Quasi-streamwise Vortices Research Articles

Analysis of the Interconnections between Classic Vortex Models of Coherent Structures Based on DNS Data

Low- and high-speed streaks (ejection, Q2, and sweep, Q4, events in quadrant analysis) are significant features of coherent structures in turbulent flow. Streak formation is closely related to turbulent structures in several vortex models, such as attached eddy models, streamwise vortex analysis models, and hairpin vortex models, which are all standard models. Vortex models are complex, whereby the relationships among the different vortex models are unclear; thus, further studies are still needed to complete our understanding of vortices. In this study, 30 sets of direct numerical simulation (DNS) data were obtained to analyze the mechanics of the formation of coherent structures. Image processing techniques and statistical analysis were used to identify and quantify streak characteristics. We used a method of vortex recognition to extract spanwise vortices in the x–z plane. Analysis of the interactions among coherent structures showed that the three standard vortex models all gave reasonably close results. The attached eddy vortex model provides a good explanation of the linear dimensions of streaky structures with respect to the water depth and Q2 and Q4 events, whereby it can be augmented to form the quasi-streamwise vortex model. The legs of a hairpin vortex envelop low-speed streaky structures and so move in the streamwise direction; lower-velocity vortex legs also gradually accumulate into a streamwise vortex. Statistical analysis allowed us to combine our present results with some previous research results to propose a mechanism for the formation of streaky structures. This study provides a deeper understanding of the interrelationships among different vortex models.

Study of wave effect on vorticity in Langmuir turbulence using wave-phase-resolved large-eddy simulation

The effects of a water surface wave on the vorticity in the turbulence underneath are studied for Langmuir turbulence using wave-phase-resolved large-eddy simulation. The simulations are performed on a dynamically evolving wave-surface-fitted grid such that the phase-resolved wave motions and their effects on the turbulence are explicitly captured. This study focuses on the vorticity structures and dynamics in Langmuir turbulence driven by a steady and co-aligned progressive wave and surface shear stress. For the first time, the detailed vorticity dynamics of the wave–turbulence interaction in Langmuir turbulence in a wave-phase-resolved frame is revealed. The wave-phase-resolved simulation provides detailed descriptions of many characteristic features of Langmuir turbulence, such as elongated quasi-streamwise vortices. The simulation also reveals the variation of the strength and the inclination angles of the vortices with the wave phase. The variation is found to be caused by the periodic stretching and tilting of the wave orbital straining motions. The cumulative effect of the wave on the wave-phase-averaged vorticity is analysed using the Lagrangian average. It is discovered that, in addition to the tilting effect induced by the Lagrangian mean shear gradient of the wave, the phase correlation between the vorticity fluctuations and the wave orbital straining is also important to the cumulative vorticity evolution. Both the fluctuation correlation effect and the mean tilting effect are found to amplify the streamwise vorticity. On the other hand, for the vertical vorticity, the fluctuation correlation effect cancels the mean tilting effect, and the net change of the vertical vorticity by the wave straining is negligible. As a result, the wave straining enhances only the streamwise vorticity and cumulatively tilts vertical vortices towards the streamwise direction. The above processes are further quantified analytically. The role of the fluctuation correlation effect in the wave-phase-averaged vorticity dynamics provides a deeper understanding of the physical processes underlying the wave–turbulence interaction in Langmuir turbulence.

Large Eddy Simulation of Turbulent Channel Flows over Rough Walls with Stochastic Roughness Height Distributions

This paper studies the 3-D turbulent channel flows over rough walls with stochastic roughness height distributions by using the large eddy simulation and the immersed boundary method. The obtained mean and fluctuating velocity profiles for the smooth and rough channel flows agree well with the available numerical and experimental results. The stochastic surface roughness is found to have a more significant influence than the uniform surface roughness on the turbulent velocity statistics and the coherent structures, with the same average roughness height. With a greater variation in the roughness height, the mean velocity and the streamwise fluctuating velocity is decreased and the spanwise velocity and the wall-normal fluctuating velocity are increased. In addition, one observes larger and more profuse quasi-streamwise vortices, hairpin vortices and elongated structures above the crest plane of the roughness array in cases of highly stochastic rough walls. However, the low-speed streaky structures are broken up locally and the ejection and sweep events are depressed by the stochastic roughness below the average roughness height. The results of this study support Townsend’s wall-similarity hypothesis for both stochastic and uniform rough wall turbulences, demonstrating that in both cases the effects of the surface roughness on the turbulent flow are limited to the rough sub-layer.

