Model comparison for single droplet fragmentation under varying accelerations

Transcription

1 , 24th European Conference on Liquid Atomization and Spray Systems, Estoril, Portugal, September 2011 F.-O. Bartz 1, D. R. Guildenbecher 2, R. Schmehl 3, R. Koch 1, H.-J. Bauer 1 and P. E. Sojka 2 1: Institut für Thermische Strömungsmaschinen, Karlsruher Institut für Technologie, Germany 2: Maurice J. Zucrow Laboratories, Department of Mechanical Engineering, Purdue University, United States 3: Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands Abstract Different models for single droplet fragmentation have been assessed for different initial accelerations in the bag and sheet-thinning regimes. The predictions have been compared to high-speed-video sequences where a single droplet is injected in a cross-flow of air. It is concluded that the modeling of the temporal evolution of the breakup process is important. This includes modeling of the droplet drag before and after droplet initiation as well as a realistic breakup time. Furthermore, if the breakup criterion is based on a critical deformation instead of a critical Weber number, predictions of trajectories are improved for transient aerodynamic loadings. Introduction The prediction of aerodynamic droplet fragmentation is important for many technical applications, as for example fuel injection in a gas turbine combustor. Most experimental and numerical studies of aerodynamic droplet fragmentation focus on the temporal evolution of the Weber number in a shock-tube or of a free falling drop in stagnant air. Furthermore, in most droplet breakup models, the characteristic breakup times as well as fragment size and velocity distributions are based on shock-loading conditions. Despite this, in the majority of natural and technical systems, the initial acceleration differs from the impulsive loading (shock tube) or constant acceleration (drop tower) cases. Limited experimental results are available which resolve the effect of varying initial accelerations on drop fragmentation, as for example the study by Schmelz and Walzel [17]. Common droplet breakup models are the Taylor-Analogy Breakup () model [10] and its extensions as well as the Wave breakup model [12]. Some models define a critical Weber number, We c, as the breakup criterion, like for example the breakup model presented by Schmehl et al [14]. The -model [10] as well as the deformation based model presented by Bartz et al. [3] have a critical deformation as breakup criterion. Other breakup models include the unified spray breakup model [6] and the stochastic breakup model of Apte et al. [2]. The goal of this investigation is to quantify the capability of various breakup models to predict the fragmentation of a single droplet in an air cross-flow. Single droplet fragmentation has been chosen, so that no interaction between droplets exists. Droplets have been injected at different axial locations along a free gas jet to obtain a variation of temporal acceleration profiles. Computed deformations and trajectories have been compared to experimental results obtained from high-speed-videos. In this paper, the investigated droplet deformation and fragmentation models will be presented first, followed by the description of the experimental and numerical setup. Subsequently, the predicted deformations and trajectories are compared to experimental results. Droplet deformation and breakup models In this section, the droplet deformation and breakup models which are to be compared are briefly presented. The Weber number, We, and the Ohnesorge number, On, are non-dimensional numbers characterizing droplet distortion and fragmentation. The Weber number is defined as the ratio of the aerodynamic forces acting on the droplet to the stabilizing surface tension force. The Ohnesorge number describes the influence of liquid viscosity: We = ρ g v 2 rel D 0 σ, On = µ d ρd D 0 σ. (1) Droplet deformation models When the aerodynamic forces on a droplet exceed a certain intensity, the droplet oscillates and distorts. For low viscosities of the liquid (On < 0.1), Hsiang and Faeth [8] reported, that the droplet drag behaviour cannot be Corresponding author: frank-oliver.bartz@kit.edu 1

