In Rayleigh–Bénard convection, the size of a flow domain and its aspect ratio $\varGamma$ (a ratio between the spatial length and height of the domain) affect the shape of the large-scale circulation. For some aspect ratios, the flow dynamics includes a three-dimensional oscillatory mode known as a jump rope vortex (JRV); however, the effects of varying aspect ratios on this mode are not well investigated. In this paper, we study these aspect ratio effects in liquid metals, for a low Prandtl number ${{Pr}}=0.03$. Direct numerical simulations and experiments are carried out for a Rayleigh number range $2.9 \times 10^4 \leq {{Ra}} \leq 1.6 \times 10^6$ and square cuboid domains with $\varGamma =2$, $2.5$, $3$ and $5$. Our study demonstrates that a repeating pattern of a JRV encountered at aspect ratio $\varGamma \approx 2.5$ is the basic structural unit that builds up to a lattice of interlaced JRVs at the largest aspect ratio. The size of the domain determines how many structural units are self-organised within the domain; the number of the realised units is expected to scale as $\varGamma ^2$ with sufficiently large and growing $\varGamma$. We find the oscillatory modes for all investigated $\varGamma$; however, they are more pronounced for $\varGamma =2.5$ and $5$. Future studies for large-aspect-ratio domains of different shapes would enhance our understanding of how the JRVs adjust and reorganise at such scaled-up geometries, and answer the question of whether they are indeed the smallest superstructure units.

We derive the evolution equation of the average uncertainty energy for periodic/homogeneous incompressible Navier–Stokes turbulence and show that uncertainty is increased by strain rate compression and decreased by strain rate stretching. We use three different direct numerical simulations (DNS) of non-decaying periodic turbulence and identify a similarity regime where (a) the production and dissipation rates of uncertainty grow together in time, (b) the parts of the uncertainty production rate accountable to average strain rate properties on the one hand and fluctuating strain rate properties on the other also grow together in time, (c) the average uncertainty energies along the three different strain rate principal axes remain constant as a ratio of the total average uncertainty energy, (d) the uncertainty energy spectrum's evolution is self-similar if normalised by the uncertainty's average uncertainty energy and characteristic length and (e) the uncertainty production rate is extremely intermittent and skewed towards extreme compression events even though the most likely uncertainty production rate is zero. Properties (a), (b) and (c) imply that the average uncertainty energy grows exponentially in this similarity time range. The Lyapunov exponent depends on both the Kolmogorov time scale and the smallest Eulerian time scale, indicating a dependence on random large-scale sweeping of dissipative eddies. In the two DNS cases of statistically stationary turbulence, this exponential growth is followed by an exponential of exponential growth, which is, in turn, followed by a linear growth in the one DNS case where the Navier–Stokes forcing also produces uncertainty.

We consider laminar, fully developed, Poiseuille flows of liquid in the Cassie state through diabatic, parallel-plate microchannels symmetrically textured with isoflux ridges. Via matched asymptotic expansions, we develop expressions for (apparent hydrodynamic) slip lengths and Nusselt numbers. Our small parameter ($\epsilon$) is the pitch of the ridges divided by the height of the microchannel. When the ridges are oriented parallel to the flow, we quantify the error in the Nusselt number expressions in the literature and provide a new closed-form result. It is accurate to $O\left (\epsilon ^2\right )$ and valid for any solid (ridge) fraction, whereas previous ones are accurate to $O\left (\epsilon ^1\right )$ and breakdown in the important limit when the solid fraction approaches zero. When the ridges are oriented transverse to the (periodically fully developed) flow, the error associated with neglecting inertial effects in the slip length is shown to be $O\left (\epsilon ^3{Re}\right )$, where ${Re}$ is the channel-scale Reynolds number based on its hydraulic diameter. The corresponding Nusselt number expressions’ accuracies are shown to depend on the Reynolds number, Péclet number and Prandtl number in addition to $\epsilon$. Manipulating the solution to the inner temperature problem encountered in the vicinity of the ridges shows that classic results for the thermal spreading resistance are better expressed in terms of polylogarithm functions.

