We revisit the problem of scalar dissipation anomaly and scaling of scalar gradients in passive scalar turbulence using theory and data from well-resolved direct numerical simulations (DNS) on grid sizes of up to 8192^3, spanning Taylor-scale Reynolds numbers Re_λ=140-1000 and Schmidt numbers Sc = 1-512. The theory is based on a joint multifractal description of longitudinal velocity increments and scalar increments, constrained by Yaglom's law and extended to gradients via a fluctuating Batchelor cutoff scale. The DNS data show that the normalized mean scalar dissipation approaches a single asymptotic value as both Re_λ and Sc increase, although larger Sc requires larger to reach this state. In the multifractal framework, this corresponds to an effective scalar Hölder exponent tending to zero, associated with sharp cliff-like scalar fronts, and saturation of inertial-range scaling scalar structure-function exponents. The joint velocity-scalar fractal dimension of the dissipative structures is inferred to approach 7/3, indicating a non-space-filling support. The framework further predicts that for fixed Re_λ, higher-order central moments of scalar gradients are independent of Sc. This prediction is confirmed by DNS data and by the collapse of standardized probability distributions of scalar-gradient across Schmidt numbers. These results suggest that the Sc-scaling of scalar gradients is dictated solely by scalar dissipation anomaly. In contrast, their Re_λ-dependence reflects strong intermittency, which can be directly related to mixed velocity-scalar structure function exponents.

A central problem in turbulence is understanding small-scale intermittency, which refers to the sporadic generation of intense fluctuations in velocity gradients and increments. These extreme events, strongly non-Gaussian in nature, govern dissipation, mixing and transport processes in virtually all turbulent flows. Yet, despite decades of study, a faithful and predictive characterisation of small scales remains elusive owing to the inherent mathematical intractability of the Navier–Stokes equations and the difficulty in resolving them in both simulations and experiments at high Reynolds numbers. Recent advances in high-resolution simulations and experiments have significantly reshaped this picture, particularly by providing precise data at high Reynolds numbers to probe the full tensorial structure and dynamics at small scales. In this article, we synthesise the current understanding of small-scale intermittency and universality, drawing on modern data from well-resolved simulations and experiments that resolve the full velocity-gradient tensor. The results show that, while prevailing intermittency theories capture several key trends, they fail to describe or account for observed asymmetries between longitudinal and transverse fluctuations or between strain and vorticity amplification. Evidence suggests that intermittency is closely tied to the dynamics and geometry of vorticity and strain fields, with non-locality playing an important role. We argue that a consistent picture has emerged, but a complete theory will require unifying the statistical scaling frameworks with the underlying dynamical mechanisms that govern gradient amplification. Additional implications of these findings are discussed, and several pressing open problems are identified for future work.

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.

How well turbulence mixes a scalar θ is governed by the scalar dissipation rate χ= 2D ||^2, making scalar gradients central to turbulent mixing. We study the structure and amplification of these gradients for passive scalars driven by a uniform mean-gradient in isotropic turbulence, using DNS at grid resolutions up to 8192^3. The Re_λ spans 140-1000, and Sc/D spans 1-512. We analyze joint statistical correlations of velocity and scalar gradients that underlie scalar-gradient amplification. Unconditional statistics reaffirm earlier observations that production of χ is dominated by nonlinear amplification of scalar gradients by strain-rate. Scalar gradients preferentially align with the most compressive strain eigenvector and remain orthogonal to vorticity, with both trends virtually independent of Re_λ and Sc. Conditional statistics reveal that this organization becomes dramatically enhanced in regions of intense scalar dissipation: scalar gradient becomes near-perfectly aligned with the most compressive eigendirection and orthogonal to other eigendirections and vorticity. This and visualizations suggest that intense scalar dissipation is organized in sheet-like structures formed in shear layers between vortex tubes, where intense strain also generally resides. However, the effective strain acting along intense scalar gradients is comparatively much weaker, indicating intense scalar dissipation arises primarily from optimal alignments rather than intense strain alone. Molecular diffusion arrests intense scalar-gradient events primarily by redistributing scalar-gradient variance away from intense structures. The contribution from imposed mean-gradient is negligible,but still imprints anisotropy directly onto smallest scales via the strain field. The statistics broadly become universal as Sc and Re_λ increases

