The rim and bag dynamics in aerodynamic droplet breakup are investigated experimentally and theoretically. Three main modes contribute to the breakup sizes in aerodynamic droplet breakup: the rim node, the remaining rim and the bag breakup modes. However, existing models only consider one mode and are, therefore, unable to predict the size distribution. The present theoretical work seeks to model the dominant breakup mechanisms of each mode and to relate these mechanisms to the size distribution. It is shown that the nodes can be modelled using either the Rayleigh–Taylor or Rayleigh–Plateau instabilities with comparable results and that the variation in the node sizes results from the variation in the amount of mass in the rim that flows into the node prior to the rim breakup. The breakup of the rim is shown to be a result of a combination of the Rayleigh–Plateau instability and a newly proposed collision mechanism, wherein the impact of the corrugated receding rim of the bag with the main rim forces the main rim to break with the same wavelength as the receding rim. The resulting size distribution of the droplet breakup is estimated assuming that the relative weighting of the breakup mechanisms for each mode follows a two-parameter gamma distribution. The volume of each geometry is used to estimate the volume weighting of the modes, giving a reasonable prediction of the size distribution resulting from aerodynamic droplet breakup.

Helicity is a quadratic inviscid conservative quantity in three-dimensional turbulent flows and is crucial for turbulent system evolution. Helicity effects have mainly been highlighted over the past few decades to explore the intrinsic mechanism of turbulent flows, while the statistical characteristics of helicity itself are nearly absent in general anisotropic turbulent flows. In this paper, we investigate the helicity statistics in turbulent channel flows with streamwise rotation at moderate rotation numbers ($Ro_{\tau }=7.5,15$ and 30) and Reynolds numbers ($Re_{\tau }=180$ and 395), including their spatial and scale distributions, anisotropy and cross-scale transfer. The appearance of a mean secondary flow in the spanwise direction corresponds to a mean streamwise vorticity, which indicates the presence of a high-helicity distribution. Numerical results reveal a regular helicity profile along the wall-normal direction, and a new peak is found in the near-wall region around $y^+=6$ of the streamwise or spanwise helicity profiles. The inter-scale helicity transfer is analysed by the filtering method, and the numerical consequences reveal that the second channel of the helicity cascade we proposed previously is dominant in contrast to the first channel. The rotation effects are explored by comparing the numerical results obtained under different rotation numbers. With increasing rotation number, more helical structures in the near-wall regions are present, with peaks of helicity profiles and fluxes coming closer to the wall. With a higher Reynolds number, their amplitudes are larger and scale-space transfer is strengthened. These systematic numerical analyses uncover the helicity distributions and transfer in wall-bounded turbulent flows.

We present a numerical study of emulsions in homogeneous and isotropic turbulence (HIT) at $Re_\lambda =137$. The problem is addressed via direct numerical simulations, where the volume of fluid is used to represent the complex features of the liquid–liquid interface. We consider a mixture of two iso-density fluids, where fluid properties are varied with the goal of understanding their role in turbulence modulation, in particular the volume fraction ($0.03<\alpha <0.5$), viscosity ratio ($0.01<\mu _d/\mu _c<100$) and large-scale Weber number ($10.6< We_\mathcal {L}<106.5$). The analysis, performed by studying integral quantities and spectral scale-by-scale analysis, reveals that energy is transported consistently from large to small scales by the interface, and no inverse cascade is observed. Furthermore, the total surface is found to be directly proportional to the amount of energy transported, while viscosity and surface tension alter the dynamic that regulates energy transport. We also observe the $-10/3$ and $-3/2$ scaling on droplet size distributions, suggesting that the dimensional arguments that led to their derivation are verified in HIT conditions.

The effects of streamwise-elongated, spanwise-periodic surface heating on a supersonic boundary-layer instability are investigated under the assumption of high Reynolds number. Our focus is on the lower-branch viscous instability and so the spanwise spacing of the elements is chosen to be of $O({\textit {Re}}^{-3/8}L)$, the wavelength of the latter, where ${\textit {Re}}$ is the Reynolds number based on $L$, the distance from the leading edge to the centre of the elements. The streamwise length is assumed to be much longer in order to simplify the mathematical description. Starting with classical triple-deck theory, the equations governing the heating-induced streaky flow are derived by appropriate rescaling. When Chapman's viscosity law is adopted, a similarity solution is found. The stability of the streaky flow, which is of a bi-global nature, is shown to be governed by a novel triple-deck structure characterised by fully compressible dynamics in the lower deck. Through asymptotic analysis, the bi-global stability is reduced to a one-dimensional eigenvalue problem, which involves only the spanwise-dependent wall temperature and wall shear. The instability modes may be viewed as a continuation of oncoming first Mack modes, but might also be considered as a new kind since they exhibit two distinctive features: strong temperature perturbation near the wall and spontaneous radiation of an acoustic wave to the far field, neither of which is shared by first Mack modes. Numerical calculations, performed for two simple patterns of spanwise-periodic heating elements, demonstrate their stabilising/destabiling effects on modes with different frequencies and spanwise wavelengths.

