A uniform distribution of roughness elements of height, $h$, enhances turbulent momentum exchanges in wall flows of thickness, $\delta$. For $\delta/h$ above a nominal threshold, this distribution of elements enhances drag, but does not induce structural response to outer-/inertial-layer turbulence. Large-scale spatial variability in element attributes, however, can induce $\delta$-scale response in the instantaneous and Reynolds-averaged flow. For the case of a large-scale heterogeneity aligned orthogonal and parallel to the primary transport direction, the Reynolds-averaged flow responds with formation of an internal boundary layer (IBL) and counter-rotating secondary cells, respectively. For oblique flow-heterogeneity alignments, Anderson (J. Fluid Mech., vol. 886, 2020, p. A15), showed that IBL formation persists for flow-heterogeneity aligned at a series of oblique angles, before abruptly collapsing and transitioning to counter-rotating secondary cells at a critical obliquity. We define flow-roughness obliquity angle with symbol, $\theta$, where $\theta = 0$ and ${\rm \pi}/2$ corresponds with streamwise and spanwise heterogeneity, respectively, while the critical obliquity angle is denoted by $\theta_c$. We have used large-eddy simulation to perform a comprehensive parametric assessment on the effect of obliquity, element height, and spacing between adjacent ‘rows’ of elements, all of which is intended to identify the aforementioned critical obliquity. We once again report the aforementioned IBL persistence. It is shown that this response is a product of successive element sheltering, which can be recorded via the element frontal area index: as obliquity angle increases, there is a critical obliquity at which downflow elements fall in the sheltering zone of upflow elements; we find that $\theta_c = 22 {\rm \pi}/ 56$.

This work addresses the effect of travelling thermal waves applied at the fluid layer surface, on the formation of global flow structures in two-dimensional (2-D) and 3-D convective systems. For a broad range of Rayleigh numbers ($10^3\leq Ra \leq 10^7$) and thermal wave frequencies ($10^{-4}\leq \varOmega \leq 10^{0}$), we investigate flows with and without imposed mean temperature gradients. Our results confirm that the travelling thermal waves can cause zonal flows, i.e. strong mean horizontal flows. We show that the zonal flows in diffusion dominated regimes are driven purely by the Reynolds stresses and end up always travelling retrograde. In convection dominated regimes, however, mean flow advection, caused by tilted convection cells, becomes dominant. This generally leads to prograde directed mean zonal flows. By means of direct numerical simulations we validate theoretical predictions made for the diffusion dominated regime. Furthermore, we make use of the linear stability analysis and explain the existence of the tilted convection cell mode. Our extensive 3-D simulations support the results for 2-D flows and thus provide further evidence for the relevance of the findings for geophysical and astrophysical systems.

The curvature instability of thin vortex rings is a parametric instability discovered from short-wavelength analysis by Hattori & Fukumoto (Phys. Fluids, vol. 15, 2003, pp. 3151–3163). A full-wavelength analysis using normal modes then followed in Fukumoto & Hattori (J. Fluid Mech., vol. 526, 2005, pp. 77–115). The present work extends these results to the case with different densities inside and outside the vortex core in the presence of surface tension. The maximum growth rate and the instability half-bandwidth are calculated from the dispersion relation given by the resonance between two Kelvin waves of $m$ and $m+1$, where $m$ is the azimuthal wavenumber. The result shows that vortex rings are unstable to resonant waves in the presence of density and surface tension. The curvature instability for the principal modes is enhanced by density variations in the small axial wavenumber regime, while the asymptote for short wavelengths is close to the constant density case. The effect of surface tension is marginal. The growth rates of non-principal modes are also examined, and long waves are most unstable.

