The flow-induced snap-through dynamics of a buckled flexible filament was explored using the penalty immersed boundary method. The effects of the filament length, bending rigidity and Reynolds number on the mode transition were systematically examined. Three different modes were observed when the aforementioned parameters were varied: an equilibrium mode, a streamwise oscillation mode and a snap-through oscillation mode. Two mode transitions occurred when the bending rigidity was lowered and the length and Reynolds number were increased: a direct transition from the equilibrium mode to the snap-through oscillation mode and the successive appearance of the three modes. An increase in transverse fluid force induced the snap-through oscillation mode. A vortex-induced vibration and a self-excited vibration occurred in the streamwise oscillation mode and the snap-through oscillation mode, respectively. A wake pattern of 2S appeared in the streamwise oscillation mode, and a pattern of 2S + 2P appeared in the snap-through oscillation mode. A hysteresis was observed near the critical Reynolds number. The hysteresis loop increased in magnitude with increasing bending rigidity. The greater energy harvesting was achieved by the larger deflection and the higher strain energy. We found that most of the strain energy was concentrated in the last half of the filament.

Direct numerical simulations (DNSs) are performed to investigate the roughness effects on the statistical properties and the large-scale coherent structures in the turbulent channel flow over three-dimensional sinusoidal rough walls. The outer-layer similarities of mean streamwise velocity and Reynolds stresses are examined by systematically varying the roughness Reynolds number $k^{+}$ and the ratio of the roughness height to the half-channel height $k / \delta$. The energy transfer mechanism of turbulent motions in the presence of roughness elements with different sizes is explored through spectral analysis of the transport equation of the two-point velocity correlation and the scale-energy path display of the generalized Kolmogorov equation. The results show that, with increasing $k^+$, the downward shift of the mean streamwise velocity profile in the logarithmic region increases and the peak intensities of turbulent Reynolds stresses decrease. At an intermediate Reynolds number ($Re_{\tau }= 1080$), the length scale and intensity of the large-scale coherent structures increase for a small roughness ($k^{+}=10$), which leads to failure of the outer-layer similarity in rough-wall turbulence, and decrease for a large roughness ($k^{+}=60$), as compared with the smooth-wall case. The existence of the small roughness ($k^{+}=10$) enhances the mechanism of inverse energy cascade from the inner-layer small-scale structures to the outer-layer large-scale structures. Correspondingly, the self-sustaining processes of the outer-layer large-scale coherent structures, including turbulent production, interscale transport, pressure transport and spatial turbulent transport, are all enhanced, whereas the large roughness ($k^{+}=60$) weakens the energy transfer between the inner and outer regions.

The hydrodynamic mechanism of drag reduction by a flexible hairy coating was explored using the penalty immersed boundary method. A two-dimensional flexible hairy coating is constituted by multiple flexible filaments. A simulation of a cylinder without a hairy coating at a Reynolds number of 100 was also performed for comparison. The results of the simulations show good agreement with the experimental data by Niu & Hu (Phys. Fluids, vol. 23, 2011, 101701), where maximum drag reduction of 22% was attained at a particular length, bending rigidity, coating density and coating angle of the hairy coating. The hydrodynamic mechanism of drag reduction was characterized in terms of the wake pattern, shape deformation and kinetic energy of the hairy coating. The effect of a non-uniform bending rigidity of the hairy coating on drag reduction was explored. A stable streamline shape of the hairy coating was found to delay the vortex formation and stabilize the recirculation zone, resulting in decreased form drag. Active flapping of the hairy coating with enhanced vortex shedding is adverse to drag reduction. A hairy coating with a stiff base and flexible trailing edge is beneficial to maintaining a stable shape.

