I. INTRODUCTION The impact of a single droplet against a solid surface, a complex and often beautiful phenomenon, is a basic component of various natu

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1 On a three-dimensional volume tracking model of droplet impact M. Bussmann, J. Mostaghimi, and S. Chandra Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario M5S 3G8 Canada Abstract A three-dimensional model has been developed, of droplet impact onto asymmetric surface geometries. The model is based on RIPPLE, and combines a xed-grid control volume discretization of the ow equations with a volume tracking algorithm to track the droplet free surface. Surface tension is modelled as a volume force acting on uid near the free surface. Contact angles are applied as a boundary condition at the contact line. The results of two scenarios are presented, of the oblique impact of a 2 mm water droplet at 1 m/s ontoa45 o incline, and of a similar impact of a droplet onto a sharp edge. Photographs are presented of such impacts, against which the numerical results are compared. The contact angle boundary condition is applied in one of two ways. For the impact onto an incline, the temporal variation of contact angles at the leading and trailing edges of the droplet was measured from photographs. This data is applied as a boundary condition to the simulation, and an interpolation scheme proposed to evaluate contact angles between the leading and trailing edges. A simpler model is then proposed, for contact angle as a function of contact line velocity, and applied to both geometries. The model requires values of only two contact angles, at a rapidly advancing and a rapidly receding contact line. Simulation results compare well with photographic data. 1

2 I. INTRODUCTION The impact of a single droplet against a solid surface, a complex and often beautiful phenomenon, is a basic component of various natural and industrial processes. As a result, many studies have examined droplet impact, to understand and improve the larger process. Rein [1] provides a comprehensive review of research in this area. Much of this work has examined the axisymmetric, or 2D, scenario of the normal impact of a droplet onto a at surface. Far less has been published of droplet impact which is not axisymmetric, and thus must be considered three-dimensionally. Such impacts are not uncommon, however, as one need only consider a rain drop striking a window pane or the impact of a droplet against an irregular surface. It is 3D droplet impact which we consider here. In particular, we present anumerical model of droplet impact onto asymmetric surfaces, and provide results of two simulations: the oblique impact of a 2 mm diameter water droplet falling at 1 m/s onto a 45 o incline, and the impact of a similar droplet onto a sharp edge. We also present photographs of such droplet impacts, against which we compare model results. The intent is to demonstrate the resultant complex phenomena, to present an approach to modelling such impacts, and to discuss some of the modelling issues which arise. What happens when a droplet strikes a surface is dependent on a variety of factors including droplet size, impact velocity, and uid and surface properties. The particular impacts we present here may be characterized as the interplay between inertial eects which dominate the early spreading of the uid, and viscous and surface tension forces which arrest the spreading and eventually bring the uid to an equilibrium conguration. Two non-dimensional quantities characterize such impacts: the ratio of inertial to viscous eects is represented by an initial Reynolds number Re o = D o V o =, and the ratio of inertial to surface tension eects by an initial Weber number We o = D o V 2 o =. V o represents the impact velocity, D o the droplet diameter prior to impact, the liquid density, the liquid viscosity, and the liquid-air surface tension. 2

3 As mentioned, most numerical modelling has focused on the problem of normal droplet impact. The rst were Harlow and Shannon [2], who used the \marker-and-cell" (MAC) nite dierence technique [3] to solve the ow equations. However, by neglecting surface tension and viscous eects, their results apply primarily to the early stages of impact when inertial eects dominate. Later MAC analyses included such eects: Foote [4] studied the collision of raindrops, and Tsurutani et al. [5] included a heat transfer model to examine the cooling of a hot surface. Trapaga and Szekely [6] used the commercial code FLOW-3D [7] to study the normal impact of a droplet under conditions typical of thermal spray or spray forming processes. FLOW-3D employs a xed-grid Eulerian approach in conjunction with the \volume of uid" (VOF) method of Hirt and Nichols [8] to solve the ow equations and track the droplet free surface. Although they considered surface tension and imposed constant contact angles as a contact line boundary condition, capillary eects proved insignicant for the impact conditions considered. Liu et al. [9] used a modied version of RIPPLE [10], also a VOF-based code, to run simulations of molten droplet impact and simultaneous solidication typical of thermal spray conditions. The objective was to predict porosity formation between the solidied uid and the substrate, albeit in an axisymmetric context. Unlike all of the previous xed-grid solution techniques, Fukai et al. [11] developed an adaptive-grid nite element model to simulate normal droplet impact. By applying dierent contact angles (representative of an advancing and a receding contact line) to the spread and subsequent recoil of a water droplet, their results compared well with experimental data, and improved upon previous results [12] which neglected contact line eects. Pasandideh- Fard et al. [13] employedavof-based model to simulate normal droplet impact for a droplet size and velocity similar to that presented here. By measuring the temporal variation of the contact angle from photographs, and imposing this data as a boundary condition on their simulation, they obtained much better results than from corresponding simulations run with a constant contact angle. Finally, Bertagnolli et al. [14] recently presented an adaptive-grid nite element model to examine the impact of molten ceramic droplets in the context of thermal spraying, neglecting wetting eects. 3

4 We are aware of only a few references to 3D numerical modelling of droplet impact onto solid surfaces. As an addendum to their study on normal droplet impact, Trapaga and Szekely [6] presented preliminary results of 3D simulations of droplet impact onto a bre, and of the simultaneous impact and subsequent interaction of two droplets to demonstrate the modelling of 3D eects. Chang and Hills [15] also used FLOW-3D to examine oblique water droplet impact in the context of sprinkler irrigation. Unfortunately, the simulations were run on such a coarse grid (cell width = D o =6) as to cast doubt on the results. More recently, Karl et al. [16] developed a 3D numerical model similar to the one presented here. However, they proceeded to examine only the axisymmetric normal impact of an ethanol droplet against a hot wall. By imposing a free slip condition between uid and solid and a contact angle of 180 o on the contact line, they modelled lm boiling beneath the droplet, with the consequent rebound of the droplet o of the surface. Their numerical results agreed well with their experimental data. The model we present here, like many of those already mentioned, is a xed-grid Eulerian model, employing a volume tracking algorithm to track uid deformation and the droplet free surface. The choice of a xed-grid technique was made for several reasons: the relative simplicity of implementation; the capability of a volume tracking method to model gross uid deformation, including breakup; and the relatively small demand on computational resources. As we will demonstrate, such a model can simulate complex uid deformation surprisingly well. Figs. 1 and 2 illustrate the impact scenarios presented in this paper. The rst is a water droplet of diameter D o = 2 mm falling downwards at V o = 1 m/s onto a plane stainless steel surface inclined at an angle =45 o from the horizontal, corresponding to Re o = 2000 and We o = 27. The second impact is a 2 mm diameter water droplet falling at 1.2 m/s onto a stainless steel edge, corresponding to Re o = 2400 and We o = 39. The height of the edge was chosen arbitrarily to be 1 mm, or D o =2, and the point of impact to be oset from the edge by " =0:25 mm. We begin with a short description of our experimental methodology, present a more detailed description of our numerical model, and nally present our results. 4

