We analyse velocity spectra from direct numerical simulations (DNS) of turbulent pipe flow up to a friction Reynolds number $ \textit{Re}_{\tau } \approx 12{\,}000$. The spectral maps naturally partition into four regions, corresponding to small- and large-scale motions, and to wall-attached eddies, and are further divided into near-wall and outer contributions. Physically grounded models are formulated for each region, with particular emphasis on the near-wall range beneath the wall-attached eddies. Integration of the spectral models yields the distribution of the streamwise velocity variance in the lower part of the logarithmic layer, which agrees well with the DNS data. The analysis shows that the emergence of an outer-layer peak in the velocity variance at sufficiently high Reynolds number arises from the increasing relative contribution of wall-attached eddies, and can be interpreted without invoking additional physical mechanisms beyond this classical framework. Extrapolation to extremely high Reynolds numbers suggests a saturation of the velocity variance, characterised by two peaks of comparable magnitude.

Flow structures directly responsible for local unsteady skin-friction generation in turbulent channel flows are identified using the newly developed canonical correlation decomposition (CCD) method. The dominant structures take the form of streamwise streaks that are spanwise-localised around the position where the skin friction is targeted and exhibit progressively shorter streamwise extents as the mode number increases. The resulting CCD spectrum shows a clear low-rank behaviour; flow reconstruction using only the first 4 CCD modes recovers more than 80 % of the examined skin-friction variation, in contrast to inefficient skin-friction reconstructions using other correlation-based methods. When the opposition control technique is used to reduce drag, the application of CCD shows that drag reduction is achieved by lifting the original streak structures and generating thinner and shorter streaks with opposite phases underneath. These findings demonstrate that CCD isolates the causally relevant flow structures governing the turbulent skin-friction generation and modification, which is expected to find use in various drag control applications in wall-bounded turbulence.

The addition of polymers to turbulent pipe flows induces significant drag reduction and fundamentally modifies turbulent flow structures. This study presents a fractal dimension analysis of polymeric turbulent pipe flows using velocity fields captured via two-dimensional particle image velocimetry in the streamwise-radial plane. Two experimental datasets were generated: one by varying the polymer concentration at a constant Reynolds number ($\textit{Re}$) and another by varying $\textit{Re}$ at a fixed polymer concentration. Friction factors were measured concurrently to quantify the extent of drag reduction. The two-dimensional fractal dimension was evaluated for isosurfaces of turbulent kinetic energy. While Newtonian turbulence exhibits a nearly constant fractal dimension at length scales exceeding a critical threshold, the introduction of polymers causes the fractal dimension to decrease monotonically with increasing concentration. Conversely, the fractal dimension remains insensitive to changes in the Reynolds number. The ratio of the critical length scale to the Kolmogorov scale varies according to both $\textit{Re}$ and polymer concentration; however, this scale ratio becomes independent of both parameters once the maximum drag reduction asymptote is reached. Spatial analysis of the one-dimensional fractal dimension across radial positions helps to further reveal the evolution of turbulence fractality. The results demonstrate that while flow inertia promotes the formation of space-filling structures, viscoelastic effects smooth these structures and transition them towards sheet-like or linear geometries. Finally, the correlation between the fractal dimension and turbulence intermittency is discussed.

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.

Turbulence-induced friction is a significant contributor to energy consumption in the fluid-transport and piping industries. Here, we describe a passive approach to suppress turbulence and reduce friction: we show that a local increase in streamwise flow curvature, combined with changing a circular cross-section to an oval, relaminarises turbulent flow in curved pipes. We exemplify this effect in a $180^{\circ }$ bend at ${\textit{Re}}_D=10\,000$ and $20\,000$ (based on bulk velocity $U_B$ and pipe diameter $D$), well above the limit for sustained turbulence in straight pipes and the linear stability limit in $180^{\circ }$ bends. Curvature inhibits streamwise Reynolds stresses, and cross-sectional modifications weaken the unstable secondary flow, together disrupting the near-wall regeneration cycle and collapsing turbulence. Simulations and experiments confirm that these geometric modifications suppress turbulence and reduce pressure loss by 53 % and 36 % compared with the baseline $180^{\circ }$ bend and a fully developed straight pipe of equal length, respectively. The results establish a passive, mechanism-based route to relaminarisation in curved pipes with implications for energy-efficient control in other wall-bounded flows with curvature.