The universality of small scales, a cornerstone of turbulence, has been nominally confirmed for low-order mean-field statistics, such as the energy spectrum. However, small scales exhibit strong intermittency, exemplified by formation of extreme events which deviate anomalously from a mean-field description. Here, we investigate the universality of small scales by analyzing extreme events of velocity gradients in different turbulent flows, viz. direct numerical simulations (DNS) of homogeneous isotropic turbulence, inhomogeneous channel flow, and laboratory measurements in a von Karman mixing tank. We demonstrate that the scaling exponents of velocity gradient moments, as function of Reynolds number (Re), are universal, in agreement with previous studies at lower Re, and further show that even proportionality constants are universal when considering one moment order as a function of another. Additionally, by comparing various unconditional and conditional statistics across different flows, we demonstrate that the structure of the velocity gradient tensor is also universal. Overall, our findings provide compelling evidence that even extreme events are universal, with profound implications for turbulence theory and modeling.

Fluid flows are intrinsically characterized via the topology and dynamics of underlying vortex lines. Turbulence in common fluids like water and air, mathematically described by the incompressible Navier-Stokes equations (INSE), engenders spontaneous self-stretching and twisting of vortex lines, generating a complex hierarchy of structures. While the INSE are routinely used to describe turbulence, their regularity remains unproven; the implicit assumption being that the self-stretching is ultimately regularized by viscosity, preventing any singularities. Here, we uncover an inviscid regularizing mechanism stemming from self-stretching itself, by analyzing the flow topology as perceived by an observer aligned with the vorticity vector undergoing amplification. While, initially, vorticity amplification occurs via increasing twisting of vortex lines, a regularizing anti-twist spontaneously emerges to prevent unbounded growth. By isolating a vortex, we additionally demonstrate the genericity of this self-regularizing anti-twist. Our work, directly linking dynamics of vortices to turbulence statistics, reveals how the Navier-Stokes dynamics avoids the development of singularities even without the aid of viscosity.

Inertial-range scaling exponents for both Lagrangian and Eulerian structure functions are obtained from direct numerical simulations of isotropic turbulence in triply periodic domains at Taylor-scale Reynolds number up to 1300. We reaffirm that transverse Eulerian scaling exponents saturate at ≈2.1 for moment orders p≥10, significantly differing from the longitudinal exponents (which are predicted to saturate at ≈7.3 for p≥30 from a recent theory). The Lagrangian scaling exponents likewise saturate at ≈2 for p≥8. The saturation of Lagrangian exponents and transverse Eulerian exponents is related by the same multifractal spectrum by utilizing the well-known frozen hypothesis to relate spatial and temporal scales. Furthermore, this spectrum is different from the known spectra for Eulerian longitudinal exponents, suggesting that Lagrangian intermittency is characterized solely by transverse Eulerian intermittency. We discuss possible implications of this outlook when extending multifractal predictions to the dissipation range, especially for Lagrangian acceleration.

Recently, Shete et al. [Phys. Rev. Fluids 7, 024601 (2022)] explored the characteristics of passive scalars in the presence of a uniform mean gradient, mixed by stationary isotropic turbulence. They concluded that at high Reynolds and Schmidt numbers, the presence of both inertial-convective and viscous-convective ranges renders the statistics of the scalar and velocity fluctuations to behave similarly. However, their data included Schmidt numbers of 0.1, 0.7, 1.0, and 7.0, only the last of which can (at best) be regarded as moderately high. Additionally, they do not consider already available data in the literature at substantially higher Schmidt number of up to 512. By including these data, we demonstrate here that the differences between velocity and scalar statistics show no vanishing trends with increasing Reynolds and Schmidt numbers, and essential differences remain intact at all Reynolds and Schmidt numbers.

