Unified multifractal description of longitudinal and transverse intermittency in fully developed turbulence

Abstract

Small-scale intermittency is a defining feature of fully developed fluid turbulence, marked by rare and extreme fluctuations of velocity increments and gradients that defy mean-field descriptions. Existing multifractal descriptions of intermittency focus primarily on longitudinal increments and gradients, despite mounting evidence that transverse components exhibit distinct and stronger intermittency. Here, we develop a unified multifractal framework that jointly prescribes longitudinal and transverse velocity increments, and extends to gradients. We derive explicit relations linking inertial-range scaling exponents of structure functions to moments of velocity gradients in dissipation range. Our results reveal that longitudinal gradient scaling is solely prescribed by longitudinal structure functions, as traditionally expected; however, transverse gradient scaling is prescribed by mixed longitudinal-transverse structure functions. Validation with high-resolution direct numerical simulations of isotropic turbulence, at Taylor-scale Reynolds number up to $1300$ demonstrates excellent agreement, paving way for a more complete and predictive description of intermittency faithful to the underlying turbulence dynamics.

Figures (4)

• Figure 1: Central moments of longitudinal and transverse velocity gradients for moment orders (a) $4$, (b) $6$, and (c) $8$, together with the best-fit power-law scalings.
• Figure 2: Longitudinal inertial-range scaling exponents $\zeta_{p,0}$ versus $p$. Integer-order exponents (blue circles) are obtained directly from structure functions and were reported previously iyer2020BS_2023, while non-integer orders (red asterisks) are inferred from velocity gradient moments using Eq. \ref{['eq:long_inv']}, for $n=2,3,4,6,8$. For reference, theoretical predictions K62SL94SreeniYakhot:2021 and the transverse exponents $\zeta_{0,p}$ are also shown.
• Figure 3: Mixed inertial-range scaling exponents $\zeta_{p-n,n}$ versus $p$ for various integer values of $n$ (see legend). Exponents at integer $p$ are obtained directly from structure functions, while non-integer orders (red asterisks) are inferred from the scaling of transverse velocity gradient moments using Eq. \ref{['eq:tran_inv']} for $n=2,4,6,8$. The dashed curve for $n=2$ corresponds to the log-normal prediction from Fig. \ref{['fig:zeta_long']} and represents $\zeta_{p-2,2}$ (which is identical to $\zeta_{p,0}$SreeniYakhot:2021).
• Figure 4: Flatness of longitudinal (top) and transverse (bottom) velocity gradients, compiled using data from various sources Ishihara07Gotoh:2022Khurshid_2023BP:2025. The best-fit power-law scaling are also marked in dashed lines. The dotted horizontal line at $3$ marks the flatness of a standard Gaussian distribution.