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Binary collision of droplets is a fundamental form of droplet interaction in the spraying flow field. In order to reveal the central collision mechanism of two gel droplets with equal diameters, an axi-symmetric form of the Navier-Stokes equations are firstly solved and the method of VOF (volume of fluid) is utilized to track the evolution of the gas-liquid free interface. Then, the numerical computation model is validated with Qian’s experimental results on Newtonian liquids. Phenomena of rebound, coalescence and reflexive separation of droplets after collision are investigated, and structures of the complicated flow fields during the collision process are also analyzed in detail. Results show that the maximum shear rate will appear at the point where the flow is redirected and accelerated. Rebound of droplets is determined by the Weber number and viscosity of the fluid together. It can be concluded that the gel droplets are easier to rebound in comparison with the base fluid droplets. The results also show that the alternant appearance along with the deformation of droplets in the radial and axial direction is the main characteristic of the droplet coalescence process, and the deformation amplitude attenuates gradually. Moreover, the reflexive separation process of droplets can be divided into three distinctive stages including the radial expansion, the recovery of the spherical shape, and the axial extension and reflexive separation. The variation trend of the kinetic energy is opposite to that of the surface energy. The maximum deformation of droplets appears in the radial expansion stage; in the case of a low Weber number, the minimum central thickness of a droplet appears later than its maximum deformation, however, this result is on the contrary in the case of a high Weber number.

Droplet collision exists widely and plays an elementary role in various spraying combustion processes of power equipment (aero engine, internal-combustion engine, and liquid rocket engine). The collision outcome affects the atomization characteristics, and substantially influences the subsequent combustion performance, because of the dependence of the vaporization and combustion of droplets on the droplet size, according to the well-known d^{2}-law. Extensive numerical and experimental work has been carried out for Newtonian fluids in the past, in order to improve atomization characteristics and combustion efficiency [

Newtonian fluid droplet collision has been investigated both experimentally and numerically for decades. Experimental studies have showed that outcome of droplet collision mainly depends on the droplet diameter _{1}, _{2} (_{1} ≤ _{2}), the initial relative velocity _{r}_{l}_{l}

Schematic of binary droplet collision.

These variables are grouped into three important dimensionless parameters in binary droplet collision: Weber (We) number and Reynolds (Re) number, and impact parameter (B), which are defined as follows:

Since most fuels are Newtonian fluids, experimental studies on droplet collision are mainly limited to such fluid [

For numerical simulations of binary droplet collisions, several numerical methods have been used for tracking the liquid-gas interface, including the Marker-And-Cell (MAC) method, the front tracking method, Lattice Boltzmann (LB) method, Level-Set (LS) method, Smoothed Particle Hydrodynamic (SPH) method and VOF method [

Focke and Bothe [

Hirt and Nichols [

The incompressible, transient flows, negligible thermal transfer, the continuity equation can be written as:

The momentum equation with surface tension can be written as:

A VOF function

According

According to mass continuity, the advection equation for the density can then be written as an equivalent advection equation for the volume fraction:

Since the simulation of two equal-sized spherical droplets collision is axisymmetric about the X axis, we only need to solve the two-dimensional, axi-symmetric form conservative equations, in order to reduce computational cost. _{0} and initial velocity _{0.} At

The physical problem and the computational domain. The size of the computational domain (

The simulant gels were used in the present investigation, and were formulated by dissolving Carbopol in water with a mass fraction of 0.3 wt %. The physical properties of the simulant gels (non-Newtonian fluids) are from [^{n}^{−1}, the density ^{−3}.

For Newtonian fluid droplet collision, the two nondimensional parameters (We and Re) have been defined [Equations (1) and (2)]. For power-law model fluid, the effective viscosity is given by Equation (11).

Substituting Equation (12) into Equation (2), we have the modified Reynolds number:

Semi-log plot of the viscosity of water based simulants and water.

The solution of the Navier–Stokes equations for droplet collision has to be validated. We carried out two test cases, one to check the gas/liquid interface motion for Newtonian fluid and the other to check the coalesced fluid radial expansion for shear-thinning fluid. In both cases, numerical simulations are to be compared to experimental results.

In order to test the gas/liquid interface motion for Newtonian fluid droplet coalescence, the calculation conditions are consistent with experimental conditions (Tetradecane droplets in nitrogen environment at atmospheric pressures. We = 32.8, Re = 210.8, B = 0.08, _{0} = 318 μm).

Sequence of coalescence collision: (

In order to validate the numerical method for shear dependent viscosity fluid droplet collision, the central binary collisions of 2.8 wt % CMC solution two droplets were studied to check the coalesced fluid radial expansion; the calculation conditions are consistent with experimental conditions.

Dimensionless diameter of collision _{0} = 300 μm, _{0} = 5 m/s.

The grid and time step independence are important for tracking the liquid-gas interface, the central binary collisions of shear dependent viscosity fluid droplet was selected for the grid and time step independence study (the calculation conditions are from [_{0}/100, _{0}/300, _{0}/400, and _{0}/500 were used for simulation, respectively. Comparing to grids with cell size equal to _{0}/300, _{0}/400, and _{0}/500, the result of grid with cell size equal to _{0}/100 showed slightly large difference in dimensionless diameter and evolution of droplet shape, however, results of fine grids with cell size equal to _{0}/300, _{0}/400, and _{0}/500 were almost identical (difference in the dimensionless diameter is less than 1%), and agreed well with experimental data. The adaptive time step was used for simulation; the simulation was run using the minimum time step equal to 1.0 × 10^{−9} s and 1.0 × 10^{−10} s, respectively. The results were almost same. Consequently, the grids with cell size equal to _{0}/300 and the minimum time step equal to 1.0 × 10^{−9} s were utilised for the present investigations.

