Numerical Studies of Droplet Deformation and Break-up

ILASS Americas 14th Annual Conference on Liquid Atomization and Spray Systems, Dearborn, MI, May 2001 Numerical Studies of Droplet Deformation and Break-up B. T. Helenbrook Department of Mechanical and Aeronautical Engineering Clarkson University, Potsdam, NY 13699 Abstract Numerical results are presented on the deformation and stability of liquid droplets in a uniform gaseous flow. The shape response of the droplets is categorized over a wide range of conditions, and a new parameter that predicts this response is identified. The stability limits of the droplets are also investigated. It is shown that the critical Weber number of break-up can be easily predicted by the onset of an exponentially growing oscillatory instability. Predicted critical Weber numbers of break-up are in good agreement with experiments. Introduction Both statistical [1] and Lagrangian [2] spray simulations require information on the behavior of individual liquid droplets in a gaseous flow. This information typically includes a drag law, an evaporation law, a droplet merging model, and a droplet break-up model. In this paper, we perform detailed numerical simulations of nonevaporating, isolated, liquid droplets to better understand droplet deformation and break-up properties. The droplet behavior is examined in a very fundamental flow, a uniform stream. There is a vast amount of experimental data for this flow obtained predominantly from drop-tower and shock-tube experiments [3]. By doing numerical simulations however, we can obtain precise information for an extremely wide range of conditions. This enables us to provide new insights into the experiments and examine conditions that are difficult to study experimentally. A well-known difficulty in studying droplet behavior is the droplet s extreme sensitivity to surface contaminants [4]. In this work, we study uncontaminated droplets. Most experimental data is to some degree affected by surface contaminants so differences between our results and the experimental data are expected. However, it is important to study uncontaminated liquid droplets because in most spray systems it is difficult or impossible to determine the degree/nature of the contamination. Numerical results for uncontaminated droplets can bound this limit of the droplet behavior. In our future work, we will perform parametric studies of contaminated droplets. There have been several previous numerical studies of droplet behavior [5, 6]. The most relevant is that of Dandy and Leal. They studied axisymmetric falling droplets and categorized the drop shapes, drag response, and flow fields at various conditions. This work extends those efforts including more than 3000 simulations over the entire range of physically relevant conditions. We focus mainly on the drop deformation response and the break-up limits. The ability to perform this study is due to a new numerical algorithm we have developed for two-phase flows which is both accurate and efficient and, in addition, can simulate drops at highly deformed conditions such as those that occur near the break-up limits. This algorithm is described in [7]. Formulation The physical problem is that of an axisymmetric liquid drop being driven through a quiescent gas by a body force such as gravity. We study the deformation and stability of the drop at its terminal velocity, U. This problem is relevant not only to systems with gravity, but also to spray systems in which the injected droplet s deformation rate is fast relative to its velocity decay rate. This will be discussed further after we introduce the non-dimensional parameters governing the problem. Assuming that the droplet velocities are small relative to the speed of sound, we approximate both the gas and the liquid as incompressible. We also neglect temperature gradients and evaporation. This is appropriate for non-combusting systems and for combusting systems in which droplet deformation

