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15.10.2 Wave Breakup Model


An alternative to the TAB model that is appropriate for high-Weber-number flows is the wave breakup model of Reitz [ 287], 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.

Use and Limitations

The wave model is appropriate for high-speed injections, where the Kelvin-Helmholtz instability is believed to dominate droplet breakup ( ${\rm We} > 100$). Because this breakup model can increase the number of computational parcels, you may wish to inject a modest number of droplets initially.

Jet Stability Analysis

The jet stability analysis described in detail by Reitz and Bracco [ 289] is presented briefly here. The analysis considers the stability of a cylindrical, viscous, liquid jet of radius $a$ issuing from a circular orifice at a velocity $v$ into a stagnant, incompressible, inviscid gas of density $\rho_2$. The liquid has a density, $\rho_1$, and viscosity, $\mu_1$, and a cylindrical polar coordinate system is used which moves with the jet. An arbitrary infinitesimal axisymmetric surface displacement of the form

 \eta = \eta_0 e^{ikz + \omega t} (15.10-26)

is imposed on the initially steady motion and it is thus desired to find the dispersion relation $\omega = \omega(k)$ which relates the real part of the growth rate, $\omega$, to its wave number, $k=2\pi/\lambda$.

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

$\displaystyle \phi_1$ $\textstyle =$ $\displaystyle C_1 I_0 (kr) e^{ikz + \omega t}$ (15.10-27)
$\displaystyle \psi_1$ $\textstyle =$ $\displaystyle C_2 I_1 (Lr) e^{ikz + \omega t}$ (15.10-28)

where $\phi_1$ and $\psi_1$ are the velocity potential and stream function, respectively, $C_1$ and $C_2$ are integration constants, $I_0$ and $I_1$ are modified Bessel functions of the first kind, $L^2 = k^2 + \omega/\nu_1$, and $\nu_1$ is the liquid kinematic viscosity [ 287]. 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 $r = a$:

 -p_{21} = -\rho_2(U - i \omega k)^2 k \eta \frac{K_0 (ka)}{K_1 (ka)} (15.10-29)

where $K_0$ and $K_1$ are modified Bessel functions of the second kind and $u$ is the relative velocity between the liquid and the gas. The linearized boundary conditions are

$\displaystyle v_1$ $\textstyle =$ $\displaystyle \frac{\partial \eta}{\partial t}$ (15.10-30)
$\displaystyle \frac{\partial u_1}{\partial r}$ $\textstyle =$ $\displaystyle -\frac{\partial v_1}{\partial z}$ (15.10-31)


 -p_1 + 2\mu_1 - \frac{\sigma}{a^2}\left(\eta + a^2 \frac{\partial^2 \eta}{\partial z^2}\right) + p_2 = 0 (15.10-32)

which are mathematical statements of the liquid kinematic free surface condition, continuity of shear stress, and continuity of normal stress, respectively. Note that $u_1$ is the axial perturbation liquid velocity, $v_1$ is the radial perturbation liquid velocity, and $\sigma$ is the surface tension. Also note that Equation  15.10-31 was obtained under the assumption that $v_2=0$.

As described by Reitz [ 287], Equations  15.10-30 and 15.10-31 can be used to eliminate the integration constants $C_1$ and $C_2$ in Equations  15.10-27 and 15.10-28. Thus, when the pressure and velocity solutions are substituted into Equation  15.10-32, the desired dispersion relation is obtained:

\omega^2 + 2 \nu_1 k^2 \omega \left[\frac{I'_1 (ka)}{I_0 (ka)... ...2}\frac{I_1 (ka)}{I_0 (ka)}\frac{I'_1 (La)}{I_1 (La)}\right] =

 \frac{\sigma k}{\rho_1 a^2}(1 - k^2 a^2)\left(\frac{L^2 -a^2... ...a^2}\right) \frac{I_1 (ka)}{I_0 (ka)}\frac{K_0 (ka)}{K_1 (ka)} (15.10-33)

As shown by Reitz [ 287], Equation  15.10-33 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  15.10-33 were generated for the maximum growth rate, $\Omega$, and the corresponding wavelength, $\Lambda$, and are given by Reitz [ 287]:

$\displaystyle \frac{\Lambda}{a}$ $\textstyle =$ $\displaystyle 9.02 \frac{(1 + 0.45{\rm Oh}^{0.5})(1 + 0.4{\rm Ta}^{0.7})}{(1 + 0.87{\rm We}_2^{1.67})^{0.6}}$ (15.10-34)
$\displaystyle \Omega \left(\frac{\rho_1 a^3}{\sigma}\right)$ $\textstyle =$ $\displaystyle \frac{(0.34 + 0.38{\rm We}_2^{1.5})}{(1 + {\rm Oh})(1 + 1.4{\rm Ta}^{0.6})}$ (15.10-35)

where ${\rm Oh} = \sqrt{{\rm We}_1}/{\rm Re}_1$ is the Ohnesorge number and ${\rm Ta} = {\rm Oh} \sqrt{{\rm We}_2}$ is the Taylor number. Furthermore, ${\rm We}_1 = \rho_1 U^2a/\sigma$ and ${\rm We}_2 = \rho_2 U^2a/\sigma$ are the liquid and gas Weber numbers, respectively, and ${\rm Re}_1 = U a/\nu_1$ is the Reynolds number.

Droplet Breakup

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,

 r = B_0 \Lambda (15.10-36)

where $B_0$ is a model constant set equal to 0.61 based on the work of Reitz [ 287]. Furthermore, the rate of change of droplet radius in the parent parcel is given by

 \frac{da}{dt} = -\frac{(a - r)}{\tau}, \; \; r \leq a (15.10-37)

where the breakup time, $\tau$, is given by

 \tau = \frac{3.726 B_1 a}{\Lambda \Omega} (15.10-38)

and $\Lambda$ and $\Omega$ are obtained from Equations  15.10-34 and 15.10-35, respectively. The breakup time constant, $B_1$, is set to a value of 1.73 as recommended by Liu et al. [ 205]. Values of $B_1$ 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  15.10-38 until the shed mass is equal to 5% of the initial parcel mass. At this time, a new parcel is created with a radius given by Equation  15.10-36. The new parcel is given the same properties as the parent parcel (i.e., 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 which determine how the gas phase interacts with the liquid droplets. For example, the breakup time constant B1 is the constant multiplying the time scale which determines how quickly the parcel will loose mass. Therefore, a larger number means that it takes longer for the particle to loose 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.

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