We explore wall-attached structures in a drag-reduced turbulent channel flow with the Navier slip boundary condition. Three-dimensional coherent structures of the streamwise velocity fluctuations (u) are examined in an effort to assess the influence of wall-attached u structures on drag reduction. We extract the u clusters from the direct numerical simulation (DNS) data; the DNS data for the no-slip condition are included for comparison. The wall-attached structures, which are physically adhered to the wall, in the logarithmic region are self-similar with their height and contribute to the presence of logarithmic behaviour. The influence of the streamwise slip on wall-attached structures is limited up to the lower bound of the logarithmic region. Although wall-attached self-similar structures (WASS) slide at the wall, the formation and hierarchy of WASS are sustained. Weakened mean shear by the streamwise slip results in a diminution in the population density of wall-attached structures within the buffer layer, leading to sparse population of WASS. In contrast, the space occupied by WASS in the fluid domain increases. The streamwise slip induces long tails in the near-wall part of WASS, reminiscent of the footprints of large-scale motions. Both a decrease in the population density of WASS and a reduction in the density of skin friction of WASS are responsible for the overall drag reduction.

Townsend's attached-eddy hypothesis (AEH) provides a theoretical description of turbulence statistics in the logarithmic region in terms of coherent motions that are self-similar with the wall-normal distance ($y$). This hypothesis was further extended by Perry and coworkers who proposed attached-eddy models that predict the coexistence of the logarithmic law in the mean velocity and streamwise turbulence intensity as well as spectral scaling for the streamwise energy spectra. The AEH can be used to predict the statistical behaviours of wall turbulence, yet revealing such behaviours has remained an elusive task because the proposed description is established within the limits of asymptotically high Reynolds numbers. Here, we show the self-similar behaviour of turbulence motions contained within wall-attached structures of streamwise velocity fluctuations using the direct numerical simulation dataset of turbulent boundary layer, channel, and pipe flows ($Re_{\tau } \approx 1000$). The physical sizes of the identified structures are geometrically self-similar in terms of height, and the associated turbulence intensity follows the logarithmic variation in all three flows. Moreover, the corresponding two-dimensional energy spectra are aligned along a linear relationship between the streamwise and spanwise wavelengths ($\lambda _x$ and $\lambda _z$, respectively) in the large-scale range ($12y < \lambda _x <3\text{--}4\delta$), which is reminiscent of self-similarity. Consequently, one-dimensional spectra obtained by integrating the two-dimensional spectra over the self-similar range show some evidence for self-similar scaling $\lambda _x \sim \lambda _z$ and the possible existence of $k_x^{-1}$ and $k_z^{-1}$ scaling regions in a similar subrange. The present results reveal that the asymptotic behaviours can be obtained by identifying the self-similar coherent structures in canonical wall turbulence, albeit in low-Reynolds-number flows.

A roughness scaling behaviour is tested by performing the direct numerical simulation (DNS) of a turbulent channel flow over three-dimensional sinusoidal rough walls. By systematically varying the roughness height ${{k}^{+}}$ and the roughness steepness S, the results for three groups of cases are considered and compared with those for flat-wall turbulence. The results show that the mean velocity and Reynolds stresses are highly dependent on both ${{k}^{+}}$ and S. To describe these specific relationships, we define a coupling scale ${{k}^{+}} S$. With this coupling scale, all the simulated data for the roughness function (${\rm \Delta} {{U}^{+}}$), the ratio of the pressure drag to the total wall resistance (${{\gamma }_{p}}$), the normalized bulk mean velocity ($U_{b}^{+}$) and the peak of the streamwise turbulent velocity fluctuations ($\overline {u_{p}^{\prime +}}$) collapse onto single curves, which shows that there is a strong direct correlation between them, i.e. ${\rm \Delta} U^{+}, \gamma _{p}, U_{b}^{+}, \overline {u_{p}^{\prime +}} \propto f(k^{+} S)$. Furthermore, a model for the prediction of wall resistance based on the roughness function can be established by defining a drag increasing ratio (DI). Accordingly, the wall resistance coefficient ${{C}_{f}}$ can be estimated directly from ${{k}^{+}}S$ of a given rough surface. These results suggest that this coupling scale provides a useful alternative to the equivalent sand grain roughness ${{k}_{s}}$.

