Whereas streamwise effective slope ($ES_{x}$) is accepted as a key topographical parameter in the context of rough-wall turbulent flows, the significance of its spanwise counterpart ($ES_{y}$) remains largely unexplored. Here, the response of turbulent channel flow over irregular, three-dimensional rough walls with systematically varied values of $ES_{y}$ is studied using direct numerical simulation. All simulations were performed at a fixed friction Reynolds number 395, corresponding to a viscous-scaled roughness height $k^{+}\approx 65.8$ (where $k$ is the mean peak-to-valley height). A surface generation algorithm is used to synthesise a set of ten irregular surfaces with specified $ES_{y}$ for three different values of $ES_{x}$. All surfaces share a common mean peak-to-valley height and are near-Gaussian, which allows this study to focus on the impact of varying $ES_{y}$, since roughness amplitude, skewness and $ES_{x}$ can be eliminated simultaneously as parameters. Based on an analysis of first- and second-order velocity statistics, as well as turbulence co-spectra and the fractional contribution of pressure and viscous drag, the study shows that $ES_{y}$ can strongly affect the roughness drag penalty – particularly for low-$ES_{x}$ surfaces. A secondary observation is that particular low-$ES_{y}$ surfaces in this study can lead to diminished levels of outer-layer similarity in both mean flow and turbulence statistics, which is attributed to insufficient scale separation between the outer length scale and the in-plane spanwise roughness wavelength.

Viscoelastic flow instability, which is compelled by elastic effects rather than inertia, can be driven to a chaotic state termed elastic turbulence (ET) manifested as strong velocity fluctuations with an algebraic decay in the frequency spectrum and increased mixing. We report the first spatiotemporally complete description of ET by considering a broad volume within a novel three-dimensional ordered porous medium, reconstructing flow at a micrometre characteristic length scale ($\text {Reynolds numbers} \ll 1$) via time-resolved microtomographic particle image velocimetry. Beyond a critical Weissenberg number of 2, we observe an elastic flow instability accompanied by an enhanced pressure drop with spectral characteristics typical of ET. Polymer chains in the ET flow state are advected along increasingly curved streamlines between pores such that they accumulate strain and generate a local flow instability evaluated per an established instability criterion based on local evaluation of elastic tensile stress and streamline curvature. The onset of ET leads to increased pore-scale resistance and positive feedback on upstream streamline curvature. ET is thus characterized by a continuous evolution between states of laminar and unstable flow: pores with unstable flow flood their adjacent peers and thus encourage straightened streamlines and flow stability across the array, while positive feedback from flow resistance on streamline curvature results in the instability propagating upstream along the array. By employing a geometrically ordered medium, we permit flow state communication between pores, yielding generalized insights highlighting the significance of spatial correlation and flow history, and thus provide new avenues for explaining the mechanisms of ET.

Using numerical simulations, we investigate the effects of Marangoni stresses induced by surfactants on the effective slip length of liquid-infused surfaces (LIS) with transverse grooves. The surfactants are assumed soluble in the external liquid shearing the surface and can adsorb onto the interfaces. Two different adsorption models are used: a classical Frumkin model and a more advanced model that better describes the decrease of surface tension for minuscule concentrations. The simulations show that LIS may face even more severe effects of surfactants than previously investigated superhydrophobic surfaces. Constructing an analytical model for the effective slip length, we can predict the critical surfactant concentration for which the slip length decreases significantly. This analytical model describes both adsorption models of LIS on a unified framework if properly adjusted. We also advance the understanding of when surfactant advection gives rise to highly skewed interfacial concentrations: the so-called partial stagnant cap regime. To a good approximation, this regime can only exist below a specific surfactant concentration given by the Marangoni number and the strength of the surfactants.

The local eddy diffusivity model for the turbulent scalar flux is widely used to understand and predict scalar transport in turbulence. However, local approximation in space and time is not always valid for actual turbulent flow. An exact non-local expression for the scalar flux was previously derived by the author using Green's function. Even though the profile of the non-local eddy diffusivity in a turbulent channel flow was evaluated, its model expression was only discussed phenomenologically. In this study, the non-local eddy diffusivity approach was validated with help of direct numerical simulation (DNS) data of homogeneous isotropic turbulence with an inhomogeneous mean scalar to propose a systematic model expression. Initially, it was verified that the non-local expression for the scalar flux agrees with the value directly obtained from the DNS. In addition, the non-local effect accounts for the overestimation of the scalar flux by the local model and the counter-gradient diffusion phenomenon. The temporal behaviour of the non-local eddy diffusivity was further evaluated using the DNS data. Its model expression being proportional to the two-point velocity correlation was proposed in a manner customary in the statistical theory of turbulence. The profile of the non-local eddy diffusivity obtained from the model expression agrees well with the DNS value, and the non-local model reproduced the scalar flux. These results substantiate the potential of the non-local eddy diffusivity model.

