The Maxey–Riley equation has been extensively used by the fluid dynamics community to study the dynamics of small inertial particles in fluid flow. However, most often, the Basset history force in this equation is neglected. Analytical solutions have almost never been attempted because of the difficulty in handling an integro-differential equation of this type. Including the Basset force in numerical solutions of particulate flows involves storage requirements which rapidly increase in time. Thus the significance of the Basset history force in the dynamics has not been understood. In this paper, we show that the Maxey–Riley equation in its entirety can be exactly mapped as a forced, time-dependent Robin boundary condition of the one-dimensional diffusion equation, and solved using the unified transform method. We obtain the exact solution for a general homogeneous time-dependent flow field, and apply it to a range of physically relevant situations. In a particle coming to a halt in a quiescent environment, the Basset history force speeds up the decay as a stretched exponential at short time while slowing it down to a power-law relaxation, ${\sim}t^{-3/2}$, at long time. A particle settling under gravity is shown to relax even more slowly to its terminal velocity (${\sim}t^{-1/2}$), whereas this relaxation would be expected to take place exponentially fast if the history term were to be neglected. An important mechanism for the growth of raindrops is by the gravitational settling of larger drops through an environment of smaller droplets, and repeatedly colliding and coalescing with them. Using our solution we estimate that the rate of growth rate of a raindrop can be grossly overestimated when history effects are not accounted for. We solve exactly for particle motion in a plane Couette flow and show that the location (and final velocity) to which a particle relaxes is different from that due to Stokes drag alone. For a general flow, our approach makes possible a numerical scheme for arbitrary but smooth flows without increasing memory demands and with spectral accuracy. We use our numerical scheme to solve an example spatially varying flow of inertial particles in the vicinity of a point vortex. We show that the critical radius for caustics formation shrinks slightly due to history effects. Our scheme opens up a method for future studies to include the Basset history term in their calculations to spectral accuracy, without astronomical storage costs. Moreover, our results indicate that the Basset history can affect dynamics significantly.

A two-dimensional model describing river morphodynamic processes under mixed-size sediment conditions is analysed with respect to its well posedness. Well posedness guarantees the existence of a unique solution continuously depending on the problem data. When a model becomes ill posed, infinitesimal perturbations to a solution grow infinitely fast. Apart from the fact that this behaviour cannot represent a physical process, numerical simulations of an ill-posed model continue to change as the grid is refined. For this reason, ill-posed models cannot be used as predictive tools. One source of ill posedness is due to the simplified description of the processes related to vertical mixing of sediment. The current analysis reveals the existence of two additional mechanisms that lead to model ill posedness: secondary flow due to the flow curvature and the effect of gravity on the sediment transport direction. When parametrising secondary flow, accounting for diffusion in the transport of secondary flow intensity is a requirement for obtaining a well-posed model. When considering the theoretical amount of diffusion, the model predicts instability of perturbations that are incompatible with the shallow water assumption. The effect of gravity on the sediment transport direction is a necessary mechanism to yield a well-posed model, but not all closure relations to account for this mechanism are valid under mixed-size sediment conditions. Numerical simulations of idealised situations confirm the results of the stability analysis and highlight the consequences of ill posedness.

The application of the Kutta–Joukowski (KJ) theorem to estimating the lift of a flying animal based on wake velocity fields often leads to significant underprediction of the lift, which is known as the wake momentum paradox. This work attempts to answer the puzzling question on whether the KJ theorem is legitimate in its use for complex viscous unsteady wakes generated by flapping wings. The limitations in applying the KJ theorem to flapping wings are quantitatively examined through numerical simulations of viscous incompressible flows over three flapping wing models. The three flapping wing models studied in this work are a flapping wing with a fixed wingspan, a flapping wing with a dynamically changing wingspan and a dihedral flapping wing. The KJ theorem fails to give a satisfactory prediction of the time-averaged lift unless an effective span length is correctly computed. We propose a wake-sectional Kutta–Joukowski (WS-KJ) model to predict the time-averaged lift, where the effective span length is computed based on the time-averaged distance between the streamwise vorticity centroids in the right and left half sides of the Trefftz plane. The WS-KJ model incorporates the spatial evolutionary effects of the complex vortex structures in the wake and significantly improves the prediction of the time-averaged lift. The physical foundation for such improvement is explored. In addition, the time-dependent amplitude and phase changes of the unsteady lift are discussed as the fluid acceleration effect.

