Modeling the splash of a droplet impacting a solid surface

Transcription

1 PHYSICS OF FLUIDS VOLUME 12, NUMBER 12 DECEMBER 2000 Modeling the splash of a droplet impacting a solid surface M. Bussmann, S. Chandra, and J. Mostaghimi a) Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada Received 18 November 1999; accepted 4 September 2000 A numerical model is used to simulate the fingering and splashing of a droplet impacting a solid surface. A methodology is presented for perturbing the velocity of fluid near the solid surface at a time shortly after impact. Simulation results are presented of the impact of molten tin, water, and heptane droplets, and compared with photographs of corresponding impacts. Agreement between simulation and experiment is good for a wide range of behaviors. An expression for a splashing threshold predicts the behavior of the molten tin. The results of water and especially heptane, however, suggest that the contact angle plays an important role, and that the expression may be applicable only to impacts characterized by a relatively low value of the Ohnesorge number. Various experimental data of the number of fingers about an impacting droplet agree well with predictions of a previously published correlation derived from application of Rayleigh Taylor instability theory American Institute of Physics. S I. INTRODUCTION Consider a droplet that falls onto a dry, flat, and relatively smooth surface. On impact, as the droplet reacts to a sudden increase in pressure, a thin axisymmetric sheet of fluid begins to jet radially outward over the solid surface. Depending on a number of factors, the leading edge of this sheet may become unstable soon after impact, resulting in the emergence of very regular azimuthal undulations. Photographs taken from beneath a transparent surface illustrate such undulations when the sheet diameter D is a fraction of the initial droplet diameter D o. 1,2 Where the undulations continue to grow they may begin to resemble fingers. If the fingers grow far enough beyond the leading edge, the Rayleigh instability leads to the pinching off of secondary droplets. In what follows, the term fingering will refer to the presence of a perturbed leading edge, whether or not the undulations resemble fingers. The term splashing will describe the formation of secondary, or satellite, droplets. As noted, the above description of fingering and splashing describes the impact of a droplet onto a relatively smooth surface. The fluid spreads, fingers, and splashes while remaining in contact with the solid surface. When the same droplet impacts a sufficiently rough surface, the expanding sheet tends to lift up off of the surface, and the subsequent behavior of the fluid changes dramatically. The most distinctive characteristic of such an impact is the formation of a crown or coronet usually associated with impact of a droplet onto a liquid surface. The milk drop coronet of Harold Edgerton 3 is a well known example of such an impact. This paper considers the normal impact of a spherical droplet onto a dry, flat surface that is sufficiently smooth such that fingering and splashing occur in the absence of a Re D ov o, We D 2 ov o, Oh D o. 1 Re and We are the dimensionless quantities used in this work. Results in the literature presented in terms of Oh have been converted according to Oh We/Re. As the following literature review will demonstrate, the related phenomena of fingering and splashing during the impact of a droplet onto a dry surface are not well understood. There is little agreement on the mechanism that initiates the perturbation at the leading edge of the expanding sheet, nor agreement on a model of the number of fingers that form. The role of surface roughness has not been satisfactorily explained, nor has the onset of crown formation been addressed. Although correlations exist to predict the onset of splashing, these have not proven universal. And the influence of surface wettability has received scant attention. This paper presents a numerical methodology for examining droplet fingering and splashing, by initiating a pertura Author to whom correspondence should be addressed. Telephone: ; Fax: Electronic mail: mostag@mie.utoronto.ca crown. The influence of the gas phase about the impacting droplet is considered to be negligible. The impact is assumed to be isothermal and the liquid phase properties are assumed to be constant. The variables that then affect such an impact include: the initial droplet diameter D o and velocity V o, liquid density, viscosity, and liquid gas surface tension, and measures of the surface roughness and wettability. Surface roughness is usually characterized by an average roughness height R a, and wettability by a dynamic contact angle d that reflects the relative values of the three interfacial tensions at a moving contact line. Nondimensionalization reduces this set of variables to only four. The usual choices are a non-dimensional average roughness R a * R a /D o, the contact angle d, and any two of the Reynolds, Weber, and Ohnesorge numbers, to represent the relative magnitudes of inertial, viscous, and surface tension forces /2000/12(12)/3121/12/$ American Institute of Physics

