Streamwise Velocity Streaks Research Articles

Turbulent spot formation in three-dimensional yukawa liquids using large-scale molecular dynamics simulation—effect of system size

We have performed classical ‘first principles’ 3D Molecular Dynamics (MD) simulation of plane-Couette flow (PCF) in a 3D Yukawa liquid. The main aim of the present investigation is to study the effect of stream-wise (flow-direction) and span-wise (transverse direction to the flow) width on the subcritical transition to turbulence in plane Couette flow (PCF), separately. In the past, taking very large-aspect ratio systems, subcritical transition to turbulence in PCF has been studied in great detail. However, the effect of stream-wise and span-wise width on the turbulent dynamics separately, have not been studied in detail. In the present work, we have investigated the effect of stream-wise and span-wise width or length and found that the stream-wise length enhances the large-scale energy, which gives rise to strong large-scale flow, and span-wise width enhances the small-scale energy, which gives rise to strong small-scale structures. In other words, topology of the turbulent spot is observed to change with the change in system size. The spectral separation between the modes governing the large and small-scale dynamics improves with the increase in system sizes. The connection between the stream-wise vortices and the stream-wise velocity streaks is investigated in details. We have found that the number stream-wise velocity streaks is one unit larger than the number of stream-wise vortices. A qualitative comparison between the results obtained from the MD simulation and the hydrodynamics experiment is presented in this article. The behavior of the system was observed to vary with the range of interactions among the dust grains.

Turbulent–turbulent transient concept in pulsating flows

The turbulence behaviour of current-dominated pulsating flows has been investigated. Direct numerical simulations have been carried out for Stokes lengths over a range of $l_s^+=5\unicode{x2013}26$ , and amplitudes spanning 90 % of the current-dominated regime, about a mean flow of $\overline {Re}=6275$ . The results show that the turbulence response in intermediate and low-frequency pulsations is governed by a multistage turbulent–turbulent transition process, which bears a strong similarity to the multistage response of non-periodic acceleration. During the early acceleration period, the flow enters a pretransition stage, in which a new laminar perturbation boundary layer forms at the wall, and the streamwise velocity streaks are stretched. If the low-speed streaks destabilise prior to the deceleration period, then the flow enters a transition stage in which the perturbation boundary layer undergoes a bypass-like transition process. A unique feature of pulsating flows is the ongoing mechanism of turbulence decay, which initiates during the deceleration period and constitutes the main transient turbulence mechanism for much of the cycle. For high-frequency pulsations, the perturbation boundary layer fails to reach the pretransition stage prior to the deceleration period. Instead, the flow alternates between two inertial stages which are characterised by two layers of amplified viscous force; one growing at the wall, and one detached and moving towards the core.

Turbulent boundary layer over porous media with wall-normal permeability

Porous walls are a widely used passive flow control technique, which shows potential in reducing skin friction and mitigating flow-introduced noise. In the present study, porous media with wall-normal permeability is applied to a flat plate to investigate its interaction with the turbulent boundary layer at the Reynolds number based on friction velocity of Reτ=225. Time-resolved planar and tomographic particle image velocimetry were employed to identify the impact on mean statistics and coherent structures. An overall skin friction reduction of 22% is achieved. The porous wall induces counter-rotating streamwise vortex pairs at the spanwise sides of each pore, leading to momentum transport and the generation of alternative low- and high-speed regions close to the wall. Slip velocity is obtained, associated with the reduction in turbulent fluctuations and Reynolds shear stress. The streamwise velocity streaks and the hairpin vortices are significantly distorted compared with the smooth wall condition due to the downwash and upwash motion, featuring a notable reduction in the number and scale of the coherent structures, in which the skin friction reduction mechanism is related to. The proper orthogonal decomposition analysis returns the most energetic unsteady modes. Although the wall-coherent mode type remains to dominate the production of turbulent fluctuations, the scale and energy content of wall-incoherent modes increase, confirming the modification of the distribution and scale of near-wall turbulent structures.