The Moore–Saffman–Tsai–Widnall (MSTW) instability is a parametric instability that arises in strained vortex columns. The strain is assumed to be weak and perpendicular to the vortex axis. In this second part of our investigation of vortex instability including density and surface tension effects, a linear stability analysis for this situation is presented. The instability is caused by resonance between two Kelvin waves with azimuthal wavenumber separated by two. The dispersion relation for Kelvin waves and resonant modes are obtained. Results show that the stationary resonant waves for $m=\pm 1$ are more unstable when the density ratio $\rho_2/\rho_1$, the ratio of vortex to ambient fluid density, approaches zero, whereas the growth rate is maximised near $\rho _2/\rho _1 =0.215$ for the resonance $(m,m+2)=(0,2)$. Surface tension suppresses the instability, but its effect is less significant than that of density. As the azimuthal wavenumber $m$ increases, the MSTW instability decays, in contrast to the curvature instability examined in Part 1 (Chang & Llewellyn Smith, J. Fluid Mech. vol. 913, 2021, A14).

We extend unsteady thin aerofoil theory to aerofoils with generalised chordwise porosity distributions by embedding the material characteristics of the porous medium into the linearised boundary condition. Application of the Plemelj formulae to the resulting boundary value problem yields a singular Fredholm–Volterra integral equation which does not admit an analytical solution. We develop a numerical solution scheme by expanding the bound vorticity distribution in terms of appropriate basis functions. Asymptotic analysis at the leading and trailing edges reveals that the appropriate basis functions are weighted Jacobi polynomials whose parameters are related to the porosity distribution. The Jacobi polynomial basis enables the construction of a numerical scheme that is accurate and rapid, in contrast to the standard choice of Chebyshev basis functions that are shown to be unsuitable for porous aerofoils. Applications of the numerical solution scheme to discontinuous porosity profiles, quasi-static problems and the separation of circulatory and non-circulatory contributions are presented. Further asymptotic analysis of the singular Fredholm–Volterra integral equation corroborates the numerical scheme and elucidates the behaviour of the unsteady solution for small or large reduced frequency in the form of scaling laws. At low frequencies, the porous resistance dominates, whereas at high frequencies, an asymptotic inner region develops near the trailing edge and the effective mass of the porous medium dominates. Analogues to the classical Theodorsen and Sears functions are computed numerically, and Fourier transform inversion of these frequency-domain functions produces porous extensions to the Wagner and Küssner functions for transient aerofoil motions or gust encounters, respectively. Results from the present analysis and its underpinning numerical framework aim to enable the unsteady aerodynamic assessment of design strategies using porosity, with implications for unsteady gust rejection, noise-reducing aerofoil design and biologically inspired flight.

We present several time-dependent analytical solutions for the incompressible Euler system with a free surface. These analytical solutions give quantitative descriptions of some physical phenomena, such as water motion or waves on a space variable topography, and can be used as reference solutions when validating numerical simulation codes. They concern fluid flows governed by Euler equations with or without the hydrostatic hypothesis including a wet/dry interface, variable density and a wide variety of boundary conditions.

Rotating Rayleigh–Bénard convection is investigated numerically with the use of an asymptotic model that captures the rapidly rotating, small Ekman number limit, $Ek \rightarrow 0$. The Prandtl number ($Pr$) and the asymptotically scaled Rayleigh number ($\widetilde {Ra} = Ra Ek^{4/3}$, where $Ra$ is the typical Rayleigh number) are varied systematically. For sufficiently vigorous convection, an inverse kinetic energy cascade leads to the formation of a pair of large-scale vortices of opposite polarity, in agreement with previous studies of rapidly rotating convection. With respect to the kinetic energy, we find a transition from convection dominated states to a state dominated by large-scale vortices at an asymptotically reduced (small-scale) Reynolds number of $\widetilde {Re} \approx 6$ ($\widetilde {Re} = Re Ek^{1/3}$, where $Re$ is the Reynolds number associated with vertical flows) for all investigated values of $Pr$. The ratio of the depth-averaged kinetic energy to the kinetic energy of the convection reaches a maximum at $\widetilde {Re} \approx 24$, then decreases as $\widetilde {Ra}$ is increased. This decrease in the relative kinetic energy of the large-scale vortices is associated with a decrease in the convective correlations with increasing Rayleigh number. The scaling behaviour of the convective flow speeds is studied; although a linear scaling of the form $\widetilde {Re} \sim \widetilde {Ra}/Pr$ is observed over a limited range in Rayleigh number and Prandtl number, a clear departure from this scaling is observed at the highest accessible values of $\widetilde {Ra}$. Calculation of the forces present in the governing equations shows that the ratio of the viscous force to the buoyancy force is an increasing function of $\widetilde {Ra}$, that approaches unity over the investigated range of parameters.