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.

Numerical study of the turbulent channel flow under space-dependent electromagnetic force control at different Reynolds numbers

In this paper, the control of turbulent channel flow by space-dependent electromagnetic force and the mechanism of drag reduction are investigated with the direct numerical simulation (DNS) methods for different Reynolds numbers. A formulation is derived to express the relation between the drag and the Reynolds shear stress. With the application of optimal electromagnetic force, the in-depth relations among characteristic structures in the flow field, mean Reynolds shear stress, and the effect of drag reduction for different Reynolds numbers are discussed. The results indicate that the maximum drag reductions can be obtained with an optimal combination of parameters for each case of different Reynolds numbers. The regular quasi-streamwise vortex structures, which appear in the flow field, have the same period with that of the electromagnetic force. These structures suppress the random velocity fluctuations, which leads to the absolute value of mean Reynolds shear stress decreasing and the distribution of that moving away from the wall. Moreover, the wave number of optimal electromagnetic force increases, and the scale of the regular quasi-streamwise vortex structures decreases as the Reynolds number increases. Therefore, the rate of drag reduction decreases with the increase in the Reynolds number since the scale of the regular quasi-streamwise vortex structures decreases.

Proposal of control laws for turbulent skin friction reduction based on resolvent analysis

This paper evaluates and modifies the so-called suboptimal control technique for turbulent skin friction reduction through a combination of low-order modelling and direct numerical simulation (DNS). In a previous study, Nakashima et al. (J. Fluid Mech., vol. 828, 2017, pp. 496–526) employed resolvent analysis to show that the efficacy of suboptimal control was mixed across spectral space when the streamwise wall shear stress (case ST) was used as a sensor signal, i.e. specific regions of spectral space showed drag increment. This observation suggests that drag reduction may be attained if control is applied selectively in spectral space. DNS results presented in the present study, however, do not show a significant effect on the flow with selective control. A posteriori analyses attribute this lack of efficacy to a much lower actuation amplitude in the simulations compared to model assumptions. Building on these observations, resolvent analysis is used to design and provide a preliminary assessment of modified control laws that also rely on sensing the streamwise wall shear stress. Control performance is then assessed by means of DNS. The proposed control laws generate as much as $10\,\%$ drag reduction, and these results are broadly consistent with resolvent-based predictions. The physical mechanisms leading to drag reduction are assessed via conditional sampling. It is shown that the new control laws effectively suppress the near-wall quasi-streamwise vortices. A physically intuitive explanation is proposed based on a separate evaluation of clockwise and anticlockwise vortices.

Vortex axis tracking by iterative propagation (VATIP): a method for analysing three-dimensional turbulent structures

Vortex is a central concept in the understanding of turbulent dynamics. Objective algorithms for the detection and extraction of vortex structures can facilitate the physical understanding of turbulence regeneration dynamics by enabling automated and quantitative analyses of these structures. Despite the wide availability of vortex identification criteria, they only label spatial regions belonging to vortices, without any information on the identity, topology and shape of individual vortices. This latter information is stored in the axis lines lining the contours of vortex tubes. In this study, a new tracking algorithm is proposed which propagates along the vortex axis lines and iteratively searches for new directions for growth. The method is validated in flow fields from transient simulations where vortices of different shapes are controllably generated. It is then applied to statistical turbulence for the analysis of vortex configurations and distributions. It is shown to reliably extract axis lines for complex three-dimensional vortices generated from the walls. A new procedure is also proposed that classifies vortices into commonly observed shapes, including quasi-streamwise vortices, hairpins, hooks and branches, based on their axis-line topology. Clustering analysis is performed on the extracted axis lines to reveal vortex organization patterns and their potential connection to large-scale motions in turbulence.

