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.

We numerically investigated the global linear instability and bifurcations in electro-thermo-convection (ETC) of a dielectric liquid confined in a two-dimensional (2-D) concentric annulus subjected to a strong unipolar injection. Seven kinds of solutions exist in this ETC system due to the complex bifurcations, i.e. saddle-node, subcritical and supercritical Hopf bifurcations. These bifurcation routes constitute at most four solution branches. Global linear instability analysis and energy analysis were conducted to explain the instability mechanism and transition of different solutions and to predict the local instability regions. The linearized lattice Boltzmann method (LLBM) for global linear instability analysis, first proposed by Pérez et al. (Theor. Comput. Fluid Dyn., vol. 31, 2017, pp. 643–664) to analyse incompressible flows, was extended here to solve the whole set of coupled linear equations, including the linear Navier–Stokes equations, the linear energy equation, Poisson's equation and the linear charge conservation equation. A multiscale analysis was also performed to recover the macroscopic linearized Navier–Stokes equations from the four different discrete lattice Boltzmann equations (LBEs). The LLBM was validated by calculating the linear critical value of 2-D natural convection; it has an error of 1.39% compared with the spectral method. Instability with global travelling wave behaviour is a unique behaviour in the annulus configuration electrothermohydrodynamic system, which may be caused by the baroclinity. Finally, the chaotic behaviour was quantitatively analysed through calculation of the fractal dimension and Lyapunov exponent.

We present numerical analyses of two-dimensional electrohydrodynamic (EHD) flows of a dielectric liquid between a wire electrode and two plate electrodes with a Poiseuille flow, using direct numerical simulation and global stability analysis. Both conduction and injection mechanisms for charge generation are considered. In this work we focused on the intensity of the cross-flow and studied the EHD flows without a cross-flow, with a weak cross-flow and with a strong cross-flow. (1) In the case without a cross-flow, we investigated its nonlinear flow structures and linear dynamics. We found that the flow in the conduction regime is steady, whereas the flow in the injection regime is oscillatory, which can be explained by a global stability analysis. (2) The EHD flow with a weak cross-flow is closely related to the flow phenomena in an electrostatic precipitator (ESP). Our analyses indicate that increasing the cross-flow intensity or the electric Reynolds number leads to a less stable flow. Based on these results, we infer that one should adopt a relatively low voltage and weak cross-flow in the wire-plate EHD flow to avoid flow instability, which may hold practical implications for ESP. (3) The case of strong cross-flow is examined to study the EHD effect on the wake flow. By comparing the conventional cylindrical wake with the EHD wake in linear and nonlinear regimes, we found that the EHD effect brings forward the vortex shedding in wake flows. Besides, the EHD effect reduces the drag coefficient when the cross-flow is weak, but increases it when it is strong.

We study the diffusive instability subject to a solid–liquid interface and a Kolmogorov flow using modal, non-modal analyses and energy analysis. It is found that the phase boundary has different effects on the threshold of diffusive convection for weak and strong salinity stratification. In the context of oceanography where the salinity Rayleigh number RS is very high, the ice–water interface has negligible influence on the onset of diffusive convection. In the presence of shear, the diffusive convection for RS < 106 tends to be inhibited and with the increase of shear intensity, the oscillatory, steady convective and Kelvin–Helmholtz instabilities will be successively dominant. For RS > 106, the shear has a destabilizing effect on the diffusive convection, due to the generation of thermohaline-sheared instability found by Radko (J. Fluid Mech., vol. 805, 2016, pp. 147–170). Non-modal analysis indicates that for realistic parameters of high-latitude oceans, with the transition of the ultimate energy source of instability from the density gradient to background current, the thermohaline-shear instability is expected to transition from oscillatory to steady instability. The initial transient amplification due to double diffusion has also been studied, which is due to the generation of overstable instability at the initial phase. For RS < 106, the optimal initial condition to achieve the maximum transient growth favours longitudinal rolls. For thermohaline-shear instability, however, it favours transverse rolls and specifically, in oscillatory thermohaline-shear instability, the transient amplification can be enhanced by the shear by one order of magnitude, thus having important influence on the stability of the system.