The shape of a sessile drop on a horizontal substrate depends upon the Bond number $Bo$ and the contact angle $\alpha$. Inspired by puddle approximations at large $Bo$ (Quéré, Rep. Prog. Phys., vol. 68, 2005, p. 2495), we address here the limit of small contact angles at fixed drop volume and arbitrary $Bo$. It readily leads to a pancake shape approximation, where the drop height and radius scale as $\alpha$ and $\alpha ^{-1/2}$, respectively, with capillary forces being appreciable only near the edge. The pancake approximation breaks down for $Bo=\textrm {ord}(\alpha ^{2/3})$. In that distinguished limit, capillary and gravitational forces are comparable throughout, and the drop height and radius scale as $\alpha ^{2/3}$ and $\alpha ^{-1/3}$, respectively. For $Bo\ll \alpha ^{2/3}$ these scalings remain, with the drop shape turning into a spherical cap. The asymptotic results are compared with a numerical solution of the exact problem.

The wake of a wall-mounted finite prism is numerically studied and characterized with an aspect ratio (height-to-width) of $1$ and varying depth ratios (length to width) of between $0.016$ and $4$ at Reynolds numbers of 50–500. The prism is immersed in a laminar boundary layer. The minimum depth ratio considered here accounts for the special case of a wall-mounted very thin prism (similar to a flat plate), which is used to establish the mechanism and evolution of the wake associated with free-end effects and the shear-layer dynamics in small aspect-ratio prisms. The onset of an unsteady wake behind a very thin prism at a Reynolds number of $200$ is characterized by symmetric shedding of hairpin-like vortices. A unique asymmetric wake pattern appears at lower depth ratios starting at a Reynolds number of 250, which transitions to an symmetric wake with increasing depth ratio. The threshold depth ratio for this symmetric transition increases with Reynolds number. The asymmetric wake results from alternate shear-layer peel-off from either side of the prism, which itself is attributed to the out-of-phase shedding of tip vortices at a lower Strouhal number ($St_{sh}/2$) that interact with the detaching side shear layers. Alternate shedding of tip vortices form secondary vortex structures that are fed by the excess vorticity resulting from shear-layer detachment from either side of the prism. Increasing the depth ratio leads to simultaneous shedding of the tip vortices, which restores the commonly observed wake symmetric patterns. Thus, we identify and characterize the formation and interaction mechanisms of symmetric and asymmetric wakes during the transition process with increasing Reynolds number for different depth-ratio prisms.

The effect of a zero-mean-flow turbulent ambient on the geometry of the turbulent/turbulent interface (TTI) of an axisymmetric jet is investigated and compared with the traditional turbulent/non-turbulent interface (TNTI). The ambient turbulence is generated using a random jet array. Orthogonal cross-sections of the jet subjected to different levels of ambient turbulence intensities and length scales are captured using planar laser-induced fluorescence. The enhanced radial transport of concentration from the jet core towards the interface and the increased scalar fluctuations within the jet caused by the ambient turbulence result in steeper mean and root-mean-square scalar conditional jumps across the TTI layer compared with those of the TNTI, respectively. When compared with the quiescent ambient, the mean effect of background turbulence is to stretch and corrugate the interface surface area as evident from the wider probability density of the interface radial position, the lowered occurrence of zero-curvature surface elements, the greater misalignment between the radial and normal unit vectors of the interface, the increased tortuosity and the increased magnitude of the fractal exponent.

We investigate the modulation of turbulence caused by the presence of finite-size dispersed particles. Bluff (isotropic) spheres versus slender (anisotropic) fibres are considered to understand the influence of the shape of the objects on altering the carrier flow. While at a fixed mass fraction – but different Stokes number – both objects provide a similar bulk effect characterized by a large-scale energy depletion, a scale-by-scale analysis of the energy transfer reveals that the alteration of the whole spectrum is intrinsically different. For bluff objects, the classical energy cascade shrinks in its extension but is unaltered in the energy content and its typical features, while for slender ones we find an alternative energy flux which is essentially mediated by the fluid–solid coupling.

