Statistics and dynamics of a liquid jet under fragmentation by a gas jet

The breakup of a liquid jet by a surrounding gas jet is studied in a coaxial configuration using high-speed back-lit imaging. This work focuses on the time dynamics and the statistics of the length of the liquid jet. The inlet velocities are varied for both fluids to obtain a wide range of gas and liquid Reynolds numbers and equivalently a wide range of gas-to-liquid dynamic pressure ratio $M$ and Weber number. The variety of scales exhibited throughout this range, exploring two breakup regimes, is covered by adapting the spatial and temporal resolutions as well as the field of view of the imaging system. An in-depth study of the distributions of the length of the liquid jet is presented, with the associated scalings for the evolution of the first three statistical moments with the relevant dimensionless parameters, fully describing the statistics through a unique function. The first two moments of the distributions are shown to be power laws of $M$ and their ratio is observed to be constant. The temporal dynamics are studied using autocorrelation functions of the length of the liquid jet. The correlation times are shown to be controlled by the gas jet, with a secondary influence of the liquid Reynolds number through a change of behavior that appears to be related to the onset of liquid turbulence. In addition, a transition between two regimes highlighted by a change of shape of both the probability density and autocorrelation functions of the liquid core length is introduced and its link to the turbulence characteristics of the gas jet and the underlying breakup mechanisms is discussed.

Figure 1

(a) Schematic view of the atomizer, including a cross section of the exit plane showing the relevant dimensions. (b) Example subsample time series of liquid core length $L_B$. (c) Detection of the liquid core length on an instantaneous image.

Figure 2

[(a)–(c)] Normalized images of the jet for three different operating conditions. (a) $M=10.7, {\mathrm{Re}}_l=2200$, and ${\mathrm{Re}}_g=71000$. (b) $M=0.7, {\mathrm{Re}}_l=8500$, and ${\mathrm{Re}}_g=71000$. (c) $M=0.4, {\mathrm{Re}}_l=2300$, and ${\mathrm{Re}}_g=14000$. Probability density functions (d) and autocorrelation functions (e) computed for the three injection conditions illustrated in (a), (b), and (c).

Figure 3

Probability density function of the liquid core length normalized by the inner liquid diameter [same condition as Fig. 2]. Gaussian, skew-Gaussian [see Eq. (1)], and Gamma functions are represented respectively in blue dotted line, black dashed line, and red dash-dotted line (parameters directly obtained from the first three statistical moments of the data).

Figure 4

Average (a) and standard deviation (b) of the liquid core length normalized by the inner liquid diameter $d_l$ as a function of the gas-to-liquid dynamic pressure ratio $M$. Membrane-breakup, fiber-type breakup, and transitional regimes are represented using circles, squares, and diamonds, respectively. Power-law fits $A{M}^n$ are shown in solid red line with (a) $A_{\mathrm{avg}}=12.8\pm 0.8$ and $n_{\mathrm{avg}}=-0.34\pm 0.04$; (b) $A_{\mathrm{STD}}=2.14\pm 0.12$ and $n_{\mathrm{STD}}=-0.30\pm 0.03$.

Figure 5

Probability density functions of the centered and normalized liquid core length ${\tilde{L}}_B=\frac{L_B-\langle L_B\rangle }{L_{B,\mathrm{STD}}}$ for every operating condition. The black curve corresponds to a skew-Gaussian function with zero average, unit standard deviation, and whose skewness is computed by averaging the skewness obtained for every experimental condition.

Figure 6

Skewness of the liquid core length as a function of the gas-to-liquid dynamic pressure ratio $M$ (a) and gas Reynolds number ${\mathrm{Re}}_g$ (c). For conditions with ${\mathrm{Re}}_g>33000$, alternatively ${\mathrm{We}}_g=75$, we compute the average skewness $\langle {\beta }_{L_B}\rangle =0.46$ and the standard deviation of the skewness, ${\beta }_{L_B,\mathrm{STD}}=0.07$. The dashed line corresponds to $\langle {\beta }_{L_B}\rangle$ and the dashed-dotted line to $\langle {\beta }_{L_B}\rangle \pm {\beta }_{L_B,\mathrm{STD}}$. Ratio of the standard deviation to the average value of the liquid core length $I_{L_B}=L_{B,\mathrm{STD}}/\langle L_B\rangle$ as a function of $M$ (b) and ${\mathrm{Re}}_g$ (d). We compute the average and standard deviation of $I_{L_B}$, respectively $\langle I_{L_B}\rangle =18.1\%$ and $I_{L_B,\mathrm{STD}}=2.6\%$. The dashed line corresponds to $\langle I_{L_B}\rangle$ and the dashed-dotted line to $\langle I_{L_B}\rangle \pm I_{L_B,\mathrm{STD}}$. The $x$ axes are divided into bins to compute averages and standard deviations of ${\beta }_{L_B}$ (c) and $I_{L_B}$ (d) to form box-and-whisker plots. Membrane-breakup, fiber-type breakup, and transitional regimes are represented using circles, squares, and diamonds, respectively.

Figure 7

Autocorrelation functions of $L_B$ as a function of the normalized time lag $\frac{\tau }{{\tau }_c}$. (a) Linear ordinate. (b) Logarithmic ordinate. The black curve corresponds to $exp(-\frac{\tau }{1.33{\tau }_c})$.

Figure 8

Correlation time ${\tau }_c$ normalized by the timescale of the gas jet $T_g$ as a function of (a) ${\mathrm{Re}}_g$ and (b) ${\mathrm{Re}}_l$. The $x$ axes are divided into bins to compute averages and standard deviations of $\frac{{\tau }_c}{T_g}$ to form box-and-whisker plots.

Figure 9

Qualitative phase diagram of breakup regimes in the $\{{\mathrm{Re}}_l;{\mathrm{We}}_g\}$ parameter space. Red circles, black squares, and blue diamonds respectively correspond to membrane-breakup, fiber-type atomization, and to conditions where both of these breakup mechanisms coexist. The dashed line shows ${\mathrm{We}}_g=75$ (equivalently ${\mathrm{Re}}_g=33000$) and the green solid line corresponds to the transition reported in Ref. [15]. Note that the Weber number is defined as ${\mathrm{We}}_g={\rho }_g{U}_g^2{d}_l/\sigma$ in this figure.

Figure 10

Snapshot of the liquid jet exiting the nozzle in a still gas environment. (a) ${\mathrm{Re}}_l=2000$, the interface of the jet remains unperturbed in the vicinity of the nozzle and the liquid jet is laminar. (b) ${\mathrm{Re}}_l=4000$, the interface of the jet only suffers from large-scale disturbances, which suggests that the liquid jet exiting the nozzle is laminar but presents some local flow perturbations. (b) ${\mathrm{Re}}_l=8000$, the interface of the jet presents small-scale corrugations, suggesting that the liquid jet exiting the nozzle is turbulent.