The three-dimensional clusters of streamwise velocity fluctuations ($u$) in turbulent boundary layers (TBLs) are explored from the perspective of the attached-eddy model, which provides a basis for understanding the asymptotic behaviours of high-Reynolds-number wall turbulence in terms of coherent structures. We extract the $u$ clusters from the direct numerical simulation data of a TBL subjected to an adverse pressure gradient ($\unicode[STIX]{x1D6FD}=1.43$). For comparison, the direct numerical simulation data of a zero-pressure-gradient TBL are included. The identified structures are decomposed into attached self-similar, attached non-self-similar, detached self-similar and detached non-self-similar motions with respect to the minimum distance from the wall ($y_{min}$) and height ($l_{y}$). The attached structures ($y_{min}\approx 0$) are the main energy-containing motions and carry approximately half of the streamwise Reynolds stress and the Reynolds shear stress in the logarithmic and outer regions. The sizes of the attached self-similar structures scale with $l_{y}$, and their population density has an inverse-scale distribution over the range $0.4\unicode[STIX]{x1D6FF}<l_{y}<0.58\unicode[STIX]{x1D6FF}$ ($\unicode[STIX]{x1D6FF}$ is the 99 % boundary layer thickness). They also contribute to the logarithmic variation of the streamwise Reynolds stress and to the presence of the $k_{z}^{-1}$ region in the pre-multiplied energy spectra ($k_{z}$ is the spanwise wavenumber), i.e. these structures are universal wall motions in the logarithmic region. The tall attached structures with $l_{y}=O(\unicode[STIX]{x1D6FF})$ are non-self-similar and responsible for the enhancement of the outer large scales under the adverse pressure gradient. They extend beyond $6\unicode[STIX]{x1D6FF}$ in the streamwise direction and penetrate deeply into the near-wall region, which is reminiscent of very-large-scale motions or superstructures. The detached self-similar structures ($y_{min}>0$ and $l_{y}>100\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$) are geometrically isotropic and mainly arise in the outer region, whereas the sizes of the detached non-self-similar structures ($y_{min}>0$ and $l_{y}<100\unicode[STIX]{x1D708}/u_{\unicode[STIX]{x1D70F}}$) scale with the Kolmogorov length scale. Here, $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $u_{\unicode[STIX]{x1D70F}}$ the friction velocity. The present study provides a new perspective on the analysis of turbulence structures in the view of the attached-eddy model.

We examine the origin of very-large-scale motions (VLSMs) in fully developed turbulent pipe flow at friction Reynolds number, $\mathit{Re}_{\unicode[STIX]{x1D70F}}=934$, using data from a direct numerical simulation. The VLSMs and the packet-like large-scale motions (LSMs) found in this study are very similar to those found in earlier studies. Three-dimensional time-evolving instantaneous fields show that one component of the process leading to the large streamwise length of VLSMs is the concatenation of adjacent streamwise LSMs caused by the continuous elongation of LSMs due to the strain component of the mean shear. Spatial organization patterns of the VLSMs and LSMs and their properties are studied by separating auto-correlation of the streamwise velocity fluctuations into the components of the VLSM and the LSM defined by low-pass/high-pass filtering in the streamwise direction. The structures of the two-point spatial correlations of the streamwise velocity component of the VLSMs and the LSMs in the streamwise-azimuthal plane are characterized by multiple maxima and complex patterns that beg explanation in terms of patterned coherent arrangements of the LSMs. Using proper orthogonal decomposition (POD), it is found that the X-shape correlation pattern of the VLSMs results from the superposition of very long helically inclined structures and streamwise-aligned structures. Further explanation of the patterns in the correlations of the VLSMs and LSMs is provided through the study of synthetically constructed arrangements of simple hairpin packet models of the LSM. Head-to-tail alignment of the model packets along streamwise and helical directions suggested by the eigenvalues of the POD creates a pair of long roll-cells centred above the logarithmic layer, and bracketing the LSMs. These roll-cells are pure kinematic consequences of the induction within the LSM packets, but they may also serve to organize smaller packets.

