The T-shaped mixer is widely used in fundamental studies of chemical engineering. Its transitional regime is well understood, whereas the turbulent dynamics has received scarce attention so far. Here, we perform direct numerical simulations of the turbulent regime for Reynolds numbers up to $\textit{Re}=2000$ at Schmidt number $\textit{Sc}=1$. Our analysis reveals two distinct stages along the mixing channel prior to relaxation toward duct flow. Near the junction, a jet-like flow forms and exhibits the approximately self-similar behaviour of transitional planar jets. Subsequently, a decay region characterised by power-law decay of turbulent kinetic energy, dissipation and scalar variance emerges. For the velocity field, the observed exponents are consistent with those of decaying turbulence in bounded domains, whereas the scalar-variance exponent is consistent with that of unbounded turbulence. We argue that this apparent discrepancy is a consequence of the mixing process progressing from the centre of the channel toward the sidewalls in the decay region, while turbulence already fills the channel cross-section entirely at the end of the jet region.The time-averaged mixing state presents error-function profiles of the scalar in the transverse direction, similar to the laminar cases, and is quantified here through a streamwise evolving effective diffusion coefficient.

In high-Reynolds-number wall-bounded flows, the inner-scaled wall-pressure variance is often represented as a logarithmic increase with frictional Reynolds number. We consider the two sources of the incompressible pressure–Poisson equation: a linear (rapid) term linked to mean shear and a nonlinear (slow) term composed of quadratic velocity fluctuations. This paper establishes a link between the sources and the coefficients in a logarithmic inner-scaled variance representation. To leading order we posit that the linear source provides a Reynolds-number-independent offset, while the nonlinear source contributes the logarithmic coefficient. The illustrative dataset is direct numerical simulation (DNS) at frictional Reynolds number $\delta ^+\approx 550$, although the principal contribution is the establishment of a mechanistic link to well-known high-$\delta ^+$ scalings of wall-bounded turbulence. Through consideration of the sources and the integral solution method of the Poisson equation, we find that the linear source contribution sits predominantly in the buffer layer and maps to the near-wall cycle. To leading order, this contribution becomes $\delta ^+$ invariant under inner scaling, thus contributing an offset in the logarithmic representation. The interfacial regions between uniform momentum zones characteristic of the inertial layer (vortical fissures) spatially localise strain and vorticity contributions and contain an increasingly large proportion of the strain and vorticity. We show that fissures act as a compact carrier for the source terms, with the nonlinear term especially prominent in these regions. By considering the inertial layer statistics, we link the changing nonlinear contribution to the $\ln \delta ^+$ growth, in agreement with previous empirical observations.

We investigate the exponential growth of uncertainty energy in three-dimensional (3-D) Navier–Stokes turbulence, emphasising the intermittent and highly localised amplification/production of uncertainty, a critical factor in understanding the predictability of turbulent systems. From the Navier–Stokes equations, one can identify some key fields contributing to the growth/decay of uncertainty-production term $P_{\varDelta }$: strain rate, vorticity and vortex deformation. The dynamics of these fields is examined in the $Q{-}R$ plane, where $Q$ and $R$ are the second and third invariants of the velocity gradient tensor, to understand their role in the evolution of uncertainty-production term $P_{\varDelta }$. We proceed by estimating committor functions across the entire spatiotemporal domain of the direct numerical simulation (DNS) of turbulence in a periodic domain at different Reynolds numbers. Our estimates of the probability of rare extreme events of the local uncertainty-production term as a function of uncertainty energy, strain rate, vorticity and vortex deformation confirm the role of strain rate in driving uncertainty. Where strain rate and vorticity are too close to their space-average values, stable probabilistic forecasts appear impossible solely based on the fields considered here.

Integral identities relating the skin-friction coefficients and the Reynolds shear stresses in annular cylindrical flows are derived. In the pressure-driven case, an identity for the radial location of the maximum streamwise velocity is obtained and, in the cylinder-driven case, an identity for the bulk velocity is found. The formulas are used to analyse existing numerical and experimental data and simplify to classical channel-flow and pipe-flow identities in the limiting cases of vanishingly small and infinitely large radius ratios.