5 II. EXPERIMENTAL METHODOLOGY The experimental methodology is similar to that originally presented in detail by Chandra and Avedisian [17] and later by Pasandideh-Fard et al. [13]. Fig. 3 illustrates the experimental apparatus. Single droplets are formed by slowly pumping distilled water through a hypodermic needle until they detach under their own weight. Droplets are uniformly 2 mm in diameter. Droplets fall onto a stainless steel surface, polished with 600 grit emery paper. The distance between the needle tip and the point of impact determines the impact velocity. The velocities considered here, 1{1.2 m/s, are suciently low that the droplets do not shatter upon impact. A single 35 mm photograph is taken of any one instant during an impact, as determined by a set time delay between droplet release and the illumination provided by a strobe of 8 s duration. The photographs of any particular instant from one droplet to the next are suciently repeatable that a complete impact sequence may be reconstructed from individual photographs of dierent droplets. Contact angles and contact diameters were measured manually from enlarged photographs. Measurements of most contact angles were reproducible to within 3 o. However, as we will show later, there were instances during oblique droplet impact where contact angles became very small, as a thin layer of uid slowly trailed the bulk of the droplet. Such angles proved more dicult to measure accurately, and thus the error associated with measurements of these angles was likely larger. III. NUMERICAL METHODOLOGY & VALIDATION We begin a discussion of our model by introducing a few simplifying assumptions. We assume that for the impact of a water droplet against a solid surface, that the ambient air about the droplet is dynamically inactive, which implies that the impact may be modelled by following the ow eld only in the liquid phase. The droplet is assumed to be spherical 5

6 at impact. The liquid is modelled as incompressible, with constant values of viscosity and surface tension. Fluid ow is assumed to be Newtonian and laminar. And nally, as a consequence of these assumptions, we assume that the only stress at the liquid free surface is a normal stress, and that any tangential stress is negligible. Equations of conservation of mass and momentum govern the uid dynamics: ~r~v =0 + ~r(~v ~V )=, 1 ~ rp + r 2 ~V + 1 ~ F b (2) where ~V represents the velocity vector, p the pressure, the density, the kinematic viscosity and ~F b any body forces acting on the uid. Boundary conditions for uid along solid surfaces are the no-slip and no-penetration conditions. At the liquid free surface, Laplace's equation species the surface tension-induced jump in the normal stress p s across the interface: p s = (3) where represents the liquid-air surface tension and the total curvature of the interface. Finally, a boundary condition is required at the contact line, the line at which the solid, liquid and gas phases meet. It is this boundary condition which introduces into the model information regarding the wettability of the solid surface. Although it is conceivable that one could formulate this boundary condition incorporating values of the solid surface tensions, such values are often inaccessible. Rather, we specify a \dynamic" contact angle d, the apparent contact angle measured at a moving contact line, which is likely a complex function of the contact line velocity. The basis for our model is RIPPLE [10], a 2D xed-grid Eulerian code written specifically for free surface ows with surface tension. In addition to a straightforward threedimensionalization, signicant improvements were incorporated into the model, including new algorithms for evaluating surface tension and for interface tracking. We focus on these 6

7 improvements in what follows. Note that we present some details in a 2D context to avoid unnecessary complexity. We hope the extension to three dimensions is fairly obvious. Eqs. (1) and (2) are discretized according to typical nite volume conventions on a rectilinear grid encompassing both the volume occupied by the droplet prior to impact as well as sucient volume to accommodate the subsequent deformation. Velocities and pressures are specied as on a traditional staggered grid [18]: velocities at the centre of cell faces, pressure at the cell centre. Fig. 4 illustrates a representative control volume in two dimensions. Eqs. (1) and (2) are solved using a two-step projection method, in which a time discretization of the momentum equation is broken up into two steps: ~~V, ~V n t =,~r(~v ~V ) n + r 2 ~V n + 1 n ~F n b (4) ~V n+1, ~ ~V t =, 1 n ~rp n+1 (5) The superscripts n and n+1 refer to the previous and current time levels respectively. In the rst step, Eq. (4), an interim velocity eld ~ ~V is computed explicitly from changes to the known eld V ~ n which result from convective, viscous and body forces acting on the uid during the timestep t. The explicit evaluation of these terms sets various limits on the size of t; the timestep chosen is the smallest of these. In the second step, Eq. (5) is combined with Eq. (1) at time level n+1 to yield an implicit Poisson equation for pressure: ~r( 1 ~ r rp n+1 )= ~ ~ ~V n t (6) Although the uid is assumed incompressible, the density is retained within the divergence operator to account for non-zero density gradients at the liquid free surface. The resulting set of linear equations in p is symmetric and positive denite; a solution is obtained at each timestep with an Incomplete Cholesky Conjugate Gradient solver (SLATEC routine DSICCG). Finally, the new velocity eld ~V n+1 is evaluated via Eq. (5). 7