In this work, we report numerical results on the flow instability and bifurcation of a viscoelastic fluid in the upstream region of a cylinder in a confined narrow channel. Two-dimensional direct numerical simulations based on the FENE-P model (the finite-extensible nonlinear elastic model with the Peterlin closure) are conducted with numerical stabilization techniques. Our results show that the macroscopic viscoelastic constitutive relation can capture the viscoelastic upstream instability reported in previous experiments for low-Reynolds-number flows. The numerical simulations reveal that the non-dimensional recirculation length (LD) is affected by the cylinder blockage ratio (BR), the Weissenberg number (Wi), the viscosity ratio (β) and the maximum polymer extension (L). Close to the onset of upstream recirculation, LD with Wi satisfy Landau-type quartic potential under certain parameter space. The bifurcation may exhibit subcritical behaviour depending on the values of L2 and β. The parameters β and L2 have nonlinear influence on the upstream recirculation length. This work contributes to our theoretical understanding of this new instability mechanism in viscoelastic wake flows.

This work reports the application of a model-free deep reinforcement learning (DRL) based flow control strategy to suppress perturbations evolving in the one-dimensional linearised Kuramoto–Sivashinsky (KS) equation and two-dimensional boundary layer flows. The former is commonly used to model the disturbance developing in flat-plate boundary layer flows. These flow systems are convectively unstable, being able to amplify the upstream disturbance, and are thus difficult to control. The control action is implemented through a volumetric force at a fixed position, and the control performance is evaluated by the reduction of perturbation amplitude downstream. We first demonstrate the effectiveness of the DRL-based control in the KS system subjected to a random upstream noise. The amplitude of perturbation monitored downstream is reduced significantly, and the learnt policy is shown to be robust to both measurement and external noise. One of our focuses is to place sensors optimally in the DRL control using the gradient-free particle swarm optimisation algorithm. After the optimisation process for different numbers of sensors, a specific eight-sensor placement is found to yield the best control performance. The optimised sensor placement in the KS equation is applied directly to control two-dimensional Blasius boundary layer flows, and can efficiently reduce the downstream perturbation energy. Via flow analyses, the control mechanism found by DRL is the opposition control. Besides, it is found that when the flow instability information is embedded in the reward function of DRL to penalise the instability, the control performance can be further improved in this convectively unstable flow.

We study numerically a sequence of eddies in two-dimensional electrohydrodynamics (EHD) flows of a dielectric liquid, driven by an electric potential difference between a hyperbolic blade electrode and a flat plate electrode (or the blade–plate configuration). The electrically driven flow impinges on the plate to generate vortices, which resemble Moffatt eddies (Moffatt, J. Fluid Mech., vol. 18, 1964, pp. 1–18). Such a phenomenon in EHD was first reported in the experimental work of Perri et al. (J. Fluid Mech., vol. 900, 2020, A12). We conduct direct numerical simulations of the EHD flow with three Moffatt-type eddies in a large computational domain at moderate electric Rayleigh numbers ($T$, quantifying the strength of the electric field). The ratios of size and intensity of the adjacent eddies are examined, and they can be compared favourably to the theoretical prediction of Moffatt; interestingly, the quantitative comparison is remarkably accurate for the two eddies in the far field. Our investigation also shows that a larger $T$ strengthens the vortex intensity, and a stronger charge diffusion effect enlarges the vortex size. A sufficiently large $T$ can further result in an oscillating flow, consistent with the experimental observation. In addition, a global stability analysis of the steady blade–plate EHD flow is conducted. The global mode is characterised in detail at different values of $T$. When $T$ is large, the confinement effect of the geometry in the centre region may lead to an increased oscillation frequency. This work contributes to the quantitative characterisation of the Moffatt-type eddies in EHD flows.

We study the two-dimensional Rayleigh–Bénard instability subject to the combined effects of a solid–liquid phase boundary and shear using linear stability theory and energy analysis. We consider two thermal states of the solid (isothermal and conducting), and two types of shear that can arise in different contexts. When the melting temperature is equal to that held at the top boundary, three instability modes can arise with the increase of Reynolds number Re that characterizes the shear intensity: a boundary mode, a mixed boundary–bulk mode and a bulk flow mode. When the melting temperature lies between the top and the bottom boundaries, the introduction of Couette flow, independent of its intensity, always leads to the mixed mode, whereas the instability with Poiseuille flow is dominated by the bulk flow mode once Re exceeds a critical value, below which the mixed mode dominates. The energy analysis suggests that there exist two mechanisms by which the shear flow affects the system: one is by inhibiting the upward heat flux and another is by absorbing energy from the perturbed hydrodynamic field. These two mechanisms can play totally different roles in different cases. Results in the high-Re regime indicate that, when Re exceeds its classical threshold, i.e. Re = 5772.2 for Poiseuille flow, the Tollmien–Schlichting instability will be dominant in the present system.

