Drop Impact on a Wet Surface: Computational Investigation of Gravity and Drop Shape

Drop Impact on a Wet Surface: Computational Investigation of Gravity and Drop Shape MURAT DINC and DONALD D. GRAY Department of Civil and Environmental Engineering West Virginia University P.O. Box 6103, Morgantown, WV 26506-6103 UNITED STATES OF AMERICA mdinc@mix.wvu.edu Abstract: - Single water drops impinging onto wet, flat, isothermal surfaces have been simulated numerically to investigate the influence of gravity level and drop shape on the resulting flow. The commercial code ANSYS 14 Fluent was employed to perform 2D-axisymmetric simulations for incompressible, laminar and unsteady flow conditions using the explicit Volume of Fluid (VOF) surface tracking method with the Piecewise Linear Interface Calculation scheme (PLIC), the Continuum Surface Force model (CSF), and adaptive mesh refinement. Simulations of a spherical water drop impacting an upward facing wet surface (Re = 6690, We = 139, h/d = 0.837, = 0 ) were performed with Solar surface gravity (g = 275 m/s 2 ), Earth surface gravity (g = 9.81 m/s 2 ), Lunar surface gravity (g = 1.68 m/s 2 ), and zero gravity (g = 0). While the general evolution of the craters was similar, the rate at which they evolved increased along with the value of g. The impact of an upward moving drop on a downward facing layer in Earth surface gravity (g = 9.81 m/s 2 ) caused a significantly different flow. Because laboratory observations have found that water drops in the size range under study deviate slightly from true spheres, two cylindrical drops were simulated. The flows produced by the cylindrical drops were sufficiently similar to spherical drop impacts that it will not be necessary to account for the much smaller deviations from sphericity which occur in reality in order to obtain realistic simulations. Key-Words: - CFD, Fluent, VOF, PISO, PLIC, CSF, gravity, drop, impact dynamics 1 Introduction The interaction of drops and sprays with wet and dry surfaces is the key feature in many engineering applications such as spray cooling, spray painting, and fuel injection systems [1]. This paper reports one step in a study of the spray cooling of heated surfaces. A first principles simulation of a spray of millions of drops is impractical using present computers. The goal of this project is to develop correlations for inclusion in a Monte Carlo spray cooling model [2] which will embody sufficient physics to yield reliable predictions while being computationally efficient enough to use as a practical design tool. The impact of liquid drops onto both dry and wet surfaces has been studied by many researchers for more than a century [3]. This paper presents computer simulations of single water drops which are spheres (unless otherwise specified) of diameter D moving at velocity U normal to initially motionless thin layers of water having thickness h. The water has density and dynamic viscosity. The ambient fluid is air at atmospheric pressure, and everything is at room temperature. The interaction between a drop and a surface may involve inertial, viscous, surface tension, and gravitational forces. A sufficient set of dimensionless numbers to describe the interaction of these forces is the Reynolds number (Re = ρud/μ), the Weber number (We = ρu 2 D/σ), and the Froude number (Fr = U 2 /gd, where g is the acceleration due to gravity). Another key dimensionless number is the relative layer thickness (h/d). This paper is focused on the effects of gravity and drop shape. 2 Computational Method The computational simulations reported here were obtained using the commercial Computational Fluid Dynamics (CFD) code ANSYS 14 Fluent on a desktop workstation. The Navier-Stokes and continuity equations were solved in 2Daxisymmetric coordinates using the finite volume method for unsteady, incompressible, laminar flow. The Volume of Fluid (VOF) multiphase model was implemented with three phases (drop liquid, layer liquid, and air) to track the interface and distinguish

the drop liquid from the liquid in the pre-existing layer even though they are physically identical. In this flow, it is essential to be able to determine the location of the free surface of the liquid. This was accomplished by using the explicit VOF method [4]. For each liquid an advective transport equation was solved for an indicator function defined as the fraction of a computational grid cell occupied by that liquid. In the present simulations, liquid 1 was drop liquid and liquid 2 was layer liquid (both were water). If the sum of these indicator functions is less than 1, the cell must contain an air-water interface at which surface tension acts. Each fluid was treated as incompressible, and volume weighted fluid properties were used in cells which contained air and water. Surface tension was incorporated into the Navier-Stokes equation by using the Continuum Surface Force (CSF) model, which accounts for the curvature of the