The centrifugal motion of the liquid within the injector creates an air core surrounded by a liquid film. The thickness of the film at the injector exit, , is related to the mass flow rate by:

(12–375)

where is the injector exit diameter, and is the effective mass flow rate, which is defined by Equation 12–357. The other unknown in Equation 12–375 is , the axial component of velocity at the injector exit. This quantity depends on internal details of the injector and is difficult to calculate from first principles. Instead, the approach of Han et al. [230] is used. The total velocity is assumed to be related to the injector pressure by

(12–376)

where is the velocity coefficient. Lefebvre [351] has noted that is a function of the injector design and injection pressure. If the swirl ports are treated as nozzles and if it is assumed that the dominant portion of the pressure drop occurs at those ports, is the expression for the discharge coefficient (). For single-phase nozzles with sharp inlet corners and ratios of 4, a typical value is 0.78 or less [370]. If the nozzles are cavitating, the value of may be as low as 0.61. Hence, 0.78 could be considered a practical upper bound for . The effect of other momentum losses is approximated by reducing by 10% to 0.7.

Physical limits on require that it be less than unity from conservation of energy, yet be large enough to permit sufficient mass flow. The requirement that the size of the air-core be non-negative implies the following constraint on the film thickness, :

(12–377)

Combining this with Equation 12–375 gives the following constraint on the axial velocity, :

(12–378)

This can be combined with Equation 12–376 and , where is the spray half-angle and is assumed to be known. This yields a constraint on :

(12–379)

Thus, Fluent uses the following expression for :

(12–380)

Assuming that is known, Equation 12–376 can be used to find . Once is determined, is found from

(12–381)

At this point, the thickness and axial component of the liquid film are known at the injector exit. The tangential component of velocity () is assumed to be equal to the radial velocity component of the liquid sheet downstream of the nozzle exit. The axial component of velocity is assumed to remain constant.

The pressure-swirl atomizer model includes the effects of the surrounding gas, liquid viscosity, and surface tension on the breakup of the liquid sheet. Details of the theoretical development of the model are given in Senecal et al. [587] and are only briefly presented here. For a more robust implementation, the gas-phase velocity is neglected in calculating the relative liquid-gas velocity and is instead set by you. This avoids having the injector parameters depend too heavily on the usually under-resolved gas-phase velocity field very near the injection location.

The model assumes that a two-dimensional, viscous, incompressible liquid sheet of thickness moves with velocity through a quiescent, inviscid, incompressible gas medium. The liquid and gas have densities of and , respectively, and the viscosity of the liquid is . A coordinate system is used that moves with the sheet, and a spectrum of infinitesimal wavy disturbances of the form

(12–382)

is imposed on the initially steady motion. The spectrum of disturbances results in fluctuating velocities and pressures for both the liquid and the gas. In Equation 12–382, is the initial wave amplitude, is the wave number, and is the complex growth rate. The most unstable disturbance has the largest value of , denoted here 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 calculated as a function of wave number.

Squire [628], Li and Tankin [365], and Hagerty and Shea [225] have shown that two solutions, or modes, exist that satisfy the governing equations subject to the boundary conditions at the upper and lower interfaces. The first solution, called the sinuous mode, has waves at the upper and lower interfaces in phase. The second solution is called the varicose mode, which has the waves at the upper and lower interfaces radians out of phase. It has been shown by numerous authors (for example, Senecal et al. [587]) that the sinuous mode dominates the growth of varicose waves for low velocities and low gas-to-liquid density ratios. In addition, it can be shown that the sinuous and varicose modes become indistinguishable for high-velocity flows. As a result, the atomization model in Ansys Fluent is based upon the growth of sinuous waves on the liquid sheet.

As derived in Senecal et al. [587], the dispersion relation for the sinuous mode is given by

(12–383)

(12–384)

where and .

Above a critical Weber number of = 27/16 (based on the liquid velocity, gas density, and sheet half-thickness), the fastest-growing waves are short. For , the wavelengths are long compared to the sheet thickness. The speed of modern high pressure fuel injection systems is high enough to ensure that the film Weber number is well above this critical limit.

An order-of-magnitude analysis using typical values shows that the terms of second order in viscosity can be neglected in comparison to the other terms in Equation 12–384. Using this assumption, Equation 12–384 reduces to

(12–385)

(12–386)

For waves that are long compared with the sheet thickness, a mechanism of sheet disintegration proposed by Dombrowski and Johns [145] is adopted. For long waves, ligaments are assumed to form from the sheet breakup process once the unstable waves reach a critical amplitude. If the surface disturbance has reached a value of at breakup, a breakup time, , can be evaluated:

(12–387)

where , the maximum growth rate, is found by numerically maximizing Equation 12–386 as a function of . The maximum is found using a binary search that checks the sign of the derivative. The sheet breaks up and ligaments will be formed at a length given by

(12–388)

where the quantity is an empirical sheet constant that you must specify. The default value of 12 was obtained theoretically by Weber [695] for liquid jets. Dombrowski and Hooper [144] showed that a value of 12 for the sheet constant agreed favorably with experimental sheet breakup lengths over a range of Weber numbers from 2 to 200.

The diameter of the ligaments formed at the point of breakup can be obtained from a mass balance. If it is assumed that the ligaments are formed from tears in the sheet twice per wavelength, the resulting diameter is given by

(12–389)

where is the wave number corresponding to the maximum growth rate, . The ligament diameter depends on the sheet thickness, which is a function of the breakup length. The film thickness is calculated from the breakup length and the radial distance from the center line to the mid-line of the sheet at the atomizer exit, :

(12–390)

This mechanism is not used for waves that are short compared to the sheet thickness. For short waves, the ligament diameter is assumed to be linearly proportional to the wavelength that breaks up the sheet,

(12–391)

where , or the ligament constant, is equal to 0.5 by default.

In either the long-wave or the short-wave case, the breakup from ligaments to droplets is assumed to behave according to Weber’s [695] analysis for capillary instability.

(12–392)

Here, is the Ohnesorge number, which is a combination of the Reynolds number and the Weber number (see Jet Stability Analysis for more details about Oh). Once has been determined from Equation 12–392, it is assumed that this droplet diameter is the most probable droplet size of a Rosin-Rammler distribution with a spread parameter of 3.5 and a default dispersion angle of 6^° (which can be modified in the user interface). The choice of spread parameter and dispersion angle is based on past modeling experience [582]. It is important to note that the spray cone angle must be specified by you when using this model.