The instability of a shear flow passing over a wavy wall with wave crests not perpendicular to the flow direction is investigated. The friction Reynolds number for the flow is large and the wave amplitude scaled on the wavelength is small compared with the viscous wall layer. The instability takes the form of a streamwise vortex of wavelength comparable to the viscous wall layer in which the basic flow adjusts to the presence of the wall. For a given wall amplitude, the instability considered is the first one to arise as the Reynolds number is increased and modes of wavelength comparable to the viscous layer grow much faster than modes of wavelength comparable to the wall wavelength. The instability is not driven by centrifugal or viscous effects but is a novel kind of cross-flow vortex instability associated with a spatially periodic flow; the existence of the instability is associated with the orientation of the wave crests. The instability is investigated for wavelengths comparable to the depth of the viscous wall layer; the limiting cases of large and small wavelengths are investigated asymptotically. At small wavenumbers and roughness heights the instability connects with disturbances of wavelength comparable to the wall wavelength. At high wavenumbers and roughness heights a new structure emerges and the disturbance moves away from the wall, that structure takes on a self-similar form with progressively faster variations in the streamwise and spanwise directions as the roughness height increases.

This paper experimentally investigates the flow field development and unsteady loading of three force-mitigating pitch manoeuvres during a transverse gust encounter. The manoeuvres are constructed using varying levels of theoretical and simulation fidelity and implemented as open-loop kinematics in a water towing tank. It is found that pitch actuation during a gust encounter results in two important changes in flow topology: (i) early detachment of the leading-edge vortex (LEV) and (ii) formation of an LEV on the pressure side of the wing upon gust exit. Each of the pitch manoeuvres is found to mitigate a significant portion of the circulatory contribution of the lift force while only manoeuvres with accurate modelling of the added-mass force are found to adequately mitigate the total lift force. The penalty of aerodynamic lift mitigation using pitch manoeuvres was a twofold increase in the pitching moment transients experienced by the wing for all cases. By quantifying changes in the vertical gust momentum before and after the encounter, lift-mitigating manoeuvres were found to reduce the disturbance to the gust's flow field, thereby reducing the momentum exchange between the gust and the wing.

An investigation of shock diffraction through a non-quiescent background medium is presented using both experimental and numerical techniques. Unlike diffracting shocks in quiescent media, a spatial distortion of the shock front occurs, producing a region of constant shock angle. An example of this process arises in the exhaust from a pulse-detonation combustor. As the background velocity is increased, such as through the inclusion of a converging nozzle at the exhaust, the spatial distortion becomes more apparent. Numerical simulations using a compressible Euler solver demonstrate that the distortion is not due to the geometrical influence of the nozzle, but rather is a function of the magnitude of the background flow velocity. The distortion is studied using a modified geometrical shock dynamics formulation which includes the background flow and is validated against experiments. A simple model is presented to predict the shock distortion angle in the weak-shock limit. Finally, the axial decay behaviour of the shock is investigated and it is shown that the advection of the shock by the background flow delays the arrival of the head and tail of the expansion characteristic at the centreline. This leads to an increase in the rate of decay of the shock Mach number as the background flow velocity is increased.

We formulate and solve a generalized inverse Navier–Stokes problem for the joint velocity field reconstruction and boundary segmentation of noisy flow velocity images. To regularize the problem, we use a Bayesian framework with Gaussian random fields. This allows us to estimate the uncertainties of the unknowns by approximating their posterior covariance with a quasi-Newton method. We first test the method for synthetic noisy images of two-dimensional (2-D) flows and observe that the method successfully reconstructs and segments the noisy synthetic images with a signal-to-noise ratio (SNR) of three. Then we conduct a magnetic resonance velocimetry (MRV) experiment to acquire images of an axisymmetric flow for low (${\simeq }6$) and high (${>}30$) SNRs. We show that the method is capable of reconstructing and segmenting the low SNR images, producing noiseless velocity fields and a smooth segmentation, with negligible errors compared with the high SNR images. This amounts to a reduction of the total scanning time by a factor of 27. At the same time, the method provides additional knowledge about the physics of the flow (e.g. pressure) and addresses the shortcomings of MRV (i.e. low spatial resolution and partial volume effects) that otherwise hinder the accurate estimation of wall shear stresses. Although the implementation of the method is restricted to 2-D steady planar and axisymmetric flows, the formulation applies immediately to three-dimensional (3-D) steady flows and naturally extends to 3-D periodic and unsteady flows.