Vortex structures and drag reduction in turbulent channel flow with the effect of space-dependent electromagnetic force

The flow of a weakly conductive fluid (i.e. seawater) can be controlled by electromagnetic forces, which holds promising applications in the drag reduction. In this paper, a fully developed weakly conductive turbulent flow controlled by a space-dependent electromagnetic force is investigated with direct numerical simulation (DNS) methods for different Reynolds numbers. With the application of electromagnetic force, the in-depth relations among characteristic structures in the flow field, mean Reynolds shear stress and the effect of drag reduction for different Reynolds numbers are discussed. The numerical results indicate that the well-organized quasi-streamwise vortex structures induced by electromagnetic force are generated with the optimal values of the amplitude and the wave number. The Reynolds shear stress is suppressed and the maximum of drag reduction is obtained with the effect of these structures. However, with the increase of Reynolds number, the wave number of optimal electromagnetic force increases, resulting in the decrease of scale of the well-organized quasi-streamwise vortex structures, which leads to the decrease of the height of these vortex structures. Therefore, the maximum rate of drag reduction decreases due to the decrease of the effective depth with the increase of Reynolds number.

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.

Coherent structures in streamwise rotating channel flow

Direct numerical simulation and large-eddy simulation are employed to study the effects of streamwise rotation on turbulent channel flow, especially the coherent structures. Two Reynolds numbers (Re = Umh/ν = 2800 and 7000) with a wide range of rotation numbers (0 ≤ Ro = 2Ωh/Um ≤ 7.5) are considered, where Ω denotes the angular velocity of the system and Um and h are the bulk velocity and one-half of the channel height, respectively. The demand on the streamwise computational domain size increases as the rotation number increases. Flow statistics are presented and discussed. Among the near-wall quasi-streamwise vortices rotating in opposite directions, the rotation of the channel tends to promote the cyclones (vortices rotating in the same direction as the channel) and suppress the anti-cyclones, which is visualized by the conditional-sampling method. Elongated large-scale vortices, which typically form in rotating homogeneous turbulence, are observed through the conditional-sampling method. Similar to the columnar vortices in homogeneous turbulence, the elongated large-scale vortices are dominated by cyclones at Ro ∼ 1, while the dominance becomes less apparent as the rotation number further increases. When rotation is suddenly imposed to fully developed turbulent flow, the integral scale along the rotating axis increases linearly at a growth rate proportional to the rotation number. What is interesting is that the cyclones of the large-scale vortices are not completely elongated along the rotating axis but slightly tilted to the spanwise direction. The cyclones prefer to develop in the lower and upper half channels, while the anti-cyclones, if they exist, occur in the central region of the channel. This preference of the large-scale vortices in different regions contributes to the secondary flow. It is revealed that the mean shear of the channel flow may have effects on the tilting of the large-scale vortical structures and the preference of their distributions in the channel.

Turbulent duct flow with polymers

We have performed direct numerical simulation of the turbulent flow of a polymer solution in a square duct, with the FENE-P model used to simulate the presence of polymers. First, a simulation at a fixed moderate Reynolds number is performed and its results compared with those of a Newtonian fluid to understand the mechanism of drag reduction and how the secondary motion, typical of the turbulent flow in non-axisymmetric ducts, is affected by polymer additives. Our study shows that the Prandtl’s secondary flow is modified by the polymers: the circulation of the streamwise main vortices increases and the location of the maximum vorticity moves towards the centre of the duct. In-plane fluctuations are reduced while the streamwise ones are enhanced in the centre of the duct and dumped in the corners due to a substantial modification of the quasi-streamwise vortices and the associated near-wall low- and high-speed streaks; these grow in size and depart from the walls, their streamwise coherence increasing. Finally, we investigated the effect of the parameters defining the viscoelastic behaviour of the flow and found that the Weissenberg number strongly influences the flow, with the cross-stream vortical structures growing in size and the in-plane velocity fluctuations reducing for increasing flow elasticity.

