We investigated experimentally the settling behaviour of vertically aligned spherical particles within various quiescent media at different release frequencies. The particles had a diameter of $d = 4$ mm and density of $\rho _s = 2200$ kg m$^{-3}$, and were released near the free surface of water, ethanol, a G60 water–glycerine mixture (60 % glycerine by weight) and oil media at frequencies of $f_P = 4$, 6 and 8 Hz, thereby allowing study of Galileo numbers, $Ga \in [16, 976]$. Particle tracking velocimetry quantified the motion of nearly 800 particles in a 600 mm high tank, and particle image velocimetry examined flow patterns around the particles. Results revealed that the centre of mass of the particle trajectories exhibited preferential in-plane motions, with significant lateral dispersion and large $Ga$ in water and ethanol, and nearly vertical paths with low $Ga$ in the G60 mixture and oil media. Varying degrees of particle separation resulted in higher terminal velocities than for a single particle. Hence, particle drag decreased in all cases, with the oil medium showing the highest drag reduction under the closest particle separation, reaching up to nearly 70 % of that for the single particle. The vertical and lateral pair dispersions, $R^2_z$ and $R^2_L$, exhibited ballistic scaling, with dependences on the initial separation, $r_0$, and the type of medium. With large $Ga$, $R^2_z$ displayed a ballistic regime followed by a slower rate, whereas with small $Ga$, $R^2_z$ maintained a consistent ballistic regime throughout settling. Finally, normalized $R^2_z$ demonstrated distinct scaling (exponent 2/3 and 1) dependent on the normalized initial separation and $Ga$.

We investigate self-excited vortex-acoustic lock-in in a bluff body combustor. The bluff body not only anchors the flame but also sheds vortices. Mutual interaction between vortex shedding and the combustor acoustic field leads to lock-in. A lower-order model for vortex shedding is coupled to the acoustic equations to obtain a set of discrete dynamical maps, which are then solved numerically. Lock-in is identified when a definite phase relationship between vortex shedding and acoustic field remains unaltered with time. The common frequency during (phase) lock-in is either close to the natural acoustic or vortex shedding frequencies, accordingly termed A- or V-lock-in, respectively. Instability and amplitude suppression occur with most parts of the A- and V-lock-in regions, respectively. Furthermore, we relate the lock-in and pre-lock-in regimes observed in our previous experiments (Singh & Mariappan, Combust. Sci. Technol., 2019, pp. 1–29) to A- and V-lock-in phenomena, respectively. Among the two lock-in phenomena, combustion instability is favoured by $1:1$ A-lock-in, whose boundaries in the parameter space can be obtained from a forced response of the vortex shedding process. Finally, we discuss the bifurcations leading to lock-in.

An experimental study of long interfacial gravity waves was conducted in a closed wave tank containing two layers of viscous immiscible fluids. The study focuses on the development in time of the mean particle drift that occurs close to the interface where the two fluids meet. From a theoretical analysis by Weber & Christensen (Eur. J. Mech.-B/Fluids, vol. 77, 2019, pp. 162–170) it is predicted that the growing drift in this region is associated with the action of the virtual wave stress. This effect has not been explored experimentally before. Interfacial waves of different amplitudes and frequencies were produced by a D-shaped paddle. Particle tracking velocimetry was used to track the time development of the Lagrangian mean drift. The finite geometry of the wave tank causes a mean return flow that is resolved by mass transport considerations. The measurements clearly demonstrate the importance of the virtual wave stress mechanism in generating wave drift currents near the interface.

