Owing to the Taylor–Proudman theorem, it is generally believed that rotating flows should have two-dimensional dynamics for rapid background rotation. Yet, we show that two infinitely long counter-rotating vertical vortices always remain unstable to three-dimensional perturbations even for large rotation rate about the vertical axis. The dominant instability consists of quasi-antisymmetric displacements of the two vortices. Its growth rate is independent of the rotation rate when it is sufficiently large, while the most amplified vertical wavelength scales like the rotation rate. Direct numerical simulations show that the instability leads ultimately to a full three-dimensional breakdown of the vortex pair. In other words, a two-dimensional vortex pair will spontaneously develop three-dimensional variations even in the limit of infinite rotation rate but the wavelength will tend to infinity. This implies that the Taylor–Proudman theorem can be strictly valid only in vertically bounded flows. The scaling for the wavelength is next generalized by showing that the typical vertical scale in rapidly rotating unbounded flows is $L_v \sim L_h/Ro$, where $L_h$ is the horizontal scale and $Ro$ the Rossby number. This scaling law is shown to derive from a self-similarity of the Navier–Stokes equations for $Ro \ll 1$. The resulting reduced equations are identical to those obtained first by Julien et al. (Theor. Comput. Fluid Dyn., vol. 11, issue 3–4, 1998, pp. 251–261) and Nazarenko & Schekochihin (J. Fluid Mech., vol. 677, 2011, pp. 134–153) by means of multiscale analyses in the cases of rotating convection or turbulence. The self-similarity demonstrated herein suggests that these reduced equations are valid for any rapidly rotating unbounded flow.

Gas volumes with an equivalent spherical diameter greater than that of the tube in which they rise by buoyancy – so-called Taylor bubbles – assume a characteristic bullet shape and ascend with a velocity that is dependent primarily on the tube diameter, and also on the physical properties of the liquid, but, remarkably, is mostly independent of the gas volume. The requirement of liquid volume conservation suggests a plausible explanation of this paradoxical feature in that the space vacated under the rising bubble must be replenished by the liquid film falling along the bubble surface. It is demonstrated by numerical means that, by limiting the bubble diameter, a cylindrical ‘cage’ of thin vertical rods coaxial with the tube permits the flow rate of this film to increase, with the result that the bubble is able to ascend with a significantly higher velocity than in an empty tube with the same diameter.

We identify an exceptional point in the numerical stability map of a combustion experiment operated with laminar premixed flames. A low-order model of the experiment allows one to investigate the interplay between the system modes and an exceptional point. The latter is located in the unstable region of the complex eigenvalue plane. Under gradual variation of the operating parameters, the branch-switching characteristic of the exceptional point can facilitate abrupt changes between strong instability and stable operation. The results also indicate that intrinsic thermoacoustic feedback plays an important role in the stability characteristics of this experiment.

Inertia-gravity waves in the atmosphere and ocean are transported and refracted by geostrophic turbulent currents. Provided that the wave group velocity is much greater than the speed of geostrophic turbulent currents, kinetic theory can be used to obtain a comprehensive statistical description of the resulting interaction (Savva et al., J. Fluid Mech., vol. 916, 2021, A6). The leading-order process is scattering of wave energy along a surface of constant frequency, $\omega$, in wavenumber space. The constant-$\omega$ surface corresponding to the linear dispersion relation of inertia-gravity waves is a cone extending to arbitrarily high wavenumbers. Thus, wave scattering by geostrophic turbulence results in a cascade of wave energy to high wavenumbers on the surface of the constant-$\omega$ cone. Solution of the kinetic equations shows establishment of a wave kinetic energy spectrum $\sim k_h^{-2}$, where $k_h$ is the horizontal wavenumber.

