Turbulent entrainment at the turbulent/non-turbulent interface (TNTI) plays an important role in understanding the turbulent diffusion. While entrainment in fully developed canonical turbulent flows has been extensively studied, the evolution of entrainment in spatially developing flows remains poorly understood. In this work, characteristics of entrainment and the effect of vortices on entrainment of the shear layer separated from a wall-mounted fence are studied by the experiment in a water channel. The shedding vortex experiences a series of stages, including generation, growth, deformation and breakdown into smaller vortices. With the development of the flow, entrainment varies correspondingly. The prograde vortex near the TNTI is found to suppress entrainment but have little effect on the detrainment process, while the retrograde vortex promotes entrainment and suppresses detrainment as well. Consequently, the local entrainment velocity is decreased by the prograde vortex and increased more significantly by the retrograde vortex. Along the streamwise direction, the time-mean entrainment velocity is smallest where the prograde vortex is strongest in the vortex deformation stage. However, the largest time-mean entrainment velocity is located where the enstrophy gradient near the TNTI is greatest after reattachment, rather than where the retrograde vortex is strongest shortly after the breakdown of the shedding vortex, because the scarcity of retrograde vortices in the vicinity of the TNTI makes their long-time cumulative contribution not as significant as their local enhancement. The present study reveals how entrainment evolves in the separated and reattaching flow, and improves our understanding of the effect of vortices on entrainment.

The turbulent boundary layer (TBL) is a widely existing flow phenomenon in nature and engineering applications. Its strong mixing effect can achieve more sufficient material mixing, heat transport, etc. The understanding of the entrainment process and mechanism of irrotational fluids entering the turbulent region can be promoted by studying the geometric and dynamic characteristics of turbulent${/}$non-turbulent interfaces (TNTI). In compressible flow, it is unclear whether the properties of TNTI will change and whether the entrainment will show different features due to the influence of compressibility. Based on the direct numerical simulation results of supersonic compressible plate TBLs with Mach number of 2.9, the geometric and dynamic characteristics of TNTI are investigated in this paper. The interface is identified by the enstrophy method, and the height, thickness, fractal dimension, enstrophy transportation and entrainment characteristics of the interface are investigated. It is found that for the enstrophy transportation in a TBL, the contribution of compressibility-related terms accounts for approximately 13.4 % of the total enstrophy transportation, which tends to transfer the enstrophy of turbulence near the interface to both directions vertical to the interface. This promotes the expansion of the turbulent region towards the non-turbulent region, and the mean height, thickness and entrainment velocity are increased by approximately 3.7 %, 7.0 % and 8.5 %, respectively, while the fractal dimension is basically unaffected. Different from the incompressible flow, the contribution of the compressibility-related terms to the entrainment velocity is independent of the local curvature, and the intense entrainment process is more likely to occur on a highly curved concave surface.

Three-dimensional flows over low-aspect-ratio rectangular flat plates (${A{\kern-4pt}R} = 1.00$–$1.50$) are investigated using tomographic and planar particle image velocimetry techniques. The chord-based Reynolds number is $5400$, and the angle of attack is fixed at $6^\circ$. This study reveals for the first time the interplay between spanwise fluid transport and downwash, both originating from the tip effects. Spanwise fluid transport promotes the formation and subsequent coherent development of leading-edge vortices, whereas downwash stabilizes the flow. Specifically, two mechanisms related to spanwise fluid transport are revealed. First, the spanwise fluid transport enhances the intensity of the reversed flow, promoting the shear layer roll-up and vortex shedding. Second, the near-wall spanwise flow interacts with the shed C-shape vortices, thereby strengthening the vortex heads. In particular, through these interactions, spanwise fluid transport can sustain the coherence of the C-shape vortices until the vortex heads split in a regular fashion. Consequently, the C-shape vortices are transformed into novel Þ-shape vortices for the plates of ${A{\kern-4pt}R} \leq 1.25$, which supplements the previously discovered transformation from C-shape to M-shape vortices for larger ${A{\kern-4pt}R}$ plates. Downstream of this novel vortex-splitting transformation, two fundamental processes contribute to the formation of hairpin vortices. The above comprehensive understanding of complete vortex evolution routine provides valuable insights into the tip effects on the formation of three-dimensional flows over low-${A{\kern-4pt}R}$ plates.

