ILASS Amerias, 19 th Annual Conerene on Liquid Atomization and Spray Systems, Toronto, Canada, May 2006 Eet o Droplet Distortion on the Drag Coeiient in Aelerated Flows Shaoping Quan, S. Gopalakrishnan and David P. Shmidt * Department o Mehanial and Industrial Engineering University o Massahusetts - Amherst Amherst, MA 01003 USA Abstrat The distortion o a liquid droplet rom its spherial shape has a proound inluene on its drag oeiient. The areas o greatest unertainty are the drag at high Reynolds and Weber numbers as well as aeleration and history eets. Though several redued models have been developed or use in spray modeling, a detailed, diret numerial simulation oers the opportunity to learn more about the underlying physis. A numerial investigation has been perormed to study the variation o the droplet drag with hange in deormation ator o the liquid drop, and to better understand the dependene on Ohnesorge number and aeleration. The transient nature o these simulations will advane the understanding o low eets on the droplet drag.
Introdution The auray o any spray model depends greatly on the orretness o the drag oeiient omputed or the liquid droplets. Experiments have shown that droplets having a Weber number signiiantly larger than unity tend to deorm, whih results in a hange in their drag harateristis. The distortion o the droplet auses the inrease in the rontal projeted area, ausing a deviation rom the ideal spherial drag harateristis. Until reently droplet distortion and its eet on drag harateristis have been studied experimentally using wind tunnels [1] and shok tube experiments [2]. A summary o these experiments and the orrelations obtained by them an be ound in [3]. These experiments are limited by the range o droplet sizes, luid densities and visosities being investigated. A numerial study is posed with no suh hallenges and oers a greater degree o ontrol in the simulation parameters. An arbitrary Lagrangian Eulerian approah with an adaptive reing sheme was used in [4] to simulate two-dimensional quasi-steady low around liquid droplets. This method was suessul in handling large deormation o the interae. Three-dimensional omputations have typially ound it diiult to reliably handle large interaial deormations without running into signiiant numerial errors. In this paper, we use a moving based approah with an adaptive reing sheme [5]. A 3D unstrutured inite volume sheme is used to perorm the simulation. The interae is traked in a Lagrangian manner in a deorming tetrahedral. This approah gives us the advantage o exatly traking the interae within the limits o disretization error. An adaptive reinement sheme is implemented in order to maintain the quality o. The adaptation involves lipping, edge ollapse and edge bisetion algorithms. An optimization based smoothing algorithm is used in onjuntion with the adaptation shemes. Here, we present a preliminary report on the eet o droplet distortion on its drag oeiient. Our approah helps us to losely orrelate the deormation o a liquid droplet to the drag oeiient. The transient nature o the simulation will help us better understand the eet o low history on the drag oeiient. At this junture, we are just beginning to investigate the parameter spae and the early results provide new insight into the drag harateristis whih have not yet been unearthed by experimental data. Governing equations and Disretization The governing equations or the moving and deorming ontrol volume are desribed by the ollowing set o equations. dv dt = v n ds (1) CS Eqn. (1) represents the onservation o volume or a distorting ontrol volume. d ρdv + ρ( ds = 0 dt u-v) n (2) CV CS Eqn. (2) denotes the onservation o mass and Eqn. (3) gives the onservation o momentum. d dt ρudv + ρu( u-v) nds = CV CS (3) T ρ dv pdv + µ ( u+ u ) nds CV CV CS In the above equations CV stands or the ontrol volume, CS the ontrol surae, u is the luid veloity, v the veloity o the ontrol surae, n is the unit vetor o the ae normal and denotes the body ore per unit mass. The two phases are assumed to be inompressible and immisible. The ontinuity and jump onditions aross the interae are given by the ollowing equations. u = u = v (4) 