A horizontal cylinder with a concave cross-section partially submerged in a liquid at a given position may permit multiple menisci around itself. The number and stabilities of the menisci are analysed, and how the menisci change during the processes of gradually hoisting and lowering the cylinder is explained by bifurcation theory. The restoring force on the concave cylinder and the rebounding potential energy (defined as the work done by the restoring force during the whole hoisting process to represent the potential rebounding capacity of a cylinder on water) are also investigated. The results show that, when the radius of the concave arc is smaller than the critical value, the concave cylinder at a given position permits multiple possible menisci. The equilibria form fold bifurcations with the position of the cylinder as the bifurcation parameter, and two successive fold bifurcations can form a one-fold hysteresis loop. The force–distance curve representing the relation between the restoring force and the position of the cylinder also has corresponding hysteresis loops, where the restoring force will jump (i.e. change discontinuously) at the bifurcation points. In contrast to a convex cylinder, a concave cylinder can have different values of the restoring force at the same height because of multiple menisci, and the values depend on whether it is hoisted or lowered. Under the condition of a fixed cross-sectional area, the optimal cross-sectional shape is determined when the maximum rebounding potential energy is reached, and it is close to the shape with the critical concave arc angle for the existence of multiple possible menisci. The cross-sections with concave parts are preferable to circular, laterally planed and corner-concave cross-sections. This paper provides an effective method of enhancing the restoring force and potential rebounding height of a robotic water strider insect or particles on the surface of water.

Pattern-forming with externally imposed symmetry is ubiquitous in nature but little studied. We present experimental studies of pattern formation and selection by spatial periodic forcing in rapidly rotating convection. When periodic topographic structures are constructed on the heated boundary, they modulate the local temperature and velocity fields. Symmetric convection patterns in the form of regular vortex lattices are observed near the onset of convection, when the periodicity of the external forcing is set close to the intrinsic vortex spacing. We show that the new patterns arise as a dynamical process of imperfect bifurcation which is well described by a Ginzburg–Landau-like model. We explore the phase diagram of buoyancy strength and periodicity of external forcing to find the optimal experimental settings for which the vortex patterns best match that of the external forcing.

In Wang & Pan (J. Fluid Mech., vol. 918, 2021, A19), the authors developed the first ensemble-based data assimilation (DA) capability for the reconstruction and forecast of ocean surface waves, namely the EnKF-HOS method coupling an ensemble Kalman filter (EnKF) and the high-order spectral (HOS) method. In this work, we continue to enrich the method by allowing it to simultaneously estimate the ocean current field, which is in general not known a priori and can (slowly) vary in both space and time. To achieve this goal, we incorporate the effect of ocean current (as unknown parameters) on waves to build the HOS-C method as the forward prediction model, and obtain a simultaneous estimation of (current) parameters and (wave) states via an iterative EnKF (IEnKF) method that is necessary to handle the complexity in this DA problem. The new algorithm, named the IEnKF-HOS-C method, is first tested in synthetic problems with various forms (steady/unsteady, uniform/non-uniform) of current. It is shown that the IEnKF-HOS-C method is able to not only estimate the current field accurately, but also boost the prediction accuracy of the wave field (even) relative to the state-of-the-art EnKF-HOS method. Finally, using real data from a shipborne radar, we show that the IEnKF-HOS-C method successfully recovers the current speed that matches the in situ measurement by a floating buoy.

In this work, we propose using an ensemble Kalman method to learn a nonlinear eddy viscosity model, represented as a tensor basis neural network, from velocity data. Data-driven turbulence models have emerged as a promising alternative to traditional models for providing closure mapping from the mean velocities to Reynolds stresses. Most data-driven models in this category need full-field Reynolds stress data for training, which not only places stringent demand on the data generation but also makes the trained model ill-conditioned and lacks robustness. This difficulty can be alleviated by incorporating the Reynolds-averaged Navier–Stokes (RANS) solver in the training process. However, this would necessitate developing adjoint solvers of the RANS model, which requires extra effort in code development and maintenance. Given this difficulty, we present an ensemble Kalman method with an adaptive step size to train a neural-network-based turbulence model by using indirect observation data. To our knowledge, this is the first such attempt in turbulence modelling. The ensemble method is first verified on the flow in a square duct, where it correctly learns the underlying turbulence models from velocity data. Then the generalizability of the learned model is evaluated on a family of separated flows over periodic hills. It is demonstrated that the turbulence model learned in one flow can predict flows in similar configurations with varying slopes.