interface [5]. Velocity and pressure coupling was achieved with the Pressure Implicit with Splitting of Operators (PISO) algorithm [6]. The sharpness of the interface was enhanced by use of the georeconstruct (Piecewise Linear Interface Calculation, PLIC) scheme [7]. Discretization of the advective terms in the momentum equations used second order upwinding. Solution convergence was assured by monitoring the mass imbalance at some points in domain and also checking the residual values of parameters, such as the velocity, far from the impact region. The initial drop diameter D, the liquid layer thickness h, and the drop velocity U were defined as initial conditions. The boundary conditions are shown in Fig.1 with a grid that shows the initial mesh. The impermeable no-slip wall boundary condition was applied on the bottom boundary, and the centerline was a symmetry boundary. Pressure outlet boundary conditions were defined on the top and maximum radius boundaries. A great improvement in computational speed was achieved without sacrificing solution accuracy by using the adaptive mesh refinement method in which the grid was automatically refined in critical areas of rapid change such as the liquid-gas interface, and coarsened in areas of little change. In each level of refinement, the mesh length (both in vertical and horizontal directions) was divided by 2. For the present domain, a uniform grid of square D/80 cells would contain 325 000 cells. By using adaptive mesh refinement with the largest cells measuring D/5 and the smallest cells measuring D/80, only 25 000 cells were needed. For validation purposes some simulations were compared to computational and experimental studies in the literature and were found to be in good agreement. As an example, Fig.2 shows a comparison with the physical experiment of [8] and the computational study of [9]. Pressure Outlet Axis Fig.1 Boundary conditions and initial mesh for 2Daxisymmetric computational domain (0.05 m x 0.05 m). t=1 ms t=3 ms t=7.5 ms t=10 ms Fig.2 Comparison of results for 4.2 mm diameter droplet impacting onto a liquid layer with a thickness of 2.1 mm: Re = 1168, We = 2009, Fr = 631, and h/d = 0.5. (Experimental images from [8] in the left column, numerical results from [9] in the middle column, and numerical results of the current study in the right column with blue representing water and red representing air.) 3 Effects of Varying Gravity Four simulations were performed for the impact of a downward moving drop onto an upward facing liquid layer. One negative gravity simulation was

made of a drop moving upward and impacting a downward facing layer. For all of these cases D = 4.48 mm, h = 3.75 mm, and U = 1.5 m/s; only the magnitude of g was varied. Table 1 displays the values of g and the dimensionless parameters for these simulations. It is seen that Fr was the only dimensionless number which varied. In all these cases, the contact angle was set to 0, and had no effect because for this contact angle the wall is always wet. Table 1 Parameters used in gravity simulations. Case Gravity (m/s 2 ) Re We Fr h/d Solar 275 6690 139 1.83 0.837 Earth 9.81 6690 139 51 0.837 0 ms 5 ms 7 ms 11 ms 9 ms 15 ms Fig.3 Single drop impact onto a wet surface in Solar gravity. (Note: Red is drop water, green is layer water, and blue is air, "ms" refers to milliseconds.) Lunar 1.68 6690 139 310 0.837 Zero 0 6690 139 0.837 Earth negative gravity -9.81 6690 139-51 0.837 Simulations were performed for upward facing layers in Solar surface gravity (Fr = 1.83), Earth surface gravity (Fr = 51), Lunar surface gravity (Fr = 310) and zero gravity (Fr = ). Solar gravity is an upper bound for applications in this stellar system and dramatizes the effects of very high g values. In all these cases, a crater with a very thin bottom layer formed and spread outward with a crown at its outer edge until it reached a maximum radius and crown height. Then the crown collapsed and the crater refilled. The liquid flowing back into the crater formed an incipient Worthington jet at its center. Splashing was not observed, as shown in Figs.3, 4, 5 and 6 for Solar, Earth, Lunar and zero gravity, respectively. The times at which the crown reached its maximum height increased as Fr increased, showing that gravity as well as surface tension caused the craters to refill. These times were 10, 34, 45, and 51 ms. The pace of events increased as Fr increased, as illustrated by the well defined Worthington jet at 11 ms in Solar gravity whereas the jet has barely begun to form at 35 ms in zero gravity. In Figs.4, 5, and 6, it appears that the drop liquid has broken into a number of discrete volumes, but this is an artefact of using a finite computational grid. Within the limits of the continuum approximation, the drop liquid must remain connected. Fig.4 Single drop impact onto a wet surface in Earth gravity. 0 ms 5 ms 16 ms 25 ms 35 ms 45 ms Fig.5 Single drop impact onto a wet surface in Lunar gravity.

