We employ direct numerical simulations to investigate the heat transfer and flow structures in turbulent Rayleigh–Bénard convection in both cylindrical cells and laterally periodic domains, spanning an unprecedentedly wide range of aspect ratios $0.075 \leqslant \varGamma \leqslant 32$. We focus on Prandtl number ${Pr}=1$ and Rayleigh numbers ${{Ra}}=2\times 10^7$ and ${{Ra}}=10^8$. In both cases, with increasing aspect ratio, the heat transfer first increases, then reaches a maximum (which is more pronounced for the cylindrical case due to confinement effects), and then slightly goes down again before it finally saturates at the large aspect ratio limit, which is achieved already at $\varGamma \approx 4$. Already for $\varGamma \gtrsim 0.75$, the heat transfers in both cylindrical and laterally periodic domains become identical. The large-$\varGamma$ limit for the volume-integrated Reynolds number and the boundary layer thicknesses are also reached at $\varGamma \approx 4$. However, while the integral flow properties converge at $\varGamma \approx 4$, the confinement of a cylindrical domain impacts the temperature and velocity variance distributions up to $\varGamma \approx 16$, as thermal superstructures cannot form close to the sidewall.

This DNS study considers transition to turbulence in plane Couette flow (pCf) with a rough stationary wall and a smooth moving wall. The roughness elements are square ribs of height $k=0.2h$ (where $h$ is the half-channel height). Two different pitch separations, $\lambda =2k$ and $10k$, are considered, i.e. d-type and k-type roughness, respectively. The transition in both rough pCf cases takes place through a stage of alternate laminar–turbulent bands aligned in an oblique fashion. However, roughness causes a shift in the transitional Reynolds number ($Re$) range. In the k-type roughness, stable bands are observed in the range $Re \in [300, 325]$, which is a downwards shift from the transitional $Re$ range for the smooth pCf ($Re \in [325,400]$). The d-type roughness, on the other hand, surprisingly shifts the transitional $Re$ range upwards to $Re \in [350,425]$. This peculiar behaviour is associated with the ability of the ribs to shed and regenerate vorticity. Large-scale components extracted using a filtering process relate to the transition bands and flow parallel to the oblique laminar–turbulent boundaries. Counter-rotating vortices are present in the turbulent regions of the flow field, which exist in tandem with the high- and low-velocity streaks. Another interesting observation is the secondary Reynolds shear stresses, $-\overline {v^{\prime }u^{\prime }}$ and $-\overline {w^{\prime }v^{\prime }}$, which are non-zero in the transitional regime, in contrast to the turbulent regime where they are negligible.

We investigate the coupling effects of the two-phase interface, viscosity ratio and density ratio of the dispersed phase to the continuous phase on the flow statistics in two-phase Taylor–Couette turbulence at a system Reynolds number of $6\times 10^3$ and a system Weber number of 10 using interface-resolved three-dimensional direct numerical simulations with the volume-of-fluid method. Our study focuses on four different scenarios: neutral droplets, low-viscosity droplets, light droplets and low-viscosity light droplets. We find that neutral droplets and low-viscosity droplets primarily contribute to drag enhancement through the two-phase interface, whereas light droplets reduce the system's drag by explicitly reducing Reynolds stress due to the density dependence of Reynolds stress. In addition, low-viscosity light droplets contribute to greater drag reduction by further reducing momentum transport near the inner cylinder and implicitly reducing Reynolds stress. While interfacial tension enhances turbulent kinetic energy (TKE) transport, drag enhancement is not strongly correlated with TKE transport for both neutral droplets and low-viscosity droplets. Light droplets primarily reduce the production term by diminishing Reynolds stress, whereas the density contrast between the phases boosts TKE transport near the inner wall. Therefore, the reduction in the dissipation rate is predominantly attributed to decreased turbulence production, causing drag reduction. For low-viscosity light droplets, the production term diminishes further, primarily due to their greater reduction in Reynolds stress, while reduced viscosity weakens the density difference's contribution to TKE transport near the inner cylinder, resulting in a more pronounced reduction in the dissipation rate and consequently stronger drag reduction. Our findings provide new insights into the physics of turbulence modulation by the dispersed phase in two-phase turbulence systems.

