Floating fluid-filled membrane breakwater (FFMB) is a temporary structure that can attenuate waves in the deep sea. In this paper the hydrodynamic performance of the FFMB is analysed by using the eigenfunction expansion boundary element method (EEBEM) and physical model experiments. A general motion equation is derived that considers both the dynamic tension and curvature of the membrane. Moreover, an integral expression for the dynamic tension is provided. On this basis, a linear model for solving wave–membrane interaction is established through the EEBEM. Newly designed physical experiments are performed to verify the model and elucidate the nonlinear characteristics of the FFMB. Following verification of the model, this paper investigates the effects of various structural parameters of the FFMB on the wave transmission coefficient, reflection coefficient, horizontal wave force, vertical wave force and dynamic tension. Furthermore, the interrelationship between the structural resonant response and the hydrodynamic performance is elucidated, and the optimal density and filling ratio of the FFMB for engineering applications are proposed. The results demonstrate that the numerical and experimental results are in good agreement, indicating that the model and the motion equation are both practical and highly accurate. By optimizing the structural parameters, the FFMB is capable of effectively attenuating waves within a specific frequency band, while minimizing the wave force.

Bluff-body wakes generally become three-dimensional (3-D) and then turbulent when the Reynolds number exceeds a few hundred. Other than an alternate shedding of the spanwise vortices behind the body and a gradual decay and annihilation of the vortices with distance downstream, whether a secondary vortex street would develop in the relatively far wake has been a long-standing argument in the literature. This argument is addressed in the present study. Specifically, direct numerical simulations and transient growth analysis are performed to examine the two-dimensional and 3-D wakes of different bluff bodies, including circular cylinder, square cylinder, diamond cylinder and rectangular cylinders with different cross-sectional aspect ratios. We found that a secondary vortex street is absent for most 3-D and turbulent wakes. The root cause is the weakening of spanwise vortices by 3-D wake instability modes and streamwise circulation/vorticity. The weakened spanwise vortices induce reduced mean shear in the intermediate wake, which then induces much smaller perturbation energy growth that is below the threshold for the emergence of a secondary vortex street. This finding suggests that the 3-D and turbulence characteristics, and the momentum, mass and heat transport in the relatively far wake of bluff bodies, would not be influenced by extra anisotropy or inhomogeneity caused by a secondary vortex street.

Two-dimensional (2-D) and three-dimensional (3-D) direct numerical simulations are conducted for flow past rectangular cylinders with various cross-sectional aspect ratios. The primary focuses are the interactions between the 2-D wake transitions in the spanwise vortex street (with distance downstream) and the 3-D wake transitions in the streamwise vortices, and the influence of both 2-D and 3-D wake transitions on the hydrodynamic forces on the cylinder. The 2-D wake transitions generally move upstream with increasing Reynolds number and decreasing aspect ratio. The corresponding reasons are explained. The 2-D wake transitions emerging close to the cylinder may directly alter the hydrodynamic forces on the cylinder, e.g. the Strouhal number, time-averaged drag coefficient and root-mean-square lift coefficient. By using specifically designed numerical cases to decompose the effects of the two 2-D transitions, it is found that the first 2-D transition from the primary to the two-layered vortex street results in reductions in the hydrodynamic forces, while the second 2-D transition to the secondary vortex street results in increases in the forces. The reduction/increase in the hydrodynamic forces becomes more significant when the transition location moves closer to the cylinder. The physical mechanisms for the influence on the hydrodynamic forces are elucidated. The upstream movement of the 2-D wake transitions also induces complex interactions between the 2-D and 3-D wake transitions (which also depends on the type of the 3-D mode). Correspondingly, the 3-D hydrodynamic forces may be governed by both 2-D and 3-D wake transitions (and their mutual influence).

This study investigates the scenario of flow past a circular cylinder in proximity to a moving wall (or equally a body translating in still fluid parallel to a stationary wall). Fifty high-fidelity three-dimensional direct numerical simulations are performed over a parameter space of turbulent Reynolds numbers (Re) of 300 to 1000 combined with gap-to-diameter ratios (G/D) of 0.2 to 3. The flow, hydrodynamic and turbulence characteristics over the (Re, G/D) parameter space are examined in detail. Small-scale rib-like mode B structures and alternate vortex shedding are observed for all cases. The streamwise location for the vortex shedding (quantified by the wake recirculation length Lr) varies strongly with both Re and G/D. The variation of Lr with Re and G/D can be explained by the spanwise circulation $\varGamma_z$ fed into the wake, where the variation trends of Lr and $\varGamma_z$ are inversely correlated. The variations of the mean drag and fluctuating lift coefficients are also governed by the variations in Lr and $\varGamma_z$. The total kinetic energy in the wake region reduces drastically as G/D reduces below 0.8, which is contributed collectively by (i) reduction in the strength of the shed vortices, (ii) downstream movement of the location of vortex shedding and (iii) associated delayed generation of streamwise vortices. The present results on a moving wall also help to explain several flow and hydrodynamic characteristics reported in the literature for a stationary wall, because the moving wall eliminates the complex wall boundary layer and retains a ‘clean’ near-wall effect.