Experiments have shown that flow in compliant microchannels can become unstable at a much lower Reynolds number than the corresponding flow in a rigid conduit. Therefore, it has been suggested that the wall's elastic compliance can be exploited towards new modalities of microscale mixing. While previous studies mainly focused on the local instability induced by the fluid–structure interactions (FSIs) in the system, we derive a one-dimensional (1-D) model to study the FSI's effect on the global instability. The proposed 1-D FSI model is tailored to long, shallow rectangular microchannels with a deformable top wall, similar to the experiments. Going beyond the usual lubrication flows analysed in these geometries, we include finite fluid inertia and couple the reduced flow equations to a novel reduced 1-D wall deformation equation. Although a quantitative comparison with previous experiments is difficult, the behaviours of the proposed model show, qualitatively, agreement with the experimental observations, and capture several key effects. Specifically, we find the critical conditions under which the inflated base state of the 1-D FSI model is linearly unstable to infinitesimal perturbations. The critical Reynolds numbers predicted are in agreement with experimental observations. The unstable modes are highly oscillatory, with frequencies close to the natural frequency of the wall, suggesting that the observed instabilities are resonance phenomena. Furthermore, during the start-up from an undeformed initial state, self-sustained oscillations can be triggered by FSI. Our modelling framework can be applied to other microfluidic systems with similar geometric scale separation under different operating conditions.

Turbulent mixed convection in channel flows with heterogeneous surfaces is studied using direct numerical simulations. The relative importance of buoyancy and shear effects, characterised by the bulk Richardson number $Ri_b$, is varied in order to cover the flow regimes of forced, mixed and natural convection, which are associated with different large-scale flow organisations. The heterogeneous surface consists of streamwise-aligned ridges, which are known to induce secondary motion in the case of forced convection. The large-scale streamwise rolls emerging under smooth-wall mixed convection conditions are significantly affected by the heterogeneous surfaces and their appearance is considerably reduced for dense ridge spacings. It is found that the formation of these rolls requires larger buoyancy forces than over smooth walls due to the additional drag induced by the ridges. Therefore, the transition from forced convection structures to rolls is delayed towards larger $Ri_b$ for spanwise heterogeneous surfaces. The influence of the heterogeneous surface on the flow organisation of mixed convection is particularly pronounced in the roll-to-cell transition range, where ridges favour the transition to convective cells at significantly lower $Ri_b$. In addition, the convective cells are observed to align perpendicular to the ridges with decreasing ridge spacing. We attribute this reorganisation to the fact that flow parallel to the ridges experience less drag than flow across the ridges, which is energetically more beneficial. Furthermore, we find that streamwise rolls exhibit a very slow dynamics for $Ri_b=1$ and $Ri_b=3.2$ when the ridge spacing is of the order of the rolls’ width. For these cases the up- and downdrafts of the rolls move slowly across the entire channel instead of being fixed in space, as observed for the smooth-wall cases.

This paper describes, from a theoretical point of view, the appearance and characteristics of granular roll waves in chute flow, and the maximal size these waves can attain for a given influx of material into the system. Granular roll waves are steady travelling wave solutions of the generalized Saint-Venant equations for flowing granular matter, appearing when the Froude number $Fr$ of the incoming flow exceeds a critical value, $Fr>Fr_{cr}$. We focus upon the phase space of the corresponding dynamical system, where the roll waves take the form of a stable limit cycle around an unstable fixed point; this limit cycle gives precise information on the size and periodicity of the roll wave. It is found that, for any given value of $Fr$, the limit cycle cannot become arbitrarily large because it is constrained by a homoclinic loop in phase space. Roll waves of larger amplitude can be generated by increasing the Froude number $Fr$.

