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.

Local instantaneous exchanges of volume, momentum and buoyancy across turbulent/non-turbulent interfaces (TNTIs) and turbulent/turbulent interfaces (TTIs) are studied using data from direct numerical simulations of a turbulent forced fountain. We apply a novel algorithm that enables independent calculation of the instantaneous local entrainment and detrainment fluxes, and therefore, for the first time, the entrainment and detrainment coefficients according to the fountain model (Bloomfield & Kerr, J. Fluid Mech., vol. 424, 2000, pp. 197–216) are determined explicitly. Across the interface between the fountain and the ambient fluid, which is a TNTI, only volume entrainment occurs, and it is well predicted by the fountain model. Across the interface between the rising upflow and falling downflow within the fountain, which is a TTI, both entrainment and detrainment of volume, momentum and buoyancy occur – with the magnitude of both entrainment and detrainment typically being large compared with the net for all exchanges. However, the model seems to be unable to capture the momentum exchanges due to its ignorance of the pressure. We find that each conditional entrainment and detrainment rate, of volume, momentum and buoyancy, can be described accurately by Gaussian profiles, while the net exchange that is the superposition of the entrainment and detrainment cannot. Moreover, the entrainment exchange rate has its maximum closer to the fountain centreline than that of detrainment, explaining the tendency for net entrainment closer to the fountain centreline and net detrainment further away.

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.

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 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.

Considerable effort has been directed towards the characterization of chiral mesoscale structures, as shown in chiral protein assemblies and carbon nanotubes. Here, we establish a thermally driven hydrodynamic description for the actuation and separation of mesoscale chiral structures in a fluid medium. Cross-flow of a Newtonian liquid with a thermal gradient gives rise to an effective torque that propels each particle of a chiral suspension according to its handedness. In turn, the chiral suspension alters the liquid flow, which thus acquires a transverse (chiral) velocity component. Since observation of the predicted effects requires a low degree of sophistication, our work provides an efficient and inexpensive approach to test and calibrate chiral particle propulsion and separation strategies.

Recent studies reveal the dramatic impact of seafloor roughness on the dynamics and stability of broad oceanic flows. These findings motivate the development of parameterizations that concisely represent the effects of small-scale bathymetric patterns in theoretical and coarse-resolution numerical circulation models. The previously reported quasi-geostrophic ‘sandpaper’ theory of flow–topography interaction a priori assumes gentle topographic slopes and weak flows with low Rossby numbers. Since such conditions are often violated in the ocean, we now proceed to formulate a more general model based on shallow-water equations. The new version of the sandpaper model is validated by comparing roughness-resolving and parametric simulations of the flow over a corrugated seamount.

How locally injected turbulence spreads in space is investigated with direct numerical simulations. We consider a turbulent flow in a long triply periodic box generated by a forcing that is localized in space. The forcing is such that it does not inject any mean momentum into the flow. We show that at long times a statistically stationary state is reached where the turbulent energy density in space fluctuates around a mean profile that peaks at the forcing location and decreases fast away from it. We measure this profile as a function of the distance from the forcing region for different values of the Reynolds number. It is shown that, as the Reynolds number is increased, it converges to a Reynolds-number-independent profile, implying that turbulence spreads due to self-advection and not due to molecular diffusion. In this limit, therefore, turbulence plays the simultaneous role of cascading the energy to smaller scales and transporting it to larger distances. The two effects are shown to be of the same order of magnitude. Thus a new turbulent state is reached where turbulent transport and turbulent cascade are equally important and control its properties.