This work studies the three-dimensional flow dynamics around a rotating circular cylinder of finite length, whose axis is positioned perpendicular to the streamwise direction. Direct numerical simulations and global stability analyses are performed within a parameter range of Reynolds number $Re=DU_\infty /\nu <500$ (based on cylinder diameter $D$, uniform incoming flow velocity $U_\infty$), length-to-diameter ratio ${\small \text{AR}}=L/D\leq 2$ and dimensionless rotation rate $\alpha =D\varOmega /2U_\infty \leq 2$ (where $\varOmega$ is rotation rate). By solving Navier–Stokes equations, we investigated the wake patterns and explored the phase diagrams of the lift and drag coefficients. For a cylinder with ${\small \text{AR}}=1$, we found that when the rotation effect is weak ($0\leq \alpha \lesssim 0.3$), the wake pattern is similar to the unsteady wake past the non-rotating finite-length cylinder, but with a new linear unstable mode competing to dominate the saturation state of the wake. The flow becomes stable for $0.3\lesssim \alpha \lesssim 0.9$ when $Re<360$. When the rotation effect is strong ($\alpha \gtrsim 0.9$), new low-frequency wake patterns with stronger oscillations emerge. Generally, the rotation effect first slightly decreases and then sharply increases the $Re$ threshold of the flow instability when $\alpha$ is relatively small, but significantly decreases the threshold at high $\alpha$ ($0.9<\alpha \leq 2$). Furthermore, the stability analyses based on the time-averaged flows and on the steady solutions demonstrate the existence of multiple unstable modes undergoing Hopf bifurcation, greatly influenced by the rotation effect. The shapes of these global eigenmodes are presented and compared, as well as their structural sensitivity, visualising the flow region important for the disturbance development with rotation. This research contributes to our understanding of the complex bluff-body wake dynamics past this critical configuration.

The recent Lagrangian relaxation towards equilibrium (LaRTE) approach (Fowler et al., J. Fluid Mech., vol. 934, 2022, A44) is a wall model for large-eddy simulations (LES) that isolates quasi-equilibrium wall-stress dynamics from non-equilibrium responses to time-varying LES inputs. Non-equilibrium physics can then be modelled separately, such as the laminar Stokes layers that form in the viscous region and generate rapid wall-stress responses to fast changes in the pressure gradient. To capture additional wall-stress contributions due to near-wall turbulent eddies, a model term motivated by the attached eddy hypothesis is proposed. The total modelled wall stress thus includes contributions from various processes operating at different time scales (i.e. the LaRTE quasi-equilibrium plus laminar and turbulent non-equilibrium wall stresses) and is called the multi-time-scale (MTS) wall model. It is applied in LES of turbulent channel flow subject to a wide range of unsteady conditions from quasi-equilibrium to non-equilibrium. Flows considered include pulsating and linearly accelerating channel flow for several forcing frequencies and acceleration rates, respectively. We also revisit the sudden spanwise pressure gradient flow (considered in Fowler et al., J. Fluid Mech., vol. 934, 2022, A44) to review how the newly introduced model features affect this flow. Results obtained with the MTS wall model show good agreement with direct numerical simulation data over a vast range of conditions in these various non-equilibrium channel flows. To further understand the MTS model, we also describe and test the instantaneous-equilibrium limit of the MTS wall model. In this limit, good wall-stress predictions are obtained with reduced model complexity but providing less complete information about the wall-stress physics.

Metachronal rowing is a biological propulsion mechanism employed by many swimming invertebrates (e.g. copepods, ctenophores, krill and shrimp). Animals that swim using this mechanism feature rows of appendages that oscillate in a coordinated wave. In this study, we used observations of a swimming ctenophore (comb jelly) to examine the hydrodynamic performance and vortex dynamics associated with metachronal rowing. We first reconstructed the beating kinematics of ctenophore appendages based on a high-speed video of a metachronally coordinated row. Following the reconstruction, two numerical models were developed and simulated using an in-house immersed-boundary-method-based computational fluid dynamics solver. The two models included the original geometry (16 appendages in a row) and a sparse geometry (8 appendages, formed by removing every other appendage along the row). We found that appendage tip vortex interactions contribute to hydrodynamic performance via a vortex-weakening mechanism. Through this mechanism, appendage tip vortices are significantly weakened during the drag-producing recovery stroke. As a result, the swimming ctenophore produces less overall drag, and its thrust-to-power ratio is significantly improved (up to 55.0 % compared with the sparse model). Our parametric study indicated that such a propulsion enhancement mechanism is less effective at higher Reynolds numbers. Simulations were also used to investigate the effects of substrate curvature on the unsteady hydrodynamics. Our results illustrated that, compared with a flat substrate, arranging appendages on a curved substrate can boost the overall thrust generation by up to 29.5 %. These findings provide new insights into the fluid dynamic principles of propulsion enhancement underlying metachronal rowing.

