Coupled population balance and large eddy simulation model for polydisperse droplet evolution in a turbulent round jet

A population balance model coupled with large eddy simulations (LES) is adapted and applied to study the evolution of oil droplets in an axisymmetric turbulent jet including the effects of droplet breakup. A key unknown in simulating secondary breakup in turbulent multiphase jets is the inflow size distribution generated within the primary breakup zone near the nozzle exit. A monodisperse injection inflow condition is commonly used for simplicity, but this choice is often unrealistic. In order to provide more realistic inlet conditions for LES, we develop a one-dimensional (1D) parcel model to predict the evolution of the dispersed phase along the jet centerline due to the combined effects of advection, radial turbulent transport, and droplet breakup due to turbulence in the regions closer to the jet nozzle that cannot be resolved using coarse LES. The model is validated with experimental data measured far from the nozzle. The 1D model is also used to generate an initial size distribution for use in a coarse-resolution LES of a turbulent jet. Number density fields for each bin of the discretized droplet size distribution are modeled using an Eulerian LES approach and scalar transport equations are solved for each bin. LES results are compared to published experimental data, with good agreement, and we examine the statistics of the velocity field and the concentration of the polydisperse oil droplet plumes for two droplet Weber numbers. We find that the centerline decay rate of the concentration for different droplet sizes is modified in the breakup dominated zone. Unlike Reynolds averaged approaches, LES also allows us to quantify size distribution variability due to turbulence. We quantify the radial and axial distributions and the variability of key quantities such as the Sauter mean diameter, total surface area, and droplet breakup timescale and explore their sensitivity to the Weber number.

Figure 1

Schematic diagram of the hybrid ODE-LES modeling approach: Between the nozzle and the end of the 1D ODE model region, the size distribution is obtained by integrating a 1D ordinary differential equation for the centerline concentrations. The results are used as inflow concentrations for the Eulerian-Eulerian LES further downstream.

Figure 2

(a) Relative number density distribution of Expt. 1 (blue square) and Expt. 2 (red circle) at $z=2$ m. The $y$ axis is scaled differently for visualization purposes. The right axis depicts the size distribution from Expt. 1 while the left axis depicts the size distribution from Expt. 2. The dashed line denotes the fit of the tail of the distribution with $F_1\left(d\right)=A_2{d}^{-4}$ and $F_2\left(d\right)=A_1{d}^{-6}$. The fitted constants are $A_1=1.28\times {10}^{-6}$ and $A_2=6.76\times {10}^{-14}$. (b) Comparison of number density distribution from the 1D ODE model for Expt. 1 () and Expt. 2 () and corresponding experimental data (blue square, red circle) at measurement location.

Figure 3

(a) Parametrized jet centerline velocity (blue line) and dissipation (red line), used as an input to the 1D ODE model. (b) Scaled centerline number concentration $n_2$: $d=18.5\mu \text{m}$ (magenta line), $5\times n_7$: $d=76\mu \text{m}$ (blue line), $10\times n_9$: $d=134\mu \text{m}$ (black line), and $10\times n_{12}$: $d=313\mu \text{m}$ (red line) as a function of downstream distance. The initial conditions for LES are determined by the concentration values at $z=10D_J$ depicted by the dashed line.

Figure 4

(a) Sketch of the simulation setup. Volume rendering of the instantaneous $14\mu \mathrm{m}$ diameter droplet concentration with the $1000\mu \mathrm{m}$ droplets visualized as dots placed randomly with density proportional to its concentration field. (b) Inlet distribution $n_i$ (number of droplets per ${\text{m}}^3$ of fluid) for LES determined by the one-dimensional model.

Figure 5

Downstream variation of half-width of the jet (blue circle, right axis) and the evolution of the inverse of the averaged centerline velocity (red square, left axis) from LES. The linear fit to the data is depicted by the black dashed line.

Figure 6

(a) Averaged axial velocity profiles as a function of normalized radial distance. (b) Averaged concentration profiles at $z/D_J=135$ (red triangle), $z/D_J=168$ (green circle), $z/D_J=211$ (blue square), and $z/D_J=243$ (magenta right triangle) as a function of self-similarity variable $r/r_{1/2}$. The dashed line denotes the DNS data [35] and the solid line represents the analytical constant eddy-viscosity solution.