The dynamics of two slender Hopf-linked vortex rings at vortex Reynolds numbers ($Re \equiv \varGamma /\nu, \mathrm {circulation/viscosity}$) $2000$, $3000$ and $4000$ is studied using direct numerical simulations of the incompressible Navier–Stokes equations. Under self-induction, the initially perpendicularly placed vortex rings approach each other and reconnect to form two separate vortex rings. The leading ring is closely cuddled and further undergoes secondary reconnection to form two even smaller rings. At high $Re$, the leading ring and the subsequent smaller rings are unstable and break up into turbulent clouds consisting of numerous even smaller-scale structures. Although the global helicity $H$ remains constant before reconnection, it increases and then rapidly decays during reconnection – both the growth and decay rates increase with $Re$. In the two higher $Re$ (i.e. 3000 and 4000) cases, $H$ further rises after the first reconnection and reaches a quasi-plateau with the asymptotic value continuously increasing with $Re$ – suggesting that $H$ for viscous flows is not conserved at very high $Re$. Further flow analysis demonstrates that significant numbers of positive and negative helical structures are simultaneously generated before and during reconnection, and their different decay rates is the main reason for the complex evolution of $H$. By examining the topological aspects of the helicity dynamics, we find that, different from $H$, the sum of link and writhe ($L_k+W_r$) continuously drop during reconnection. Our results also clearly demonstrate that the twist, which increases with $Re$, plays a significant role in the helicity dynamics, particularly at high $Re$.

A smooth body, adverse pressure gradient (APG), turbulent boundary layer (TBL) separation is experimentally studied. The geometry features canonical TBL development prior to encountering a smooth, two-dimensional convex ramp geometry of finite span onto which a streamwise APG that is fully adjustable is imposed. Both large- and small-scale separations are studied, and all data are archived on the NASA Turbulence Modeling Resource website. This paper describes the large-scale separation case with focus on the surface topography and topology of both separation and reattachment. Despite the spanwise uniform approach TBL and ramp geometry, the separation is highly three-dimensional but the reattachment is spanwise uniform. The surface flow topology is characterized by the ‘owl-face pattern of the fourth kind’ – found to be highly repeatable over multiple experiments. This ubiquitous topology has been reported for a variety of flows including inclined bodies of revolution. It is demonstrated that the APG and the secondary flow associated with the sidewall–ramp juncture is responsible for the formation of the surface separation patterns.

This work provides the derivation of a closed macroscopic model for immiscible two-phase, incompressible, Newtonian and isothermal creeping steady flow in a rigid and homogeneous porous medium without considering three-phase contact. The mass and momentum upscaled equations are obtained from the pore-scale Stokes equations, adopting a two-domain approach where the two fluid phases are separated by an interface. The average mass equations result from using the classical volume averaging method. A Green's formula and the adjoint Green's function velocity pair problems are used to obtain the pore-scale velocity solutions that are averaged to obtain the upscaled momentum balance equations. The macroscopic model is based on the assumptions of scale separation and the existence of a periodic representative elementary volume allowing a local description as usually postulated for upscaling. The macroscopic momentum equation in each phase includes the generalized Darcy-like dominant and viscous coupling terms and, importantly, an additional compensation term that accounts for surface tension effects to momentum transfer that is, otherwise, incompletely captured by the Darcy terms. This interfacial term, as well as the dominant and viscous coupling permeability tensors, can be predicted from the solutions of two associated closure problems that coincide with those reported in the literature. The relevance of the compensation term and the upscaled model validity are assessed by comparisons with direct numerical simulations in a model two-dimensional periodic structure. Upscaled model predictions are found to be in excellent agreement with direct numerical simulations.