and break-up occurs in a zone with nearly constant temperature while evaporation occurs downstream in the vicinity of the flame. We can approximate the above conditions as two fluids with constant densities, ρ l and ρ g, and viscosities, µ l and µ g, separated by an interface with constant surface tension, σ. The subscripts denote either liquid or gas. Both fluids must satisfy the axisymmetric form of the incompressible Navier- Stokes equations. At the interface, we enforce the conditions that the interfacial mass flux is zero and that the jump in stress across the interface is balanced by surface tension. For the mathematical form of the governing equations and the interface conditions, see [7]. Physical Parameters If we non-dimensionalize the problem, there are four independent parameters. We choose the liquid to gas density ratio ρ l /ρ g, the liquid to gas dynamic viscosity ratio µ l /µ g, the Weber number W e = ρ g Ud 2 /σ, and the Ohnesogre number, Oh = µ l / ρ l σd to describe the problem where d is the volume equivalent diameter of the droplet. The body force on the droplet does not appear in any of the independent parameters because we have assumed the drops are at their terminal velocity. At the terminal velocity, the body force and the drop velocity are not independent. For a given pair of fluids such as hexane/air, the liquid to gas density ratio varies mainly with the gas density which is function of the ambient pressure and temperature. For combustion systems, the pressures of interest are between atmospheric and the critical pressure of the mixture, and the ambient temperature range is between ambient and approximately 2500 K, the approximate adiabatic flame temperature. To investigate the effect of density ratio over this range of conditions, we study density ratios of 5, 50, and 500. The liquid to gas viscosity ratio is primarily a function of the temperature. If we assume that the gas temperature varies between ambient and 2500 K while the liquid temperature is fixed near the boiling temperature, the viscosity ratio varies between 5 and 15. We study the values 5, 10, and 15. It is somewhat inconsistent with our formulation to assume that the gas temperature is much greater than the liquid temperature because we have neglected temperature gradients, but by examining this range we can bound the effect of viscosity ratio. Results are obtained by fixing the Ohnesogre number and increasing the Weber number from small values, usually 0.1 or less, up to the critical Weber number of break-up. The method of determining the critical Weber number of break-up will be discussed in the results. For fixed fluid properties, the Ohnesogre number only varies with drop diameter, thus increasing the Weber number with Oh constant corresponds to an experiment in which drops of a fixed size are driven through a flow at increasing velocities. For a given set of conditions, the simulations determine the steady drop shape, the flow field in and around the drop, and the required body force to hold the drop at the specified conditions. In some cases, a more traditional unsteady simulation is performed where the body force is fixed and we let the drop evolve to its terminal state. The maximum gas-phase Reynolds number, ρ g Ud/µ g, we study is 200. Beyond 200, the flow probably becomes non-axisymmetric based on results for flow over a sphere [8]. This also limits the smallest Ohnesogre number case (largest drop size) for which we can study drop break-up; for Oh < ρ l /ρ g µ l /µ g 1/200, the Reynolds number exceeds 200 before the Weber number reaches unity. In the results, the Ohnesgore number is increased from this minimum value by factors of 10 0.2 over a range of 2 to 3 orders of magnitude. Time Scale Analysis To justify our statement that the behavior of falling droplets is predictive of injected drops, we analyze the time scales in the problem. We begin by transforming the governing equations to a coordinate system that moves with the center of mass of the droplet. This transformation results in one additional term in the momentum equation for the axial velocity equation [6], ρ k U c / t, where U c is the center of mass velocity of the drop. If the time scale associated with this term is large relative to the time scale of the droplet response, we can neglect it. Physically, this means that the drop will respond in a quasi-steady manner to the instantaneous relative velocity between the drop and the gas even though this velocity is changing with time. To determine the velocity decay rate, we assume that the drag on the drop is similar to that on a solid sphere. In this case, we can approximate the drag using the Stokes law, C D = 24/Re where C D is the coefficient of drag. There are more accurate curve-fits for drag on a sphere, but for our purposes, Stokes law is sufficient. Based on this drag, we arrive at an exponential velocity decay rate for the droplet of 18(ρ g /ρ l )ν g /d 2 where ν g is the kinematic

