We perform a comprehensive linear non-modal stability analysis of the Rayleigh–Bénard convection with and without a Poiseuille/Couette flow in Oldroyd-B fluids. In the absence of shear flow, unlike the Newtonian case in which the perturbation energy decays monotonically with time, the interaction between temperature gradient and polymeric stresses can surprisingly cause a transient growth up to 104. This transient growth is maximized at the Hopf bifurcation when the stationary instability dominant in the weakly elastic regime transitions to the oscillatory instability dominant in the strongly elastic regime. In the presence of a Poiseuille/Couette flow, the streamwise-uniform disturbances may achieve the greatest energy amplification, and similar to the pure bounded shear flows, Gmax ∝ Re2 and tmax ∝ Re, where Gmax is the maximum energy growth, tmax the time to attain Gmax, Re the Reynolds number. It is noteworthy that there exist two peaks during the transient energy growth at high-Re cases. Different from the first one which is less affected by the temperature gradient and elasticity, the second peak, at which the disturbance energy is the largest, is simultaneously determined by the temperature gradient, elasticity and shear intensity. Specifically, the polymeric stresses field absorbs energy from the temperature field and base flow, which is partially transferred into the perturbed hydrodynamic field eventually, driving the transient amplification of the perturbed wall-normal vorticity.

We present an experimental study of convection–evaporation of a pool of water evaporating into a quiescent atmosphere. The temperature difference between the bottom of the pool and the surrounding air, as well as the water layer’s aspect ratio $\varGamma$, are systematically varied. Compared with classical Rayleigh–Bénard convection (RBC), this configuration involves a free-surface mechanical upper boundary and a mixed thermal upper boundary in contact with a poorly conducting air layer: evaporation extracts latent heat from the liquid and injects lighter vapour into the air, while radiation adds further cooling. As a result, neither temperature nor heat flux is fixed at the water–air interface, but they are instead strongly coupled. To characterise the respective contributions of convection, evaporation and radiation, we perform three sets of experiments: convection–evaporation, evaporation without bottom heating and convection without evaporation. High-resolution infrared imaging reveals multiple scales of convection at the surface: small hot plumes, cold sheet-like plumes and a large-scale circulation. The latter is constrained by the tank geometry for $\varGamma \lesssim 12$, but several turbulent superstructures develop for larger $\varGamma$. This is reminiscent of RBC but with different temperature statistics, due to the mixed boundary condition. Scaling laws are derived for interfacial transfers and mean surface temperature. Evaporation dominates heat extraction, accounting for 60 %–70 % of the flux, while radiation contributes 15 %–20 %. The release of vapour further enhances coupling between the liquid and air layers. When evaporation is blocked, radiation becomes dominant (70 %–80 %).These results have important implications for industrial and natural systems.

The linear instability of the flow in a pipe subjected to a wavelike transpiration velocity at the walls is considered. The fully nonlinear problem is formulated at high Reynolds numbers and small transpiration velocities. Solutions of the nonlinear system describing the bifurcation of disturbances caused by the transpiration are calculated and a complex bifurcation structure is uncovered with several nonlinear states possible at some transpiration amplitudes. The symmetries and structure of the nonlinear solutions are discussed.

We study the dynamic interaction of two gravity currents in a confined porous layer, one heavier and one lighter, partly inspired by the practice of geological $\mathrm {CO}_2$ sequestration in oil fields. Two coupled nonlinear advective-diffusive equations are derived to describe the time evolution of the profile shape of both the upper (lighter) and lower (heavier) currents. At early times, the upper and lower currents remain separated and propagate independently. As time progresses, the currents approach each other and start to interact. We have identified eight different regimes of gravity current interaction at late times, impacted by four dimensionless parameters, representing the flow rate partition, ratio of buoyancy over the injection force, and the viscosity contrasts between the two injecting and displaced fluids. By defining appropriate similarity variables at either the early or late times, the governing partial differential equations (PDEs) reduce to different ordinary differential equations (ODEs), corresponding to the classic nonlinear diffusion solutions at early times and eight different self-similar solutions at late times when the currents attach to each other. It is of interest to note that in four of the eight regimes of late-time interaction (regimes 2, 6–8), self-similar solutions can be constructed by combining appropriately the three basic solutions (i.e. shock, rarefaction and travelling wave solutions) identified during single fluid injection in confined porous layers. In the four other regimes (regimes 1, 3–5), implicit solutions in the form of logarithm or error functions are obtained due to the influence of flow confinement and interaction of gravity currents. Potential implications of the model and solutions are also briefly discussed in the context of ${\rm CO}_2$-water co-flooding for simultaneous ${\rm CO}_2$ sequestration and oil recovery.

