Statistical properties of two-dimensional elastic turbulence

We numerically investigate the spatial and temporal statistical properties of a dilute polymer solution in the elastic turbulence regime, i.e., in the chaotic flow state occurring at vanishing Reynolds and high Weissenberg numbers. We aim at elucidating the relations between measurements of flow properties performed in the spatial domain with the ones taken in the temporal domain, which is a key point for the interpretation of experimental results on elastic turbulence and to discuss the validity of Taylor's hypothesis. To this end, we carry out extensive direct numerical simulations of the two-dimensional Kolmogorov flow of an Oldroyd-B viscoelastic fluid. Static pointlike numerical probes are placed at different locations in the flow, particularly at the extrema of mean flow amplitude. The results in the fully developed elastic turbulence regime reveal large velocity fluctuations, as compared to the mean flow, leading to a partial breakdown of Taylor's frozen-field hypothesis. While second-order statistics, probed by spectra and structure functions, display consistent scaling behaviors in the spatial and temporal domains, the third-order statistics highlight robust differences. In particular the temporal analysis fails to capture the skewness of streamwise longitudinal velocity increments. Finally, we assess both the degree of statistical inhomogeneity and isotropy of the flow turbulent fluctuations as a function of scale. While the system is only weakly nonhomogenous in the cross-stream direction, it is found to be highly anisotropic at all scales.

Figure 1

Positions of the different probes in the Kolmogorov flow setup. The large square box represents the spatial domain with size $[0,2\pi ]\times [0,2\pi ]$, while the solid violet line represents the mean velocity profile $U\left(y\right)=Ucos\left(ky\right)$, which has local maxima, minima, and nodes denoted $U_{\max }, U_{\min }$, and $U_{\mathrm{zero}}$. The positions of the pointlike temporal probes $T_{\mathrm{probes}}$ are denoted by colored bullet symbols, while the positions of local spatial probes are indicated by the horizontal dashed lines.

Figure 2

Instantaneous snapshots of $u_x^{\prime }(x,y)$ (a), $u_y^{\prime }(x,y)$ (b), ${\partial }_x{u}_y^{\prime }$ (d), ${\partial }_y{u}_x^{\prime }$ (e), and ${\partial }_x{u}_x^{\prime }$ (f). Panel (c) shows the vector-field representation of the fluctuating velocity ${\mathit{u}}^{\prime }$ at the same instant of time as the other panels; here the color codes the intensity of the fluctuating vorticity ${\zeta }^{\prime }$.

Figure 3

Time series of $u_x^{\prime }$ (a) and $u_y^{\prime }$ (b), normalized by the mean flow amplitude $U$, at $\mathrm{Wi}=24.9$, as recorded by three different probes at the $U_{\max }, U_{\mathrm{zero}}$, and $U_{\min }$ locations. Horizontal dotted lines indicate $\overline{u_{x,y}^{\prime }}\pm {\sigma }_{u_{x,y}^{\prime }}$, where the overbar denotes a temporal average and ${\sigma }_{u_{x,y}^{\prime }}$ the standard deviation with respect to it. To ease visualization, only a short subset of the full data set (which reaches $t/\tau =800$) is shown.

Figure 4

Normalized temporal (a) and spatial (b) autocorrelation functions of $u_x^{\prime }$ (left axes) and $u_y^{\prime }$ (right axes). In both (a) and (b) data from probes where the mean flow is maximum, zero, and minimum are shown. The Weissenberg number is $\mathrm{Wi}=24.9$.

Figure 5

Probability distribution functions of (a) $u_x^{\prime }/{\sigma }_{u_x^{\prime }}$ and (b) $u_y^{\prime }/{\sigma }_{u_y^{\prime }}$ (with ${\sigma }_{u_i^{\prime }}$ the rms values of $u_i^{\prime }$ and $i=x,y$) for $\mathrm{Wi}=24.9$. Red (circle) and blue (triangle) points respectively correspond to the PDFs for temporal and spatial probes, using data collected where the mean flow amplitude is maximum and minimum. Aqua color points (solid line) stand for the PDFs from global spatial data from 1000 instantaneous snapshots of velocity fields.