In this work, the characteristics of intense streamwise velocity and temperature fluctuation structures, including spatial and temporal properties, are investigated in compressible turbulent boundary layer flows at Mach number 2.25. To resolve the evolution process and capture large-scale structures that advect over long distances, two sets of flow field snapshots for each case, with varying domain lengths and temporal resolutions, are stored, amounting to up to 50 TB. Regions with high fluctuations are identified as structures using a clustering method, and connections between structures in consecutive flow fields are established, forming a large graph that reveals the complete temporal evolution relationship. Spatial characteristics are analysed, providing evidence for self-similarity in the logarithmic layer at this Mach number. Extra plate-like temperature structures show up extremely near the wall in the quasi-adiabatic case and their variation with different wall conditions are summarised. Conditional average associated with negative velocity fluctuations structures are also conducted, revealing a distinct lift-up of the temperature structures. Furthermore, temporal characteristics are examined, providing a foundational understanding of the structures’ kinematic behaviours. Positive velocity structures and negative temperature structures are advected faster than their corresponding counterparts with opposite signs, but all are quite close to the mean streamwise velocity except in the near-wall region. The lifetime of structures in the logarithmic layer exhibits an almost linear relationship with their height scale. Finally, the temporal and spatial evolution of the structures is described, offering a perspective to probe structure characteristics at any stage of their life cycle.

We study bidirectional one-dimensional (1-D) shallow-water waves within a class of Boussinesq equations, including the integrable Kaup–Boussinesq (KB) equation and a truncated-dispersion variant, which serves as a representative non-integrable model. For these two systems, the normal-form transformation yields an interaction coefficient of the same general structure, differing only through the dispersion relation. We derive this coefficient and numerically confirm that it vanishes on the resonant manifold for the KB equation, as expected from integrability. In contrast, the non-integrable model admits a non-vanishing interaction coefficient consistent with broken integrability, producing a non-trivial wave kinetic equation (WKE), which is the first known in a 1-D shallow-water setting. The resulting WKE is non-homogeneous in nature due to the non-homogeneity of the corresponding dispersion relation; however, approximate Kolmogorov–Zakharov solutions can be derived in a novel way under certain approximations. Numerical experiments in two settings validate the kinetic predictions and elucidate the underlying dynamics: (i) in free-evolution cases of the KB equation, despite complete integrability and the invariance of the discrete nonlinear spectrum guaranteed by isospectrality, an initial arbitrary wavenumber spectrum undergoes substantial evolution driven by quasi-resonant triad interactions and (ii) in forced-dissipated cases of the non-integrable equation, we find stationary power-law spectra that agree with the theoretical predictions.

To investigate the spanwise length scales of turbulent motions essential for sustaining near-wall turbulence, we perform numerical experiments in direct numerical simulations of full-scale turbulent channel flows by filtering out motions wider than a prescribed cutoff wavelength $\lambda _{zf}$. As $\lambda _{zf}$ decreases, near-wall short vortical structures are suppressed, while long streaks are amplified. When $\lambda _{zf}$ is reduced to approximately 200 wall units, the flow relaminarises, indicating that at least turbulent motions twice those of near-wall coherent structures are necessary. To elucidate the underlying mechanism, we derive transport equations for Reynolds shear stresses associated separately with long streaks and short vortical structures, and decompose nonlinear transport terms into contributions from spanwise-wide motions, long streaks, short vortical structures and their interactions. Spanwise-wide motions enhance the energy production of short vortical structures by supplying energy, via nonlinear interactions, to vertical velocity fluctuations; thus, their removal suppresses these structures. Conversely, nonlinear interactions involving spanwise-wide motions act as a major energy sink for long streamwise fluctuations, so their absence amplifies streaks. However, long streaks ultimately depend on short vortical structures, as their energy is extracted from the mean shear by vertical velocity fluctuations, which draw energy from spanwise velocity fluctuations sustained by nonlinear interactions involving both short structures and spanwise-wide motions. Therefore, when short structures are strongly suppressed at sufficiently small $\lambda _{zf}$, the flow relaminarises. These findings highlight the central role of short vortical structures and their dependence on ‘inactive’ spanwise-wide turbulent motions, and reveal multiple interconnected energy cycles governing near-wall coherent structures.