2 3122 Phys. Fluids, Vol. 12, No. 12, December 2000 Bussmann, Chandra, and Mostaghimi bation of the flow shortly after the impact of a droplet. The three-dimensional 3D numerical model has been presented previously. 4 Numerical results of the effect of impact velocity, contact angle, and roughness are compared with photographs of corresponding impacts. Various data of the number of fingers about an impacting droplet are presented as evidence that the perturbation may be the result of the Rayleigh Taylor instability acting at the leading edge. Although some results corroborate a published expression for a splashing threshold, other results are contradictory. II. LITERATURE REVIEW The first study of droplet fingering and splashing was that of Worthington 5 7 published more than a century ago. Utilizing an experimental setup that is conceptually similar to techniques still in use today, to trigger a spark at some instant during the impact of a droplet, Worthington repeatedly observed and then sketched the fingering and splashing of large milk and mercury droplets impacting onto smooth glass plates. Worthington observed no crown formation; noted that the number of fingers increased with both droplet size and fall height; observed the merging of fingers at or soon after maximum spread; and noted that fingering was more pronounced for fluids that did not wet the substrate mercury on smoked glass than for fluids that did milk on clean glass. Engel 8 and Levin and Hobbes 9 observed a strong correlation between increasing surface roughness and the tendency of droplets to splash upwards and form a crown. Impact without crown formation was only observed when the impact surface was characterized as very smooth. 9 Stow and Hadfield 10 examined splashing as a function of measured roughness, and reported that splashing only occurred for values of a splash parameter K We 0.5 Re 0.25 greater than a threshold value K c that was strictly a function of the surface roughness. K c varied dramatically with small values of R a * and approached an asymptotic value as roughness increased. Stow and Hadfield noted that the scale of roughness necessary to initiate splashing was much smaller than the characteristic height of the fluid sheet that developed soon after impact; proposed that the role of roughness was to initiate an instability in the sheet; and suggested that the proposed threshold would likely not apply to surface roughnesses comparable to the height of the sheet. Mundo et al. 11 confirmed the asymptotic relationship of splashing with surface roughness by examining the impact of small water droplets onto relatively rough surfaces (0.019 R a * 1.3). Impact was oblique to the surface, yet by using the normal velocity component to evaluate We and Re, the onset of splashing was consistently observed at K c Cossali et al. 12 then correlated the data of Stow and Hadfield, 10 Mundo et al., 11 and of Yarin and Weiss, 13 who studied the impact of a train of droplets onto a solid surface, with the following relationship: K 1.6 c R a * The splashing threshold has also been correlated with a critical Weber number We c. Wachters and Westerling 14 observed the onset of splashing at We c 80 for a number of different liquids impacting a polished gold surface. And Range and Feuillebois 2 recently considered the influence of roughness on droplet impact at low values of Oh, corresponding to a weak influence of viscosity. They correlated the onset of splashing with the following expression: We c a ln b R a *, where the parameters a and b differed with each liquid solid combination. Little has been written of the number of fingers that appear about an impacting droplet. More than two decades ago, Allen 15 published a brief calculation based on an application of Rayleigh Taylor RT instability theory. RT is a fingering instability that occurs at an interface between two fluids of different densities, when the lighter fluid pushes the heavier fluid. 16 A linear analysis of the RT instability provides the following expression for a fastest growing wavelength: 3 2 a H L, a represents the acceleration of the interface, which is negative as the droplet spreads, and H and L represent the densities of the heavy and light fluids, respectively. When applied to the interface between a liquid droplet and the ambient gas phase, H L. Returning to Allen, 15 he obtained a reasonable prediction of the number of fingers visible about the rim of an ink blot by estimating the acceleration of the interface as a V 2 o /(D max /2), with D max the measured diameter of the ink blot, and N D max /. Bhola and Chandra 17 recently revisited the work of Allen. 15 They estimated the acceleration somewhat differently, as a V 2 o /D o, and introduced an analytical expression for the maximum fluid spread, D max, derived from consideration of an integral energy balance. 18 The resulting expression was a function of the splash parameter K: N We0.5 Re 0.25 K Equation 5 accurately predicted the number of fingers observed about the perimeter of impacting wax droplets, and later offered a good prediction of the number of fingers about molten tin droplets impacting a hot surface. 19 Marmanis and Thoroddsen 20 counted fingers about the blots left by droplets of a variety of liquids impacting a paper surface. They obtained a correlation of the form N (We 0.25 Re 0.5 ) Data gleaned from the work of Worthington 7 of mercury droplets impacting a glass surface deviated significantly from their own data, which the authors speculated might be related to the difference in the wettability of mercury on glass as compared to other liquids on paper. Finally, a recent paper by Kim et al. 21 applied an RTbased linear perturbation theory to examine the stability of 3 4

3 Phys. Fluids, Vol. 12, No. 12, December 2000 Modeling the splash of a droplet impacting a solid surface 3123 the sheet of fluid that jets radially outward beneath an impacting droplet shortly after impact. Their results predicted a variation of the fastest growing wavelength with time. However, for reasonable assumptions of when the fingers first appear, the model yielded estimates of N in line with experimental observations. 