We present an analytical study of two-dimensional flow in a wedge driven by a time-dependent surface heat flux as a model problem to understand buoyancy-induced cross-shore flow. Besides the turbulent Prandtl number and the relevant Rayleigh number, both assumed to be of order unity, the solution is seen to depend on the geometry through a parameter $\beta$, representing the bottom slope. An analytic solution is sought in the asymptotic limit $\beta \ll 1$ for a water layer bounded by an adiabatic bottom surface subject to a harmonic heat flux on the upper surface. The analysis reveals that the motion at leading order can be expressed as the sum of a harmonic component and a steady component, the latter driven by nonlinear advection. This steady-streaming motion includes a near-shore vortex with associated recirculating motion that can affect cross-shore transport and dispersion in coastal environments. The analytical solution is compared with numerical solutions of the complete conservation equations for small values of $\beta$. Excellent quantitative agreement is found for values of the Rayleigh number below a critical value at which the periodic solution undergoes a period-doubling bifurcation, leading to the establishment of thermal-instability cells that dominate the offshore flow dynamics, while the near-shore dynamics remains well described by the analytical solution. The analysis illustrates that a periodic heat input that leads to a vertically inhomogeneous temperature distribution can result in residual motion, net heat fluxes and persistent temperature structure in the cross-shore direction.

As two particles approach each other, the continuum lubrication force diverges, with decreasing separation preventing contact. However, for separations comparable to the mean free path of the gas, $\lambda$, non-continuum effects cause the lubrication force to diverge more slowly with decreasing separation distance, allowing for contact in finite time. The first study of this phenomenon was done by Sundararajakumar & Koch (J. Fluid Mech., vol. 313, 1996, pp. 238–308) for two particles moving along their line of centres. We extend their normal motion study to include tangential motions. For small Knudsen number $Kn=\lambda / a$, where $a$ is the harmonic mean of the two particle radii, we use a matched asymptotic expansion technique to obtain the non-continuum forces and torques for tangential motions of spheres separated by distances within the lubrication regime that are at or below the mean free path of the gas. The hydrodynamic resistivity functions are fitted to provide a uniformly valid approximation that smoothly transitions between the continuum multipole and non-continuum lubrication expressions for the forces and torques as the minimum gap between the particles $h_0$ varies from values of $O(a)$ to values of $O(\lambda )$. These functions, in combination with the result by Sundararajakumar & Koch (J. Fluid Mech., vol. 313, 1996, pp. 238–308) and the classical work by Jeffrey & Onishi (J. Fluid Mech., vol. 139, 1984, pp. 261–290), yield a complete formulation for the hydrodynamic interactions of two spheres at all separations, from non-interacting spheres in the extreme far field through all the transitions that occur up to contact. We apply the new formulation to the classical case of a particle settling parallel to a vertical wall. The continuum Stokes equation predicts a settling speed that decreases with decreasing gap separation and vanishes at contact, whereas the non-continuum model developed herein predicts a finite settling speed at contact.

Thin, viscous liquid films subjected to impact events can deform. Here we investigate free-surface oil-film deformations that arise owing to the build up of air under the impacting and rebouncing of water drops. Using digital holographic microscopy, we measure the three-dimensional surface topography of the deformed film immediately after the drop rebound, with a resolution down to 20 nm. We first discuss how the film is initially deformed during impact, as a function of film thickness, film viscosity and drop impact speed. Subsequently, we describe the slow relaxation process of the deformed film after the rebound. Scaling laws for the broadening of the width and the decay of the amplitude of the perturbations are obtained experimentally and found to be in excellent agreement with the results from a lubrication analysis. We finally arrive at a detailed spatio–temporal description of the oil-film deformations that arise during the impact and rebouncing of water drops.

A phenomenological model is proposed to describe the deformation and orientation dynamics of finite-sized bubbles in both quiescent and turbulent aqueous media. This model extends and generalizes a previous work that is limited to only the viscous deformation of neutrally buoyant droplets, conducted by Maffettone & Minale (J. Non-Newtonian Fluid Mech., vol. 78, 1998, pp. 227–241), into a high Reynolds number regime where the bubble deformation is dominated by flow inertia. By deliberately dividing flow inertia into contributions from the slip velocity and velocity gradients, a new formulation for bubble deformation is constructed and validated against two experiments designed to capture the deformation and orientation dynamics of bubbles simultaneously with two types of surrounding flows. The relative importance of each deformation mechanism is measured by its respective dimensionless coefficient, which can be isolated and evaluated independently through several experimental constraints without multi-variable fitting, and the results agree with the model predictions well. The acquired coefficients imply that bubbles reorient through body rotation as they rise in water at rest but through deformation along a different direction in turbulence. Finally, we provide suggestions on how to implement the proposed framework for characterizing the dynamics of deformable bubbles/drops in simulations.