viscosity of the gas. To determine the deformation time scale of a liquid droplet, we examine an isolated liquid droplet with no gas effects. The primary mode of oscillation of an isolated liquid droplet can be modeled by a damped spring mass system. The characteristic rates are then λ = 16ν l d 2 ± (16νl d 2 ) 2 64σ ρ l d 3 (1) The constant multiplying the σ term is chosen such that for the inviscid case the oscillation frequency is the same as that predicted by linear analysis [9] for the primary mode of oscillation. We determined the constant multiplying the ν l term by performing numerical calculations of the decay rate of the primary oscillation mode of a viscous drop. This model faithfully represents small-amplitude, primary-mode droplet oscillation for all values of ρ l, d, σ, and µ l. The model shows that when Oh = 1/2, the drops are critically damped. For Oh less than 1/2 the decay rate of drop oscillations is 16ν l /d 2. If we divide this by the velocity decay rate of a drop in a gas we arrive at 16/18 µ l /µ g. Thus, in the small Ohnesogre number limit, the drops will respond in a quasi-steady manner to the flow if the liquid to gas viscosity ratio is large. For the conditions we are studying this is a reasonable leading order approximation. When the Ohnesogre number is much greater than one, we can expand equation (1) to determine the decay rate of a perturbation as 2σ/(dµ l ). Dividing by the velocity decay rate, we get the ratio Oh 2 /9 µ l /µ g. Thus, when the Ohnesogre number is of order µ l /µ g, the quasi-steady approximation breaks down. In this limit, our simulations will only be relevant to falling liquid droplets and not to injected droplets. Results The solutions are calculated on a trapezoidal domain given by the r, z points (0,-10), (0,15), (10,7.5), and (10,12.5) with the drop positioned at r, z = (0, 0). At the lower boundary of the domain an inflow condition is enforced with a nondimensional velocity of unity. At the right and upper boundaries, a no-stress condition is enforced. We have performed drag calculations for flow over a sphere with the boundaries at various distances from the sphere in [7]. With the boundaries given, the change in drag due to increasing the distance of the boundaries by 25% is less than a percent. Deformation Response We begin the analysis by categorizing the drop deformation response. This will help us to understand the break-up behavior. Figure 1 shows the three most prevalent drop shapes: prolate, oblate, and dimpled. The gas flow direction is from the bottom to the top of the figure. The first two shapes are obviously mutually exclusive while the third can occur in both prolate and oblate drops. Figure 1: Prolate, oblate and dimpled drop shapes. The transition between prolate and oblate can be characterized by the ratio of magnitude of the dynamic pressures in the liquid to the magnitude of the dynamic pressures in the gas. The dynamic pressure in the gas is of order ρ g U 2 with high pressures occurring at the leading and trailing edge of the drop and low pressures occurring at the equator. This tends to cause oblate shapes. Inside the drop, internal circulation also causes high pressures at the leading and trailing edges. This opposes the effects of the gas and tends to cause prolate drop shapes. To estimate the magnitude of the dynamic pressures in the liquid, we must estimate the magnitude of the internal circulation velocity. Stokes solution for flow around a spherical liquid drop predicts that the internal circulation velocity will be of magnitude U/(2 + 2µ l /µ g ). Comparing this to our numerical results, we find that it is fairly predictive even for cases that have large Reynolds numbers. Thus, the liquid to gas dynamic pressure ratio is approximately P l/g = ρ l ρ g ( µg µ l ) 2 (2) where we have assumed the viscosity ratio is large

relative to one. Figure 2 shows the length to width aspect ratio of all calculations performed versus P l/g. Each vertical line of data points consists of all the Oh and W e conditions simulated for a single pair of parameters, ρ l /ρ g and µ l /µ g. There are 9 vertical lines in all corresponding to the 3 x 3 matrix of density ratios and viscosity ratios studied. From this figure we see that P l/g is a reasonable predictor of the prolate to oblate transition; cases with P l/g less than one are predominantly oblate while those with P l/g greater than one are prolate. This is true independent of the Weber number and the Ohnesogre number although the magnitude of the deviation from spherical is dependent on these parameters. Length / Width Ratio 5 4.5 4 3.5 3 2.5 2 1.5 inary calculations with a contaminant model and shown that the internal circulation pattern is radically different. In some cases, this causes the drop shape to change from prolate when uncontaminated to oblate when contaminated. Thus, the