1 There are three further studies of droplet fingering and splashing that deserve mention. Scheller and Bousfield 22 studied the effect of viscosity on the spreading of droplets during impact, while maintaining relatively constant values of surface tension. They reported that splashing occurred only in a viscosity window, above and below which droplets did not splash. Prunet-Foch et al. 23 and Vignes-Adler et al. 24 studied the impact of both pure water and emulsion oil water droplets. They noted a marked difference in the shape of fingers extending beyond the contact line, which they suggested was related to the relative motion of the dispersed phase within the continuous phase of the emulsion droplets. Thoroddsen and Sakakibara 1 examined the very early development of fingers during impact of watersurfactant droplets onto smooth glass plates. They speculated that initiation of the instability did not occur after impact, but actually just before, as fluid at the bottom of the droplet decelerated in response to the compression of air between the droplet and the surface. Numerical studies of droplet impact have dealt almost exclusively with the two-dimensional 2D, or axisymmetric, problem of a droplet impacting normal to a solid surface. The assumption of axisymmetry precludes the consideration of flow in the azimuthal direction, and thus the consideration of fingering and splashing. Fully 3D models of droplet impact have appeared only recently. The results of this paper are based on the 3D model of Bussmann et al. 4 Utilizing a volume tracking methodology to track the liquid gas interface, and with a simple model of the variation of contact angle with contact line velocity, the 3D model yielded good predictions of gross fluid deformation during droplet impact onto an incline and onto an edge. Gueyffier and Zaleski 25 and Rieber and Frohn 26,27 have developed similar 3D models to simulate droplet impact onto a liquid surface. The choice of a liquid surface eliminates the influence of the contact angle, and decreases the influence of underlying surface roughness. The results of these groups are similar, predicting the formation of a crown and the detachment of droplets. The significant difference between the two models is in the approach to initiating a perturbation. Gueyffier and Zaleski imposed a small harmonic perturbation to the shape of the droplet prior to impact. The amplitude of the perturbation required to produce fingering atop the crown varied between and of D o. Rieber and Frohn 26,27 applied two different perturbations to obtain realistic results. A first paper 26 presented results where a small perturbation was applied to the liquid film. More realistic results were presented in a second paper 27 in which a random disturbance with a Gaussian distribution was applied to all initial velocities within the film and the droplet. Although the standard deviation of the disturbance appeared rather large up to 0.5V o ), the authors noted that viscous and surface tension forces quickly reduced the energy of the disturbances. III. METHODOLOGY A. Numerical methodology The numerical model has been presented previously. 4 The following is a brief overview of its main features. Equations of conservation of mass and momentum govern the flow within the liquid phase following impact: V 0, 6 V t VV 1 p 2 V 1 F b. 7 V represents the velocity vector, p the pressure, and F b any body forces that act on the fluid. Boundary conditions include: the surface tension-induced pressure jump at the droplet surface, p s, where represents the total curvature of the interface; a zero tangential stress condition at the surface; and the dynamic contact angle d imposed at the contact line, where the solid, liquid, and gas phases meet. The model is an enhanced three-dimensionalization of RIPPLE, 28 a 2D Eulerian fixed-mesh fluid dynamics code developed specifically for free surface flows. Many details of the discretization are similar to those presented by Kothe et al. 28 Equations 6 and 7 are discretized in a typical control volume formulation. Convective, viscous, and surface tension effects are evaluated explicitly at each time step, followed by an implicit evaluation of pressure to enforce local mass conservation. The computational domain is discretized by a rectangular mesh large enough to encompass the droplet before impact and to capture the subsequent fluid deformation. The model also includes a piecewise linear volume tracking algorithm to track the deformation of the free surface. The interface is defined by a scalar function f that represents fluid volume, equal to one within the liquid phase and zero without. In discretized form, a volume fraction f i, j,k represents the fraction of the volume of a cell that contains liquid: equal to one if the cell is full, zero if empty, and 0 f i, j,k 1 if the cell contains a portion of the free surface, deemed an interface cell. Since f is passively advected by the flow, f satisfies f V f 0. t 8 The 3D model incorporates the advection algorithm of Youngs 29 to solve Eq. 8 geometrically. At each time step, the algorithm consists of two steps. First, the free surface is reconstructed by locating a plane within each interface cell corresponding to f i, j,k and the unit normal to the surface, nˆ i, j,k. The second step is a geometric evaluation of volume fluxes to obtain an updated volume fraction field. A distinct characteristic of the model is the treatment of surface tension, which is accounted for as a body force acting on fluid in the vicinity of the free surface according to the Continuum Surface Force CSF model of Brackbill et al. 30

4 3124 Phys. Fluids, Vol. 12, No. 12, December 2000 Bussmann, Chandra, and Mostaghimi F ST x S r nˆ r x r ds. 9 is the Dirac delta function and the integration is performed over some area of free surface S. Surface tension is then incorporated into the flow equations as a component of the body force F b in Eq. 7. We discretize Eq. 9 by introducing a finite kernel 2h, acting over a radius 2h, to approximate. A discrete surface tension force F STi, j,k is evaluated for each interface cell: F STi, j,k i, j,k A i, j,k i, j,k nˆ i, j,k. 10 A i, j,k represents the surface area contained within the cell, evaluated by Youngs advection algorithm. i, j,k is the volume of the cell. The F STi, j,k are then convolved over the radius 2h to obtain a force field F STi, j,k diffused over cells in the vicinity of the free surface F i, j,k g i, j,k l,m,n F STl,m,n 2h x i, j,k r l,m,n l,m,n. 11 Note the introduction of a weighting function g i, j,k 2 f i, j,k that transforms the volumetric force into a body force acting on non-empty cells near the interface. Estimates of nˆ i, j,k and i, j,k, which are geometric characteristics of the interface, are obtained from the volume fractions: nˆ f f, nˆ. 12 The volume fraction representation of the interface is steep, with the transition occurring over no more than one or two cell widths. As a result, better estimates of f are obtained by evaluating the gradient of a convolved, or diffused, f i, j,k field. We employ the same kernel 2h to convolve both f i, j,k and F STi, x j,k 2h 1 cos x 2h /c x 2h, 13 0 x 2h where c normalizes the kernel c 32 3 h3 2 6 /. 