The effects of compressibility on the statistics and coherent structures of a temporally developing mixing layer are studied using numerical simulations at convective Mach numbers ranging from $M_c=0.2$ to $1.8$ and at Taylor Reynolds numbers up to 290. As the convective Mach number increases, the streamwise dissipation becomes more effective to suppress the turbulent kinetic energy. At $M_c=1.8$, the streamwise dissipation increases much faster than the other two components in the transition region, even larger than pressure–strain redistribution, correlating with the streamwise elongated vortical structures at a higher level of compressibility. We confirm the existence of the large-scale high- and low-speed structures in the mixing layers, which accompany the spanwise Kelvin–Helmholtz rollers at low convective Mach number and dominate the mixing layer at higher convective Mach number. Conditional statistics demonstrate that the large-scale low-speed structures are lifted upwards by a pair of counter-rotating quasi-streamwise rollers flanking the structures. The small-scale vortical structures have an apparent preference for clustering into the top of the low-speed regions, which is directly associated with high-shearing motions on top of the low-speed structures. The high-speed structures statistically exhibit central symmetry with the low-speed structures. The statistics and dynamics of large-scale high- and low-speed structures in the compressible mixing layers resemble those in the outer region of the turbulent boundary layers, which reveals the universality of the large-scale structures in free shear and wall-bounded turbulence. A conceptual model is introduced for the large-scale high- and low-speed structures in turbulent mixing layers.

The finite-amplitude instability of the buoyancy-driven boundary layer is considered on a vertical plate immersed in a thermally stratified ambient medium, where the wall and surrounding fluid have different temperature gradients. Although the linear stability in this configuration has been investigated, the finite-amplitude solution arising from the critical instability has been studied only for specific parameter values. We extend this by using the amplitude expansion method. The primary bifurcations to the two-dimensional least unstable mode for different temperature gradient ratios ($0 \leqslant \lambda \leqslant 10$) and Prandtl numbers ($10^{-1} \leqslant Pr \leqslant 10^{4}$) are investigated. Only supercritical bifurcations are found to occur when $0 \leqslant \lambda < 2$ and $Pr \leqslant 2800$, while subcritical bifurcations are also found for larger values of temperature gradient ratio and Prandtl number. Analysis of the contribution of the nonlinear terms in the Landau coefficient reveals that the interaction of the modification of the mean flow and second harmonic for velocity with the fundamental mode for temperature plays an important role in subcritical bifurcation. Based on the Landau equation, the threshold amplitude of the nonlinear equilibrium solution is discussed as well. These encouraging results should be helpful for understanding such a buoyancy-driven flow system.

We derive from first principles analytic relations for the second- and third-order moments of $\boldsymbol{\mathsf{m}}$, the spatial gradient of fluid velocity $\boldsymbol{u}$, $\boldsymbol{\mathsf{m}} = \nabla \boldsymbol{u}$, in compressible turbulence, which generalize known relations in incompressible flows. These relations, although derived for homogeneous flows, hold approximately for a mixing layer. We also discuss how to apply these relations to determine all the second- and third-order moments of the velocity gradient experimentally for isotropic compressible turbulence.

In this study, we present a three-dimensional numerical model for the interaction of flow with submerged flexible vegetation, based on a large-eddy simulation and the immersed boundary method. The model innovatively realises the interaction between the flow and highly flexible vegetation with clustered leaves. Besides being a three-dimensional model of motion with full degrees of freedom, this study improves the consideration of the motion of the vegetation in all directions, and in addition the energy and momentum transfer in the spanwise direction. Furthermore, we perform a flume experiment for the flow with submerged flexible vegetation, the results of which are used to validate the simulation effects of the numerical model. It is found that the numerical model can effectively simulate the velocity profiles and the movement of vegetation induced by the flow. Using the model to analyse the flow–vegetation interaction, we find that the movement of vegetation is closely related to the flow velocity. As the flow velocity increases, both the offset angle and the vegetation swaying amplitude increase. Compared to vertical rigid vegetation, the tilting of flexible vegetation does not significantly change the velocity difference and the magnitude of the turbulent kinetic energy between the inside and the outside of the vegetation canopy, but it does weaken the disturbance to flow, thus reducing the resistance to flow. However, the swaying of vegetation dose significantly increase the velocity difference between the inside and the outside of the canopy. It forms Kelvin–Helmholtz hairpin vortices intensifying the turbulence production, and enhancing the disturbance and resistance to flow.

