Experimental analysis of Lagrangian paths of drops generated by liquid/liquid sprays

Abstract

In the present study, Lagrangian measurements of drop motion of a two-phase spray are analyzed to estimate the collision, fragmentation, size and velocity field of the drops. An experimental setup is designed to record fluorescence images of drops, illuminated by a planar laser sheet. The recorded images are then processed with a commercially available Lagrangian tracking software, Track. The methodology for particle identification, tracking and size approximation in Track in the near interface region of spray is stated. With the stated methodology, the information about size, velocity, fragmentation, collision and change in the shape of the drops is obtained. The effectiveness of the developed methodology is tested against synthetically produced images and is then applied to a liquid/liquid jet. Similar behaviors for dispersed phase fragmentation–collision and mass fragmented along axial location normalized by breakup length of the jet are reported.

Appendices

Appendix 1: Image generation

A three-dimensional domain with dimensions 120 pixels (width) $\times$ 200 pixels length in flow direction $\times$ 40 units depth with N number of the particle initiated at random locations. The domain is set such that the particles going out the domain are entering again in the domain. The particle is reflected from the side walls (directions perpendicular to flow direction). The initial velocity of the particles at random locations is also initiated with the random velocity in any given interval. The initialized random velocity follows a uniform (Gaussian) distribution in the given interval. The velocity is positive in the flow direction. The initiated particles are allowed to flow with zero acceleration until there is collision or fragmentation. In order to model the fragmentation of particles, a fragmentation factor is assigned to each particle. This factor consists of a random constant (in interval 0 to 1) and is proportional to kinetic energy. As the flow progress, this factor increases and the particle is fragmented into two particles when the value of the fragmentation factor reaches a threshold value. The fragmentation threshold value of 5 is used for all cases. The radius and velocities of new particles are prescribed randomly while following the conservation of mass and momentum. For collision, distances between every two particles are calculated and compared with particle radius at each time to estimate their contact. Two particles are merged as soon as two particles are found in contact. The mass and momentum of the merging particle are also conserved. Since the synthetic images are used just to examine the procedure of collision and fragmentation, the shape of the particle is kept spherical at all the times, even while fragmentation and collision. To replicate the experimental condition, a Gaussian thickness of particle illumination along depth is included. For generating images, an averaged intensity along the depth is assigned to generate two-dimensional images.

Appendix 2: Volume fraction and Eulerian particle velocity from Track

The Lagrangian information of particles can be used to obtain information about the volume fraction. Using the information about the location of centroid, width (W) and height (H) of an identified drop, the pixels occupied by the drop are assigned with the phase indicator function such that, $Y = 1 \text { if the drop is present in pixel else } Y=0$. While plotting the phase indicator function we also use the approximation that its shape is approximated by an ellipse with it borders satisfying the equation $\frac{x_{pix}^2}{(0.5 \cdot W)^2} + \frac{y_{pix}^2}{(0.5 \cdot H)^2} =1$. Using the indicator defined, the average volume fraction ${\bar{Y}}$ over 1500 images for each pixel (${\bar{Y}}(i,j)=\frac{1}{1500}\sum _{n=1}^{1500} Y(i,j;n)$). Using the Lagrangian velocity information, Eulerian information of velocity is also obtained. With Track, the particle velocity is directly obtained by substituting pixels with the velocity of particle occupied by it.

Appendix 3: Eulerian particle velocity from optical flow

The Farnebäck (2003) optical flow methodology module in Matlab with an interrogative window size of 5 pixels is used. The velocity from optical flow ($U_{of}$) is obtained by solving Eq. 2 with the method stated by Farnebäck (2003) optical flow as,

$$\begin{aligned} \frac{\partial g}{\partial t} + \nabla _{12} \cdot \left( g \langle U_{of} \rangle \right)&=0. \end{aligned}$$

(2)

In Eq. 2, g is the intensity measured by detector and $\nabla _{12}$ and is the spatial gradient in two directions. Liu and Shen (2008) in their detailed derivation for optical flow with laser sheet-induced fluorescence have stated the expression for $U_{of}$ as, $U_{of}= \frac{\int \phi U_{12} dV}{\int \phi dV}$ where $\phi$ is the scalar concentration in flow (e.g., dye), or particle number per unit total volume for particulate flows. Since fluorescence is recorded only at the pixels occupied by the particle, therefore $\int \phi U_{12} dV = U_p$ and $\int \phi dV ={\bar{Y}}$, here $U_p$ is the average velocity of particle and ${\bar{Y}}$ is the volume fraction in the interrogation window. ${\bar{Y}}$ can be calculated with the procedure stated in “Appendix 2”. Using the simplifications stated, the particle velocity from Optical flow can be stated as $U_p = U_{of} \cdot {\bar{Y}}$.