14 A final note concerns the numerical treatment of fluid breakup. In a recent review of the breakup of free surface flows, Eggers 31 acknowledges the challenge of numerically modeling fluid breakup, and suggests that no best approach has yet emerged. Volume tracking techniques such as the one implemented here rely on a finite mesh resolution to dissolve and break interfaces inherently. Schemes which explicitly track the interface can resolve much thinner interfaces, but then require an ad hoc prescription to break interfaces and thus model flows beyond breakup. FIG. 1. Comparison of top views of the 0.93 m/s impact of a 1.5 mm diameter heptane droplet onto a stainless steel surface, as a function of mesh resolution. B. Validation The results of various validations of the model have been presented previously, 4 and demonstrate a capability to accurately predict complicated asymmetric flows. Further to these results, we consider how well the 3D model reproduces 2D results. In particular, we consider an axisymmetric impact in the absence of an applied perturbation. Figure 1 presents results of 3D simulations of the normal 0.93 m/s impact of a 1.5 mm diameter heptane droplet onto a stainless steel surface. A constant value of d 32 Ref. 32 was applied to the simulations. These simulations and all others presented in this paper were run on a uniform mesh. The mesh resolution is specified as the number of cells per initial droplet radius cpr. Figure 1 presents the results of simulations run at three different resolutions, reflecting a progressive halving of the cell volume. Inspection of Fig. 1 reveals a squaring of the droplet at a resolution of 16 cpr that diminishes at finer resolutions. Figure 2 presents the variation of spread factor D/D o with time, evaluated both along the gridlines and along the diagonal, and compared to the data of Chandra and Avedisian. 32 Figure 2 confirms that fluid spreads more quickly along the diagonal, and that this occurs even at 25

5 Phys. Fluids, Vol. 12, No. 12, December 2000 Modeling the splash of a droplet impacting a solid surface 3125 FIG. 3. Typical views of a droplet at t* 0.1, when the perturbation is applied, and at t* 0.2. FIG. 2. Spread factor versus time for the 0.93 m/s impact of a 1.5 mm diameter heptane droplet onto a stainless steel surface. Lines correspond to measured along gridlines and along the diagonal, as a function of mesh resolution. cpr, although Fig. 1 shows that the droplets are becoming more symmetric. The reason for the squaring is related to the difficulty in specifying velocities as a boundary condition at a curved interface defined on a cartesian mesh. The simulation of an axisymmetric impact magnifies these errors, which introduce a systematic bias to the boundary velocities. The result is a subtle movement of fluid from along the gridlines towards the diagonal. The reduction of the error with mesh refinement is first order. For a more detailed analysis of this problem, see Bussmann. 33 C. Initiating a perturbation As was mentioned earlier, the leading edge of the fluid sheet beneath an impacting droplet may show evidence of an instability shortly after the moment of impact, 1,2 at a time when the contact line diameter D D o. Ideally, one would like to model such behavior by imposing a random perturbation on the velocity field and/or the interface profile at or just after the moment of impact, just as surface roughness seemingly perturbs the fluid at the base of a droplet on impact. The model would predict the subsequent growth rate of a spectrum of frequencies, the emergence of a fastest growing wavelength, and simulate the growth of fingers and the possible detachment of satellite droplets. In reality, computational resources are finite. Sharp 16 points out that most numerical studies of the Rayleigh Taylor instability consider the growth of a single frequency disturbance. This is also the approach employed by two of the aforementioned studies of droplet impact onto a liquid surface. Gueyffier and Zaleski 25 perturbed the shape of a droplet prior to impact according to R o,p D o 0.5 A p cos 12 A p cos R o,p represents the perturbed initial radius, A p the amplitude of the perturbation, and and the two angular coordinates associated with the spherical coordinate system. The simulation yielded 12 fingers atop a crown, corresponding to the frequency of the imposed perturbation. Rieber and Frohn 26 also imposed a perturbation of known but unspecified frequency, but onto the liquid surface profile, to obtain physically realistic results. Rieber and Frohn s other work 27 more closely approaches the ideal. A perturbation with Gaussian distribution was applied to all initial velocities within the computational domain. Rieber and Frohn demonstrated that the number of fingers that appeared numerically corresponded to the number predicted by Rayleigh instability theory 34 applied to the breakdown of the liquid torus atop the crown. Unfortunately, applying such a perturbation is unlikely to succeed when modeling the impact of a droplet onto a solid surface, because of how quickly the instability manifests itself beneath the droplet. When a droplet impacts a liquid surface, the fingers appear atop the crown, far later than when fingers appear when the droplet impacts a solid surface. Like the models of Gueyffier and Zaleski 25 and Rieber and Frohn, 26,27 the