15.10.1 Taylor Analogy Breakup (TAB) Model

Introduction

The Taylor analogy breakup (TAB) model is a classic method for calculating droplet breakup, which is applicable to many engineering sprays. This method is based upon Taylor's analogy [ 347] between an oscillating and distorting droplet and a spring mass system. Table  15.10.1 illustrates the analogous components.

Table 15.10.1: Comparison of a Spring-Mass System to a Distorting Droplet

Spring-Mass System	Distorting and Oscillating Droplet
restoring force of spring	surface tension forces
external force	droplet drag force
damping force	droplet viscosity forces

The resulting TAB model equation set, which governs the oscillating and distorting droplet, can be solved to determine the droplet oscillation and distortion at any given time. As described in detail below, when the droplet oscillations grow to a critical value the "parent'' droplet will break up into a number of smaller "child'' droplets. As a droplet is distorted from a spherical shape, the drag coefficient changes. A drag model that incorporates the distorting droplet effects is available in ANSYS FLUENT. See Section  15.3.5 for details.

Use and Limitations

The TAB model is best for low-Weber-number sprays. Extremely high-Weber-number sprays result in shattering of droplets, which is not described well by the spring-mass analogy.

Droplet Distortion

The equation governing a damped, forced oscillator is [ 256]

where $x$ is the displacement of the droplet equator from its spherical (undisturbed) position. The coefficients of this equation are taken from Taylor's analogy:

$$\frac{F}{m} = C_F \frac{\rho_g u^2}{\rho_l r}$$ (15.10-2)

$$\frac{k}{m} = C_k \frac{\sigma}{\rho_l r^3}$$ (15.10-3)

$$\frac{d}{m} = C_d \frac{\mu_l}{\rho_l r^2}$$ (15.10-4)

where $\rho_l$ and $\rho_g$ are the discrete phase and continuous phase densities, $u$ is the relative velocity of the droplet, $r$ is the undisturbed droplet radius, $\sigma$ is the droplet surface tension, and $\mu_l$ is the droplet viscosity. The dimensionless constants $C_F$, $C_k$, and $C_d$ will be defined later.

The droplet is assumed to break up if the distortion grows to a critical ratio of the droplet radius. This breakup requirement is given as

Where $C_b$ is a constant equal to 0.5, if breakup is assumed to occur when the distortion is equal to half the droplet radius, i.e., oscillations at the north and south pole with this amplitude will meet at the droplet center. This implicitly assumes that the droplet is undergoing only one (fundamental) oscillation mode. Equation  15.10-1 is nondimensionalized by setting $y=x/(C_b r)$ and substituting the relationships in Equations  15.10-2- 15.10-4:

where breakup now occurs for $y>1$. For under-damped droplets, the equation governing $y$ can easily be determined from Equation  15.10-6 if the relative velocity is assumed to be constant:

where

$${\rm We} = \frac{\rho_g u^2 r}{\sigma}$$ (15.10-8)

$${\rm We}_c = \frac{C_F}{C_k C_b} {\rm We}$$ (15.10-9)

$$y_0 = y(0)$$ (15.10-10)

$$\frac{dy_0}{dt} = \frac{dy}{dt}(0)$$ (15.10-11)

$$\frac{1}{t_d} = \frac{C_d}{2} \frac{\mu_l}{\rho_l r^2}$$ (15.10-12)

$$\omega^2 = C_k \frac{\sigma}{\rho_l r^3} - \frac{1}{t_d^2}$$ (15.10-13)

In Equation  15.10-7, $u$ is the relative velocity between the droplet and the gas phase and We is the droplet Weber number, a dimensionless parameter defined as the ratio of aerodynamic forces to surface tension forces. The droplet oscillation frequency is represented by $\omega$. The default value of $y_0$ is 0, based upon the work of Liu et al. [ 205]. The constants have been chosen to match experiments and theory [ 174]:

$$\begin{eqnarray*} C_k & = & 8 \\ C_d & = & 5 \\ C_F & = & \frac{1}{3} \\ \end{eqnarray*}$$

If Equation  15.10-7 is solved for all droplets, those with $y>1$ are assumed to break up. The size and velocity of the new child droplets must be determined.

Size of Child Droplets

The size of the child droplets is determined by equating the energy of the parent droplet to the combined energy of the child droplets. The energy of the parent droplet is [ 256]

where $K$ is the ratio of the total energy in distortion and oscillation to the energy in the fundamental mode, of the order ( $\frac{10}{3}$). The child droplets are assumed to be nondistorted and nonoscillating. Thus, the energy of the child droplets can be shown to be

where $r_{32}$ is the Sauter mean radius of the droplet size distribution. $r_{32}$ can be found by equating the energy of the parent and child droplets (i.e., Equations  15.10-14 and 15.10-15), setting $y=1$, and $\omega^2 = 8\sigma/\rho_l r^3$:

Once the size of the child droplets is determined, the number of child droplets can easily be determined by mass conservation.

Velocity of Child Droplets

The TAB model allows for a velocity component normal to the parent droplet velocity to be imposed upon the child droplets. When breakup occurs, the equator of the parent droplet is traveling at a velocity of $dx/dt = C_b r (dy/dt)$. Therefore, the child droplets will have a velocity normal to the parent droplet velocity given by

where $C_v$ is a constant of order (1).

Droplet Breakup

To model droplet breakup, the TAB model first determines the amplitude for an undamped oscillation $(t_d \approx \infty$) for each droplet at time step $n$ using the following:

According to Equation  15.10-18, breakup is possible only if the following condition is satisfied:

This is the limiting case, as damping will only reduce the chance of breakup. If a droplet fails the above criterion, breakup does not occur. The only additional calculations required then, are to update $y$ using a discretized form of Equation  15.10-7 and its derivative, which are both based on work done by O'Rourke and Amsden [ 256]:

All of the constants in these expressions are assumed to be constant throughout the time step.

If the criterion of Equation  15.10-19 is met, then breakup is possible. The breakup time, $t_{\rm bu}$, must be determined to see if breakup occurs within the time step $\Delta t$. The value of $t_{\rm bu}$ is set to the time required for oscillations to grow sufficiently large that the magnitude of the droplet distortion, $y$, is equal to unity. The breakup time is determined under the assumption that the droplet oscillation is undamped for its first period. The breakup time is therefore the smallest root greater than $t^n$ of an undamped version of Equation  15.10-7:

where

and

If $t_{\rm bu} > t^{n+1}$ , then breakup will not occur during the current time step, and $y$ and $(dy/dt)$ are updated by Equations  15.10-20 and 15.10-21. The breakup calculation then continues with the next droplet. Conversely, if $t^n < t_{\rm bu} < t^{n+1}$, then breakup will occur and the child droplet radii are determined by Equation  15.10-16. The number of child droplets, $N$, is determined by mass conservation:

It is assumed that the child droplets are neither distorted nor oscillating; i.e., $y = (dy/dt) = 0$. The child droplets are represented by a number of child parcels which are created from the original parcel. These child parcels are distributed equally along the equator of the parent droplet in a plane normal to the parent relative velocity vector. The diameter of each of the child parcels is sampled from a Rosin Rammler distribution based on the Sauter mean radius (Equation  15.10-16) and a spread parameter of 3.5.

A velocity component normal to the relative velocity vector, with magnitude computed by Equation  15.10-17, is imposed upon the child droplets. It is decomposed at the equator into components pointing radially outward.

A large number of child parcels ensures a smooth distribution of particle diameters and source terms which is needed when simulating, for example, evaporating sprays.