Primary breakup for diesel fuel jet

The primary breakup of a high speed jet is studied numerically using the Front Tracking method [BoLiuGli10]. We also study the flow in the nozzle, to determine the level of cavitation within the nozzle and the turbulence occuring at the nozzle exit.

Atomization of a high speed jet has been studied extensively in theory [ReiBra79a] [ReiBra82] and experiments [MacTatPow02] [LinPacHal06]

The flow leading to liquid jet breakup is divided roughly into four parameter regions, according to the flow Reynolds and Weber numbers [Lef89] . Here we are concerned with high speed jets, well within the fourth of these regimes, called the second wind driven regime. Possible mechanisms that contribute to jet breakup and atomization in this regime include the nozzle geometry, cavitation and turbulence within the nozzle, aerodynamic instabilities and relaxation of the boundary layer as the fluid flows out of the nozzle. For the flow parameters considered here, the jet, as it leaves the nozzle, is slightly supersonic relative to the ambient air. We model the flows as compressible. Cavitation is defined as the formation of vapor bubbles in a region where the pressure of the liquid falls sufficiently below its vapor pressure, a value given by the tensile strength of the liquid. Cavitation occurs inside the injector nozzle, especially within a corner turning flow near the nozzle inlet. It is observed in simulations there and in the liquid core. Experimental studies suggest that cavitation enhances spray breakup.

Two phases (liquid and vapor) are involved in the nozzle cavitation, as well as two equations of state (EOS), or two branches of a common EOS. The free surface between vapor and liquid as well as that between the liquid and the ambient gas (with its own EOS) are modeled in FronTier. We use the the cavitation model proposed in [XuKimGli05]. This model involves two numerical parameters, the cavitation bubble size and spacing at the time of bubble formation, and one physical parameter, the critical pressure for vapor bubble formation. The nozzle cavitation bubbles are first produced at a sharp corner. Further downstream, the cavitation films extend in the flow direction along the nozzle wall. This is consistent with experiments with large Reynolds number [BauShiBus01].

Figure 1: The cavitation surface inside the nozzle at a quasi-steady state (3D above, 2D below).
3D Cavitation
2D caviitation

Our simulations support the following mechanism of breakup. Nozzle turbulence drives the initial jet instability. Once initilized, these modes grow into liquid films due to a Kelvin-Helmholtz instability. The films are themselves unstable, and become ligaments, Finally the ligaments break up into droplets, iu a Rayleigh instability mode.

We measure droplet size and distribution in our simulations. The log-normal PDF model for droplet size (Slater mean diameter) is effective in describing the early stages of jet breakup [Yoo05].

Parameters for 3D jet breakup simulation
fluid density 0.66 g/cc density ratio 10
ambient density 0.66 g/cc Reynolds number 20,380
nozzle diameter 0.02cm Weber number 2200
jet velocity 200 m/s Ohnesorge number 0.0023
surface tension 2.4 N/m sq. fluid viscosity 0.00136 Poise
ampient pressure 40 bar Mesh spacing 2 microns

We perform a 3D simulation in the computational domain 3R x 3R x 40R, where R = 0.01 cm is the nozzle radius. Only a quarter of the jet is simulated considering the rotational symmetry of the jet. The parameters in our simulation is summarized in the above table. The Weber number and density ratio have been decreased by factors of 5 and 4 respectively from Parker et.al.'s experiments [ParRai98] to make our simulation feasible. The jet is in the second wind induced regime with these parameters. The whole region is discretized by using a uniform cartesian mesh of 120 x 120 x 1600 cells. At the beginning of computations, the domain is filled with high pressure gas. The liquid jet is injected from the left side of the domain with a turbulent inlet perturbation. The inlet turbulent velocity is given by a filter based generator with a prescribed length scale [KleSadJan03]. In our simulation, the radical integral length scale Λ = 0.38 R is from the nozzle flow simulation in analyzed separately. The turbulence intensity is 0.056 of the mean inlet velocity. Reflecting boundary conditions are used on the planes y=0 and z=0. All other boundary conditions are flow through.

Fig. [2 Left] shows snapshots of the simulation result. In the snapshots, the quarter jet surface is reflected to form a whole jet for a better visulization. The jet has an clear intact core near the nozzle exit, which is noticed in the X-ray image [PowYuePoo01a]. The surface instability is first initialized by the inflow turbulence. Then, these instabilities grow into films due to the Kelvin-Helmholtz instability. Finally, the films are unstable and break up into filaments, which further break up into droplets. Liquid films, filaments and droplets can be observed from Fig. [2 Right] which shows a detail of the jet surface in the end of the simulation.

We evaluate the distribution of droplet diameters in the simulation. In Fig. [3], Probability Density Function (PDF) and its log normal fit are plotted togather. The PDF follows a log normal distribution as expected by many experiments [Lef89]. We compare the resulting SMD with Wu et.al.'s correlation [WuFae93], 
SMD/\Lambda
[1+0.04(\rho_g/\rho)(\bar{u}_0/\bar{v}_0^{\prime})^2(\Lambda/SMD)^{2/3}]^{3/5} 
= 76(We_{fΛ})^{-0.69} , here u0 and v0' are the averaged streamwise velocity and rms cross-stream velocity in the nozzle exit, Λ is the turbulence length scale, We is the liquid Weber number with Λ as length scale. The SMD from the above formula is 23 microns which is close to 20 microns from the simulation. Fig.[3] also shows the SMD from the simulation and those from Wu et.al.'s experiments along with the correlation. We also study the range of the droplet diameter distribution. The ratio of the mass median diameter (MMD) to the SMD is proposed to describe the width of the droplet distribution [Lef89]. Many authors found the relation MMD/SMD = 1.2 to hold after primary breakup for direct injection nozzles [Sim77] [WuFae93]. We obtain MMD/SMD=1.23 from the simulation, which means the droplet diameter distribution is nearly equal but slightly wider than that in experiments.

Figure 2: Snapshots of 3d jet surface.
Figure 3: Left: Droplet diameter distribution from the 3d simulation. Right: The SMD from the simulation and the correlation from [WuFae93].