The understanding of the entrainment mechanism of synthetic jets can help optimise the synthetic jet actuators in engineering applications. It is generally believed that vortex rings or strong velocity fluctuations in the near field of the synthetic jet are responsible for its enhanced entrainment. However, in recent years, it has been found that the enhanced entrainment of the synthetic jet may be caused by the instability or the vortex ring breakdown in the transition region. To shed new light on this issue, synthetic jets with different Reynolds numbers and dimensionless stroke lengths are investigated with time-resolved two-dimensional particle image velocimetry. Based on the analyses of velocity triple-decomposition, Fourier mode decomposition and phase-averaged $\lambda _{ci}D/U_0$ field, the streamwise positions of the vortex ring breakdown are determined for the synthetic jets, and the entrainment coefficient can be divided into three components, i.e. the coherent turbulent kinetic energy production, the random turbulent kinetic energy production and the shape of the velocity profile. It is found that the entrainment coefficient is dominated by the component related to the random turbulent kinetic energy production, and reaches its peak value at the position of vortex ring breakdown. The results obtained in different cases show a strong correlation between vortex ring breakdown and entrainment enhancement. From the perspective of instantaneous snapshot, the mechanism of vortex ring breakdown enhanced entrainment is revealed, that is, vortex ring breakdown enhanced the small-scale vortex near the turbulent/non-turbulent interface, resulting in an increase of enstrophy production, and thus enhanced local entrainment.

The near wake of a hemisphere immersed in a laminar boundary layer is studied utilizing time-resolved tomographic particle image velocimetry (TPIV). Focus is placed on the three-dimensional vortical structures and the formation details of hairpin vortices before the onset of transition. The three-dimensional instantaneous pressure field of the hemisphere wake is reconstructed for better understanding the flow mechanism. Experiments are carried out with Reynolds number $Re_{r}=1370$, based on the hemisphere radius $R$. Features of periodicity of the near wake are analysed using proper orthogonal decomposition and Fourier transformation. The velocity fluctuation in the wall-normal direction is shown to be crucial to the formation of hairpin vortices in the near wake. By investigating the transport of mass and vorticity, and the correlation between pressure and hairpin vortex strength, the formation mechanism is revealed clearly. Specifically, the main hairpin vortices (MHVs) are formed within the reaction of outer high-speed flow and near-wall flow. The formation of the head portion is followed by the leg portion formation. The shedding of the MHVs is highly correlated with the pressure, as well as the pressure gradient in the wall-normal direction. For the side hairpin vortices (SHVs), the leg portion is formed first, followed by the generation of the head portion thanks to induction of the re-oriented standing vortices. The generation of the SHVs can be regarded as the downstream bridging of the standing vortices, similar to the generation of hairpin vortices due to the connection of streamwise vortices in turbulent boundary layers.

The three-dimensional flow over a low-aspect-ratio (low-$A\!R$) trapezoidal plate is investigated experimentally with a focus on how the tip effects impact the structure and dynamics of the separation bubble. The chord-based Reynolds number is $5800$, and the angle of attack varies from $4^\circ$ to $10^\circ$. Once the flow separates, the separation bubble emerges and features a swallow-tailed structure that shrinks near the midspan, which is first found for the flows over low-$A\!R$ plates. This structure develops into the conventional single-tailed structure as the angle of attack increases. Moreover, the vortex shedding within the swallow-tailed separation bubble is restored from multiple asynchronously measured local velocity fields. It is revealed that the leading-edge vortex undergoes the novel transformation from a C-shape vortex into an M-shape vortex. This vortex transformation stems from the mass transport of the near-wall spanwise flow, which affects the fluid motion on the windward side of the C-shape vortex head, strengthening and accelerating the vortex head. The strengthened vortex head facilitates the entrainment of high-momentum fluid from the outer flow. This is responsible for the formation of the swallow-tailed structure. These findings help to fill the gaps left by the downwash at low angles of attack for low-$A\!R$ wings, and are of value in improving the cruising and gliding performance of micro-air vehicles.

Synthetic jets have received extensive attention due to their superior mixing property. However, its mechanisms have not been investigated from the perspective of the turbulent/non-turbulent interface (TNTI). To shed new light on this issue, the entrainment and TNTI properties of a synthetic jet are experimentally investigated and compared with a continuous jet at $Re_j = 3150$. The fuzzy clustering method is applied to select an appropriate vorticity threshold to detect the TNTI. Statistically, it is revealed that the entrainment coefficients of the two jets significantly differ in the near field, while they become almost identical in the far field. Instead of the vortex ring, the ‘breakdown of the vortex ring’ enhances the entrainment in the present synthetic jet. Instantaneously, the TNTI more violently fluctuates in the near field of the synthetic jet, which leads to a larger fluctuation in the TNTI radial position, a higher fractal dimension and enhanced local entrainment. Moreover, the transition of the probability density function of the TNTI orientation from the unimodal distribution in the near field to the bimodal distribution in the far field is found in both jets. The multi-scale analysis reveals a new mechanism for the bimodal distribution caused by the TNTI-thickness-scale structures.