This study extends the data-driven quasilinear approximation (DQLA) (Holford, Lee & Hwang 2024 J. Fluid Mech., vol. 980, A12) to compressible turbulent channel flow. The DQLA employs the eddy viscosity enhanced linearised compressible Navier–Stokes operator (Chen et al. 2023 J. Fluid Mech., vol. 962, A7), driven by stochastic forcing whose streamwise weights are determined through self-similarity assimilated from an incompressible direct numerical simulation (DNS) database, and spanwise weights are obtained by minimising discrepancies in the Reynolds stresses between the mean and the fluctuation equations. Without any compressible DNS input, the extended DQLA reproduces turbulence intensities and energy spectra in close quantitative agreement with DNS up to bulk Mach number $\textit{Ma}_b=1.5$, and exhibits a consistent collapse of turbulence statistics across Mach numbers when expressed in semilocal units at the same centreline semilocal friction Reynolds number $\textit{Re}_{\tau c}^*$. This collapse is largely inherited from the scaling properties of the mean flow and is preserved through DQLA, consistent with Morkovin’s hypothesis. At higher Mach number ($\textit{Ma}_b=3.0$), systematic deviations emerge, reflecting the limitations of the present modelling assumptions, particularly the neglect of nonlinear terms associated with density fluctuations. The results demonstrate the predictive capability of DQLA up to moderate Mach numbers, and establish its potential as a physically interpretable and computationally efficient framework for exploring compressible wall-bounded turbulence at high Reynolds numbers.

The Kolmogorov’s refined similarity hypothesis (RSH), which plays a fundamental role in studying turbulence intermittency, establishes a connection between locally averaged energy dissipation rates and statistics of velocity increments. This work extends the RSH framework from hydrodynamic (HD) turbulence to magnetohydrodynamic (MHD) turbulence. In analogy to the form of the RSH in HD turbulence and the third-order law in MHD turbulence (Politano & Pouquet, Geophys. Res. Lett., vol. 25, 1998, pp. 273–276), simple and mixed forms of RSH in MHD turbulence are proposed. These two forms are systematically verified and compared with datasets from direct numerical simulations. The results reveal that the simple form satisfies the predictions of Kolmogorov’s RSH, with respect to the scaling exponents of the local variables and the probability density functions (PDFs) of the Elsässer increments. The range of the local Reynolds number over which the RSH is satisfied is wider for the simple form than for the mixed form. The PDFs of the normalised Elsässer increments reasonably follow Gaussian and Rayleigh distributions for the simple and mixed forms, respectively. The effects of the external forcing and mean magnetic fields are studied, and these do not alter the performance of the two forms. However, cross-helicity can change the RSH constant for the simple form, while this constant for the mixed form does not vary. This study advances understanding of intermittency, multifractality, anomalous scaling and cascade dynamics in MHD turbulence.