8 Discretization of most boundary conditions is straightforward. No-slip conditions are applied by dening \ctitious" velocities within solid cells adjacent to uid cells. Normal velocities at solid surfaces, and the pressure within cells empty of uid, are simply dened as zero. Finally, the tangential stress condition at the droplet surface is implemented at cell faces between empty cells adjacent to an interface cell, by setting to zero values of and In addition to solving the ow equations within the liquid, the numerical model must also track the location of the liquid free surface. Various approaches exist to tracking a sharp discontinuity through a ow eld: the approach chosen is the rst-order accurate 3D volume tracking method of Youngs [19] in place of the Hirt-Nichols algorithm [8] implemented in RIPPLE. Although the Hirt-Nichols algorithm can be three-dimensionalized, Youngs' algorithm is a more sophisticated and accurate approach. A recent comparison of various 2D algorithms [20], including Hirt-Nichols and Youngs' equivalent 2D method [21], demonstrated a signicant dierence in the accuracy of the two approaches. Consider a function f dened in a continuous domain as: f = 8 >< >: 1 within the liquid phase 0 without (7) For a cell (i; j; k) of volume i;j;k, a\volume fraction" f i;j;k is dened as: f i;j;k = 1 Z fd (8) i;j;k i;j;k and a corresponding cell density i;j;k, which appears in the discretization of Eq. (6), is evaluated as: i;j;k = f f i;j;k (9) where f represents the (constant) value of the liquid density. Obviously, f i;j;k = 1 for a cell lled with liquid and f i;j;k = 0 for an empty call. When 0 < f i;j;k < 1, the cell is deemed to contain a portion of the free surface and is termed an \interface cell." Note that unlike f, the integrated quantity f i;j;k no longer contains information regarding the exact location 8

9 of the interface. This is, in fact, the primary drawback of volume tracking as an interface tracking method, and becomes problematic when dealing with surface tension and contact angles, as will be discussed. On the other hand, volume tracking is relatively simple to implement even in three dimensions, retains this simplicity regardless of the complexity of the interface geometry, conserves mass (or volume, since the uid is incompressible) exactly, and demands only a modest computational resource beyond that required by the ow solver. Since the function f is passively advected with the ow, f satises the +( ~V ~r)f =0 (10) Given the volumetric nature of f i;j;k and in order to maintain a sharp interface, the discretization of Eq. (10) requires special treatment. As with most other volume tracking algorithms, Youngs' algorithm consists of two steps: an approximate reconstruction of the interface followed by a geometric evaluation of volume uxes across cell faces. The interface is reconstructed by locating a plane within each interface cell, corresponding exactly to the volume fraction f i;j;k and to an estimate of the orientation of the interface, specied as a unit normal vector ^n i;j;k directed into the liquid phase. A discussion of the evaluation of ^n i;j;k is left to later in this paper. In two dimensions such aninterface is simply a line crossing a cell; in three dimensions the line becomes a three- to six-sided polygon, depending on how the plane slices the cell. To illustrate in two dimensions, Fig. 5(b) portrays the volume fractions corresponding to the exact (albeit unknown) interface of Fig. 5(a). Note that nothing guarantees that interface planes be contiguous. The position of the interface within each cell and the new velocities at the cell faces are then used to determine volume uxes across each face during the timestep. Fig. 5(c) illustrates such a ux across one face of a cell. Volume uxes are evaluated one direction at a time, always followed by aninterim interface reconstruction. Directional bias is minimized by alternating the order of advection from one timestep to the next. The original RIPPLE code was primarily a vehicle for introducing the \Continuum Surface Force," or CSF, model [22] as a novel approach toevaluating surface tension. The CSF 9

10 model reformulates surface tension into an equivalent volume force ~F ST : ~F ST (~x) = Z S (~r) ^n(~r) (~x, ~r) d~r (11) where is the Dirac delta function and the integration is performed over some area of free surface S. Surface tension is then incorporated into the ow equations simply as a component of the body force ~ F b in Eq. (2). Discretization of Eq. (11) requires an approximation to which spreads the surface tension force over uid in the vicinity of the surface. Unfortunately, the original discretization of Eq. (11) resulted in a surface tension force distribution which induced spurious uid motion near free surfaces [22]. Simulations of 2D droplet impact with the original surface tension model showed very unrealistic oscillations of the uid, especially as the droplet approached an equilibrium shape. Other discretizations of Eq. (11) have been proposed more recently [23,24] and were tested in both two and three dimensions (note that we are maintaining a 2D code, with algorithms equivalent to the 3D version). The accuracy of our simulations improved dramatically when we incorporated these improvements. What follows is an overview of the surface tension algorithm in our numerical model. Let 2h represent the kernel chosen to approximate, where the subscript 2h represents the nite radius over which 2h is non-zero. With 2h, Eq. (11) may bewritten: ~F ST (~x) Z S (~r)^n(~r) 2h (~x, ~r) d~r (12) Eq. (12) is now discretized as follows. We evaluate a local volumetric surface tension force for each interface cell: ~F STi;j;k = i;j;k A i;j;k i;j;k ^n i;j;k (13) where A i;j;k is the free surface area contained within the cell, evaluated as the area of the planar interface determined by Youngs' volume tracking algorithm. ~ F STi;j;k is then convolved with 2h to obtain a smoothed force eld ~ ~F STi;j;k : ~~F i;j;k = g i;j;k X l;m;n ~F STl;m;n 2h (~x i;j;k, ~r l;m;n ) l;m;n (14) 10

11 Note the introduction of a weighting function g i;j;k, dened as: g i;j;k = f i;j;k f (15) where f = 1=2 represents the average value of f. The role of g i;j;k is to transform the volumetric surface tension force into a uniform body force, and consequently forces ~ ~F STi;j;k = 0 for empty cells. Eq. (14) is simply a summation over cells within a radius 2h of ~x i;j;k, and can be applied to any grid structure. This allows for the possibility, for example, of applying the surface tension model to an adaptively rened grid. The resulting summation in Eq. (14) will then envelope more cells in a rened region than in an unrened region, which may allow for decreasing the radius 2h with grid renement. Finally, when the grid is uniform, a more accurate Eq. (14) is possible by evaluating the kernel contribution exactly: X ~~F i;j;k = g i;j;k ( ~F STl;m;n 2h (~x i;j;k, ~r ) d~r ) (16) Z i;j;k l;m;n What remains is to evaluate ^n i;j;k, required by the volume tracking algorithm to reconstruct the interface, and essential to the accurate evaluation of F ~ STi;j;k, especially since is evaluated as: =, ~ r^n (17) In a continuous domain, ^n = ~ rf j~rfj (18) But given the volumetric nature of f i;j;k, a simple algebraic discretization of Eq. (18) leads to poor estimates of ^n i;j;k. In two dimensions, complex geometric algorithms have been devised to evaluate ^n and [25,26]. There are no obvious extensions of these algorithms to three dimensions. Instead, the approach implemented in our model comes from a suggestion contained within the original CSF formulation [22]. Analogous to spreading the surface tension force 11

