Clustering and entrainment effects on the evaporation of dilute droplets in a turbulent jet

The evaporation of droplets within turbulent sprays involves unsteady, multiscale, and multiphase processes which make its comprehension and modeling capabilities still limited. The present work aims to investigate the dynamics of droplet vaporization within a turbulent spatial developing jet in dilute, nonreacting conditions. We address the problem considering a turbulent jet laden with acetone droplets and using the direct numerical simulation framework based on a hybrid Eulerian-Lagrangian approach and the point droplet approximation. A detailed statistical analysis of both phases is presented. In particular, we show how crucial is the preferential sampling of the vapor phase induced by the inhomogeneous localization of the droplets through the flow. Strong droplet preferential segregation develops suddenly downstream from the inflow section both within the turbulent core and the jet mixing layer. Two distinct mechanisms have been found to drive this phenomenon: the inertial small-scale clustering in the jet core and the intermittent dynamics of droplets across the turbulent-nonturbulent interface in the mixing layer, where dry air entrainment occurs. These phenomenologies strongly affect the overall vaporization process and lead to an impressive widening of the droplet size and vaporization rate distributions in the downstream evolution of the turbulent spray.

Figure 1

A sketch of the 3D cylindrical domain and the turbulent periodic pipe. A representative ensemble of the whole droplet population is plotted with black points. The colors contour the vapor mass fraction field within the jet and the axial instantaneous velocity of the turbulent pipe, respectively.

Figure 2

(a) Mean axial velocity profile of the turbulent pipe expressed in internal coordinates. $U^{+}=U_z/u_{\tau }$ and $y^{+}=y{u}_{\tau }/\nu$ with $u_{\tau }=\sqrt{{\tau }_w/\rho }$ the friction velocity, $U_z=\langle u_z\rangle$ the mean axial velocity, and $y=R-r$ the distance to the wall. The plot is log-linear. Within the viscous sublayer the velocity profile follows the linear law $U^{+}=y^{+}$. In the outer layer the log law $U^{+}=1/kln(y+)+B$ provides a good fit of the mean velocity profile with $k=2.8$ and $B=4.5$. The red square-dotted line represents the present DNS data, and the black triangles correspond to the DNS reported in Eggels et al. [39]. (b) Root mean square of the azimuthal $u_{t,\mathrm{rms}}^{+}$, radial $u_{r,\mathrm{rms}}^{+}$, and axial $u_{z,\mathrm{rms}}^{+}$ velocities of the turbulent pipe expressed in internal coordinates vs $r/D$. The numerical data set is compared to the fully turbulent pipe DNS of Eggels et al. [39] reported by symbols. (c) Inverse of the mean centerline axial velocity of the unladen jet at $\text{Re}=3000, U_0/U_c$, with $U_0$ the jet bulk velocity at the inflow section. In the far field of the jet, $z/R>20$, the centerline axial velocity follows a linear trend, $U_0/U_c\simeq 1/\left(2B\right)(z/R-z_0/R)$. The decaying constant estimated from the interpolation of the data set in the far field is $B\simeq 7.5$ while the virtual origin of the jet is found to be $z_0/R\simeq 0.25$.

