Elliptical tracers in two-dimensional, homogeneous, isotropic fluid turbulence: The statistics of alignment, rotation, and nematic order

We study the statistical properties of orientation and rotation dynamics of elliptical tracer particles in two-dimensional, homogeneous, and isotropic turbulence by direct numerical simulations. We consider both the cases in which the turbulent flow is generated by forcing at large and intermediate length scales. We show that the two cases are qualitatively different. For large-scale forcing, the spatial distribution of particle orientations forms large-scale structures, which are absent for intermediate-scale forcing. The alignment with the local directions of the flow is much weaker in the latter case than in the former. For intermediate-scale forcing, the statistics of rotation rates depends weakly on the Reynolds number and on the aspect ratio of particles. In contrast with what is observed in three-dimensional turbulence, in two dimensions the mean-square rotation rate increases as the aspect ratio increases.

Figure 1

(a) Pseudocolor plot of $\omega$ at $t=17.5$ for run $\mathtt{A}\mathtt{3}$; (b) particle positions and orientations at $t=17.5$ for run $\mathtt{A}\mathtt{3}$ and $\gamma =1$; (c) pseudocolor plot of $\omega$ at $t=17.5$ for run $\mathtt{B}\mathtt{3}$; (d) particle positions and orientations at $t=17.5$ for run $\mathtt{B}\mathtt{3}$ and $\gamma =1$. The number of particles shown in (b) and (d) is $2\times {10}^3$. For the spatiotemporal evolution of these plots see Ref. [29].

Figure 2

PDFs of the angle between $\mathit{p}$ and $\nabla \times \mathit{\omega }$. Left: Run $\mathtt{A}\mathtt{1}$ (solid, blue curve), run $\mathtt{A}\mathtt{2}$ (dashed, red curve), and run $\mathtt{A}\mathtt{3}$ (dot-dashed, black curve) for $\gamma =1$. The inset shows the same PDF for run $\mathtt{A}\mathtt{3}$ and different values of $\gamma$ (from top to bottom, $\gamma =1$, 3/4, 1/2, and 1/4). Right: Run $\mathtt{B}\mathtt{1}$ (solid, blue curve), run $\mathtt{B}\mathtt{2}$ (dashed, red curve), and run $\mathtt{B}\mathtt{3}$ (dot-dashed, black curve) for $\gamma =1$. The inset shows the same PDF for run $\mathtt{B}\mathtt{3}$ and different values of $\gamma$ (from top to bottom, $\gamma =1,3/4,1/2,\mathrm{and}1/4$).

Figure 3

PDFs of the angle $\beta$ between $\mathit{p}$ and ${\mathit{e}}_1$ for $\gamma =1$ and different ${\mathrm{Re}}_{\lambda }$ in case (A) (left) and in case (B) (right). The insets show the PDF of $\beta$ for different values of $\gamma$ for runs $\mathtt{A}\mathtt{3}$ (left) and $\mathtt{B}\mathtt{3}$ (right). The color code is the same as in Fig. 2.

Figure 4

Single-time two-point correlation function of $M$ as a function of the space separation rescaled by the correlation length. Left: Run $\mathtt{A}\mathtt{1}$ (solid, blue curve), run $\mathtt{A}\mathtt{2}$ (dashed, red curve), and run $\mathtt{A}\mathtt{3}$ (dot-dashed, black curve) for $\gamma =1$. Right: Run $\mathtt{B}\mathtt{1}$ (solid, blue curve), run $\mathtt{B}\mathtt{2}$ (dashed, red curve), run $\mathtt{B}\mathtt{3}$ (dot-dashed, black curve) for $\gamma =1$. The insets show the correlation length as a function of $\gamma$; the color code is the same as in the main plots.

Figure 5

Autocorrelation function of $\mathit{p}$ as a function of the time separation rescaled by ${\tau }_{\eta }$ for different ${\mathrm{Re}}_{\lambda }$ in case (A) (left) and in case (B) (right). The insets show the correlation time rescaled by ${\tau }_{\eta }$ as a function of $\gamma$ for different ${\mathrm{Re}}_{\lambda }$. The color codes are the same as in Fig. 4.

Figure 6

PDFs of $\dot{\theta }$ rescaled by ${\dot{\theta }}_{\mathrm{rms}}=\sqrt{\langle {\dot{\theta }}^2\rangle }$ for $\gamma =1$ and different ${\mathrm{Re}}_{\lambda }$ in case (A) (left) and in case (B) (right). The insets show the mean-square rotation rate multiplied by ${\tau }_{\eta }^2$ as a function of $\gamma$. The color codes are the same as in Fig. 4.