The stability of an incompressible boundary layer flow over a wing in the presence of free stream turbulence (FST) has been investigated by means of direct numerical simulations and compared with the linearised boundary layer equations. Four different FST conditions have been considered, which are characterised by their turbulence intensity levels and length scales. In all cases the perturbed flow develops into elongated disturbances of high and low streamwise velocity inside the boundary layer, where their spacing has been found to be strongly dependent on the scales of the incoming free stream vorticity. The breakdown of these streaks into turbulent spots from local secondary instabilities is also observed, presenting the same development as the ones reported in flat plate experiments. The disturbance growth, characterised by its root mean squares value, is found to depend not only on the turbulence level, but also on the FST length scales. Particularly, higher disturbance growth is observed for our cases with larger length scales. This behaviour is attributed to the preferred wavenumbers that can exhibit maximum transient growth. We study this boundary layer preference by projection of the flow fields at the leading edge onto optimal disturbances. Our results demonstrate that optimal disturbance growth is the main cause of growth of disturbances on the wing boundary layer.

A fully nonlinear non-dispersive energy balance for surfzone waves is derived based on the nonlinear shallow water equations to study the nearshore dynamics of infragravity (IG) waves. Based on simulations of waves on a relatively moderate and mild beach slope with a non-hydrostatic wave-flow model (SWASH), the new theory shows that spatial gradients in IG energy flux are nearly completely balanced by the combined effect of bottom stresses and predominantly nonlinear triad interactions. The new balance confirms many features of existing weakly nonlinear theories, and yields an improved description in the inner surfzone where waves become highly nonlinear. A gain of IG energy flux throughout the shoaling and outer surfzones is driven by triad interactions between IG waves and pairs of sea-swell (SS) waves. The IG energy flux decreased in the inner surfzone, primarily through an energy cascade to the swell-band and superharmonic frequencies where wave energy is ultimately dissipated. Dissipation by bottom friction was weak on both slopes. The IG wave breaking, characterized by triads between three IG or two IG waves and one SS wave, was significant only deep inside the surfzone of the mild slope. Even though IG waves broke on the mild slope, nonlinear interactions between IG waves and pairs of SS waves were responsible for at least half of the net IG flux loss.

The mixing-layer analogy is due to Raupach, Finnigan & Brunet (Boundary-Layer Meteorol., vol. 25, 1996, pp. 351–382). In the analogy, the flow in the roughness sublayer of a homogeneous deep vegetation canopy boundary layer is analogous to a plane mixing layer rather than a surface layer. Evidence for the analogy includes the inflected velocity profile, which resembles the velocity profile in a plane mixing layer, and, most notably, the following estimate as a result of the Kelvin–Helmholtz instability: $\varLambda _x=8.1L_s$, where $\varLambda _x$ is the spacing of the large-scale eddies, and $L_s$ is the shear length. The mixing-layer analogy has been very successful in vegetation canopy flow research, but has received only limited support in urban-canopy flow research. This work revisits Raupach et al.'s mixing-layer analogy, and we present the evidence for the mixing-layer analogy in urban-canopy flows: the exponential velocity profile in the canopy layer, i.e. $(U-U_c)/(U_h-U_c)=\exp (z/L_m)$, and $L_m\sim [(U_h/U_c-1)(U_h/U_c+3)]^{-1}$. Here, $z$ is the vertical coordinate, $L_m$ is the attenuation length and is a measure of the largest eddy in the canopy layer, $U_h$ is the wind speed at the canopy crest and $U_c$ is the velocity in the inactive layer. We conduct direct numerical simulations of various deep homogeneous urban-canopy flows and test the above two scalings. We also discuss why Raupach et al.'s analogy has not seen as many successes in urban-canopy flows as in vegetation canopy flows.