12 to uid in the vicinity of the free surface, better estimates of ^n i;j;k are obtained by evaluating the gradient of a smoothed f i;j;k, equivalent to employing a spatially-weighted gradient operator to evaluate ~rf. In practice, we employ the same 2h for smoothing f i;j;k as for smoothing ~F STi;j;k. ^n i;j;k are rst evaluated at cell vertices, to accommodate the evaluation of the cell-centred i;j;k ; cell-centred ^n i;j;k are then evaluated as an average of eight vertex values. The particular 2h chosen for the model is a radially-symmetric variation of a widely-used kernel proposed by Peskin [27]: 2h (~x) = 8 >< >: 1 + cos( j~xj 2h ) =c j~xj 2h 0 j~xj > 2h (19) where c normalizes the kernel: c = 32 3 h3 ( 2, 6)= (20) The reason we modify Peskin's kernel is found in work by Aleinov and Puckett [23] which demonstrates that radial symmetry appears to be an attractive attribute of 2h. All results presented in this paper were run on a uniform grid, with 2h = 3 x, thereby limiting 2 convolutions to a 4x4x4 stencil. This value of 2h, albeit arbitrary, was chosen for several reasons: the prohibitive computational cost of convolving over larger stencils; the diculty in evaluating convolutions near solid surfaces, to be discussed; and the argument presented by Kothe et al. [24] in favour of selecting the minimum 2h necessary to reduce noise associated with the evaluation of the curvature eld, without unduly extending the surface tension force. Note too that the small stencil blurs the distinction between dierent 2h, thereby reducing the inuence of kernel shape on model results. To illustrate the relationship between 2h and the evaluation of ~rf, we present the expression for the x-component of ~rf evaluated at the lower left vertex of the cell (i; j) illustrated in Fig. 4: f xi, 1 2 ;j, 1 2 = 1 2x f :3679 (f i;j + f i;j,1, f i,1;j, f i,1;j,1 ) 12

13 + :1510 (f i+1;j + f i+1;j,1, f i,2;j, f i,2;j,1 ) + :0949 (f i;j+1 + f i;j,2, f i,1;j+1, f i,1;j,2 ) + :0281 (f i+1;j+1 +f i+1;j,2, f i,2;j+1, f i,2;j,2 )g (21) A corresponding equation may be written for f yi, 2 1 ;j, ; the expression for ~rf in 3D is similar, 1 2 but includes 64 terms with dierent coecients. A dierent choice of 2h yields somewhat dierent coecients; increasing 2h adds further terms to the equation. Eq. (21) may be used to evaluate ^n at any vertex at least two cell widths from a solid surface. And at vertices along the boundary, normals ^n i;j;k are imposed to reect the contact angle d boundary condition; Fig. 6 provides an illustration. The remaining vertices, one cell width removed from a solid surface, require special treatment. The approach we take is to assign pseudo volume fractions to solid cells adjacent to the uid, in order to evaluate Eq. (21). In 2D, we implemented a geometric scheme to evaluate these volume fractions in a manner consistent with the value of d. The inuence of this scheme on global results, however, was insignicant when compared with a simpler approach of mirroring values of f i;j;k from uid cells into adjacent solid cells. Since an extension of our geometric scheme to 3D appeared hopelessly complicated, the simpler scheme was adopted. Evaluation of ~ ~F i;j;k is handled in the same way, by mirroring values of ~F ST from uid cells into adjacent solid cells. In this way, the volumetric force is conserved, in that the fraction of ~ F ST convolved out of the uid and into the boundaries is exactly reintroduced from the boundaries into the uid. Initial conditions for a droplet impact simulation are a specied droplet diameter and location to calculate an initial volume fraction eld, and an impact velocity applied to all droplet uid. The contact line treatment immediately following impact is the same as that just described, despite the short length of the contact line. The accuracy of the surface tension calculations likely suers as a result, but this is completely oset by the insignicance of such eects at a time when inertial eects dominate. Finally, much has been written of the apparent contradiction of a contact line moving 13

14 along a no-slip solid surface. Analytical solutions of the Navier-Stokes equations (subject to various simplications) yield a force singularity at a contact line unless a slip condition is imposed near the line [28]. Numerical models which explicitly track the free surface also require that a slip boundary condition be imposed on any contact line velocities [11]. This turns out not to be an issue for this model, precisely because it does not explicitly track the free surface, nor does it solve for contact line velocities. Instead, since velocities are specied at cell faces, the nearest velocity to the contact line is specied one half cell height above the solid surface. Again, Fig. 6 provides an illustration. It is this non-zero velocity which is then used to move uid near the contact line at each timestep. Before presenting the results of impact simulations, we present the results of three tests of individual components of the model. Fig. 7 illustrates plots of the interface reconstruction of a droplet (resolved by 10 uniform cells per radius, hereafter designated \cpr") before and after a 7.5 diameter translation, to assess the volume tracking algorithm and calculations of ^n. The gure was generated by plotting the actual polygons calculated by the advection algorithm. An exact velocity eld ~V = (u; v; w) = (3; 2; 1) was imposed on a uniform grid; the translation was divided into 400 uniform timesteps, with a maximum Courant number ( ut ) of 0.3. Qualitatively, the only signicant distortion of the droplet is a slight x \squaring" along grid axes; otherwise, the algorithm advects the droplet well. Quantitatively, we evaluated an error E f =0:040, dened as: E f = P i;j;k jf i;j;k, f e i;j;kj P i;j;k i (22) f i;j;k and fi;j;k e represent calculated and expected volume fractions following translation; the summations included only cells containing (or expected to contain) uid. Fig. 8 illustrates the inuence of mesh renement on the evaluation of the surface tensioninduced pressure jump p s across the surface of a static droplet. Droplets were dened at various resolutions, and for each resolution centred at ve dierent locations within a cell. Surface tension forces were calculated and the pressure equation solved, for one timestep. The calculated pressure jump was then evaluated as: 14