Linear response analysis of supersonic turbulent channel flows with a large parameter space

In this work, the linear responses of turbulent mean flow to both harmonic and stochastic forcing are investigated for supersonic channel flow. Well-established universal relations are utilized to obtain efficiently the mean profiles with a large parameter space, with the bulk Mach number up to 5 and the friction Reynolds number up to$10^4$, so a systematic parameter study is feasible. The most amplified structure takes the form of streamwise velocity and temperature streaks forced optimally by the streamwise vortices. The outer peak of the pre-multiplied energy amplification corresponds to the large-scale motion, whose spanwise wavelength ($\lambda _z^+$) is very insensitive to compressibility effects. In contrast, the classic inner peak representing small-scale near-wall motions disappears for the stochastic response with increasing Mach number. Meanwhile, the small-scale motions become much less coherent. A decomposition of the forcing identifies different effects of the incompressible counterpart and the thermodynamic components. Wall-cooling effects, arising with high Mach number, increase the spacing of the most amplified near-wall streaks; the spacing becomes nearly invariant with Mach number if expressed in semi-local units. Meanwhile, the coherence of stochastic response with$\lambda _z^+>90$is enhanced, but on the other hand, with$\lambda _z^+<90$it is decreased. The geometrical self-similarity of the response in the mid-$\lambda _z$range is still roughly satisfied, insensitive to Mach number. Finally, theoretical analyses of the perturbation equations are presented to help with understanding the scaling of energy amplification.

Drag increase and turbulence augmentation in two-way coupled particle-laden wall-bounded flows

The exact regularized point particle method is used to characterize the turbulence modulation in two-way momentum-coupled direct numerical simulations of a turbulent pipe flow. The turbulence modification is parametrized by the particle Stokes number, the mass loading, and the particle-to-fluid density ratio. The data show that in the wide region of the parameter space addressed in the present paper, the overall friction drag is either increased or unaltered by the particles with respect to the uncoupled case. In the cases where the wall friction is enhanced, the fluid velocity fluctuations show a substantial modification in the viscous sub-layer and in the buffer layer. These effects are associated with a modified turbulent momentum flux toward the wall. The particles suppress the turbulent fluctuations in the buffer region and concurrently provide extra stress in the viscous sub-layer. The sum of the turbulent stress and the extra stress is larger than the Newtonian turbulent stress, thus explaining the drag increase. The non-trivial turbulence/particles interaction turns out in a clear alteration of the near-wall flow structures. The streamwise velocity streaks lose their spatial coherence when two-way coupling effects are predominant. This is associated with a shift of the streamwise vortices toward the center of the pipe and with the concurrent presence of small-scale and relatively more intense vortical structures near the wall.

Coherent organizational states in turbulent pipe flow at moderate Reynolds numbers

Turbulent pipe flow is still an essentially open area of research, boosted in the last two decades by considerable progress achieved on both the experimental and numerical frontiers, mainly related to the identification and characterization of coherent structures as basic building blocks of turbulence. It has been a challenging task, however, to detect and visualize these coherent states. We address, by means of stereoscopic particle image velocimetry, that issue with the help of a large diameter (6 in.) pipe loop, which allowed us to probe for coherent states at various moderate Reynolds numbers (5300 < Re < 29 000) of the single-phase Newtonian flow. Although these states have been observed at flow regimes around laminar–turbulent transition (Re ≈ 2300) and also at high Reynolds number pipe flow (Re ≈ 35 000), at moderate Reynolds numbers, their existence had not been observed yet by experiment. By conditionally averaging the flow fields with respect to their dominant azimuthal wavenumber of streamwise velocity streaks, we have been able to uncover the existence of ten well-defined coherent flow patterns. It turns out, as a remarkable phenomenon, that their occurrence probabilities and the total number of dominant modes do not essentially change as the Reynolds number is varied. Their occurrence probabilities are noted to be reasonably well described by a Poisson distribution, which suggests that low-speed streaks are created as a Poisson process on the pipe circular geometry.

Structure of turbulence in temporal planar jets

A detailed analysis of the structure of turbulence in a temporal planar turbulent jet is reported. Instantaneous snapshots of the flow and three-dimensional spatial correlation functions are considered. It is found that the flow is characterized by large-scale spanwise vortices whose motion is felt in the entire flow field. Superimposed to this large-scale motion, a hierarchy of turbulent structures is present. The most coherent ones take the form of quasi-streamwise vortices and high and low streamwise velocity streaks. The topology of these interacting structures is analyzed by quantitatively addressing their shape and size in the different flow regions. Such information is recognized to be relevant for a structural description of the otherwise disorganized motion in turbulent free-shear flows and can be used for the assessment of models based on coherent structure assumptions. Finally, the resulting scenario provides a phenomenological description of the elementary processes at the basis of turbulence in free-shear flows.