Large-eddy simulations of the unsteady flows around rectangular prisms with chord-to-depth ratios (B/D) ranging from 3 to 12 are carried out at a Reynolds number of 1000. A particular focus of the study is the physical mechanisms governing the global instability of the flow. The coherent structures and velocity spectra reveal that large-scale leading-edge vortices (L vortices) are formed by the coalescence of Kelvin–Helmholtz rollers. Based on dynamic mode decomposition, the interactions between the L vortex and the trailing-edge vortex (T vortex) at different B/D values are revealed. It is found that the phase difference between the L and T vortices is the critical factor promoting a stepwise increase in the Strouhal number with increasing B/D. According to the phase analysis, there are two types of pressure feedback-loop mechanisms maintaining the self-sustained oscillations. When B/D = 4–5, the feedback loop covers the separation region, and the global instability is controlled by the impinging shear-layer instability. When B/D = 3 and 6–12, the feedback loop covers the entire chord, and the global instability is controlled by the impinging leading-edge vortex shedding instability. Self-sustained oscillations of the shear layer still exist after a splitter plate is placed in the near wake, indicating that the T vortex shedding is not essential in triggering the global instability. Nevertheless, with the participation of the T vortex, the primary instability mode may be reselected due to the upper and lower limits of the shedding frequency imposed by the T vortex.

The turbulent kinetic energy and energy dissipation rate in the wake of a circular cylinder are examined at a Reynolds number of 1000. The turbulence characteristics are quantified using direct numerical simulation, which provides a comprehensive dataset that is almost impossible to acquire from physical experiments. The energy dissipation rate is decomposed into the components due to the mean flow, the coherent primary vortices and the remainder. It is found that the remainder component, which develops only in a three-dimensional turbulent wake and resides mainly in the regions of vortices, accounts for 95 % and 97 % of the total dissipation rate for 10 and 20 cylinder diameters downstream of the cylinder, respectively (while the remainder accounts for 62 % and 83 % of the total turbulent kinetic energy). Based on the remainder component, the validity of local isotropy, local axisymmetry, local homogeneity and homogeneity in the y–z plane for the turbulent dissipation in the wake is examined. The analysis reveals that the turbulent dissipation is largely locally homogeneous, but not locally isotropic or axisymmetric, even after the annihilation of the primary vortex street. In addition, the performances of the four corresponding surrogates to the true dissipation rate are evaluated. Owing to the general validity of local homogeneity, the surrogates of local homogeneity and homogeneity in the y–z plane perform well. Although local axisymmetry does not hold, the corresponding surrogate performs well, because errors from different terms largely cancel out. However, the surrogate of local isotropy generally underestimates the true dissipation rate.

Three-dimensional (3-D) wake transition for flow past a diamond cylinder is investigated numerically. Detailed 3-D direct numerical simulations (DNS) show that the wake is represented by mode A with global vortex dislocations for Reynolds numbers Re = 121–150, followed by a mode swapping between modes A and B for Re = 160–210 and increasingly disordered mode B for Re ≥ 220. In the mode swapping regime, different characteristics of the dislocation and non-dislocation periods are revealed by decomposing flow properties (e.g. the root-mean-square lift coefficient) into the values corresponding to the dislocation and non-dislocation periods. Such decomposition helps to explain some major differences observed for the cases of a diamond and a circular cylinder. In addition to DNS, Floquet stability analyses are conducted to identify the 3-D wake instability modes of a diamond cylinder up to Re = 300. Phase-averaged base flow is used to eliminate the quantitative uncertainties induced by the aperiodic secondary vortex street of the base flow. Interestingly, a subharmonic instability mode is identified at Re ≥ 285, whereas mode B is absent. The origin of the subharmonic mode is explained. The disagreement between the DNS and the Floquet analysis regarding the existence of mode B and the subharmonic mode is also explained. It is found that the natural 3-D flow involves complex interactions between the streamwise and spanwise vortices, as well as between the 3-D wake transition and the two-dimensional base-flow transition, which excite mode B and suppress the subharmonic mode.