model presented here is a fixedmesh model that cannot sharply resolve the contact line just after the moment of impact. The circumference of the contact line then is too short relative to the mesh spacing to adequately resolve even a single frequency. As a result, a somewhat different approach has been implemented to initiate an instability. Instead of imposing a disturbance at the moment of impact, simulations are run unperturbed to t* V o t/d o 0.1. At this time D is typically just less than D o, and the contact line circumference is of sufficient length to adequately resolve an imposed perturbation. The simulation is stopped, a perturbation is applied to the radial component of the velocity field near the solid surface the x and y components illustrated in Fig. 3, and the simulation is restarted. The perturbation is of the form V r,p V r,up 1 A p exp B p z cos N, 16 2 D o where the subscripts r, p, and up refer to radial, perturbed and unperturbed quantities. A p is the amplitude of the perturbation at z 0, B p is the rate of decay of the perturbation away from the solid surface, and N is the number of fingers to be imposed. The philosophy behind this approach to perturbing the flow is admittedly pragmatic. The numerical model may be unsuitable for examining the very earliest moments of impact beneath the droplet, but is very capable of yielding accurate predictions of macroscopic flow behavior. The objec-

6 3126 Phys. Fluids, Vol. 12, No. 12, December 2000 Bussmann, Chandra, and Mostaghimi used, the perturbation affected only velocities in the first two or three cells above the solid surface. Figure 4 illustrates the decay of the exponential term. FIG. 4. Decay of the perturbation. tive is to instantaneously affect the simulation in a way that reflects the growth of an instability up to t* 0.1. Although the amplitude of the instability will vary with Re and We, the results of this paper will demonstrate that the same perturbation can be applied to simulations of different impacts and yet accurately predict very different behaviors. Several comments are in order. Although the perturbation is not applied at the moment of impact, t* 0.1 is nonetheless early in the usual lifetime of an impact. Figure 3 illustrates the typical spread of a droplet at t* 0.1, and a representative view of a droplet an equal time later. The perturbation is applied only to the radial component of velocity, and thus initiates no movement of fluid upwards off of the surface. Crown formation is thus neither initiated nor observed numerically. This limits this approach to modeling the fingering and splashing of droplets onto relatively smooth surfaces. The exponential term of Eq. 16 focuses the perturbation onto fluid very near the surface, where the instability is observed. A value of B p 4000 was applied to all simulations presented here. For the mesh resolutions D. The number of fingers The perturbation requires the specification of N, the number of fingers. For all of the results presented in this paper, the number could be obtained from photographs of corresponding impacts. To be predictive, however, the model requires an independent estimate of N. The following is presented in support of Eq. 5, which was obtained from a simple application of Rayleigh Taylor instability theory. 17 Table I presents various experimental data of the number of fingers about an impacting droplet at maximum spread, as well as the corresponding number predicted by Eq. 5. The data are of impacts characterized by a range of Re and We, and in most instances Eq. 5 offers a good prediction of N. Note, however, that Eq. 5 is strictly an expression for the number of fingers likely to appear, and incorporates no physics related to whether or not fingers appear at all. Thus Eq. 5 predicts a finite, if small, number of fingers even when none are observed experimentally, and of course predicts no limit to the number of fingers as the splash factor K. An alternative means of applying Rayleigh Taylor theory to the impact of a droplet is to introduce time-varying expressions for D and a into N D/. The resulting equation would then indicate the number of fingers likely to be initiated at any instant: N D a On purely geometric grounds, by assuming that droplet volume displaced by a solid surface is transferred to the periphery of a droplet spreading on a surface resulting in the shape of a truncated sphere atop a cylinder, 18,21 the early spread of a droplet can be shown to vary as D (D o V o t) 1/2. The constant of proportionality varies depending on the assumed geometry of the fluid at the periphery of the droplet. Marengo et al., 37 for example, confirmed this TABLE I. Comparison of predicted and observed numbers of fingers N at maximum spread. Note that the wax and NaNO 3 data are of molten droplets solidifying on a cold surface. Material D o mm V o m/s Re We K N K/4 3 N expt Reference tin , , , , wax water , this work dyed water , , , ,100 2, mercury , NaNO ,700 1,

7 Phys. Fluids, Vol. 12, No. 12, December 2000 Modeling the splash of a droplet impacting a solid surface 3127 TABLE II. Summary of simulations. Tin Tin Tin Water Heptane D o mm V o m/s Re We Oh d ( ) varying 32 K N ( K/4 3) FIG. 5. Variation of N with D/D o for the 2 m/s impact of a 2.7 mm diameter tin droplet onto a hot stainless steel surface, via Eq. 18, with Neither of these observations proves that the Rayleigh Taylor instability is at work beneath an impacting droplet. The observations do, however, provide more evidence that the conjecture is plausible, and lend credence to the analyses of Allen, 15 Bhola and Chandra, 17 and Kim et al. 21 IV. RESULTS AND DISCUSSION relationship of D(t) by fitting data of the early spread of a water droplet to D t Thus, let D t. Differentiating twice with respect to t to obtain an expression for a, and substituting into Eq. 17 yields an expression for N as a function of either t or D N 3/2 96 t 1/ D 1/2. 