We investigate the role of pressure, via its Hessian tensor ${\boldsymbol {H}}$ , on amplification of vorticity and strain-rate and contrast it with other inviscid nonlinear mechanisms. Results are obtained from direct numerical simulations of isotropic turbulence with Taylor-scale Reynolds number in the range 140–1300. Decomposing ${\boldsymbol {H}}$ into local isotropic ( ${\boldsymbol {H}}^{I}$ ) and non-local deviatoric ( ${\boldsymbol {H}}^{D}$ ) components reveals that ${\boldsymbol {H}}^{I}$ depletes vortex stretching, whereas ${\boldsymbol {H}}^{D}$ enables it, with the former slightly stronger. The resulting inhibition is significantly weaker than the nonlinear mechanism which always enables vortex stretching. However, in regions of intense vorticity, identified using conditional statistics, contribution from ${\boldsymbol {H}}$ prevails over nonlinearity, leading to overall depletion of vortex stretching. We also observe near-perfect alignment between vorticity and the eigenvector of ${\boldsymbol {H}}$ corresponding to the smallest eigenvalue, which conforms with well-known vortex-tubes. We discuss the connection between this depletion, essentially due to (local) ${\boldsymbol {H}}^{I}$ , and recently identified self-attenuation mechanism (Buaria et al. , Nat. Commun. , vol. 11, 2020, p. 5852), whereby intense vorticity is locally attenuated through inviscid effects. In contrast, the influence of ${\boldsymbol {H}}$ on strain-amplification is weak. It opposes strain self-amplification, together with vortex stretching, but its effect is much weaker than vortex stretching. Correspondingly, the eigenvectors of strain and ${\boldsymbol {H}}$ do not exhibit any strong alignments. For all results, the dependence on Reynolds number is very weak. In addition to the fundamental insights, our work provides useful data and validation benchmarks for future modelling endeavours, for instance in Lagrangian modelling of velocity gradient dynamics, where conditional ${\boldsymbol {H}}$ is explicitly modelled.

Recently, Shete et al. [Phys. Rev. Fluids 7, 024601 (2022)] explored the characteristics of passive scalars in the presence of a uniform mean gradient, mixed by stationary isotropic turbulence. They concluded that at high Reynolds and Schmidt numbers, the presence of both inertial-convective and viscous-convective ranges, renders the statistics of the scalar and velocity fluctuations to behave similarly. However, their data included Schmidt numbers of 0.1, 0.7, 1.0 and 7.0, only the last of which can (at best) be regarded as moderately high. Additionally, they do not consider already available data in the literature at substantially higher Schmidt number of up to 512. By including these data, we demonstrate here that the differences between velocity and scalar statistics show no vanishing trends with increasing Reynolds and Schmidt numbers, and essential differences remain in tact at all Reynolds and Schmidt numbers.

We study the properties of various Eulerian contributions to fluid particle acceleration by using well-resolved direct numerical simulations of isotropic turbulence, with the grid resolution as high as $12288^3$ and the Taylor-scale Reynolds number $R_\lambda$ in the range between 140 and 1300. The variance of convective acceleration, when normalized by Kolmogorov scales, increases linearly with $R_\lambda$, consistent with simple theoretical arguments, but very strongly differing from phenomenological predictions of Kolmogorov's hypothesis as well as Eulerian multifractal models. The scaling of the local acceleration is also linear $R_\lambda$ to the leading order, but more complex in detail. The strong cancellation between the local and convective acceleration -- faithful to the random sweeping hypothesis -- results in the variance of the Lagrangian acceleration increasing only as $R_\lambda^{0.25}$, as recently shown by Buaria \& Sreenivasan [Phys. Rev. Lett. 128, 234502 (2022)]. The acceleration variance is dominated by irrotational pressure gradient contributions, whose variance also follows an $R_\lambda^{0.25}$ scaling; the solenoidal viscous contributions are relatively small and follow a $R_\lambda^{0.13}$, consistent with Eulerian multifractal predictions.