Figure 3

(a) Evolution over time of the radius of a single water droplet in a dry air laminar jet. The ambient pressure and temperature are set to $p=101300\text{Pa}$ and $T=273.15\text{K}$. The inlet radius and the bulk velocity of the jet are, respectively, $R=5\times {10}^{-3}\text{m}$ and $U_0=1.7\mathrm{m}/\mathrm{s}$. The droplet is injected at the local carrier phase velocity. The initial droplet radius is set to $r_{d,0}=5\mu \text{m}$ and the temperature to $T_{d,0}=273.15\text{K}$. The droplet radius and the time are scaled by the droplet initial radius and the reference time scale $t_0=R/U_0$, respectively. The continuous line represents the analytical solution computed by the Spalding d-square law, $r_d/r_{d,0}=\sqrt{1-kt/t_0}$, where $k=\left(\rho \mathrm{Sh}{R}^2\right)/\left(\mathrm{Re}{\rho }_l\mathrm{Sc}{r}_{d,0}^2\right)ln(1+B_m)$. The symbols represent the numerical results. (b) Temperature evolution over time of a water droplet freely falling in air at pressure $p=101300\text{Pa}$, temperature $T=301.45\text{K}$, and relative humidity $\chi =0.22$. Droplet is initially at rest with an initial temperature, $T_{d,0}$, equal to the environmental air one. The temperature is scaled by the initial droplet temperature, $T_{d,0}$, while time is scaled by droplet thermal relaxation time, ${\tau }_t$. This latter is defined as the time required by droplet temperature to change by the 63% of its total change between initial temperature and regime temperature. The continuous line represents the result of the simulation while the symbols report an experimental data set [41]. The regime temperature of droplet corresponds to the ventilated wet bulb temperature at prescribed environmental pressure and actual temperature, which is $T_{wb}=288.15\text{K}$.

Figure 4

(a) A radial-axial slice of the turbulent spray. The black points represent a subset of the whole droplet population constituted by droplets located within a distance $h/R=0.01$ from the slice plane. Each point size is proportional to the corresponding droplet radius (scale factor 100). The carrier phase is contoured according to the instantaneous vapor mass fraction field, $Y_v$, which is bounded between 0 and 0.18, the former corresponding to the dry condition and the latter to the 99% saturation level prescribed at inlet. (b) Enlargements of two different jet regions centered at $z/R=20$ (lower panel) and $z/R=30$ (upper panel).

Figure 5

(a) Mean liquid mass fraction, $\mathrm{\Psi }=m_l/m_g$, where $m_l$ and $m_g$ are the mean mass of liquid acetone and air inside each mesh cell, respectively. It should be noted that each isoline of the mass fraction field is also an isoline of the liquid volume fraction field, $\mathrm{\Phi }=\mathrm{\Psi }{\rho }_g/{\rho }_l$, with ${\rho }_l/{\rho }_g$ the mean density ratio. The labels show different distances from the jet inlet section, $z/R$, in correspondence of which the 50%, 90%, 99%, and 99.9% of the injected liquid mass is evaporated. (b) Average saturation field, $S=Y_v/Y_{v,s}$, where $Y_v$ is the actual vapor mass fraction field and $Y_{v,s}=Y_{v,s}(p_0,T)$ is the value of vapor mass fraction corresponding to the local saturation condition.

Figure 6

(a) Mean droplet radius rescaled by the droplet initial radius $r_{d,0}=6\mu \text{m}$. (b) Mean droplet vaporization rate divided by the reference scale defined as ${\dot{m}}_{d,0}=m_{d,0}/{\tau }_{d,0}$ with $m_{d,0}$ the initial droplet mass and ${\tau }_{d,0}$ the initial droplet relaxation time.

Figure 7

(a) Mean gas phase temperature, $T$, rescaled by the injection temperature, $T_0$. (b) Mean droplet temperature, $T_d$, rescaled by droplet initial temperature, $T_{d,0}$. The reference temperature scales are equal, $T_d=T_{d,0}$.

Figure 8

(a) Mean nondimensional mass source term, $S_m^{*}=S_m{R}^3{\tau }_{d,0}/m_{d,0}$, with $R^3$ the inlet radius, and $m_{d,0}$ and ${\tau }_{d,0}$ the initial mass and relaxation time of the monodisperse droplets, respectively. (b) Mean nondimensional sensible energy source term, $S_{e,s}^{*}=(S_e-L_v^0{S}_m)R^3{\tau }_{d,0}/\left(m_{d,0}{c}_l{T}_{d,0}\right)$, with $L_v^0$ the latent heat of vaporization, $T_{d,0}$ the initial droplet temperature, and $c_l$ the liquid-specific heat capacity.

