Wall-pressure and velocity statistics in the turbulent boundary layer (TBL) on a cambered controlled-diffusion aerofoil at $8^{\circ }$ incidence, a Mach number of 0.25 and a chord-based Reynolds number ${Re}_c=1.5\times 10^{5}$ are analysed at four locations on the suction side with zero and adverse pressure gradients (ZPG and APG), characterised by increasing Reynolds numbers based on momentum thickness, ${Re}_{\theta }=319$, 390, 877 and $1036$. The strong APG yields a highly non-equilibrium TBL at the trailing edge that significantly affects the turbulent flow statistics. Different normalisations of the full wall-pressure statistics involved in trailing-edge noise are analysed for the first time in such strong APG with convex curvature, and compared with available experimental and numerical data. Good overall agreement is found in the ZPG region, and most results obtained in previous APG TBL can be extended to the present highly non-equilibrium case. The presence of strong APG augments the intensity of wall-pressure fluctuations noticeably at low frequencies, shortens the streamwise and broadens the spanwise coherence of wall-pressure fluctuations in both time and space, and significantly reduces the convection velocity. The wall-pressure power spectral density are found to scale with the displacement thickness, the Zaragola–Smits velocity and the root-mean-squared pressure, the latter possibly being replaced by the local maximum Reynolds shear stress. The other two key parameters to trailing-edge noise modelling, the spanwise coherence length and the convection velocity, rather scale with displacement thickness and friction velocity, respectively.

The essence of the maximum drag reduction (MDR) state of viscoelastic drag-reducing turbulence (DRT) is still under debate, which mainly holds two different types of views: the marginal state of inertial turbulence (IT) and elasto-inertial turbulence (EIT). To further promote its understanding, this paper conducts a large number of direct numerical simulations of DRT at a modest Reynolds number Re with $Re = 6000$ for the FENE-P model that covers a wide range of flow states and focuses on the problem of how nonlinear extension affects the nature of MDR by varying the maximum extension length $L$ of polymers. It demonstrates that the essence of the MDR state can be both IT and EIT, where $L$ is somehow an important parameter in determining the dominant dynamics. Moreover, there exists a critical $L_c$ under which the minimum flow drag can be achieved in the MDR state even exceeding the suggested MDR limit. Systematic analyses on the statistical properties, energy spectrum, characteristic structures and underly dynamics show that the dominant dynamics of the MDR state gradually shift from IT-related to EIT-related dynamics with an increase of $L$. The above effects can be explained by the effective elasticity introduced by different $L$ at a fixed Weissenberg number (Wi) as well as the excitation of pure EIT. It indicates that larger $L$ introduces more effective elasticity and is favourable to EIT excitation. Therefore, we argue that the MDR state is still dominated by IT-related dynamics for the case of small $L$, but replaced by EIT-related dynamics at high $L$. The obtained results can harmonize the seemingly controversial viewpoints on the dominant dynamics of the MDR state and also provide some ideas for breaking through the MDR limit, such as searching for a polymer solution with a proper molecular length and concentration.

Coupling surface deformations with active stresses in two-dimensional nematic liquid crystal films leads to a rich area of investigation, particularly in biological fluid mechanics across multiple scales from tissue mechanics to cell membrane mechanics. In Al-Izzi & Morris (J. Fluid Mech., vol. 957, 2023, A4), the authors derive the complete set of governing equations for such systems. Their results provide an extended theoretical framework with which active nematic fluid films with in-plane flow and out-of-plane deformation can be analysed. To illustrate the potential applications of this framework, a few specific biologically inspired examples are discussed.

This work presents the first experimental characterization of the flow field in the vicinity of periodically spaced discrete roughness elements (DRE) in a swept wing boundary layer. The time-averaged velocity fields are acquired in a volumetric domain by high-resolution dual-pulse tomographic particle tracking velocimetry. Investigation of the stationary flow topology indicates that the near-element flow region is dominated by high- and low-speed streaks. The boundary layer spectral content is inferred by spatial fast Fourier transform (FFT) analysis of the spanwise velocity signal, characterizing the chordwise behaviour of individual disturbance modes. The two signature features of transient growth, namely algebraic growth and exponential decay, are identified in the chordwise evolution of the disturbance energy associated with higher harmonics of the primary stationary mode. A transient decay process is instead identified in the near-wake region just aft of each DRE, similar to the wake relaxation effect previously observed in two-dimensional boundary layer flows. The transient decay regime is found to condition the onset and initial amplitude of modal crossflow instabilities. Within the critical DRE amplitude range (i.e. affecting boundary layer transition without causing flow tripping) the transient disturbances are strongly receptive to the spanwise spacing and diameter of the elements, which drive the modal energy distribution within the spatial spectra. In the super-critical amplitude forcing (i.e. causing flow tripping) the near-element stationary flow topology is dominated by the development of a high-speed and strongly fluctuating region closely aligned with the DRE wake. Therefore, elevated shears and unsteady disturbances affect the near-element flow development. Combined with the harmonic modes transient growth these instabilities initiate a laminar streak structure breakdown and a bypass transition process.