This work explains a scaling law of the first Landau coefficient of the derived Ginzburg–Landau equation in the weakly nonlinear analysis of axisymmetric viscoelastic pipe flows in the large-Weissenberg-number ($Wi$) limit, recently reported in Wan et al. (J. Fluid Mech., vol. 929, 2021, A16). Using an asymptotic method, we derive a reduced system, which captures the characteristics of the linear centre-mode instability near the critical condition in the large-$Wi$ limit. Based on the reduced system we then conduct a weakly nonlinear analysis using a multiple-scale expansion method, which readily explains the aforementioned scaling law of the Landau coefficient and some other scaling laws. Particularly, the equilibrium amplitude of disturbance near linear critical conditions is found to scale as $Wi^{-1/2}$, which may be of interest to experimentalists. The current analysis reduces the numbers of parameters and unknowns and exemplifies an approach to studying the viscoelastic flow at large $Wi$, which could shed new light on the understanding of its nonlinear dynamics.

It is recently found that viscoelastic pipe flows can be linearly unstable, leading to the possibility of a supercritical transition route, in contrast to Newtonian pipe flows. Such an instability is referred to as the centre mode, which was studied numerically by Chaudhary et al. (J. Fluid Mech., vol. 908, 2021, p. A11) based on an Oldroyd-B model. In this paper, we are interested in expanding the parameter space investigated and exploring the asymptotic scalings related to this centre instability in the Oldroyd-B viscoelastic pipe flow. It is found from the asymptotic analysis that the centre mode exhibits a three-layered asymptotic structure in the radial direction, a wall layer, a main layer and a central layer, which are driven by pure viscosity, axial and/or radial pressure gradient, and a combined effect of viscosity and elasticity, respectively. Depending on the relations of the control parameters, two regimes, the long-wavelength and short-wavelength centre instabilities, emerge, for which the central-layer thicknesses are of different orders of magnitude. Our large-Reynolds-number asymptotic predictions are compared to the numerical solutions of the original eigenvalue system, and favourable agreement is achieved, especially when the parameters approach their individual limits. In addition to revealing the dominant factors and their balances, the asymptotic analysis describes the instability system by reducing the number of control parameters, and furthermore explaining the collapse of the numerical results for different re-scalings.

This paper presents results of three-dimensional direct numerical simulations (DNS) and global linear stability analyses of a viscous incompressible flow past a finite-length cylinder with two free flat ends. The cylindrical axis is normal to the streamwise direction. The work focuses on the effects of aspect ratios (in the range of $0.5\leq {\small \text{AR}} \leq 2$, cylinder length over diameter) and Reynolds numbers ($Re\leq 1000$ based on cylinder diameter and uniform incoming velocity) on the onset of vortex shedding in this flow. All important flow patterns have been identified and studied, especially as ${\small \text{AR}}$ changes. The appearance of a steady wake pattern when ${\small \text{AR}} \leq 1.75$ has not been discussed earlier in the literature for this flow. Linear stability analyses based on the time-mean flow has been applied to understand the Hopf bifurcation past which vortex shedding happens. The nonlinear DNS results indicate that there are two vortex shedding patterns at different $Re$, one is transient and the other is nonlinearly saturated. The vortex-shedding frequencies of these two flow patterns correspond to the eigenfrequencies of the two global modes in the stability analysis of the time-mean flow. Wherever possible, we compare the results of our analyses to those of the flows past other short-${\small \text{AR}}$ bluff bodies in order that our discussions bear more general meanings.