On the structure of the self-sustaining cycle in separating and reattaching flows

The separating and reattaching flows and the wake of a finite rectangular plate are studied by means of direct numerical simulation data. The large amount of information provided by the numerical approach is exploited here to address the multi-scale features of the flow and to assess the self-sustaining mechanisms that form the basis of the main unsteadinesses of the flows. We first analyse the statistically dominant flow structures by means of three-dimensional spatial correlation functions. The developed flow is found to be statistically dominated by quasi-streamwise vortices and streamwise velocity streaks as a result of flow motions induced by hairpin-like structures. On the other hand, the reverse flow within the separated region is found to be characterized by spanwise vortices. We then study the spectral properties of the flow. Given the strongly inhomogeneous nature of the flow, the spectral analysis has been conducted along two selected streamtraces of the mean velocity field. This approach allows us to study the spectral evolution of the flow along its paths. Two well-separated characteristic scales are identified in the near-wall reverse flow and in the leading-edge shear layer. The first is recognized to represent trains of small-scale structures triggering the leading-edge shear layer, whereas the second is found to be related to a very large-scale phenomenon that embraces the entire flow field. A picture of the self-sustaining mechanisms of the flow is then derived. It is shown that very-large-scale fluctuations of the pressure field alternate between promoting and suppressing the reverse flow within the separation region. Driven by these large-scale dynamics, packages of small-scale motions trigger the leading-edge shear layers, which in turn created them, alternating in the top and bottom sides of the rectangular plate with a relatively long period of inversion, thus closing the self-sustaining cycle.

Turbulent channel flow of an elastoviscoplastic fluid

We present numerical simulations of laminar and turbulent channel flow of an elastoviscoplastic fluid. The non-Newtonian flow is simulated by solving the full incompressible Navier–Stokes equations coupled with the evolution equation for the elastoviscoplastic stress tensor. The laminar simulations are carried out for a wide range of Reynolds numbers, Bingham numbers and ratios of the fluid and total viscosity, while the turbulent flow simulations are performed at a fixed bulk Reynolds number equal to 2800 and weak elasticity. We show that in the laminar flow regime the friction factor increases monotonically with the Bingham number (yield stress) and decreases with the viscosity ratio, while in the turbulent regime the friction factor is almost independent of the viscosity ratio and decreases with the Bingham number, until the flow eventually returns to a fully laminar condition for large enough yield stresses. Three main regimes are found in the turbulent case, depending on the Bingham number: for low values, the friction Reynolds number and the turbulent flow statistics only slightly differ from those of a Newtonian fluid; for intermediate values of the Bingham number, the fluctuations increase and the inertial equilibrium range is lost. Finally, for higher values the flow completely laminarizes. These different behaviours are associated with a progressive increases of the volume where the fluid is not yielded, growing from the centreline towards the walls as the Bingham number increases. The unyielded region interacts with the near-wall structures, forming preferentially above the high-speed streaks. In particular, the near-wall streaks and the associated quasi-streamwise vortices are strongly enhanced in an highly elastoviscoplastic fluid and the flow becomes more correlated in the streamwise direction.

Exact coherent states with hairpin-like vortex structure in channel flow

Hairpin vortices are widely studied as an important structural aspect of wall turbulence. The present work describes, for the first time, nonlinear travelling wave solutions to the Navier–Stokes equations in the channel flow geometry – exact coherent states (ECS) – that display hairpin-like vortex structure. This solution family comes into existence at a saddle-node bifurcation at Reynolds number $Re=666$. At the bifurcation, the solution has a highly symmetric quasi-streamwise vortex structure similar to that reported for previously studied ECS. With increasing distance from the bifurcation, however, both the upper and lower branch solutions develop a vortical structure characteristic of hairpins: a spanwise-oriented ‘head’ near the channel centreplane where the mean shear vanishes connected to counter-rotating quasi-streamwise ‘legs’ that extend toward the channel wall. At $Re=1800$, the upper branch solution has mean and Reynolds shear-stress profiles that closely resemble those of turbulent mean profiles in the same domain.