15 p c s = P i;j;k f i;j;k p i;j;k P i;j;k f i;j;k (23) and an error dened as: E p = pc s 4=D o, 1 (24) The symbols in Fig. 8 indicate the error associated with a droplet centred at a cell vertex (which was the case for all simulation results presented in this paper); the corresponding bars indicate the range of errors calculated at the dierent cell centres. There is a certain randomness to the error: as the droplet centre moves, values of f i;j;k shift, some interface cells are lled and others created. Nevertheless, convergence varies between rst and second order. The nal test examines the inuence of the value of 2h on the uniformity of ~ ~ F ST, as evidenced by the kinetic energy of the ow (so-called \parasite" currents [29]) induced in an initially static drop as a result of noise in the surface tension forces. Results were generated for a initially static 2 mm diameter water droplet resolved by 10 cpr, with simulations run for 100 timesteps to a physical time of 1 ms. Results are illustrated in Fig. 9. Not smoothing the F ~ ST at all, corresponding to 2h = 1 x, induced motion several times stronger 2 than smoothing ~F ST with 2h = 3 x. Although larger stencils yielded further incremental 2 reductions in the kinetic energy, all subsequent simulations presented here employed 2h = 3x for the reasons mentioned above. 2 IV. DROPLET IMPACT RESULTS We present simulation and experimental results for two 3D impacts: the 1 m/s impact of a 2 mm diameter water droplet onto a 45 o incline, and the 1.2 m/s impact of a similar droplet onto a sharp edge. Simulations were run on a uniform grid (x i =y j =z k ). Both simulations considered only one half of the droplet, exploiting the planar symmetry of the respective geometries. Corresponding simulations of full droplets yielded exactly symmetric results. Run time of each simulation on an SGI Indigo 2 workstation was on the order of a 15

16 few hours. Unlike the droplets illustrated in Fig. 7, the remaining gures were generated by plotting the f = 1=2 contour, resulting in depictions which more closely resembled actual photographs. Figs. 10 and 11 illustrate photographs and corresponding numerical views of the 45 o droplet impact, with the times to the right of the gures measured from the moment of impact. Fig. 11 clearly reveals the condition of the surface, including its inuence on the contact line, particularly at 7 ms, where the contact line appears slightly asymmetric. As well, close examination of the photographs of Fig. 11 corresponding to 5, 7 and 10 ms reveals a barely visible outline of uid beyond the contact line, a thin lm left behind on the solid surface. By 20 ms, the lm is no longer visible. From the complete set of photographs similar to those of Fig. 10, we measured contact angles at the bottom and top of the contact line, which we shall refer to as the \leading" and \trailing" contact angles, ` and t respectively. Fig. 12 illustrates the variation of these angles with time. The early behaviour of the droplet (1 ms) is very similar to that of normal droplet impact, with a symmetric jetting of uid about the point ofimpact, and large contact angles about the entire contact line. The symmetry is short-lived, however, and by 3 ms, the droplet is moving down the incline with ` t. By 7 ms, much of the uid has accumulated near the bottom of the contact line, while the remainder of the uid is spread thinly near the top of the droplet, slowly sliding down the incline. At 9 ms, Fig. 12 shows a momentary decrease in ` with an accompanying increase in t. Close examination of photographs about this point indicates that this behaviour corresponds to the point at which the contact line at the leading edge stops advancing, forcing a brief oscillation of the uid. From 9 ms onward capillary eects dominate, and the nal photographs reveal a droplet slowly approaching the equilibrium position depicted at 20 ms. For our simulation, we applied the measured variation of ` and t depicted in Fig. 12 as the boundary condition at the leading and trailing points on the contact line. We then tested various interpolation schemes between these two points to determine contact angles about the contact line. Results varied little between schemes for the rst 6{8 ms of impact, 16

17 when inertial eects dominate. Beyond 8mssimulation results were strongly inuenced by the form of the interpolation. We present our nal interpolation scheme now, which while admittedly ad hoc, we believe captures the correct physics. The basis for our interpolation is the realization that various parts of the contact line are either advancing or receding at dierent times during impact, and that although the contact line at the leading edge comes to rest well before the contact line at the trailing edge, nevertheless ` and t are likely representative of contact angles at any point at which the contact line is advancing or receding respectively. Put another way, the variation of contact angle with contact line velocity is not linear, but rather quickly approaches asymptotic advancing or receding values as the magnitude of the contact line velocity increases. To implement such aninterpolation in our model is dicult, however, because as outlined previously, the model does not track the contact line, nor evaluates a true contact line velocity. As an alternative, we chose to evaluate velocities V CL near the contact line one half cell height above the solid surface, and deemed these indicative of the true contact line velocities. Fig. 6 provides a 2D illustration. We then imposed ` and t at points about the perimeter which were obviously advancing or receding. The criterion for this determination was that V CL >V o =10, or 0.1 m/s. This value is arbitrary, but was chosen to be large enough to leave little doubt of the movement of the contact line, given the uncertainty inherent in our estimates of contact line velocity. The nal step then was to assign contact angles to those points on the perimeter not advancing or receding quickly (V CL < 0:1 m/s), by interpolating between the nearest known contact angles. Fig. 13 illustrates the interpolation scheme. By locating the instantaneous centre of the droplet midway between the leading and trailing edges, contact angles were interpolated linearly with the angle between the nearest known values. We ran simulations at grid resolutions ranging from eight to 16 cpr. Figs. 10 and 11 illustrate the results of the 10 cpr simulation, which predicts droplet shape surprisingly well, given what is a relatively coarse implementation of the contact line boundary condition. In particular, the view at 7 ms in Fig. 11 provides an example of the rationale behind our 17

18 particular contact angle interpolation: the entire thin collar of uid which surrounds much of the droplet is receding and likely characterized by a contact angle approximately equal to t. Fig. 14 illustrates the same view as Fig. 11, but compares simulations run at 10 and 16 cpr, and reveals good qualitative agreement. One quantitative measure of impact is the spread factor, measured from the leading to the trailing edge of the droplet and scaled by D o, illustrated in Fig. 15. Agreement with experiment at all resolutions is relatively good, although convergence is not exact. Examination of simulation data revealed a gradually varying inuence of the cuto velocity chosen for the contact angle model: in particular, our estimates of contact line velocities evaluated at ner resolutions were of smaller magnitude than corresponding velocities at coarser resolutions, resulting in fewer specied contact angles and more interpolation. We were unable, however, to devise a scaling for the cuto velocity with grid renement which would lead to absolute convergence of. A feature of the variation of which was not captured numerically was the \bump" in the curve at 14 ms. Close examination of photographs about this instant revealed that while the leading edge of the droplet remained pinned at the same position as at 9 ms, the trailing edge momentarily travelled back up the incline, driven by uid oscillating back from the leading edge. Numerically, estimated contact line velocities at 14 ms were less than the cuto velocity all along the contact line. The variation of d was thus reduced to a simple interpolation between ` and t at the leading and trailing edges, clearly insucient to predict the bump. Fig. 16 illustrates the computed variation of droplet kinetic energy with time. Results are very similar at all grid resolutions. Note too, however, that kinetic energy at all grid resolutions never reaches absolute zero, but asymptotes to approximately 0.8% of the initial kinetic energy regardless of grid resolution, evidence of the parasite currents presented in Fig. 9. The contact angle model just described relies on measured values of ` and t, and is thus not particularly useful for predictive purposes. A simpler model, however, of contact 18

