An alternative to the TAB model that is appropriate for high-Weber-number flows is the wave breakup model of Reitz [547], which considers the breakup of the droplets to be induced by the relative velocity between the gas and liquid phases. The model assumes that the time of breakup and the resulting droplet size are related to the fastest-growing Kelvin-Helmholtz instability, derived from the jet stability analysis described below. The wavelength and growth rate of this instability are used to predict details of the newly-formed droplets.

The wave model is appropriate for high-speed injections, where the Kelvin-Helmholtz instability is believed to dominate droplet breakup (). Because this breakup model can increase the number of computational parcels, you may want to inject a modest number of droplets initially.

The jet stability analysis described in detail by Reitz and Bracco [549] is presented briefly here. The analysis considers the stability of a cylindrical, viscous, liquid jet of radius issuing from a circular orifice at a velocity into a stagnant, incompressible, inviscid gas of density . The liquid has a density, , and viscosity, , and a cylindrical polar coordinate system is used, which moves with the jet. An arbitrary infinitesimal axisymmetric surface displacement of the form

(12–425)

is imposed on the initially steady motion and it is therefore desired to find the dispersion relation which relates the real part of the growth rate, , to its wave number, .

In order to determine the dispersion relation, the linearized equations for the hydrodynamics of the liquid are solved assuming wave solutions of the form

(12–426)

(12–427)

where and are the velocity potential and stream function, respectively, and are integration constants, and are modified Bessel functions of the first kind, , and is the liquid kinematic viscosity [547]. The liquid pressure is obtained from the inviscid part of the liquid equations. In addition, the inviscid gas equations can be solved to obtain the fluctuating gas pressure at :

(12–428)

where and are modified Bessel functions of the second kind and is the relative velocity between the liquid and the gas. The linearized boundary conditions are

(12–429)

(12–430)

and

(12–431)

which are mathematical statements of the liquid kinematic free surface condition, continuity of shear stress, and continuity of normal stress, respectively. Note that is the axial perturbation liquid velocity, is the radial perturbation liquid velocity, and is the surface tension. Also note that Equation 12–430 was obtained under the assumption that .

As described by Reitz [547], Equation 12–429 and Equation 12–430 can be used to eliminate the integration constants and in Equation 12–426 and Equation 12–427. Thus, when the pressure and velocity solutions are substituted into Equation 12–431, the desired dispersion relation is obtained:

(12–432)

As shown by Reitz [547], Equation 12–432 predicts that a maximum growth rate (or most unstable wave) exists for a given set of flow conditions. Curve fits of numerical solutions to Equation 12–432 were generated for the maximum growth rate, , and the corresponding wavelength, , and are given by Reitz [547]:

(12–433)

(12–434)

where is the Ohnesorge number and is the Taylor number. Furthermore, and are the liquid and gas Weber numbers, respectively, and is the Reynolds number.

In the wave model, breakup of droplet parcels is calculated by assuming that the radius of the newly-formed droplets is proportional to the wavelength of the fastest-growing unstable surface wave on the parent droplet. In other words,

(12–435)

where is a model constant set equal to 0.61 based on the work of Reitz [547]. Furthermore, the rate of change of droplet radius in the parent parcel is given by

(12–436)

where the breakup time, , is given by

(12–437)

and and are obtained from Equation 12–433 and Equation 12–434, respectively. The breakup time constant, , is set to a value of 1.73 as recommended by Liu et al. [385]. Values of can range between 1 and 60, depending on the injector characterization.

In the wave model, mass is accumulated from the parent drop at a rate given by Equation 12–437 until the shed mass is equal to 5% of the initial parcel mass. At this time, a new parcel is created. The child particle diameter is given by Equation 12–435. The new parcel is given the same properties as the parent parcel (that is, temperature, material, position, etc.) with the exception of radius and velocity. The new parcel is given a component of velocity randomly selected in the plane orthogonal to the direction vector of the parent parcel, and the momentum of the parent parcel is adjusted so that momentum is conserved. The velocity magnitude of the new parcel is the same as the parent parcel.

You must also specify the model constants that determine how the gas phase interacts with the liquid droplets. For example, the breakup time constant B1 is the constant multiplying the time scale that determines how quickly the parcel will lose mass. Therefore, a larger number means that it takes longer for the particle to lose a given amount. A larger number for B1 in the context of interaction with the gas phase would mean that the interaction with the subgrid is less intense. B0 is the constant for the drop size and is generally taken to be 0.61.

In the initial breakup of a cylindrical liquid jet at low gas Weber numbers (Rayleigh regime), Equation 12–435 can predict droplet diameters that are larger than the initial (jet) diameter. In that case, the size of the droplets is calculated from the fastest growing wavelength, which determines the spacing of the initial droplets, and mass conservation considerations. You can suppress this accounting for the Rayleigh regime as described in Breakup in the Fluent User's Guide.