The influence of an applied magnetic field on the collisional plasma Richtmyer–Meshkov instability (RMI) is investigated through numerical simulation. The instability is studied within the five-moment multifluid plasma model without any simplifying assumptions such as infinite speed of light, negligible electron inertia or quasineutrality. The plasma is composed of ion and electron fluids, and elastic collisions are modelled with the Braginskii transport coefficients. A collisional regime is investigated and the magnetic field is applied in the direction of shock propagation, which is perpendicular to the density interface. The primary instability is influenced by several terms affecting the evolution of circulation, the most significant of which are the baroclinic, magnetic field torque and intraspecies collisional terms. The applied magnetic field results in a reduction of interface perturbation growth, agreeing qualitatively with previous numerical simulations for the case of an ideal multifluid plasma RMI. The only major difference in the present case's instability mitigation by applied magnetic field, relative to the ideal case with applied magnetic field, is that the elastic collisions replace and obstruct the secondary vorticity suppression mechanism through collisional dissipation of vorticity. Additionally the collisions, influenced by the combination of self-generated and the applied magnetic field, introduce anisotropy to the problem. The primary suppression mechanism for the RMI is unchanged relative to the ideal case, i.e. the magnetic field torque resisting baroclinic deposition of vorticity in the ion fluid.

Rotation and orientation of non-spherical particles in a fluid flow depend on the hydrodynamic torque they experience. However, little is known about the effect of the fluid inertial torque on the dynamics of tiny inertial spheroids in turbulent channel flows, as only Jeffery torque has been considered in previous studies by point-particle direct numerical simulations. In this study, we investigate the rotation and orientation of tiny spheroids with both fluid inertial torque and Jeffery torque in a turbulent channel flow. By comparing with the case in the absence of fluid inertial torque, we find that the rotational and orientational dynamics of spheroids is significantly affected by the fluid inertial torque when the Stokes number, which is non-dimensionalized by fluid viscous time scale, is larger than the critical value $St_c\approx 2$, indicating that the fluid inertial torque is non-negligible for most particle cases considered in earlier studies. In contrast to the earlier findings considering only Jeffery torque (Challabotla et al., J. Fluid Mech., vol. 776, 2015, p. R2), we find that prolate (oblate) spheroids with a large Stokes number tend to tumble (spin) in the streamwise–wall-normal plane in a thinner region near the wall due to the presence of the fluid inertial torque. Approaching the channel centre, the flow shear gradually vanishes, but the velocity difference between local fluid and particles is still pronounced and increasing as particle inertia grows. As a result, in the core region, fluid inertial torque is dominant and drives the particles to align with its broad side normal to the streamwise direction rather than a random orientation observed in earlier studies without fluid inertial torque. Meanwhile, the presence of fluid inertial torque enhances the tumbling rates of spheroids in the core region. In addition, the effect of fluid inertial force on the dynamics of spheroids is also examined in this study, but the results indicate the effect of fluid inertial force is weak. Our findings imply the importance of fluid inertial torque in modelling the dynamics of inertial non-spherical particles in turbulent channel flows.

The unsteady hydrodynamic drag exerted on an oscillating sphere near a planar wall is addressed experimentally, theoretically and numerically. The experiments are performed by using colloidal-probe atomic force microscopy in thermal noise mode. The resonance frequencies and quality factors are extracted from the measurement of the power spectrum density of the probe oscillation for a broad range of gap distances and Womersley numbers. The shift in the resonance frequency of the colloidal probe as the probe goes close to a solid wall infers the wall-induced variations of the effective mass of the probe. Interestingly, a crossover from a positive to a negative shift is observed as the Womersley number increases. In order to rationalize the results, the confined unsteady Stokes equation is solved numerically using a finite-element method, as well as asymptotic calculations. The in-phase and out-of-phase terms of the hydrodynamic drag acting on the sphere are obtained and agree well with the experimental results. All together, the experimental, theoretical and numerical results show that the hydrodynamic force felt by an immersed sphere oscillating near a wall is highly dependent on the Womersley number.

