Refined similarity hypothesis using three-dimensional local averages

The refined similarity hypotheses of Kolmogorov, regarded as an important ingredient of intermittent turbulence, has been tested in the past using one-dimensional data and plausible surrogates of energy dissipation. We employ data from direct numerical simulations, at the microscale Reynolds number $R_{\lambda }\sim 650$, on a periodic box of ${4096}^3$ grid points to test the hypotheses using three-dimensional averages. In particular, we study the small-scale properties of the stochastic variable $V=\mathrm{\Delta }u\left(r\right)/{\left(r{\epsilon }_r\right)}^{1/3}$, where $\mathrm{\Delta }u\left(r\right)$ is the longitudinal velocity increment and ${\epsilon }_r$ is the dissipation rate averaged over a three-dimensional volume of linear size $r$. We show that $V$ is universal in the inertial subrange. In the dissipation range, the statistics of $V$ are shown to depend solely on a local Reynolds number.

Figure 1

Cube of edge length $r$ used for the calculation of $V\left(r\right)$. Line segments AD, AE, and AB denote edges along which $V$ [Eq. (13)] is calculated in the $X, Y$, and $Z$ directions, respectively.

Figure 2

Second-order moments of 3D local averages ($\times$) and 1D local averages ($*$) of dissipation for (a) $k_{\max }\eta =1.4$ and (b) $k_{\max }\eta =5.7$, both at $R_{\lambda }\sim 240$. Horizontal solid lines correspond to second-order dissipation moment ($\langle {\epsilon }^2\rangle /{\langle \epsilon \rangle }^2$).

Figure 3

PDF of local 3D dissipation (${\epsilon }_r$) and 1D dissipation (${\widehat{\epsilon }}_r$) at $R_{\lambda }\sim 240$ for (a) ${\epsilon }_r, k_{\max }\eta =1.4$; (b) ${\widehat{\epsilon }}_r, k_{\max }\eta =1.4$; (c) ${\epsilon }_r, k_{\max }\eta =5.7$; and (d) ${\widehat{\epsilon }}_r, k_{\max }\eta =5.7$. Curves in each frame correspond to scale separations $r/\eta \approx 2$ ($+$), 4 ($\times$), 8 ($*$), 16 ($\square$), and 32 ($\blacksquare$), respectively. Dashed curves correspond to PDF of pointwise dissipation ($\epsilon /\langle \epsilon \rangle$).

Figure 4

Compensated energy spectra at ${4096}^3, R_{\lambda }\sim 650$. (a) Longitudinal ($\times$) and transverse ($\square$) spectra as functions of wave-number magnitude in the $k_1$ direction correspond to $\alpha =1,2$, respectively. The solid horizontal line at $C_k=0.53$ is the inertial range constant for longitudinal spectrum from experiments (Ref. [17]). The corresponding constant for the transverse spectrum is given by the dotted line at $4C_k/3$. (b) Three-dimensional energy spectrum as a function of wave-number magnitude $k$. The dotted horizontal line at $55C_k/18$ shows the Kolmogorov constant.

Figure 5

Normalized third-order velocity structure function at ${4096}^3, R_{\lambda }\sim 650$, averaged over the three Cartesian directions. (a) Log-log scales to check small-$r$ behavior; the dashed line corresponds to $r^2$ to check small-$r$ slope. (b) Log-linear scales to assess inertial range extent. The dotted line at $0.8$ for comparison with Eq. (1). The inertial range (IR) is taken as $30<r/\eta <300$. Ratio of the integral scale to the Kolmogorov scale, $L/\eta \approx 1000$.

Figure 6

Correlation coefficient as a function of spatial separation between ${\left(r{\epsilon }_r\right)}^{1/3}$ and the quantity $\beta$, where $\beta$ is $\mathrm{\Delta }u\left(r\right)$ ($*$) and $\left|\mathrm{\Delta }u\right(r\left)\right|$ ($\times$) for $R_{\lambda }\sim 650, {4096}^3$. Correlation coefficient is defined as $\text{corr}(x,y)=\langle (x-\langle x\rangle )(y-\langle y\rangle )\rangle /{\sigma }_x{\sigma }_y$, where ${\sigma }_x$ and ${\sigma }_y$ are the standard deviations of $x$ and $y$. The inertial range (IR) is marked for reference.