Piezoelectric macro-fibre composite (MFC) actuators are employed onto the flexible leeward surface of an airfoil for active control. Time-resolved aerodynamic forces, membrane deformations and flow fields are synchronously measured at low Reynolds number (Re = 6 × 104). Mean aerodynamics show that the actively controlled airfoil can achieve lift-enhancement and drag-reduction simultaneously in the angle of attack range of 10° ≤ α ≤ 14°, where the rigid airfoil encounters stall. The maximum increments of lift and lift-to-drag ratio are 27.1 % and 126 % at the reduced actuation frequency of ${f^ + } = 3.52$. The unsteady coupling features are further analysed at α = 12°, where the maximum lift-enhancement occurs. It is newly discovered that the membrane vibrations and flow fields are locked into half of the actuation frequency when ${f^ + } > 3$. The shift of the dominant vibration mode from bending to inclining is the reason for the novel ‘half-frequency lock-in’ phenomenon. To the fluid–structure interaction, there are three characteristic frequencies for the actively controlled airfoil: $S{t_1} = 0.5{f^ + }$, $S{t_2} = {f^ + }$, and $S{t_3} = 1.5{f^ + }$. Here, St1 and its harmonics (St2, St3) are coupled with the natural frequencies of the leading-edge shear layer, resulting in the generation of multi-scale flow structures and suppression of flow separation. The lift presents comparable dominant frequencies between St1 and St3, which means the instantaneous lift is determined by the flow structures of St1 and St3. The local membrane bulge and dent affect the instantaneous swirl strength of flow structures near the maximum vibration amplitude location, which is the main reason for the variation of instantaneous lift.

Physical representation of coherent motions in wall-bounded turbulence satisfying Townsend's attached-eddy hypothesis (AEH) is still an open question to be debated. Here, we apply proper orthogonal decomposition (POD) to instantaneous snapshots of $u$ component velocity fields in the streamwise–wall-normal plane of three turbulent boundary layers, which cover a frictional Reynolds number ($Re$) of $Re_\tau \approx 1000{\sim} 4000$ and were either measured by large-field-of-view planar particle image velocimetry or calculated by direct numerical simulation. The $u$ component POD modes characterized by streamwise recurring large-scale motions are decomposed into wall-attached and wall-detached parts depending on the wall-attachment condition. The former, i.e. wall-attached POD eddies, are believed to be the prime statistical representations of attached eddies. The reason lies in three aspects. Firstly, wall-attached POD eddies are geometrically self-similar and statistically independent. Secondly, the wall-normal variation of the variance of reduced-order streamwise velocity components $\tilde {u}_{AE}$, which is only contributed by wall-attached POD eddies, presents logarithmic decay in the inertial flow region. Thirdly, analysis on the scaling behaviour of the high-even-order moment, moment generating function, as well as longitudinal two-point structure function, all indicate a quasi-Gaussian behaviour of $\tilde {u}_{AE}$. None of these AEH-predicted behaviours are evident when the contribution of superstructure-like POD eddies are considered. Furthermore, wall-detached POD eddies are found to also present quasi-Gaussian behaviour. They might be treated as passive fragments of wall-attached ones.

The modulations of high/low-speed large-scale motions (H/L-SMs) on the turbulent/non-turbulent interface (TNTI) and turbulent entrainment are investigated in turbulent boundary layers via both experimental and numerical studies. The spanwise locations of large-scale motions can be locked by the spanwise heterogeneity, so the boundary layers over such a configuration are investigated first as an instructive case. In the engulfment process, it is found that irrotational ‘bubbles’ near the TNTI are more likely to originate from engulfment, while bubbles far from the TNTI could be produced by the local turbulence itself. Additionally, H-SMs are found to enhance the engulfment by the sweep flow. In the nibbling process, a competition relationship is observed: L-SMs induce stronger instantaneous ‘nibbling’ events by transporting more fluids towards the TNTI, while the H-SMs induced a more distorted TNTI. Consequently, the integral nibbling flux is greater above H-SMs. Furthermore, the explored mechanisms are verified to be insensitive to the wall shapes such as smooth and homogeneous roughness walls, which demonstrates that these modulations are universal for turbulent boundary layers. Finally, a conceptual modulation model is proposed to illustrate the entrainment process above the large-scale motions.