The action of temporal heating modulation on Marangoni flows in a droplet on a liquid substrate is investigated. The problem is studied numerically in the framework of long-wave amplitude equations and precursor model. It is shown that temporal modulation can lead to a change of the droplet shape. Specifically, rhombic droplets have been obtained. A modulated cooling from below can lead to periodic or quasiperiodic oscillations or the droplet's decomposition.

Linear hydroelastic waves propagating in a frozen channel are investigated. The channel has a rectangular cross-section, finite depth and infinite length. The liquid in the channel is an inviscid, incompressible liquid and covered with ice. The ice is modelled as a thin elastic plate of variable thickness clamped to the channel walls. The thickness is constant along the channel length and varying across it. The flow induced by ice deflections is potential. The problem reduced to a problem of wave profiles across the channel and was solved using a piecewise linear approximation of a shape of the thickness. Normal modes are calculated to ensure continuous deflections, slopes, bending stresses and shear forces in the ice plate. Two thickness distributions are studied: in Case I, the thickness is constant at a middle segment and linearly increases at edge segments over the channel’s width; in Case II, the thickness linearly decreases at the edge segments. In Case I, there is one segment with the thin part of the ice cover where, as expected, the oscillations of the ice plate will be concentrated. In Case II, there are two such areas, separated by the middle segment with the thick part of the ice cover. Dispersion relations, phase and group velocities, wave profiles and strain distributions in the ice plate are studied. Results show that the properties of periodic hydroelastic waves are significantly influenced by the ice thickness distribution across the channel.

The present study aims to demonstrate the application of the oblique temperature gradient (OTG) to manipulate the buoyancy instabilities and resulting mixing. To accomplish this, we consider a model consisting of a liquid layer supported by a perfectly conducting bottom wall from below and the upper surface is exposed to ambient inert gas in a shallow container. The stability analysis employs the pseudospectral method and perturbation energy budget approach. The imposed OTG consists of a negative vertical temperature gradient (VTG), which leads to unstable density stratification, and a negative horizontal temperature gradient (HTG) component responsible for the Hadley circulation, which can be utilised to control instabilities. Without the HTG, a Rayleigh–Bénard mode of instability exists due to the imposed VTG termed as a VTG mode. The imposed HTG induces a linear VTG term and a quintic polynomial VTG term in the bottom wall-normal coordinate $y$. The induced linear VTG term reinforces the imposed VTG, while the induced quintic polynomial VTG term opposes the imposed VTG. For the vanishing Biot number ($Bi$), the induced quintic VTG term stabilises the VTG mode for the Prandtl number, $Pr=7$. However, for $Pr=0.01$, the Reynolds stresses associated with the Hadley circulation destabilise the VTG mode. For non-zero $Bi$, new streamwise and spanwise instability modes arise for $Pr=7$ due to the induced linear VTG term. Physical arguments show that the unstable density stratification near the gas–liquid interface caused by the synergistic effect of the imposed VTG and the induced linear VTG term lead to the existence of the predicted new modes. The present study thus shows that the strength of the HTG can be utilised to control the instability modes and critical parameters, thereby demonstrating the utility of the OTG in manipulating the buoyancy instabilities.