On the role of turbulent large-scale streaks in generating sediment ridges

The role of turbulent large-scale streaks or large-scale motions in forming subaqueous sediment ridges on an initially flat sediment bed is investigated with the aid of particle resolved direct numerical simulations of open channel flow at bulk Reynolds numbers up to 9500. The regular arrangement of quasi-streamwise ridges and troughs at a characteristic spanwise spacing between 1 and 1.5 times the mean fluid height is found to be a consequence of the spanwise organisation of turbulence in large-scale streamwise velocity streaks. Ridges predominantly appear in regions of weaker erosion below large-scale low-speed streaks and vice versa for troughs. The interaction between the dynamics of the large-scale streaks in the bulk flow and the evolution of sediment ridges on the sediment bed is best described as ‘top-down’ process, as the arrangement of the sediment bedforms is seen to adapt to changes in the outer flow with a time delay of several bulk time units. The observed ‘top-down’ interaction between the outer flow and the bed agrees fairly well with the conceptual model on causality in canonical channel flows proposed by Jiménez (J. Fluid Mech., vol. 842, 2018, P1, § 5.6). Mean secondary currents of Prandtl's second kind of comparable intensity and lateral spacing are found over developed sediment ridges and in single-phase smooth-wall channels alike in averages over ${O}(10)$ bulk time units. This indicates that the secondary flow commonly observed together with sediment ridges is the statistical footprint of the regularly organised large-scale streaks.

Influence of gas expansion on the propagation of a premixed flame in a spatially periodic shear flow

It has been previously demonstrated that thermal gas expansion might have a role in boundary layer flashback of premixed turbulent flames [Gruber et al., J Fluid Mech 2012], inducing local flow-reversal in the boundary layer's low-velocity streaks on the reactants’ side of the flame and facilitating its upstream propagation. We perform a two-dimensional numerical investigation of the interaction between a periodic shear flow and a laminar premixed flame. The periodic shear is a simplified model for the oncoming prolonged streamwise velocity streaks with alternating regions of high and low velocities found in turbulent boundary layers in the vicinity of the walls. The parametric study focuses on the amplitude and wavelength of the periodic shear flow and on the gas expansion ratio (unburnt-to-burnt density ratio). With the increase of the amplitudes of the periodic shear flow and of the gas expansion, the curved flame velocity increases monotonically. The flame velocity dependence on the periodic shear wavelength is non-monotonic, which is consistent with previous theoretical studies of curved premixed flame velocity. The flame shape that is initially formed by the oncoming periodic shear appears to be metastable. At a later stage of the flame propagation, the flame shape transforms into the stationary one dominated by the Darrieus-Landau instability.

Mechanisms of particle preferential concentration induced by secondary motions in a dilute turbulent square duct flow

Particle-laden turbulent square duct flows at Reτ = 300 (based on the duct half-width and the mean friction velocity) are investigated using direct numerical simulation with one-way coupled Lagrangian particle tracking. Four particle-to-fluid density ratios are considered with the corresponding shear Stokes number St+ = 0.31, 25, 125, and 260. Particle motion is governed by drag, lift, added-mass, and pressure gradient forces. The main purpose of this work is to examine the effect of the turbulence-driven secondary flows on particle preferential accumulation and their dependence on the Stokes number. Results obtained indicate that the cross-stream secondary motions encourage inertial particles to accumulate preferentially in the duct corners, where the maximum of the cross-sectional particle concentration occurs. The extent of accumulation here is strongly dependent on the Stokes number, with the greatest accumulation found at St+ = 25. Interestingly, the maximum of the intensity of the secondary particle velocity along the corner bisector is also achieved at St+ = 25, whereas in the region adjacent to the wall, it is found to decrease with a particle Stokes number. Additionally, it is observed that the higher inertia particles are more easily trapped in the stagnation zone of secondary flows with low turbulence intensity in the corner region. In the near-wall region, the heavier particles (St+ ≥ 25) are prone to reside and form elongated clusters along the low-speed streamwise velocity streaks, with this trend less pronounced with the increasing Stokes number. Along the wall, away from the corner where the secondary motion is attenuated, particle accumulation is dominated by the near-wall coherent vortices. This phenomenon is further discussed using a region-based correlation analysis between the particle spatial distribution and local flow topology. An in-depth particle dynamic analysis determines that the average cross-sectional drag force resulting from the secondary flow is mainly responsible for the particle motion throughout the duct cross section, which tends to push particles away from the walls in the near-wall region but shows the exact opposite trend in the bulk flow region. Moreover, the pressure gradient force also plays an important role for low-inertia particles. As the Stokes number is increased, the lift force becomes progressively dominant in the viscous sublayer, acting to pull particles toward the corners and walls of the duct.