This work studies the application of a reinforcement learning (RL)-based flow control strategy to the flow past a cylinder confined between two walls to suppress vortex shedding. The control action is blowing and suction of two synthetic jets on the cylinder. The theme of this study is to investigate how to use and embed physical information of the flow in the RL-based control. First, global linear stability and sensitivity analyses based on the time-mean flow and the steady flow (which is a solution to the Navier–Stokes equations) are conducted in a range of blockage ratios and Reynolds numbers. It is found that the most sensitive region in the wake extends itself when either parameter increases in the parameter range we investigated here. Then, we use these physical results to help design RL-based control policies. We find that the controlled wake converges to the unstable steady base flow, where the vortex shedding can be successfully suppressed. A persistent oscillating control seems necessary to maintain this unstable state. The RL algorithm is able to outperform a gradient-based optimisation method (optimised in a certain period of time) in the long run. Furthermore, when the flow stability information is embedded in the reward function to penalise the instability, the controlled flow may become more stable. Finally, according to the sensitivity analyses, the control is most efficient when the probes are placed in the most sensitive region. The control can be successful even when few probes are properly placed in this manner.

In this work we study the convection flow driven by a heat-releasing concentration field which itself is stably stratified. The heat-releasing rate is linearly proportional to concentration. Linear stability analysis is conducted to determine the critical heat-releasing rate for given fluid properties and concentration differences. The most unstable mode associated with the critical heat-releasing rate can be oscillatory for a large concentration Rayleigh number, i.e. the non-dimensionalized concentration difference and large Schmidt number, i.e. the ratio of viscosity to diffusivity of the concentration component. Fully developed flows are then investigated by direct numerical simulations. Flow structures near the bottom plate have larger horizontal scales than those near the top plate. The concentration in the bulk is almost constant and takes a similar value for all the explored parameters, which results in the convective flux increasing linearly with height. To explain the dependences of the global transport properties, we extend the unifying theory of the Rayleigh–Bénard convection to the current system and develop scaling laws for the global fluxes. The numerical results can be described accurately by the theoretical model.

Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently found to be linearly unstable by Garg et al. (Phys. Rev. Lett., vol. 121, 2018, 024502). From a nonlinear point of view, this means that the flow can transition to turbulence supercritically, in contrast to the subcritical Newtonian pipe flows. Experimental evidence of subcritical and supercritical bifurcations of viscoelastic pipe flows have been reported, but these nonlinear phenomena have not been examined theoretically. In this work, we study the weakly nonlinear stability of this flow by performing a multiple-scale expansion of the disturbance around linear critical conditions. The perturbed parameter is the Reynolds number with the others being unperturbed. A third-order Ginzburg–Landau equation is derived with its coefficient indicating the bifurcation type of the flow. After exploring a large parameter space, we found that polymer concentration plays an important role: at high polymer concentrations (or small solvent-to-solution viscosity ratio $\beta \lessapprox 0.785$), the nonlinearity stabilizes the flow, indicating that the flow will bifurcate supercritically, while at low polymer concentrations ($\beta \gtrapprox 0.785$), the flow bifurcation is subcritical. The results agree qualitatively with experimental observations where critical $\beta \approx 0.855$. The pipe flow of upper convected Maxwell fluids can be linearly unstable and its bifurcation type is also supercritical. At a fixed value of $\beta$, the Landau coefficient scales with the inverse of the Weissenberg number ($Wi$) when $Wi$ is sufficiently large. The present analysis provides a theoretical understanding of the recent studies on the supercritical and subcritical routes to the elasto-inertial turbulence in viscoelastic pipe flows.

We investigate deterministic and stochastic bifurcations in electroconvecitve flows of a dielectric liquid confined between two parallel plates subjected to a strong unipolar injection by direct numerical simulations. A long-standing discrepancy of linear instability criteria between the experiment and theory exists in this flow. We here test the hypothesis that the discrepancy may be related to the inhomogeneity in ion-exchange membranes used in experiments, contrasted by the homogeneous ion injection assumed in theoretical and numerical analyses. To study this effect, we consider stochastic boundary conditions around linear criticality and first bifurcations in this flow. For a complete presentation of flow bifurcations, deterministic bifurcation analysis (without stochasticity) is first performed to investigate primary bifurcations in this flow by progressively increasing the strength of electric field. Lyapunov spectrum and dimension are calculated and probed to characterise the chaotic motion therein. Our results confirm the high dimensionality of chaos in electroconvective flows and reveal for the first time that its chaos is extensive in a range of finite-sized systems. We then conduct stochastic bifurcation analyses by considering random perturbations in the boundary conditions of charge density and electric potential. Owing to the subcritical nature of electroconvective flows, the linear instability criteria under stochastic boundaries are closer to the experimental values than former theoretical and numerical results (assuming the homogeneous charge injection) for different levels of stochasticity, which confirms the hypothesis aforementioned. Furthermore, stochasticity can also enhance the efficiency of ionic transport.