Using numerical simulations, we investigate the near-field temporal variability of axisymmetric helium plumes as a function of inlet-based Richardson (${Ri}_0$) and Reynolds (${Re}_0$) numbers. Previous studies have shown that ${Ri}_0$ plays a leading-order role in determining the frequency at which large-scale vortices are produced (commonly called the ‘puffing’ frequency). By contrast, ${Re}_0$ dictates the strength of localized gradients, which are important during the transition from laminar to turbulent flow. The simulations presented here span a range of ${Ri}_0$ and ${Re}_0$, and use adaptive mesh refinement to achieve high spatial resolutions. We find that as ${Re}_0$ increases for a given ${Ri}_0$, the puffing motion undergoes a transition at a critical ${Re}_0$, marking the onset of chaotic dynamics. Moreover, the critical ${Re}_0$ decreases as ${Ri}_0$ increases. When the puffing instability is non-chaotic, time series of different variables are well-correlated, exhibiting only modest changes in the dynamics (including period doubling and flapping). Once the flow becomes chaotic, denser ambient fluid penetrates the core of the plume, similar to penetrating ‘spikes’ formed by Rayleigh–Taylor instabilities, leading to only moderately correlated flow variables. These changes result in a non-trivial dependence of the puffing frequency on ${Re}_0$. Specifically, at sufficiently low ${Re}_0$, the puffing frequency falls below the prediction from Wimer et al. (J. Fluid Mech., vol. 895, 2020). As ${Re}_0$ increases beyond the critical ${Re}_0$, the puffing frequency increases and then drops back down to the predicted scaling. The dependence of the puffing frequency on ${Re}_0$ provides a possible explanation for previously observed changes in the scaling of the puffing frequency for high ${Ri}_0$.

Hydroelastic experiments have been carried out in a free surface water channel to study the influence of the cross-section shape on the mechanical energy transfer from a uniform flow to a self-excited crossflow oscillating prism. Seventeen cross-section shapes were tested with the same mechanical properties (say, mass ratio and mechanical damping) and aspect ratio. Then, the study was focused on the role of the cross-section shape and reduced velocity on the prism dynamics and energy transfer from the fluid flow. It has been found that the geometric centroid of the semi-cross-section and the afterbody length of the cross-section represent a simple and effective metric to characterise the energy transfer. Two related dimensionless geometric parameters are introduced. The dependence of the energy transfer with these two parameters is studied and used to design successfully ‘in advance’ cross-section shapes with high energy transfer efficiencies. It has been observed from particle image velocimetry measurements that large energy transfer shapes are those in which, under strong oscillations, the flow follows the cross-section contour of one side and is fully detached on the other side.

We study the collisions in a gaseous medium of a dilute bidisperse suspension of non-Brownian spherical particles sedimenting along the flow axis of a simple shear flow. Continuum lubrication forces prevent particles from coming into contact in a finite time, thus collisions can occur only due to attractive interactions such as the van der Waals force. However, in a low-pressure medium, the lubrication forces are weaker than their continuum counterparts and allow particle pairs to collide, even without any attractive forces. The Knudsen number, defined as the ratio of the mean free path of the medium to the mean radius of the interacting spheres, captures the significance of non-continuum interactions. We use uniformly valid hydrodynamic mobility functions, reflecting non-continuum lubrication at small separations and full continuum hydrodynamic interactions at moderate to larger separations. Due to the nature of the pair trajectory topology, the collision efficiency vanishes at a critical Knudsen number when simple shear flow alone drives the dynamics. Thus we perform collision calculations where particles experience the combined effects of van der Waals attraction and non-continuum hydrodynamics; van der Waals interactions enable collisions below the critical Knudsen number. Next, we calculate the collision efficiency for coupled differential sedimentation and simple shear driven motion in the presence of van der Waals interaction and non-continuum hydrodynamics. Finally, we explore the role of small particle inertia on relative trajectories and collision efficiencies in a non-continuum gas subject to a simple shear flow, ignoring the van der Waals force and gravity.

A straight cylindrical duct is considered containing an axial mean flow that is uniform everywhere except within a boundary layer near the wall, which need not be thin. Within this boundary layer the mean flow varies parabolically. The linearized Euler equations are Fourier transformed to give the Pridmore-Brown equation, for which the Green's function is constructed using Frobenius series. The critical layer gives a non-modal contribution from the continuous spectrum branch cut, and dominates the downstream pressure perturbation in certain cases, particularly for thicker boundary layers. The continuous spectrum branch cut is also found to stabilize what are otherwise convectively unstable modes by hiding them behind the branch cut. Overall, the contribution from the critical layer is found to give a neutrally stable non-modal wave when the source is located within the sheared flow region, and to decay algebraically along the duct as $O(x^{-{5}/{2}})$ for a source located with the uniform flow region. The Frobenius expansion, in addition to being numerically accurate close to the critical layer where other numerical methods lose accuracy, is also able to locate modal poles hidden behind the branch cut, which other methods are unable to find; this includes the stabilized hydrodynamic instability. Matlab code is provided to compute the Green's function.

