A new two-equation model for gravity-driven liquid film flow based on the long-wave expansion has been derived. The novelty of the model consists in using a base velocity profile combining parabolic (Ruyer-Quil & Manneville, Eur. Phys. J. B, vol. 15, issue 2, 2000, pp. 357–369) and ellipse (Usha et al., Phys. Fluids, vol. 32, issue 1, 2020, 013603) profile functions in the wall-normal coordinate. The dependence on a free parameter $A$ related to the eccentricity of an ellipse serves as an adjustable parameter. The resulting models are consistent at $O(\varepsilon )$ for inertia terms and at $O(\varepsilon ^2)$ for viscous diffusion effects, and predict accurately the primary instability. Appropriate tuning of the adjustable parameter helps to recover accurate predictions for the asymptotic wave celerity of nonlinear solitary waves. Further, the model is shown to capture the closed separation vortices that can form underneath the troughs of precursory capillary ripples.

Understanding the fluid dynamics associated with a circular cylinder oscillating normal to a plane wall is important for safe design of offshore infrastructure, such as power cables and pipeline risers. This paper investigates the fluid dynamics of an oscillating cylinder with no imposed incident current experimentally using flow visualisation and force measurements where the ratio of the cylinder Reynolds number ($Re$) to Keulegan–Carpenter number ($KC$) is $\beta =500$ and $KC$ varies between 2 and 12. The minimum distance between the cylinder and wall was between 12.5 % and $50\,\%$ of the diameter. Across this parameter space three primary vortex flow regimes were observed: (i) for $KC\leq 5$, the flow field is approximately symmetric about the cylinder centreline and the velocity field between the cylinder and the wall resembled a pumping flow in phase with cylinder motion, which is well predicted by potential theory for most of the cycle; (ii) for $5< KC<8$, the flow field is increasingly asymmetric but with frequent switching of the side associated with vortex shedding; and (iii) for $KC\geq 8$, the flow field is consistently asymmetric due to vortex shedding. The in-line force increases when the cylinder is near the wall due to dynamic pressures associated with pumping. This increase can be estimated using potential theory superimposed onto the force time history for an isolated cylinder at the same $KC$ and $Re$. This study complements recent numerical modelling focused on low Reynolds number conditions and provides important insights into the fluid mechanics associated with trenching beneath cable and pipeline risers.

The Lagrangian-mean motion of fluid particles induced by horizontally localized small-amplitude wavepackets of vertically trapped inertia–gravity waves is computed analytically, at second order in wave amplitude, and the results are supported by direct nonlinear numerical simulations. The leading-order motion is assumed to be inertia–gravity waves, which is applicable to oceanic mesoscale flows in regions where wave activity is as strong as or stronger than the balanced flow. The analytical computation is based on time-dependent asymptotic wave–mean interaction theory, and the numerical simulation uses a Galerkin-truncated $f$-plane nonlinear hydrostatic Boussinesq model that retains the barotropic mode and two baroclinic modes (vertical wavenumbers 0, $m$ and $2m$), this being the minimal set on which consistent wave–mean interactions can take place. Two novel dynamical effects are revealed: First, we find that the barotropic component robustly dominates the Lagrangian-mean flow response, which is contrary to earlier findings for the same problem. Second, we discovered a new wavepacket regime in which the baroclinic mean-flow response consists of the persistent radiation of resonantly forced secondary internal waves. The latter effect occurs in an oceanically accessible parameter regime.

We consider the role of gravity in solute transport when a thin droplet evaporates. Under the physically relevant assumptions that the contact line is pinned and the solutal Péclet number, ${Pe}$, is large, we identify two asymptotic regimes that depend on the size of the Bond number, ${Bo}$. When ${Bo} = O(1)$ as ${Pe}\rightarrow \infty$, the asymptotic structure of solute transport follows directly from the surface-tension-dominated regime, whereby advection drives solute towards the contact line, only to be countered by local diffusive effects, leading to the formation of the famous ‘coffee ring.’ In the distinguished limit in which ${Bo} = O({Pe}^{4/3})$ as ${Pe}\rightarrow \infty$, this interplay between advection and diffusion takes place alongside that between surface tension and gravity. In each regime, we perform a systematic asymptotic analysis of the solute transport and compare our predictions to numerical simulations. We identify the effect of gravity on the nascent coffee ring, providing quantitative predictions of the size, location and shape of the solute mass profile. In particular, for a fixed Péclet number, as the effect of gravity increases, the coffee ring is diminished in height and situated further from the contact line. Furthermore, for certain values of ${Bo}$, ${Pe}$ and the evaporation time, a secondary peak may exist inside the classical coffee ring. The onset of this secondary peak is linked to the change in type of the critical point in the solute mass profile at the droplet centre. Both the onset and the peak characteristics are shown to be independent of ${Pe}$.