P l/g transition may be only observable in contaminant-free experiments. The dimpled shape is defined by a concave region at the rear of the drop. This phenomenon is primarily due to viscous effects and occurs when the Capillary number, µ g U/σ = W e/re is O(1). To confirm this, on Figure 3 we plot the Capillary number vs. Ohnesogre number of all the calculated points that have a dimpled shape with a dark square and the remaining cases with a gray triangle. A line of constant Weber number is also shown on the plot, this is given by Ca = W e Oh µ g /µ l ρl /ρ g = W e Pl/g Oh. For Oh < P 1/2 l/g, with increasing flow velocity the Weber number will exceed unity before the Capillary number. Thus, the drops will tend to break-up before the Capillary number exceeds unity. This explains why there are no calculations of stable droplets in the upper-left quadrant of the figure. For Oh > P 1/2 l/g we see dimpling when the Capillary number becomes O(1). 1 0.5 10-1 10 0 10 1 ρ l / ρ g (µ g / µ l ) 2 Figure 2: Length to width ratio versus the parameter P l/g With increasing Weber number, the prolate drops continue to increase in aspect ratio. In some cases, we find stable axisymmetric solutions with aspect ratios of two or greater. This leads us to question the validity of the axisymmetric assumption. Simple experiments of a falling solid ellipsoid have shown that the ellipsoid tends to align itself with its largest cross-sectional area normal to the flow. Thus, we suspect that the prolate cases are not axisymmetrically stable but rather will tumble as they move through the gas. For this reason, we have not tried to use the axisymmetric simulations to determine the break-up limits of prolate drops. According to these results, there should be qualitatively different behaviors for P l/g greater or less than one, however this has not been observed in experiments. This is most likely due to the effect of contaminants. We have performed some prelim- Capillary Number 10 0 10-1 10-2 10-3 We = c 10-3 10-2 10-1 10 0 10 1 Oh Figure 3: Capillary number of drops with dimpled shape. There are two distinct trends in the appearance of dimpling: one showing Ca 1 over a range of Oh and the other showing a linear increase in Ca number from 0.1 to 1.0. This a secondary effect due to P l/g. The cases with P l/g less than one tend to be oblate which flattens the back of the drop and makes it more likely to dimple. These are the cases that give the linear increase in Ca from 0.1

to 1.0. The cases with P l/g greater than one tend to be prolate which makes it harder for the end to become dimpled due to the increased opposing curvature. In this case, the Ca must be approximately one independent of Oh. Stability Limits Having classified the deformation behavior, we now examine the stability limits of the droplets. We do not study any prolate cases, P l/g > 1, because these probably do not remain axisymmetric. The stability limits are determined by increasing the Weber number of the drop until the drop becomes unstable. Figure 4 shows two examples of this for ρ l /ρ g = 50, µ l /µ g = 15, and Oh = 0.063, 0.25. The axes of the plot are the length to width ratio of the drop and the Weber number. For these conditions, P l/g = 0.2 and the drops become oblate as the Weber number is increased. For the case of Oh = 0.063, at W e 14.0 there is transition in stability and the drop begins to oscillate. We have simulated the growth of these oscillations up to a point at which the minimum droplet length is less than 0.2. Some improvements in the numerical algorithm are needed to be able to continue the simulation. However, up to this point the amplitude grows exponentially which leads us to believe that this transition in stability is an accurate predictor of the break-up point of the droplet. Length/Width Ratio 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Oh = 0.25 Oh = 0.063 0 5 10 Weber Number Figure 4: Deformation response characterized by the length to width ratio versus Weber number for ρ l /ρ g = 50, µ l /µ g = 15 and Oh = 0.063, 0.25. The triangle denotes onset of instability The second case, Oh = 0.25, provides further confirmation of this statement. In this case, as the Weber number is increased the solution reaches a turning point. At the turning point solutions are obtained by specifying a body force and then determining the terminal Weber number. As the body force is increased, the terminal velocity initially increases as expected, but beyond the turning point it decreases with increased body force which indicates that the increase in drag with body force is