We study the effect of surface texture on an overlying turbulent flow for the case of textures made of an alternating slip/no-slip pattern, a common model for superhydrophobic surfaces, but also a particularly simple form of texture. For texture sizes $L^+ \gtrsim 25$, we have previously reported that, even though the texture effectively imposes homogeneous slip boundary conditions on the overlying, background turbulence, this is not its sole effect. The effective conditions only produce an origin offset on the background turbulence, which remains otherwise smooth-wall-like. For actual textures, however, as their size increases from $L^+ \gtrsim 25$ the flow progressively departs from this smooth-wall-like regime, resulting in additional shear Reynolds stress and increased drag, in a non-homogeneous fashion that could not be reproduced by the effective boundary conditions. This paper focuses on the underlying physical mechanism of this phenomenon. We argue that it is caused by the nonlinear interaction of the texture-coherent flow, directly induced by the surface topology, and the background turbulence, as it acts directly on the latter and alters it. This does not occur at the boundary where effective conditions are imposed, but within the overlying flow itself, where the interaction acts as a forcing on the governing equations of the background turbulence, and takes the form of cross-advective terms between the latter and the texture-coherent flow. We show this by conducting simulations where we remove the texture and introduce additional, forcing terms in the Navier–Stokes equations, in addition to the equivalent homogeneous slip boundary conditions. The forcing terms capture the effect of the nonlinear interaction on the background turbulence without the need to resolve the surface texture. We show that, when the forcing terms are derived accounting for the amplitude modulation of the texture-coherent flow by the background turbulence, they quantitatively capture the changes in the flow up to texture sizes $L^+ \approx 70{-}100$. This includes not just the roughness function but also the changes in the flow statistics and structure.

Turbulent circular pipe flows subjected to axial system rotation are studied using direct numerical simulations (DNS) for a wide range of rotation numbers of $Ro_b = 0\unicode{x2013}20$ at a fixed Reynolds number. To ensure that energetic turbulent eddy motions are captured at high rotation numbers, long pipes up to $L_z = 180{\rm \pi} R$ are used in DNS. Two types of energy-containing flow structures have been observed. The first type is hairpin structures that are characteristic of the turbulent boundary layer developing over the pipe wall for both non-rotating and axially rotating flows. The second type is Taylor columns forming at moderate and high rotation numbers. Based on the study of two-point autocorrelation coefficients, it is observed that Taylor columns exhibit quasi-periods in both axial and azimuthal directions. According to the premultiplied spectra, Taylor columns feature one single characteristic axial length scale at the moderate rotation numbers but two at high rotation numbers. It is discovered that the axial system rotation suppresses the sweep events systematically and impedes the formation of hairpin structures. As the rotation number is increased, the turbulence kinetic energy held by Taylor columns enhances rapidly associated with significant increases in their axial length scales.

The friction drag of the axial flow along the outer surface of a cylinder varies with the cylinder radius and flow conditions. This study included direct numerical simulations of the axial turbulent flow along a circular cylinder under different conditions for obtaining the turbulence statistics and wall friction coefficient. Then the characteristics of velocity streaks were observed from a geometrical perspective of turbulence structures around the circular cylinder, and compared with the characteristics of the turbulence structures in a boundary layer on a flat plate. The results showed that the velocity streak spacing and the distance between the velocity streak and the cylinder surface in the viscous length scale do not vary substantively with the radius of the cylinder, and are the same as those of the turbulent flow along a flat plate. Therefore, they can be considered geometrical characteristics of the turbulence structure independent of the cylinder radius. Moreover, the friction coefficient per pair of high- and low-speed velocity streaks is the same as that of flat-plate turbulent flow, independent of the cylinder radius, and can be regarded as a dynamical characteristic for a pair of velocity streaks. Two equations were derived based on the characteristics of wall turbulence. The characteristics of the turbulence predicted by the two formulae were consistent with the simulation results. Consequently, we showed that the wall friction coefficient and number of the velocity streak pairs, which are statistical and structural characteristics of wall turbulence, can be predicted appropriately by specifying the radius Reynolds number.