We investigate the equal-height, two-dimensional second-order structure function of the streamwise velocity fluctuation $u$ in turbulent channel flow. Specifically, at a given wall-normal location $z$ we consider $S_2(r_x,r_y;z)=\langle [u(x+r_x,y+r_y,z)-u(x,y,z)]^2\rangle$, where $(r_x,r_y)$ are the streamwise and spanwise separations in a wall-parallel plane and $\langle \boldsymbol{\cdot }\rangle$ denotes wall-parallel spatial averaging. Focusing on separations within the logarithmic region, we build on Townsend’s attached-eddy hypothesis and its hierarchical random additive-process formulation to introduce an attached-eddy elliptic approximation, in which the two-point geometry is represented by an elliptic in-plane footprint. The model yields a compact prediction for $S_2(r_x,r_y;z)$: its one-dimensional projections recover the canonical logarithmic forms with an asymptotically common slope, and its iso-$S_2$ contours at fixed $z$ are predicted to be self-similar ellipses whose semi-axes scale linearly with $z$ while maintaining an approximately constant aspect ratio (with mild finite-Reynolds-number effects). These predictions are assessed using direct numerical simulations at friction Reynolds numbers $Re_\tau =1000$ and $5200$, where $Re_\tau =u_\tau \delta /\nu$, and $u_\tau$ is the friction velocity, $\nu$ is the kinematic viscosity and $\delta$ is the channel half-height. When expressed in terms of $r_i/z$ ($i=x,y$), the projected structure functions collapse across wall-normal locations and vary linearly with $\ln (r_i/z)$. At fixed $z$, the iso-$S_2$ contours in the $(r_x,r_y)$ plane are well described by ellipses characterised by a single anisotropy parameter $R=b/a$, where $a$ and $b$ are the ellipse semi-axes in the $r_x$ and $r_y$ directions, respectively. A simple level-dependent parameterisation of $a$ and $b$ provides an accurate and parsimonious reconstruction of the two-dimensional fields. Collectively, these results establish a geometrically transparent and quantitatively predictive framework for the two-dimensional organisation of wall-attached motions in the logarithmic region, extending attached-eddy concepts beyond traditional one-dimensional formulations.

The evolution of small perturbations applied to turbulent Couette flow is examined over a long time horizon, until the perturbed flow field becomes completely decorrelated from the original state. To elucidate the fundamental physical processes involved, we focus on the minimal flow unit, where the dynamics of coherent structures is well understood in terms of the self-sustaining process (Hamilton et al. J. Fluid Mech. vol. 287, 1995, pp. 317–348). As expected, in the short term, perturbations exhibit exponential growth governed by the leading Lyapunov exponent. This mechanism is driven by the streamwise-dependent flow, which is known to involve intense turbulent dissipation events within the self-sustaining process, consistent with previous findings. Beyond the initial exponential phase, we observe a slow, sustained growth of perturbations over a long period – spanning tens of integral time scales – before eventual saturation. During this stage, the perturbation energy increases approximately linearly with time. While this behaviour resembles observations and predictions in other turbulent flows, the underlying physical process here is fundamentally different. Specifically, the perturbation field during this period is dominated by streaky structures and the growth mechanism is linked to the saturation of the wall-normal streak length scale at the largest dimension permitted by the flow geometry (i.e. the channel height). Finally, an evaluation of the dominant production term components reveals that the well-known lift-up effect is primarily responsible for the growth of these streaky perturbations.

Estimation of instantaneous and statistical properties of wall-bounded turbulence across different Reynolds numbers has significant applications in scientific research and engineering, where the key challenge is to identify the Reynolds number invariance underlying the varying flow properties as Reynolds number changes. In this study, we develop a refined inner–outer interaction model (IOIM) that isolates the footprints of large-scale structures populating within a prescribed wall-normal range $y_{R}^+ \in [ y_{R1}^+ , y_{R2}^+ )$. When the reference layers are located within $ 80 \leqslant y_{R1}^+ \lt y_{R2}^+ \leqslant 0.3\textit{Re}_\tau$, the energy spectral densities of the isolated large-scale structures, including the wall-attached and wall-detached eddies, are found to be independent of Reynolds numbers with direct numerical simulations spanning $ \textit{Re}_\tau \sim O(10^2)$–$O(10^3)$. The spectral energy densities of the near-wall small-scale motions also show little Reynolds number dependence. The increase in near-wall turbulent kinetic energy with Reynolds number is explained from the identified Reynolds number invariance of coherent structures. Meanwhile, certain amplitude and scale modulation effects caused by large-scale structures are found to make the distribution patterns of fluctuation amplitudes and scales of near-wall structures more susceptible to Reynolds number influence. The findings of this study provide insights for the developments of predictive models for wall-bounded turbulence across different Reynolds numbers.