This study identifies and explains the formation mechanisms (FMs) of the secondary vortex street in the far wake of a circular cylinder. Direct numerical simulation and transient growth analysis are performed for a Reynolds number (Re) in the range of 100–200. Unlike most of the earlier studies, which have attributed the formation of the secondary vortices to either the hydrodynamic instability of the mean flow (FM1) or a merging of the two-layered vortices (FM2), the present study demonstrates that both FM1 and FM2 are at play. Specifically, FM1 is observed for Re = 150–160, whereas FM2 is observed for Re = 200. Interestingly, both FM1 and FM2 exist (in an alternate manner) for Re = 170 and 180. The underlying physical mechanism controlling the manifestation of either FM1 or FM2 is explored. In short, secondary vortices that emerge after and before the annihilation of the two-layered vortices are formed from FM1 and FM2, respectively. The variation in the streamwise locations for the emergence of the secondary vortices with Re is quantified and physically explained. Fundamentally, the emergence of the secondary vortices through either FM1 or FM2 is induced by the convective instability of the shear layers in the intermediate wake, which is further induced by an obvious increase in the shear rate of the shear layers as the wake gradually transitions from a primary vortex street to a two-layered vortex street. The general conclusions drawn from a circular cylinder are expected to be applicable to other bluff bodies.

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Instabilities and flow characteristics in the far wake of a circular cylinder are examined through direct numerical simulations. The transitions to the two-layered and secondary vortex streets are quantified by a new method based on the time-averaged transverse velocity field. Two processes for the transition to the secondary vortex street are observed: (i) the merging of two same-sign vortices over a range of low Reynolds numbers ($Re$) between 200 and 300, and (ii) the pairing of two opposite-sign vortices, followed by the merging of the paired vortices into subsequent vortices, over a range of $Re$ between 400 and 1000. Single vortices may be generated between the merging cycles due to mismatch of the vortices. The irregular merging process results in flow irregularity and an additional frequency signal $f_{2}$ (in addition to the primary vortex shedding frequency $f_{1}$) in the two-layered and secondary vortex streets. In particular, a gradual energy transfer from $f_{1}$ to $f_{2}$ with distance downstream is observed in the two-layered vortex street prior to the merging. The frequency spectra of $f_{2}$ are broad-band for $Re=200$–300 but become increasingly sharp-peaked with increasing $Re$ because the vortex merging process becomes increasingly regular. The ratio of the sharp-peaked frequencies $f_{2}$ and $f_{1}$ is equal to the ratio of the numbers of vortices observed after and before the merging. The general conclusions drawn from a circular cylinder are expected to be applicable to other bluff bodies.

In this paper we consider the degree-wise effect of a second step for a random walk on a graph. We prove that under the configuration model, for any fixed degree sequence the probability of exceeding a given degree threshold is smaller after two steps than after one. This builds on recent work of Kramer et al. (2016) regarding the friendship paradox under random walks.

This paper studies the friendship paradox for weighted and directed networks, from a probabilistic perspective. We consolidate and extend recent results of Cao and Ross and Kramer, Cutler and Radcliffe, to weighted networks. Friendship paradox results for directed networks are given; connections to detailed balance are considered.

Three-dimensional (3-D) wake transition for flow past a square cylinder aligned with sides perpendicular and parallel to the approaching flow is investigated using direct numerical simulation. The secondary wake instability, namely a Mode A instability, occurs at a Reynolds number ($Re$) of 165.7. A gradual wake transition from Mode A* (i.e. Mode A with vortex dislocations) to Mode B is observed over a range of $Re$ from 185 to 210, within which the probability of occurrence of vortex dislocations decreases monotonically with increasing $Re$. The characteristics of the Strouhal–Reynolds number relationship are analysed. At the onset of Mode A*, a sudden drop of the 3-D Strouhal number from its two-dimensional counterpart is observed, which is due to the subcritical nature of the Mode A* instability. A continuous 3-D Strouhal–Reynolds number curve is observed over the mode swapping regime, since Mode A* and Mode B have extremely close vortex shedding frequencies and therefore only a single merged peak is observed in the frequency spectrum. The existence of hysteresis for the Mode A and Mode B wake instabilities is examined. The unconfined Mode A and Mode B wake instabilities are hysteretic and non-hysteretic, respectively. However, a spanwise confined Mode A could be non-hysteretic. It is proposed that the existence of hysteresis at a wake instability can be identified by examining the sudden/gradual variation of the 3-D flow properties at the onset of the wake instability, with sudden and gradual variations corresponding to hysteretic (subcritical) and non-hysteretic (supercritical) flows, respectively.