In offshore offloading operations, two vessels in a side-by-side configuration experience actions of both ambient water waves and liquid sloshing in internal tanks. Under the excitation of water waves, complex multibody motions are induced, resulting in liquid sloshing in tanks, and concurrently liquid sloshing can feedback to affect the vessels’ motions. The interaction between waves and two barges in a side-by-side configuration coupled with liquid sloshing effects is investigated for a fixed–free arrangement. A numerical model is developed based on the boundary element method to deal with complex wave induced multibody motions coupled with liquid sloshing in internal tanks. Due to the presence of a narrow gap between two vessels, gap resonance may occur, and a damping surface is introduced to suppress an unrealistic response near resonance. Concurrently, physical experiments with and without liquid sloshing effects are carried out. In-depth discussions on motion characteristics are given, and Stokes and non-Stokes natural frequencies associated with liquid sloshing are discussed. The significance of the present study is twofold. Firstly, the experimental measurements provide reference results for validations of numerical simulations. Secondly, this work gives an insight into wave induced motions with liquid sloshing effects under different wave headings which affect vessel operational safety.

Dynamics of two-dimensional flow past a rigid flat plate with a trailing closed flexible filament acting as a deformable afterbody are investigated numerically by an immersed boundary-lattice Boltzmann method for the fluid flow and a finite element method for the filament motion. The effects of Reynolds number ($Re$) and length ratio ($Lr$) on the flow patterns and dynamics of the rigid-flexible coupling system are studied. Based on our numerical results, five typical state modes have been identified in $Lr\unicode{x2013}Re$ plane in terms of the filament shape and corresponding dynamics, i.e. static deformation, micro-vibration, multi-frequency flapping, periodic flapping and chaotic flapping modes, respectively. Benefiting from the passive flow control by using the flexible filament as a deformable afterbody, the coupled system may enjoy a significant drag reduction (up to $22\,\%$) compared with bare plate scenarios ($Lr=1$). Maximum drag reduction achieved at $L_{c,{min}} \in [1.8, 2]$ is often accompanied by the onset of the system state transition. The flow characteristic and its relation to the change in hydrodynamic drag are further explored in order to reveal the underlying mechanisms of the counterintuitive dynamical behaviour of the coupled system. The scaling laws for the form drag and the friction drag, which arise from the pressure and viscous effects, respectively, are proposed to estimate the overall drag acting on the system. The results obtained in the present study may shed some light on understanding the dynamical behaviour of rigid-flexible coupling systems.

The essence of sub-critical transition of oscillatory boundary-layer flows is the non-modal growth of finite-amplitude disturbances. The current understanding of the mechanisms of the orderly and bypass transitions of oscillatory boundary-layer flows is limited. The present study adopts optimisation approaches to predict the maximum energy amplification of two- and three-dimensional perturbations in response to the optimal initial disturbance with or without external forcing. A series of direct numerical simulations are also performed to compare with the results obtained from the stability analyses. In particular, the optimal initial perturbation similar to a Tollmien–Schlichting (T–S) wave yields the largest transient growth under the combined effects of the Orr mechanism and inflectional point instability. With a considerable level of two-dimensional disturbance, the vortex tube nonlinearly develops from the T–S-like wave, and then either deforms into a $\varLambda$-vortex in the near-wall region or rolls up to the free shear region. The further burst of turbulence can follow the first pathway as K-type transition or the second one as vortex tube breakdown due to the elliptical instability. Additionally, non-modal growth can initiate the inception of streaky structures by favourable three-dimensional initial perturbations and/or forcing. The secondary instabilities responsible for the streak breakdown are classified as the varicose (symmetric) and sinuous (anti-symmetric) modes. Under a sufficiently high level of three-dimensional disturbance, the bypass transition is predominantly characterised by the formation of the sinuous mode and turbulent spots, which leads to the suppression of inflection point instability.

We report a laboratory study on the scattering, energy dissipation and mean flow induced by internal gravity waves incident upon slopes with varying surface roughness. The experiment was performed in a rectangular box filled with thermally stratified water. The roughness of the slope surface, $\lambda$, defined as the height of a roughness element over its base width, and the off-criticality $\gamma =(\alpha -\beta )/\beta$, with $\alpha$ and $\beta$ being the angles of the incident wave and the slope, are used as two control parameters. The distribution of energy dissipation in the direction normal to the slope is found to be more uniform in the rough surface cases. Counter-intuitively, both the maximum value in the dissipation profile and the total energy dissipation near the slope are reduced by surface roughness under most circumstances. The measured peak width (the full width at half-maximum of the peaks) of the dissipation profile is found to be broadened significantly in the rough surface cases. We also observed that there exists a non-zero optimal off-criticality ($\gamma =0.17$ for the present measurement resolution) for the normalized average dissipation and total dissipation, which may be due to the strongest wave energy near the slope at this $\gamma$. Unlike surface roughness, the off-criticality has a small effect on the distribution of energy dissipation. Moreover, surface roughness is also found to change the structure of the scattering-induced mean flow and enhance its strength. The present study provides new perspectives on how the surface roughness on topographic features influences energy dissipation.