Bypass transition in flow over a flat plate triggered by a pair of dielectric-barrier-discharge plasma actuators mounted on the plate surface and aligned in the streamwise direction is investigated. A four-species plasma–fluid model is used to model the electrohydrodynamic force generated by the plasma actuation. A pair of counter-rotating streamwise vortices is created downstream of the actuators, leading to the formation of a high-speed streak in the centre and two low-speed streaks on each side. As the length of actuators increases, more momentum is added to the boundary layer and eventually a turbulent wedge is generated at an almost fixed location. With large spanwise distance between the actuators (wide layout), direct numerical simulations indicate that the low-speed streaks on both sides lose secondary stability via an inclined varicose-like mode simultaneously, leaving a symmetric perturbation pattern with respect to the centre of the actuators. Further downstream, the perturbations are tilted by the mean shear of the high- and low-speed streaks and consequently a ‘W’-shaped pattern is observed. When the pair of plasma actuators is placed closer (narrow layout) in the spanwise direction, the mean shear in the centre becomes stronger and secondary instability first occurs on the high-speed streak with an asymmetric pattern. Inclined varicose-like and sinuous-like instabilities coexist in the following breakdown of the negative streaks on the side and the perturbations remain asymmetric with respect to the centre. Here the tilting of disturbances is dominated by the mean shear in the centre and the perturbations display a ‘V’ shape. Linear analysis techniques, including biglobal stability and transient growth, are performed to further examine the fluid physics; the aforementioned phenomena at narrow and wide layouts, such as the secondary instabilities, the ‘V’ and ‘W’ shapes, and the symmetric and asymmetric breakdown, are all observed.

Representing complex flows by evolving vortex structures is an important principle in many investigations of wall-bounded turbulence. The practice of this principle benefits from the bi-directional transformation between the velocity field and the corresponding vortex field. While the velocity-to-vortex transformation could be implemented by various vortex identification criteria, few efforts have been devoted to the inverse process. This work develops a linear reconstruction method, which allows an effective reconstruction for the velocity field of wall turbulence based on a given vortex field. The vortex field is defined as a vector field by combining the swirl strength and the real eigenvector of the velocity gradient tensor. The reconstructed velocity fields are calculated by convolution operations on the vortex fields, with the kernel functions derived by the field-based linear stochastic estimation. The method can effectively recover the turbulent motions in a large scale range, showing clear advantages over the Biot–Savart formula in the near-wall region. The method is also employed to investigate the inducing effects of vortices at different heights. The wall-bounding effect on the induced motions is observed from the contribution spectra of vortices. The higher-order moments of the reconstructed streamwise velocity component present larger deviations from the original data, which is discussed and explained reasonably. At last, the vortex fields filtered by prescribed thresholds are employed to reconstruct the velocity fields. It is found that the strongest vortex components occupying 5 % of the total volume can reasonably recover the main flow features including both the near-wall streaks and the large-scale motions.

Characteristics of particle clustering in a particle-laden turbulent boundary layer at a moderate frictional Reynolds number ($Re_\tau =5500$) are experimentally investigated based on Voronoï analysis. High-inertia sand grains with a large viscous-time-based Stokes number, i.e. $St_p=10^2 - 10^3$, are used as the dispersed phase. The bulk particle volume fraction is $\varPhi _V\sim O(10^{-5})$. Two-dimensional velocity fields of the fluid and particle phases in the streamwise–wall-normal plane are simultaneously measured via particle image/tracking velocimetry. Under the net sedimentation condition, a self-similarity of the geometries of particle clusters is clearly seen in the log layer. This can be characterized by the $-5/3$ power law of the probability density functions of the particle cluster areas and the $y$-scaling of their first- and second-order moments. Considering the self-similarity of wall-attached structures in the attached eddy hypothesis, we propose a conceptual model in which a large proportion of particle clusters prefer to reside on the back ridges of low-momentum attached $u$-structures due to the high-strain effect caused by converging sweep–ejection events. Such a scenario can be partly evidenced by the conditional streamwise spectra of the streamwise and wall-normal fluid-phase velocity components being conditioned inside cluster-occupied regions, which present a distinct $y$-scaling in the log layer. Furthermore, the fluid-phase sweep and ejection events inside particle clusters are observed to be weaker than the cases outside cluster-occupied regions, and the local rate of strain inside cluster-occupied regions is relatively higher than in all unconditional statistics. Finally, conditional statistics reveal that those sand grains with relatively small slip velocities are prone to be captured by low-momentum turbulent motions to form particle clusters.