Bayesian based estimation of turbulent flow fields from lidar observations in a conventionally neutral atmospheric boundary layer

In this paper, we consider the reconstruction of 3D turbulent flow fields from a time series of lidar data in a conventionally neutral atmospheric boundary layer (CNBL). For the reconstruction we use the maximum a posteriori estimate of the flow field. This corresponds to an optimization problem, with a cost function that has two contributions; a first term originating from the prior belief on the probability of having a certain turbulent flow field without any observations. Flow field fluctuations are assumed normally distributed and thus statistically fully determined by the mean and two-point covariance of the velocity field. The second term, is related to the likelyhood of the observations, influenced by model and measurement uncertainties. The two-point covariance is computed and found to be significantly altered by the Coriolis force, breaking up longer streamwise velocity streaks and veering spanwise structures by ∼ 45° with respect to the mean flow direction. For the reconstruction, we consider two different scanning modes, a plan position indicator (PPI) mode and a trajectory which is based on a Lissajous curve. For the PPI scanning mode we find that the mean squared error of the reconstructed velocity field is around 10% of the background variance in the scanning plane, and quickly increases outside this region. The Lissajous curve on the other hand attains an average error of 40% over the scanning region, which spans almost the whole BL height.

On the friction drag reduction mechanism of streamwise wall fluctuations

Understanding how to decrease the friction drag exerted by a fluid on a solid surface is becoming increasingly important to address key societal challenges, such as decreasing the carbon footprint of transport. Well-established techniques are not yet available for friction drag reduction. Direct numerical simulation results obtained by Józsa et al. (2019) previously indicated that a passive compliant wall can decrease friction drag by sustaining the drag reduction mechanism of an active control strategy. The proposed compliant wall is driven by wall shear stress fluctuations and responds with streamwise wall velocity fluctuations. The present study aims to clarify the underlying physical mechanism enabling the drag reduction of these active and passive control techniques. Analysis of turbulence statistics and flow fields reveals that both compliant wall and active control amplify streamwise velocity streaks in the viscous sublayer. By doing so, these control methods counteract dominant spanwise vorticity fluctuations in the near-wall region. The lowered vorticity fluctuations lead to an overall weakening of vortical structures which then mitigates momentum transfer and results in lower friction drag. These results might underpin the further development and practical implementation of these control strategies.