A low-order vortex model has been developed for analysing the unsteady aerodynamics of airfoils. The model employs an infinitely thin vortex sheet in place of the attached boundary layer and a sheet of point vortices for the shed shear layer. The strength and direction of the vortex sheet shed at the airfoil trailing edge are determined by an unsteady Kutta condition. The roll-up of the ambient shear layer is represented by a unique point vortex, which is consistently fed circulation by the last point vortex of the free vortex sheet. The model's dimensionality is reduced by using three tuning parameters to balance representational accuracy and computational efficiency. The performance of the model is evaluated through experiments involving impulsively started and heaving and pitching airfoils. The model accurately captures the dynamics of the development and evolution of the shed vortical structure while requiring minimal computational resources. The validity of the model is confirmed through comparison with experimental force measurements and a baseline unsteady panel method that does not transfer circulation in the free vortex sheet.

We investigate the shallow flow of viscous fluid into and out of a channel whose gap width increases as a power law ($x^n$), where $x$ is the downstream axis. The fluid flows slowly, while injected at a rate in the form of $t^\alpha$, where $t$ is time and $\alpha$ is a constant. The invading fluid has a higher viscosity than the ambient fluid, thus avoiding Saffman–Taylor instability. Similarity solutions of the first kind for the outflow problem are found using approximations of lubrication theory. Zheng et al. (J. Fluid Mech., vol. 747, 2014, pp. 218–246) studied the deep-channel case and found divergent behaviour of the similarity variable as $n\rightarrow 1$ and $n\rightarrow 3$, when fluid flows into and out of the channel, respectively. No divergence is found in the shallow case presented here up to the breakdown of the geometric assumption. The characteristic equilibration time for the numerically simulated constant-volume flow to converge to the similarity solution is calculated assuming an inverse dependence on the ratio disagreement between the current front using the method of lines. An inverse power dependence between equilibration time and ratio disagreement is found for channels of different powers. A similarity solution of the second kind for the inflow problem is found using the phase-plane formalism and the bisection method. An exponential decay relationship is found between $n$ and the degree $\delta$ of the similarity variable $xt^{-\delta }$, which does not show any divergent behaviour for large $n$. An asymptotic behaviour is found for $\delta$ that approaches $1/2$ for $n\gg 1$.

We investigate jumping of sessile droplets from a solid surface in ambient oil using modulated electrowetting actuation. We focus on the case in which the electrowetting effect is activated to cause droplet spreading and then deactivated exactly at the moment the droplet reaches its maximum deformation. By systematically varying the control parameters such as the droplet radius, liquid viscosity and applied voltage, we provide detailed characterisation of the resulting behaviours including a comprehensive phase diagram separating detachment from non-detachment behaviours, as well as how the detach velocity and detach time, i.e. duration leading to detachment, depend on the control parameters. We then construct a theoretical model predicting the detachment condition using energy conservation principles. We finally validate our theoretical analysis by experimental data obtained in the explored ranges of the control parameters.

The stratified inclined duct (SID) sustains an exchange flow in a long, gently sloping duct as a model for continuously forced density-stratified flows such as those found in estuaries. Experiments have shown that the emergence of interfacial waves and their transition to turbulence as the tilt angle is increased appears to be linked to a threshold in the exchange flow rate given by inviscid two-layer hydraulics. We uncover these hydraulic mechanisms by (i) using recent direct numerical simulations (DNS) providing full flow data in the key flow regimes (Zhu et al., J. Fluid Mech., vol. 969, 2023, A20), (ii) averaging these DNS into two layers, and (iii) using an inviscid two-layer shallow-water and instability theory to diagnose interfacial wave behaviour and provide physical insight. The laminar flow is subcritical and stable throughout the duct and hydraulically controlled at the ends of the duct. As the tilt is increased, the flow becomes supercritical everywhere and unstable to long waves. An internal jump featuring stationary waves first appears near the centre of the duct, then leads to larger-amplitude travelling waves, and to stronger jumps, wave breaking and intermittent turbulence at the largest tilt angle. Long waves described by the (nonlinear) shallow-water equation are interpreted locally as linear waves on a two-layer parallel base flow described by the Taylor–Goldstein equation. This link helps us to interpret long-wave instability and contrast it with short-wave (e.g. Kelvin–Helmholtz) instability. Our results suggest a transition to turbulence in SID through long-wave instability relying on vertical confinement by the top and bottom walls.