This study analyses the stability characteristics of the shear layer vortices (SLV) in a reacting jet in crossflow, analysing effects of flame position, momentum flux ratio ($J$) and density ratio ($S$). It utilizes 40 kHz particle image velocimetry to characterize the dominant SLV frequencies, streamwise evolution and convective/global stability characteristics for three different canonical configurations, one non-reacting and two reacting (‘R1’ and ‘R2’). In the non-reacting case, both convective and global instability is observed, depending upon $S$ and $J$. Qualitatively similar $S$ dependencies occur for the R1 reacting case where the radial flame position lies outside the jet shear layer, albeit with slower SLV growth rates. When the flame lies inside the jet shear layer, the R2 reacting case, a qualitatively different behaviour is observed, as vorticity concentration in the shear layers is suppressed almost completely. Finally, we show that frequency and stability characteristics of the non-reacting and R1 cases can be scaled in a unified manner using a counter-current shear layer model. This model relates these SLV behaviours to a vorticity layer thickness, a velocity scale and an effective density ratio (noting that there are three distinct densities associated with the jet, the crossflow and the burned gases). These parameters were extracted from the data and used to collapse the frequency scaling, and to explain the transition to self-excited oscillatory behaviour.

Interaction of supercritical granular flow with obstacles in a confined channel generates shock waves characterised by a nearly parabolic front of agitated grains in the outer region and a heap of static grains in the inner region. The inner static heap results from granular collapse due to high volume fraction and enhanced collision rate near the obstacle. The present work reports interesting flow structures when granular shock waves are formed on an array of three identical triangular obstacles placed in a rectangular channel at different spacings. It is observed that spacing has a profound influence, resulting in three types of flow structures. Through dimensional analysis, it is found that the normalised shock stand-off distance primarily depends on the Froude number, $Fr$, and the normalised spacing between the wedges. The experimental data show a strong dependence on these parameters. The normalised shock stand-off distance decreases linearly for small $Fr$ and asymptotically approaches a small value at high $Fr$. The presence of a new stagnant dome-like structure results in a non-intuitive behaviour of shock stand-off with the wedge spacing. These features are discussed in detail using high-resolution shadowgraphy and the velocity field from particle image velocimetry.

In this paper, we put forward a hypothesis for turbulent kinetic energy, Reynolds stresses and scalar variance in wall-bounded turbulent flows, whereby these quantities, when normalized with the kinematic viscosity, mean turbulent energy dissipation rate and scalar dissipation rate, are independent of the Reynolds and Péclet numbers when they are sufficiently large. In particular, there exist two scaling ranges: (i) an inertial-convective range at sufficiently large distance from the wall over which a $2/3$ power-law scaling emerges for all quantities mentioned above; (ii) a viscous-convective range between the viscous-diffusive and inertial-convective ranges at large Prandtl number over which the normalized scalar variance is constant. The relatively large amount of available wall turbulence data either provides reasonably good support for this hypothesis or at least exhibits a trend that is consistent with the predictions of this hypothesis. The relationship between the proposed scaling and the traditional wall scaling is discussed. Possible ultimate statistical states of wall turbulence are also proposed.

The vortex shedding topology of a heavy pendulum oscillating in a dense fluid is investigated using time-resolved three-dimensional particle tracking velocimetry (tr-3-D-PTV). A series of experiments with eight different solid to fluid mass ratios $m^*$ in the range $[1.14, 14.95]$ and corresponding Reynolds numbers of up to $Re \sim O(10^4)$ was conducted. The period of oscillation depends heavily on $m^*$. The relation between amplitude decay and oscillation frequency is non-monotonic, having a damping optimum at $m^* \approx 2.50$. Moreover, a novel digital object tracking (DOT) method using vorticity-magnitude iso-surfaces is implemented to analyse vortical structures. A similar vortex shedding topology is observed for various mass ratios $m^*$. Our observations show that first, a vortex ring in the pendulum's wake is formed. Soon after, the initial ring breaks down to two clearly distinguishable structures of similar size. One of the two vortices remains on the circular path of the pendulum, while the other detaches, propagates downwards, and eventually dissipates. The time when the first vortex is shed, and its initial propagation velocity, depend on $m^*$ and the momentum imparted by the spherical bob. The findings further show good agreement between the experimentally determined vortex shedding frequency and the theoretical vortex shedding time scale based on the Strouhal number.