Mean dynamics and transition to turbulence in oscillatory channel flow

The mean dynamics in oscillatory channel flow is examined to investigate the dynamical mechanisms underlying the transition to turbulence in oscillatory wall-bounded flow. The analyses employ direct numerical simulation data acquired at three Stokes Reynolds numbers: $Re_{s}=648$, 801 and 1009, where the lower $Re_{s}$ flow is transitional over the entire cycle and the two higher $Re_{s}$ flows exhibit flow characteristics similar to steady turbulent wall-bounded flow during part of the cycle. The flow evolution over a half-period of oscillation for all three $Re_{s}$ is as follows: near-wall streamwise velocity streaks develop during the early accelerating portion of the cycle; then at some later point in the cycle that depends on $Re_{s}$, the near-wall streaks breakdown (demarking the onset of the nonlinear development stage), and the near-wall Reynolds stress grows explosively; the Reynolds stress remains elevated for part of the cycle before diminishing (yet remaining finite) during the late decelerating portion of the cycle. This process is then repeated indefinitely. The present findings demonstrate that transition to turbulence occurs when the nonlinear development stage begins during the accelerating portion of the cycle. This crucially leads to the diminishing importance of the centreline momentum source, the emergence of a locally accelerating/decelerating internal layer centred about the edge of the Stokes layer and the wall-normal rearrangement of the mean forces prior to the start of the decelerating portion of the cycle. The rearrangement of mean forces culminates in a four layer structure in the mean balance of forces. This is significant on a number of accounts since empirical and theoretical evidence suggests that the formation of a four layer structure is an important characteristic of a self-similar hierarchal structure that underlies logarithmic dependence of the mean velocity profile in steady turbulent wall-bounded flows (Klewicki et al., J. Fluid Mech., vol. 638, 2009, pp. 73–93). When the nonlinear development stage begins during the decelerating portion of the cycle (i.e. at $Re_{s}=648$), a four layer structure is not observed in the mean balance of forces and the flow remains weakly transitional over the entire cycle.

Transient growth in turbulent particle-laden channel flow

Linear transient growth of optimal perturbations in particle-laden turbulent channel flow is investigated in this work. The problem is formulated in the framework of a Eulerian–Eulerian approach, employing two-way coupling between fine particles and fluid flow. The model is first validated in laminar cases, after which the transient growth of coherent perturbations in turbulent channel flow is investigated, where the mean particle concentration distribution is obtained by direct numerical simulation. It is shown that the optimal small-scale structures for particles are streamwise streaks just below the optimal streamwise velocity streaks, as was previously found in numerical simulations of particle-laden channel flow. This indicates that the optimal growth of perturbations is a dominant mechanism for the distribution of particles in the near-wall region. The current study also considers the transient growth of small- and large-scale perturbations at relatively high Reynolds numbers, which reveals that the optimal large-scale structures for particles are in the near-wall region while the optimal large-scale structures for fluid enter the outer region.

Large-eddy simulation of an open-channel flow bounded by a semi-dense rigid filamentous canopy: Scaling and flow structure

We have carried out a large-eddy simulation of a turbulent open-channel flow over a marginally dense, fully submerged, rigid canopy. The canopy is made of a set of rigid, slender cylinders normally mounted on a solid wall. The flow in the canopy is resolved stem-by-stem by means of an immersed boundary method. It is found that the flow behavior can be categorized according to the velocity distribution into two separate spatial regions: the canopy itself and the outer region above it. Within the region occupied by the canopy elements, the velocity magnitude is found to be related to the local shear stress. Outside the canopy, a logarithmic velocity profile matching the canonical turbulent open-channel flow over rough walls is recovered albeit the strong manipulation exerted by the canopy on the buffer layer. In the innermost layer, the presence of the stems is responsible for redistributing the local momentum fluctuations from a streamwise to a spanwise leading component, inhibiting the survival of the wall streamwise velocity streaks. On the other hand, the outer region presents a structure very similar to the well-known logarithmic boundary layer with the presence of large and energetic streamwise velocity streaks generated by a system of quasistreamwise vortices. These vortices strongly contribute to the intracanopy fluctuations through vigorous sweep and ejection events that affect all the region occupied by the stems. Consistent with the results of previous investigations [H. Nepf, “Flow and transport in regions with aquatic vegetation,” Annu. Rev. Fluid Mech. 44, 123–142 (2012)], it is found that the inflection point in the mean velocity profile, produced by the drag discontinuity at the canopy tip, promotes the appearance of another system of spanwise oriented vorticity structures. However, different from previous results reported in the literature [J. Finnigan, “Turbulence in plant canopies,” Annu. Rev. Fluid Mech. 32, 519–571 (2000)], in our simulations, the presence of alternating head up–head down hairpin vortices generated by a mutual induction of the counter-rotating spanwise vortices is not observed. Instead, we advocate that the modulation of the spanwise coherent vorticity is due to the action of the external logarithmic layer structures (i.e., the outer streamwise vortices that penetrate the canopy) rather than by upwash and downwash motions induced by the mutual interaction of the spanwise rollers.