Motivated by nuclear safety issues, we study the heat transfers in a thin cylindrical fluid layer with imposed fluxes at the bottom and top surfaces (not necessarily equal) and a fixed temperature on the sides. We combine direct numerical simulations and a theoretical approach to derive scaling laws for the mean temperature and for the temperature difference between the top and bottom of the system. We find two asymptotic scaling laws depending on the flux ratio between the upper and lower boundaries. The first one is controlled by heat transfer to the side, for which we recover scaling laws characteristic of natural convection (Batchelor, Q. Appl. Maths, vol. 12, 1954, pp. 209–233). The second one is driven by vertical heat transfers analogous to Rayleigh–Bénard convection (Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56). We show that the system is inherently inhomogeneous, and that the heat transfer results from a superposition of both asymptotic regimes. Keeping in mind nuclear safety models, we also derive a one-dimensional model of the radial temperature profile based on a detailed analysis of the flow structure, hence providing a way to relate this profile to the imposed boundary conditions.

The influence of roughness spacing on boundary layer transition over distributed roughness elements is studied using direct numerical simulation and global stability analysis, and compared with isolated roughness elements at the same Reynolds number $Re_h=U_eh/\nu$ ($U_e$ is the boundary layer edge velocity, h is roughness height and $\nu$ is the kinetic viscosity of the fluid). Small spanwise spacing ($\lambda _z=2.5h$) inhibits the formation of counter-rotating vortex pairs (CVP) and, as a result, hairpin vortices are not generated and the downstream shear layer is steady. For $\lambda _z=5h$, the CVP and hairpin vortices are induced by the first row of roughness, perturbing the downstream shear layer and causing transition. The temporal periodicity of the primary hairpin vortices appears to be independent of the streamwise spacing. Distributed roughness leads to a lower critical roughness Reynolds number for instability to occur and a more significant breakdown of the boundary layer compared with isolated roughness. When the streamwise spacing is $\lambda _x=5h$, the high-momentum fluid barely moves downward into the cavities and the wake flow has little impact on the following roughness elements. The leading unstable varicose mode is associated with the central low-speed streaks along the aligned roughness elements, and its frequency is close to the shedding frequency of hairpin vortices in the isolated case. For larger streamwise spacing ($\lambda _x=10h$), two distinct modes are obtained from global stability analysis. The first mode shows varicose symmetry, corresponding to the primary hairpin vortex shedding induced by the first-row roughness. The high-speed streaks formed in the longitudinal grooves are also found to be unstable and interact with the varicose mode. The second mode is a sinuous mode with lower frequency, induced as the wake flow of the first-row roughness runs into the second row. It extracts most energy from the spanwise shear between the high- and low-speed streaks.

If a flat, horizontal, plate settles onto a flat surface, it is known that the gap $h$ decreases with time $t$ as a power law: $h\sim t^{-1/2}$. We consider what happens if the plate is not initially horizontal, and/or the centre of mass is not symmetrically positioned: does one edge contact the surface in finite time, or does the plate approach the horizontal without making contact? The dynamics of this system is analysed and shown to be remarkably complex. We find that, depending upon the initial position of the plate and the position of the centre of force, the plate might either make contact in finite time or settle progressively without ever making contact. Our results show an excellent agreement between analytical exact solutions, asymptotic solutions and numerical studies of the lubrication equations.

The present study investigates the flame dynamics of a contactless burning fuel droplet under free fall subjected to a co-flow. The dynamic external relative flow established due to co-flow and droplet acceleration results in a series of droplet flame transitions. Different flame structures were observed, including a wake flame, reversed wake flame and enveloped flame. Following ignition, the droplet is allowed to fall through the central tube of a co-flow arrangement, and, at its exit, the droplet flame encounters the co-flow. The wake flame, which was established based on the droplet's instantaneous velocity of descent, encounters the abrupt relative velocity jump due to the co-flow. This causes the droplet flame to go through various transitions as it approaches equilibrium with the surrounding flow. Once it equilibrates, the droplet flame evolves in response to the instantaneous relative flow velocity. The droplet flame evolves by altering both its shape and the stabilization mechanism. Two stabilization mechanisms were identified for the droplet wake flame: edge-flame stabilization and bluff-body stabilization. The stabilization mechanism for different flame structures and the transition events have been theoretically analysed, and the relation between flame shape evolution and flow velocity has been determined based on the flow-field characteristics at the corresponding Re (Reynolds number) range. Furthermore, these correlations are employed in a mathematical formulation based on the spring–mass analogy, which predicts the droplet flame evolution after encountering the co-flow, including all the transition events.

