Similarity of dissipation and enstrophy in particle-induced small-scale turbulence

Small-scale turbulence is the most promising place to establish proper turbulence theory. An understanding of the relationships between the two descriptors of small-scale turbulence, namely, enstrophy and dissipation, is one of the longstanding research topics. Meanwhile, current knowledge on this point is controversial because past observations are inconsistent with the existing theoretical arguments. We present a different approach to investigate this problem by enhancing small-scale fluctuations with finite-size particles. In this particle-augmented turbulent flow, several common observations about small-scale turbulence are obtained with recent high-Reynolds-number studies, though the operating conditions are quite different. Specifically, it is found that the discrepancy between the intermittency of enstrophy and dissipation is reduced when small-scale motions become intense. In addition, the spatial distributions of intense enstrophy and dissipation are found to be correlated. Lastly, a quantitative relationship for enstrophy and dissipation is observed in these intense events. The similarity of enstrophy and dissipation in all of these aspects does not occur in low-Reynolds-number turbulence, and the current common findings could imply potential universality for intense small-scale turbulence. The conditional probability density functions of pressure Laplacian and velocity derivatives are also analyzed to further understand why dissipation and enstrophy become similar when small-scale motions become strong.

Figure 1

PDFs of normalized dissipation rate and normalized enstrophy. Solid lines correspond to the normalized dissipation rate; dashed lines correspond to normalized enstrophy. (a)–(d) Cases SP, P1, P2, and P3, respectively.

Figure 2

Temporal evolution of the spatial mean values of dissipation rate (lines) and enstrophy (scatters).

Figure 3

PDFs of normalized dissipation and enstrophy for (a)–(d) the single-phase case (SP) and (e)–(h) the particle-laden case (P3) at different times. (a),(e) $2.8{\tau }_0$; (b),(f) $3.5{\tau }_0$; (c),(g) $4.2{\tau }_0$; (d),(h) $4.9{\tau }_0$.

Figure 4

Scatterplot of fourth-order moments of normalized dissipation and enstrophy. Each point stands for the spatial averages of enstrophy and dissipation at a time snapshot. Colors of the symbols indicate the time. Inset: The temporal development of Taylor-scale Reynolds number ${\mathrm{Re}}_{\lambda }$.

Figure 5

Contour plots of local dissipation and enstrophy (both dimensionless). (a)–(d) The 2D slice views of dissipation; (e)–(h) views of enstrophy. Contours for cases SP, P1, P2, and P3 are displayed on different columns, respectively.

Figure 6

Joint PDFs of normalized dissipation rate and normalized enstrophy. (a) Case SP, (b) case P1, (c) case P2, and (d) case P3. The slope of the gray dashed lines in the first quadrants is 1.

Figure 7

Joint PDFs of normalized dissipation and normalized enstrophy for (a)–(d) the single-phase case (SP) and (e)–(h) the particle-laden case (P3) at different times. (a),(e) $2.8{\tau }_0$; (b),(f) $3.5{\tau }_0$; (c),(g) $4.2{\tau }_0$; (d),(h) $4.9{\tau }_0$.

Figure 8

Comparison of PDFs conditioned on intense events ($\varepsilon {}^{\prime }\ge 10$ and $\mathrm{\Omega }{}^{\prime }\ge 10$) and on full domain (unconditional) for case P3. (a) PDFs of ${\nabla }^{2}p$; (b) PDFs of ${\nabla }^{2}p/R_{ij}{R}_{ij}$. Insets: The corresponding PDFs for the single-phase case (SP).

Figure 9

PDFs of the velocity derivatives for case P3. (a) Joint PDF of $\partial u_1/\partial x_2$ and $\partial u_2/\partial x_1$. (b) Unconditional and conditional PDFs of term $B$.