The disturbance flow field in a hypersonic boundary layer excited by particle impingement was investigated with a focus on the first stage of the laminar-to-turbulent transition process, namely the receptivity process. A previously validated direct numerical simulation approach adopting disturbance flow tracking is used to simulate the particle-induced transition process. Particle impingement generates a highly complex disturbance flow field that can be characterised by a wide range of frequencies and wavenumbers. After providing some insight about the spectral characteristics of the disturbance flow field in the frequency and wavenumber domains, biorthogonal decomposition is employed to reveal the composition of the disturbance flow field consisting of different continuous and discrete eigenmodes that are triggered through particle impingement. The disturbance flow characteristics for different frequency and wavenumber pairs are discussed where large contributions in the disturbance flow spectrum are observed in the vicinity of the impingement location. A significant amount of the disturbance energy is diverted into the free stream leading to large coefficients of projection for the slow and fast acoustic branches while contributions to the entropy and vorticity branches are negligible. In addition to the continuous acoustic spectra, the first-, second- and other higher-order Mack modes are activated and provide large contributions to the disturbance flow field inside the boundary layer. Finally, it is demonstrated that the disturbance flow field in the vicinity of the impingement location can be reconstructed with a maximum relative error of $2.3\,\%$ by employing a theoretical biorthogonal eigenfunction system expansion and by considering contributions from fast and slow acoustic waves and at most four discrete modes only.

Indirect noise is a significant contributor to aircraft engine noise, which needs to be minimized in the design of aircraft engines. Indirect noise is caused by the acceleration of flow inhomogeneities through a nozzle. High-fidelity simulations showed that some flow inhomogeneities can be chemically reacting when they leave the combustor and enter the nozzle (Giusti et al., Trans. ASME J. Engng Gas Turbines Power, vol. 141, issue 1, 2019). The state-of-the-art models, however, are limited to chemically non-reacting (frozen) flows. In this work, first, we propose a low-order model to predict indirect noise in nozzle flows with reacting inhomogeneities. Second, we identify the physical sources of sound, which generate indirect noise via two physical mechanisms: (i) chemical reaction generates compositional perturbations, thereby adding to compositional noise; and (ii) exothermic reaction generates entropy perturbations. Third, we numerically compute the nozzle transfer functions for different frequency ranges (Helmholtz numbers) and reaction rates (Damköhler numbers) in subsonic flows with hydrogen and methane inhomogeneities. Fourth, we extend the model to supersonic flows. We find that hydrogen inhomogeneities have a larger impact on indirect noise than methane inhomogeneities. Both the Damköhler number and the Helmholtz number markedly influence the phase and magnitude of the transmitted and reflected waves, which affect sound generation and thermoacoustic stability. This work provides a physics-based low-order model which can open new opportunities for predicting noise emissions and instabilities in aeronautical gas turbines with multi-physics flows.

Understanding particle drift in suspension flows is of the highest importance in numerous engineering applications where particles need to be separated and filtered out from the suspending fluid. Commonly known drift mechanisms such as the Magnus force, Saffman force and Segré–Silberberg effect all arise only due to inertia of the fluid, with similar effects on all non-spherical particle shapes. In this work, we present a new shape-selective lateral drift mechanism, arising from particle inertia rather than fluid inertia, for ellipsoidal particles in a parabolic velocity profile. We show that the new drift is caused by an intermittent tumbling rotational motion in the local shear flow together with translational inertia of the particle, while rotational inertia is negligible. We find that the drift is maximal when particle inertial forces are of approximately the same order of magnitude as viscous forces, and that both extremely light and extremely heavy particles have negligible drift. Furthermore, since tumbling motion is not a stable rotational state for inertial oblate spheroids (nor for spheres), this new drift only applies to prolate spheroids or tri-axial ellipsoids. Finally, the drift is compared with the effect of gravity acting in the directions parallel and normal to the flow. The new drift mechanism is stronger than gravitational effects as long as gravity is less than a critical value. The critical gravity is highest (i.e. the new drift mechanism dominates over gravitationally induced drift mechanisms) when gravity acts parallel to the flow and the particles are small.

We experimentally and theoretically investigate the dynamics of a partially wetting water droplet subject to a two-dimensional high-speed jet of air blowing perpendicularly to the substrate. When the jet velocity is above a critical value, the droplet evolves under wind and splits into two secondary drops. In addition to droplet splitting, we observe depinning of the droplet on one side when the jet is applied at a small distance from the initial centre of the droplet. In parallel with systematic experiments, we develop a mathematical model to compute the coupled evolution of the droplet and an idealised stagnation-point flow. Our simplified lubrication model yields a criterion for the critical jet velocity, as well as the time scale of the droplet breakup, in qualitative agreement with the experiments.