The anomalous scaling of passive scalar fluctuations is experimentally investigated in turbulent pipe flow with a Taylor-microscale Péclet number of $\mathcal{O}(10^5)$, where the turbulence is known to deviate from the homogeneous isotropic turbulence. The scalar structure functions and intermittency in the mixing are examined. The experimental results consolidate that the scaling exponents of scalar structure functions saturate at high-order even moments, evidenced previously in homogeneous isotropic turbulence with a Taylor-microscale Péclet number of $\mathcal{O}(10^3)$. The saturation scaling exponent decreases to approach unity as the Taylor-microscale Péclet number increases. This saturation scaling exponent is further corroborated by the fractal codimension of sharp scalar fronts.

Planar particle image velocimetry (PIV) measurements were conducted to investigate turbulent flows through a square duct roughened by transverse rectangular ribs of four blockage ratios (${\textit{Br}}=0.1$, 0.15, 0.2 and 0.25) at a bulk Reynolds number of ${\textit{Re}}_b = 9400$. In contrast to the classical two-dimensional (2-D) rib-roughened boundary-layer flows, the turbulent flow studied here is intrinsically three-dimensional (3-D) and inhomogeneous, complicated by not only the internal shear layers (ISLs) triggered by the rib crests, but also the intense interaction of the four boundary layers developing over duct sidewalls. It is observed that turbulent motions near the rib crest are mainly dominated by the ejection and sweep events. As the blockage ratio increases, the magnitudes of Reynolds stresses near the rib crest increase significantly attributed to enhanced sweep events and large-scale flapping motions. The results of temporal auto-correlations and spatial two-point auto-correlations show that both temporal and spatial integral scales of turbulence structures are dominated by the streamwise velocity fluctuations, which increase as the rib height increases. Based on proper orthogonal decomposition (POD) analyses, it is interesting to observe that the ISL near the rib crest is dominated by both the low- and high-frequency flapping motions characteristic of the first POD mode.

Direct numerical simulations of turbulence in a flexible pipe with imposed standing-wave vibration are performed to reveal the flow dynamics inside an oscillating pipe. We choose the parameters of standing-wave vibration with small amplitude as the most unstable mode in flow-induced free vibration. The flow is driven under the condition of constant mass flow rate, with the bulk Reynolds number, based on the bulk velocity and pipe diameter, being ${\textit{Re}}_b$ = 5300. In response to the imposed vibration, the evolution of the flow inside manifests obvious space–time-dependent characteristics. Specifically, the streamwise velocity fluctuation is enhanced downstream of the crest – the convex region on the internal pipe wall – an event often accompanied by localised flow separation. Meanwhile, the two other components of velocity fluctuation are augmented downstream of the trough – the concave region of the wall’s sinusoidal undulation. This is attributed to the wall deformation, which forces a redistribution of turbulent kinetic energy among the components. The latter process gives rise to a high-level fluctuation of wall shear stresses, leading to the intermittent variation of the drag force in that region. In addition, secondary flow emerges in the form of a typical counter-rotating vortex pair due to the bending of pipe, with the vortex cores located near the wall. The temporal variation of the magnitude of secondary flow lags slightly behind the pipe vibration and its maximum occurs closer to the node where the pipe displacement is consistently zero. Moreover, the secondary flow intensity increases with the increasing of steepness and a slight drag reduction can be achieved with relatively low-wavenumber vibration.

Triply periodic minimal surfaces (TPMS)-based media (a type of metamaterial) are defined by mathematical expressions, which are amenable to additive manufacturing, and are finding increasing practical applications owing to their porous nature. We present experimental pressure drop measurements for a range of velocities spanning laminar to turbulent regimes for three TPMS geometries – gyroid, primitive and body-centred cubic (BCC) – with different porosity, unit cell length and surface finish. Dimensional Darcy and Forchheimer permeabilities are estimated via quadratic fitting for the gyroid geometry, which closely resembles random packed porous media. Subsequently, the non-dimensional drag (${\kern-0.5pt}f$) is plotted against Reynolds number ($Re$) yielding distinct curves for each case. The lack of collapse stems from varying definitions of pore diameter, complicating comparisons across porous media (not just TPMS). Therefore, a method is developed to estimate an equivalent hydraulic diameter $d_{{H\hbox{-}\textit{equ}}}$ from pressure drop data by matching the laminar drag $f$ of packed spheres via the Ergun equation, allowing the collapse of all porous media $f-Re$ curves in the laminar regime. The value of $d_{ {H\hbox{-}\textit{equ}}}$ is related to the ‘true’ Darcy permeability defined strictly in the linear regime (unlike permeability from quadratic fitting). We observe an approximate linear relationship between the $d_{ {H\hbox{-}\textit{equ}}}$ and the hydraulic diameter for self-similar TPMS configurations. The common basis of $d_{ {H\hbox{-}\textit{equ}}}$ allows intercomparison of TPMS geometries, and shows that BCC achieves significant drag reduction compared with packed spheres in the turbulent regime partially because of their open tube-like structure, whereas some configurations show drag increase. Although gyroid can be represented using the traditional quadratic drag law, primitive and BCC show an increase in $f$ with increasing $Re$ immediately before transitioning to fully turbulent regime – akin to rough-wall pipe flows, likely owing to their periodic streamwise elongated open structures.