Very-large-scale motions (VLSMs) and large-scale motions (LSMs) coexist at moderate Reynolds numbers in a very long open channel flow. Direct numerical simulations two-way coupled with inertial particles are analysed using spectral information to investigate the modulation of VLSMs. In the wall-normal direction, particle distributions (mean/preferential concentration) exhibit two distinct behaviours in the inner flow and outer flow, corresponding to two highly anisotropic turbulent structures, LSMs and VLSMs. This results in particle inertia’s non-monotonic effects on the VLSMs: low inertia (based on the inner scale) and high inertia (based on the outer scale) both strengthen the VLSMs, whereas moderate and very high inertia have little influence. Through conditional tests, low- and high-inertia particles enhance VLSMs following two distinct routes. Low-inertia particles promote VLSMs indirectly through the enhancement of the regeneration cycle (the self-sustaining mechanism of LSMs) in the inner region, whereas high-inertia particles enhance the VLSM directly through contribution to the Reynolds shear stress at similar temporal scales in the outer region. This understanding also provides more general insight into inner–outer interaction in high-Reynolds-number, wall-bounded flows.

Thermocapillary convection has always been one of the most important research topics in microgravity fluid physics. A space experimental study on the thermocapillary convection in an open annular liquid pool – a typical thermocapillary flow system – has been conducted on the SJ-10 satellite of China. This space experiment has observed the spatial temperature distribution of the liquid free surface using an infrared thermal imager, obtained the flow pattern transition process, analysed the oscillation characteristics and revealed the instability mechanism of themocapillary convection. The shape effects on the flow instability are researched by changing the volume ratio, Vr, which denotes the ratio of the liquid volume to the volume of the cylindrical gap between the walls. The volume ratio effect has been focused on for the first time. For a certain volume ratio, the flow pattern would transform from the steady state to the oscillation state accompanied by directional propagating hydrothermal waves with increasing temperature difference. In addition, the significant influences of the volume ratio on the critical conditions and wavenumber selection have been analysed in detail.

A weakly compressible flow direct numerical simulation of a controlled-diffusion aerofoil at $8^{\circ }$ geometrical angle of attack, a chord-based Reynolds number of $Re_{c}=150\,000$ and a Mach number of $M=0.25$ based on the free-stream velocity relevant to many industrial applications was conducted to improve the understanding of the impact of the pressure gradient on the development of turbulent structures. The evolution equations for the two invariants $Q$ and $R$ of the velocity-gradient tensor have been studied at various locations along the aerofoil chord on its suction side. The shape of the mean evolution of the velocity-gradient tensor invariants were found to vary strongly when the flow encounters favourable, zero and adverse pressure gradients and as well for different wall-normal locations. The coupling between the pressure-Hessian tensor and the velocity-gradient tensor was found to be the major factor that causes these changes and is greatly influenced by the mean pressure-gradient condition and the wall-normal distance. Striking differences exist from the mean trajectories of this coupling at least in the log layer and outer layer subject to different mean pressure gradients. The nonlinearity and viscous diffusion effects keep their respective invariant characters regardless of the pressure-gradient effects and wall-normal locations. The wall and the mean adverse pressure gradient were both found to suppress the vortical stretching features of the flow. These features are of great importance for the development of future turbulence models on wall-bounded flows, especially on surfaces with significant curvature such as cambered aerofoils and blades for which significant mean pressure gradients exist.

Discrete unstable modes of hypersonic laminar boundary layers, obtained from an eigenvalue analysis, provide insight into key transition scenarios. The character of such modes near the leading edge is often identified with the corresponding asymptotic free-stream behaviour of acoustic, vortical or entropic (thermal) content, which we designate fluid-thermodynamic (FT) components. In downstream regions, however, this direct one-to-one correspondence between discrete modes and FT components does not hold, since FT components interact in well-defined ways with the basic state and with each other (even under linear scenarios). In the present work, we perform an FT decomposition of discrete modes using momentum potential theory, to yield a physics-based analysis that complements linear stability theory in the linear regime, and seamlessly extends to the nonlinear domain where direct numerical simulations are appropriate. Linear and nonlinear saturated disturbance effects, different forcing types and wall thermal conditions are considered, with emphasis on phenomena occurring near stability-mode synchronization locations. The results show that, in the linear regime, each discrete mode contains all FT components, whose relative amplitudes vary with streamwise distance. Vortical components are always the largest, followed by thermal and acoustic components. These latter two show distinct fore and aft signatures near mode synchronization. The vortical component displays a series of rope-shaped recirculation-cell patterns across the generalized inflection point. However, both acoustic and thermal components display ‘trapped’ structures. The former contains an alternating monopole array between the wall and the critical layer, while the latter is confined to an undulating region between the wall and a wavy locus straddling the generalized inflection point. Nonlinear saturation in the region of Mack-mode growth further strengthens the rope-shaped structures in the vortical component and higher harmonics appear, whose form and location depend on the specific component. Wall cooling modifies the eigenfunctions such that the acoustic component accounts for more of its composition, consistent with its destabilization. Analysis of energy interactions among the FT components indicates that, even though the vorticity component is the largest, the thermal component induces the most significant source term for the growth of acoustic perturbations, possibly due to the trapped nature of both.