A new model of the mean temperature profiles within incompressible wall-bounded turbulence is developed based on a relationship between the momentum and thermal eddy diffusivities for a wide range of Prandtl numbers ($ \textit{Pr}$). Flow media with $ \textit{Pr}$ and Schmidt numbers ($ \textit{Sc}$) other than unity are of great engineering importance, and the proposed model for passive scalar mean profiles is able to suitably account for $ \textit{Pr}$ and $ \textit{Sc}$ variations and their effects. Existing models have a higher degree of error at the lower $ \textit{Pr}$ ranges due to significant inaccuracies of the modelled thermal eddy diffusivity in this region. Considering incompressible wall-bounded turbulence at $0.007 \leqslant \textit{Pr} \leqslant 10$, the proposed methodology reduces the average error to just around $4\,\%$ across all cases considered, lower than previously proposed models due to its ability to capture low $ \textit{Pr}$ behaviour, showcasing a new temperature–velocity relationship that can account for variations in $ \textit{Pr}$ for all the cases considered.

Obtaining accurate field statistics continues to be one of the major challenges in turbulence theory and modelling. From the various existing modelling approaches, multifractal models have been successful in capturing intermittency in velocity gradient and increment distributions. Moreover, superstatistical models from non-equilibrium statistical mechanics have shown the capacity to model probability density functions (p.d.f.s) of various statistical turbulent quantities as ensembles of simpler stochastic processes. Here, we present an approach that generates field statistics in the form of a characteristic functional by promoting a model for multifractal increment statistics to an ensemble of Gaussian fields. By carefully designing the correlation function and the corresponding weight of each subensemble, we are able to define a functional that exhibits multifractal two-point inertial-range and dissipation-range statistics, and that blends into realistic large-scale behaviour. Additionally, the method is capable of producing multifractal statistics with any of the widely used singularity spectra. We characterise the fidelity of our approach through comparisons to literature results from direct numerical simulations. Overall, our framework thereby bridges between three different perspectives: superstatistics, multifractals and functional approaches to turbulence.

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.

Turbulent pipe flow is of substantial importance in practical applications, and it remains challenging to depict the characteristic complex multiscale dynamics by a unified theoretical framework, hindered by its inherent intermittency. Inspired by a recent study of velocity circulation in turbulent channel flows from Duan, Chen & Sreenivasan (2025 J. Fluid Mech., vol. 1009, p. R4), in this study, we investigate the statistical characteristics of velocity circulation (or equally the area integral of wall-normal vorticity) over rectangular loops in concentric cylindrical shells, parallel to the pipe wall. The statistics are implemented using direct numerical simulation data at friction Reynolds numbers of $ \textit{Re}_\tau =1057$ and $2000$. Close to the pipe wall, the circulation in the inertial range resides on space-filling unifractal sets, with the Hölder exponent smaller than Kolmogorov’s $4/3$. Away from the pipe wall, the circulation displays bifractal characteristics and the Hölder exponents for high moment orders are very close to those reported in channel flows and homogeneous isotropic turbulence. The circulation statistics are only dependent on the area enclosed by the loops, and are invariant to the loop aspect ratio, once both edge lengths of the loops are in the inertial range.

We perform numerical simulations of forced homogeneous isotropic turbulence over a range of bulk viscosities, Reynolds numbers and Mach numbers to investigate the scaling of key flow statistics. Using the Helmholtz decomposition, we analyse the scalings of Favre-averaged turbulent kinetic energy (TKE), root-mean-square (r.m.s.) pressure, pressure dilatation, dilatational dissipation and higher-order velocity-gradient moments. Additionally, new models are proposed for the pressure-dilatation term and the bulk-viscosity dependence of dilatational dissipation. Although the solenoidal and dilatational components of the Favre-averaged TKE are not strictly orthogonal, our numerical results demonstrate that their ratio is well approximated by the squared ratio of the corresponding r.m.s. velocities. The r.m.s. pressure approaches the pseudo-sound scaling as bulk viscosity increases. Within the Donzis r.m.s. pressure model (Donzis & John 2020 Phys. Rev. Fluids 5(8), 084609), we find that the solenoidal contribution becomes dominant for large bulk viscosity. Pressure dilatation is found to depart systematically from pseudo-sound predictions: without bulk viscosity it favours transfer from kinetic to internal energy, while finite bulk viscosity can reverse this transfer at high Mach numbers. The scaling exponent of dilatational dissipation is shown to vary with bulk viscosity, enabling a corrected model for its exponent and prefactor. Velocity-gradient skewness and flatness reveal that the onset of shocklet-induced divergence is delayed with increasing bulk viscosity and may be suppressed entirely. The results extend recent velocity-ratio-based scaling frameworks and provide modelling insights into compressible turbulence.