Turbulence in fluid flows is characterized by a wide range of interacting scales. Since the scale range increases as some power of the flow Reynolds number, a faithful simulation of the entire scale range is prohibitively expensive at high Reynolds numbers. The most expensive aspect concerns the small-scale motions; thus, major emphasis is placed on understanding and modeling them, taking advantage of their putative universality. In this work, using physics-informed deep learning methods, we present a modeling framework to capture and predict the small-scale dynamics of turbulence, via the velocity gradient tensor. The model is based on obtaining functional closures for the pressure Hessian and viscous Laplacian contributions as functions of velocity gradient tensor. This task is accomplished using deep neural networks that are consistent with physical constraints and explicitly incorporate Reynolds number dependence to account for small-scale intermittency. We then utilize a massive direct numerical simulation database, spanning two orders of magnitude in the large-scale Reynolds number, for training and validation. The model learns from low to moderate Reynolds numbers and successfully predicts velocity gradient statistics at both seen and higher (unseen) Reynolds numbers. The success of our present approach demonstrates the viability of deep learning over traditional modeling approaches in capturing and predicting small-scale features of turbulence.

We investigate the role of pressure, via its Hessian tensor 𝐇, on amplification of vorticity and strain-rate and contrast it with other inviscid nonlinear mechanisms. Results are obtained from direct numerical simulations of isotropic turbulence with Taylor-scale Reynolds number in the range 140-1300. Decomposing 𝐇 into local isotropic (𝐇^ I) and nonlocal deviatoric (𝐇^ D) components reveals that 𝐇^ I depletes vortex stretching (VS), whereas 𝐇^ D enables it, with the former slightly stronger. The resulting inhibition is significantly weaker than the nonlinear mechanism which always enables VS. However, in regions of intense vorticity, identified using conditional statistics, contribution from 𝐇 dominates over nonlinearity, leading to overall depletion of VS. We also observe near-perfect alignment between vorticity and the eigenvector of 𝐇 corresponding to the smallest eigenvalue, which conforms with well-known vortex-tubes. We discuss the connection between this depletion, essentially due to (local) 𝐇^ I, and recently identified self-attenuation mechanism [Buaria et al. Nat. Commun. 11:5852 (2020)], whereby intense vorticity is locally attenuated through inviscid effects. In contrast, the influence of 𝐇 on strain-amplification is weak. It opposes strain self-amplification, together with VS, but its effect is much weaker than VS. Correspondingly, the eigenvectors of strain and 𝐇 do not exhibit any strong alignments. For all results, the dependence on Reynolds number is very weak. In addition to the fundamental insights, our work provides useful data and validation benchmarks for future modeling endeavors, for instance in Lagrangian modeling of velocity gradient dynamics, where conditional 𝐇 is explicitly modeled.

The role of initial separation for inertial range scaling of turbulent two-particle dispersion is reassessed in light of recent results. (slightly expanded version of Comment published in Physical Review Letters).

The scaling of acceleration statistics in turbulence is examined by combining data from the literature with new data from well-resolved direct numerical simulations of isotropic turbulence, significantly extending the Reynolds number range. The acceleration variance at higher Reynolds numbers departs from previous predictions based on multifractal models, which characterize Lagrangian intermittency as an extension of Eulerian intermittency. The disagreement is even more prominent for higher-order moments of the acceleration. Instead, starting from a known exact relation, we relate the scaling of acceleration variance to that of Eulerian fourth-order velocity gradient and velocity increment statistics. This prediction is in excellent agreement with the variance data. Our Letter highlights the need for models that consider Lagrangian intermittency independent of the Eulerian counterpart.

An efficient approach for extracting 3D local averages in spherical subdomains is proposed and applied to study the intermittency of small-scale velocity and scalar fields in direct numerical simulations of isotropic turbulence. We focus on the inertial-range scaling exponents of locally averaged energy dissipation rate, enstrophy and scalar dissipation rate corresponding to the mixing of a passive scalar $\theta$ in the presence of a uniform mean gradient. The Taylor-scale Reynolds number $R_\lambda$ goes up to $1300$, and the Schmidt number $Sc$ up to $512$ (albeit at smaller $R_\lambda$). The intermittency exponent of the energy dissipation rate is $\mu \approx 0.23$, whereas that of enstrophy is slightly larger; trends with $R_\lambda$ suggest that this will be the case even at extremely large $R_\lambda$. The intermittency exponent of the scalar dissipation rate is $\mu_\theta \approx 0.35$ for $Sc=1$. These findings are in essential agreement with previously reported results in the literature. We further show that $\mu_\theta$ decreases monotonically with increasing $Sc$, either as $1/\log Sc$ or a weak power law, suggesting that $\mu_\theta \to 0$ as $Sc \to \infty$, reaffirming recent results on the breakdown of scalar dissipation anomaly in this limit.