This paper investigates the transient characteristics of uniform momentum zones (UMZs) in a rapidly accelerating turbulent pipe flow using direct numerical simulation datasets starting from an initial friction Reynolds number ($Re_{\tau 0}) = 500$ up to a final friction Reynolds number ($Re_{\tau 1}) = 670$. Instantaneous UMZs are identified following the identification methodology proposed by Adrian et al. (2000 J. Fluid Mech. vol. 422, pp. 1–54). The present results reveal that, as the flow rapidly accelerates, the average number of UMZs drops. However, as the flow recovers, it is regained. This result is complemented by the temporal evolution of the average number of internal shear layers. The temporal evolution of UMZs reveals that UMZs sustain their hierarchical flow arrangement with slower zones near the wall and faster zones away from the wall throughout the rapid turbulent flow acceleration. The results show that UMZs speed up during the inertial and pre-transition phases, and progressively slow down during the transition and core-relaxation stages. It is also revealed that UMZs near the wall respond first to flow instability and show earlier signs of recovery based on UMZ kinematic results. Finally, the dominant quadrant behaviour of Reynolds shear stress within UMZs has been investigated. It is found that, prior to the flow excursion, the UMZs nearest to the wall are always $Q2$ dominated, while the rest of the UMZs are always $Q4$ dominated. This behaviour is detected to not change during and after the flow excursion, suggesting that this is a characteristic behaviour of UMZs in accelerating turbulent wall-bounded flows.

The convection velocity in high-Reynolds-number pipe flow was investigated using two-point correlations obtained from two laser Doppler velocimetry systems. The Reynolds number ranged from ${\textit{Re}}_{{\tau}}=3000$ to 20 800, and profiles were obtained from $y/R=0.002$ up to the pipe centre, where $R$ is the pipe radius. This study examines the scaling behaviour of convection velocity profiles derived from raw velocity signals, and the convection characteristics of very large-scale motions (VLSMs) and large-scale motions extracted via scale-separated or time-resolved velocity signals. The profiles show that convection velocities from raw signals exceed the local mean velocity near the wall and gradually approach it toward the centre. These profiles can be scaled using inner variables, namely $y^+$ and $\Delta x^+$, where $\Delta x^+$ represents the measurement distance. Scale-separated convection velocities for VLSM-scale structures – defined as those larger than $5R$ – were higher than the unfiltered values and remained nearly constant up to $y^+ \leq 2000$ at ${\textit{Re}}_{{\tau}} \approx 20\,000$. In this constant region, the convection velocity of VLSMs scaled well with the bulk velocity $U_{\textit{b}}$, taking values of approximately $0.85U_{\textit{b}}$. Furthermore, analysis of the time-resolved data highlights that, when applying Taylor’s frozen turbulence hypothesis, it is essential to consider both the scale dependence and the temporal fluctuations of the convection velocity, which reflect the underlying spatio-temporal dynamics of the flow structures. The present study provides valuable data for discussions on converting frequency-domain measurements into wavenumber space using Taylor’s hypothesis.