Lagrangian averaging has been shown to be more effective than the Eulerian mean in separating waves from slow dynamics in two time scale flows. It also appears in many reduced models that capture the wave feedback on the slow flow. Its calculation, however, requires tracking particles in time, which imposes several difficulties in grid-based numerical simulations or estimation from fixed-point measurements. To circumvent these difficulties, we propose a grid-based iterative method to calculate the Lagrangian mean without tracking particles in time, which also reduces computation, memory footprint and communication between processors in parallelised numerical models. To assess the accuracy of this method several examples are examined and discussed. We also explore an application of this method in the context of shallow-water equations by quantifying the validity of wave-averaged geostrophic balance – a modified form of geostrophic balance accounting for the effect of strong waves on slow dynamics.

We explore the two-dimensional flow past a cylinder undergoing forced transversal oscillations with the aim of elucidating the evolution of the topology of the wake under variation of the forcing amplitude at a Reynolds number of 100. In particular, we study the change from two single vortices (2S mode) to a pair and a single vortex ($\text {P}+ \text {S}$ mode) being shed per period. Matharu et al. (J. Fluid Mech., vol. 918, 2021, p. A21) showed that a dynamical symmetry-breaking pitchfork bifurcation plays a key role in this transition. We show that in addition to this bifurcation, a number of topological bifurcations in the vorticity field occur, both on the symmetric 2S branch and on the asymmetric $\text {P}+ \text {S}$ branch. The topological bifurcations are cusp bifurcations where an extremum of vorticity is created or destroyed. To describe the effect of the topological bifurcations we introduce an extended symbolic classification of the wake modes to account for the spatial variations of the vortex patterns that occur in the transition process. We identify four amplitude values that define critical stages in the transition, and provide a complete qualitative picture of the transition from 2S to $\text {P}+ \text {S}$ mode. We confirm the robustness of the observed transition process by simulations at Reynolds number 80.

The interactions between the near-wall streaks and the large-scale motions (LSMs) of the outer region of wall-bounded turbulent flows are investigated. The co-supporting hypothesis of Toh & Itano (J. Fluid Mech., vol. 524, 2005, pp. 249–262) is checked in full-scale channels at low to moderate Reynolds numbers, from two points of view. To study the top-down influence of the outer structures on the spanwise motion of the near-wall streaks, a method inspired by particle-image velocimetry is used to track the spanwise position of the streaks. Their spanwise advection velocity is found to be affected by the hierarchy of large-scale circulations in the logarithmic layer, but their spanwise streak density is only weakly related to the LSMs. The evidence suggests that a top-down influence exists and drives the drift of the streaks in the spanwise direction, as suggested by Toh & Itano (J. Fluid Mech., vol. 524, 2005, pp. 249–262), but that the hypothesised streak accumulation rarely occurs. Numerical experiments at $Re_{\tau }\thickapprox 535$ are then performed to clarify the role of the near-wall streaks in the generation and preservation of the outer LSMs. The results show that the merger of the near-wall streaks is only weakly correlated with the generation of the LSMs, and that removing the near-wall roots of the LSMs does not affect the evolution of their outer region. It is concluded that the bottom-up influence from the near-wall streaks is not essential for the LSM generation and preservation, also weakening the evidence for the co-supporting hypothesis.

In two dimensions, the problem governing a homogeneous phoretic swimmer of circular cross-section is ill-posed because of the logarithmic divergence associated with a purely diffusive solute transport. We address here the well-posed problem that is devised by introducing a slight inhomogeneity in the interfacial chemical activity. With the radial symmetry being perturbed, phoretic motion is animated by diffusio-osmosis. Solute advection, associated with that motion, becomes comparable to diffusion at large distances. The singular problem associated with that scale disparity is analysed using matched asymptotic expansions for arbitrary values of the Damköhler number $\textit {Da}$ and the intrinsic Péclet number $\textit {Pe}$. Asymptotic matching provides an implicit equation for the particle velocity in terms of these two parameters. The velocity exhibits a non-trivial dependence upon the sign $M$ of the slip coefficient. For $M=-1$, we observe the appearance of several solutions beyond a $\textit {Da}$-dependent critical value of $\textit {Pe}$. We also address the respective limits of small and large $\textit {Da}$ for fixed $\textit {Pe}$ and arbitrary inhomogeneity, and illuminate their linkage to the limit of weak inhomogeneity.