Figure 7

Conditional PDF of $V\left(r\right)$ for a given spatial separation. The separation distance $r/\eta$, the number of curves, and the minimum and maximum ${\text{Re}}_r$ in each frame are as follows: (a) 35, 10, 21, and 71; (b) 70, 10, 71, and 247; (c) 139, 10, 260, and 595; (d) 279, 10, 823, and 1886. Curves $A, B$, and $C$ in each frame correspond to the three lowest values of ${\text{Re}}_r$ (in ascending order). In (b) and (c), where inertial effects dominate (Fig. 5), the three labeled curves corresponding to the three lowest ${\text{Re}}_r$ are distinct from the others (at higher ${\text{Re}}_r$), which collapse. In frames (a) and (d), where noninertial effects are more significant, the collapse of the unmarked curves is less pronounced. The dashed curve is the Gaussian distribution with zero mean and unity variance.

Figure 8

Logarithm of the mean of $\left|\mathrm{\Delta }u\right(r\left)\right|$ conditioned on the local velocity scale $U_r={\left(r{\epsilon }_r\right)}^{1/3}$ as a function of the logarithm of $U_r$. Symbols ($+$), ($\times$), ($*$), and ($\square$) correspond to inertial range separations $r/\eta =35,70,139$, and 279, respectively. The dashed line has a slope of 1.

Figure 9

Correlation coefficient as a function of spatial separation between ${\left(r{\epsilon }_r\right)}^{1/3}$ and the quantity $\gamma$, where $\gamma$ is either $V\left(r\right)$ ($*$) or $\left|V\right(r\left)\right|$ ($\times$), at $R_{\lambda }\sim 650, {4096}^3$. The inertial range (IR) is marked for reference.

Figure 10

Second moment of $V$ at ${4096}^3, R_{\lambda }\sim 650$. (a) Log-log scales to verify small-$r$ behavior; the dashed line corresponds to $r^{4/3}$ to check the small-$r$ slope. (b) Log-linear scales to check inertial range behavior; the dotted line at $2.13$ is for comparison with corresponding $K41$ result. The inertial range (IR) extent from Fig. 5 is shown for reference.

Figure 11

Third moment of $V$ at ${4096}^3, R_{\lambda }\sim 650$ [Eq. (19)]. (a) Log-log scales to check small-$r$ behavior; the dashed line corresponds to $r^2$ to check the small-$r$ slope. (b) Log-linear scales to check inertial range behavior; the dotted line at $0.8$ shows the $K41$ plateau. The inertial range (IR) extent from Fig. 5 is shown for reference.

Figure 12

Conditional PDF of $V\left(r\right)$ in the small-scale range for a given spatial separation. The separation distance $r$ (fixed for each frame) and the minimum and maximum values of ${\text{Re}}_r$ are as follows: (a) $r/\eta =2, {\text{Re}}_r$ ranging from $0.2$ to $4.4$; (b) $r/\eta =4, {\text{Re}}_r$ ranging from $0.5$ to $12.9$. In each frame, there are ten curves, each corresponding to a different ${\text{Re}}_r$, increasing in the direction of the arrows. Dashed curve is the standard Gaussian distribution.

Figure 13

The logarithm of the expectation $\left|\mathrm{\Delta }u\right(r\left)\right|$ conditioned on the velocity scale $U_r={\left(r{\epsilon }_r\right)}^{1/3}$ as a function of the logarithm of $U_r$ for different spatial separations in the small-$r$ range. Symbols ($+$), ($\times$), ($*$), ($\square$), and ($◯$) correspond to spatial separations $r/\eta =1,2,4,8$, and 16, respectively. The dashed line has a slope of $1.5$.

Figure 14

Conditional PDF of $V\left(r\right)$ for a given local Reynolds number ${\text{Re}}_r$, for different spatial separations in the small-scale range. (a) ${\text{Re}}_r\approx 27$ and (b) ${\text{Re}}_r\approx 39$. Symbols ($+$), ($\times$), and ($\square$) correspond to spatial separations $r/\eta =4,8$, and 17, respectively. The exact values of ${\text{Re}}_r$ are within $12\%$ of each other in each panel. The dashed curve is the standard Gaussian distribution.

Figure 15

The logarithm of the expectation of $r\left|\mathrm{\Delta }u\right(r\left)\right|$ conditioned on $r{U}_r [U_r={\left(r{\epsilon }_r\right)}^{1/3}]$ as a function of the logarithm of $r{U}_r$ for different spatial separations in the small-$r$ range. Symbols ($+$), ($\times$), ($*$), ($\square$), and ($◯$) correspond to spatial separations $r/\eta =1,2,4,8$, and 16, respectively.