TAB Distortion and Breakup Model

The Taylor analogy breakup (TAB) model is based on Taylor's analogy. The analogy represents a distorting droplet as a damped spring-mass system; it considers only the fundamental mode of oscillation of the droplet. The displacement and velocity of the mass in the spring-mass system correspond to representative distortion and rate of distortion quantities for the droplet.

The TAB model equations govern the droplet oscillation and distortion [680]. When the droplet oscillations reach a critical value, breakup replaces the parent particles with child particles whose diameter is chosen from a Rosin-Rammler distribution. By default, the breakup event creates no parcels—the original parcel is retained, but the particle diameter changes to the new child value. However, new parcels can optionally be created, for which the droplet diameter is chosen independently. This feature can be useful in modeling the generation of a distribution of droplet sizes through breakup. Parcels that are involved in a breakup event are given a lateral velocity proportional to the kinetic energy of the droplet oscillation at the instant of breakup. This velocity tends to generate a spreading effect from an injector which, to some extent, makes a cone injector unnecessary.

Despite being based on a single mode of oscillation in the vibrational regime, the TAB model reproduces the same characteristic time-scales in low and high Weber number limits as the Reitz-Diwakar model. Typically, however, the TAB model is used at low Weber numbers; hollow-cone gasoline sprays are an example of a preferred application for this model [647]. Outside its range of validity, the model tends to underpredict droplet sizes.

Droplet Distortion

The goal of the TAB distortion model is to calculate the instantaneous displacement x of the droplet equator from its equilibrium position.

This displacement is illustrated in the following diagram (after Baumgarten [647]).

For simplicity of analysis, this displacement normalizes to the droplet diameter

Figure 1. EQUATION_DISPLAY

y = \frac{2 x}{C_{b} D_{p}}

(3110)

in which $C_{b}$ is an empirical constant. The Taylor analogy between a distorting droplet and a damped spring-mass system [711] gives a second-order ordinary differential equation governing y

Figure 2. EQUATION_DISPLAY

\frac{ρ_{l} D_{p}^{2} \ddot{y}}{4} = \frac{C_{k} σ}{D_{p}} (\frac{W e}{W e_{c r i t}} - 2 y) - C_{d} μ_{l} \dot{y}

(3111)

where

σ

and

μ_{l}

are respectively the surface tension and viscosity of the droplet and

W e

its Weber number, Eqn. (3093).

C_{d}

C_{k}

and

W e_{c r i t}

are empirical constants with default values 5, 8 and 12 respectively.

C_{b}

does not appear in Eqn. (3111): it is not a necessary parameter to determine y. Eqn. (3111) is integrated for each droplet from injection, with initial conditions

y = \dot{y} = 0

Droplet Breakup

The TAB breakup model relies on the TAB distortion model to calculate the normalized droplet distortion y.

When y reaches unity (Eqn. (3110)), the droplet is assumed to be critically distorted; a breakup event is then triggered. The Sauter mean diameter $D_{32}$ of the post-breakup droplets is related to the diameter $D_{p}$ of the parent droplet through an energy balance [680], giving

Figure 3. EQUATION_DISPLAY

\frac{D_{p}}{D_{32}} = 1 + \frac{K C_{b}^{2} C_{k}}{5} + \frac{3 C_{b}^{2} m_{p}}{4 σ π} (\frac{K}{5} - \frac{C_{v}^{2}}{6}) {\dot{y}}^{2}

(3112)

Here, $C_{k}$ and $C_{b}$ are inherited from the TAB distortion model, with $C_{b}$ taking the default value 0.5. $K$ is the ratio of the total energy in distortion and oscillation to the energy in the fundamental mode, default 10/3. In Simcenter STAR-CCM+, the diameter of child droplets is obtained by sampling a Rosin-Rammler distribution with the Sauter mean diameter given by Eqn. (3112) and a user-specified exponent (spread). This is done separately for the parent parcel and, optionally, each child parcel.

Post-breakup parcels are also given a velocity normal to the original velocity of the parent parcel. This is proportional to their rate of distortion at the instant of breakup ${\dot{y}}_{b}$

Figure 4. EQUATION_DISPLAY

v_{⊥} = \frac{1}{2} C_{v} C_{b} D_{p} {\dot{y}}_{b}

(3113)

In both Eqn. (3112) and Eqn. (3113), $C_{v}$ is the normal velocity coefficient. This coefficient can be tuned to produce a known spray angle; alternatively the spray angle may be specified directly at a cone injector, in which case $C_{v}$ should be set to zero.