Simulations of elastic turbulence, the chaotic flow of highly elastic and inertialess polymer solutions, are plagued by numerical difficulties: the chaotically advected polymer conformation tensor develops extremely large gradients and can lose its positive-definiteness, which triggers numerical instabilities. While efforts to tackle these issues have produced a plethora of specialized techniques – tensor decompositions, artificial diffusion, and shock-capturing advection schemes – we still lack an unambiguous route to accurate and efficient simulations. In this work, we show that even when a simulation is numerically stable, maintaining positive-definiteness and displaying the expected chaotic fluctuations, it can still suffer from errors significant enough to distort the large-scale dynamics and flow structures. We focus on two-dimensional simulations of the Oldroyd-B and FENE-P equations, driven by a large-scale cellular body forcing. We first compare two positivity-preserving decompositions of the conformation tensor: symmetric square root (SSR) and Cholesky with a logarithmic transformation (Cholesky-log). While both simulations yield chaotic flows, only the latter preserves the pattern of the forcing, i.e. its fluctuating vortical cells remain ordered in a lattice. In contrast, the SSR simulation exhibits distorted vortical cells that shrink, expand and reorient constantly. To identify the accurate simulation, we appeal to a hitherto overlooked mathematical bound on the determinant of the conformation tensor, which unequivocally rejects the SSR simulation. Importantly, the accuracy of the Cholesky-log simulation is shown to arise from the logarithmic transformation. We also consider local artificial diffusion, a potential low-cost alternative to high-order advection schemes. Unfortunately, the artificially enhanced diffusive smearing of polymer stress in regions of intense stretching substantially modifies the global dynamics. We then show how the spurious large-scale motions, identified here, contaminate predictions of scalar mixing. Finally, we discuss the effects of spatial resolution, which controls the steepness of gradients in a non-diffusive simulation.

Non-helical turbulence within a linear shear flow has demonstrated efficient amplification of large-scale magnetic fields in numerical simulations, but its precise mechanism remains elusive. The incoherent $\alpha$ mechanism proposes that a zero-mean fluctuating transport coefficient $\alpha$ (linked to kinetic helicity) in the shear flow is a candidate driver. Previous renovating-flow models have proposed that the correlation time of helicity fluctuations must be sufficiently extended to overcome turbulent magnetic diffusivity, yet only empirical validation of this concept has been obtained. In this study, we conduct direct numerical simulations of weakly compressible non-helical hydrodynamic turbulence. We scrutinize the correlation times of velocity and kinetic helicity fluctuations in distinct flow configurations, including rotation, shearing and Keplerian flows, as well as the shearing burgulence counterpart. Our findings indicate that rotation contributes to a prolonged correlation time of helicity compared with velocity, particularly notable in auto-correlations of both volume-averaged quantities and individual Fourier modes due to the formation of large-scale vortices. In contrast, moderate shear strength does not exhibit significant scale separation, with shear flows elongating vortices in the shear direction. Shearing burgulence, characterized by shorter helicity correlation times, appears less conducive to hosting the incoherent $\alpha$ effect. Notably, at modest shear rates, only Keplerian flows exhibit sufficiently coherent helicity fluctuations, in contrast to shearing flows. However, the relative strength of helicity fluctuations compared with turbulent diffusivity is significantly lower, raising doubts about the viability of the incoherent $\alpha$ effect as a potential dynamo driver in the subsonic flows examined in this study.

The dimensional transition in turbulent jets of a shear-thinning fluid is studied via direct numerical simulations. Our findings reveal that under vertical confinement, the flow exhibits a unique mixed-dimensional (or 2.5-dimensional) state, where large-scale two-dimensional and small-scale three-dimensional structures coexist. This transition from three-dimensional turbulence near the inlet to two-dimensional dynamics downstream is dictated by the level of confinement: weak confinement guarantees turbulence to remain three-dimensional, whereas strong confinement forces the transition to two dimensions; the mixed-dimensional state is observed for moderate confinement and it emerges as soon as flow scales are larger than the vertical length. In this scenario, we observed that the mixed-dimensional state is an overall more energetic state, and it shows a multi-cascade process, where the direct cascade of energy at small scales and the direct cascade of enstrophy at large scales coexist. The results provide insights into the complex dynamics of confined turbulent flows, relevant in both natural and industrial settings.