Analysis of convection driven by spatial periodic heating shows a wealth of responses due to Prandtl number dependence. The primary convection is delineated by the heating wavenumber and the secondary convection is characterized by the critical wavenumber established by an instability process. The resulting two-wavenumber dynamical system involves spatial parametric resonance leading to the wavenumber lock-in and the Rayleigh–Bénard (RB) instability producing unlocked states. Heating conditions leading to the co-existence of both types of states have been identified. Transition from the locked to the unlocked states produces states with a wide range of wavelengths and diverse patterns of movement. The possible secondary states are driven by a competition between the RB mechanism and the spatial parametric resonance. The relative strengths of these mechanisms change with Prandtl number $Pr$ resulting in four types of system responses. In the type A response occurring for $Pr>0.4$, parametric resonance dominates for the heating wavenumber $\alpha < 4.5$ resulting in the pattern lock-in between the primary and secondary convections whereas RB occurs for $\alpha > 10$. There is a wealth of possible convection patterns in the in-between zone where a small change of $\alpha$ results in a major change of flow pattern. In the type B response, which occurs for $0.19< Pr< 0.4$, the RB effect dominates eliminating the lock-in. The type C response, which occurs for $0.08 < Pr< 0.19$, is similar to type A but stronger spatial modulation extends the range of dominance of parametric resonance up to $\alpha =7$ while the RB effect dominates for $\alpha >10$ as in the type A response. A wealth of possible patterns occurs in the transition zone ($7<\alpha <10$). In the type D response, which occurs for $Pr<0.08$, the strong spatial modulation results in the formation of two separate critical stability curves, one resulting from the dominance of the spatial parametric resonance leading to the lock-in effect, and the other one corresponding to the dominance of the RB effect producing unlocked states. No continuous transition between both states can occur. Conditions where both distinct states can arise simultaneously were identified. Morphing between different patterns of secondary convection may occur in response to small changes in the heating pattern in types A and C, while such processes are not possible in types B and D.

The objective of this work is to investigate the interaction of turbulence with the nonlinear processes of particle nucleation and growth that occur in reaction crystallisation, also known as precipitation. A validated methodology for coupling the population balance equation with direct numerical simulation of turbulent flows is employed for simulating an experiment conducted by Schwarzer et al. (Chem. Engng Sci., vol. 61, no. 1, 2006, pp. 167–181), where barium sulphate nanoparticles are formed by mixing and reaction of barium chloride and sulphate acid in a T-mixer, with the spatial resolution resolved down to the Kolmogorov scale. A unity Schmidt number is assumed, since at present it is not possible to resolve the Batchelor scale for realistic Schmidt numbers (order of 1000 or more). The probability density function, filtered averages and spatial distribution of time and length scales are all examined in order to shed light on the interplay of turbulence and precipitation. Separate Damköhler numbers are defined for nucleation and growth and both are found to be close to unity, indicating that the process is neither mixing nor kinetics controlled. The nucleation length scales are also evaluated and compared with the Kolmogorov scale to show the importance of resolving nucleation bursts. In addition, zones of different rate-determining mechanisms are identified. The ultimate aim of precipitation is to obtain control over the product particle size distribution, and the present study elucidates the synergistic or competing roles of mixing, nucleation and growth on the process outcome and discusses the implications for modelling.

A magnetorheological fluid, which consists of magnetic particles suspended in a viscous fluid, flows freely with well-dispersed particles in the absence of a magnetic field, but particle aggregation results in flow cessation when a field is applied. The mechanism of dynamical arrest is examined by analysing interactions between magnetic particles in a magnetic field subject to a shear flow. An isolated spherical magnetic particle undergoes a transition between a rotating state at low magnetic field and a static orientation at high magnetic field. The effect of interactions for spherical dipolar and polarisable particles with static orientation is examined for an unbounded dilute viscous suspension. There are magnetic interactions due to the magnetic field disturbance at one particle caused by the dipole moment of another, hydrodynamic interactions due to the antisymmetric force moment of a non-rotating particle in a shear flow, and a modification of the magnetic field due to the particle magnetic moment density. When there is a concentration variation, the torque balance condition results in a disturbance to the orientation of the particle magnetic moment. The net force and the drift velocity due to these disturbances is calculated, and the collective motion generated is equivalent to an anisotropic diffusion process. When the magnetic field is in the flow plane, the diffusion coefficients in the two directions perpendicular to the field direction are negative, implying that concentration fluctuations are unstable in these directions. This instability could initiate field-induced dynamical arrest in a magnetorheological fluid.