The time-dependent flow driven by electromagnetic forcing of an electrolytic fluid in the gap of a concentric spheres set-up is studied experimentally and theoretically. The driving Lorentz force is generated by the interaction of an alternating current radially injected through electrodes located at the equatorial zone of the spheres and a dipolar magnetic field produced by a permanent magnet inside the inner sphere. Experimentally, the time-dependent flows were explored in the laminar regime with a Reynolds number ${Re} = 640$ and different forcing frequencies, which resulted in oscillatory Reynolds numbers ranging from $28$ to $2820$. Velocity profiles in the equatorial line between spheres were obtained with particle image velocimetry. Given the symmetry of the problem at the equatorial plane, asymptotic and approximate solutions for the azimuthal velocity are obtained for the limiting cases of low-${Re}_{\omega }$ (in real arguments) and high-${Re}_{\omega }$ (in complex arguments). Furthermore, a general methodology is proposed in such a way that an exact solution for the problem is obtained. The analytical solutions reproduce the main characteristic behaviour of the flow. An estimation of the oscillatory boundary layer due to the electromagnetic forcing is obtained through the exact solution. A full three-dimensional numerical model, that introduces the dipolar magnetic field and the radial dependency of the applied current, is able to quantitatively reproduce both the analytical solutions and the experimental measurements. Additionally, numerical results show a resonant behaviour of the flow when the forcing frequency is approximately ${Re}_{\omega } \approx 560$.

The fluid–structure--sound interaction in reducing the flow-induced noise of a circular cylinder with a back flexible splitter plate is investigated experimentally. The rigid splitter plate (RSP) and the flexible splitter plate (FSP) are individually attached to the rear of the cylinder with a diameter D = 30 mm. Three parameters are considered: the flexibility of the plates, the length of the plates (L = 0.5 D ~ 6.0 D) and the Reynolds numbers (Re = 3.83×104 ~ 9.57 × 104). Both RSP and FSP achieve the optimal reduction in the Aeolian tone at L/D = 1.0 regardless of Re, where the RSP results in a reduction of 14 dB on average and the FSP a superior reduction of 18 dB. Subsequently, the velocity fields are measured with two-dimensional particle image velocimetry for L/D = 1.0. The dynamic deformation of the FSP is captured with a single-camera three-dimensional digital image correlation system. The results show that the splitter plate elongates the recirculation region length (Lr) significantly, and it is preferable to normalize the streamwise coordinate x with Lr instead of D in this investigation. It is determined that the Aeolian tone is mainly induced by the fluctuations in the near wake region of 1 < x/Lr < 3. The splitter plates effectively suppress the overall fluctuations originating from vortex shedding in the wake, thereby reducing the Aeolian tone successfully. The movement of the FSP, which is dominated by low-frequency vibrations with broadband and a relatively energetic periodic vibration, has important effects on the secondary and high-frequency noise.

A weakly nonlinear investigation of sinuous instabilities in a planar viscoelastic liquid sheet having corotational Jeffrey's rheological model is performed. Analysis predicts that viscoelastic properties may exhibit a non-monotonic dual effect depending upon their range and velocity ratios. At velocity ratios of 2 and 2.15, an increase in elasticity stabilizes the sheet for elasticity number ranging from 0.1 to 4 and from 0.1 to 1, respectively. Beyond this range, elasticity produces a destabilizing effect on the sheet. However, at higher velocity ratios of 2.50 and 2.75, an increase in elasticity only destabilizes the liquid sheet. The effect of time constant ratio at different velocity ratios is opposite to that of elasticity number. An increase in time constant ratio destabilizes the sheet at velocity ratio of 2, whereas it stabilizes the sheet for relatively higher velocity ratios of 2.50 and 2.75. At intermediate velocity ratio of 2.15, two regimes of time constant ratio are identified in the range 0.1 to 0.4 and 0.4 to 0.9, representing stabilizing and destabilizing effect of time constant ratio, respectively. The nonlinear interaction between the viscoelastic sheet and surrounding gases may enhance or dampen the second-order amplitude. The contribution of second-order amplitude to sheet breakup is much higher than that of linear growth rate and is responsible for the dual effect of viscoelastic properties. Finally, the size distribution of droplets formed after primary breakup is investigated using maximum entropy formulation. Results reveal that an increase in elasticity number and time constant ratio produces finer and larger droplets, respectively.