Exact mathematical expressions are derived to predict the exponent $p$ observed in non-equilibrium turbulence, where the classical dissipation law is replaced by a new dissipation scaling law $C_{\varepsilon } \sim \textit{Re}_{\lambda }^p$. Here, $ \textit{Re}_{\lambda }$ is the Taylor-based Reynolds number and $C_{\varepsilon } = \varepsilon L_{11} / u^{\prime 3}$ is the non-dimensional dissipation rate, defined by the viscous dissipation rate, $\varepsilon$, longitudinal integral scale, $L_{11}$, and root-mean-square of the velocity fluctuations $u^{\prime} = \sqrt {\overline {u^{\prime 2}}}$ (Vassilicos, Annu. Rev. Fluid Mech., vol. 47, 2015, pp. 95–114). Assuming homogeneous and isotropic turbulence, it is shown that the exact value of $p$ involves only first-order derivatives of these variables; however, at very high Reynolds numbers, and under particularly strong changes in the power input of the external forcing (without changing the shape of the forcing spectrum), the exact expression simplifies to $p = 3\pi / 4\alpha L_{110} - 5 / 2$, where $L_{110}$ is the initial value of the longitudinal integral scale and $\alpha$ represents an effective forcing wavenumber. Thus, the main finding is that only large-scale effects are involved in the imposition of the non-equilibrium dissipation scaling law. The results are compared with direct numerical simulation (DNS) results of isotropic turbulence under abruptly changing forcing conditions and with experimental data of non-equilibrium decaying isotropic turbulence, showing consistent results.

The flow in a rapidly rotating cylinder forced by the harmonic oscillations of a small sphere along the rotation axis is explored numerically. For oscillation frequencies less than twice the cylinder rotation frequency, the forced response flows feature conical shear layers emitted from the critical latitudes of the sphere. These latitudes are where the characteristics of the hyperbolic system, arrived at by ignoring nonlinear, viscous and forcing terms in the governing equations, are tangential to the sphere. These conical shear layers vary continuously with the forcing frequency so long as it remains inertial. At certain values of the forcing frequency, linear inviscid inertial modes of the cylinder are resonated. Of all possible inertial modes, only those whose symmetries are compatible with the symmetry of the forced system are resonated. This all occurs even in the linear limit of vanishingly small forcing amplitude. As the forcing amplitude is increased, nonlinearity leads to non-harmonic oscillations and a non-zero mean flow which features a Taylor columnar structure extending from the sphere to the two endwalls in an axially invariant fashion.

Recent studies have shown that, in coastal waters where water depth decreases significantly due to rapid bathymetric changes, the non-equilibrium dynamics (NED) substantially increases the occurrence probability of extreme (rogue) waves. Nevertheless, research on depth-induced NED has been predominantly confined to unidirectional irregular waves, while the role of directionality remains largely unexplored. The scarce studies on multidirectional waves mainly rely on numerical simulations and have yielded conflicting results. In this work, we report on an experimental investigation of wave directionality on the depth-induced non-equilibrium wave statistics. High-order statistical moments, skewness and kurtosis, are used as proxies for the non-equilibrium wave response. Our results indicate that the directional spreading has a minor effect on decreasing the maximum values of these statistical moments. In contrast, the incidence direction plays a significant role in the non-equilibrium wave response, which is attributed to the effective bottom slope.

This paper discusses the propagation of coastal currents generated by a river outflow using a 1 ${1}/{2}$-layer, quasigeostrophic model, following Johnson et al. (2017) (JSM17). The model incorporates two key physical processes: Kelvin-wave-generated flow and vortical advection along the coast. We extend JSM17 by deriving a fully nonlinear, long-wave, dispersive equation governing the evolution of the coastal current width. Numerical solutions show that, at large times, the flow behaviour divides naturally into three regimes: a steady outflow region, intermediate regions consisting of constant-width steady currents and unsteady propagating fronts leading the current. The widths of the steady currents depend strongly on dispersion when the constant outflow potential-vorticity anomaly is negative. Simulations using contour dynamics show that the dispersive equation captures the full quasigeostrophic behaviour more closely than JSM17 and give accurate bounds on the widths of the steady currents.