Figure 7

Radial distributions of concentration fluctuation root-mean square at $z/D_J=135$ (red triangle), $z/D_J=168$ (green circle), $z/D_J=211$ (blue square), and $z/D_J=243$ (magenta right triangle), normalized by centerline mean concentration, as a function of $r/r_{1/2}$.

Figure 8

Instantaneous snapshots of concentration fields at the midplane of the jet plotted in logarithmic scale for different droplet sizes. The domain has been cropped at $z/D_J=500$ for visualization purposes.

Figure 9

Evolution of Sauter mean diameter $D_{32}$ as a function of downstream distance from the nozzle for SIM 1 (blue square) and SIM 2 (red circle). The $D_{32}$ curves from the ODE model for both cases are depicted by a solid line.

Figure 10

(a) Comparison of centerline droplet size distribution from experimental data (blue square, right axis) at $z/D_J=666$ with extended LES results (also right axis, red dashed line). The latter is obtained by solving Eq. (15) using the LES data as inlet condition at $z/D_J=333$ (left axis) as initial condition (these LES data at $z/D_J=333$ are shown by the top red dashed line). Error bars display the r.m.s. at $z/D_J=333$ due to turbulence. The 1D ODE model applied between $z/D_J=2$ and 333 is depicted by (circle, left axis). (b) Comparison of droplet size distribution from SIM 2 (blue dotted line) with 1D ODE model (circle) and SIM 1 size distribution (red dashed line) at $z/D_J=333$.

Figure 11

Decay of centerline concentration of different droplet sizes as a function of downstream distance from the nozzle. The symbols represent the LES evolution for (a) SIM 1 and (b) SIM 2. The symbols are $d=14\mu \mathrm{m}$ (red triangle), $d=100\mu \mathrm{m}$ (green circle), $d=550\mu \mathrm{m}$ (blue square), and $d=3\mathrm{mm}$ (magenta right triangle). The total concentration from SIM 1 and SIM 2 is represented by the black dashed line.

Figure 12

Evolution of the inverse centerline concentration for SIM 1(open symbols) and SIM 2 (closed symbols). The symbols are $d=14\mu \mathrm{m}$ (red triangle), $d=100\mu \mathrm{m}$ (green circle), $d=550\mu \mathrm{m}$ (blue square), and $d=3\mathrm{mm}$ (magenta right triangle).

Figure 13

The top panel depicts the radial distribution of the averaged Sauter mean diameter $D_{32}$ normalized by its centerline value while the bottom panel depicts the normalized standard deviation at $z/D_J=135$ (red line), $z/D_J=168$ (green dashed line), $z/D_J=211$ (blue dotted line), and $z/D_J=243$ (magenta dashed line) for (a) SIM 1 and (b) SIM 2.

Figure 14

The top panel depicts the radial distribution of the averaged total surface area $\tilde{A}$ normalized by its centerline value while the bottom panel depicts the normalized standard deviation at $z/D_J=135$ (red line), $z/D_J=168$ (green dashed line), $z/D_J=211$ (blue dotted line), and $z/D_J=243$ (magenta dashed line) for (a) SIM 1 and (b) SIM 2.

Figure 15

Evolution of the inverse breakup timescale with downstream distance for (a) SIM 1 and (b) SIM 2. The lines are $d=14\mu \mathrm{m}$ (red line), $d=100\mu \mathrm{m}$ (green dashed line), $d=550\mu \mathrm{m}$ (blue dotted line), $d=2261\mu \mathrm{m}$ (magenta dashed line), and $d=3000\mu \mathrm{m}$ (black dashed line).

Figure 16

The top panel depicts the radial distribution of the averaged inverse breakup timescale ${\tilde{t}}_i={\tilde{S}}_{b,i}/{\tilde{n}}_i$ normalized by its centerline value while the bottom panel depicts the normalized standard deviation for (a) SIM 1 and (b) SIM 2. The lines are $d=14\mu \mathrm{m}$ (red line), $d=100\mu \mathrm{m}$ (green dashed line), $d=550\mu \mathrm{m}$ (blue dotted line), and $d=3000\mu \mathrm{m}$ (magenta dashed line) at $z/D_J=70$.