Forward and Inverse Energy Cascade in Fluid Turbulence Adhere to Kolmogorov’s Refined Similarity Hypothesis

We study fluctuations of the local energy cascade rate ${\mathrm{\Phi }}_{\ell }$ in turbulent flows at scales ($\ell$) in the inertial range. According to the Kolmogorov refined similarity hypothesis (KRSH), relevant statistical properties of ${\mathrm{\Phi }}_{\ell }$ should depend on ${\epsilon }_{\ell }$, the viscous dissipation rate locally averaged over a sphere of size $\ell$, rather than on the global average dissipation. However, the validity of KRSH applied to ${\mathrm{\Phi }}_{\ell }$ has not yet been tested from data. Conditional averages such as $⟨{\mathrm{\Phi }}_{\ell }|{\epsilon }_{\ell }⟩$ as well as of higher-order moments are measured from direct numerical simulations data, and results clearly adhere to the predictions from KRSH. Remarkably, the same is true when considering forward (${\mathrm{\Phi }}_{\ell }>0$) and inverse (${\mathrm{\Phi }}_{\ell }<0$) cascade events separately. Measured ratios of forward and inverse cascade probability densities conditioned on ${\epsilon }_{\ell }$ also confirm the applicability of the KRSH to analysis of the fluctuation relation from nonequilibrium thermodynamics.

Figure 1

(a) Spatial distribution of local dissipation rate normalized by $⟨\epsilon ⟩$ on a plane in a small subset of isotropic turbulence at $R_{\lambda }=1250$. The left portion shows ${\epsilon }_{\ell }$ distribution obtained from spherical filtering. (b) Close-up portion of panel (a) also showing a sphere with a diameter $\ell =45\eta$ marked as the black circle. The black dash arrow represents $\mathbf{r}$ separating the two points $+$ and $-$.

Figure 2

Conditional averages of terms in the scale-integrated local KH equation based on ${\epsilon }_{\ell }$, i.e., $Z={\mathrm{\Phi }}_{\ell }$ (black symbols and lines), $Z=P_{\ell }$ (yellow symbols and lines), $Z=D_{\ell }$ (blue symbols and lines). The red dashed line indicates ${\epsilon }_{\ell }$. Different symbols denote different scales $\ell /L=0.012$ [triangles, $(30\eta )$], 0.018 [circles, $(45\eta )$], and 0.024 [squares, $(60\eta )$]. Data is from DNS at $R_{\lambda }=1,250$ (solid symbols) and $R_{\lambda }=430$ at $\ell /L=0.092(45\eta )$ (open circles), for which $Z=\tilde{d}{k}_{\ell }/dt$ (purple diamonds) is included.

Figure 3

Conditional averaged $Z={\mathrm{\Phi }}_{\ell }^q$ (symbols) for the isotropic8192 (a) and isotropic1024 (b) datasets, and $Z=W_{\ell }^n$ (dashed lines for the isotropic1024 dataset), for $\ell =45\eta$, plotted as function of ${\epsilon }_{\ell }$. Results for $q$, $n=1$, 2, 3 are shown in black, blue, and green, respectively. All terms display linear trends with ${\epsilon }_{\ell }$, consistent with the KRSH.

Figure 4

Conditionally averaged positive (forward cascade, full symbols, $Z={\mathrm{\Phi }}_{\ell }$, cond: ${\mathrm{\Phi }}_{\ell }>0$) and negative (inverse cascade, empty symbols, $Z={\mathrm{\Phi }}_{\ell }$, cond: ${\mathrm{\Phi }}_{\ell }<0$) for the isotropic8192 dataset, and for three scales $\ell$ (symbols are the same as in Fig. 2). The green stars show the standard $4/5$-law quantity based on longitudinal structure function, i.e., for $Z=(-5/4\ell )(\delta u_j{\widehat{n}}_j{)}^3$, and no additional condition beyond ${\epsilon }_{\ell }$. Blue symbols is $\mathrm{Pr}({\mathrm{\Phi }}_{\ell }>0|{\epsilon }_{\ell })/\mathrm{Pr}({\mathrm{\Phi }}_{\ell }<0|{\epsilon }_{\ell })$ ($\approx 2.0$).

Figure 5

Conditional PDFs of ${\mathrm{\Psi }}_{\ell }$, for $\ell =45\eta$ conditioned on ranges in bins centered at ${\epsilon }_{\ell }/⟨\epsilon ⟩=0.15$ (black), 0.45, 0.75, 1.05, 1.8, 3.0, 4.2, and 5.4 (yellow). The gray dashed lines have slopes $=2$ (left) and $-1$ (right). A natural logarithm is used.

Figure 6

Test of conditional FR for different values of ${\epsilon }_{\ell }/⟨\epsilon ⟩$ (same line colors as in Fig. 5). The gray line has slope $=1$.