We investigate the three-dimensional surface pattern and nonlinear dynamics of a condensing liquid layer suspended from a cooled substrate and in contact with a mixture of its vapour and an inert gas below. A vapour boundary layer (VBL) is introduced, to which the changes in gaseous composition and temperature are assumed to be confined. An interfacial transport equation is derived, which incorporates the physical effects of convection and diffusion of vapour within the VBL, coupled with a long-wave evolution equation for the location of the free surface. This work extends the study of Kanatani (J. Fluid Mech., vol. 732, 2013, pp. 128–149) on a sessile evaporating film to the Rayleigh–Taylor unstable condensing/evaporating case with nonlinear analyses which also accounts for the effect of vapour recoil due to mass transfer on interfacial evolution and that of gravity combined with buoyancy on the internal convection of pendent drops in a condensate layer. The coupled nonlinear evolutionary system is referred to as a $1.5$-sided model. It can be reduced to the conventional one-sided model when phase change is limited by processes in the liquid. An extended basic state is obtained, whose stability is investigated with pseudo-steady linear theory and time-dependent nonlinear simulations. With the one-sided model, the influences of vapour recoil and Marangoni effects are illustrated with three representative cases. In the one-sided simulations with a random perturbation, the interface is prone to finite-time rupture and the surface patterns feature isolated droplets when vapour recoil is significant, while it becomes more regular and even without rupture as vapour recoil is weakened relative to the Marangoni effect. This suggests that, in the absence of the convection and diffusion of vapour, the destabilizations of vapour recoil and negative gravity could prevail over the stabilizations of thermocapillarity, capillarity, viscous dissipation and mass gain. With an unsaturated initial interface concentration, $\tilde {x}_{A,I0}$, the $1.5$-sided model indicates that the liquid layer can be stabilized to a quasi-hexagon pattern and the Rayleigh–Taylor-driven rupture can be suppressed with the effects of vapour convection and diffusion near the interface. However, the initial dynamics is in contrast to the case with a saturated $\tilde {x}_{A,I0}$, where transition from weak evaporation to a condensation-dominated regime is seen in the later stage. The viewpoint of stability competition offers vital evidence for an induced Marangoni stabilization, which is a quintessential characteristic of the $1.5$-sided model. Comparisons of the theory and simulations with available experiments are included throughout.

We present a study of absolute and convective instabilities in electrohydrodynamic flow subjected to a Poiseuille flow (EHD-Poiseuille). The electric field is imposed on two infinite flat plates filled with a non-conducting dielectric fluid with unipolar ion injection. Mathematically, the dispersion relation of the linearised problem is studied based on the asymptotic response of an impulse disturbance imposed on the base EHD-Poiseuille flow. Transverse, longitudinal and oblique rolls are investigated to identify the saddle point satisfying the pinching condition in the corresponding complex wavenumber space. It is found that when the ratio of Coulomb force to viscous force increases, the transverse rolls can transit from convective instability to absolute instability. The ratio of hydrodynamic mobility to electric mobility, which exerts negligible effect on the linear stability criterion when the cross-flow is small, has significant influence on the convective–absolute instability transition, especially when the ratio is small. As we change the value of the mobility ratio, a saddle point shift phenomenon occurs in the case of transverse rolls. The unstable longitudinal rolls are convectively unstable as long as there is a cross-flow, a result which is deduced from a one-mode Galerkin approximation. Longitudinal rolls have a larger growth rate than transverse rolls except for a small cross-flow. Finally, regarding the oblique rolls, a numerical search for the saddle point simultaneously in the complex streamwise and transverse wavenumber spaces always yields an absolute transverse wavenumber of zero, implying that oblique rolls give way to transverse rolls when the flow is unstable.