A Universal Relation Between Intermittency and Dissipation in Turbulence

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.

💡 Research Summary

This paper presents an experimental investigation into the relationship between two fundamental constants in turbulence theory: the dissipation constant (C_ε) and the intermittency factor (μ). The study challenges the traditional view of these parameters as independent and potentially universal values by examining a broad class of non-ideal, inhomogeneous turbulent flows, including wakes behind porous and solid objects, grid-generated turbulence, and an axisymmetric jet.

The core methodology relied on extensive hot-wire anemometry measurements, converting temporal signals to spatial data via Taylor’s hypothesis. A key aspect of the analysis was a rigorous filtering process. From a large dataset, only velocity time series exhibiting consistent scaling behavior—such as a sufficiently long inertial range in the energy spectrum, near-Gaussian statistics at large scales, and a highly linear evolution of the scale-dependent shape factor Λ²(r)—were selected for final analysis. This resulted in 312 high-quality data points, each representing a specific flow condition.

The individual analysis of C_ε and μ confirmed prior observations that neither constant converges to a universal value across different flows or Reynolds numbers (Re_λ). C_ε showed a dependence on inflow conditions, scaling linearly with √(Re_G)/Re_λ for wake flows, consistent with non-equilibrium cascade theories. Similarly, μ exhibited significant scatter without a clear asymptotic trend.

The groundbreaking finding emerges when these two parameters are examined together. The data reveals a clear inverse proportionality between μ and C_ε. While each varies independently, their product, α = μ * C_ε, remains remarkably constant (α ≈ 0.106) across all studied flows, Reynolds numbers, and boundary conditions. The probability distribution of α is Gaussian with a small standard deviation, strongly suggesting that this combined parameter, rather than C_ε or μ alone, may embody a universal feature of turbulent cascades.

The authors further investigate connections to other parameters, namely the exponent (γ) of the energy spectrum’s power-law correction and the Kolmogorov constant (C_k). However, no equally robust or simple relationship is found between these and either C_ε or μ.

In conclusion, this work empirically discovers a previously unknown scaling relation that bridges large-scale (energetic) and small-scale (dissipative/intermittent) properties of turbulence. The inverse relationship between the dissipation constant and the intermittency factor suggests a trade-off within the energy cascade process: flows with a higher overall cascade efficiency (higher C_ε) develop less intense intermittency (lower μ), and vice versa, such that their product is invariant. This new empirical law provides a fresh constraint for turbulence theories and a potential novel avenue for modeling complex turbulent flows by linking two of their most critical descriptors.