19 angle versus contact line velocity, d = d (V CL ), can be based on a knowledge only of the asymptotic values of d associated with rapidly advancing and receding contact lines. We designate these angles as a and r respectively, not be confused with the thermodynamic advancing and receding angles associated with contact angle hysteresis. The proposed model is depicted in Fig. 17, and is similar to that of Fukai et al. [11] who imposed measured advancing and receding angles onto a 2D simulation of droplet impact. For water droplets on a stainless steel surface, we measured these angles from photographs: a = 110 o, r =40 o. For contact line velocities less than V e, d = e, an equilibrium contact angle. This is of course the inverse of contact angle hysteresis, for which a range of contact angles exist at which the contact line velocity is zero. The reason for formulating the relationship in this manner is related to the residual uid motion even at equilibrium: imposing e for all jv CL j < V e dampens the inuence of such uctuations on the evaluation of contact angles, and leads to smooth variations of contact angle along the contact line. For the simulation presented, V a =,V r =0:1 m/s, V e =0:05 m/s and e =75 o. Fig. 14 illustrates a qualitative comparison between results obtained with this model and the previous one, and demonstrates good agreement. Figs. 15 and 16 also illustrate the variation of and kinetic energy with time. The only signicant dierence between the results of the two contact angle models applied to the 45 o impact is the approach to an equilibrium position. The simpler model assumes a single value of an equilibrium contact angle, in contrast to the known dierence of 30 o between the leading and trailing edges (see Fig. 12). The result is a somewhat dierent equilibrium shape, evidenced by the nal plot in the sequence presented in Fig. 14. Of course, nothing precludes formulating a slightly more sophisticated contact angle model, in which e is allowed to vary linearly between the measured values at the leading and trailing edges, albeit at the cost of introducing two additional parameters (which would vary with the inclination angle of the surface). We ran simulations with such a model: as one would expect, the equilibrium shape was much nearer that pictured in Fig. 14(b) than Fig. 14(c). 19

20 Finally, Figs. 18 and 19 illustrate photographs and numerical views of the impact of a droplet onto an edge, again generated with the simpler contact angle model. Note that the photographs of Fig. 18 were taken from a slight inclination, such that some reection of the droplet in the surface is visible, especially at the outer edges of the droplet. The deformation of uid is dramatic, with the droplet breaking into two. Agreement between simulation and experiment is good, especially during the rst few milliseconds of impact. In fact, on the top of the edge, the model accurately predicts the entire impact, to a good estimate of the nal equilibrium shape of the upper droplet and a smooth contact line. The only signicant discrepancy between experiment and simulation is at the bottom of the edge, as the simulation predicts uid wetting the edge, in contrast to the corresponding photographs. It is likely, however, that our contact angle model may be inappropriate when applied to a vertical surface. Values of a and r were measured from contact lines on a horizontal surface. The setup of the experimental apparatus did not allow us to photograph and measure contact angles on the vertical surface, and thus the validity of our contact angle model along this surface is uncertain. V. CONCLUSIONS A3Dnumerical model of droplet impact has been developed, to predict uid deformation and possible breakup during impact onto asymmetric surfaces. The model is an Eulerian xed-grid algorithm, utilizing a volume tracking approach to track uid deformation and the droplet free surface. Surface tension is modelled as a volume force, acting on uid in the vicinity of the free surface. A contact angle is specied as a boundary condition along the contact line. Simulations of impact onto an inclined plane and onto a sharp edge were presented, as well as corresponding photographs to validate the results. Imposing measured values of contact angles on the simulation of the oblique impact yielded good agreement between simulation and experiment. A simpler model of contact angle versus contact line velocity 20

21 was proposed, based primarily on knowledge of the advancing and receding angles associated with a rapidly moving contact line. Results of simulations of both impact scenarios with this simpler model yielded reasonable agreement with experiment. Regarding the applicability of the model, the use of a volume tracking approach for surface tracking, of a corresponding volume-based surface tension model, and relatively coarse models for the variation of contact angle about the contact line, preclude using this model to predict near equilibrium ows and equilibrium shapes of sessile drops. However, note that accurate models of contact angle versus signicant contact line velocity are not available. For impacts characterized by signicant inertial and/or viscous eects, the model accurately predicts uid deformation during droplet impact. 21

22 REFERENCES [1] M. Rein, \Phenomena of liquid drop impact on solid and liquid surfaces," Fluid Dyn. Res. 12, 61 (1993). [2] F. H. Harlow and J. P. Shannon, \The splash of a liquid droplet," J. Appl. Phys. 38, 3855 (1967). [3] J. E. Welch, F. H. Harlow J.P. Shannon and B. J. Daly, \The MAC method," Technical Report LA-3425, LANL, [4] G. B. Foote, \The water drop rebound problem: Dynamics of collision," J. Atmos. Sci. 32, 390 (1975). [5] K. Tsurutani, M. Yao, J. Senda and H. Fujimoto, \Numerical analysis of the deformation process of a droplet impinging upon awall," JSME Int. J. Ser. II 33, 555 (1990). [6] G. Trapaga and J. Szekely, \Mathematical modeling of the isothermal impingement of liquid droplets in spraying processes," Metal. Trans. B 22, 901 (1991). [7] \FLOW-3D: Computational modeling power for scientists and engineers", Technical Report FSI , Flow Science, Inc. (San Diego, CA), [8] C. W. Hirt and B. D. Nichols, \Volume of uid (VOF) method for the dynamics of free boundaries," J. Comput. Phys. 39, 201 (1981). [9] H. Liu, W. Cai, R. H. Rangel and E. J. Lavernia, \Numerical and experimental study of porosity evolution during plasma spray deposition of W," in Science and Technology of Rapid Solidication and Processing (Kluwer Academic Publishers, 1995), pp [10] D. B. Kothe, R. C. Mjolsness and M. D. Torrey, \RIPPLE: A computer program for incompressible ows with free surfaces," Technical Report LA MS, LANL, [11] J. Fukai, Y. Shiiba, T. Yamamoto, O. Miyatake, D. Poulikakos, C. M. Megaridis and Z. Zhao, \Wetting eects on the spreading of a liquid droplet colliding with a at surface: 22