The nonlinear free-surface response of moonpools with recesses is investigated through both experimental and theoretical analyses. A theoretical model is developed to compute the natural frequencies using linearised potential flow theory and eigenfunction expansions. Four moonpool configurations with varying recess lengths are examined experimentally. The analysis reveals that larger recess lengths correspond to increasingly pronounced nonlinear responses. It is also shown that, for an incident wave group with suitable frequency content, the linear moonpool response can be significantly smaller than the second- and third-harmonic components. This effect is attributed to super-harmonic secondary resonance, characterised by $n \omega =\omega _{pq}$ ($n\geqslant 2$ and $p+q\geqslant 1$), where $n$ denotes the super-harmonic order, $\omega$ is the excitation frequency, and $p$ and $q$ are the longitudinal and transverse mode numbers, respectively. Here, $\omega_{pq}$ represents the sloshing frequency of the three-dimensional moonpool. Furthermore, it is found that, as the primary responses increase, cross-flow instability can lead to secondary resonance in non-symmetric modes. This occurs because the double and triple frequencies of the base mode approach the transverse or diagonal sloshing frequencies. Additionally, hard-spring Duffing effects for secondary resonance induced by super-harmonics are observed in cases with recesses, becoming more pronounced as the recess length increases, particularly when $h/l\lt 0.3368$, where $h$ is the water depth above the recess and $l$ is the moonpool length.

We study the effect of gas rarefaction on the interaction of small thermodynamic non-uniformities with a finite body. Considering a two-dimensional set-up, the initial system state is modelled as slight perturbations over its uniform density and temperature fields, prescribed in the vicinity of a thin plate. The problem is analysed in the collisionless limit and complemented by direct simulation Monte Carlo computations to cover the entire range of gas rarefaction rates. The high-Knudsen ‘sink-like’ and ‘source-like’ propagation patterns observed in the density- and temperature-driven set-ups, respectively, are discussed, together with the impact of specular (smooth) and diffuse (isothermal) wall reflections. At highly rarefied conditions, the solid body obstructs part of the gas domain, preventing the propagation of acoustic disturbances therein. With decreasing gas rarefaction, the acoustic field penetrates the obscured area via the effect of molecular collisions. Inspecting the near-field description, the propagation of flow disturbances along the plate surface is examined, and the acoustic force on the body is computed. In the thermally excited case, both normal- and shear-force components change sign at late times, attracting the plate towards the initial perturbation location. With reducing gas rarefaction, the shear force diminishes while the normal force sharply increases due to the decrease in signal decay. Finally, we apply the analysis to study the impact of gas rarefaction on acoustic reciprocity. Notably, acoustic reciprocity does not hold at non-continuum conditions over non-specular surfaces, where boundary reflections propagate in the presence of few molecular collisions, insufficient to retain reciprocal symmetry.

The propulsion speed of spheroidal squirmers was obtained by Keller & Wu (J. Fluid Mech., 1977, vol. 80, p. A31). It has become the benchmark to investigate the effect of shape on the propulsion of ciliated microorganisms. However, their study focused on translational motion whereas many biologically relevant organisms also experience rotational (or swirling) motion. We derive an analytical expression for the angular velocity of a swirling spheroidal squirmer. Our analysis reveals that spheroidal squirmers rotate faster than their spherical counterparts in Newtonian fluids. We also determine the contribution of the second azimuthal mode to the power dissipation generated by a spheroidal squirmer, and uncover a behaviour uniquely distinct from the power dissipation of a strictly translating swimmer.

A reactive control strategy is implemented to attenuate the streaks formed on a wing boundary layer due to free-stream turbulence (FST). Numerical simulations are performed on a section of a NACA0008 profile, considering its leading edge, while forced by FST with turbulence intensities of 0.5 % and 2.5 %. The controller is composed of localised sensors and actuators, with the control law consisting of a linear quadratic Gaussian regulator designed on a reduced-order model based only on the impulse responses of the system. Three configurations are evaluated by considering three different numbers of sensors/actuators along the spanwise direction. It is found that all configurations are effective in damping the streaks inside the boundary layer, whose effect is sustained downstream of the objective function location. However, distinct behaviours are observed when comparing the capability of the controllers with delay transition, where the best performance is attained for the case with larger number of sensors/actuators. This is attributed to the effectiveness of the controller in damping the streaks that will later break down, which in this case are associated with relatively short spanwise wavelength. This observation is confirmed by analysing the stability of the flow before the appearance of turbulent spots. Our results suggest that for an effective transition delay, efforts should not only be put into control of streaks with average spanwise wavelength, but also in the short spanwise wavelength associated with breakdown.