2 represented by the drag of a sphere for We > 0.6. Rather, the change of droplet cross-section leads to a drag which differs from that of a sphere and can therefore significantly influence droplet trajectories. Liu et al. [9] proposed to account for the change of the droplet drag by interpolating the drag coefficient between the value of a sphere and a disk shaped droplet. In the following, the dynamic drag model [9] and the NL model [1, 16] are presented. Both models are restricted to spheroidal droplet shapes. Since both models are described in different notations, the equation for the dynamic drag model will be transformed to the notation of Schmehl [1, 16]. An equator coordinate,y = D/D 0, is introduced to represent the droplet deformation, with D being the cross-stream andd 0 the volume equivalent diameter. In the dynamic drag model proposed by Liu et al. [9] the differential equation of the Taylor Analogy Breakup () model [10] is solved. The model uses the analogy between oscillating and distorted drops and a spring mass system. The differential equation of the model as proposed by O Rourke and Amsden [10] has been reformulated according to the notation of Schmehl [1, 16], witht σ = t/t σ being the time scale normalized by the characteristic time of free shape oscillationst σ = D0 3 ρ d/σ: d 2 y dt 2 σ = 20On dy dt σ 64 (y 1)+2C 2, We. (2) The terms on the right hand side describe the influence of the droplet viscosity (damping, first derivative), the surface tension (spring, linear term) and the aerodynamic force (external force). O Rourke and Amsden [10] matched their constants to a critical Weber number, We c = 12, when the critical deformation, y c = 1., is reached. This is equivalent to: C 2, = 2 3. (3) This constant is used in the current investigation. Schmehl [1, 16] developed the Non-Linear-Taylor-Analogy-Breakup (NL) model, which is an non-linear extension of the -model. It accounts for effects of larger deformations, where the linear theory is not applicable. The following differential equation is solved [1, 16], where ds/dy is the rate of surface change, and S 0 the initial surface of the undeformed droplet: π y 6 π d 2 y dt 2 σ = ( ) dy π 2 +16y 7 40On 1 dy dt σ y ds dt σ S 0 dy + 2C 2 We. (4) y A value of the aerodynamic constantc 2 = 1.03 is chosen for the current investigation. Schmehl [1, 16] matched this value for a critical Weber number, We c = 13, and a critical deformation of y c = 1.8. This value increases the influence of the aerodynamic forces compared to the -model (Eq. (3)). It can also be observed, that the constant factor of the viscous term is twice as high for the -model. After calculating the deformation, y, the drag coefficient of a spheroid with an arbitrary equator coordinate, y, is then interpolated between the drag coefficient of a sphere and a disk shaped droplet to account for the influence of droplet deformation. Droplet fragmentation models The droplet fragmentation models to be compared in this paper are described in this section. The -model [10] and deformation based model [3] have a critical deformation as breakup criterion. The model proposed by Schmehl et al. [14] has a critical Weber number as breakup criterion. Within the model presented by Schmehl et al. [14], characteristic times are calculated after a critical Weber number is exceeded. The initiation time and breakup times are calculated using correlations based on experiments. Then, the deformation and breakup process is described using empirical correlations. The droplet drag is linearly interpolated between the values of a spherical droplet and a disk shaped droplet during initiation. The child drops are generated at one discrete instance of time using a root-normal distribution which is based on an empirical Sauter mean diameter derived from experiments. The deformation based fragmentation model, which has been presented by Bartz et al. [3], is based on the principals of the work of Schmehl [16]. It uses a critical deformation, y c = 1.8, as the breakup criterion. The NL-model is used to calculate the droplet deformation until a certain deformation betweeny =1.8 andy =2.1 is reached, depending on the aerodynamic loading. Since the droplet shape deviates from the spheroidal shape 2

3 after the droplet initiation, which is usually assumed by dynamic deformation models, the droplet deformation and breakup process are then modeled based on empirical data. A corresponding shock-loading state is calculated when the critical deformation is reached. This is needed, because this model relies on empirical data obtained from experiments in shock-tubes; however, the aerodynamic loading in complex flows deviates from the shock loading. A detailed droplet fragmentation model is implemented relying on experimental breakup times taken from [11] and fragment sizes as reported in [4], [] and [7]. The model [10] is based on the analogy between the forces acting on a droplet leading to its deformation and the forces in a spring mass system. The differential Eq. (2) is solved until a critical deformation,y c = 1., is reached. Fragments are generated once this critical deformation is exceeded. O Rourke and Amsden [10] use an energy conversation between the parent droplet and the child droplets to predict the sizes of the child droplets. In the Wave breakup model [12] breakup times and fragment sizes are calculated according to the fastest growing Kelvin-Helmholtz instability [1]. Hence, this breakup model is only suited for higher Weber numbers. Droplets are stripped off of the parent droplet according to a time constant, τ Wave [12], with a being the parent droplet radius, and r the fragment radius: da dt = a r, forr a, r = B 0 Λ max. () τ The time constant is given by a τ = 3.726B1, Λ max Ω max whereω max is the maximum growth rate andλ max the corresponding wavelength. Liu et al. [9] suggest a breakup time constant B1 = 1.73 by matching trajectories to experimental results for Weber numbers between 18 and 319. The constantb 0 is set to 0.61 according to [12]. These constants are also used in the current investigation. al and numerical setup The experimental setup consists of a vertical injection of single ethanol droplets into a continuous cross flow of air (Fig. 1a) at ambient conditions. This setup has been chosen to establish an aerodynamic loading of the droplet which mimics combustor conditions, different from experiments in shock-tubes. Single droplets are injected instead of a droplet chain, in order to avoid interaction between droplets. Droplets have been injected downwards at different locations downstream of the air nozzle to reproduce different aerodynamic loadings, as can be observed in Fig. 1b. Here, the axial distance from the air nozzle exit is given by x, and y is defined as the distance from the air nozzle centerline in the opposite direction of the gravity vector. Axial velocity contours are displayed as solid lines. The droplet starting positions are marked. The droplet injection at the location x 1 (position A) is close to the nozzle exit. The injection at the locationx 2 (position B) is further downstream, and therefore a more gradual temporal evolution of the relative velocity between the air and the droplet is present. The air velocity field has been measured by PIV and LDV. Good agreement exists between measured and predicted axial velocities 10 mm downstream of the nozzle. The process of fragmentation has been recorded with a Vision Research Phantom v7.3 high-speed-video camera at frames per second. The air flow has been computed with ANSYS Fluent The linear pressure-strain Reynolds-stress model has been used to describe the influence of air turbulence. One-way coupled Lagrangian particle tracking was used. Droplet fragmentation has been modeled using the models presented by Schmehl et al. [14], the deformation based model [3], as well as the [10] and Wave Breakup model[12]. The calculation of droplet fragmentation has been done with Ladrop [13] using the first two models and the NL model to calculate subcritical droplet deformation. ANSYS Fluent has been used to predict fragmentation with the and Wave Breakup models in combination with the dynamic drag model [9]. The number of fragments produced from a parent droplet has been set to five for the model and fifty for the deformation based model. The influence of turbulent droplet dispersion in the calculation has been neglected. Therefore, only the mean calculated trajectories of the parent droplets are illustrated. Predicted droplet deformations and trajectories are compared to high-speed-video sequences for the bag breakup and plume-sheet-thinning regime. Comparison of droplet trajectory and deformation In this section, the temporal evolution of predicted droplet deformation as well as droplet trajectories are compared to experimental results. The temporal evolution of the instantaneous Weber number based on the initial droplet diameter for the presented cases is illustrated in Fig. 2. One loading with a stronger temporal increase of 3 (6)

4 drop injection honeycomb and mesh g R0 30 x 2, position B air R70.1 air nozzle 2.4 y [mm] x 1, position A (a) (b) x [mm] Figure 1: (a) al setup, dimensions in mm, and (b) droplet injection locations and axial velocity contours (CFD) in m/s for plume-sheet-thinning. We and one with a more gradual loading is realized for fragmentation in the bag breakup as well as in the plumesheet-thinning regime by injecting at two different downstream locations. The instantaneous Weber number is derived from experimental high-speed-videos and PIV data for position A, according to Eq. (1). For position B, the relative velocities were calculated using CFD and high-speed-videos results. The same droplet sizes and liquid properties are utilized for all cases. With the material properties of ethanol at room temperature and droplet diameters of 1.7 mm, a characteristic time of free shape oscillations oft σ 0.01 s is obtained. In the following, the results of the plume-sheet-thinning cases will be presented first, followed by the ones for the bag breakup regime instantaneous W e Bag breakup, position A Bag breakup, position B Sheet-thinning, position A Sheet-thinning, position B t[s] Figure 2: Transient Weber number,t σ 0.01 s Fragmentation in the plume-sheet thinning regime The deformation and breakup models will be compared for fragmentation in the plume-sheet-thinning regime. Only the comparison of the breakup models for position B will be discussed in detail. The temporal evolution of the aerodynamic loading differs more from the shock-tube loading for the injection at position B compared to position A. The process of fragmentation is depicted in Figs.3a,3b and 3c for this case. As shown in Fig. 2, the maximum 4