Pulsatile fluid flows through straight pipes undergo a sudden transition to turbulence that is extremely difficult to predict. The difficulty stems here from the linear Floquet stability of the laminar flow up to large Reynolds numbers, well above experimental observations of turbulent flow. This makes the instability problem fully nonlinear and thus dependent on the shape and amplitude of the flow perturbation, in addition to the Reynolds and Womersley numbers and the pulsation amplitude. This problem can be tackled by optimising over the space of all admissible perturbations to the laminar flow. In this paper, we present an adjoint optimisation code, based on a GPU implementation of the pseudo-spectral Navier–Stokes solver nspipe, which incorporates an automatic, optimal checkpointing strategy. We leverage this code to show that the flow is susceptible to two distinct instability routes: one in the deceleration phase, where the flow is prone to oblique instabilities, and another during the acceleration phase with similar mechanisms as in steady pipe flow. Instability is energetically more likely in the deceleration phase. Specifically, localised oblique perturbations can optimally exploit nonlinear effects to gain over nine orders of magnitude in energy at a peak Reynolds number of ${\textit{Re}}_{\textit{max}}\approx 4000$. These oblique perturbations saturate into regular flow patterns that decay in the acceleration phase or break down to turbulence depending on the flow parameters. In the acceleration phase, optimal perturbations are substantially less amplified, but generally trigger turbulence if their amplitude is sufficiently large.

Linearly stable shear flows first transition to turbulence in the form of localised patches. At low Reynolds numbers, these turbulent patches tend to suddenly decay, following a memoryless process typical of rare events. How far in advance their decay can be forecasted is still unknown. We perform massive ensembles of simulations of pipe flow and a reduced-order model of shear flows (Moehlis et al. 2004 New J. Phys. vol. 6, issues 1, p. 56) and determine the first moment in time at which decay becomes fully predictable, subject to a given magnitude of the uncertainty on the flow state. By extensively sampling the chaotic sets, we find that, as one goes back in time from the point of inevitable decay, predictability degrades at greatly varying speeds. However, a well-defined (average) rate of predictability loss can be computed. This rate is independent of the uncertainty and also of the type of rare event, i.e. it applies to decay and to other extreme events. We leverage our databases to define thresholds that approximately separate phase-space regions of distinct decay predictability. Our study has implications for the development of predictive models, in particular it sets their theoretical limits. It also opens avenues to study the causes of extreme events in turbulent flows: a state which is predictable to produce an extreme event is causal to it from a probabilistic perspective.

We analyse the long-time dynamics of trajectories within the stability boundary between laminar and turbulent square duct flow. If not constrained to a symmetric subspace, the edge trajectories exhibit a chaotic dynamics characterised by a sequence of alternating quiescent phases and intense bursting episodes. The dynamics reflects the different stages of the well-known near-wall streak–vortex interaction. Most of the time, the edge states feature a single streak with a number of flanking vortices attached to one of the four surrounding walls. The initially straight streak undergoes a linear instability and eventually breaks in an intense bursting event. At the same time, the downstream vortices give rise to a new low-speed streak at one of the neighbouring walls, thereby causing the turbulent activity to ‘switch’ from one wall to the other. If the edge dynamics is restricted to a single or twofold mirror-symmetric subspace, the bursting and wall-switching episodes become self-recurrent in time, representing the first periodic orbits found in square duct flow. In contrast to the chaotic edge states in the non-symmetric case, the imposed symmetries enforce analogue bursting cycles to simultaneously appear at two parallel opposing walls in a mirror-symmetric configuration. Both the localisation of turbulent activity to one or two walls and the wall-switching dynamics are shown to be common phenomena in marginally turbulent duct flows. We argue that such episodes represent transient visits of marginally turbulent trajectories to some of the edge states detected here.

Space–time correlations of velocity and high-Schmidt-number ($Sc \approx 2000$) passive scalar fields are investigated in turbulent pipe flow using particle image velocimetry and planar laser-induced fluorescence, respectively. Both the velocity and scalar fields exhibit characteristic elliptical patterns in their respective space–time correlations. The elliptic approximation model, originally developed for the velocity field, is applied to estimate convection and sweeping velocities for both fields. In both fields, the convection velocity decreases, while the sweeping velocity increases, along the pipe radius. The convection velocity ratio between the scalar and velocity fields shows that high-Schmidt-number scalar fluctuations are advected faster than the velocity fluctuations. Similarly, the sweeping velocity of the scalar fluctuations is found to be larger than that of the velocity fluctuations. Furthermore, the high-Schmidt-number scalar is found to decorrelate more rapidly than the corresponding velocity, with the scalar Taylor microscale distinctly smaller than the velocity Taylor microscale.