The motion of a disk in a Langmuir film bounding a liquid substrate is a classical hydrodynamic problem, dating back to Saffman (J. Fluid Mech., vol. 73, 1976, p. 593) who focused upon the singular problem of translation at large Boussinesq number, ${\textit {Bq}}\gg 1$. A semianalytic solution of the dual integral equations governing the flow at arbitrary ${\textit {Bq}}$ was devised by Hughes et al. (J. Fluid Mech., vol. 110, 1981, p. 349). When degenerated to the inviscid-film limit ${\textit {Bq}}\to 0$, it produces the value $8$ for the dimensionless translational drag, which is $50\,\%$ larger than the classical $16/3$-value corresponding to a free surface. While that enhancement has been attributed to surface incompressibility, the mathematical reasoning underlying the anomaly has never been fully elucidated. Here we address the inviscid limit ${\textit {Bq}}\to 0$ from the outset, revealing a singular mechanism where half of the drag is contributed by the surface pressure. We proceed beyond that limit, considering a nearly inviscid film. A naïve attempt to calculate the drag correction using the reciprocal theorem fails due to an edge singularity of the leading-order flow. We identify the formation of a boundary layer about the edge of the disk, where the flow is primarily in the azimuthal direction with surface and substrate stresses being asymptotically comparable. Utilising the reciprocal theorem in a fluid domain tailored to the asymptotic topology of the problem produces the drag correction $(8\,{\textit {Bq}}/{\rm \pi} ) [ \ln (2/{\textit {Bq}}) + \gamma _E+1]$, $\gamma _E$ being the Euler–Mascheroni constant.

A cylinder immersed in a current and free to translate along a circular arc is considered to investigate the impact of path curvature on the flow-induced vibrations (FIV) occurring without structural restoring force. Path curvature magnitude ($\kappa$, inverse of path radius non-dimensionalized by the body diameter $D$) is varied from $0$ (transverse rectilinear path) to $20$, over a wide range of values of the structure to displaced fluid mass ratio, $m^\star \in [0.05,10]$. The exploration is carried out numerically at subcritical and postcritical values of the Reynolds number ($Re$, based on $D$ and the inflow velocity), i.e. below and above the critical value $47$ for the onset of flow unsteadiness when the body is fixed, up to $100$. Path curvature triggers a desynchronized regime of the flow–body system in addition to the synchronized regime typical of vortex-induced vibrations, and alters the composition of fluid forcing. The most prominent effect uncovered here is, however, a global enhancement of FIV, with three principal results: (i) vibrations and flow unsteadiness are found to arise at lower subcritical $Re$ along a curved path, down to $19.5$ versus $31$ for $\kappa =0$; (ii) the $m^\star$ range where substantial responses develop is considerably extended and encompasses the entire interval under study, which contrasts with the narrow band of low $m^\star$ identified for $\kappa =0$; (iii) the vibrations are amplified, $+45\,\%$ relative to the peak amplitude measured along a rectilinear path at $Re =100$.

We study turbulent flow in open channels with a free surface and rectangular cross-section, for various Reynolds numbers and duct aspect ratios. Direct numerical simulations are used to obtain accurate characterization of the secondary motions, which are found to be more intense than in closed ducts, and to scale with the bulk, rather than with the friction velocity. A notable feature of open-duct flows is the presence of a velocity dip, namely the peak velocity is achieved at some depth underneath the free surface. We find that the depth of the velocity peak increases with the Reynolds number, and correspondingly the flow becomes more symmetric with respect to the horizontal midplane. This is also confirmed from the change of the topology of the secondary motions, which exhibit a strong corner circulation at the free-surface/wall corners at low Reynolds number, which, however, weakens at higher $Re$. The structure of the mean velocity field is such that the log law applies with good approximation in the direction normal to the nearest wall, which allows us to explain why predictive friction formulae based on the hydraulic diameter concept are successful. Additional analysis shows that the secondary motions account for a large fraction of the frictional drag (up to $15$ %).