Measurement data sets are presented for the turbulence intensity profile of three velocity components ($u$, $v$ and $w$) and turbulent kinetic energy (TKE, $k$) over a wide range of Reynolds numbers from $Re_\tau =990$ to 20 750 in a pipe flow. The turbulence intensity profiles of the $u$- and $w$-component show logarithmic behaviour, and that of the $v$-component shows a constant region at high Reynolds numbers, $Re_\tau >10\,000$. Furthermore, a logarithmic region is also observed in the TKE profile at $y/R=0.055\unicode{x2013}0.25$. The Reynolds number dependences of peak values of $u$-, $w$-component and TKE fit to both a logarithmic law (Marusic et al., Phys. Rev. Fluids, vol. 2, 2017, 100502) and an asymptotic law (Chen and Sreenivasan, J. Fluid Mech., vol. 908, 2020, R3), within the uncertainty of measurement. The Reynolds number dependence of the bulk TKE $k^+_{bulk}$, which is the total amount of TKE in the cross-sectional area of the pipe also fits to both laws. When the asymptotic law is applied to the $k^+_{bulk}$, it asymptotically increases to the finite value $k^+_{bulk}=11$ as the Reynolds number increases. The contribution ratio $\langle u'^2\rangle /k$ reaches a plateau, and the value tends to be constant within $100< y^+<1000$ at $Re_\tau >10\ 000$. Therefore, the local balance of each velocity component also indicates asymptotic behaviour. The contribution ratios are balanced in this region at high Reynolds numbers as $\langle u'^2\rangle /k\simeq 1.25$, $\langle w'^2\rangle /k\simeq 0.5$ and $\langle v'^2\rangle /k \simeq 0.25$.

Flux conditions are semi-analytical expressions that can be used to determine the flux at the grounding line of marine ice sheets. In the glaciology literature, such flux conditions have initially been established for the Weertman and Coulomb friction laws. However, the extension to more general and complex friction laws, such as the Budd friction law, for which basal friction depends on both the basal velocity and the effective pressure, is a topic of recent research. Several studies have also shown that hybrid friction laws, which consider a transition between a power-law friction far from the grounding line and a plastic behaviour close to it, were good candidates for improved modelling of marine ice sheets. In this article, we show that the flux conditions previously derived for the Weertman and Coulomb friction laws can be generalised to flux conditions for the Budd friction law with two different effective-pressure models. In doing so, we build a bridge between the results obtained for these two friction laws. We provide a justification for the existence and uniqueness of a solution to the boundary-layer problem based on asymptotic developments. We also generalise our results to hybrid friction laws, based on a parametrisation of the flux condition. Finally, we discuss the validity of the assumptions made during the derivation, and we provide additional explicit expressions for the flux that stay valid when the bedrock slopes are important or when the friction coefficients are relatively small.

We study the spectral energy transfer due to wave–triad interactions in the Garrett–Munk spectrum of internal gravity waves based on a numerical evaluation of the collision integral in the wave kinetic equation. Our numerical evaluation builds on the reduction of the collision integral on the resonant manifold for a horizontally isotropic spectrum. We evaluate directly the downscale energy flux available for ocean mixing, whose value is in close agreement with the finescale parameterization. We further decompose the energy transfer into contributions from different mechanisms, including local interactions and three types of non-local interactions, namely parametric subharmonic instability, elastic scattering (ES) and induced diffusion (ID). Through analysis on the role of each mechanism, we resolve two long-standing paradoxes regarding the mechanism for forward cascade in frequency and zero ID flux for the GM76 spectrum. In addition, our analysis estimates the contribution of each mechanism to the energy transfer in each spectral direction, and reveals new understanding of the importance of local interactions and ES in the energy transfer.

The gravity flow of a granular material between two vertical walls separated by a width $2W$ is simulated using the discrete element method (DEM). Periodic boundary conditions are applied in the flow (vertical) and the other horizontal directions. The mass flow rate is controlled by specifying the average solids fraction $\bar {\phi }$, the ratio of the volume of the particles to the volume of the channel. A steady fully developed state can be achieved for a narrow range of $\bar {\phi }$, $\bar {\phi }_{max} \geq \bar {\phi } \geq \bar {\phi }_{cr}$, and the material is in free fall for $\bar {\phi } < \bar {\phi }_{min}$. For an intermediate range of $\bar {\phi }$ ($\bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$), there are oscillations in the horizontal coordinate of the centre of mass, velocity components and stress. As $\bar {\phi }$ decreases in the range $\bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$, the amplitude of the oscillations increases proportional to $\sqrt {\bar {\phi }_{cr} - \bar {\phi }}$ and the frequency appears to tend to a non-zero value as $\bar {\phi } \rightarrow \bar {\phi }_{cr}$, indicating a supercritical Hopf bifurcation. The relation between the dominant frequency and the higher harmonics of the position, velocity and stress fluctuations are explained using the momentum balance. It is found that dissipation in the inter-particle and particle–wall contacts is critical for the presence of an oscillatory state.