The fluid–structure--sound interaction in reducing the flow-induced noise of a circular cylinder with a back flexible splitter plate is investigated experimentally. The rigid splitter plate (RSP) and the flexible splitter plate (FSP) are individually attached to the rear of the cylinder with a diameter D = 30 mm. Three parameters are considered: the flexibility of the plates, the length of the plates (L = 0.5 D ~ 6.0 D) and the Reynolds numbers (Re = 3.83×104 ~ 9.57 × 104). Both RSP and FSP achieve the optimal reduction in the Aeolian tone at L/D = 1.0 regardless of Re, where the RSP results in a reduction of 14 dB on average and the FSP a superior reduction of 18 dB. Subsequently, the velocity fields are measured with two-dimensional particle image velocimetry for L/D = 1.0. The dynamic deformation of the FSP is captured with a single-camera three-dimensional digital image correlation system. The results show that the splitter plate elongates the recirculation region length (Lr) significantly, and it is preferable to normalize the streamwise coordinate x with Lr instead of D in this investigation. It is determined that the Aeolian tone is mainly induced by the fluctuations in the near wake region of 1 < x/Lr < 3. The splitter plates effectively suppress the overall fluctuations originating from vortex shedding in the wake, thereby reducing the Aeolian tone successfully. The movement of the FSP, which is dominated by low-frequency vibrations with broadband and a relatively energetic periodic vibration, has important effects on the secondary and high-frequency noise.

Student's t test is valid for statistical inference under the normality assumption or asymptotically. By contrast, although the bootstrap t test was proposed in 1993, it is seldom adopted in medical research. We aim to demonstrate that the bootstrap t test outperforms Student's t test under normality in data. Using random data samples from normal distributions, we evaluated the testing performance, in terms of true-positive rate (TPR) and false-positive rate and diagnostic abilities, in terms of the area under the curve (AUC), of the bootstrap t test and Student's t test. We explore the AUC of both tests with varying sample size and coefficient of variation. We compare the testing outcomes using the COVID-19 serial interval (SI) data in Shenzhen and Hong Kong, China, for demonstration. With fixed TPR, the bootstrap t test maintained the equivalent accuracy in TPR, but significantly improved the true-negative rate from the Student's t test. With varying TPR, the diagnostic ability of bootstrap t test outperformed or equivalently performed as Student's t test in terms of the AUC. The equivalent performances are possible but rarely occur in practice. We find that the bootstrap t test outperforms by successfully detecting the difference in COVID-19 SI, which is defined as the time interval between consecutive transmission generations, due to sex and non-pharmaceutical interventions against the Student's t test. We demonstrated that the bootstrap t test outperforms Student's t test, and it is recommended to replace Student's t test in medical data analysis regardless of sample size.

The role of large-scale motions (LSMs) in energy transfer is investigated by analysing wall-parallel velocity fields at low-to-moderate Reynolds number ($Re_{\tau }=1200\text {--}3500$), which are obtained via a two-dimensional (2-D) particle image velocimetry measurement with large field-of-view. Two types of energy flux, i.e. local interscale energy flux and in-plane spatial energy flux are inspected in detail. Targeting the energy transfer in large-scale regime, an anisotropic filter is designed based on the zero-crossing scale boundary in a 2-D energy transfer spectrum, across which the net energy flux is the maximum. This ‘optimal’ energy flux boundary separates the scale space into an energy donating large-scale part and an energy receiving small-scale one. The crossover energy flux, as well as the associated flow field structures, are studied by conditional statistics and linear stochastic estimation, in which the statistical spanwise symmetry is deliberately broken by designing special velocity gradient conditions for event probing. A strong connection between large-scale energy flux events and LSMs are found. Namely, forward scatter events have higher probability to reside on the wavy flank of low-momentum LSMs, if compared with the scenario of being clamped in the middle of two streamwise-aligned high- and low-momentum LSMs (Natrajan & Christensen, Phys. Fluids, vol. 18, issue 6, 2006, pp. 299–325). Meanwhile, pairs of positive and negative spatial transfer events tend to locate inside LSMs. It is thus argued that the meandering nature of LSMs, which forms the necessary velocity gradient, might play a determining role in the process of large-scale energy transfer. The spatial correlation between them is then schematized in a conceptual model, which explains most of the present observations.