Wall turbulence is a ubiquitous phenomenon in nature and engineering applications, yet predicting such turbulence is difficult due to its complexity. High-Reynolds-number turbulence arises in most practical flows, and is particularly complicated because of its wide range of scales. Although the attached-eddy hypothesis postulated by Townsend can be used to predict turbulence intensities and serves as a unified theory for the asymptotic behaviours of turbulence, the presence of coherent structures that contribute to the logarithmic behaviours has not been observed in instantaneous flow fields. Here, we demonstrate the logarithmic region of the turbulence intensity by identifying wall-attached structures of the velocity fluctuations ($u_{i}$) through the direct numerical simulation of a moderate-Reynolds-number boundary layer ($Re_{\unicode[STIX]{x1D70F}}\approx 1000$). The wall-attached structures are self-similar with respect to their heights ($l_{y}$), and in particular the population density of the streamwise component ($u$) scales inversely with $l_{y}$, reminiscent of the hierarchy of attached eddies. The turbulence intensities contained within the wall-parallel components ($u$ and $w$) exhibit the logarithmic behaviour. The tall attached structures ($l_{y}^{+}>100$) of $u$ are composed of multiple uniform momentum zones (UMZs) with long streamwise extents, whereas those of the cross-stream components ($v$ and $w$) are relatively short with a comparable width, suggesting the presence of tall vortical structures associated with multiple UMZs. The magnitude of the near-wall peak observed in the streamwise turbulent intensity increases with increasing $l_{y}$, reflecting the nested hierarchies of the attached $u$ structures. These findings suggest that the identified structures are prime candidates for Townsend’s attached-eddy hypothesis and that they can serve as cornerstones for understanding the multiscale phenomena of high-Reynolds-number boundary layers.

Direct numerical simulation of a turbulent boundary layer (TBL) subjected to a moderate adverse pressure gradient (APG, $\unicode[STIX]{x1D6FD}=1.45$) is performed to explore the contribution of large scales to the skin friction, where $\unicode[STIX]{x1D6FD}$ is the Clauser pressure gradient parameter. The Reynolds number based on the momentum thickness develops from $Re_{\unicode[STIX]{x1D703}}\approx 110$ to 6000 with an equilibrium region in $Re_{\unicode[STIX]{x1D703}}=4000$–5500. The spanwise wavelength ($\unicode[STIX]{x1D706}_{z}$) spectra of the streamwise and spanwise velocity fluctuations show that the large-scale energy is significantly enhanced throughout the boundary layer. We quantify the superposition and amplitude modulation effects of these enhanced large scales on the skin friction coefficient ($C_{f}$) by employing two approaches: (i) spanwise co-spectra of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$; (ii) conditionally averaged $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$. The velocity–vorticity correlations $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ are related to the advective transport and the vortex stretching, respectively. Although $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ negatively contributes to $C_{f}$, the positive contribution of the large scales ($\unicode[STIX]{x1D706}_{z}>0.5\unicode[STIX]{x1D6FF}$) is observed in the co-spectra of weighted $\langle v\unicode[STIX]{x1D714}_{z}\rangle$. For the co-spectra of weighted $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$, we observe an outer peak at $\unicode[STIX]{x1D706}_{z}\approx 0.75\unicode[STIX]{x1D6FF}$ and the superposition of the large scales in the buffer region, leading to the enhancement of $C_{f}$. The magnitude of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ depends on the large-scale streamwise velocity fluctuations ($u_{L}$). In particular, the negative-$u_{L}$ events amplify $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ in the outer region, and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ is enhanced by the positive-$u_{L}$ events. As a result, the skin friction induced by $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ increases in the present APG TBL.