We study the inertial migration of a torque-free neutrally buoyant sphere in wall-bounded plane Couette flow over a wide range of channel Reynolds numbers $Re_c$ in the limit of small particle Reynolds number ($Re_p\ll 1$) and confinement ratio ($\lambda \ll 1$). Here, $Re_c = V_{wall}H/\nu$, where $H$ denotes the separation between the channel walls, $V_\text {wall}$ denotes the speed of the moving wall, and $\nu$ is the kinematic viscosity of the Newtonian suspending fluid. Also, $\lambda = a/H$, where $a$ is the sphere radius, with $Re_p=\lambda ^2 Re_c$. The channel centreline is found to be the only (stable) equilibrium below a critical $Re_c$ ($\approx 148$), consistent with the predictions of earlier small-$Re_c$ analyses. A supercritical pitchfork bifurcation at the critical $Re_c$ creates a pair of stable off-centre equilibria, located symmetrically with respect to the centreline, with the original centreline equilibrium becoming unstable simultaneously. The new equilibria migrate wall-ward with increasing $Re_c$. In contrast to the inference based on recent computations, the aforementioned bifurcation occurs for arbitrarily small $Re_p$ provided that $\lambda$ is sufficiently small. An analogous bifurcation occurs in the two-dimensional scenario, that is, for a circular cylinder suspended freely in plane Couette flow, with the critical $Re_c$ being approximately $110$.

We study the effects of buoyancy, surface-tension gradients and phase boundary on the stability of a layer of water that is confined between air at the top and a layer of ice at the bottom. The temperature of the overlying air and flux condition at the free surface of the water layer are such that the layer is susceptible to both thermal and thermocapillary instabilities. We perform a linear stability analysis to identify these modes of instability and investigate the effects of the phase boundary on them. We find that with increasing thickness of the ice layer, the critical Rayleigh and Marangoni numbers for the instabilities are found to first decrease and then asymptote to constant values for ice thicknesses much larger than the thickness of the water layer. In the case of thermocapillary instability, we find that the thickness of the ice layer has negligible influence on the stability threshold for dimensionless wavenumber $k \gg 1$, and that the presence of an unstably stratified liquid layer significantly alters the stability threshold for $k = O (1)$. Furthermore, the inclusion of Marangoni stresses reduces the stability threshold of the thermal instability.

The rheological behaviour of dense suspensions of ideally conductive particles in the presence of both electric field and shear flow is studied using large-scale numerical simulations. Under the action of an electric field, these particles are known to undergo dipolophoresis (DIP), which is the combination of two nonlinear electrokinetic phenomena: induced-charge electrophoresis (ICEP) and dielectrophoresis (DEP). For ideally conductive particles, ICEP is predominant over DEP, resulting in transient pairing dynamics. The shear viscosity and first and second normal stress differences $N_1$ and $N_2$ of such suspensions are examined over a range of volume fractions $15\,\% \leq \phi \leq 50\,\%$ as a function of Mason number $Mn$, which measures the relative importance of viscous shear stress over electrokinetic-driven stress. For $Mn < 1$ or low shear rates, the DIP is shown to dominate the dynamics, resulting in a relatively low-viscosity state. The positive $N_1$ and negative $N_2$ are observed at $\phi < 30\,\%$, which is similar to Brownian suspensions, while their signs are reversed at $\phi \ge 30\,\%$. For $Mn \ge 1$, the shear thickening starts to arise at $\phi \ge 30\,\%$, and an almost five-fold increase in viscosity occurs at $\phi = 50\,\%$. Both $N_1$ and $N_2$ are negative for $Mn \gg 1$ at all volume fractions considered. We illuminate the transition in rheological behaviours from DIP to shear dominance around $Mn = 1$ in connection to suspension microstructure and dynamics. Lastly, our findings reveal the potential use of nonlinear electrokinetics as a means of active rheology control for such suspensions.