Coherent structures in the linearized impulse response of turbulent channel flow

We study the evolution of velocity fluctuations due to an isolated spatio-temporal impulse using the linearized Navier–Stokes equations. The impulse is introduced as an external body force in incompressible channel flow at $Re_{\unicode[STIX]{x1D70F}}=10\,000$. Velocity fluctuations are defined about the turbulent mean velocity profile. A turbulent eddy viscosity is added to the equations to fix the mean velocity as an exact solution, which also serves to model the dissipative effects of the background turbulence on large-scale fluctuations. An impulsive body force produces flow fields that evolve into coherent structures containing long streamwise velocity streaks that are flanked by quasi-streamwise vortices; some of these impulses produce hairpin vortices. As these vortex–streak structures evolve, they grow in size to be nominally self-similar geometrically with an aspect ratio (streamwise to wall-normal) of approximately 10, while their kinetic energy density decays monotonically. The topology of the vortex–streak structures is not sensitive to the location of the impulse, but is dependent on the direction of the impulsive body force. All of these vortex–streak structures are attached to the wall, and their Reynolds stresses collapse when scaled by distance from the wall, consistent with Townsend’s attached-eddy hypothesis.

Flow structures in a shallow channel with lateral bed-roughness variation

Highly heterogeneous floodplains can give rise to secondary flow structures responsible for the bulk of lateral momentum exchange. Quantifying the redistribution of momentum is required to predict lateral profiles of flow velocity and the associated water level in a river. In the work herein, we focus on studying secondary flow structures and the momentum redistribution associated with a lateral bed-roughness variation in a channel with low relative submergence of the roughness elements, h=k = 3, 2 and 1.5, where h is the flow depth and k is the roughness height. A series of laboratory experiments were performed in a flume containing rows of cubes. They were arranged in two types of regular patterns, with higher and lower frontal density, and placed side by side such that the bed roughness varies in the lateral direction. The measurements were performed using stereoscopic PIV in a vertical cross plane spanning between the two roughness types. The time-averaged and turbulence statistics of the three components of the velocity field were analyzed. First, we focus on the intensity of the secondary currents. As the flow becomes shallower (lower relative submergence), the cross-stream velocity normalized by the streamwise velocity increases. A large-scale secondary current at the border between the two roughnesses as observed in [1] (though in their case between smooth and rough regions) appears for h=k = 3. As h=k decreases, this structure reaches to the same size as the secondary flow generated by the roughness elements. Also, the discharge distribution between the two sides of the channel becomes less uniform with decreasing h=k. In this sense, the relative importance of the roughness difference increases with decreasing water depth. Moreover, higher discharge is observed on the side with higher equivalent sand roughness, contrary to what is observed for smooth-to-rough transition [1, 2]. Time series of the streamwise velocity fluctuations are calculated using Taylor’s “frozen turbulence” hypothesis. In this representation, streamwise velocity streaks are apparent for h=k = 3, but they appear to lose coherence for the most shallow case of h=k=1.5.

Streamwise Vortices and Velocity Streaks in a Locally Drag-Reduced Turbulent Boundary Layer

This work aims to understand the changes associated with the near-wall streaky structures in a turbulent boundary layer (TBL) where the local skin-friction drag is substantially reduced. The Reynolds number is R e 𝜃 = 1000 based on the momentum thickness or R e τ = 440 based on the friction velocity of the uncontrolled flow. The TBL is perturbed via a local surface oscillation produced by an array of spanwise-aligned piezo-ceramic (PZT) actuators and measurements are made in two orthogonal planes using particle image velocimetry (PIV). Data analyses are conducted using the vortex detection, streaky structure identification, spatial correlation and proper orthogonal decomposition (POD) techniques. It is found that the streaky structures are greatly modified in the near-wall region. Firstly, the near-wall streamwise vortices are increased in number and swirling strength but decreased in size, and are associated with greatly altered velocity correlations. Secondly, the velocity streaks grow in number and strength but contract in width and spacing, exhibiting a regular spatial arrangement. Other aspects of the streaky structures are also characterized; they include the spanwise gradient of the longitudinal fluctuating velocity and both streamwise and spanwise integral length scales. The POD analysis indicates that the turbulent kinetic energy of the streaky structures is reduced. When possible, our results are compared with those obtained by other control techniques such as a spanwise-wall oscillation, a spanwise oscillatory Lorentz force and a transverse traveling wave.