Most previous research has handled the task of ship type recognition by exploring hand-craft ship features, which may fail to distinguish ships with similar visual appearances. This situation motivates us to propose a novel deep learning based ship type recognition framework which we have named coarse-to-fine cascaded convolution neural network (CFCCNN). First, the proposed CFCCNN framework formats the input training ship images and data, and provides trainable input data for the hidden layers of the CFCCNN. Second, the coarse and fine steps are run in a nesting manner to explore discriminative features for different ship types. More specifically, the coarse step is trained in a similar manner to the traditional convolution neural network, while the fine step introduces regularisation mechanisms to extract more intrinsic ship features, and fine tunes parameter settings to obtain better recognition performance. Finally, we evaluate the performance of the CFCCNN model for recognising the most common types of merchant ship (oil tanker, container, LNG tanker, chemical carrier, general cargo, bulk carrier, etc.). The experimental results show that the proposed framework obtains better recognition performance than the conventional methods of ship type recognition.

The flow over a square cylinder controlled by a slot synthetic jet positioned at the front surface is investigated experimentally at different excitation frequencies. The Reynolds number based on the free-stream velocity and the side length of the square cylinder is 1000. The flow visualization was conducted using the laser-induced fluorescence technique. The velocity fields upstream and downstream of the square cylinder were measured synchronously with the two-dimensional time-resolved particle image velocimetry technique. Both the evolution of vortex structures and the characteristic frequencies of upstream and downstream flow fields are presented. The flow dynamics vary significantly with the excitation frequency at a fixed stroke length. During one excitation cycle, the synthetic jet vortex pair deflects to one side and later swings to the other side at a quite small excitation frequency of $f_{e}/f_{0}=0.6$, while it only deflects toward one side and does not turn to the other side at $f_{e}/f_{0}=1.0$. Compared with the natural case, the wake characteristics for the above two cases are not changed much by the synthetic jet adopted. At a moderate excitation frequency of $f_{e}/f_{0}=2.0$, the synthetic jet deflects upwards and downwards alternatively. The upstream flow field has a dominant frequency identical to half of the excitation frequency. Under the perturbations of the synthetic jet, two wake vortex pairs are formed per shedding cycle with a shedding frequency equal to that of the square cylinder without control. At a higher excitation frequency of $f_{e}/f_{0}=3.4$, the synthetic jet keeps deflecting to one side, and the upstream flow field is governed by the excitation frequency. The flow separation on the deflected side is suppressed effectively, and no periodic vortex shedding can be observed in the wake. Statistically, the velocity profiles also change with control. The recirculation bubble length in the wake is shortened, and the time-averaged velocity fluctuation is weakened remarkably. The control effects of the synthetic jet and the continuous jet are compared in this paper when placed at the front surface of a square cylinder.

Vortex structures are very popular research objects in turbulent boundary layers (TBLs) because of their prime importance in turbulence modelling. This work performs a tomographic particle image velocimetry measurement on the near-wall region ($y<0.1\unicode[STIX]{x1D6FF}$) of TBLs at three Reynolds numbers $Re_{\unicode[STIX]{x1D70F}}=1238$, 2286 and 3081. The main attention is paid to the wall-normal evolution of the vortex geometries and topologies. The vortex is identified with swirl strength ($\unicode[STIX]{x1D706}_{ci}$), and its orientation is recognized by using the real eigenvector of the velocity gradient tensor. The vortex inclination angles in the streamwise–wall-normal plane and in the streamwise–spanwise plane as functions of wall-normal positions are investigated, which provide useful information to speculate on the three-dimensional shape of the vortex tubes in a TBL. The difference between the orientations of vorticity and swirl is discussed and their inherent relationship is revealed based on the governing equation of vorticity. Linear stochastic estimation (LSE) is further deployed to directly extract three-dimensional vortex models. The LSE velocity fields for ejection events happening at different wall-normal positions shed light on the evolution of the topologies for the vortices dominating ejection events. LSE based on a centred prograde spanwise vortex provides a typical packet model, which indicates that the population density of the packets in a TBL is large enough to leave footprints in conditionally averaged flow fields. This work should help to settle the severe debate on the existence of packet structures and also lays some foundation for the TBL model theory.