We derive expressions relating the entrainment fluxes of momentum and kinetic energy, relative to the mass flux entrained into a turbulent wake exposed to a turbulent background. These expressions contain correlations between the entrainment velocity and the turbulent fluctuations within the background. We perform high-resolution, simultaneous particle image velocimetry and planar laser-induced fluorescence experiments, and observe these correlations to be negligible in the far wake, such that momentum and kinetic energy are entrained into the wake with the same relative efficiency to mass as from an idealised, non-turbulent background. This is a useful result in the context of modelling, since the entrainment hypothesis (Turner, J. Fluid Mech., vol. 173, 1986, pp. 431–471) can still be used to model the entrainment of momentum and kinetic energy. Nevertheless, the entrainment rate of mass is shown to vary spatially, and with the specific nature of the background turbulence, so this in turn drives a spatial/background-turbulence-specific entrainment rate of momentum/kinetic energy. Contrastingly, in the near wake, whilst momentum is entrained from a turbulent background with the same relative efficiency to mass as for an idealised non-turbulent background, this is not the case for kinetic energy. Owing to the sum of multiple positive, small-valued correlations between the fluctuations in the background and the entrainment velocity, kinetic energy is entrained more efficiently than in the idealised case. This includes entrainment from a non-turbulent background, where small correlations are observed between the irrotational background fluctuations and the entrainment velocity. Evidence is also presented that the entrainment velocity scales with the Kolmogorov velocity scale when the background is turbulent.

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.

Compound bubbles with a liquid coating in another continuous immiscible bulk phase are ubiquitous in a wide range of natural and industrial processes. Their formation, rise and ultimate bursting at the air–liquid interface play crucial roles in the transport and fate of natural organic matter and contaminants. However, the dynamics of compound bubbles has not received considerable attention until recently. Here, inspired by our previous work (Yang et al., Nat. Phys., vol. 19, 2023, pp. 884–890), we investigate the entrainment of daughter oil droplets in bulk water produced by a bursting oil-coated bubble. We document that the size of the entrained daughter oil droplet is affected by the oil coating fraction and the bulk liquid properties. We rationalize this observation by balancing the viscous force exerted by the extensional flow produced by bubble bursting with the capillary force resisting the deformation of the oil coating, and considering the subsequent end-pinching process which finally entrains the daughter oil droplets. We propose a scaling analysis for the daughter oil droplet size that well captures the experimental results for a wide range of oil coating fractions and Ohnesorge numbers of the bulk liquid. In addition, we discuss the non-monotonic variation of daughter droplet size with the Ohnesorge number, and show the eventual absence of daughter droplets because of the strong viscous effect in the high-Ohnesorge-number regime. Our findings may advance the fundamental understanding of compound bubble bursting and provide guidance and modelling constraints for bubble-mediated contaminant transport in liquids.

Invariant solutions of the Navier–Stokes equations play an important role in the spatiotemporally chaotic dynamics of turbulent shear flows. Despite the significance of these solutions, their identification remains a computational challenge, rendering many solutions inaccessible and thus hindering progress towards a dynamical description of turbulence in terms of invariant solutions. We compute equilibria of three-dimensional wall-bounded shear flows using an adjoint-based matrix-free variational approach. To address the challenge of computing pressure in the presence of solid walls, we develop a formulation that circumvents the explicit construction of pressure and instead employs the influence matrix method. Together with a data-driven convergence acceleration technique based on dynamic mode decomposition, this yields a practically feasible alternative to state-of-the-art Newton methods for converging equilibrium solutions. We compute multiple equilibria of plane Couette flow starting from inaccurate guesses extracted from a turbulent time series. The variational method outperforms Newton(-hookstep) iterations in converging successfully from poor initial guesses, suggesting a larger convergence radius.

Existing experimental results show that swirling flames in annular combustors respond with a different gain to acoustic azimuthal modes rotating in either the clockwise or anti-clockwise direction. The ratio $R$ of these two gains is introduced, with $R=1$ being the conventional case of flames responding the same to the two forcing directions. To allow a difference in response to the different directions ($R\neq 1$), a multiple-input single-output azimuthal flame describing function is successfully implemented in a quaternion valued low-order model of an annular combustion chamber in the current work. Theoretical studies have explored this kind of symmetry breaking between the two acoustic wave directions in the past, but it has not been backed by experimental data. One of the main features of the new model proposed in this work is the potential difference in mode shapes between the acoustic and the heat release rate modes, which has recently been observed experimentally. This results in a gain-dependent equation for the nature of the mode, which has a significant influence on the fixed points of the system. For example, one of the spinning solutions and the standing solution can disappear through a saddle node bifurcation as the parameters are varied. The presence of only a single direction for the spinning solution matches experimental observations better than the conventional models, and the proposed model is shown to qualitatively describe experimental measurements well.

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.