The influence of near-wall density and viscosity gradients on turbulence in channel flows

The influence of near-wall density and viscosity gradients on near-wall turbulence in a channel is studied by means of direct numerical simulation of the low-Mach-number approximation of the Navier–Stokes equations. Different constitutive relations for density $\unicode[STIX]{x1D70C}$ and viscosity $\unicode[STIX]{x1D707}$ as a function of temperature are used in order to mimic a wide range of fluid behaviours and to develop a generalised framework for studying turbulence modulations in variable-property flows. Instead of scaling the velocity solely based on local density, as done for the van Driest transformation, we derive an extension of the scaling that is based on gradients of the semilocal Reynolds number, defined as $Re_{\unicode[STIX]{x1D70F}}^{\star }\equiv Re_{\unicode[STIX]{x1D70F}}\sqrt{(\overline{\unicode[STIX]{x1D70C}}/\overline{\unicode[STIX]{x1D70C}}_{w})}/(\overline{\unicode[STIX]{x1D707}}/\overline{\unicode[STIX]{x1D707}}_{w})$ (the bar and subscript $w$ denote Reynolds averaging and wall value respectively, while $Re_{\unicode[STIX]{x1D70F}}$ is the friction Reynolds number based on wall values). This extension of the van Driest transformation is able to collapse velocity profiles for flows with near-wall property gradients as a function of the semilocal wall coordinate. However, flow quantities like mixing length, turbulence anisotropy and turbulent vorticity fluctuations do not show a universal scaling very close to the wall. This is attributed to turbulence modulations, which play a crucial role in the evolution of turbulent structures and turbulence energy transfer. We therefore investigate the characteristics of streamwise velocity streaks and quasistreamwise vortices and find that, similarly to turbulence statistics, the turbulent structures are also strongly governed by $Re_{\unicode[STIX]{x1D70F}}^{\star }$ profiles and that their dependence on individual density and viscosity profiles is minor. Flows with near-wall gradients in $Re_{\unicode[STIX]{x1D70F}}^{\star }$ ($\text{d}Re_{\unicode[STIX]{x1D70F}}^{\star }/\text{d}y\neq 0$) show significant changes in inclination and tilting angles of quasistreamwise vortices. These structural changes are responsible for the observed modulation of the Reynolds stress generation mechanism and the inter-component energy transfer in flows with strong near-wall $Re_{\unicode[STIX]{x1D70F}}^{\star }$ gradients.

Near-wall turbulence

The current state of knowledge about the structure of wall-bounded turbulent flows is reviewed, with emphasis on the layers near the wall in which shear is dominant, and particularly on the logarithmic layer. It is shown that the shear interacts with scales whose size is larger than about one third of their distance to the wall, but that smaller ones, and in particular the vorticity, decouple from the shear and become roughly isotropic away from the wall. In the buffer and viscous layers, the dominant structures carrying turbulent energy are the streamwise velocity streaks, and the vortices organize both the dissipation and the momentum transfer. Farther from the wall, the velocity remains organized in streaks, although much larger ones than in the buffer layer, but the vortices lose their role regarding the Reynolds stresses. That function is taken over by wall-attached turbulent eddies with sizes and lifetimes proportional to their heights. Two kinds of eddies have been studied in some detail: vortex clusters, and ejections and sweeps. Both can be classified into a detached background, and a geometrically self-similar wall-attached family. The latter is responsible for most of the momentum transfer, and is organized into composite structures that can be used as models for the attached-eddy hierarchy hypothesized by Townsend [“Equilibrium layers and wall turbulence,” J. Fluid Mech. 11, 97–120 (1961)]. The detached component seems to be common to many turbulent flows, and is roughly isotropic. Using a variety of techniques, including direct tracking of the structures, it is shown that an important characteristic of wall-bounded turbulence is temporally intermittent bursting, which is present at all distances from the wall, and in other shear flows. Its properties and time scales are reviewed, and it is shown that bursting is an important part of the production of turbulent energy from the mean shear. It is also shown that a linearized model captures many of its characteristics.