1n 2n n t 0 u = (5) 1 1 R1 R2 µ u u i j ( + ) in j 0 x j x τ = i p = σ ( + ) + 2 µ ( ) u n n (6) (7) where is assumed to be the onstant surae tension, R 1 and R 2 denote the radii o urvature and is the tangential vetor. Eqn. (4) states that the normal veloities o both the luids at the interae are equal and same as the normal veloity o the moving. The ontinuity in the tangential veloity aross the interae is given by Eqn. (5). The pressure jump balaned by the surae tension is denoted by Eqn. (6). The ontinuity o the visous shear stresses aross the interae is desribed by Eqn. (7). The governing equations (1),(2) and (3) are disretized as given in [6]. V n + 1 n V = U (8) t ell aes ρ n+ 1 n 1 n n V + ρv + ρ ( U U ) = 0 (9) t ell aes
ρ n+ 1 n+ 1 n 1 n n n u V + ρ uv + ρ u ( U U ) CG T = V ( p ρ g r ) + µ ( u+ u ) n A CG V ( g r ) ρ ell aes ell aes t (10) where V is the ell volume, U denotes the ae lux, U the equivalent lux o the ae, A is the ae area, r CG denotes the position o ell enter and g is gravitational ore per unit mass. Subsript n is the time step. Simulation Setup In the numerial investigations both the luids are assumed to be inompressible, to have onstant densities g and l, and onstant visosities, g and l or the gas and liquid phases respetively. The surae tension at the interae o the two luids is assumed to be a onstant. The deormation ator D is deined as the ratio o the shortest axis to that o the longest axis o the deormed droplet. The Reynolds number is based on the relative veloity and the gas phase properties and is given by, ρg ( U g Ul ) 2r0 Reg = (11) µ g The general drag ore on the droplet is given by the Basset-Boussinesq-Oseen (BBO) equation, whih is too omplex or use in applied CFD simulations with large numbers o parels. The drag ore inludes the aerodynami steady-state drag, added mass ores, pressure gradient ores, and a history integral. However, in typial CFD analyses where the droplet is onsidered to be a point mass, numerous terms o this equation may be negleted [3]. Typially, only the steady state drag is used in order to deine a oeiient o drag. 8 d U l η r0 3 d t (12) C D = 2 ( U U ) g l where = ρ l / ρ g is the density ratio between liquid and gas phases. This deinition presumes that drag is equal to the steady state value and that the droplet may be represented as a sphere. The ollowing detailed simulations will report the aeleration nondimensionalized as in Eqn. (12). These values o C D are what a typial CFD ode would have to use in order to predit droplet aeleration orretly. These results are not intended to validate the orretness o Eqn. (12) or general use. Case 1. This simulation is setup to mimi a shok tube experiment. A stationary liquid droplet is suddenly put under the inluene o a gaseous low and is impulsively aelerated. The density and visosity ratios maintained between the liquid phase and the gas phase are 50. The omputational domain is box with a spherial droplet o radius r o loated at the enter o the box. The length o the box is 48r o, the width and height are 16r o. Boundary eets are negleted sine the box is large ompared to spherial droplet. An inlet boundary ondition is enored on the right side o the domain and an exit boundary ondition is set on the let side o the domain. Initially the veloity in the liquid phase is set to zero and a onstant veloity is given to the gas phase in order to ahieve a uniorm veloity ield. Sine the veloity is not ontinuous aross the interae large veloity errors will our as shown in Figure 1. To alleviate this problem the simulation is run with the droplet being treated as a ixed solid until the veloity jump is smoothed out by the visosity as shown in Figure 2. 1.79 5.37 8.95 12.53 16.10 19.68 23.26 26.84 Figure 1: Initial veloity ield with stream untion disontinuities aross the interae Y X Y X 5.00 4.33 3.67 3.00 2.33 1.67 1.00 0.34 Figure 2: Veloity ield smoothed out by visosity. Case 2. Here a wind tunnel experiment [1] is simulated in whih a droplet is injeted in a onverging wind tunnel and then subjeted to an aelerating low. It is Z Z