We consider steady, fully-developed flows of deformable, inelastic grains driven by gravity between identical bumpy walls. Using constitutive relations from extended kinetic theory (EKT) for the erodible bed near the centreline and the collisional flow between the surfaces of the bed and the walls, we calculate the fields of mean velocity, fluctuation velocity and solid volume fraction across the chute. We consider both situations in which the solid volume fraction at and near the centreline is high enough to form a bed and when it is not. We compare results predicted by EKT with recent discrete element simulations results, and obtain very good agreement.

Hydraulic fractures propagating at depth are subjected to buoyant forces caused by the density contrast between fluid and solid. This paper is concerned with the analysis of the transition from an initially radial fracture towards an elongated buoyant growth – a critical topic for understanding the extent of vertical hydraulic fractures in the upper Earth crust. Using fully coupled numerical simulations and scaling arguments, we show that a single dimensionless number governs buoyant hydraulic fracture growth, namely the dimensionless viscosity of a radial hydraulic fracture at the time when buoyancy becomes of order 1. It quantifies whether the transition to buoyancy occurs when the growth of the radial hydraulic fracture is either still in the regime dominated by viscous flow dissipation or already in the regime where fracture energy dissipation dominates. A family of fracture shapes emerge at late time from finger-like (toughness regime) to inverted elongated cudgel-like (viscous regime). Three-dimensional toughness-dominated buoyant fractures exhibit a finger-like shape with a constant-volume toughness-dominated head and a viscous tail having a constant uniform horizontal breadth: there is no further horizontal growth past the onset of buoyancy. However, if the transition to buoyancy occurs while in the viscosity-dominated regime, both vertical and horizontal growths continue to match scaling arguments. As soon as the fracture toughness is not strictly zero, horizontal growth stops when the dimensionless horizontal toughness becomes of order 1. The horizontal breadth follows the predicted scaling.

Three-dimensionalities in the wake of flow past a circular cylinder executing sinusoidal rotary oscillations about its axis is studied experimentally. The results of water tunnel experiments on a rotationally oscillating cylinder for Reynolds number of 250 with varying amplitude and forcing frequency are discussed. Qualitative studies using hydrogen bubble and laser-induced fluorescence flow visualisation techniques are performed. Observation made for oscillating amplitude, $\theta _{0} = {\rm \pi}/4$ and $\theta _{0}=3{\rm \pi} /4$, and a low normalised forcing frequency, $FR$, of 0.75 and 0.5, respectively, confirmed a mode having a spanwise non-dimensional wavelength of $\sim$1.8 which is also observed for a rotating cylinder. On increasing forcing frequency this mode disappears and a new mode with a bean-shaped structure and a much smaller spanwise normalised wavelength of $\sim$0.8 appears at an $FR$ of 1 and an oscillation amplitude of ${\rm \pi} /2$. This mode remains almost stable until a forcing frequency of $FR=1.4$. At higher forcing frequency, $FR=2.75$, and oscillation amplitude of $3{\rm \pi} /4$, a mode with cellular structure and a normalised spanwise wavelength of $\sim$1.6 is identified. The cells in this mode flatten up with increasing downstream distance and are shed alternately with respect to the adjacent cell. Certain combinations of forcing parameters resulted in a forced two-dimensionality of the wake. Quantitative studies using hot-wire measurements and particle image velocimetry confirm the presence of these modes and wake characteristics. Wake mode map in the forcing frequency and amplitude plane is presented showing regions of newly discovered modes and wake lock-on boundaries.

An analysis of laminar natural convection in inclined slots subjected to patterned heating has been performed. The imposed heating takes a simple form characterized by a single Fourier mode combined with uniform heating. It is shown that periodic heating applied at the lower plate produces no net flow when the slot is either horizontal or vertical, but a net upward flow is generated when the slot is tilted. Periodic heating applied at the upper plate produces net downward flow in the inclined situation. The addition of uniform heating promotes the upward flow while cooling has the opposite effect. There is a critical inclination angle at which the maximum net flow rate is greatest. Dynamic and thermal boundary layers are present when the wavenumber of the imposed heating is large. The use of heating at both plates, with the same wavenumber, leads to a flow dominated by the plate exposed to a more intense heating; when the two plates are heated equally no net flow is observed irrespective of the inclination angle. Changes of the relative positions of the two patterns can change the net flow rate by up to 50 %. The intensity of the flow increases with reduction of the Prandtl number. If the heating applied to the plates is of different wavelength, but of the same intensity, a wide range of behaviours of the flow system is possible. The details of this response are sensitive to the ratio of the two wavenumbers.