5 Weber number is We max = 9. for this case. This Weber number is the maximum instantaneous Weber number during the lifetime of the droplet. The droplet is first deformed to a disk shape (initiation, Fig. 3a). Subsequently, disintegration starts (Fig. 3b). In the presented case, a plume-core droplet remains, which disintegrates afterwards (Fig. 3c). (a) initiation (b) sheet fragmentation (c) fragmentation of plume-core Figure 3: Breakup process, We max = 9. The comparisons between experiment and predictions are shown in Fig. 4a and 4b, in terms of the temporal evolution of the droplet deformation. The deformations as recorded in the experiment (circles) are plotted until fragmentation begins. The droplet deformation, y = D/D 0, increases due to the aerodynamic forces when the droplet crosses the shear-layer. The droplet deforms to a disk shape until initiation, and continues to deform to a shape, which can not be represented by spheroids. Deformations calculated with the model (solid line) are in good agreement with experimental results. Fragmentation in the -model takes place once a critical deformation of y c, =1. is reached, at one instance of time. In the experimentally observed high-speed-videos, however, the fragmentation begins at higher deformations and is a temporal process. In the deformation based model [3], the NL model is used to calculate the droplet deformation until a deformation of y = 2.1 is exceeded (for We > 20). Afterwards, the deformation is modeled based on empirical data. A stagnation of the deformation after the droplet reaches initiation can be observed. Then, a constant gradient of the deformation is assumed until the end of fragmentation. Deformations predicted with the NL model are overestimated in comparison to the experiment. Uncertainties in the determination of the droplet cross-stream diameter in the experiment exist during the initial deformation, because the droplet cross-stream diameter in the experiment has been evaluated normal to the air-flow. This assumption neglects the influence of the droplet velocity vector on the relative velocity vector and, therefore, on the resulting aerodynamic force. The relative velocity vector is determined as the difference between the air velocity vector and droplet velocity vector. The influence of the droplet velocity vector might not be negligible before the droplet crosses the shear-layer. However, this should be negligible once the droplet crosses the shear-layer. Additionally, the droplet rotates in the experiment after crossing the shear layer. This might happen, since the lower half of the droplet experiences higher accelerations than the upper half, which might lead to a tilted droplet, where the longest equator distance differs from the cross-stream diameter normal to the air-flow. In the deformation based model [3], a stagnation of the droplet cross-stream diameter after the droplet initiation is assumed, based on experimental observations in the bag breakup regime. This stagnation of the droplet deformation was not observed in the experiment. However, the gradient, dy/dt, after this constant phase of deformation is in good agreement with experimental results for position A. To study the influence of the coefficient C 2, a calculation with the deformation based model and a value of C 2 = 2/3, which is identical to that of the model, was performed. The calculated deformation (dotted line) agrees to the experimental results in the range of uncertainty. In the following, the predicted trajectories for the injection of the droplet at position B are compared to highspeed-videos. Figure shows the trajectory as predicted by the model (dotted white line) at times corresponding to the experimentally observed initiation and during fragmentation. The model predicts breakup at the location shown by the marker (white cross), and after this point the predicted trajectory of the fragments are

6 shown as multiple dotted lines. In addition, the background of Figure shows the experimentally observed droplet at the instant in time corresponding to the predicted fragment locations (white circles). The sizes of the predicted droplet diameters are plotted to scale. These are very small for the prediction by the model and thus difficult to see. The scale of axial velocity contours of the air (black solid line) is given in m/s. The predicted fragmentation occurs earlier than in the high-speed-video, at an instance of time, where the droplet in the high-speed-video is still at the stage of initiation (compare to Fig. 3a). Also, in the model fragmentation occurs at one discrete time, which does not correspond to the experimental results, where