In this work, the classical Prandtl relation for the skin-friction law of incompressible turbulent channel and pipe flows is generalised to compressible cases. Specifically, based on the law of the wall and asymptotic analysis, a skin-friction transformation is proposed to map the skin-friction law of compressible turbulent channel and pipe flows to the classical Prandtl relation. It has been theoretically proven that the skin-friction coefficient $C_{\!f,i}$ and the bulk Reynolds number $\textit{Re}_{b,i}$ for compressible turbulent channel and pipe flows, where the subscript $i$ denotes the transformed quantity obtained from the proposed skin-friction transformation, adhere to the Prandtl relation, expressed as $\sqrt {2/C_{\!f,i}}\propto \ln (\textit{Re}_{b,i}\sqrt {C_{\!f,i}/2})$. Moreover, it is quantitatively verified that the transformed $C_{\!f,i}$ and $\textit{Re}_{b,i}$, obtained from direct numerical simulations (DNS) of compressible turbulent channel flows with bulk Mach numbers ranging from $0.2$ to $4$, and friction Reynolds numbers from $200$ to $2000$, elegantly collapse into the Prandtl relation for the incompressible skin-friction law. Additionally, the transformed $C_{\!f,i}$ and $\textit{Re}_{b,i}$ from DNS of compressible turbulent pipe flows, with bulk Mach numbers ranging from $1.5$ to $3$, and friction Reynolds numbers from $200$ to $1000$, are also unified with the Prandtl relation for the incompressible skin-friction law.

We present computations of individual mode-to-mode energy transfers from direct numerical simulations of homogeneous isotropic turbulence. Unlike previous approaches based on shell-filtered velocity fields, this method distinguishes between the energy exchanged by each pair of modes within a triad. We introduce a potential function based on the energy content of the modes involved, and show that it predicts the distribution of intense energy transfers in the vicinity of the sampling mode considered. By performing simulations with forcing applied at intermediate wavenumbers, we demonstrate that the region of most intense transfers is determined by the spectral location of the energy-containing scales rather than by the local or non-local character of the triad. Direct energy exchanges with the energy-containing range are suppressed by geometric constraints from the divergence-free condition, but persist as residuals when the sampling mode is close to the energy-containing scales. The comparison with an estimator derived from eddy-damped quasi-normal Markovian theory shows good agreement and recovers the forward, scale-local nature of energy transfer consistent with the cascade picture.

Similarities and differences between Kolmogorov scale-by-scale equilibria/non-equilibria for velocity and scalar fields are investigated in the intermediate layer of a fully developed turbulent channel flow with a passive scalar/temperature field driven by a uniform heat source. The analysis is based on intermediate asymptotics and direct numerical simulations at different Prandtl numbers lower than unity. Similarly to what happens to the velocity fluctuations, for the fluctuating scalar field Kolmogorov scale-by-scale equilibrium is achieved asymptotically around a length scale $r_{\textit{min}}$, which is located below the inertial range. The length scale $r_{\textit{min}}$ and the ratio between the inter-scale transfer and dissipation rates at $r_{\textit{min}}$ vary following power laws of the Prandtl number, with exponents determined by matched asymptotics based on the hypothesis of homogeneous two-point physics in non-homogeneous turbulence. The inter-scale transfer rates of turbulent kinetic energy and passive scalar variance are globally similar but show evident differences when their aligned/anti-aligned contributions are considered.