A fish may gain hydrodynamic benefits from being a member of a school. Inspired by fish schools, a two-dimensional simulation was performed for flexible fins propelled in tandem, diagonal, triangular and diamond configurations. The flow-mediated interactions between the flexible fins were analysed by using an immersed boundary method. A transverse heaving motion was prescribed on the leading edge of each fin, and other posterior parts passively adapted to the surrounding fluid as a result of the fluid–flexible-body interaction. The flexible fins were allowed to actively adjust their relative positions in the horizontal direction. The four basic stable configurations are spontaneously formed and self-sustained purely by the vortex–vortex and vortex–body interactions. The hydrodynamic benefits depend greatly on the local positions of the members. For the same heaving motion prescribed on the leading edge, the input power of the following fin in the stable tandem and diagonal configurations is lower by 14 % and 6 %, respectively, than that of the leading fin. The following fin in the diagonal formation can keep pace with the leading fin even for reduced heaving amplitudes because of the help of the leader via their shared fluid environment, where its required input power is reduced by 21 %. The heaving amplitudes of the trailing fins are reduced to optimize the propulsive efficiency, and the average efficiencies in the triangular and diamond configurations increase by up to 14 % and 19 %, respectively, over that of the isolated swimmer. The propulsive efficiencies are enhanced by 22 % for the fins in the second row and by 36 % for the fin in the third row by decreasing the heaving amplitude in the diamond formation.

It has often been proposed that the formation of large-scale motion (or bulges) is a consequence of successive mergers and/or growth of near-wall hairpin vortices. In the present study, we report our direct observation that large-scale motion is generated by an instability of an ‘amplified’ streaky motion in the outer region (i.e. very-large-scale motion). We design a numerical experiment in turbulent channel flow up to $Re_{\unicode[STIX]{x1D70F}}\simeq 2000$ where a streamwise-uniform streaky motion is artificially driven by body forcing in the outer region computed from the previous linear theory (Hwang & Cossu, J. Fluid Mech., vol. 664, 2015, pp. 51–73). As the forcing amplitude is increased, it is found that an energetic streamwise vortical structure emerges at a streamwise wavelength of $\unicode[STIX]{x1D706}_{x}/h\simeq 1{-}5$ ($h$ is the half-height of the channel). The application of dynamic mode decomposition and the examination of turbulence statistics reveal that this structure is a consequence of the sinuous-mode instability of the streak, a subprocess of the self-sustaining mechanism of the large-scale outer structures. It is also found that the statistical features of the vortical structure are remarkably similar to those of the large-scale motion in the outer region. Finally, it is proposed that the largest streamwise length of the streak instability determines the streamwise length scale of very-large-scale motion.

Direct numerical simulation data of a turbulent boundary layer ($Re_{\unicode[STIX]{x1D70F}}=1000$) were used to investigate the large-scale influences on the vortical structures that contribute to the local skin friction. The amplitudes of the streamwise and wall-normal swirling strengths ($\unicode[STIX]{x1D706}_{x}$ and $\unicode[STIX]{x1D706}_{y}$) were conditionally sampled by measuring the large-scale streamwise velocity fluctuations ($u_{l}$). In the near-wall region, the amplitudes of $\unicode[STIX]{x1D706}_{x}$ and $\unicode[STIX]{x1D706}_{y}$ decreased under negative $u_{l}$ rather than under positive $u_{l}$. This behaviour arose from the spanwise motions within the footprints of the large-scale low-speed ($u_{l}<0$) and high-speed structures ($u_{l}>0$). The intense spanwise motions under the footprint of positive $u_{l}$ noticeably strengthened the small-scale spanwise velocity fluctuations ($w_{s}$) below the centre of the near-wall vortical structures as compared to $w_{s}$ within the footprint of negative $u_{l}$. The streamwise and wall-normal components were attenuated or amplified around the modulated vortical motions, which in turn led to the dependence of the swirling strength on the $u_{l}$ event. We quantified the contribution of the modulated vortical motions $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$, which were related to a change-of-scale effect due to the vortex-stretching force, to the local skin friction. In the near-wall region, intense values of $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ were observed for positive $u_{l}$. By contrast, these values were low for negative $u_{l}$, in connection with the amplification of $w_{s}$ and $\unicode[STIX]{x1D706}_{y}$ by the strong spanwise motions of the positive $u_{l}$. The resultant skin friction induced by the amplified vortical motions within $u_{l}^{+}>2$ was responsible for 15 % of the total skin friction generated by the change-of-scale effect. Finally, we applied this analysis to a drag-reduced flow and found that the amplified vortical motions within the footprint of positive $u_{l}$ were markedly diminished, which ultimately contributed to the total drag reduction.