fragmentation is a temporal process. Despite of that, a good agreement of the child droplet trajectories is observed. However, the child droplets predicted with the model are moving faster downstream than the droplets in the experiment (Fig., right). The trajectories as predicted with the Wave breakup model are shown in Fig. 6. The fragmentation starts earlier and ends before the fragmentation in the experiment. In contrast to the model, fragments are generated continuously over a finite time. Compared to the model, a bigger spread of the child droplets can be observed, as well as a better temporal representation of the breakup process. It should however be noted that the breakup time is directly proportional to the constantb1, and therefore dependent on the user input. Figure 7 shows the trajectories as predicted by the model of Schmehl et al. [14]. In this model, the characteristic initiation and breakup times are based on the We that is computed at the time step after We c is exceeded. This leads to an underestimation of We max and, therefore, an overestimation of the initiation and breakup times. As a result, no breakup is predicted in the field of view of Fig. 7. The droplet location is also not represented well. In Fig. 8, the fragmentation process predicted by the deformation based model is compared to the high-speedvideos. This model better represents the physical process of fragmentation compared to all other models. Good agreement between the predicted and measured droplet location and shape is seen at initiation. Fragments are generated continuously during fragmentation, as seen in Fig. 8, right. The drag coefficient of the core drop (large drop in Fig. 8, right) is overestimated, which results in an overestimation of the acceleration of the core droplet and the fragments. This effect may be explained by the observation that the NL model overestimates the initial deformation as shown in Fig. 4b. Furthermore, during breakup, a constantly growing deformation is modeled, until the end of droplet fragmentation. The drag coefficient is calculated using the value of a disk. The validity of these assumptions has to be assessed in more detail. In the calculations for position A (not shown in this paper), a better agreement of the temporal behavior of the droplet location was observed for this model. 2.4 y 4 3 deformation based deformation based,c 2 = 2 3 y deformation based deformation based,c 2 = y c, y c, t[s] (a) Position A, We max = t[s] (b) Position B, We max = 9. Figure 4: Comparison of deformation for plume-sheet-thinning, critical deformation of model marked with red cross Fragmentation in the bag breakup regime The results for droplet fragmentation in the bag breakup regime are discussed in this section. During bag breakup, the droplet deforms to a disk shape. Afterwards a liquid bag is formed. This bag then disintegrates, followed by fragmentation of the ring. The temporal evolution of the deformation for injection at positions A and B is illustrated in Fig 9a and 9b. 6

7 Figure : Predicted trajectories (dottet line), -model, at initiation (left) and during fragmentation (right). Mean gas velocity contours displayed as solid lines (labels in m/s), marker located at predicted breakup location. We max = 9.. Figure 6: Predicted trajectories, Wave breakup model, at initiation (left) and during fragmentation (right). Mean gas velocity contours displayed as solid lines (labels in m/s), We max = 9.. Figure 7: Predicted trajectories, Schmehl et al model, at initiation (left) and during fragmentation (right). Mean gas velocity contours displayed as solid lines (labels in m/s), We max = 9.. 7

8 Figure 8: Predicted trajectories, deformation based model, at initiation (left) and during fragmentation (right). Mean gas velocity contours displayed as solid lines (labels in m/s), We max = 9.. The experimentally obtained deformations (circles) are plotted until the beginning of fragmentation of the bag. Predictions of the deformations using the model (solid line) underestimate the temporal gradient of the deformation for position A (Fig 9a). Fragmentation is modeled after the droplet exceeds the critical deformation of y c = 1.. Contrary, a good agreement of the temporal gradient of the deformation exists with the deformation based model. The phase of constant deformation could, however, not be observed in the experiment. For position B (Fig. 9b), the deformation as predicted with the model did not exceed the critical deformation of y c = 1.. Hence, the droplet will not be fragmented. This could be due to non-linearities, that are not captured by the model. Also, the influence of the aerodynamic forces might be underestimated due to the choice of the constantc 2. This constant is matched by O Rourke and Amsden [10] based on a critical deformation, y c = 1., at the critical Weber number, We c = 12. Furthermore, the critical deformation might