deviation from a sphere. Simulations were performed for single cylinders of water with a diameter D = 4.48 mm (the same as for the previously described spherical drops) and lengths L of 4.48 and 2.24 mm impinging onto static water layers with thickness h = 3.75 mm in Earth gravity. Table 2 shows the parameters for these cases, where D is the length scale used to calculate Re, We, and Fr. A contact angle of 0 was specified in all these cases. Table 2 Parameters used to study drop shape. Fig.6 Single drop impact onto a wet surface in zero gravity. The negative gravity case of an upward moving drop hitting a downward facing liquid layer in Earth surface gravity (Fr = 51.5) is shown in Fig.7. There was still no splashing, but the impact excited a Rayleigh-Taylor instability in which the crown of the impact crater grew large as it fell away from the surface. The crater continued to spread until most of the liquid fell from the surface. Fig.7 Single drop impact onto a wet surface in negative gravity on Earth. The ceiling is at the bottom of the figure. 4 Effects of Drop Shape High speed videos obtained in the experimental phase of this research project have confirmed that drops in the 4-mm size range are seldom truly spherical. Instead they quiver about a spherical mean. In order to determine if deviations from sphericity would have a significant effect on the drop impact flow, computations were performed for cylindrical drops which represent an extreme Shape Re We Fr h/d L/D Sphere 6690 139 51 0.837 NA Full cylinder 6690 139 51 0.837 1 Half cylinder 6690 139 51 0.837 0.5 Results for the spherical shape have been presented in Fig.4. The results for the full cylinder and half cylinder are given in Figs.8 and 9, respectively. The results are generally similar, but there are some differences. At 5 ms one very small air bubble was observed on the centreline for the sphere, but a much larger one was observed with the full cylinder, and a ring bubble was observed for the half cylinder. The more prominent bubble formation under the cylinders may be attributed to the greater difficulty for the air to flow out from beneath the bottom of the drop when the bottom is flat. At each time, the crater diameter and crown height were greatest for the full cylinder and least for the half cylinder. These results are not surprising given that the volumes of the full and half cylinders are 150% and 75% of the sphere. At 45 ms, the Worthington jet was thinnest for the full cylinder and fattest for the half cylinder. Because the differences in shape between a sphere and a cylinder are far more extreme than the differences observed among actual drops in the lab, it can be concluded that it is not necessary to simulate realistic nonspherical drops. Simulations of spherical drops are sufficiently accurate.