The effects of different geometries of two-dimensional (2-D) roughness elements in a zero pressure gradient (ZPG) turbulent boundary layer (TBL) on turbulence statistics and drag coefficient are assessed using single hot-wire anemometry. Three kinds of 2-D roughness are used: (i) circular rods with two different heights, $k= 1.6$ and 2.4 mm, and five different streamwise spacing of $s_{x}= 6k$ to $24k$, (ii) three-dimensional (3-D) printed triangular ribs with heights of $k= 1.6$ mm and spacing of $s_{x}= 8k$ and (iii) computerized numerical control (CNC) machined sinewave surfaces with two different heights, $k= 1.6$ and 2.4 mm, and spacing of $s_{x}= 8k$. These roughnesses cover a wide range of ratios of the boundary layer thickness to the roughness height ($23 < \delta /k < 41$), where $\delta$ is the boundary layer thickness. All roughnesses cause a downward shift on the wall-unit normalised streamwise mean velocity profile when compared with the smooth wall profiles agreeing with the literature, with a maximum downward shift observed for $s_{x}= 8k$. In the fully rough regime, the drag coefficient becomes independent of the Reynolds number. Changing the roughness height while maintaining the same spacing ratio $s_{x}/k$ exhibits little influence on the drag coefficient in the fully rough regime. On the other hand, the effective slope $(ES)$ and the height skewness $(k_{sk})$ appear to be major surface roughness parameters that affect the drag coefficient. These parameters are used in a new expression for $k_{s}$, the equivalent sand grain roughness, developed for 2-D uniformly distributed roughness in the fully rough regime.

We examine the gravity-driven flow of thin films of viscous fluid of power-law rheology, lubricated by another power-law viscous fluid from below. Such flows are relevant to a range of geophysical and industrial settings, including the flow of ice sheets and fast-flowing ice streams lubricated from below by a layer of subglacial till. We model both layers using lubrication theory in two-dimensional and axisymmetric settings, in the limit in which vertical shear provides the dominant resistance to the flow. The flow is self-similar if the power-law exponents, describing the rheology of the two layers of fluid, are equal. We examine the similarity solutions in both geometries and describe the flow in terms of four distinct flow regimes ranging from thin films of viscous fluid coating a more viscous fluid from above, to thin layers of fluid lubricating a more viscous fluid from below. In contrast to the former scenario, a thin film of a low-viscosity fluid strikingly alters the dynamics of a more viscous fluid when it lubricates it from below: the overlying layer thins, the upper surface gradients lessen and most of the shear is confined to the lower layer. Such features amplify, and this flow regime becomes increasingly dominant when the viscous fluids are shear thinning, like the deformation of glacial ice on the large scale. This flow regime is most relevant to the flow of lubricated ice sheets, which thin and accelerate, forming fast-flowing ice streams, when they are well lubricated from below.

We examine the stability of radially spreading, gravity-driven thin films of power-law fluids, lubricated from below by another power-law viscous fluid. Such flows are susceptible to a viscous fingering instability, also known as a non-porous viscous fingering instability, when a less viscous fluid intrudes beneath a more viscous fluid. In contrast to the Saffman–Taylor instability, such instabilities originate from a jump in hydrostatic pressure gradient across the intrusion front, associated with gradients in the upper surface. These are stabilised by buoyancy forces associated with the lower layer near its nose, and all instabilities are suppressed above a critical density difference. We find that shear-thinning flows are more prone to instability than Newtonian and shear-thickening flows. Lower consistency ratios are sufficient for the onset of instability of shear-thinning flows, and the stabilising influences of buoyancy forces are suppressed. As such, higher density differences are required to suppress the instability completely.

The vortex dynamics of leading-edge vortices on plunging high-aspect-ratio (AR = 10) wings and airfoils were investigated by means of volumetric velocity measurements, numerical simulations and stability analysis to understand the deformation of the leading-edge vortex filament and spanwise instabilities. The vortex filaments on both the wing and airfoil exhibit spanwise waves, but with different origins. The presence of a wing-tip causes the leg of the vortex to remain attached to the wing upper surface, while the initial deformation of the filament near the wing tip resembles a helical vortex. The essential features can be modelled as the deformation of an initially L-shaped semi-infinite vortex column. In contrast, the instability of the vortices is well captured by the instability of counter-rotating vortex pairs, which are formed either by the trailing-edge vortices or the secondary vortices rolled-up from the wing surface. The wavelengths observed in the experiments and simulations are in agreement with the stability analysis of counter-rotating vortex pairs of unequal strength.