The exchange of tracer mass in shallow waters between the mainstream and a recirculating flow to the side is examined in this paper over a wider range of Froude numbers than in any previous studies. We used a well-calibrated weighted essentially non-oscillatory scheme to capture the shock waves while maintaining the stability of the computation. The radiation of the waves suppressed the turbulence and the entrainment of surrounding fluids into a mixing layer. Shock waves began to form in the recirculating flow at a mainstream Froude number of $Fr_{o}\simeq 3$. The effect of the shock waves was a sudden increase in lateral mixing between the mainstream and the recirculating flow, leading to a corresponding sharp drop in a retention-time coefficient. These simulation results for the effect of the waves on mixing in shallow waters at large Froude numbers were consistent with the available data obtained from laboratory experiments.

Turbulence is investigated in the lee of an open-cell metal foam layer. In contrast to canonical grids, metal foams are locally irregular but statistically isotropic. The solid matrix is characterised by two lengths, the ligament thickness $d_f$ and the pore diameter $d_p$. A direct numerical simulation is conducted on a realistic metal foam geometry for which $d_f/d_p = 0.14$ and the porous layer thickness is five times the pore diameter. The Reynolds number based on the pore size is ${\textit {Re}}_{d_p} = 4000$, corresponding to a Taylor-scale Reynolds number ${\textit {Re}}_\lambda \approx 80$. Closer to the foam than two pore diameters, the pressure and turbulent transports of turbulent kinetic energy are non-negligible. In the same region, ${\textit {Re}}_\lambda$ undergoes a steep decrease whereas the dissipation coefficient $C_{\epsilon }$ increases like ${\textit {Re}}_\lambda ^{-1}$. At larger distances from the porous layer, the classical grid turbulence situation is recovered, where the mean advection of turbulent kinetic energy equals dissipation. This entails a power-law decay of turbulent quantities and characteristic lengths. The decaying exponents of integral, Taylor and Kolmogorov scales are close to one-half, indicating that the turbulence simulated here differs from Saffman turbulence. Analysis of the scaling exponents of structure functions and the decorrelation length of dissipation reveals that small-scale fluctuations are weakly intermittent.

We elucidate the vortex dynamics of flows past a flapping plate using the vortex-surface field (VSF) and develop models for estimating thrust from shedding vortex surfaces in wakes. The VSF evolution is calculated from numerical simulation using the immersed boundary method. The VSF visualization reveals that a spoon-like vortex surface dominated by tip vortex lines is formed and periodically shed into the wake owing to the alternating upstroke and downstroke of the flapping plate. We simplify the finite-domain impulse theory based on a particular vortex surface. The simplified theory demonstrates that the force on the plate is only dependent on the vortical impulse and Lamb-vector integral of the vortex surface enclosing the plate. Then, we propose a time-averaged thrust model from near-wake discrete vortex surfaces, where the incorporation of the Lamb-vector integral significantly improves the model estimation from the impulse model. Furthermore, we estimate the mean thrust based on two arbitrary vortex surfaces in the far wake from the linear impulse decay of periodically shedding vortex surfaces, which provides a possible approach to infer the state of the moving body in experimental investigation and practical applications.