18 Note the weak dependence of N on time. If D t 2/3, which implies that the droplet spreads more rapidly than predicted by the above geometric arguments, then Eq. 18 would lose its dependence on time. However, as Kim et al. 21 point out, although little data exists of the very early spread of a droplet, due to the difficulty of measurement, the available results do appear to support the t dependence. Consider now the data of two impacts applied to Eq. 18. The first data are of a simulation presented later in this paper Fig. 7, the 2 m/s impact of a 2.7 mm diameter molten tin droplet onto a hot surface. A fit of the early spread data yields Figure 5 illustrates the resulting variation of N with D/D o. Equation 5 predicts N 20, a few more than the 15 fingers observed experimentally at maximum spread. If the fingering beneath this impacting droplet is a manifestation of the Rayleigh Taylor instability, then Fig. 5 suggests that fingering begins no later than D/D o 0.2. The second set of data is that of Marengo et al., 37 of the impact of a water droplet onto a PVC surface, characterized by We 88. They present the early variation of contact line velocity V CL with time in their Fig. 5. The authors point out a sudden decrease of velocity at 52 s attributed to the instant of lamella ejection, when the expanding fluid sheet first appears from beneath the bulk of the droplet. Marengo et al. only mention that the velocity variation scales with t 0.52 ; inspection of the data from the article suggests a relationship like V CL 0.065t Integrating and differentiating to obtain D and a, and substituting into Eq. 17 yields N 13 at t 52 s. The corresponding evaluation of Eq. 5 yields a similar N 11. The remainder of this paper presents results of simulations of droplet fingering and splashing, and examines the role of impact velocity, contact angle, and surface roughness. Photographs of droplet impact were obtained by techniques described previously. 18,19,32,35 A summary of the simulations is presented in Table II. A. The effect of impact velocity The effect of impact velocity was examined by considering the impact of a 2.7 mm diameter molten tin droplet onto a hot stainless steel surface. Simulation results and corresponding photographs are presented in Figs. 6 8, corresponding to impact at 1, 2, and 3 m/s. Note that the photographs in these figures often display both the droplet and its reflection in the stainless steel surface. Simulations were run of one quarter of a droplet with appropriate boundary conditions applied at the two planes of symmetry. This imposed a symmetry on the results, with the number of fingers N necessarily a multiple of four. The alternative, of simulating an entire droplet, requires an order of magnitude increase in simulation run time. The decision was thus made to maximize the resolution. Simulations of molten tin impact at 2 and 3 m/s were run at a resolution of 32 cpr, the simulation of the 1 m/s impact at 20 cpr. A perturbation amplitude A p 0.25 was applied to each of the simulations. A constant value of the contact angle d 140 was specified, as measured from photographs. Regarding the presentation of the numerical results in these and subsequent figures, the underlying grid is spaced by the initial droplet diameter D o to afford a sense of the extent of fluid spread. The results of the 1 m/s impact, with N 8 fingers imposed on the simulation, are presented in Fig. 6. The numerical results are axisymmetric through t 3.0 ms, with no evidence of the perturbation. An asymmetry appears only at 6.0 ms after the fluid has spread to a maximum extent, and begun to recoil towards the point of impact. Beyond 6 ms, the simulation results deviate somewhat from the experimental results, but do reflect in general terms the actual behavior: A

8 3128 Phys. Fluids, Vol. 12, No. 12, December 2000 Bussmann, Chandra, and Mostaghimi FIG. 6. Photographs and simulation views of the 1 m/s impact of a 2.7 mm diameter molten tin droplet onto a hot stainless steel surface, with A p 0.25, N 8. Numbers at the right indicate milliseconds following impact. dramatic recoil of fluid leading to a bouncing of the fluid up off of the surface, and the pinching off of a small droplet. Results of the 2 m/s impact are illustrated in Fig. 7 and present a very different outcome. N 20 fingers were imposed and are visible soon after impact. The fingers tend to grow slowly with time as the droplet spreads and then more dramatically as the fluid recoils. This is the simulation that was most affected by the problem discussed earlier, the movement of fluid azimuthally from along the axes towards the diagonals. The reason relates to the 20 fingers that were imposed, which results in a finger placed on either side of the diagonal. The movement of fluid towards the diagonal initiates the collapse of the two fingers into one, already visible at 1.9 ms and certainly at 3.1 ms. Obviously the asymmetry of the simulation is undesirable, but coincidentally corresponds to the asymmetry of the FIG. 7. Photographs and simulation views of the 2 m/s impact of a 2.7 mm diameter molten tin droplet onto a hot stainless steel surface, with A p 0.25, N 20. Numbers at the right indicate milliseconds following impact. actual impact, portrayed most clearly by the photograph at 5.1 ms. The merging of the fingers disrupts the subsequent behavior of the fluid and leads to the large variation in finger thickness by 7.1 ms. The formation of satellite droplets at 9.1 ms is not observed experimentally. Nonetheless, Fig. 7 reveals a good agreement between the numerical results and the photographs. Finally, Fig. 8 portrays the results of the 3 m/s impact that results in splashing. While the first two simulations were run with a value of N approximately equal to the number observed experimentally, only 24 fingers were imposed on the 3 m/sec simulation, for reasons related to available computing capacity. Equation 5 predicts 34 fingers, approximately the number observed experimentally. The agreement between simulation and photographs is nonetheless good.