Two-dimensional horizontally periodic Rayleigh–Bénard convection between stress-free boundaries displays two distinct types of states, depending on the initial conditions. Roll states are composed of pairs of counter-rotating convection rolls. Windy states are dominated by strong horizontal wind (also called zonal flow) that is vertically sheared, precludes convection rolls and suppresses heat transport. Windy states occur only when the Rayleigh number $Ra$ is sufficiently above the onset of convection. At intermediate $Ra$ values, windy states can be induced by suitable initial conditions, but they undergo a transition to roll states after finite lifetimes. At larger $Ra$ values, where windy states have been observed for the full duration of simulations, it is unknown whether they represent chaotic attractors or only metastable states that would eventually undergo a transition to roll states. We study this question using direct numerical simulations of a fluid with a Prandtl number of 10 in a layer whose horizontal period is eight times its height. At each of seven $Ra$ values between $9\times 10^6$ and $2.25\times 10^7$ we have carried out 200 or more simulations, all from initial conditions leading to windy convection with finite lifetimes. The lifetime statistics at each $Ra$ indicate a memoryless process with survival probability decreasing exponentially in time. The mean lifetimes grow with $Ra$ approximately as $Ra^4$. This analysis provides no $Ra$ value at which windy convection becomes stable; it might remain metastable at larger $Ra$ with extremely long lifetimes.

The transition to turbulence in conduits is among the longest-standing problems in fluid mechanics. Challenges in producing or saving energy hinge on understanding promotion or suppression of turbulence. While a global picture based on an intrinsically 3-D subcritical mechanism is emerging for 3-D turbulence, subcritical turbulence is yet to even be observed when flows approach two dimensions, e.g. under intense rotation or magnetic fields. Here, stability analysis and direct numerical simulations demonstrate a subcritical quasi-two-dimensional (quasi-2-D) transition from laminar flow to turbulence, via a radically different 2-D mechanism to the 3-D case, driven by nonlinear Tollmien–Schlichting waves. This alternative scenario calls for a new line of thought on the transition to turbulence and should inspire new strategies to control transition in rotating devices and nuclear fusion reactor blankets.

The dynamics of suspensions of particles has been an active area of research since Einstein first calculated the leading-order correction to the viscosity of a suspension of spherical particles (Einstein, Proc. R. Soc., vol. A102, 1906, pp. 161–179). Since then, researchers have strived to develop an accurate description of the behaviours of suspensions that goes beyond just leading order in the particle volume fraction. Here, we consider the low-Reynolds-number behaviour of a suspension of spherical particles. Working from the Green's functions for the flow due to a single particle, we derive a continuum-level description of the dynamics of suspensions. Our analysis corrects an error in the derivation of these equations in the work of Jackson (Chem. Engng Sci., vol. 52, 1997, pp. 2457–2469) and leads to stable equations of motion for the particles and fluid. In addition, our resulting equations naturally give the sedimentation speed for suspended particles and correct a separate error in the calculation by Batchelor (J. Fluid Mech., vol. 52, 1972, pp. 245–268). Using the pair-correlation function for hard spheres, we are able to compute the sedimentation speed out to seventh order in the volume fraction, which agrees with experimental data up to 30 %–35 %, and also get higher-order corrections to the suspension viscosity, which agree with experiments up to $\sim$15 %. Then, using the pair distribution for spheres in shear flow, we find alterations to both the first and second normal stresses.

We investigate experimentally the two-dimensional flow of a shear-thickening suspension around a rotating cylinder to which a constant torque is applied. While for low torques both the drag and the flow are steady and close to those for a Newtonian fluid, above the onset torque for discontinuous shear thickening the average velocity of the cylinder saturates and large periodic oscillations of the cylinder velocity are observed. The oscillations result from a hydrodynamic instability of the flow: slow-acceleration phases are followed by high-deceleration phases, triggered by the propagation of a thickening front, and so on. The slow-acceleration phases set the oscillation period, which is limited by the cylinder inertia and inversely proportional to the applied torque. Combined analyses of the cylinder motion and the flow reveal that the front typically nucleates when the shear rate at the cylinder surface reaches the discontinuous shear-thickening threshold. In addition, the characteristics (duration, stress) of the deceleration are set by the interplay between the thickening front propagation and the suspension and cylinder inertiae or the container size. Since for a slow acceleration the shear rate at the cylinder surface is essentially the cylinder angular velocity, this description of the unsteadiness elucidates the saturation of the average velocity. More generally, it illustrates how the hydrodynamics of a shear-thickening suspension with a strongly re-entrant rheology can lead to a marginally re-entrant, although steep, drag curve.