As studied earlier [_{0} = 262 μm, We = 1.5, Re = 1.7) shows a sequence of photographs from the present simulations. From this figure, the collision process can be divided into two stages (the fluid radial expansion, recovery of the spherical shape and separation). As the droplets approach each other, pressure is built up in the gap (

(

As the two droplets approach each other, the gas between them is squeezed out, and thus two vortex ring are formed, as shown in

Then, the droplets start to bounce and the gap widens, leading to dynamic viscosity decreasing. Simultaneously, the vortices change rotational direction, as ambient gas accelerates to fill in the gap between the two separating droplets (

In order to investigate difference between gel propellant (non-Newtonian fluids) droplet collision and base fluid (Newtonian fluids) droplet collision, base fluid (water) droplet collision was studied with the same conditions, and the outcome is coalescence. We can conclude that the fluid viscosity influences the outcome of droplet collisions. The reason is that high dynamic viscosity exists within droplets at the time of their maximum deformation (

Dynamic viscosity (Pa·s) and velocity field evolution for bouncing collision.

According to the binary collision theory of Newtonian fluids, increasing Weber number leads to droplet coalescence. Based on the theory, we conducted numerical simulations on collision of two gel propellant droplets, increasing Weber number. _{0} = 262 μm, We = 12.0, Re = 8.6) illustrates a sequence of images from these simulations. When the two droplets start to touch each other, a gaseous film is formed, leading to liquid surfaces deformation and flattening (

Time evolution for coalescence collision.

A sheet of gas jet between the two approaching droplets is formed, which is the same as droplets bouncing, and on either side of the jet a vortex ring is also formed. The flow is redirected and accelerated in the region near the interface, which leads to decreasing viscosity (

Dynamic viscosity (Pa·s) and velocity field evolution for coalescence collision.

Increasing Weber number leads to droplet reflexive separation. The difference between droplet reflexive separation and coalescence is that the two droplets coalesce almost immediately after their initial contact; subsequently coalescence of the two initial droplets is followed by reflexive separation, and several satellite droplets are formed. _{0} = 302 μm, We = 82.3, Re = 37.6) shows three stages during process of droplet reflexive separation. At the radial expansion stage of the merging droplet, the two droplets coalesce almost immediately after contact, and continue to expand in the radial direction, in such a way as to form a boundary ring with a thin connecting liquid disc inside (

Time evolution for reflexive separation collision.

(^{−1}), (^{−1}) and (

The total energy of the droplet collision system consists of the surface energy (SE), kinetic energy (KE), and cumulative viscous dissipation (DE). The SE and KE at an instant are nondimensionalized by initial SE and KE value, respectively.

Temporal evolution of surface and kinetic energy for (

The analysis on the energy budget indicates that the maximum deformation occurs in the stage of radial expansion, namely at the moment this stage is over. In order to investigate the evolution of deformations, two parameters in binary droplet collision, the dimensionless diameter, _{0} and _{0}, respectively. _{0} is the initial droplet diameter;

Schematic of fluid radial expansion.

Comparing

Time evolution of the dimensionless diameter

Binary collision of gel propellant droplets, which plays an important role in dense spray combustion process, was investigated numerically and theoretically. The VOF methodology was utilized to track the evolution of the gas-liquid free interface. Phenomena of rebound, coalescence and reflexive separation of droplets after collision are investigated, and the structure of the complicated flow fields during the collision process is also analyzed in detail. The main results can be summarized as follows:

The VOF methodology is capable of predicting the details of complex flow configurations, like the evolution of the gas-liquid free interface, gas bubbles entrapment and coalescence, and ligament formation.

The maximum shear rate occurs at the point where the flow is redirected and accelerated, and minimum effective viscosity occurs at the corresponding point. Rebound of droplets is determined by the Weber number and viscosity of the fluid together. At the time of maximum droplet deformation, the fluid within droplets is at the stagnant state, and dynamic viscosity increases, leading to easier rebound in comparison with the base fluid droplets. The alternant appearance along with the deformation of droplets in the radial and axial direction is the main characteristic of the droplet coalescence process, and the deformation amplitude attenuates gradually. Three distinctive stages (radial expansion, recovery of the spherical shape, and the axial extension and reflexive separation) were identified for reflexive separation process of droplets.

During the rebound process of droplets, the kinetic energy decreases remarkably, whilst the surface energy increases slightly. The kinetic energy and surface energy take on opposite variation trends for the process of droplet coalescence. The influence of Weber number on variation of surface energy is more remarkable, while the influence on variation of kinetic energy is small.

In the case of a low Weber number, the radial velocity of rim reverses its direction towards the center of the disc; the flow within the thin disc continues to accumulate in the boundary ring, which results in the minimum central thickness of a droplet appearance later than its maximum deformation. However, this result is contrary to the case of a high Weber number, because the dimensionless center thickness

This work was supported partially by the National Science Foundation of China under Grant No.11172327. The authors also give great acknowledges and thanks to the College of Aerospace Science and Engineering at National University of Defense Technology for support of their work.