An investigation of strong backflow events at the interface of air–water systems using large-eddy simulation

ABSTRACT A large-eddy simulation of a counter-current gas–liquid flow is performed. At the flat interface where the different fluids meet, continuity of momentum and momentum fluxes are enforced following the work of Lombardi et al. [Direct numerical simulation of near-interface turbulence in coupled gas-liquid flow. Phys Fluids. 1996;8(6):1643–1665]. The increase in vertical vorticity fluctuations near the interface increases mixing, reducing the thickness of the inner region of the boundary layer. Such increase reduces shear while allowing for more frequent backflow motions in the inner region, being this phenomenon stronger on water. Due to the higher inertia of water these backflow motions are ultimately responsible for the streaky structure of shear stresses seen along the interface. The present study shows that such bimodality in the streamwise velocities is also seen in the angle distribution of vorticity relative to the interface, where such angles are linked to the presence of interface-connected and quasi-streamwise vortex cores. Finally, it is shown that backflow events on the interface shear stresses correlate with coupled ‘strong’ ejections in the near interface region despite the disparagingly different near-interface streamwise velocity distributions on the near interface boundary layers.

Manipulation of near-wall turbulence by surface slip and permeability

We study the effect on near-wall turbulence of tangential slip and wall-normal transpiration, typically produced by textured surfaces and other surface manipulations. For this, we conduct direct numerical simulations (DNSs) with different virtual origins for the different velocity components. The different origins result in a relative wall-normal displacement of the near-wall, quasi-streamwise vortices with respect to the mean flow, which in turn produces a change in drag. The objective of this work is to extend the existing understanding on how these virtual origins affect the flow. In the literature, the virtual origins for the tangential velocities are typically characterised by slip boundary conditions, while the wall-normal velocity is assumed to be zero at the boundary plane. Here we explore different techniques to define and implement the three virtual origins, with special emphasis on the wall-normal one. We investigate impedance conditions relating the wall-normal velocity to the pressure, and linear relations between the velocity components and their wall-normal gradients, as is typically done to impose slip conditions. These models are first tested to represent a smooth wall below the boundary plane, with all virtual origins equal, and later for different tangential and wall-normal origins. Our results confirm that the change in drag is determined by the offset between the origins perceived by mean flow and the quasi-streamwise vortices or, more generally, the near-wall turbulent cycle. The origin for the latter, however, is not set by the spanwise virtual origin alone, as previously proposed, but by a combination of the spanwise and wall-normal origins, and mainly determined by the shallowest of the two. These observations allow us to extend the existing expression to predict the change in drag, accounting for the wall-normal effect when the transpiration is not negligible.

Large-scale coherent structures of suspended dust concentration in the neutral atmospheric surface layer: A large-eddy simulation study

Dust particles can remain suspended in the atmospheric boundary layer, motions of which are primarily determined by turbulent diffusion and gravitational settling. Little is known about the spatial organizations of suspended dust concentration and how turbulent coherent motions contribute to the vertical transport of dust particles. Numerous studies in recent years have revealed that large- and very-large-scale motions in the logarithmic region of laboratory-scale turbulent boundary layers also exist in the high Reynolds number atmospheric boundary layer, but their influence on dust transport is still unclear. In this study, numerical simulations of dust transport in a neutral atmospheric boundary layer based on an Eulerian modeling approach and large-eddy simulation technique are performed to investigate the coherent structures of dust concentration. The instantaneous fields confirm the existence of very long meandering streaks of dust concentration, with alternating high- and low-concentration regions. A strong negative correlation between the streamwise velocity and concentration and a mild positive correlation between the vertical velocity and concentration are observed. The spatial length scales and inclination angles of concentration structures are determined, compared with their flow counterparts. The conditionally averaged fields vividly depict that high- and low-concentration events are accompanied by a pair of counter-rotating quasi-streamwise vortices, with a downwash inside the low-concentration region and an upwash inside the high-concentration region. Through the quadrant analysis, it is indicated that the vertical dust transport is closely related to the large-scale roll modes, and ejections in high-concentration regions are the major mechanisms for the upward motions of dust particles.

