paper id iclass06-264-*** breakup of a 2d drop in cross flow...araz sarchami 1, ali jafari 2 and...

ICLASS-2006 Aug.27-Sept.1, 2006, Kyoto, Japan

1. Introduction

Breakup of a circular drop (2D drop) is of interest, since it resembles that of a Jet in Cross Flow (JICF). Therefore, a detailed study of this problem can provide information on the breakup mechanisms and even the droplet size distributions corresponding to the JICF. Jet in cross flow studies have mainly focused on the experiments and derivation of simple correlation for prediction of trajectory of the liquid jet. Only a few numerical studies have been performed. The majority of these numerical investigations (Liu et al., 2000, Zuo, 2002, and Madhabushi, 2003) have focused on Lagrangian tracking of blobs and their breakup into droplets. In this approach, a droplet (blob) size distribution is assumed or calculated based on simple correlations which do not represent the complex physics of the problem. Then, these droplets are tracked in the Lagrangian frame and their secondary breakup and dispersion in cross flow is determined.

Although much information is provided by the previous studies, still many details of the breakup and spraying in JICF configuration remain unclear. Here, we present another modeling paradigm for solution of JICF accounting for all the important effects. 2. Problem Approach

Our approach is based on simulation of full Navier-Stokes equations, in Eulerian frame, on fine grids resolving all the important flow scales for the first time. In this approach, no simplification is done and all important effects can be taken into account.

Resolving a wide range of scales needs a very fine grid. Even for an efficient, parallel code, it would take very long time (in the order of weeks) to complete a 3D simulation. This limits the applicability of 3D simulation for a wide range of parameters. Rather, as mentioned by Aalburg et al. (2005), a 2D model problem can be utilized to investigate JICF problem for many different cases.

We assume that the deformation of a jet in a cross flow is equivalent to the deformation of its cross section. As mentioned by Mazallon et al. (1999) and Aalburg (2002),

the 2D problem of flow over a circular jet is an appropriate model for the 3D problem. The dramatic decrease in run time for 2D compared to 3D makes it an interesting method for investigating the jet surface deformations, vortex generation and liquid-air interaction.

3. Numerical Simulation

We have solved the unsteady, incompressible Navier-Stokes equations in two dimensions and with fluid interfaces. A volume-of-fluid, VOF, method along with a piecewise linear interface calculation, PLIC, is used to capture the fluid interfaces. It is assumed that the velocity field is continuous across the interface, but there is a pressure jump at the interface due to the presence of the surface tension. The governing equations describing this problem are:

0=∂∂

i

ixu (1)

0=∂∂

+∂∂

ii x

FutF (2)

and

ij

ist

ij

jii gx

uiF

xp

xuu

tu

ρμρρ

+∂

∂++

∂∂

−=∂

∂+

∂∂

2

2ˆ. (3)

where, ui’s are the velocity components, and t and xi are time and space coordinates, F is volume fraction of fluid, which is zero where only fluid 2 exists and is one where only fluid 1 exits, p is pressure, i is unit vector in ith direction and ρ and μ are the mixture density and absolute viscosity and they depend on the densities and viscosities of each fluid as:

)( 212 ρρρρ −+= F (4) and

)( 212 μμμμ −+= F (5) where, ρ1 and ρ2, μ1 and μ2 are densities and viscosities of fluids 1 and 2, respectively.

For the surface tension forces, we use Continuum

Paper ID iclass06-264-*** Breakup of a 2D Drop in Cross Flow

Araz Sarchami 1, Ali Jafari 2 and Nasser Ashgriz 3

1University of Toronto, [email protected] 2University of Toronto, [email protected]

3 University of Toronto, [email protected]

ABSTRACT A numerical investigation of the deformation and breakup of a circular drop in a high speed air flow is performed. The full unsteady, incompressible Navier-Stokes equations, in Eulerian frame, on fine grids resolving the most important flow scales for this problem are solved. A volume-of-fluid, VOF, method along with a piecewise linear interface calculation, PLIC, is used to capture the fluid interfaces. The interplay between flow momentum and pressure and liquid resistance creates a combined shearing and stripping mechanism for the droplet breakup into smaller droplets. Effect of the gas density, gas velocity and droplet diameter on the drop deformation and its breakup are studied. Keywords: Jet in Cross Flow, Breakup, Drople, Eulerian frame, VOF

Surface Stress (CSS) model:

).(F

FFFst∇

∇⊗∇−∇∇= IF σ (6)

where, I is the unit tensor. In order to investigate the breakup of a jet in a cross

flow in two dimensional space, we start with the initial diameter of the jet as it exists from the orifice. Two orifice diameters of D=0.46mm and 0.56 mm are modeled. The liquid properties used are: density of 800 kg/m3, viscosity of 0.00132 kg/m.s, and the surface tension of 0.0277 kg/s2, which are properties of Jet-A fuel. The gas density is chosen as that of air at 30 psi pressure, which becomes 2.4 kg/m3. A higher gas density of 4.7 kg/m3 is also considered to investigate the effect of higher flow pressure.