We present direct numerical simulations of a three-layer Rayleigh–Taylor instability (RTI) problem with a configuration based on the experiments of Suchandra & Ranjan (J. Fluid Mech., vol. 974, 2023, A35) and Jacobs & Dalziel (J. Fluid Mech., vol. 542, 2005, pp. 251–279). The problem consists of a layer of light fluid between two layers of heavy fluid with an Atwood number of 0.3. These simulations are first validated through comparison with available experimental data. The validated simulations are then utilized to analyse statistics in this three-component flow. First, length scales are examined utilizing spectra and two-point spatial correlations of velocity and species concentration fluctuations. Next, joint probability density functions (p.d.f.s) of species concentration are compared against several model p.d.f.s representing generalizations of the bivariate beta distribution. Notably, the joint p.d.f.s do not appear to be accurately described by a Dirichlet distribution, indicating the marginal distributions do not conform to a beta distribution. Finally, similarity of the present configuration to three-component mixing found in inertial confinement fusion (ICF) applications is exploited to develop and validate an improved model for the impact of multicomponent mixing on thermonuclear (TN) reaction rates. A single time instant from the present simulations is chosen for a TN burn calculation under the hypothetical assumption of ICF materials and temperatures. Total TN output from this second calculation is then compared against the prediction of the improved model. The new model is found to accurately predict TN reaction rates in both premixed and non-premixed configurations.

The research on elasto-inertial turbulence (EIT), a new type of turbulent flow, has reached the stage of identifying the minimal flow unit (MFU). On this issue, direct numerical simulations of FENE-P fluid flow in two-dimensional channels with variable sizes are conducted in this study. We demonstrate with the increase of channel length that the simulated flow experiences several different flow patterns, and there exists an MFU for EIT to be self-sustained. At Weissenberg number ($Wi$) higher than the one required to excite EIT, when the channel length is relatively small, a steady arrowhead regime (SAR) flow structure and a laminar-like friction coefficient is achieved. However, as the channel length increases, the flow can fully develop into EIT characterized with high flow drag. Close to the size of the MFU, the simulated flow behaves intermittently between the SAR state with low drag and EIT state with high drag. The flow falling back to ‘laminar flow’ is caused by the insufficient channel size below the MFU. Furthermore, we give the relationship between the value of the MFU and the effective $Wi$, and explain its physical reasons. Moreover, the intermittent flow regime obtained based on the MFU gives us an opportunity to look into the origin and exciting process of EIT. Through capturing the onset process of EIT, we observed that EIT originates from the sheet-like extension structure located near the wall, which is maybe related to the wall mode rather than the centre mode. The fracture and regeneration of this sheet-like structure is the key mechanism for the self-sustaining of EIT.

We report direct numerical simulations results of the rough-wall channel, focusing on roughness with high $k_{rms}/k_a$ statistics but small to negative $Sk$ statistics, and we study the implications of this new dataset on rough-wall modelling. Here, $k_{rms}$ is the root mean square, $k_a$ is the first-order moment of roughness height, and $Sk$ is the skewness. The effects of packing density, skewness and arrangement of roughness elements on mean streamwise velocity, equivalent roughness height ($z_0$) and Reynolds and dispersive stresses have been studied. We demonstrate that two-point correlation lengths of roughness height statistics play an important role in characterizing rough surfaces with identical moments of roughness height but different arrangements of roughness elements. Analysis of the present as well as historical data suggests that the task of rough-wall modelling is to identify geometric parameters that distinguish the rough surfaces within the calibration dataset. We demonstrate a novel feature selection procedure to determine these parameters. Further, since there is no finite set of roughness statistics that distinguish between all rough surfaces, we argue that obtaining a universal rough-wall model for making equivalent sand-grain roughness ($k_s$) predictions would be challenging, and that each rough-wall model would have its applicable range. This motivates the development of group-based rough-wall models. The applicability of multi-variate polynomial regression and feedforward neural networks for building such group-based rough-wall models using the selected features has been shown.