The stabilization of oblique detonation waves (ODWs) in an engine combustor is important for the successful applications of oblique detonation engines, and comprehensively understanding the effects of the inviscid reflection of ODWs on their stabilization and the relevant mechanisms is imperative to overall combustor design. In this study, the flow fields of ODW reflections in a space-confined combustor are numerically studied by solving the two-dimensional time-dependent multispecies Euler equations in combination with a detailed hydrogen combustion mechanism. The inviscid Mach reflections of ODWs before an expansion corner are emphasized with different flight Mach numbers, Ma, and different dimensionless reflection locations, ζ ≥ 0 (ζ = 0: the ODW reflects precisely at the expansion corner; ζ > 0: the ODW reflects off the wall before the expansion corner). Two kinds of destabilization phenomena of the inviscid Mach reflection of an ODW induced by different mechanisms are found, namely wave-induced destabilization at large ζ > 0 for moderate (not very low) Ma and inherent destabilization at any ζ > 0 for low Ma. Wave-induced destabilization is attributed to the incompatibility between the pressure ratio across the Mach stem and its relative propagation speed, which is triggered by the action of the secondary reflected shock wave or the transmitted Mach stem on the subsonic zone behind the Mach stem. Inherent destabilization is demonstrated through an in-depth theoretical analysis and is attributed to geometric choking of the flow behind the Mach stem.

We report and analyse the results of extensive discrete element method simulations of three-dimensional gravity driven flows of cohesionless granular media over an erodible bed, the whole being confined between two flat and frictional sidewalls. We focus on the role of sidewalls by performing simulations for different gap widths ($W$) between the two confining sidewalls: from $5$ to $30$ grain sizes ($d$). Our results indicate the existence of two distinct regimes: regime I for flow angles smaller than the critical angle $\theta_c\approx 40^\circ$ and regime II at flow angles larger than $\theta_c$. Regime I corresponds to dense flows whereas flows belonging to regime II exhibit a strong variation of the volume fraction through the depth. Three relevant lengths are identified in the system: $W$ the gap between sidewalls, $l$ the length characterizing the vertical variation of the volume fraction and $h$ a characteristic length associated with the vertical variation of the streamwise velocity. Using these lengths we can rescale the profiles of various flow properties (e.g. streamwise velocity, granular temperature, particle rotation…). In regime II, in contrast to regime I, $l$ and $h$ have a similar behaviour. As a consequence, the rescaled profiles in regime II only involve $h$ (or equivalently $l$) and $W$. Other dissimilarities exist between regimes I and II. In particular, the scaling of the flow rate with $h$ (at fixed $W$) differs in the two regimes, although they display a similar scaling with $W$ (at fixed flow angle).

Recently, the nature of viscoelastic drag-reducing turbulence (DRT), especially the maximum drag reduction (MDR) state, has become a focus of controversy. It has long been regarded as polymer-modulated inertial turbulence (IT), but is challenged by the newly proposed concept of elasto-inertial turbulence (EIT). This study is to repicture DRT in parallel plane channels by introducing dynamics of EIT through statistical, structural and budget analysis for a series of flow regimes from the onset of drag reduction to EIT. Some underlying mechanistic links between DRT and EIT are revealed. Energy conversion between velocity fluctuations and polymers as well as pressure redistribution effects are of particular concern, based on which a new energy self-sustaining process (SSP) of DRT is repictured. The numerical results indicate that at low Reynolds number ($Re$), weak IT flow is replaced by a laminar regime before the barrier of EIT dynamics is established with the increase of elasticity, whereas, at moderate $Re$, EIT-related SSP can get involved and survive from being relaminarized. This further explains the reason why relaminarization phenomenon is observed for low $Re$ while the flow directly enters MDR and EIT at moderate $Re$. Moreover, with the proposed energy picture, the newly discovered phenomenon that streamwise velocity fluctuations lag behind those in the wall-normal direction can be well explained. The repictured SSP certainly justifies the conjecture that IT nature is gradually replaced by that of EIT in DRT with the increase of elasticity.