The domain size is 10D×10D, with a grid resolution of 512 by 512. This results in a resolution of 25 cells per radius. Periodic boundary conditions are used everywhere for faster convergence. Because of the periodic conditions, we need to initialize the velocity field. We have used an initial flow field around a cylinder as the initial velocity input as shown in Fig.(1). The initial droplet velocity is set to zero. The computational times for the cases reported here are 4-6 hours on a single processor computer.

A droplet sizing algorithm is developed and used here. It detects the droplet as a cluster of points using volume fraction values. It finds the connectivities between the cells occupied by the droplet and calculates the droplet size by adding the volume fractions of the cells belonging to the specific drop.

Fig.(1) Initial Velocity field

4. Results Figure 2 shows the breakup of a jet in a gas flow with a velocity of 47 m/s and at 30 psi, which corresponds to density of 2.4 kg/m3. The initial jet diameter is 0.46 mm. This figure shows that the jet deforms, stretches from its sides due to the gas shear, and forms thin ligaments. Thin ligaments are later broken into small droplets. There is a small asymmetry in the results, which is due to the effect of the small drops on the parent drop. As the number of small drops increases, it is difficult to obtain perfect symmetry in the flow. After the break up of the ligaments into small droplets, because of the strong vortex behind the parent drop, small droplets are moved back to the vicinity of the parent drop. Some of these droplets collide with the parent drop. Such collisions are the main cause of asymmetry in the results. Figure 3 shows the results for the drop breakup with the same conditions as in Fig. 2, but with a larger jet diameter of 0.56 mm. The mechanism of the drop breakup has not changed. Droplet breaks by shear stretching. In this case, higher number of droplets is produced. Also, the asymmetry of the problem is increased, because there are more small droplets. In this case, the lateral dispersion of the drop (jet) is more than that of the smaller drop (jet). In this case, the drop expands around 5D for 0.46 mm jet and around 7D for the 0.56 mm jet. This can be attributed to the larger vortex behind the jet in the larger diameter jet. Figure 4 shows results for an increase in the density, or the gas pressure, which increases the gas momentum relative to the jet. Therefore, the jet deforms faster and it has a larger wake. More number of droplets are produced compared to the lower gas density case. Here, the lateral dispersion is less than the low density case. An interesting point to notice here is that in higher density conditions, more ligaments and larger droplets are formed compared to the lower density case. Figures 5-7 show the results for a higher gas velocity of 95 m/s. Because of the higher velocity, the vortices behind the parent drop are stronger; as a result, a higher number of small droplets hit the parent drop causing more asymmetry in the flow field and liquid distribution. Lateral dispersion is wider due to stronger shear stretching from the sides of the parent drop. As expected, more droplets are formed compared to the lower velocity case. Figures 8-10 present the highest velocity case of 190 m/s. Due to the high velocity and shear, ligaments break earlier and with less stretching than that of lower velocity case. More droplets are formed and also the dispersion in this case is the maximum among all cases. Because of the higher impact velocities of the smaller droplets into the liquid body, the asymmetry is also more dramatic than the previous cases. Figure (11) shows the initial period of the droplet deformation. This figure clearly shows that the flow goes around the drop causing the drop to deform. The velocity difference between the drop and the surrounding gas results in a large shear on the drop surface. The shear effect causes the formation of two small ligaments at the sides of the drop. The formation of these ligaments redirects the flow such that a large vortex is developed on the downstream side of the drop.