We calculate the steady acoustic flow – the steady drift of fluid mass or acoustic streaming appearing along the path of an acoustic stimulus – in porous media. In particular, we suggest a mechanism to explain acoustic contributions to mass transport in porous media at geological, unit operation and lab-on-a-chip length scales. We study several cases of steady acoustic flow for a planar acoustic wave whose wavelength is large compared with the pore size. We commence our analysis at the ideal limit of same acoustic properties in the solid and fluid. The effective flow may then be treated intuitively according to the Darcy equation for flow through porous media in addition to a correction for the average azimuth of the pores compared with the acoustic path. We further consider the framework of a rigid porous frame, where the presence of a flow forcing mechanism resulting from the viscous dissipation of the acoustic wave at the solid surface of the pores hinders the intuitive application of the Darcy equation. However, we show that the steady acoustic flow in this case may be written as a quasi-Darcy-type equation. The analysis is conducted by a detailed calculation of the transport of mass through cylindrical pores of similar size but arbitrary azimuth compared with the acoustic path. We consider large, medium and small pore diameter limits relative to the viscous penetration length of the acoustic wave near the pore surface.

Adjoint-based sensitivity analysis is routinely used today to assess efficiently the effect of open-loop control on the linear stability properties of unstable flows. Sensitivity maps identify regions where small-amplitude control is the most effective, i.e. yields the largest first-order (linear) eigenvalue variation. In this study an adjoint method is proposed for computing a second-order (quadratic) sensitivity operator, and applied to the flow past a circular cylinder, controlled with a steady body force or a passive device model. Maps of second-order eigenvalue variations are obtained, without computing controlled base flows and eigenmodes. For finite control amplitudes, the second-order analysis improves the accuracy of the first-order prediction, and informs about its range of validity, and whether it underestimates or overestimates the actual eigenvalue variation. Regions are identified where control has little or no first-order effect but a second-order effect. In the cylinder wake, the effect of a control cylinder tends to be underestimated by the first-order sensitivity, and including second-order effects yields larger regions of flow restabilisation. Second-order effects can be decomposed into two mechanisms: second-order base flow modification, and interaction between first-order modifications of the base flow and eigenmode. Both contribute equally in general in sensitive regions of the cylinder wake. Exploiting the second-order sensitivity operator, the optimal control maximising the total second-order stabilisation is computed via a quadratic eigenvalue problem. The approach is applicable to other types of control (e.g. wall blowing/suction and shape deformation) and other eigenvalue problems (e.g. amplification of time-harmonic perturbations, or resolvent gain, in stable flows).

Investigation on the shock-induced finite-thickness fluid-layer evolution is very desirable but remains a challenge because it not only involves both the Richtmyer–Meshkov instability (RMI) and the Rayleigh–Taylor instability (RTI), but also strongly depends on the waves reverberated inside the layer. We experimentally and theoretically examined the evolution of a shocked $\textrm {SF}_6$ gas layer with a finite thickness surrounded by air. Specifically, three kinds of quasi-one-dimensional $\textrm {SF}_6$ gas layers with different layer thicknesses are generated to study the wave patterns and interface motions, and six types of quasi-two-dimensional $\textrm {SF}_6$ gas layers with diverse layer thicknesses and amplitude combinations are created to explore the interfacial instabilities of the layer. When the initial fluid layer is thin, the two interfaces of the layer coalesce at a late time. The present study is the first to report that except for the RMI induced by a shock wave on the two interfaces, the rarefaction waves (RW) inside the fluid layer induce the additional RTI and decompression effect on the first interface, and the compression waves (CW) inside the fluid layer cause the additional Rayleigh–Taylor stabilisation (RTS) and compression effect on the second interface. A general one-dimensional theory is established to describe the motions of the two interfaces. Linear and nonlinear models are successfully established by considering the interface-coupling effect on the RMI and the additional interfacial instabilities induced by these waves inside the heavy fluid layer. The established models predict well the perturbation growths on the two interfaces at all regimes.