Figure 6

Probability distribution functions of acceleration fluctuations ${\partial }_t{u}_i^{\prime }/{\sigma }_{{\partial }_t{u}_i^{\prime }}$ [red points (circle), from temporal probes], and of velocity-gradient fluctuations ${\partial }_x{u}_i^{\prime }/{\sigma }_{{\partial }_x{u}_i^{\prime }}$, measured from both spatial probes [blue points (triangle)] and from global spatial data [aqua color points (solid line)], for $\mathrm{Wi}=24.9$. All variables are scaled by their rms value $\sigma$. Panels (a) and (b) refer to the $x$ and $y$ components of velocity, respectively. The temporal and spatial probes are located where the mean flow is maximum and minimum, the global statistics make use of data from 1000 instantaneous velocity fields.

Figure 7

Third-order statistics ${\left|z\right|}^{3}P\left(z\right)$ for $z=u_x^{\prime }/{\sigma }_{u_x^{\prime }}$ (a) and $z=u_y^{\prime }/{\sigma }_{u_y^{\prime }}$ (b), for $\mathrm{Wi}=24.9$. Red (circle) and blue (triangle) points respectively correspond to data from temporal and spatial probes, located where the mean flow is maximum and minimum; aqua color points (solid line) correspond to global spatial data. All spatial statistics are further averaged over time using 1000 instantaneous velocity fields.

Figure 8

Third-order statistics ${\left|z\right|}^{3}P\left(z\right)$ for $z={\partial }_t{u}_i^{\prime }$ (local accelerations) and $z={\partial }_x{u}_i^{\prime }$ (longitudinal gradients) for $\mathrm{Wi}=24.9$. Panel (a) refers to the streamwise velocity component ($i=x$), panel (b) to the cross-stream one ($i=y$). Red (circle) and blue (triangle) points respectively correspond to data from temporal and spatial probes, located where the mean flow is maximum and minimum; aqua color points (solid line) correspond to global spatial data. All spatial statistics are further averaged over time using 1000 instantaneous velocity fields.

Figure 9

Spectra of velocity fluctuations $u_x^{\prime }$ (a) and $u_y^{\prime }$ (b) from temporal and spatial data from probes located where the mean flow is maximum and minimum for $\mathrm{Wi}=24.9$. The bottom horizontal axis refers to the spatial spectra, the top one refers to temporal spectra, after conversion, where $\omega =2\pi f, f$ is the frequency and $U$ the mean flow intensity. All spectra in (a) and (b) are normalized by their maximum values.

Figure 10

Second-order velocity structure functions, for $\mathrm{Wi}=24.9$, versus the spatial increment (either measured directly, $X$, or via the mean flow intensity, $TU$). Panel (a) refers to $u_x^{\prime }$, panel (b) refers to $u_y^{\prime }$. Red (circle) and blue (square) points respectively correspond to data from temporal and spatial probes, located where the mean flow is maximum and minimum; aqua color points (solid line) are for global statistics. Spatial statistics are further averaged over time using 1000 instantaneous velocity fields.

Figure 11

Isotropy ratio, $R=S_{u_x^{\prime }}^{\left(2\right)}/S_{u_y^{\prime }}^{\left(2\right)}$, versus the spatial increment (either measured directly, $X$, or via the mean flow intensity, $TU$), for $\mathrm{Wi}=24.9$. Red (circle) and blue (square) points respectively correspond to data from temporal and spatial probes, located where the mean flow is maximum and minimum; aqua color points (solid line) are for global statistics. All spatial statistics are further averaged over time using 1000 instantaneous velocity fields. The horizontal dashed gray line corresponds to the value of ${\sigma }_{u_x^{\prime }}^2/{\sigma }_{u_y^{\prime }}^2$ obtained from temporal probes (Table 2). The vertical long-short dashed black line represents the largest possible increment, $L_0/2=\pi$. The vertical dashed black line represents the length scale corresponding to a time $t=\tau$. Error bars are estimated by dividing the data sets into two equal parts.

Figure 12

Scale-dependent skewness $A_i=S_{u_i^{\prime }}^{\left(3\right)}\left(X\right)/{\left[S_{u_i^{\prime }}^{\left(2\right)}\left(X\right)\right]}^{3/2}$ versus the spatial increment (either measured directly, $X$, or via the mean flow intensity, $TU$). Panel (a) refers to $u_x^{\prime }$ and panel (b) refers to $u_y^{\prime }$. Red (circle) and blue (square) points respectively correspond to data from temporal and spatial probes, located where the mean flow is maximum and minimum; aqua color points (solid line) are for global statistics. All spatial statistics are further averaged over time using 1000 instantaneous velocity fields. For all the data shown the error bar, estimated by dividing the data sets into two equal parts, is of the order of the point size.