The turbulent wake past a square-back Ahmed body in close proximity to the ground experiences random side-to-side switching between two asymmetric positions, a phenomenon known as bi-modality. It has been observed to be sensitive to the dynamics of the upstream boundary layers formed along the body surfaces. Close to the body fore end, these separate and reattach, with hairpin vortices emanating from the reattachment points and growing along the surfaces before breaking down upstream of the base. This study uses wall-resolved large eddy simulations to investigate the effect of using suction to suppress these upstream boundary layer separations on the wake bi-modality. It is seen that, in the unforced flow (in the absence of suction), the smaller top and side surface vortices resulting from breakdown interact as they convect downstream. Steady suction is confirmed to suppress the boundary layer separations on the different body surfaces. When the boundary layer separations on the two side (vertical) surfaces are suppressed, it is found that horizontal bi-modality is completely inhibited with weak vertical asymmetry preserved. The interaction of the small top/side surface vortices is interrupted, damping boundary layer fluctuations just upstream of the base. Applying suction on different combinations of side/top/bottom boundary layer separations is found to have different effects on the underbody flow and the wake vertical balance, with bi-modality suppression dependent on side surface suction. This confirms that bi-modality is triggered, at least in part, by boundary layer disturbances on the surfaces perpendicular to the switching direction.

This paper investigates the steady-state pattern evolution of symmetric Faraday waves excited in a brimful cylindrical container when driving parameters much exceed critical thresholds. In such liquid systems, parametric surface responses are typically considered as the resonant superposition of unstable standing waves. A modified free-surface synthetic Schlieren method is employed to obtain full three-dimensional spatial reconstructions of instantaneous surface patterns. Multi-azimuth structures and localized travelling waves during the small-elevation phases of the oscillation cycle give rise to modal decomposition in the form of $\nu$-basis modes. Two-step surface-fitting results provide insight into the spatiotemporal characteristics of dominant wave components and corresponding harmonics in the experimental observations. Arithmetic combination of modal indices and uniform frequency distributions reveal the nonlinear mechanisms behind pattern formation and the primary pathways of energy transfer. Taking the hypothetical surface manifestation of multiple azimuths as the modal solutions, a linear stability analysis of the inviscid system is utilised to calculate fundamental resonance tongues (FRTs) with non-overlapping bottoms, which correspond to subharmonic or harmonic $\nu$-basis modes induced by surface instability at the air–liquid interface. Close relationships between experimental observations and corresponding FRTs provide qualitative verification of dominant modes identified using surface-fitting results. This supports the validity and rationality of the applied $\nu$-basis modes.

Algorithms for reconstructing and predicting nonlinear ocean wave fields from remote measurements are presented. Three types of synthetic observations are used to quantify the influence of remote measurement modulation mechanisms on the algorithms’ performance. First, the observations correspond to randomly distributed surface elevations. Then, they are related to a marine radar model – the second type takes the wave shadowing modulation into account whereas the third one also includes the tilt modulation. The observations are numerically generated based on unidirectional waves of various steepness values. Linear and weakly nonlinear prediction algorithms based on analytical models are considered, as well as a highly nonlinear algorithm relying on the high-order spectral (HOS) method. Reconstructing surfaces from shadowed observations is found to have an impact limited to the non-visible regions, while tilt modulation affects the reconstruction more generally due to the indirect, more complex extraction of wave information. It is shown that the accuracy of the surface reconstruction mainly depends on the correct modelling of the wave shape nonlinearities. Modelling the nonlinear correction of the dispersion relation, in particular the frequency-dependent wave phase effects in the case of irregular waves, substantially improves the prediction. The suitability of the algorithms for severe wave conditions in finite depth and using non-perfect observations is assessed through wave tank experiments. It shows that only the third-order HOS solution predicts the right amplitude and phase of an emerging extreme wave, emphasizing the relevance of the corresponding physical modelling.