Numerical simulations in two space dimensions are used to examine the dynamics, transport and equilibrium behaviours of a neutrally buoyant circular object immersed in an active suspension within a larger circular container. The continuum model of Gao et al. (Phys. Rev. Fluids, vol. 2, issue 9, 2017, 093302) represents the suspension of non-interacting, immotile, extensor-type microscopic agents that have a direction and strength, and align in response to strain rate. Such a suspension is well known to be unstable above an activity strength threshold, which depends upon the length scale of the confinement. Introducing the object leads to additional phenomenology. It can confine fluid between it and the container wall, which suppresses local suspension activity. However, its motion also correlates strain rates near its surface, which induce a correspondingly correlated active-stress response. Depending on the suspension activity strength, these mechanisms lead to either an attraction toward or a repulsion away from the container wall. In addition, a persistent propagating behaviour is found for modest activity strength, which provides a mechanism for long-range transport. When activity is so weak that the mobility of the object is essential to support suspension instability and sustain flow, all flow terminates when its mobility is diminished as it nears the container wall. If activity strength is scheduled in time, then these mechanisms could be used to perform relatively complex tasks with simple active agents.

Three-dimensional steady-state Arnold–Beltrami–Childress (ABC) flow has a chaotic Lagrangian structure, and also satisfies the Navier–Stokes (NS) equations with an external force per unit mass. It is well known that, although trajectories of a chaotic system have sensitive dependence on initial conditions, i.e. the famous ‘butterfly effect’, their statistical properties are often insensitive to small disturbances. This kind of chaos (such as governed by the Lorenz equations) is called normal-chaos. However, a new concept, i.e. ultra-chaos, has been reported recently, whose statistics are unstable to tiny disturbances. Thus, ultra-chaos represents higher disorder than normal chaos. In this paper, we illustrate that ultra-chaos widely exists in Lagrangian trajectories of fluid particles in steady-state ABC flow. Moreover, solving the NS equation when $Re=50$ with the ABC flow plus a very small disturbance as the initial condition, it is found that trajectories of nearly all fluid particles become ultra-chaotic when the transition from laminar to turbulence occurs. These numerical experiments and facts highly suggest that ultra-chaos should have a relationship with turbulence. This paper identifies differences between ultra-chaos and sensitivity of statistics to parameters. Possible relationships between ultra-chaos and the Poincaré section, ultra-chaos and ergodicity/non-ergodicity, etc., are discussed. The concept of ultra-chaos opens a new perspective of chaos, the Poincaré section, ergodicity/non-ergodicity, turbulence and their inter-relationships.

This paper presents and compares two different approaches to solving the problem of wave propagation across a large finite periodic array of surface-piercing vertical barriers. Both approaches are formulated in terms of a pair of integral equations, one exact and based on a spacing $\delta > 0$ between adjacent barriers and the other approximate and based on a continuum model formally developed by using homogenisation methods for small $\delta$. It is shown that the approximate method is simpler to evaluate than the exact method which requires eigenvalues and eigenmodes related to propagation in an equivalent infinite periodic array of barriers. In both methods, the numerical effort required to solve problems is independent of the size of the array. The comparison between the two methods allows us to draw important conclusions about the validity of homogenisation models of plate array metamaterial devices. The practical interest in this problem stems from the result that for an array of barriers there exists a critical value of radian frequency, $\omega _c$, dependent on $\delta$, below which waves propagate through the array and above which it results in wave decay. When $\delta \to 0$, the critical frequency is given by $\omega _c = \sqrt {g/d}$, where $d$ is the plate submergence and $g$ is the acceleration due to gravity, which relates to the resonance in narrow channels and is an example of local resonance, studied extensively in metamaterials. The results have implications for proposed schemes to harness energy from ocean waves and other problems related to rainbow trapping and rainbow reflection.

The flow of three non-Newtonian fluids, comprising polymer and surfactant additives, in a periodically constricted tube (PCT) are experimentally compared. The radius of the tube walls is sinusoidal with respect to the streamwise direction. The three fluids are aqueous solutions of flexible polymers, rigid biopolymers and surfactants, which are typically used for drag-reduction in turbulent flows. Steady shear viscosity measurements demonstrate that rigid and flexible polymer solutions are shear-thinning, while surfactant solutions have a Newtonian and water-like shear viscosity. Capillary driven extensional rheology demonstrates that only flexible polymer solutions produce elastocapillary thinning. Particle shadow velocimetry is used to measure the velocity of each flow within the PCT at five Reynolds numbers spanning roughly 0.5 to 300. Relative to the Newtonian flows, rigid polymer solutions exhibit a blunt velocity profile. Flexible polymer solutions demonstrate a distinct chevron-shaped velocity contour and zones of opposing vorticity when the Deborah number exceeds 0.1. Using the vorticity transport equation, it is revealed that the opposing vorticity zones are coupled with a non-Newtonian torque. The PCT reveals that the surfactant solutions have similar non-Newtonian features as flexible polymer solutions – those being a chevron velocity pattern, opposing vorticity and a finite non-Newtonian torque. This observation is of practical importance since conventional shear and extensional rheometric measurements are not capable of demonstrating non-Newtonian features of the surfactant solutions. The investigation demonstrates that the PCT serves as a viable geometry for showing the non-Newtonian traits of dilute surfactant solutions.