vary, depending on the initial acceleration. The same effect is seen for predictions with the deformation based model with C 2 = Here, the critical deformation, y c,def = 1.8, is not exceeded. By increasing the aerodynamic coefficient to a value of C 2 = 4/3, the influence of the aerodynamic forces on the droplet are increased as well, as can be seen in Eq. (4) and Eq. (2). WhenC 2 = 4/3 is used, the critical deformation,y c,def, is exceeded. For bag-breakup, the child droplets exhibit a smaller dispersion compared to sheet-thinning. Therefore, the different cases studied have been compiled into one figure. The trajectories are compared for position A in Fig. 10a. The trajectory of the droplet in the experiment is plotted as a solid line with circles. The time when the fragmentation of the bag begins,t 1, and a timet 2 during the fragmentation of the rim are marked. In this way, the temporal evolution of the trajectories can be compared. This can give an indication of the drag coefficient. In the background, the experimental image of the droplet at the timet 2 is displayed. The trajectory predicted with the model of Schmehl et al. [14] is displayed as a dotted line. As stated before, in this model the characteristic times are calculated based on the instantaneous Weber number at the time step when We c is first exceeded. The breakup time is overestimated, and fragmentation begins at a location which is not in the field of view. Trajectories calculated with the model are displayed as solid lines. It was observed in Fig. 9a, that the droplet exceeded the critical deformation at an early stage and then fragments are released. The trajectories deviate from the experimental trajectory. The calculation using the wave breakup model did not predict fragmentation. As stated earlier, the physics behind this model is only valid for high We. The droplet trajectory as well as the temporal evolution of the trajectory predicted with the deformation based model showed a better agreement to experimental results than predictions with the other models. This might be due to the modeling of the drag during the process of fragmentation. A slight under prediction of the droplet deformation until t 1 could be the cause for the deviation of the trajectory (c. f. Fig. 9a). In this model, droplet fragmentation is implemented at a discrete instance of time for bag breakup. This time is based on the time, when most of the mass is disintegrated. This breakup time is overestimated. The comparison of the trajectories for position B is illustrated in Fig. 10b. As discussed before, the predictions with the model did not predict fragmentation. Also, no fragmentation is observed with calculations with the 8

9 t 1 t2 ILASS Europe 2011 deformation based model andc 2 = Therefore, the trajectories are displayed for calculations withc 2 = 4/3. A good agreement with the experimental trajectory can be observed. The droplet is, however, moving faster downstream than in the experiment. This indicates that the drag of the droplet is overestimated. The model proposed by Schmehl et al. [14] predicts a trajectory that is closer to the experiment than the ones predicted with the and Wave Breakup model. This could be due to the prescription of the deformation and drag coefficient in the model of Schmehl et al. [14] during the initiation phase, after We c is exceeded. The drag coefficient is linearly interpolated in time between the value when We c is exceeded and the drag coefficient of a disk, at initiation. 4 deformation based 4 deformation based deformation based,c 2 = 4/3 y 3 y 3 2 y c, 2 y c,def y c, t [s] (a) Position A, We max = t [s] (b) Position B, We max = 1 Figure 9: Comparison of deformation for bag breakup Schmehl et al. (2000) deformation based model Wave Breakup Schmehl et al. (2000) deformation based model,c 2 = 4/3 Wave Breakup t 1 t 2 t 1 t2 t1 t 1 t 1,def t 2 t2,def t 2 (a) Position A, We max = 1 (b) Position B, We max = 1 Figure 10: Comparison of trajectories for bag breakup, marker denote droplet position at t 1 (fragmentation of the bag) andt 2 (fragmentation of the rim); high-speed-image at t 2 in the background Conclusions In the current investigation droplet breakup models were evaluated. The predictions of droplet deformations and trajectories of a single droplet injection in a cross-flow were compared to experimental results obtained with a high-speed-video camera. Trajectories were predicted with the -model [10], the Wave breakup model [12], the deformation based model [3] as well as the model presented by Schmehl et al. [14]. Droplet deformations 9