Turbulence accounts for most of the energy losses associated with the pumping of fluids in pipes. Pulsatile drivings can reduce the drag and energy consumption required to supply a desired mass flux, when compared with steady driving. However, not all pulsation waveforms yield reductions. Here, we compute drag- and energy-optimal driving waveforms using direct numerical simulations and a gradient-free black-box optimisation framework. Specifically, we show that Bayesian optimisation is vastly superior to ordinary gradient-based methods in terms of computational efficiency and robustness, due to its ability to deal with noisy objective functions, as they naturally arise from the finite-time averaging of turbulent flows. We identify optimal waveforms for three Reynolds numbers and two Womersley numbers. At a Reynolds number of $8600$ and a Womersley number of 10, optimal waveforms reduce total energy consumption by 22 % and drag by 37 %. These reductions are rooted in the suppression of turbulence prior to the acceleration phase, the resulting delay in turbulence onset, and the radial localisation of turbulent kinetic energy and production towards the pipe centre. Our results pinpoint that the predominant, steady operation mode of pumping fluids through pipes is far from optimal.

This study considers the global instability of unidirectional flows through single, and double, bifurcation models using linear stability and direct numerical simulation (DNS). The motivation is respiratory flows, so we consider flow in both directions, through two geometries. We identify conditions (quantified by the Reynolds number, ${Re}=U^*D/\nu$, where $U^*$ is the peak centreline velocity, $D$ is the primary pipe diameter and $\nu$ is the kinematic viscosity) where temporal fluctuations occur using DNS. We calculate the linear stability of the steady flows, identifying the critical Reynolds number and leading unstable modes. For flows from single to double pipe, the critical Reynolds number is dependent on the number of bifurcations in the domain, but the mode structures are similar, with growth observed in regions dominated by longitudinal vortices formed by the centrifugal imbalance of flows passing through curved bifurcations. Flows in the opposite direction, from double to single pipe, also depend on the number of bifurcations in the domain. The flow through the double-bifurcation case undergoes two spatial symmetry-breaking bifurcations, altering the mode structure and critical Reynolds number. In all cases, the critical Reynolds number closely matches with temporal fluctuations observed from DNS, suggesting transition is the result of a linear instability, similar to other curved geometries like toroidal and helical pipes. We compare the frequencies of the modes with the frequencies observed from DNS, finding a close match during both initial and saturated flows. These results are important for understanding respiratory flows where turbulent mixing and streaming contribute to gas transport.

We consider the efficiency of turbulence, a dimensionless parameter that characterises the fraction of the input energy stored in a turbulent flow field. We first show that the inverse of the efficiency provides an upper bound for the dimensionless energy injection in a turbulent flow. We analyse the efficiency of turbulence for different flows using numerical and experimental data. Our analysis suggests that efficiency is bounded from above, and, in some cases, saturates following a power law reminiscent of phase transitions and bifurcations. We show that for the von Kármán flow the efficiency saturation is insensitive to the details of the forcing impellers. In the case of Rayleigh–Bénard convection, we show that within the Grossmann and Lohse model, the efficiency saturates in the inviscid limit, while the dimensionless kinetic energy injection/dissipation goes to zero. In the case of pipe flow, we show that saturation of the efficiency cannot be excluded, but would be incompatible with the Prandtl law of the drag friction coefficient. Furthermore, if the power-law behaviour holds for the efficiency saturation, it can explain the kinetic energy and the energy dissipation defect laws proposed for shear flows. Efficiency saturation is an interesting empirical property of turbulence that may help in evaluating the ‘closeness’ of experimental and numerical data to the true turbulent regime, wherein the kinetic energy saturates to its inviscid limit.

The vertical heated pipe is widely used in thermal engineering applications, as buoyancy can help drive a flow, but several flow regimes are possible: shear-driven turbulence, laminarised flow and convective turbulence. Steady velocity fields that maximise heat transfer have previously been calculated for heated pipe flow, but were calculated independently of buoyancy forces, and hence independently of the flow regime and time-dependent dynamics of the flow. In this work, a variational method is applied to find an optimal body force of limited magnitude $A_0$ that maximises heat transfer for the vertical arrangement, with the velocity field constrained by the full governing equations. In our calculations, mostly at Reynolds number ${\textit{Re}}=3000$, it is found that streamwise-independent rolls remain optimal, as in previous steady optimisations, but that the optimal number of rolls and their radial position is dependent on the flow regime. Surprisingly, while it is generally assumed that turbulence enhances heat transfer, for the strongly forced case, time dependence typically leads to a reduction. Beyond offering potential improvement through the targeting of the roll configuration for this application, wider implications are that optimisations under the steady flow assumption may overestimate improvements in heat transfer, and that strategies that simply aim to induce turbulence may not necessarily be efficient in enhancing heat transfer either. Including time dependence and the full governing equations in the optimisation is challenging but offers further enhancement and improved reliability in prediction.