23 Experiment and modeling," Phys. Fluids 7, 236 (1995). [12] J. Fukai, Z. Zhao, D. Poulikakos, C. M. Megaridis and O. Miyatake, \Modeling of the deformation of a liquid droplet impinging upon a at surface," Phys. Fluids A 5, 2588 (1993). [13] M. Pasandideh-Fard, Y. M. Qiao, S. Chandra and J. Mostaghimi, \Capillary eects during droplet impact on a solid surface," Phys. Fluids 8, 650 (1996). [14] M. Bertagnolli, M. Marchese, G. Jacucci, I. St. Doltsinis and S. Noelting, \Thermomechanical simulation of the splashing of ceramic droplets on a rigid substrate," J. Comput. Phys. 133, 205 (1997). [15] W.-J. Chang and D. J. Hills, \Sprinkler droplet eects on inltration. I: Impact Simulation," J. Irrig. and Drain. Engrg. 119, 142 (1993). [16] A. Karl, K. Anders, M. Rieber and A. Frohn, \Deformation of liquid droplets during collisions with hot walls: Experiment and numerical results," Part. Part. Syst. Charact. 13, 186 (1996). [17] S. Chandra and C. T. Avedisian, \On the collision of a droplet with a solid surface," Proc. R. Soc. Lond. A 432, 13 (1991). [18] S. V. Patankar, Numerical Heat Transfer and Fluid Flow (McGraw-Hill, New York, 1980). [19] D. L. Youngs, \An interface tracking method for a 3D Eulerian hydrodynamics code," Technical Report 44/92/35, AWRE, [20] M. Rudman, \Volume-tracking methods for interfacial ow calculations," Int. J. Numer. Methods Fluids 24, 671 (1997). [21] D. L. Youngs, \Time-dependent multi-material ow with large uid distortion," in Numerical Methods for Fluid Dynamics, edited by K. W. Morton and M. J. Baines (Aca- 23

24 demic Press, New York, 1982), pp [22] J. U. Brackbill, D. B. Kothe and C. Zemach, \A continuum method for modeling surface tension," J. Comput. Phys. 100, 335 (1992). [23] I. Aleinov and E. G. Puckett, \Computing surface tension with high-order kernels," in Proceedings of the 6th International Symposium on Computational Fluid Dynamics, [24] D. B. Kothe, W. J. Rider, S. J. Mosso, J. S. Brock and J. I. Hochstein, \Volume tracking of interfaces having surface tension in two and three dimensions," Technical Report , AIAA, [25] A. J. Chorin, \Curvature and solidication," J. Comput. Phys. 57, 472 (1985). [26] J. Y. Poo and N. Ashgriz, \A computational method for determining curvatures," J. Comput. Phys. 84, 483 (1989). [27] C. S. Peskin, \Numerical analysis of blood ow in the heart," J. Comput. Phys. 25, 220 (1977). [28] E. B. Dussan V. and S. H. Davis, \On the motion of a uid-uid interface along a solid surface," J. Fluid Mech. 65, 71 (1974). [29] B. Lafaurie, C. Nardone, R. Scardovelli, S. Zaleski and G. Zanetti, \Modelling merging and fragmentation in multiphase ows with SURFER," J. Comput. Phys. 113, 134 (1994). 24

25 List of Figures 1 Droplet impact onto an incline Droplet impact onto an edge Experimental apparatus A 2D control volume, with velocities specied at cell faces, pressure at the cell centre The volume tracking method in two dimensions. (a) The exact liquid interface. (b) The corresponding volume fractions and planar interfaces. (c) With velocity u positive, the shaded region to the right of the dotted line is advected into the neighbouring cell during the timestep t Control volumes adjacent to a solid boundary. Unit normals ^n at solid boundary vertices are oriented to reect the contact angle boundary condition. Note that u, one half cell height above the solid surface, is the best estimate of the true contact line velocity Translation test of volume advection algorithm. Plots illustrate surface reconstruction of a 10 cell per radius droplet before and after a 7.5 diameter translation % error versus grid density for the pressure jump p s across the surface of a static drop. Symbols correspond to the error evaluated for a droplet centred at a cell vertex; bars indicate the range of errors evaluated for a droplet centred at ve locations within a cell Kinetic energy (scaled by initial surface energy) versus time for a static 2 mm diameter water drop dened by 10 cells per radius, for smoothed (2h = 3x) 2 and unsmoothed (2h = 1x) ~F 2 ST

26 10 Prole or side view (as illustrated in Fig. 1) of the impact of a 2 mm diameter water droplet at 1 m/s onto a 45 o stainless steel incline. Photographs at left, simulation results, corresponding to 10 cells per droplet radius, at right. Times at right measured from the moment of impact. Note that the photographs show both the droplet and its reection in the surface Normal view (perpendicular to surface) of the impact of a 2 mm diameter water droplet at 1 m/s ontoa45 o stainless steel incline. Photographs at left, simulation results, corresponding to 10 cells per droplet radius, at right Leading and trailing edge contact angles measured from photographs of the impact of a 2 mm diameter water droplet at 1 m/s onto a 45 o stainless steel incline Evaluation of contact angles. Unknown contact angles evaluated by linear interpolation with angle (measured about the midpoint of the droplet centreline) between nearest known values Comparison of normal views of the impact of a 2 mm diameter water droplet at 1 m/s onto a 45 o incline: (a) known d at 16 cells per radius; (b) known d at 10 cells per radius; (c) simpler d model at 10 cells per radius Spread factor versus time for the impact of a 2 mm diameter water droplet at 1 m/s ontoa45 o stainless steel incline. measured from the leading to the trailing edge, scaled by the droplet diameter. Lines correspond to simulations run at dierent grid densities, specied as the number of cells per droplet radius (cpr) Kinetic energy (scaled by initial kinetic energy) versus time for the impact of a 2 mm diameter water droplet at 1 m/s onto a45 o stainless steel incline. Lines correspond to simulations run at dierent grid densities, specied as the number of cells per droplet radius (cpr) Model of dynamic contact angle d versus an estimate of the contact line velocity V CL