greater than the linear increase in force on the drop. This unexpected phenomenon indicates that there are no stable steady solutions for a single drop beyond the turning point. Thus, we can take the value of the Weber number at the turning point, 12.5, as the critical Weber number of break-up. Further increases in the body force result in a transition to unstable oscillations as observed for Oh = 0.063. The value of the Weber number at this point is within 2% of the turning point value so either is a reasonable prediction of the break-up limit. Figure 5 shows the critical Weber number of drop stability versus Ohnesogre number. Four cases are shown, ρ l /ρ g, µ l /µ g = (50,15), (5,15), (5,10) and (5,5). There is some scatter in the data because we have only determined the stability boundary to approximately 10% accuracy. The data shows several distinct trends. First, for Oh < 1, the critical Weber number is between 10 and 14 and there is very little sensitivity to the density ratio and viscosity ratio. This is in very good agreement with both shock-tube studies and drop-tower studies [3] of drop break-up. For Oh > µ g /µ l ρl /ρ g, the Capillary number exceeds unity before the Weber number. Thus, for large Oh the Capillary number, Ca = W e Oh µ g /µ l ρl /ρ g is the correct parameter. This explains the sensitivity to the density ratio and viscosity ratio in this limit. If we examine the results in terms of the Capillary number, for Oh > 3 they begin to collapse on Ca 2.0 which confirms that the Capillary number is the relevant parameter in this limit. The break-up modes for Ohnesogre number greater or less than unity are different. For small Oh, the drop flattens then transitions to unstable oscillations. For moderate Oh, the drop develops a broad dimple at the rear of the drop and transitions to unstable oscillations. For large Oh the dimple at the rear of the drop becomes very sharp and difficult to resolve numerically. Because of this, we are not able to determine whether there is a transition to oscillation or not. This limits the maximum Oh for which we can determine the stability limits. If we compare the large Oh trends to those observed in shock-tube experiments, we see that they are very different. The shock tube experiments pre-

References Weber Number 13 12 11 10 9 8 7 6 5 4 3 2 1 ρ l /ρ g, µ l /µ g 50, 15 5, 15 5, 10 5, 5 10-1 10 0 10 1 Ohnesogre Number Figure 5: Stability limits of liquid droplets. dict that the critical Weber number increases at large Oh while our numerical experiments predict that the critical Weber number decreases. As discussed previously, this is because for large Oh the velocity decay rate of the drop is fast relative to the deformation rate. In this limit, the velocity of the drop will decay before the drop has a chance to break-up, and the quasi-steady response is not relevant. For falling droplets however, the large Oh limits are correct. [1] M. R. Archambault and C. F. Edwards. In Proceedings of the Eighth International Conference on Liquid Atomization and Spray Systems (ICLASS), pages 996 1003, Pasadena, CA, July 2000. [2] J. C. Oefelein. In Proceedings of ILASS- America s 12th Annual Meeting, May 1999. [3] L. P. Hsiang and G. M. Faeth. Int. J. Multiphase Flow, 18(5):635 652, 1992. [4] R. Clift, J. R. Grace, and M. E. Weber. Bubbles, Drops, and Particles. Academic Press, 1978. [5] D. S. Dandy and L. G. Leal. J. Fluid Mech., 208:161 192, 1989. [6] R. J. Haywood, M. Renksizbulut, and G. D. Raithby. Numer. Heat Transfer, Part A, 26:253 272, 1994. [7] B. T. Helenbrook. accepted to Comp. Meth. App. Mech. Eng., 2001. [8] T. A. Johnson and V. C. Patel. J. Fluid Mech., 378:19 70, 1999. [9] H. Lamb. Hydrodynamics. Dover Publications, New York, 1945. Conclusions We have categorized the behavior of uncontaminated liquid droplets over a wide range of parameters. This has allowed us to obtain new insight into drop behavior including the identification of a non-dimensional number, P l/g, that characterizes the deformation and break-up modes of an uncontaminated liquid droplet. We have also seen unexpected new physical phenomena such as the nonmonotonic response of the droplet terminal velocity to body force. Most importantly, we have established that a numerical approach can accurately predict the stability limits of liquid droplets and used this method to provide new insight into dropstability in the large and small Ohnesogre number limits. This approach will eventually allow us to answer many difficult questions about drop break-up such as the effect of unsteady flows, contaminants, droplet time-history etc...