Effects of the interface position of water-air flow on turbulent wall structures

Direct numerical simulations (DNS) are performed to investigate the effect of the interface position of water-air flow on turbulence statistics and flow structures in wall-bounded water-air turbulent flow through a straight channel. Water depths of 90 and 180 viscous wall units, referred to as shallow-water and deep-water cases, respectively, are examined. Water-to-air density and viscosity ratios of 831.7 and 55.56 are considered to model a realistic flow condition at temperature of 25 oC and pressure of 1 atm. The Reynolds number and Froude number are set to 180 and 1.22 x 10-4, respectively, for both shallow-water and deep-water case, based on the friction velocity at the bottom wall, the half depth of the channel, and water density and viscosity. The Navier-Stokes equations are solved using a timesplitting projection method on an octree grid structure, while the deformation of the interface between water and air is computed using a volume-of-fluid method. With the presence of the water-air interface, velocity profiles in deep-water and shallow-water cases are found to slightly deviate from the log-law profile for a single phase turbulent flow in a channel. The deviation is magnified when the interface is placed closer to the log-law region. Turbulent velocity fluctuations in the water stream are found to be associated with quasi-streamwise vortices and hairpin vortices. The quasi-streamwise vortices which are attached close to the wall are found in both deep-water and shallow-water cases. However, the hairpin vortices of which leading portions are lifted away from the wall are found to be diminished in the shallow-water case while they are clearly observed in the deep-water case.

Visualization and assessment of turbulent coherent structures in laminarescent boundary layers

Direct numerical simulation (DNS) with high spatial/temporal resolution of spatially developing turbulent boundary layers subject to very strong favorable pressure gradient (FPG) has been performed. The sudden acceleration imposed on the flow provokes a significant attenuation of turbulent intensities (Araya et al. in J Fluid Mech 775:189–200, 2015), particularly due to the dominance of pressure forces over approximately frozen Reynolds stresses, Narasimha and Sreenivasan (Adv Appl Mech 19:221–309, 1979). In this article, visualization of the DNS velocity field is carried out in order to evaluate the effect of very strong FPG on elementary coherent structures, such as low- and high-speed streaks, sweeps and ejections, quasi-streamwise and horseshoe vortices, and large-scale motions. An important symbiosis between quasi-streamwise vortices and low-speed streaks (or regeneration cycle) has been observed in the FPG zone. The destabilizing shear layers upstream of vortex heads in strong accelerated flows appear as very thin zones developing in the buffer region ( $$20<y^+< 30$$ ). It seems that decaying $$uv'$$ in strong FPG flows (Araya et al. 2015) is principally due to the annihilation of sweeps, while ejections are responsible for the turbulence residual.

Optimal heat transfer enhancement in plane Couette flow

Optimal heat transfer enhancement has been explored theoretically in plane Couette flow. The vector field (referred to as the ‘velocity’) to be optimised is time independent and divergence free, and temperature is determined in terms of the velocity as a solution to an advection-diffusion equation. The Prandtl number is set to unity, and consistent boundary conditions are imposed on the velocity and the temperature fields. The excess of a wall heat flux (or equivalently total scalar dissipation) over total energy dissipation is taken as an objective functional, and by using a variational method the Euler–Lagrange equations are derived, which are solved numerically to obtain the optimal states in the sense of maximisation of the functional. The laminar conductive field is an optimal state at low Reynolds number$Re\sim 10^{0}$. At higher Reynolds number$Re\sim 10^{1}$, however, the optimal state exhibits a streamwise-independent two-dimensional velocity field. The two-dimensional field consists of large-scale circulation rolls that play a role in heat transfer enhancement with respect to the conductive state as in thermal convection. A further increase of the Reynolds number leads to a three-dimensional optimal state at$Re\gtrsim 10^{2}$. In the three-dimensional velocity field there appear smaller-scale hierarchical quasi-streamwise vortex tubes near the walls in addition to the large-scale rolls. The streamwise vortices are tilted in the spanwise direction so that they may produce the anticyclonic vorticity antiparallel to the mean-shear vorticity, bringing about significant three-dimensionality. The isotherms wrapped around the tilted anticyclonic vortices undergo the cross-axial shear of the mean flow, so that the spacing of the wrapped isotherms is narrower and so the temperature gradient is steeper than those around a purely streamwise (two-dimensional) vortex tube, intensifying scalar dissipation and so a wall heat flux. Moreover, the tilted anticyclonic vortices induce the flow towards the wall to push low- (or high-) temperature fluids on the hot (or cold) wall, enhancing a wall heat flux. The optimised three-dimensional velocity fields achieve a much higher wall heat flux and much lower energy dissipation than those of plane Couette turbulence.