A framework for the budget of Reynolds stress, temperature and density variations, turbulent mass flux and turbulent heat flux is presented for buoyancy-generated turbulent flows with a focus on turbulent plumes. The dynamical interactions between the turbulence generated from the velocity and thermodynamic fluctuations in thermal and heated gas plumes is studied. A large-eddy simulation tool has been used to simulate heavier and lighter than air plumes released from a circular heated source for high Reynolds number (Re). The budget equations are developed using a Favre-averaging approach. Both heated air plumes and heated gas plumes, namely, heated ${\rm SO}_2$ and heated ${\rm CH}_4$ plumes, are included in the study. The study focuses on addressing key questions – does the turbulence kinetic energy (TKE) spectrum follow the classical inertial $-5/3$ spectrum or is there a $-3$ buoyancy regime with a different scaling law? Fundamental pathways for generation of turbulence through velocity fluctuations, and temperature and density fluctuations are discussed. Analysis of the TKE and scalar variance budget equations has demonstrated that multiple mechanisms dominate the transport of Reynolds stresses, and in particular that correlations between the density and the velocity dominates the turbulence production mechanisms. Entrainment is connected to the turbulent transport and pressure-dilation processes.

This study presents a class of steadily translating two-dimensional approximately circular vortices. The proposed solutions take the form of a superposition of two nearly concentric vorticity patches with zero net vorticity. Exact quasi-monopolar solutions of this type are found for propagation speeds that are much less than typical azimuthal velocities. While all V-states obtained are shown to be formally unstable, a large subset of configurations is characterized by very low growth rates. Fully nonlinear simulations reveal that such nearly stable eddies can propagate large distances from the point of origin while maintaining their structure and intensity. Therefore, the proposed solutions can serve as models of geophysical vortices that are known to drift relative to the ambient fluid, often exhibiting remarkable longevity and resilience to external perturbations.

We propose an efficient method to reconstruct the turbulent flow field in a neutrally stratified atmospheric boundary layer using large-eddy simulation (LES) and a series of lidar measurements. The reconstruction is formulated as a strong four-dimensional variational data assimilation problem, which involves optimizing two competing terms that contribute in the objective functional. The first term is a likelihood term, while the second contains the initial background distribution of turbulent velocity fluctuations and works as a regularization term. However, computing and storing the full background covariance tensor in turbulent flows is time consuming and resource intensive. In the current work, we investigate the possibility of replacing the complex background tensor by simple analytical approximations based on spectral tensors such as the Hunt–Graham–Wilson (HGW) model (Boundary-Layer Meteorol., vol. 85, 1997, pp. 35–52) or the Mann model (J. Fluid Mech., vol. 273, 1994, pp. 141–168). Afterwards, the problem is solved using a quasi-Newton algorithm and preconditioned to enhance the convergence rate. We test the method using virtual lidar measurements collected on a fine reference LES. Results show a super-linear convergence rate of the optimization algorithm to a local minimum and very good agreement between virtual lidar measurements and reconstruction in the scanning region. Furthermore, we demonstrate that incorporating the Saffman energy spectrum ($E(k) \sim k^2$ where E is the energy spectrum and k is the magnitude of the wavenumber vector) at low wavenumbers into the Mann spectral tensor yields a longer streamwise correlation length, resulting in reduced reconstruction error when compared with the Batchelor spectrum ($E(k) \sim k^4$). Finally, we observe that using the HGW model or Mann model with a Saffman spectrum yields similar results.

We prove a duality between the functional forms of the Faxén formulas associated with a particle of a given shape and material composition and the corresponding singularity solutions for the velocity disturbances induced by that particle, and extend it to the case of systems with coupled transport processes, enabling the solution of a large family of problems via Faxén methods. Prior approaches to constructing proofs of duality of Faxén formulas and Stokes-flow singularities relied on knowledge of all boundary conditions on all particle surfaces, viz. the Lorentz reciprocal theorem approach. We recognized that, in order to bypass the complexity of boundary conditions one can instead invoke energy methods that give reciprocity between operators rather than between specific stress and velocity fields. We derive reciprocal relations between operators, from which we demonstrate that the Faxén/singularity duality is a consequence of a generalized reciprocal relation between conjugate thermodynamic variables. We use our reciprocal relations to derive expressions for the hydrodynamic force on a sphere of arbitrary composition, the hydrodynamic stresslet exerted on a deformable droplet in an arbitrary velocity field, the phoretic force exerted on a rigid particle in the thin double-layer limit in response to arbitrary externally imposed field and the total stresslet on a charged particle in an arbitrary velocity field, i.e. an electroviscous Faxén law.