Two-dimensional homogeneous isotropic fluid turbulence with polymer additives

We carry out an extensive and high-resolution direct numerical simulation of homogeneous, isotropic turbulence in two-dimensional fluid films with air-drag-induced friction and with polymer additives. Our study reveals that the polymers (a) reduce the total fluid energy, enstrophy, and palinstrophy; (b) modify the fluid energy spectrum in both inverse- and forward-cascade régimes; (c) reduce small-scale intermittency; (d) suppress regions of high vorticity and strain rate; and (e) stretch in strain-dominated regions. We compare our results with earlier experimental studies and propose new experiments.

Figure 1

(a) Plots versus time $t/{\tau }_e$ of the total kinetic energy $\mathcal{E}$ of the fluid (top panel), the enstrophy $\Omega$ (middle panel), and the palinstrophy $\mathcal{P}$ (bottom panel) for $c=0$ [upper curve, (blue) circles, run R7] and $c=0.4$ (lower, black curve, run R7). (b) Log-log (base 10) plots of the energy spectra $E\left(k\right)$ versus $k$ for $c=0.2$ [(red) triangles, run R10] and $c=0$ [(blue) circles, run R10]. (c) Polymer contribution to the scale-dependent viscosity $\Delta \nu \left(k\right)$ versus $k$ for $c=0.2$ [(red) line, run R10]; $\Delta \nu \left(k\right)=0$ is shown as the dashed black line. (d) Energy flux $\Pi \left(k\right)$ versus $k$ for $c=0.2$ [dashed (red) line, run R10] and $c=0$ [solid (blue) line, run R10].

Figure 2

(a) Plots of the second-order velocity structure function $S_2\left(r\right)$ versus $r$ for $c=0$ [(blue) circles, run R7] and $c=0.2$ [(green) asterisks, run R7]; the line with slope 2 is shown for comparison. (b) Plots of the hyperflatness $F_6\left(r\right)$ versus $r$ for $c=0$ [(blue) circles, run R7] and $c=0.2$ [(green) asterisks, run R7].

Figure 3

(a) Log-log (base 10) plots of the energy spectra $E\left(k\right)$ versus $k$ for $c=0$ [(blue) circles, run R6], $c=0.2$ [(red) triangles, run R6], and $c=0.4$ (black squares, run R6); plots of $E\left(k\right)$ versus $c$ for $\mathcal{W}i=1.53$ and $k=1$ (bottom-left inset), $\mathcal{W}i=1.53$ and $k=30$ (top-left inset), and $\mathcal{W}i=1.53$ and $k=100$ (top-right inset). (b) Log-log (base 10) plots of $E\left(k\right)$ versus $k$ for $\mathcal{W}i=2.26$ [(blue) circles, run R2], $\mathcal{W}i=4.52$ [(red) triangles, run R2], and $\mathcal{W}i=9.04$ (black squares, run R2); plots of $E\left(k\right)$ versus ${\tau }_P$ for $c=0.4$ and $k=1$ (bottom-left inset) and $c=0.4$ and $k=100$ (top-right inset).

Figure 4

(a) Log-log (base 10) plots, for $c=0.2$ and $\mathcal{W}i=2.91$, of $E\left(k\right)$ versus $k$ for $L=100$ [(red) triangles, run R8] and $L=10$ [(green) asterisks, run R9]; $E\left(k\right)$ for $c=0$ [(blue) circles, run R8]. (b) Plots, for $L=100$ [(red) triangles, run R8] and $L=10$ [(green) asterisks, run R9], of the scale-dependent correction to the viscosity $\Delta \nu \left(k\right)$ versus $k$.

Figure 5

Plots with comparisons of (a) the energy, (b) the energy spectra, and (c) the PDF of $\Lambda$ from our DNS of the NS equations with the scale-dependent viscosity ${\nu }_e\left(k\right)$ (black squares) and from the NS + FENE-P run, R7 [(green) asterisks]. [We calculate ${\nu }_e\left(k\right)\equiv \nu +\Delta \nu \left(k\right)$ by substituting our data from run R7 into Eq. (6).] For reference, we also show plots of all these quantities for the NS equation with conventional, scale-independent viscosity [(blue) circles].

Figure 6

PDFs of the scaled polymer extensions $P(r_P/L)$ versus $r_P/L$ for $c=0.1$ and $L=100$ [(red) triangles, run R8], $c=0.4$ (black squares, run R8), $c=0.1$ and $L=10$ [(green) asterisks, run R9], and $c=0.1$ and $L=6$ [(brown) circles, run R1].

Figure 7

Probability distribution functions (PDFs) of (a) the Okubo-Weiss parameter $\Lambda$ for run R7, (b) ${\sigma }^2$ for run R7, and (c) ${\omega }^2$. Inset: PDF of the velocity component $u_x$ for $c=0$ [(blue) circles, run R7] with a fit $(1/2)exp(-u_x^2/12.5)$ [solid (blue) line] and for $c=0.2$ [(green) asterisks, run R7] with a fit $(1/2.65)exp(-u_x^2/20)$ [solid (green) line]; note that the addition of polymers reduces the rms value of $u_x$.

Figure 8

(a) Conditional PDF of $\left(r_P/L\right)$ conditioned on $\Lambda$ for run R9; (b) pseudocolor plot of $\Lambda$ superimposed on a contour plot of $r_P^2$ for run R10.