A surfactant-laden liquid film that lines the inside of an oscillating spherical cap is considered as a model of lung alveoli. Pulmonary surfactant solubility is described by Langmuir adsorption kinetics, modified by incorporating the intrinsic compressibility of the adsorbed monolayer. A novel boundary condition, supported by experimental data and scaling arguments, is applied at the rim. The condition enforces mass conservation of water and surfactant by matching the ‘large-scale’ dynamics of the alveolus to ‘small-scale’ equilibrium over mid-alveolar septa of small but finite thickness. Linear and weakly nonlinear analysis around the conditions in a non-oscillating cap indicates that the occurrence of shearing motion in the liquid is related to the non-zero film thickness over the rim, and shearing velocity at the interface is predicted an order-of-magnitude lower than the velocity of radial oscillation. Marangoni stresses dominate the interfacial dynamics, but capillary stresses affect significantly the interior flow field. In particular, they produce spatial modulations in flow rate, surface concentration of surfactant and wall shear stress, whose length scale varies with $Ca^{-1/3}$, i.e. is determined by a balance between capillary and viscous forces. Non-zero adsorption kinetics modifies at first order only the amplitude and phase of surface concentration, but affects all other variables at second order. In particular, it sets a steady drift of surfactant away from the alveolus and towards the rim. Finally, an attempt is made to relate the present predictions to physiological findings about air flow and particle deposition inside alveoli, and about shear stress-inflicted damage in diseased lungs.

We report the breakup dynamics of a magnetically active (ferrofluid) droplet in a T-shaped Lab on a Chip (LOC) device under the modulation of a non-uniform magnetic field. We adhere to high-speed imaging modalities for the experimental quantification of the droplet splitting phenomenon, while the underlying phenomenon is supported by the numerical results in a qualitative manner as well. On reaching the T-junction divergence, the droplet engulfs the intersection fully and eventually deforms into the dumbbell-shaped form, making its bulges move towards the branches of the junction. We observe that the asymmetric distribution of the magnetic force lines, acting over the T-junction divergence, induces an accelerating motion to the left of the moving bulge (since the magnet is placed adjacent to the left branch). We show that the non-uniform force field gradient allows the formation of a hump-like structure inside the left moving bulge, which triggers the onset of augmented convection in its flow field. We reveal that this augmented internal convection developed in the left moving volume/bulge, on becoming coupled to the various involved time scales of the flow field, leads to the asymmetric splitting of the droplet into two sister droplets. Our analysis establishes that, at the critical strength of the applied forcing, as realized by the critical magnetic Bond number, the flow time scale becomes minimum at the left branch of the channel, leading to the formation of larger sized sister droplets therein. Inferences of the present analysis, which demonstrates a plausible means of independently controlling the size of the sister droplet by manoeuvring the applied force field gradient, will provide a potential solution for rapid droplet splitting, which typically finds significant importance in point-of-care diagnostics.

We consider the use of sparsity-promoting norms in obtaining localised forcing structures from resolvent analysis. By formulating the optimal forcing problem as a Riemannian optimisation, we are able to maximise cost functionals whilst maintaining a unit-energy forcing. Taking the cost functional to be the energy norm of the driven response results in a traditional resolvent analysis and is solvable by a singular value decomposition (SVD). By modifying this cost functional with the $L_1$-norm, we target spatially localised structures that provide an efficient amplification in the energy of the response. We showcase this optimisation procedure on two flows: plane Poiseuille flow at Reynolds number $Re=4000$, and turbulent flow past a NACA 0012 aerofoil at $Re=23\,000$. In both cases, the optimisation yields sparse forcing modes that maintain important features of the structures arising from an SVD in order to provide a gain in energy. These results showcase the benefits of utilising a sparsity-promoting resolvent formulation to uncover sparse forcings, specifically with a view to using them as actuation locations for flow control.