The present study offers a twofold contribution on counter-gradient transport (CGT) of turbulent scalar flux. First, by examining turbulent scalar mixing through synchronized particle image velocimetry and planar laser-induced fluorescence on an inclined jet in cross-flow, we clarify the previously unexplained phenomenon of CGT, revealing key flow structures, their spatial distribution and modelling implications. Statistical analysis identifies two distinct CGT regions: local cross-gradient transport in the windward shear layer and non-local effects near the wall after injection. These behaviours are driven by specific flow structures, namely Kelvin–Helmholtz vortices (local) and wake vortices (non-local), suggesting that scalar flux can be decomposed into a gradient-type term for gradient diffusion and a term for large-eddy stirring. Second, we propose a new approach for reconstruction of turbulent mean flow and scalar fields using continuous adjoint data assimilation (DA). By rectifying model-form errors through anisotropic correction under observational constraints, our DA model minimizes discrepancies between experimental measurements and numerical predictions. As expected, the introduced forcing term effectively identifies regions where traditional models fall short, particularly in the jet centreline and near-wall regions, thereby enhancing the accuracy of the mean scalar field. These enhancements occur not only within the observation region but also in unseen regions, underscoring present DA approach's reliability and practicality for reproducing mean flow behaviours from limited data. These findings lay a solid foundation for adjoint-based model-consistent data-driven methods, offering promising potential for accurately predicting complex flow scenarios like film cooling.

Direct numerical simulations of spanwise-rotating turbulent channel flow with a parabolic bump on the bottom wall are employed to investigate the effects of rotation on flow separation. Four rotation rates, $Ro_b := 2\varOmega H/U_b = \pm 0.42$, $\pm$1.0, are compared with the non-rotating scenario. The mild adverse pressure gradient induced by the lee side of the bump allows for a variable pressure-induced separation. The separation region is reduced (increased) when the bump is on the anti-cyclonic (cyclonic) side of the channel, compared with the non-rotating separation. The total drag is reduced in all rotating cases. Through several mechanisms, rotation alters the onset of separation, reattachment and wake recovery. The mean momentum deficit is found to be the key. A physical interpretation of the ratio between the system rotation and mean shear vorticity, $S:=\varOmega /\varOmega _s$, provides the mechanisms regarding stability thresholds $S=-0.5$ and $-$1. The rotation effects are explained accordingly, with reference to the dynamics of several flow structures. For anti-cyclonic separation, particularly, the interaction between the Taylor–Görtler vortices and hairpin vortices of wall-bounded turbulence is proven to be responsible for the breakdown of the separating shear layer. A generalized argument is made regarding the essential role of near-wall deceleration and resultant ejection of enhanced hairpin vortices in destabilizing an anti-cyclonic flow. This mechanism is anticipated to have broad impacts on other applications in analogy to rotating shear flows, such as thermal convection and boundary layers over concave walls.

A permeable disk serves as a simplified model for the conversion of wind energy by a horizontal axis wind turbine. In this study, we investigate how inflow turbulence intensity (TI), $I_\infty$, and inflow turbulence integral length scale, $L_\infty$, influence the flow recovery in the wake, the capability of a permeable disk in extracting turbulence kinetic energy (TKE) of the incoming flow, and the statistics of wake-added turbulence using large-eddy simulation. The simulated inflows include various TIs (i.e. $I_\infty =2.5\,\%$–$25\,\%$) and integral length scales (i.e. $L_\infty / D =0.5$–$2.0$) for two thrust coefficients. Simulation results show that both inflow TI and integral length scale influence flow recovery via enhanced ejections and sweeps across the wake boundary, with the former strongly affecting the position where the wake starts to recover and the latter mainly on the recovery rate. Moreover, it is shown that increasing $I_\infty$ and $L_\infty$ increases the TKE extraction by the disk, occurring mainly at scales ($s$) greater than $0.5D$ and frequencies depending on the inflow integral length scale. As for the wake-added TKE, the inflow TI mainly affects its intensity, while the inflow integral length scale affects both its intensity and the sensitive frequencies, with the spectral distributions in scale space ($s$) being similar and the peak located around $s/D=1.0$ for the considered inflows.

Turbulent flows over rough surfaces can be encountered in a wide range of engineering applications. Despite the progress made after several decades of studies, the prediction of drag and roughness function from the surface geometrical parameters remains an open question. Several methods have shown encouraging results. However, they lack generality and present some scatter in the data. In this paper we propose a new parameter, the effective distribution ($ED$), which lays foundation on the effective slope with some changes to take into account the sheltering effect of large roughness elements and the drag induced by pinnacles higher than the average roughness elements. To develop this new correlation between geometrical features of the wall and the drag, we performed a set of simulations of the turbulent flow over a rough surface made of triangular elements varying their height and spatial distribution. The $ED$ correlates quite well both with the drag and the roughness function for a wide range of cases having different mean roughness height, skewness and kurtosis. To further validate the $ED$, and assessing how it can be generalized to real rough wall, an irregular wall made from the superposition of random sinusoidal function was considered. Results were consistent with the correlation here presented.

