Small-scale flow topologies in decaying isotropic turbulence laden with finite-size droplets

The topology of the fine-scale motions in decaying isotropic turbulence laden with droplets of super-Kolmogorov size is investigated using results from direct numerical simulations. The invariants of the velocity-gradient, rate-of-strain, and rate-of-rotation tensors are computed in the carrier phase. The joint probability density functions of the invariants are calculated and conditioned on different distances from the droplet surface. The results show that outside the viscous region near the interface, the flow topologies favor stable focus/stretching and unstable node/saddle/saddle structures, which is in agreement with those found in canonical homogeneous isotropic turbulence. Inside the viscous layer at the droplet surface, the flow topologies shift from a preference for high-enstrophy/low-dissipation motions to favoring low-enstrophy/high-dissipation. At the droplet surface, there is a strong tendency for boundary-layer-like and vortex-sheet flow topologies in which the strain and rotation rates are positively correlated. An interesting observation is that the shapes of the invariant distributions at the droplet surface are remarkably similar to those reported in the viscous sublayer of turbulent wall-bounded flows. Also, the results show that the smallest hydrodynamic length scale of the carrier fluid turbulence is located at the droplet interface and that this length scale is two to three times smaller than that of the surrounding bulk flow. From a computational viewpoint, this suggests a more stringent spatial resolution requirement for the direct numerical simulation of finite-size droplets in isotropic turbulence than its single-phase counterpart.

Figure 1

Topological classification of local flow fields (streamlines) for an observer traveling with the flow on the $R_A$ versus $Q_A$ diagram (a): upper-left, stable focus/stretching (SFS); upper right, unstable focus/compressing (UFC); lower left, stable node/saddle/saddle (SN/S/S); lower right, unstable node/saddle/saddle (UN/S/S). Lines in $(R_S,Q_S)$-space corresponding to different ratios of principal strains ${\lambda }_1:{\lambda }_2:{\lambda }_3$ (b): $2:-1:-1$, axisymmetric contraction; $1:0:-1$, two-dimensional straining limit; $1:1:-2$, axisymmetric expansion.

Figure 2

Instantaneous contours in the $x-y$ plane of (a) the VoF field, $C=C(\mathbf{x},t)$, and (b) the level-set field, $\varphi =\varphi (\mathbf{x},t)$, for case C at $t=2.5$.

Figure 3

Dissipation rate conditionally averaged on distance from the interface ($\langle {\varepsilon }^{\prime }|\varphi \rangle$) for (a–h) cases ${\mathrm{A}}^{★}$ to H and various times from $t=1$ to 6. The intensity of the line decreases as time increases as indicated in panel (a).

Figure 4

Time evolution of the interfacial viscous length scale in the carrier phase ${\delta }_{{\nu }_c,\mathrm{\Sigma }}\left(t\right)$ normalized by the mean viscous length scale of the carrier phase ${\delta }_{\nu ,c}\left(t\right)$ for varying (a) Weber number, (b) density ratio, and (c) viscosity ratio.

Figure 5

Schematic illustrating the layers near the droplet interface used for conditional averaging.

Figure 6

Joint PDFs of (a, d, g, j) $Q_A^{+}$ versus $R_A^{+}$, (b, e, h, k) $Q_S^{+}$ versus $R_S^{+}$, and (c, f, i, l) $-Q_S^{+}$ versus $Q_{\mathrm{\Omega }}^{+}$ for case C conditioned on different distances from the interface: (a–c) ${\varphi }^{+}=10$, (d–f) ${\varphi }^{+}=5$, (g–i) ${\varphi }^{+}=2$, and (j–l) ${\varphi }^{+}=1$.

Figure 7

Instantaneous contours in a subregion of the $x-y$ plane of $-Q_S^{+}+Q_{\mathrm{\Omega }}^{+}$ ($=A_{ij}{A}_{ij}/2$) for case C at $t=2.5$. The droplet interface is marked by a black line and the velocity vectors are projected onto the $x-y$ plane at every 12th grid point.

Figure 8

Joint PDFs of (a–c) $Q_A^{+}$ versus $R_A^{+}$, (d–f) $Q_S^{+}$ versus $R_S^{+}$, and (g–h) $-Q_S^{+}$ versus $Q_{\mathrm{\Omega }}^{+}$ for cases of increasing Weber number ${\text{We}}_{\mathrm{rms}}=$ (a, d, g) 0.1, (b, e, h) 1, and (c, f, i) 5 (cases B–D) conditioned on ${\varphi }^{+}=1$.

Figure 9

Joint PDFs of (a–c) $Q_A^{+}$ versus $R_A^{+}$, (d–f) $Q_S^{+}$ versus $R_S^{+}$, and (g–h) $-Q_S^{+}$ versus $Q_{\mathrm{\Omega }}^{+}$ for cases of increasing density ratio $\phi =$ (a,d,g) 1, (b, e, h) 10, and (c, f, i) 100 (cases E, C, F) conditioned on ${\varphi }^{+}=1$.

Figure 10

Joint PDFs of (a–c) $Q_A^{+}$ versus $R_A^{+}$, (d–f) $Q_S^{+}$ versus $R_S^{+}$, and (g–h) $-Q_S^{+}$ versus $Q_{\mathrm{\Omega }}^{+}$ for cases of increasing viscosity ratio $\gamma =$ (a, d, g) 1, (b, e, h) 10, and (c, f, i) 100 (cases G, C, H) conditioned on ${\varphi }^{+}=1$.