0 ms 16 ms 5 ms 25 ms 35 ms 45 ms Fig.8 Full cylinder drop (D = 4.48 mm, L = 4.48 mm) impact onto a thin liquid layer in Earth gravity. 0 ms 5 ms 16 ms 25 ms 35 ms 45 ms The commercial CFD code ANSYS 14 Fluent has been used to study the impact of single drops of water onto shallow water layers. The simulations were for 2D-axisymmetric domains with adaptive mesh refinement. The models used in Fluent were the explicit Volume of Fluid method, the Pressure Implicit with Splitting of Operators algorithm, the Piecewise Linear Interface Calculation scheme, and the Continuum Surface Force model. Simulations of drop impacts were run for gravitational accelerations ranging from extremely large to zero (Solar to zero). While the general evolution of the craters was similar, the rate at which they evolved increased along with the value of g. The impact of an upward moving drop on a downward facing layer caused a significantly different flow, as it excited a Rayleigh Taylor instability which caused the liquid to fall off of the ceiling. Laboratory observations have found that water drops in the 4-mm size range are not true spheres when they impact a liquid layer. To explore the importance of nonspherical drop shapes, two cylindrical drops were simulated. While there were definite differences compared to the impact of a sphere, the overall similarity of the flows produced by radically different drop shapes implies that the much smaller deviations from sphericity which occur in reality need not be considered in order to obtain realistic simulations. Fig.9 Half cylinder drop (D = 4.48 mm, L = 2.24 mm) impact onto a thin liquid layer in Earth gravity. 5 Conclusions The goal of the present research project is to create a model for spray cooling that is sufficiently accurate to yield reliable predictions, yet sufficiently simple to be a practical design tool. Kreitzer and Kuhlman [2] have formulated a preliminary version of such a model based on a Monte Carlo approach. In order to upgrade this model to the required level of physical realism, more accurate correlation equations for such parameters as maximum crater radius and minimum layer depth must be determined and incorporated. Both experimental and computational methods are being used to obtain these correlations. The simulations presented here are early results from the computational study. Much additional work remains before the needed correlations are finalized. Acknowledgements The authors thank their experimentalist collaborators Dr. John M. Kuhlman, Nicholas L. Hillen, and J. Steven Taylor for many stimulating discussions. The authors gratefully acknowledge the financial support of this work under NASA Cooperative Agreement NNX10AN0YA. The views and conclusions are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements of NASA or the West Virginia NASA Space Grant Consortium. References: [1] Silk, E. A., Golliher, E. L., Selvam, R. P., Spray Cooling Heat Transfer: Technology Overview and Assessment of Future Challenges for Micro-Gravity Application, Energy Conversion and Management, Vol.49, 2008, pp. 453 468.

[2] Kreitzer, P. J., Kuhlman, J. M., Spray Cooling Droplet Impingement Model, Paper AIAA- 2010-4500, AIAA 10th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, Chicago, IL, June 28-July 1, 2010. [3] Yarin, A. L., Drop Impact Dynamics: Splashing, Spreading, Receding, Bouncing, Annual Review of Fluid Mechanics, Vol.38, 2006, pp. 159-192. [4] Hirt, C. W., Nichols, B. D., Volume of Fluid (VOF) Method for the Dynamics of Free Boundaries, Journal of Computational Physics, Vol.39, 1981, pp. 201-225. [5] Brackbill, J. U., Kothe, D. B., Zemach, C., A Continuum Method for Modeling Surface Tension, Journal of Computational Physics, Vol.100, 1992, pp. 335-354. [6] Issa, R. I., Solution of Implicitly Discretized Fluid Flow Equations by Operator Splitting, Journal of Computational Physics, Vol.62, 1986, pp. 40-65. [7] Rider, W. J., Kothe, D. B., Reconstructing Volume Tracking, Journal of Computational Physics, Vol.141, 1998, pp. 112-152. [8] Wang, A.-B., Chen, C.-C., Splashing Impact of a Single Drop onto very Thin Liquid Films, Physics of Fluids, Vol.12, No.9, 2000, pp. 2155-2158. [9] Asadi, S., Passandideh-Fard, M., A Computational Study on Droplet Impingement onto a Thin Liquid Film, The Arabian Journal for Science and Engineering, Vol.34, No.2B, 2009, pp. 505-517.