A steady two-dimensional boundary layer subject to an adverse streamwise pressure gradient usually separates. In this paper, we investigate how free-stream vortical disturbances (FSVD) of moderate level prevent the separation in such a boundary layer over a plate or concave wall. The focus is on physically realisable FSVD with sufficiently long wavelength (low frequency) as they have the most significant impact on the boundary layer. The FSVD intensity $\epsilon$ is taken to be small but nevertheless strong enough that the streaks or Görtler vortices generated in the boundary layer are fully nonlinear and can alter the mean-flow profile by an order-one amount. The excitation and evolution of streaks and Görtler vortices are governed by the nonlinear unsteady boundary-region equations supplemented by appropriate initial (upstream) and boundary (far-field) conditions, which describe appropriately the action of FSVD on the boundary layer. The flow variables are decomposed into two parts: the steady spanwise-averaged and the unsteady or spanwise-varying components. These two parts are coupled and are computed simultaneously. Numerical results show that the separation is eliminated when the FSVD level exceeds a critical intensity $\epsilon _c$. It is inferred that the strong nonlinear mean-flow distortion associated with the nonlinear streaks or Görtler vortices prevents the separation. The critical FSVD intensity $\epsilon _c$ depends on the streamwise curvature, the pressure gradient and the frequency of FSVD. The value of $\epsilon _c$ decreases significantly with the Görtler number, indicating that concave curvature inhibits separation. A higher $\epsilon _c$ is required to prevent the separation in the case of stronger adverse pressure gradient. Interestingly, unsteady FSVD with low frequencies are found to be more effective than steady ones in suppressing the separation.

We investigate the modes of deformation of an initially spherical bubble immersed in a homogeneous and isotropic turbulent background flow. We perform direct numerical simulations of the two-phase incompressible Navier–Stokes equations, considering a low-density bubble in the high-density turbulent flow at various Weber numbers (the ratio of turbulent and surface tension forces) using the air–water density ratio. We discuss a theoretical framework for the bubble deformation in a turbulent flow using a spherical harmonic decomposition. We propose, for each mode of bubble deformation, a forcing term given by the statistics of velocity and pressure fluctuations, evaluated on a sphere of the same radius. This approach formally relates the bubble deformation and the background turbulent velocity fluctuations, in the limit of small deformations. The growth of the total surface deformation and of each individual mode is computed from the direct numerical simulations using an appropriate Voronoi decomposition of the bubble surface. We show that two successive temporal regimes occur: the first regime corresponds to deformations driven only by inertial forces, with the interface deformation growing linearly in time, in agreement with the model predictions, whereas the second regime results from a balance between inertial forces and surface tension. The transition time between the two regimes is given by the period of the first Rayleigh mode of bubble oscillation. We discuss how our approach can be used to relate the bubble lifetime to the turbulence statistics and eventually show that at high Weber numbers, bubble lifetime can be deduced from the statistics of turbulent fluctuations at the bubble scale.

The buoyancy-driven motion of two identical gas bubbles released in line in a liquid at rest is examined with the help of highly resolved simulations, focusing on moderately inertial regimes in which the path of an isolated bubble is vertical. Assuming first an axisymmetric evolution, equilibrium configurations of the bubble pair are determined as a function of the buoyancy-to-viscous and buoyancy-to-capillary force ratios which define the Galilei ($Ga$) and Bond ($Bo$) numbers of the system, respectively. The three-dimensional solutions reveal that this axisymmetric equilibrium is actually never reached. Instead, provided $Bo$ stands below a critical $Ga$-dependent threshold determining the onset of coalescence, two markedly different evolutions are observed. At the lower end of the explored $(Ga, Bo)$-range, the tandem follows a drafting–kissing–tumbling scenario, which eventually yields a planar side-by-side motion. For larger $Ga$, the trailing bubble drifts laterally and gets out of the wake of the leading bubble, barely altering the path of the latter. In this second scenario, the late configuration is characterized by a significant inclination of the tandem ranging from $30^\circ$ to $40^\circ$ with respect to the vertical. Bubble deformation has a major influence on the evolution of the system. It controls the magnitude of vortical effects in the wake of the leading bubble, hence the strength of the attractive force acting on the trailing bubble. It also governs the strength and even the sign of the lateral force acting on this bubble, a mechanism of particular importance when the tandem is released with a small angular deviation.