omputationally very expensive and ineiient to perorm a omplete diret numerial simulation o a liquid droplet in a wind tunnel. In this setup as well the omputational domain is a box with a liquid droplet entered in it. The omputational domain is translated along with the liquid droplet in suh a manner that the droplet is always entered. An inlet boundary ondition is enored on the let side on the domain. The initial ondition imposed on the domain is similar to that o ase 1. The same proedure to smooth veloities at the interae is ollowed. The veloity inlet boundary ondition is ontinuously varied with time aording the rules o low in onverging nozzles. An exit ondition is presribed to the right o the domain and the remaining walls are modeled as slip walls. In this ase the density and visosity ratios maintained are 100 between the liquid phase and the gas phase. Results As the droplet is impulsively aelerated by the gaseous low the initial Reynolds number and Weber number are 40. During the aeleration o the droplet, it ontinuously deorms, the shape o the liquid droplet at various times is given in Figure 7. The deormation rate o the droplet is initially high and redues as the relative veloity between the gas and the droplet dereases. The variation o the deormation ator with time is given in Figure 3. The Reynolds number based on the gas veloity goes down as the liquid phase athes up with gas phase. This results in the redution on the drag oeiient over time. Figure 4 and Figure 5 onirm this trend. These simulation results prove that the droplet deormation signiiantly aets the drag harateristis. Figure 4: Reynolds number vs. time Figure 5: Coeiient o Drag based on initial drop radius vs. Reynolds number Figure 6: Coeiient o Drag based on rontal projetion area vs. Reynolds number Figure 3: Variation o deormation ator with time. The C D is omputed on the basis o the initial drop radius where as C is the drag oeiient whih is * D based on the rontal projetion area. The volume o the liquid droplet will remain onstant; thereore the rontal projetion area an be alulated using the deormation ator data by assuming the droplet to be a prolate spheroid. It is evident that the drag predited is muh higher than the standard drag measured by experiments but orrelates well with experimental data rom [2]. One o the reasons or the higher drag oeiient is the larger
deormation predited by the simulation ompared to the experiment. The numerial investigations validate the belie that the unsteady drag in aelerating lows is larger than steady drag. For the seond ase where the droplet is injeted in a wind tunnel, the initial trends show a similar pattern where the predited drag oeiients are higher than the ones suggested by urrently aepted drag models. This needs to be irmly established by running the simulation over a larger time rame. Conlusions Numerial investigations were perormed to study the drag harateristis o deorming droplets in aelerating lows. The inreased rontal area due to the deormation auses an inrease in the drag oeiient whih is signiiantly higher than the values proposed by onventional drag models. These preliminary simulation results onirm this upward trend in the predition o droplet drag. Further investigations are needed to study a larger range o low onditions and the related history eets. Aknowledgements This work was perormed with the support o the Oie o Naval Researh under ontrat N00014-02-1-0507. We also thank J. Blair Perot at University o Massahusetts Amherst or sharing hardware and sotware resoures. Reerenes 1. Luxord, G.,Hammond, D.W., and Ivey, P., Forty Seond AIAA Aerospae and Sienes Meeting and Exhibit, Reno, Nevada, USA, January 2004. 2. Temkin, S., and Kim, S.S., Journal o Fluid Mehanis 96:133-157 (1980). 3. Clit, R., Grae, J.R., and Weber, M.E., Bubbles, Drops and Partiles, Aademi Press, 1978. 4. Helenbrook, B.T., and Edwards, C.F., International Journal o Multiphase Flow 28:1631-1657 (2002) 5. Dai, M., Quan, S., and Shmidt, D.P., 17 th Annual Conerene o ILASS-Amerias: Institute o Liquid Atomization and Spray Systems, Arlington, Virginia, USA, May, 2004. 6. Perot, J.B., and Nallapati, R., Journal o Computational Physis 184:192-214 (2003).
Figure 6: Droplet shapes at times: 0.0, 2.5, 5.0, 7.0, 10.0, 12.0 and 15.0