Figure 9

(a) Droplet clustering index, $K$. (b) Evolution over the jet axis of the mean droplet Stokes number based on the Kolmogorov dissipative scale, ${\text{St}}_{\eta }={\tau }_d/{\tau }_{\eta }$. The turbulent dissipation is computed in correspondence of each mesh node in the range $0<r/R<0.2$. The local dissipative time scale is then adopted in order to estimate the Stokes number of droplets located within the correspondent cell. The average of these values is then considered as the mean droplet Stokes number on the jet axis at a given distance from the inflow section, $z/R$.

Figure 10

Radial profiles of the average vapor mass fraction field at four different axial distances from the origin: (a) $z/R=10$, (b) $z/R=20$, (c) $z/R=30$, and (d) $z/R=40$. Each plot shows the enstrophy-threshold conditional average, $Y_{V,EC}$, the droplet-presence conditional average, $Y_{V,DC}$, and the unconditional Eulerian one, $Y_v$. The enstrophy-threshold conditional average is calculated by sampling the vapor mass fraction only over turbulent core events ($I=1$), that is, when local enstrophy exceeds a fixed threshold. $Y_{V,DC}$ is the vapor concentration field obtained by a conditional average on the droplet presence in a given point.

Figure 11

(a) Radial and axial positions of two droplets “1” and “2”: the former has completely evaporated around $t/t_0\simeq 45$, the latter at $t/t_0\simeq 85$. (b) Square droplet radius, $r_d^2/r_{d,0}^2$, and saturation, $S_d$, of the two droplets 1 and 2.

Figure 12

(a) Probability density function of the nondimensional droplet vaporization length, $z_e/R$, with $R$ the jet inlet radius. (b) Probability density function of the nondimensional droplet vaporization time, $t_e/t_0$, with $t_0$ the reference time scale, $t_0=R/U_0$. The PDFs are computed over the entire droplet population injected into the computational domain. See the text for detailed definition.

Figure 13

JPDF of the vaporization lengths and times of the droplets computed over the whole droplet population. See the text for detailed definition.

Figure 14

JPDF of the droplet axial velocity, $u_z/U_0$, and radial position, $r/R$, computed over the droplets that have a vaporization length $z_e/R<10$.

Figure 15

PDF of Lagrangian observables. (a) PDF of nondimensional droplet radius, $r_d/r_{d,0}$, where $r_{d,0}$ is the initial radius of the injected droplets. (b) PDF of the saturation field at droplet surface, $S_d=Y_{v,s}/Y_v$, where $Y_{v,s}=Y_{v,s}(T_d,p)$ is the vapor mass fraction at saturation computed as a function of the droplet actual temperature and the carrier phase thermodynamics pressure, $p_0$. $Y_v$ is the actual vapor mass fraction in the carrier gaseous mixture evaluated at droplet position. (c) PDF of nondimensional droplet temperature, $T_d/T_{d,0}$, where $T_{d,0}$ is the initial temperature of the injected droplets. (d) PDF of nondimensional droplet vaporization rate, $-\dot{r_d}{\tau }_{d,0}/r_{d,0}$, where ${\tau }_{d,0}$ and $r_{d,0}$ are the initial relaxation time and radius of the injected droplets. The PDF plots (a), (b), and (c) are log-linear, and plot (d) is log-log.

Figure 16

Joint probability density functions of liquid phase volume fraction, $\mathrm{\Phi }$, versus droplet vaporization rate, $d{r}_d/dt$ (a and b) and saturation field sampled by droplets, $S$ (c and d). Both statistics are computed at two different axial distances from the jet inflow, $z/R=10$ and $z/R=20$. Droplet vaporization rate is nondimensional. The reference scale is defined as $r_{d,0}/{\tau }_{d,0}$, with $r_{d,0}$ and ${\tau }_{d,0}$ initial droplet radius and relaxation time.

Figure 17

Joint probability density function of droplet square diameter and flight time, $jf(t,d_d^2)$. Also the mean square droplet diameter as a function of the flight time, $\langle d_d^2/d_{d,0}^2\rangle (t/t_0)$, and the mean droplet flight time as a function of the square droplet diameter, $\langle t/t_0\rangle (d_d^2/d_{d,0}^2)$.