Linear interaction analysis (LIA) predicts that isotropic vortical turbulence interacting with a normal shock wave at Mach 2 or greater produces a post-shock Reynolds stress tensor whose transverse components $R_{22}$ and $R_{33}$ are greater than the streamwise component $R_{11}$. However, computational and experimental studies have consistently reported Reynolds stress tensors with $R_{11}>R_{22}$ in this regime. This discrepancy is primarily due to nonlinear terms which have been neglected in the linear analysis. However, if the effect of the nonlinear terms is to move the anisotropy from a state with $R_{22}>R_{11}$, through isotropy, to a state of qualitatively opposite anisotropy, this would seem contrary to the usual return-to-isotropy effect. Here, we explain the redistribution of energy between the transverse and streamwise Reynolds stresses by considering the wavelengths of the vortical waves emitted by the interaction. Our study shows that the transverse Reynolds stresses are concentrated at smaller length scales than the streamwise stress. In turn, the evolution of the modes associated with $R_{22}$ occurs on a faster time scale, and their energy is more quickly transferred away, allowing $R_{11}$ to dominate. The overall effect is that the global anisotropy quickly attains a state where $R_{11}>R_{22}$. Furthermore, a quantitative model is developed based on this understanding of the flow physics. The model uses LIA along with an eddy viscosity approximation for the nonlinear interactions in order to quantify the redistribution of energy between the transverse and streamwise Reynolds stresses. The model predicts anisotropy levels approximately matching those observed in direct numerical simulation data.

Flow past a circular cylinder and a downstream sphere is simulated numerically for Reynolds numbers between 100 and 300 for cylinder-to-sphere gap ratios between 0 and 3. The aim of this study is to investigate the influence of the sphere on the vortex shedding of the cylinder, and the wake-induced forces on the sphere. If vortices are shed from the cylinder upstream of the sphere, vortices are disconnected after they pass the sphere. This type of vortex disconnection is defined as sphere-induced vortex disconnection. The key finding is the so-called sphere-induced vortex dislocation (SIVDL) which occurs within the vortex formation region. The main characteristics of SIVDL include the adhesion of vortices to the cylinder at dislocation positions, and periodical occurrences of SIVDL. The periodic occurrence of SIVDL makes the frequency of vortex shedding change over time. Another interesting phenomenon is the reattachment of the separated shear layers from the cylinder to the sphere, which occurs when the sphere–cylinder gap is small. This leads to the formation of vortices without disconnection downstream of the sphere that it is similar to the case of tandem cylinders. In all cases, vortices are inclined, and their angle of inclination increases when SIVDL occurs. SIVDL reduces the frequency of the lift coefficient and increases the standard deviation of the drag coefficient of the sphere. The periodic occurrence of SIVDL causes the beating of the lift coefficient along with large-amplitude, low-frequency oscillations of the sphere's drag coefficient.

The wake of an Ahmed body may be divided into high- and low-drag regimes where the rear slant angle (φ) is in the ranges of 12.5°–30° and larger than 30°, respectively. This work aims to gain a relatively thorough understanding of unsteady predominant coherent structures around an Ahmed body of φ = 35° in the low-drag regime. Extensive hot-wire, wall pressure, flow visualization and particle image velocimetry measurements have been conducted at Reynolds number $Re \in [0.3,2.7] \times {10^5}$, based on the square root of the model's frontal area. A total of five distinct Strouhal numbers have been identified in the wake. One of them, Stw ≈ 0.30, is captured behind the vertical base, which is associated with the structures that emanate from the upper recirculation bubble and pinch off from the lower bubble, respectively. It is found that Stw scales with a characteristic length αS, which reflects physically the bubble size, and the Strouhal number $St_w^ + $ based on αS is a constant 0.20, irrespective of the value of φ. A corner vortex rolling upstream is observed near the lower end of the slanted surface, whose formation mechanism and dynamical role are discussed. The Reynolds-number effect on the flow is also documented. Based on the present and previously reported data, a conceptual flow structure model is proposed for a low-drag Ahmed body, including both steady and unsteady coherent structures around the body.