The Strouhal–Reynolds number ($St{-}Re$) relationship for flow past a circular cylinder in the low $Re$ range of $Re\leqslant 1000$ is investigated through two- (2D) and three-dimensional (3D) direct numerical simulations (DNS). An improved method is proposed for the determination of the separating velocity and the wake width to allow for a better estimation of the wake Strouhal number $St^{\ast }$. For $Re$ in the extended laminar regime calculated by 2D DNS, the $St^{\ast }$ values are found to be more uniform than the original $St$ for the 2D flow. It is also found that the $St^{\ast }$ values for the 2D and 3D flows agree well in the laminar regime of $Re$ up to approximately 270. In addition, uniform $St^{\ast }$ values are also obtained for different mode A and mode B flow structures triggered artificially by using different cylinder span lengths in DNS. It is demonstrated that the drop in $St$ (with respect to its 2D counterpart) with the development of different 3D wake structures is due to the decrease in the separating velocity and the increase in the wake width for a 3D flow, rather than the existence of a particular wake structure such as pure mode A or vortex dislocation. However, as the wake flow becomes increasingly turbulent with further increase in $Re$, the $St^{\ast }$ value for the 3D flow increases gradually and deviates from its 2D counterpart, since for turbulent 3D flows the vortex shedding frequency scales on a length smaller than the wake width.

Three-dimensional (3D) wake transition for a circular cylinder placed near to a moving wall is investigated using direct numerical simulation (DNS). The study covers a parameter space spanning a gap ratio $(G/D)\geqslant 0.3$ and Reynolds number ($Re$) up to 325. The wake transition regimes in the parameter space are mapped out. It is found that vortex dislocation associated with Mode A is completely suppressed at $G/D$ smaller than approximately 1.0. The suppression of vortex dislocation is believed to be due to the confinement of the Mode A streamwise vortices by the plane wall, which suppresses the excess growth and local dislocation of any Mode A vortex loop. Detailed wake transition is examined at $G/D=0.4$, where the wake transition sequence is ‘two-dimensional (2D) $\rightarrow$ ordered Mode A $\rightarrow$ mode swapping (without dislocations) $\rightarrow$ Mode B’. Relatively strong three-dimensionality is found at $Re=160{-}220$ as the wake is dominated by large-scale structure of ordered Mode A, and also at $Re\geqslant 285$, where Mode B becomes increasingly disordered. A local reduction in three-dimensionality is observed at $Re=225{-}275$, where the wake is dominated by finer-scale structure of a mixture of ordered Modes A and B. Corresponding variations in the vortex shedding frequency and hydrodynamic forces are also investigated.

Two-dimensional (2D) and three-dimensional (3D) instabilities in the wake of a circular cylinder placed near to a moving wall are investigated using direct numerical simulation (DNS). The study covers a parameter space spanning a non-dimensional gap ratio ($G^{\ast }$) between 0.1 to 19.5 and Reynolds number ($Re$) up to 300. Variations in the flow characteristics with $Re$ and $G^{\ast }$ are studied, and their correlations with the hydrodynamic forces on the cylinder are investigated. It is also found that the monotonic increase of the critical $Re$ for 2D instability ($Re_{cr2D}$) with decreasing $G^{\ast }$ is influenced by variations in the mean flow rate around the cylinder, the confinement of the near-wake flow by the plane wall and the characteristics of the shear layer formed above the moving wall directly below the cylinder. The first factor destabilizes the wake flow at a moderate $G^{\ast }$ while the latter two factors stabilize the wake flow with decreasing $G^{\ast }$. In terms of 3D instability, the flow transition sequence of ‘2D steady $\rightarrow$ 3D steady $\rightarrow$ 3D unsteady’ for small gap ratios is analysed at $G^{\ast }=0.2$. It is found that the 3D steady and 3D unsteady flows are triggered by Mode C instability due to wall proximity. However, the Mode C structure is not sustained indefinitely, since interference with the shear layer leads to other 3D steady and unsteady flow structures.

This paper presents three-dimensional (3D) direct numerical simulations (DNS) of flow past a circular cylinder over a range of Reynolds number ($Re$) up to 300. The gradual wake transition process from mode A* (i.e. mode A with large-scale vortex dislocations) to mode B is well captured over a range of $Re$ from 230 to 260. The mode swapping process is investigated in detail with the aid of numerical flow visualization. It is found that the mode B structures in the transition process are developed based on the streamwise vortices of mode A or A* which destabilize the braid shear layer region. For each case within the transition range, the transient mode swapping process consists of dislocation and non-dislocation cycles. With the increase of $Re$, it becomes more difficult to trigger dislocations from the pure mode A structure and form a dislocation cycle, and each dislocation stage becomes shorter in duration, resulting in a continuous decrease in the probability of occurrence of mode A* and a continuous increase in the probability of occurrence of mode B. The occurrence of mode A* results in a relatively strong flow three-dimensionality. A critical condition is confirmed at approximately $Re=265{-}270$, where the weakest flow three-dimensionality is observed, marking a transition from the disappearance of mode A* to the emergence of increasingly disordered mode B structures.