The effect of large-scale motions (LSMs) on the turbulent/non-turbulent (T/NT) interface is examined in a turbulent boundary layer. Using flow fields from direct numerical simulation, the shape of the interface and near-interface statistics are evaluated conditional on the position of the LSM. The T/NT interface is identified using the vorticity magnitude and a streak detection algorithm is adopted to identify and track the LSMs. Two-point correlation and spectral analysis of variations in the interface height show that the spatial undulation of the interface is longer in streamwise wavelength than the boundary-layer thickness, and grows with the Reynolds number in a similar manner to the LSMs. The average variation in the interface height was evaluated conditional on the position of the LSMs. The result provides statistical evidence that the interface is locally modulated by the LSMs in both the streamwise and spanwise directions. The modulation is different when the coherent structure is high- versus low-speed motion: high-speed structures lead to a wedge-shaped deformation of the T/NT interface, which causes an anti-correlation between the angles of the interface and the internal shear layer. On the other hand, low-speed structures are correlated with crests in the interface. Finally, the sudden changes in turbulence statistics across the interface are in line with the changes in the population of low-speed structures, which consist of slower mean streamwise velocity and stronger turbulence than the high-speed counterparts.

Direct numerical simulation data from turbulent pipe and channel flows at $\mathit{Re}_{\unicode[STIX]{x1D70F}}\approx 930$ are used to investigate their statistical difference by focusing on large-scale motions (LSMs). The contribution to the bulk production of turbulent kinetic energy shows marked differences in the overlap and core regions. These discrepancies arise from the dominant contributions of the LSMs ($\unicode[STIX]{x1D706}_{x}>3\unicode[STIX]{x1D6FF}$) to the Reynolds shear stress in the channel flow. The spectrum of the net Reynolds shear force reveals that the LSMs accelerate the mean flow in the overlap region. The net force spectrum is further decomposed into the spectra of velocity–vorticity correlations, $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$, which are related to the advective vorticity transport and the change-of-scale effect, respectively. The dominance of large-scale accelerating motions (LSAMs) in the overlap region of the channel flow is due to the contribution of $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ at longer wavelengths ($\unicode[STIX]{x1D706}_{x}>3\unicode[STIX]{x1D6FF}$), The LSAMs are related to the long low-speed regions, and these regions are longer and wider in the channel flow than in the pipe flow. Due to the pipe curvature, the spanwise size of the LSMs is restricted by neighbouring LSMs and the spanwise velocity fluctuations are reduced. The contribution of $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ to the acceleration is prominent in the channel flow, leading to the dominance of the LSAMs associated with the change-of-scale effect.

Schooling behaviours among self-propelled animals can benefit propulsion. Inspired by the schooling behaviours of swimming jellyfish, flexible bodies that self-propel through a paddling-based motion were modelled in a tandem configuration. This present study explored the hydrodynamic patterns generated by the interactions between two flexible bodies and the surrounding fluid in the framework of the penalty immersed boundary method. The hydrodynamic patterns produced in the wake revealed flow-mediated interactions between two tandem propulsors, including vortex–vortex and vortex–body interactions. Two tandem flexible propulsors paddling with identical amplitude and frequency produced stable configurations as a result of the flow-mediated interactions. Both the upstream and downstream propulsors benefited from the tandem configuration in terms of the locomotion velocity and the cost, compared with an isolated propulsion system. The interactions were examined as a function of the initial gap distance and the phase difference in the paddling frequency. The equilibrium gap distance between two propulsors remained constant, regardless of the initial gap distance, although it did depend on the phase difference in the paddling frequency.

