Multiscale anisotropic fluctuations in sheared turbulence with multiple states

We use high-resolution direct numerical simulations to study the anisotropic contents of a turbulent, statistically homogeneous flow with random transitions among multiple energy containing states. We decompose the velocity correlation functions on different sectors of the three-dimensional group of rotations, $\text{SO}\left(3\right)$, using a high-precision quadrature. Scaling properties of anisotropic components of longitudinal and transverse velocity fluctuations are accurately measured at changing Reynolds numbers. We show that independently of the anisotropic content of the energy containing eddies, small-scale turbulent fluctuations recover isotropy and universality faster than previously reported in experimental and numerical studies. The discrepancies are ascribed to the presence of highly anisotropic contributions that have either been neglected or measured with less accuracy in the foregoing works. Furthermore, the anomalous anisotropic scaling exponents are devoid of any sign of saturation with increasing order. Our study paves the way to systematically assess persistence of anisotropy in high-Reynolds-number flows.

Figure 1

Evolution of $I_3$ and $I_2$ in the Lumley triangle ($\times$) for the ${2048}^3$ RKF. The laminar state given by the forcing configuration is shown by ($▴$). The inset shows steady-state evolution of $I_2$ ($+$) and $I_3$ ($*$) as functions of $t/T_E$. The reference line at zero is the isotropic state. Typical isocontours of the velocity magnitude in the 1C ($I_{+}$) and 2C ($I_{-}$) regions are also shown.

Figure 2

A log-log plot of $|S_{j,m}^{(3,L)}|$ vs $r$ for different $(j,m)$ sectors at $R_{\lambda }=450$. Symbols correspond to sectors $(0,0)$ ($\square$), $(2,0)$ ($○$), $(4,4)$ ($▵$), and $(6,6)$ ($▿$), respectively. The inset shows the corresponding logarithmic local slopes for sectors $(0,0)$ and $(4,4)$ at $R_{\lambda }=300$ (closed symbols) and $R_{\lambda }=450$ (open symbols). The horizontal line at 1.0 is the exact result for sector $(0,0)$ [41].

Figure 3

A lin-log plot of the undecomposed compensated structure function $-5S^{(3,L)}\left(\mathbf{r}\right)/4r\epsilon$ at $R_{\lambda }=450$ along the Cartesian directions $\widehat{x}$ ($○$), $\widehat{y}$ ($▵$), and $\widehat{z}$ ($◊$). The corresponding projection on the isotropic sector $-5S_{(0,0)}^{(3,L)}\left(r\right)Y_{00}/4r\epsilon$ is given by ($\blacksquare$). The inertial range (IR) is taken as that range within $5\%$ of the exact IR result, shown by the dashed line at unity.

Figure 4

(a) A log-log plot of $S_{j,m}^{(p,L)}{|}_{I_{+}}$ vs $S_{j,m}^{(p,L)}{|}_{I_{-}}$ at various orders $p$ and sectors $(j,m)$. (b) Same as (a) but for the transverse structure functions. Universality with respect to large-scale conditioning corresponds to slope 1 as shown by the dashed straight lines. For clarity, curves are offset upward.

Figure 5

Summary of scaling exponents of $S_{j,m}^{(p,L)}$ (open symbols) and $S_{j,m}^{(p,T)}$ (closed symbols) vs order $p$ for sectors $j\le 6$. Error bars indicate $95\%$ confidence intervals. Data from experiments for $j=2$ ($*$) are given for comparison [18, 30].