10 predicted with the model [10] and the deformation based model [3] were compared to experimentally obtained deformations. It is concluded, that the modeling of the droplet deformation before and after the droplet deforms to a disk shape (initiation) is important for the accurate prediction of trajectories. The droplet shape before the initiation can be approximated by a spheroid, as modeled in the [10] and deformation based model [3]. After the droplet deforms to a disk, the droplet cross-stream cross-section still increases, which can influence the droplet drag. This effect is captured by the deformation based model [3]. The prediction of the trajectories is significantly improved in comparison to other models where deformation beyond the initial spheroidal shape is not considered. However, deviations between the predicted and measured deformations still exist. The usage of realistic breakup times is important for the prediction of the droplet dispersion. The breakup times vary depending on the breakup regime. This is accounted for in the model of Schmehl et al. [14] and in the deformation based model [3]. These models rely on breakup times which are derived from experimental data. In the model of Schmehl et al. [14] the breakup time is based on the instantaneous Weber number at the time step, when the critical Weber number is first exceeded. This led to an overestimation of the breakup time in the investigated cases. In the deformation based model [3], the breakup times are calculated when the deformation exceeds the critical deformation. This improved the agreement with the breakup time in the experiment. In the model [10], on the contrary to both previously mentioned models, fragments are generated right after the critical deformation, y c, is exceeded. In the investigated cases, this led to fragmentation before the droplet in the experiment reached the disk shape. Using this assumption, droplet trajectories might be predicted inaccurately in flows with strong gradients and high turbulence. To fully assess the quality of the breakup models, predicted droplet size and liquid volume flux distributions should be evaluated and compared to experimental results. Future work on the deformation based model [3] will be focused on an improved description of droplet deformation and drag before and during fragmentation. Acknowledgments The stay of the first author at Purdue University has been sponsored by the Karlsruhe House of Young Scientists (KHYS). The work described within this paper was mainly supported and funded by the German Federal Department of Economy and Labour under contract 20T0802 ("EffMaTec") as part of the German national aeronautical research programme. The support is gratefully acknowledged. Nomenclature a parent drop radius (Wave breakup model) [m] B 1 constant (Wave breakup model) [-] C 2 aerodynamic coefficient [-] D cross stream diameter [m] D 0 (initial) volume equivalent diameter [m] r fragment radius (Wave breakup model) [m] T non-dimensional time [-] t time [s] t σ characteristic time of free shape oscillations [s] v rel relative velocity between gas and droplet [m s -1 ] y non-dimensional equator coordinate [-] Λ wavelength of fastest growing wave [m] µ dynamic viscosity [Pa s] Ω maximum growth rate [s -1 ] ρ density [kg s -3 ] σ surface tension [N m -1 ] τ breakup time Wave breakup model [s] Characteristic numbers On = µ d / ρ d D 0 σ Ohnesorge number We = ρ g vrel 2 D 0/σ Weber number 10

11 Subscripts g gas c critical d disperse phase g gas max maximum rel relative Taylor-Analogy-Breakup model def deformation based model Wave Wave breakup model References [1] ANSYS, Inc., ANSYS Fluent Theory Guide, Release [2] Apte, S. V., Gorokhovski, M. and Moin, P. International Journal of Multiphase Flow 29(9): (2003). [3] Bartz, F.-O., Schmehl, R., Koch, R. and Bauer, H.-J., 23rd Annual Conference on Liquid Atomization and Spray Systems, Brno, Czech Republic, September [4] Chou, W.-H., Hsiang, L.-P. and Faeth, G.M., International Journal of Multiphase Flow 23(4): (1997). [] Chou, W.-H. and Faeth, G.M. International Journal of Multiphase Flow 24(6): (1998). [6] Chryssakis, C. and Assanis, D. N., Atomization and Sprays 18(1):1-2 (2008). [7] Dai, Z. and Faeth, G.M., International Journal of Multiphase Flow 27(2): (2001). [8] Hsiang, L.P., and Faeth, G.M., Int. Journal of Multiphase Flow 21(4):4-60 (199). [9] Liu, A., Mather, D. and Reitz, R.D. SAE Technical Paper (1993). [10] O Rourke, P.-J., and Amsden, A.A., SAE Technical Paper (1987). [11] Pilch, M. and Erdman, C.A., International Journal of Multiphase Flow 13(6): (1987). [12] Reitz, R.D, Atomization and Spray Technology 3: (1987). [13] Schmehl, R., Rosskamp, H., Willmann, M. and Wittig, S., International Journal of Heat Fluid Flow 20:20-29 (1999). [14] Schmehl, R., Maier, G. and Wittig, S., Eighth International Conference on Liquid Atomization and Spray Systems, Pasadena, CA, USA, July 2000, pp [1] Schmehl, R., 18th Annual Conference on Liquid Atomization and Spray Systems, Zaragoza, Spain, September 2002, pp [16] Schmehl, R., Ph.D. Thesis, Institut für Thermische Strömungsmaschinen, Universität Karlsruhe, Germany, [17] Schmelz, F., and Walzel, P., Atomization and Sprays 13: (2003). 11