In this paper, we develop a large-Reynolds-number asymptotic theory to describe the impact of a localised roughness element on oncoming inviscid first and second Mack modes in supersonic or hypersonic boundary layers. The height and width of the roughness are assumed to be of $O(R^{-1/4}\delta ^{*})$ and $O(R^{1/4}\delta ^{*})$, respectively, such that the induced mean-flow distortion is described by the triple-deck formalism, where $R$ is the Reynolds number based on the local boundary-layer displacement thickness $\delta ^{*}$. As the wavelength of the inviscid Mack mode is comparable with $\delta ^{*}$, its interaction with the roughness forms a multi-scale problem. The Mack mode in the bulk of the boundary layer is described by the inviscid Rayleigh equation, whose evolution near the roughness is formulated by use of the solvability condition. It is found that the dominant roughness effect is attributed to both the interaction of the oncoming perturbation with the mean-flow distortion in the main layer and the inhomogeneous forcing from the curved wall (Stokes layer). This theory enables us to probe the scattering process when the frequency approaches the synchronisation frequency, which is recognised as the critical site distinguishing the destabilising and stabilising roles of the roughness. An improved asymptotic theory is also developed, which increases the accuracy of the asymptotic prediction, especially at the intersection frequency of the first and second modes. We also carry out harmonic linearised Navier–Stokes calculations and direct numerical simulations to confirm the accuracy of the asymptotic predictions, and favourable agreements are obtained even when the roughness height is a quarter of the nominal boundary-layer thickness.

Internal wave groups are modified by reflection from a uniform slope. At small values of the azimuth angle, θI, of a group approaching a slope the group's volume is decreased on reflection, but at sufficiently large θI volume increases. Group shape is also generally changed on reflection. Wave breaking can be driven by resonant interactions between the incident and reflecting waves or through convective and Kelvin–Helmholtz instability (CI and KHI) of steepening internal waves. Breaking through resonant interactions can occur only in a boundary layer within which incident and reflected groups co-exist. This imposes conditions that depend on the dimensions of incident groups, which may be severe requiring extremely large aspect ratios and implying that in practice the breaking of internal waves in groups incident on or reflected from a slope through CI and KHI often contributes more than resonant wave–wave interactions to mixing near a sloping boundary. Wave breaking associated with self-induced instability of waves in groups will have a greater and possibly more extensive effect on the boundary layer over a slope when the direction in which groups of waves continue to break is close to the up-slope direction of the boundary in the plane of the waves, the effective slope. This is possible only at the small slope angles which are typical of the oceanic continental slopes. Over these slopes internal wave breaking may contribute significantly to the diapycnal mixing of the deep ocean.

Recent simulations indicate that streamwise-preferential porous materials have the potential to reduce drag in wall-bounded turbulent flows (Gómez-de-Segura & García-Mayoral, J. Fluid Mech., vol. 875, 2019, pp. 124–172). This paper extends the resolvent formulation to study the effect of such anisotropic permeable substrates on turbulent channel flow. Under the resolvent formulation, the Fourier-transformed Navier–Stokes equations are interpreted as a linear forcing–response system. The nonlinear terms are considered the endogenous forcing in the system that gives rise to a velocity and pressure response. A gain-based decomposition of the forcing–response transfer function – the resolvent operator – identifies response modes (resolvent modes) that are known to reproduce important structural and statistical features of wall-bounded turbulent flows. The effect of permeable substrates is introduced in this framework using the volume-averaged Navier–Stokes equations and a generalized form of Darcy's law. Substrates with high streamwise permeability and low spanwise permeability are found to suppress the forcing–response gain for the resolvent mode that serves as a surrogate for the energetic near-wall cycle. This reduction in mode gain is shown to be consistent with the drag reduction trends predicted by theory and observed in numerical simulations. Simulation results indicate that drag reduction is limited by the emergence of spanwise rollers resembling Kelvin–Helmholtz vortices beyond a threshold value of wall-normal permeability. The resolvent framework also predicts the conditions in which such energetic spanwise-coherent rollers emerge. These findings suggest that a limited set of resolvent modes can serve as the building blocks for computationally efficient models that enable the design and optimization of permeable substrates for passive turbulence control.