Direct numerical simulation data of turbulent channel flow ($Re_{{\it\tau}}=930$) are used to investigate the statistics of long motions of streamwise velocity fluctuations ($u$), and the interaction of these structures with the near-wall disturbances, which is facilitated by their associated large-scale circulations. In the log layer, the negative-$u$ structures are organized into longer streamwise extent (${>}3{\it\delta}$) in comparison to the positive-$u$ counterparts. Near the wall, the footprint of negative-$u$ structures is relatively narrow in comparison to the footprint of positive-$u$ structures. This difference is due to the opposite spanwise motions in the vicinity of the footprints, which are either congregative or dispersive depending on the circulation of the outer roll cells. Conditional sampling of the footprints shows that the spanwise velocity fluctuations ($w$) are significantly enhanced by the dispersive motions of high-speed structures. On the other hand, the near-wall congregative motions of negative-$u$ structures generate relatively weak $w$ but intense negative-$u$ regions due, in part, to the spanwise collective migration of near-wall streaks. The concentrated near-wall regions of negative-$u$ upwell during the merging of the outer long scales – an effect that is demonstrated using statistical analysis of the merging process. This leads to a reduction of the convection speed of downstream negative-$u$ structures and thus promotes the merging with upstream ones. These top-down and bottom-up interactions enhance the spatial coherence of long negative-$u$ structures in the log region.

The active flapping motions of fish and cetaceans generate both propulsive and manoeuvring forces. The tail fin motions of the majority of fish can essentially be viewed as a combined pitch-and-heave motion. Downstream bodies are strongly influenced by the vortices shed from an upstream body. To investigate the interactions between flexible bodies and vortices, the present study examined tandem flexible flags in a viscous flow by using an improved version of the immersed boundary method. The upstream flag underwent passive flapping in a uniform flow while the downstream flag flapped according to a prescribed pitching and heaving motion of the leading edge. The influences of the active flapping motion on the system dynamics were examined in detail, including the frequency, the phase angle, the bending coefficient and the amplitudes of the pitching and heaving motion. The variation of the drag coefficient of the downstream flag was explored together with the instantaneous vorticity contours and the body shapes. Both the slalom mode and the interception mode were identified according to the vortex–flexible body interactions, corresponding to the low- and high-drag situations, respectively. The underlying mechanism was discussed and compared with previous studies.

Direct numerical simulations were carried out to investigate the spatial features of large- and very-large-scale motions (LSMs and VLSMs) in a turbulent channel flow ($\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_{\tau }=930$). A streak detection method based on the streamwise velocity fluctuations was used to individually trace the cores of LSMs and VLSMs. We found that both the LSM and VLSM populations were large. Several of the wall-attached LSMs stretched toward the outer regions of the channel. The VLSMs consisted of inclined outer LSMs and near-wall streaks. The number of outer LSMs increased linearly with the streamwise length of the VLSMs. The temporal features of the low-speed streaks in the outer region revealed that growing and merging events dominated the large-scale (1–$3\delta $) structures. The VLSMs $({>}3\delta )$ were primarily created by merging events, and the statistical analysis of these events supported that the merging of large-scale upstream structures contributed to the formation of VLSMs. Because the local convection velocity is proportional to the streamwise velocity fluctuations, the streamwise-aligned structures of the positive- and negative-$u$ patches suggested a primary mechanism underlying the merging events. The alignment of the positive- and negative-$u$ structures may be an essential prerequisite for the formation of VLSMs.