Building upon the intrinsic properties of Navier-Stokes dynamics, namely the prevalence of intense vortical structures and the interrelationship between vorticity and strain rate, we propose a simple framework to quantify the extreme events and the smallest scales of turbulence. We demonstrate that our approach is in excellent agreement with the best available data from direct numerical simulations of isotropic turbulence, with Taylor-scale Reynolds numbers up to 1300. We additionally highlight a shortcoming of prevailing intermittency models due to their disconnection from the observed correlation between vorticity and strain. Our work accentuates the importance of this correlation as a crucial step in developing an accurate understanding of intermittency in turbulence.

An efficient approach for extracting 3D local averages in spherical subdomains is proposed and applied to study the intermittency of small-scale velocity and scalar fields in direct numerical simulations of isotropic turbulence. We focus on the inertial-range scaling exponents of locally averaged energy dissipation rate, enstrophy and scalar dissipation rate corresponding to the mixing of a passive scalar θ in the presence of a uniform mean gradient. The Taylor-scale Reynolds number R λ goes up to 1300, and the Schmidt number Sc up to 512 (albeit at smaller R λ ). The intermittency exponent of the energy dissipation rate is µ ≈ 0 . 23, whereas that of enstrophy is slightly larger; trends with R λ suggest that this will be the case even at extremely large R λ . The intermittency exponent of the scalar dissipation rate is µ θ ≈ 0 . 35 for Sc = 1. These findings are in essential agreement with previously reported results in the literature. We further show that µ θ decreases monotonically with increasing Sc , either as 1 / log Sc or a weak power law, suggesting that µ θ → 0 as Sc → ∞ , reaffirming recent results on the breakdown of scalar dissipation anomaly in this limit.

Intense fluctuations of energy dissipation rate in turbulent flows result from the self-amplification of strain rate via a quadratic nonlinearity, with contributions from vorticity (via the vortex stretching mechanism) and pressure-Hessian-which are analysed here using direct numerical simulations of isotropic turbulence on up to [Formula: see text] grid points, and Taylor-scale Reynolds numbers in the range 140-1300. We extract the statistics involved in amplification of strain and condition them on the magnitude of strain. We find that strain is self-amplified by the quadratic nonlinearity, and depleted via vortex stretching, whereas pressure-Hessian acts to redistribute strain fluctuations towards the mean-field and hence depletes intense strain. Analysing the intense fluctuations of strain in terms of its eigenvalues reveals that the net amplification is solely produced by the third eigenvalue, resulting in strong compressive action. By contrast, the self-amplification acts to deplete the other two eigenvalues, whereas vortex stretching acts to amplify them, with both effects cancelling each other almost perfectly. The effect of the pressure-Hessian for each eigenvalue is qualitatively similar to that of vortex stretching, but significantly weaker in magnitude. Our results conform with the familiar notion that intense strain is organized in sheet-like structures, which are in the vicinity of, but never overlap with tube-like regions of intense vorticity due to fundamental differences in their amplifying mechanisms. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Passive scalars advected by three-dimensional Navier-Stokes turbulence exhibit a fundamental anomaly in odd-order moments because of the characteristic ramp-cliff structures, violating small-scale isotropy. We use data from direct numerical simulations with grid resolution of up to 8192^{3} at high Péclet numbers to understand this anomaly as the scalar diffusivity, D, diminishes, or as the Schmidt number, Sc=ν/D, increases; here ν is the kinematic viscosity of the fluid. The microscale Reynolds number varies from 140 to 650 and Sc varies from 1 to 512. A simple model for the ramp-cliff structures is developed and shown to characterize the scalar derivative statistics very well. It accurately captures how the small-scale isotropy is restored in the large-Sc limit, and additionally suggests a possible correction to the Batchelor length scale as the relevant smallest scale in the scalar field.