27 18 Prole view of the impact of a 2 mm diameter water droplet at 1.2 m/s onto a 1 mm high stainless steel edge. Photographs at left, simulation results, corresponding to 10 cells per droplet radius, at right Perspective view of the impact of a2mmdiameter water droplet at 1.2 m/s onto a 1 mm high stainless steel edge. Photographs at left, simulation results, corresponding to 10 cells per droplet radius, at right

28 d o V o α Droplet impact onto an incline. Figure 1 M. Bussmann et al. Physics of Fluids 28

29 d o ε V o d o /2 Droplet impact onto an edge. Figure 2 M. Bussmann et al. Physics of Fluids 29

30 time delay circuit syringe pump aluminum frame 35 mm camera flash unit lens diffuser optical interruptor falling droplet test surface optical table Experimental apparatus. Figure 3 M. Bussmann et al. Physics of Fluids 30

31 y j+1 x x v i, j j u i-1, j p i, j u i, j y v i, j-1 i-1 i j-1 i+1 A 2D control volume, with velocities specied at cell faces, pressure at the cell centre. Figure 4 M. Bussmann et al. Physics of Fluids 31

32 0 0.1 uδt u liquid (a) (b) (c) The volume tracking method in two dimensions. (a) The exact liquid interface. (b) The corresponding volume fractions and planar interfaces. (c) With velocity u positive, the shaded region to the right of the dotted line is advected into the neighbouring cell during the timestep t. Figure 5 M. Bussmann et al. Physics of Fluids 32

33 v u n n Control volumes adjacent to a solid boundary. Unit normals ^n at solid boundary vertices are oriented to reect the contact angle boundary condition. Note that u, one half cell height above the solid surface, is the best estimate of the true contact line velocity. Figure 6 M. Bussmann et al. Physics of Fluids 33

34 Translation test of volume advection algorithm. Plots illustrate surface reconstruction of a 10 cell per radius droplet before and after a 7.5 diameter translation. Figure 7 M. Bussmann et al. Physics of Fluids 34

35 % error E P cells / radius % error versus grid density for the pressure jump p s across the surface of a static drop. Symbols correspond to the error evaluated for a droplet centred at a cell vertex; bars indicate the range of errors evaluated for a droplet centred at ve locations within a cell. Figure 8 M. Bussmann et al. Physics of Fluids 35

36 h= x/2 tke/πd o 2 γ h=3 x/ time [ms] Kinetic energy (scaled by initial surface energy) versus time for a static 2 mm diameter water drop dened by 10 cells per radius, for smoothed (2h = 3 2 x) and unsmoothed (2h = 1 2 x) ~ F ST. Figure 9 M. Bussmann et al. Physics of Fluids 36

37 1 ms 3 ms 5 ms 7 ms 10 ms 20 ms Prole or side view (as illustrated in Fig. 1) of the impact of a 2 mm diameter water droplet at 1 m/s onto a 45 o stainless steel incline. Photographs at left, simulation results, corresponding to 10 cells per droplet radius, at right. Times at right measured from the moment of impact. Note that the photographs show both the droplet and its reection in the surface. Figure 10 M. Bussmann et al. Physics of Fluids 37

38 1 ms 3 ms 5 ms 7 ms 10 ms 20 ms Normal view (perpendicular to surface) of the impact of a 2 mm diameter water droplet at 1 m/s ontoa45 o stainless steel incline. Photographs at left, simulation results, corresponding to 10 cells per droplet radius, at right. Figure 11 M. Bussmann et al. Physics of Fluids 38

39 150 contact angle leading edge trailing edge time [ms] Leading and trailing edge contact angles measured from photographs of the impact of a 2 mm diameter water droplet at 1 m/s ontoa45 o stainless steel incline. Figure 12 M. Bussmann et al. Physics of Fluids 39

40 C L β Evaluation of contact angles. Unknown contact angles evaluated by linear interpolation with angle (measured about the midpoint of the droplet centreline) between nearest known values. Figure 13 M. Bussmann et al. Physics of Fluids 40

41 (a) (b) (c) 1 ms 3 ms 5 ms 7 ms 10 ms 20 ms Comparison of normal views of the impact of a2mmdiameter water droplet at 1 m/s onto a 45 o incline: (a) known d at 16 cells per radius; (b) known d at 10 cells per radius; (c) simpler d model at 10 cells per radius. Figure 14 M. Bussmann et al. Physics of Fluids 41

42 spread factor ξ experiment 8 cpr 10 cpr 12.5 cpr 16 cpr 10 cpr (simpler θ d ) time [ms] Spread factor versus time for the impact of a 2 mm diameter water droplet at 1 m/s onto a 45 o stainless steel incline. measured from the leading to the trailing edge, scaled by the droplet diameter. Lines correspond to simulations run at dierent grid densities, specied as the number of cells per droplet radius (cpr). Figure 15 M. Bussmann et al. Physics of Fluids 42

43 ke/ke o cpr 10 cpr 12.5 cpr 16 cpr 10 cpr (simpler θ d ) time [ms] Kinetic energy (scaled by initial kinetic energy) versus time for the impact of a 2 mm diameter water droplet at 1 m/s onto a 45 o stainless steel incline. Lines correspond to simulations run at dierent grid densities, specied as the number of cells per droplet radius (cpr). Figure 16 M. Bussmann et al. Physics of Fluids 43

44 θ d θ a θ e θ r V r -V e V e V a V CL Model of dynamic contact angle d versus an estimate of the contact line velocity V CL. Figure 17 M. Bussmann et al. Physics of Fluids 44

45 0.8 ms 1.4 ms 2.0 ms 3.0 ms 6.0 ms 16.0 ms Prole view of the impact of a 2 mm diameter water droplet at 1.2 m/s onto a 1 mm high stainless steel edge. Photographs at left, simulation results, corresponding to 10 cells per droplet radius, at right. Figure 18 M. Bussmann et al. Physics of Fluids 45

46 0.8 ms 1.4 ms 2.0 ms 3.0 ms 6.0 ms 16.0 ms Perspective view of the impact of a 2 mm diameter water droplet at 1.2 m/s onto a 1 mm high stainless steel edge. Photographs at left, simulation results, corresponding to 10 cells per droplet radius, at right. Figure 19 M. Bussmann et al. Physics of Fluids 46