We study the effects of externally imposed shear and buoyancy driven flows on the stability of a solid–liquid interface. A linear stability analysis of shear and buoyancy-driven flow of a melt over its solid phase shows that buoyancy is the only destabilizing factor and that the regime of shear flow here, by inhibiting vertical motions and hence the upward heat flux, stabilizes the system. It is also shown that all perturbations to the solid–liquid interface decay at a very modest shear flow strength. However, at much larger shear-flow strength, where flow instabilities coupled with buoyancy might enhance vertical motions, a re-entrant instability may arise.

Results of comprehensive particle image velocimetry measurements investigating the dynamics of turbulent jets in a rotating fluid are presented. It is observed that background system rotation induces a time-periodic formation–breakdown cycle of the jets. The flow dynamics associated with this process is studied in detail. It is found that the frequency of the cycle increases linearly with the background rotation rate. The data show that the onset of the breakdown phase and of the reformation phase of the cycle can be characterized in terms of a local Rossby number employing an internal velocity and a length scale of the jet. The critical values for this local Rossby number, for onset of breakdown and reformation, scale linearly with a global Rossby number based on the flow conditions at the source. The analysis of the experimental data suggests centrifugal instability as the potential origin of the formation–breakdown cycle.

Townsend (The Structure of Turbulent Shear Flow, 1976, Cambridge University Press) proposed a structural model for the logarithmic layer (log layer) of wall turbulence at high Reynolds numbers, where the dominant momentum-carrying motions are organised into a multiscale population of eddies attached to the wall. In the attached-eddy framework, the relevant length and velocity scales of the wall-attached eddies are the friction velocity and the distance to the wall. In the present work, we hypothesise that the momentum-carrying eddies are controlled by the mean momentum flux and mean shear with no explicit reference to the distance to the wall and propose new characteristic velocity, length and time scales consistent with this argument. Our hypothesis is supported by direct numerical simulation of turbulent channel flows driven by non-uniform body forces and modified mean velocity profiles, where the resulting outer-layer flow structures are substantially altered to accommodate the new mean momentum transfer. The proposed scaling is further corroborated by simulations where the no-slip wall is replaced by a Robin boundary condition for the three velocity components, allowing for substantial wall-normal transpiration at all length scales. We show that the outer-layer one-point statistics and spectra of this channel with transpiration agree quantitatively with those of its wall-bounded counterpart. The results reveal that the wall-parallel no-slip condition is not required to recover classic wall-bounded turbulence far from the wall and, more importantly, neither is the impermeability condition at the wall.

We investigate the stability of a thin Newtonian fluid spreading on a horizontal cylinder under the action of gravity. The capillary ridge forming at the advancing front is known to be unstable with respect to spanwise perturbations, resulting in the formation of fingers. In contrast to the classic case of a flow over an inclined plane, the gravity components along a cylindrical substrate vary in space and the draining flow is time-dependent, making a modal stability analysis inappropriate. A linear optimal transient growth analysis is instead performed to find the optimal spanwise wavenumber. We not only consider the optimal perturbations of the initial film thickness, as commonly done in the literature, but also the optimal topographical perturbations of the substrate, which are of significant practical relevance. We found that, in both cases, the optimal gains are obtained when the perturbation structures are the least affected by the time horizon. The optimal spanwise wavenumber is found to be dependent on the front location, due to the dependence of the characteristic length of the capillary ridge on its polar location.

We study and compare instability mechanisms of a Mach 5.65 laminar boundary layer tripped by an isolated diamond-shaped trip and by an array of diamond-shaped trips using direct numerical simulations. A low-Reynolds-number experiment, consisting of the trip array (Semper & Bowersox, AIAA J., vol. 55 (3), 2017, pp. 808–817), is used to validate our simulations. Three dynamically prominent flow structures are observed in both trip configurations. These flow structures are the upstream vortex system, the shock system, and the downstream shear layers/counter-rotating streamwise vortices that originate from the top and sides of the trips. Analysis of the power spectral density of pressure reveals the source of instability to be an interaction between the shear layers and the counter-rotating streamwise vortices downstream of both trip configurations. The interaction leads to the formation of hairpin-like structures that eventually break down to turbulent flow. This finding contrasts with that of an isolated cylindrical trip (Subbareddy et al., J. Fluid Mech., vol. 748, 2014, pp. 848–878) where the upstream vortex system is found to be the source of instability. Therefore, the shape of a trip plays an important role in the instability mechanism. Furthermore, dynamic mode decomposition (Rowley et al., J. Fluid Mech., vol. 641, 2009, pp. 115–127; Schmid, J. Fluid Mech., vol. 656, 2010, pp. 5–28) of three-dimensional snapshots of pressure fluctuations unveil globally dominant modes consistent with the power spectral density analysis in both diamond-shaped trip configurations.