We present direct numerical simulation (DNS) and modelling of incompressible, turbulent, generalized Couette–Poiseuille flow. A particular example is specified by spherical coordinates $(Re,\theta,\phi )$, where $Re = 6000$ is a global Reynolds number, $\phi$ denotes the angle between the moving plate, velocity-difference vector and the volume-flow vector and $\tan \theta$ specifies the ratio of the mean volume-flow speed to the plate speed. The limits $\phi \to 0^\circ$ and $\phi \to 90^\circ$ give alignment and orthogonality, respectively, while $\theta \to 0^\circ,\ \theta \to 90^\circ$ correspond respectively to pure Couette flow in the $x$ direction and pure Poiseuille flow at angle $\phi$ to the $x$ axis. Competition between the Couette-flow shear and the forced volume flow produces a mean-velocity profile with directional twist between the confining walls. Resultant mean-speed profiles relative to each wall generally show a log-like region. An empirical flow model is constructed based on component log and log-wake velocity profiles relative to the two walls. This gives predictions of four independent components of shear stress and also mean-velocity profiles as functions of $(Re,\theta,\phi )$. The model captures DNS results including the mean-flow twist. Premultiplied energy spectra are obtained for symmetric flows with $\phi =90^\circ$. With increasing $\theta$, the energy peak gradually moves in the direction of increasing $k_x$ and decreasing $k_z$. Rotation of the energy spectrum produced by the faster moving velocity near the wall is also observed. Rapid weakening of a spike maxima in the Couette-type flow regime indicates attenuation of large-scale roll structures, which is also shown in the $Q$-criterion visualization of a three-dimensional time-averaged flow.

High-fidelity simulations are conducted to investigate the turbulent boundary layers around a finite-span NACA0012 wing with a rounded wing-tip geometry at a chord-based Reynolds number of $Re_c=200\,000$ and at various angles of attack up to $10^\circ$. The study aims to discern the differences between the boundary layers on the finite-span wing and those on infinite-span wings at equivalent angles of attack. The finite-span boundary layers exhibit: (i) an altered streamwise and a non-zero spanwise pressure gradient as a result of the variable downwash induced by the wing-tip vortices (an inviscid effect typical of finite-span wings); (ii) differences in the flow history at different wall-normal distances, caused by the variable flow angle in the wall-normal direction (due to constant pressure gradients and variable momentum normal to the wall); (iii) laminar flow entrainment into the turbulent boundary layers near the wing tip (due to a laminar–turbulent interface); and (iv) variations in boundary layer thickness across the span, attributed to the variable wall-normal velocity in that direction (a primarily inviscid effect). These physical effects are then used to explain the differences in the Reynolds stress profiles and other boundary layer quantities, including the reduced near-wall peak of the streamwise Reynolds stress and the elevated Reynolds stress levels near the boundary layer edge, both observed in the finite-span wings. Other aspects of the flow, such as the downstream development of wing-tip vortices and their interactions with the surrounding flow, are reserved for future investigations.

The first two parts of this study showed by experiment and by large-eddy simulation that, in mixing layers formed of processions of spanwise-oriented vortex structures, there is a fundamental change in the dynamics of the large-structure growth at what has been called the ‘mixing’ transition. This third part examines the relationship of the post-transition ‘organised’ flow to the mixing layer of classical turbulence theory. Further large-eddy simulations are presented which, like some of the experiments reported in Part 1, have captured the mixing layer in both its classical and organised turbulent states, thereby allowing them to be characterised separately. The simulation results are then used to calibrate model-free integral analyses of the dynamics of the mixing layer's spatial growth. It is shown that the organised and classical flows are alternative self-preserving turbulent flow states involving fundamentally different exchanges of mass, momentum and energy, one or other of which emerges naturally in particular realisations depending on the initial conditions.