered aboard a spacecraft and routinely simulated in brief drop tower test facilities.^{2,3} Quantitative knowledge on jump time, velocity, and regime as functions of puddle volume and contact angle establishes design guides for further investigations. Our future work also includes investigation of the effects of viscous dissipation, contact angle hysteresis, and substrate profile on the puddle jump process.

## ACKNOWLEDGMENTS

We would like to thank the U.S. National Science Foundation for supporting the code development via Award No. CBET-1701339 and NASA via Cooperative Agreement No. NNX16AC38G.

## REFERENCES

_{2}-based coatings for self-cleaning and anti-fogging