Transport of solutes through channels with rough boundaries is abundant in natural and engineered settings. However, it is not known currently what the consequences of an abruptly alternating boundary are for the solute dispersion, in particular when advected by inertial fluid flow. To investigate this, we compute numerically the time-asymptotic longitudinal dispersion coefficient of a passive solute advected by fluid flow through a two-dimensional channel with square boundary roughness. We determine how the effective diffusion coefficient depends on the boundary amplitude, Péclet number and Reynolds number. For creeping flow, the effective diffusion coefficient is found to be enhanced significantly through the recirculation zones. Increasing fluid inertia reduces the effective diffusion coefficient by up to a factor of two for high Péclet numbers. We interpret this behaviour by analysing residence times computed from Lagrangian particle simulations.

We explore the interior stratification that develops in a reservoir of fluid in which there is a localised source of destabilising (positive) buoyancy and a distributed source of stabilising (negative) buoyancy at the surface of the fluid, combined with an interior turbulent diffusivity. The source of destabilising buoyancy forms a descending turbulent plume that entrains ambient fluid, and this, coupled with the interior turbulent diffusivity, leads to mixing of the interior fluid. When the ratio $Pe$ of the plume source volume flux to the diffusive flux across the whole reservoir is large, the reservoir is well mixed except for a narrow surface boundary layer, while for smaller $Pe$, the reservoir is more stratified, consistent with the results of Hughes et al. (J. Fluid Mech., vol. 581, 2007, pp. 251–276) and Hughes & Griffiths (Annu. Rev. Fluid Mech., vol. 40, issue 1, 2008, pp. 185–208). We explore the evolution of the system when the source buoyancy flux varies in time, exploring both sudden changes and oscillatory changes in the forcing. Provided that the time scale of the change in the forcing is comparable to or smaller than the response time of the reservoir, as the source buoyancy flux decreases the plume may become trapped in the upper part of the reservoir, leading to a much shallower mixed zone. However, in the case of a source whose strength oscillates in time, as the source buoyancy flux increases, again the deeper water becomes involved in the mixing and circulation. We discuss briefly how these results may provide insight into the impact of changes in the forcing on mixing in deep ocean waters.

Foams and bubbles formed in liquid mixtures have lifetimes longer by several orders of magnitude than in pure liquids of similar viscosities. We have shown recently that this effect results from slight differences in molecular concentrations between bulk and surfaces, which give rise to a thickness-dependent surface tension of liquid films. We present a quantitative description of the enhanced stability of liquid films in binary mixtures, based on experimental data and theoretical analysis. Experiments were performed with mixtures of different natures and compositions: foams of stationary heights were obtained by continuous injection of gas, on one hand, and single bubbles were swollen under the surface of a liquid bath, on the other hand. Remarkably, the lifetimes measured in both experiments exhibit the same variations with mixture composition, and follow a power law with the microscopic length that characterises the amplitude of the thickness-dependent surface tension. In addition, the lifetimes vary with the squared film thicknesses at the onset of bursting. We suggest that two stages occur between the birth of liquid films and their rupture. We show how the thickness-dependent surface tension allows an equilibrium shape to be reached at the end of a first stretching stage. We give an analytical description of this shape, which is fully consistent with experimental findings. We suggest a possible mechanism for final rupture of the film, and discuss it in light of existing theoretical predictions. Finally, we compare the properties of liquid mixtures and surfactant solutions, and in particular their surface rheology.

We perform numerical simulations of the interaction between a vortex ring and a free surface, and provide a new interpretation of the mechanism by which the vortex ring connects to the free surface. Large vorticity gradients at the free surface result in the diffusion of surface-tangential vorticity out of the fluid. This is accompanied by the diffusion of opposite-signed surface-normal vorticity away from the connection line, along the free surface, which results in the attachment of the vortex ring to the free surface. Compared to existing descriptions, this interpretation explains better how the solenoidal property that vortex lines do not end in the fluid is maintained. By including an interface vortex sheet at the free surface, the important property of vorticity conservation is maintained throughout the interaction. The upper part of the vortex ring simply diffuses out of the fluid, and into the interface vortex sheet, with the total circulation remaining constant.