9 Phys. Fluids, Vol. 12, No. 12, December 2000 Modeling the splash of a droplet impacting a solid surface 3129 FIG. 9. Photographs and simulation views of the 4 m/s impact of a2mm diameter water droplet onto a stainless steel surface, with A p 0.25, N 24. Numbers at the right indicate milliseconds following impact. FIG. 8. Photographs and simulation views of the 3 m/s impact of a 2.7 mm diameter molten tin droplet onto a hot stainless steel surface, with A p 0.25, N 24. Numbers at the right indicate milliseconds following impact. Fingers advance rapidly ahead of the advancing contact line. The simulation shows splashing to begin at about maximum spread, when overall fluid velocity is very small, yielding satellite droplets that have little velocity after pinch off and tend to remain where they are deposited. Photographs demonstrate a little bit of splashing at 3.2 ms, but significant numbers of satellite droplets only at 4.5 ms. Numerically, the 24 fingers generate 24 satellite droplets. It is not clear from the photographs whether every finger pinches off, or whether there are instances of a finger pinching off twice, once early and again during recoil. The simulation result is somewhat more organized in its placement of satellite droplets in a ring about the point of impact; the photographs show more of a scattering of satellite droplets. How well are these results predicted by the correlation of Cossali et al., 12 Eq. 2, for the splashing threshold K c as a function of R a *? Solving Eq. 2 for the threshold impact velocity of a 2.7 mm diameter tin droplet impacting a surface with R a * yields V o 2.13 m/sec. Although neither experimental nor simulation results were obtained at this velocity, inspection of Figs. 7 and 8 suggests that this value seems reasonable, and that the correlation applies to these impacts. B. The effect of the contact angle Results are now presented of two other impacts, of water and heptane droplets, that differ significantly from the tin droplet impacts. Table II summarizes the parameters that characterize these impacts. Figure 9 presents photographs and simulation results of the 4 m/sec impact of a 2 mm diameter water droplet onto the same stainless steel surface used for the tin impacts. Equation 5 predicts 29 fingers at maximum spread. Photographs of the droplet shortly after impact suggest that there are more than 40 fingers. Again, N 24 fingers were initially imposed on the simulation, limited by the available computing capacity. Where the impact of a tin or heptane droplet is charac-

10 3130 Phys. Fluids, Vol. 12, No. 12, December 2000 Bussmann, Chandra, and Mostaghimi FIG. 10. Time variation of the average value of d from a simulation of the 4 m/s impact of a2mmdiameter water droplet onto a stainless steel surface. FIG. 11. Photographs and simulation views of the 4 m/s impact of a2mm diameter heptane droplet onto a stainless steel surface, with A p 0.25, N 24. Numbers at the right indicate milliseconds following impact. terized by a relatively constant value of the dynamic contact angle, the value of d varies dramatically for a water droplet impacting onto a stainless steel surface. The advancing value of d is approximately 110 and the receding value, observed during recoil, is 40. We imposed this variation on the simulation with a contact angle model presented previously. 4 The model relates d to the contact line velocity V CL, assuming only a knowledge of the advancing and receding angles. The time variation of the average value of d calculated by the simulation is presented in Fig. 10, and demonstrates the transition from the advancing angle during droplet spread to the much smaller receding angle. Despite a value of the splash factor K 198, greater than the splashing threshold defined by Eq. 2, Fig. 9 illustrates an impact seemingly far from splashing. Fingers do form about the droplet perimeter during spreading, and appear to grow through t 1.5 ms, while d remains large. The fingers that are predicted numerically appear somewhat more pronounced than the fingers observed experimentally, which may be related to the difference in the number of fingers. By 2 ms, the shape of the fingers has begun to change as d decreases. This is especially evident from the simulation results. By 3 ms, when d 40, the distinct fingers are slowly disappearing. By 4 ms, little evidence remains of the fingers as the fluid slowly recoils. Figure 11 illustrates the 4 m/sec impact of a2mmdiameter heptane droplet onto a stainless steel surface, with d Characterized by a splash factor K 349, much larger even than the 3 m/sec tin impact, the photographs show absolutely no fingering, while the simulation results show only subtle signs of the 24 fingers imposed Eq. 5 predicts 50 fingers. The results are presented only to t 1 ms, when the simulation was stopped for lack of resolution of the very thin sheet of fluid at the droplet perimeter. The lack of resolution is also responsible for what appear to be holes in the liquid sheet at 1 ms. How does one explain the dramatic difference between the behavior of the tin droplets, which finger and splash according to the correlation of Cossali et al., 12 and the behavior of the water and heptane droplets, which appear to finger less as K increases? There are two significant differences between the tin and water/heptane impacts. The first is the value of the contact angle. Smaller values of d appear to inhibit finger growth, evidenced most clearly by the dynamic behavior of the water droplet. Heptane, characterized by a constant value of d similar to the receding contact angle of water, shows absolutely no tendency to finger. Would heptane finger and splash if d 32? Figure 12 illustrates simulation results of an impact similar to that illustrated in Fig. 11, but with d 148. The large value of d obviously affects the impact, lifting the edge of the sheet up off the surface as the fluid spreads. The rim of fluid then breaks up, although the number of fingers appears not to correspond to the 24 fingers initiated. This simulation was FIG. 12. Views of a simulation similar to that presented in Fig. 11, but with d 148.