Figure (12) shows further stretching of the ligament and its breakup into small droplets. Figures 11-12 indicate that the breakup of a 2D drop, and therefore, the breakup of a liquid jet in a cross flow is due to the shearing of the droplet surface. These results clearly show that the breakup models which are based on the Kelvin Helmholtz type of instabilities are not correct. Those models use the wavelength of the instability between two fluids. The present results show that the breakup is due to the formation of a ligament sandwiched between two fluids and the breakup process hardly resembles the K-H instability. Figure (13) shows that the number of the droplets at the downstream side of the parent drop are large enough that they can collide and possibly coalesce with each other. A very interesting finding from these results is shown in Fig. (14). This figure shows that droplets are dispersed behind the parent drop. Therefore, the large number of droplets, which are usually observed in the atomization of jets in cross flow, mainly come from the breakup of the edge ligaments. These drops later disperse downstream of the drop. Previously, it was assumed that the droplet shatters into many pieces forming drops everywhere downstream of the jet. It is clear now that this assumption is not correct, and at least for the cases studies here, the drops come from the shear thinning of the edge ligaments. Figures (15) and (16) shows the formation of another set of ligaments. These ligaments are thicker than the primary ligaments. We can divide the breakup into two modes. One is due to the formation of the primary ligaments and the other is due to the formation of a secondary ligament. The diameter of the primary ligaments is governed by the boundary layer thickness at the surface of the initially stationary droplet at the start of the process. However, the thickness of the secondary ligament is governed by the thickness of the deformed main drop. Figure (15) clearly shows the formation of thick ligaments. Although the secondary ligaments are thicker, they later stretch further after breakup and form thinner ligaments. However, this process depends on where the ligaments are located at. If there are caught into a vortex, they may stretch and become thinner, but if they are located in a more random flow, they breakup into drops forming relatively larger drops. This process is shown in Figs. (17-19). Figure (20) shows the breakup of the leftover mass from the parent drop. This final atomization generates the largest drops.

5. Conclusion

The following conclusions have been obtained from this study:

1. For this flow regime, shear breakup is dominant. Therefore, modeling the breakup based on Kelvin-Helmholtz type of instability may not be proper.

2. The vortex formed downstream of the jet is one of the main determining factors for droplet and ligament dispersion.

3. The ligaments are mainly result of edge breakup, not shattering of the jet exposed to high speed air.

6. REFERENCES 1. Liu, F., Smallwood, GJ, and Gülder, ÖL, Numerical

Study of Breakup Processes of Water Jet Injected into a Cross Air Flow, Proc. 8th International Conference on Liquid Atomization and Spray Systems ICLAS 2000, July 2000, pp. 67-74, Pasadena, CA, USA.

2. Zuo, B, Black, DL, and Crocker, DS, Fuel Atomization and drop breakup models for advanced combustion and drop breakup models for advanced combustion CFD codes, AIAA-2002-4175, 38th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, July 2002, Indianapolis, IN, USA.

3. Madabhushi, RK, A model for numerical simulation of breakup of a liquid jet in crossflow, Atomization and Sprays, Issue 4, 2003.

4. Mazallon, J, Dai, Z, and Faeth, GM, Primary breakup of nonturbulent round liquid jets in gas cross flows, Atomization and Sprays, vol. 9, no. 3, pp. 291-311, 1999.

5. Aalburg, C, van Leer, B, Faeth, GM, and Sallam, KA, Properties of nonturbulent round liquid jets in uniform gaseous cross flows, Atomization and Sprays, vol. 15, pp. 271-294, 2005.

6. Becker J, Hassa C, Plain jet kerosene injection into high temperature, high pressure crossflow with and without filmer plate, 8th International Conference on Liquid Atomization and Spray Systems ICLAS 2000, July 2000, pp. 67-74, Pasadena, CA, USA.

Fig.(2) Diameter= 0.46 mm, velocity=47 m/s, density=2.4 kg/m3

Figure (11) Initial period of the droplet deformation.

Diameter= 0.46 mm, Velocity=47 m/s, Density=2.4 kg/m3

Figure (12) Stretching and breakup of the primary ligament .


Figure (13) Collision and coalescence of droplets on the downstream side of the parent drop. Diameter= 0.46 mm, Gas

velocity=47 m/s, Gas density=2.4 kg/m3

Figure (14) Dispersion of small droplets behind the parent drop.

Diameter= 0.46 mm, Gas velocity=47 m/s, Gas density=2.4 kg/m3

Figure (15) Secondary ligament formation.


Figure (16) Breakup of the secondary ligaments.


Figure (17) Breakup of the secondary ligaments.


Figure (18) Formation of larger secondary drops.


Figure (19) Secondary droplet breakup.


Figure (20) Final breakup of the parent drop


paper id iclass06-264-*** breakup of a 2d drop in cross flow...araz sarchami 1, ali jafari 2 and...

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