Recently, the phenomena of streaming suppression and relocation of inhomogeneous miscible fluids under acoustic fields were explained using the hypothesis on mean Eulerian pressure. In this work, we derive the expression for the acoustic body force without relying on any prior assumptions regarding the second-order Eulerian pressure. We present a theory of nonlinear acoustics for inhomogeneous fluids from first principles, which explains streaming suppression and acoustic relocation in both miscible and immiscible inhomogeneous fluids inside a microchannel. This theory predicts the relocation of higher impedance fluids to pressure nodes of the standing wave, which agrees with recent experiments.

The effects of wall-normal vibration on bypass transition in a boundary-layer flow over a flat plate with roughness elements in the form of circular cylinders are investigated using direct numerical simulations (DNS), linear Floquet analyses and dynamic mode decompositions (DMD). The vibration of the plate strengthens the streamwise vorticity, consequently enhancing the velocity streaks and reducing the critical Reynolds number for transition. A map is constructed to identify the coupling effect of the vibration amplitude and Reynolds number on transition. Among all investigated combinations of height and diameter of roughness, the critical Reynolds numbers at different vibration amplitude $A$ can be unified by a scaling function of $(1-10A)$. Two instability modes are identified in the vibration-induced transition process: a wake mode immediately downstream of the roughness, related to the inviscid Kelvin–Helmholtz instability of the wall-normal shear in the wake; and a streak mode, in response to the spanwise shear of the streaky flow, occurring further downstream. The first one results in the generation of central hairpin vortices which then feed the second one. Further development of the streak instability leads to two arrays of hairpin vortices. Results from linear Floquet analyses and DMD further confirm the two modes of instability observed in DNS. A quantitative study suggests that the amplification of vibration-induced disturbance by the base shear dominates the production of streamwise vorticity and subsequently the hairpin vortices.

The slender body theory is extended to accommodate cases where a wing, in generally non-uniform motion, sideslips at sufficiently large angle to make the flow separate from one of its long edges but not from any part of the other. In those cases, an approximately triangular wake forms to the side of the wing, rendering the pressure loads on the wing not only asymmetrical, but also dependent on the history of the wing's motion. The theory can have a wide range of possible applications, from estimating lift and drag of a slender wing in pronouncedly asymmetric flight, through analysing an aero-elastic stability of that wing, to estimating thrust, lift and power of a sea snake that swims at an angle to its body axis. Coherence of the theory is demonstrated through numerous numerical simulations.

Extensions of Rayleigh–Bloch waves above the cutoff frequency are studied via the discrete spectrum of a transfer operator for a channel containing a single cylinder with quasi-periodic side-wall conditions. Above the cutoff, the Rayleigh–Bloch wavenumber becomes complex valued and an additional wavenumber appears. For small- to intermediate-radius values, the extended Rayleigh–Bloch waves are shown to connect the Neumann and Dirichlet trapped modes before embedding in the continuous spectrum. A homotopy method involving an artificial damping term is proposed to identify the discrete spectrum close to the embedding. Moreover, Rayleigh–Bloch waves vanish beyond some frequency but reappear at higher frequencies for small and large cylinders. The existence and properties of the Rayleigh–Bloch waves are connected with finite-array resonances.

Turbulence enhances the wall shear stress in boundary layers, significantly increasing the drag on streamlined bodies. Other flow features such as free stream pressure gradients and streamwise boundary layer growth also strongly influence the local skin friction. In this paper, an angular momentum integral (AMI) equation is introduced to quantify these effects by representing them as torques that alter the shape of the mean velocity profile. This approach uniquely isolates the skin friction of a Blasius boundary layer in a single term that depends only on the Reynolds number most relevant to the flow's engineering context, so that other torques are interpreted as augmentations relative to the laminar case having the same Reynolds number. The AMI equation for external flows shares this key property with the so-called FIK relation for internal flows (Fukagata et al., Phys. Fluids, vol. 14, 2002, pp. L73–L76). Without a geometrically imposed boundary layer thickness, the length scale in the Reynolds number for the AMI equation may be chosen freely. After a brief demonstration using Falkner–Skan boundary layers, the AMI equation is applied as a diagnostic tool on four transitional and turbulent boundary layer direct numerical simulation datasets. Regions of negative wall-normal velocity are shown to play a key role in limiting the peak skin friction during the late stages of transition, and the relative strengths of terms in the AMI equation become independent of the transition mechanism a very short distance into the fully turbulent regime. The AMI equation establishes an intuitive, extensible framework for interpreting the impact of turbulence and flow control strategies on boundary layer skin friction.