11 Phys. Fluids, Vol. 12, No. 12, December 2000 Modeling the splash of a droplet impacting a solid surface 3131 run at 32 cpr; a simulation run at a finer resolution yielded very similar results. Regardless of resolution, however, the rim of fluid is very thin and poorly resolved, which may explain the breakup. Further evidence of the role of the contact angle is provided by the results of Chandra and Avedisian. 32 Their Fig. 11 illustrates the influence of surface temperature T s on the impact of a heptane droplet onto a stainless steel surface. The contact angle increases with T s, from d 32 at room temperature to 180 when the surface reaches the Leidenfrost temperature the onset of pure film boiling. Photographs corresponding to T s 195 C, just below the Leidenfrost temperature, show 15 fingers about the droplet, while none are visible when T s 160 C ( d 80 ). While Eq. 5 underpredicts this number of fingers when the boiling point 98.4 C properties of heptane are used, the correlation predicts 16 fingers when the properties correspond to saturated heptane at 195 C. 38 The results also provide evidence that the initiation of fingering requires contact with the solid surface. Photographs of impact at T s 205 C, above the Leidenfrost temperature when a vapor film separates the liquid from the solid surface, show no evidence of fingering. It is unlikely that d is solely responsible for the difference in the fingering behavior between tin and water heptane. Although a large value of d obviously affects the heptane impact, the resulting behavior is nonetheless very different from that of the tin impacts. This leads then to consideration of the other significant difference between the tin and water/heptane impacts. The values of Re are of the same order of magnitude, but the values of We differ significantly see Table II. Thus, the influence of surface tension relative to viscosity for the water and heptane impacts is much smaller than for tin. This relationship may be expressed by the aforementioned Ohnesorge number Oh / D o. While surface tension appears to magnify the small undulations of the surface profile that appear shortly after impact, viscosity will tend to dampen the growth of fingers. For the fluids considered here, Oh tin Oh water, Oh heptane. It would appear that there may be an upper limit to the value of the Ohnesorge number beyond which fingering and splashing is not observed, at least in the absence of crown formation. C. The effect of surface roughness Finally, consider the influence of surface roughness on droplet fingering and splashing. Figure 13 presents photographs of the 4 m/sec impact of a 2 mm diameter water droplet onto three different stainless steel surfaces, characterized by roughness values R a 0.065, 0.16, and 0.22 m. The effect of roughness is dramatic, with the impact onto the smoothest surface almost axisymmetric. The two other surfaces, prepared by polishing with 320 and 80 grit emery paper, are rougher than the surface of Fig. 9 (R a 0.09 m, yet the shape and amplitude of the fingers that result from impact onto any of these three surfaces are very similar. The only difference between impacts is a gradual decrease of the number of fingers as roughness increases, particularly during the early spread of fluid. FIG. 13. The influence of roughness on the 4 m/s impact of a2mmdiameter water droplet onto a stainless steel surface. Figure 13 suggests that the magnitude of surface roughness is related to the strength of the perturbation of fluid at the moment of impact. This may be an avenue for introducing surface roughness into the numerical model, by relating the perturbation amplitude A p to the roughness R a *. Such an approach, however, would only reflect the influence of surface roughness at the moment of impact, and not, as may be the case, on the subsequent flow. Figure 14 illustrates the results of two simulations of the 2 m/sec impact of a 2.7 mm diameter tin droplet, similar to that illustrated in Fig. 7, but with A p 0.15 and As one would expect, the magnitude of fingering increases with A p. However, no attempt was made to articulate a relationship between R a * and A p. FIG. 14. The influence of A p on fingering. Simulation views of the 2 m/s impact of a 2.7 mm diameter molten tin droplet onto a hot stainless steel surface, with N 20, and A p 0.15 and Numbers at the right indicate milliseconds following impact.

12 3132 Phys. Fluids, Vol. 12, No. 12, December 2000 Bussmann, Chandra, and Mostaghimi V. CONCLUSIONS A methodology has been presented for considering the fingering and splashing of a droplet impacting a solid surface. Numerical results have been compared with photographs of corresponding impacts and demonstrate good agreement. Published data is also presented in support of a previously published correlation 17 for the number of fingers about an impacting droplet, based on a simple application of Rayleigh Taylor instability theory. The correlation yields good predictions for a range of behaviors. The influence of impact velocity is clearly evidenced by the results of molten tin droplets impacting a hot surface. Behavior varies from no fingering at 1 m/s to fingering at 2 m/s and fingering that leads to the detachment of satellite droplets splashing at 3 m/sec. A correlation for the splashing threshold 12 accurately predicts the onset of splashing. Water and heptane droplets demonstrate less of a tendency to finger and splash, despite being characterized by similar values of a splash parameter K We 0.5 Re Results suggest that fingering decreases as the liquid wets the solid surface, indicated by a decreasing value of the contact angle. Results also suggest that for a given Re there is an upper limit to the value of We beyond which the fingering intensity decreases. ACKNOWLEDGMENT The authors would like to thank Monika Muñoz for taking the water and heptane photographs. 1 S. T. Thoroddsen and J. Sakakibara, Evolution of the fingering pattern of an impacting drop, Phys. Fluids 10, K. Range and F. Feuillebois, Influence of surface roughness on liquid drop impact, J. Colloid Interface Sci. 203, H. E. Edgerton and J. R. Killian, Jr., Flash! Seeing the Unseen by Ultra High-speed Photography Boston C.T. Branford Co., Boston, M. Bussmann, J. Mostaghimi, and S. 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