A Universal Relation Between Intermittency and Dissipation in Turbulence

TL;DR

The paper investigates whether universal turbulence constants exist for non-ideal flows by analyzing a large set of 1D velocity time series from wakes, grid turbulence, and an axisymmetric jet. It computes $C_ ext{ε}$ and the intermittency factor $μ$ from the data and finds that neither is universal individually; however, the product $α = μ C_ ext{ε}$ is approximately constant across flows and Reynolds numbers. The authors interpret this as a link between large-scale energy input and small-scale intermittency, consistent with a Vaschy-Buckingham $Π$-theorem framework, and provide a thermodynamic picture in which faster cascades correspond to higher dissipation but lower efficiency. This relation offers a new, practical principle for modeling non-ideal turbulent flows and suggests that combined turbulence constants may capture universality better than single-parameter constants.

Abstract

Fundamental quantities of turbulent flows, such as the dissipation constant $C_\varepsilon$ and the intermittency factor $μ$, are examined in relation to each other for a broader class of non-ideal turbulent flows. In the context of the energy cascade, it is known that $C_\varepsilon$ reflects its basic overall properties, while $μ$ quantifies the intermittency that emerges throughout the cascade. Using an extensive hot-wire dataset of turbulent wakes, grid-generated turbulence, and an axisymmetric jet, we individually analyze these quantities as one-dimensional surrogates of the energy cascade, considering only data that exhibit consistent scaling behavior. We find that $μ$ is inversely proportional to $C_\varepsilon$, offering a new empirical principle that bridges the gap between large and small scales in arbitrary turbulent flows.

Figures (18)

• Figure 1: Schematic representation of the spatial expansion of the turbulence cascade: the blue curve shows the energy spectrum, with energy transported to smaller scales, while the red curve illustrates the evolution of the shape factor $\Lambda^2(r)$ for normalized velocity increments. Note that in this sketch, $\Lambda_0^2$ is approximately zero.
• Figure 2: a) $\mu$ as a function of $Re_\lambda$ for the highly restricted data. The red line corresponds to a commonly accepted value for $\mu$ for homogeneous isotropic turbulence arneodo1996structure. b) $C_\varepsilon$ versus $\sqrt{Re_G}/Re_\lambda$ for the highly restricted data. In general, the symbols in the legend are identical for all figures throughout this manuscript and correspond to the configurations in table I in the appendix SM. For laminar inflow, squared markers are used. For the regular grid and the active grid, markers are shaped as circles and triangles, respectively. For cylinders as generators, the markers contains a "$/$" (case names start with "C") while for disks a "$\backslash$" (case names start with "D") is used. Hollow circular markers means no object and is equivalent to grid turbulence (case names start with "G"). The "x" marker indicates a free jet (case names start with "J"). $N_v$ indicates the number of velocity time series shown in this plot.
• Figure 3: a) $\mu$ as a function of $C_\varepsilon$ for the highly restricted data. The red line indicates a commonly accepted value for $\mu$ for homogeneous isotropic turbulence arneodo1996structure, and the black line corresponds to a least-square fit ($R^2=0.86$ with the fitted values of $0.106 \pm 0.002$ and $0.022 \pm 0.009$). b) $\alpha$ versus $Re_\lambda$ for the highly restricted data with the PDF of $\alpha$ values $p(\alpha)$. The red line indicates the result for $\alpha$ from the fit from a) while the black solid line represents the actual mean value of the ensemble of $\alpha$ values. Additionally, two black dashed lines indicate the corresponding standard deviations $\sigma$ from the mean. For both figures, the symbols and corresponding configurations are shown and explained in figure \ref{['fig:zeroth_combined']} and table I in the appendix SM. $N_v$ indicates the number of velocity time series shown in this plot.
• Figure 4: a) $\gamma$ as a function of $C_\varepsilon$ and b) $C_k$ versus $\gamma$ for the highly restricted data. The red lines indicate both the commonly accepted value for $C_k$ and $\gamma$ for HIT sreenivasan1995universality, while the black lines correspond to a least-square fit for a) and b) with $R^2$ being $0.8$ and $0.99$, respectively. Once more, the symbols and corresponding configurations are shown and explained in figure \ref{['fig:zeroth_combined']} and table I in the appendix SM. $N_v$ indicates the number of velocity time series shown in this plot.
• Figure 5: Relations of the intermittency factor $\mu$, the mean dissipation rate $\varepsilon$ and the dissipation constant $C_\varepsilon$.