The centrifugal motion of the liquid within the injector creates a liquid film surrounding an air core. The thickness of the film, t _f, is related to the mass flow rate, , by

(6–53)

where is the liquid density, u the axial component of velocity at the injector exit, d ₀ the injector hole diameter. u is related to the total velocity U by

(6–54)

where the cone half-angle is a user-specified input parameter for a specific injector. The total velocity U is related to the pressure drop across the injector exit by

(6–55)

Based on similarity considerations between swirl ports and nozzles, the discharge coefficient k _v is set to a fixed value, 0.7, but to guarantee that the size of the air core is non-negative, the following expression is used for k _v,

(6–56)

The sheet-breakup model assumes that a two-dimensional, viscous, incompressible liquid sheet of thickness 2h moves with velocity U through a quiescent, inviscid, incompressible gas medium. A spectrum of infinitesimal disturbances of the form

(6–57)

is imposed on the initially steady motion and produces fluctuating velocities and pressures for both the liquid and the gas, where is the initial wave amplitude, is the wave number, and is the complex growth rate of the surface disturbances. The most unstable disturbance has the largest value of , denoted by , and is assumed to be responsible for sheet breakup. Thus, it is desired to obtain a dispersion relation from which the most unstable disturbance can be deduced.

Two solutions that satisfy the liquid governing equations subject to the boundary conditions at the upper and lower interfaces [86] . For the first solution, called the sinuous mode, the waves at both surfaces are exactly in phase. For the second solution, the so-called varicose mode, the waves are radians out of phase. Typically the sinuous mode is sufficient for representing engine conditions, and thus a simplified form of the dispersion relation for pressure-swirl atomizers is used:

(6–58)

Here is the liquid kinematic viscosity, Q is the gas/liquid density ratio , and is the surface tension.

Once the unstable waves on the sheet surface grow to a critical amplitude, ligaments are formed due to the sheet breakup. The breakup time for this process is formulated based on an analogy with the breakup length of cylindrical liquid jets, that is,

(6–59)

where is the critical amplitude at breakup, Ω is found by numerically maximizing Equation 6–58 as a function of k. The corresponding breakup length L is estimated by

(6–60)

where the quantity is given a constant value 12. Based on a mass balance, the resulting ligament diameter at the point of breakup is derived as

(6–61)

where K _s is the wave number corresponding to the maximum growth rate, Ω. Based on the assumption that the sheet is in the form of a cone with its vertex at a point behind the injector orifice, the sheet half-thickness h at the breakup position L, is approximately

(6–62)

where

(6–63)

Assuming that breakup occurs when the amplitude of the most unstable wave is equal to the radius of the ligament d_L, then a mass balance gives the drop size d _D

(6–64)

where the most unstable wave number K_L is given by

(6–65)

based on an analogy to Weber’s result for growing waves on cylindrical, viscous liquid columns.

As a consequence of the sheet breakup process described above, fuel droplets are introduced into the computational domain with certain initial conditions. Subsequently, the droplets are subject to secondary breakup (modeled using the Taylor-Analog-Breakup model), collision and coalescence, aerodynamic drag, vaporization, and wall impingement. The models used to describe the atomization physics are described in the following sections.