This paper focuses on finding a closed-loop strategy to reduce the drag of a cylinder in laminar flow conditions. Deep reinforcement learning algorithms have been implemented to discover efficient control schemes, using two synthetic jets located on the cylinder's poles as actuators and pressure sensors in the wake of the cylinder as feedback observation. The present work focuses on the efficiency and robustness of the identified control strategy and introduces a novel algorithm (S-PPO-CMA) to optimise the sensor layout. An energy-efficient control strategy reducing drag by $18.4\,\%$ at a Reynolds number of $120$ is obtained. This control policy is shown to be robust both to a Reynolds-number variation in the range $[100;216]$ and to measurement noise, for signal-to-noise ratios as low as $0.2$ with negligible impact on performance. Along with a systematic study on sensor number and location, the proposed sparsity-seeking algorithm has achieved a successful optimisation to a reduced five-sensor layout while keeping state-of-the-art performance. These results further highlight the interesting possibilities of reinforcement learning for active flow control and pave the way to efficient, robust and practical implementations of these control techniques in experimental or industrial systems.

In this paper the capabilities of the turbulence-resolving Eulerian–Eulerian two-phase flow model to predict the suspension of mono-dispersed finite-sized solid particles in a boundary layer flow are investigated. For heavier-than-fluid particles, having settling velocity of the order of the bed friction velocity, the two-fluid model significantly under-estimates the turbulent dispersion of particles. It is hypothesized that finite-size effects are important and a correction model for the drag law is proposed. This model is based on the assumption that the turbulent flow scales larger than the particle diameter will contribute to the resolved relative velocity between the two phases, whereas eddies smaller than the particle diameter will have two effects: (i) they will reduce the particle response time by adding a sub-particle scale eddy viscosity to the drag coefficient, and (ii) they will contribute to increase the production of granular temperature. Integrating finite-size effects allows us to quantitatively predict the concentration profile for heavier-than-fluid particles without any tuning parameter. The proposed modification of the two-fluid model extends its range of applicability to tackle particles having a size belonging to the inertial range of turbulence and allows us to envision more complex applications in terms of flow forcing conditions, i.e. sheet flow, wave-driven transport, turbidity currents and/or flow geometries, i.e. ripples, dunes, scour.

Wave shape (e.g. wave skewness and asymmetry) impacts sediment transport, remote sensing and ship safety. Previous work showed that wind affects wave shape in intermediate and deep water. Here, we investigate the effect of wind on wave shape in shallow water through a wind-induced surface pressure for different wind speeds and directions to provide the first theoretical description of wind-induced shape changes. A multiple-scale analysis of long waves propagating over a shallow, flat bottom and forced by a Jeffreys-type surface pressure yields a forward or backward Korteweg–de Vries (KdV)–Burgers equation for the wave profile, depending on the wind direction. The evolution of a symmetric, solitary-wave initial condition is calculated numerically. The resulting wave grows (decays) for onshore (offshore) wind and becomes asymmetric, with the rear face showing the largest shape changes. The wave profile's deviation from a reference solitary wave is primarily a bound wave and trailing, dispersive, decaying tail. The onshore wind increases the wave's energy and skewness with time while decreasing the wave's asymmetry, with the opposite holding for offshore wind. The corresponding wind speeds are shown to be physically realistic, and the shape changes are explained as slow growth followed by rapid evolution according to the unforced KdV equation.

Membrane filtration is a vital industrial process, with applications including air purification and blood filtration. In this paper, we study the optimal design for a concertinaed filtration membrane composed of angled porous membranes and dead ends. The geometry of the filter motivates a lubrication scaling for the flow, leading to a system of coupled (modified) Reynolds equations. By analysing this reduced system, we examine how the filter performance depends on the angle, position, thickness and permeance of the membrane, using a combination of numerical and asymptotic approaches, the latter in the limit of a slightly angled membrane. We find that, for a membrane of fixed angle and physical properties, there can exist multiple membrane positions that maximise the flux for an applied pressure difference. More generally, this shows how coupled problems in lubrication flow can exhibit non-trivial bifurcating optima. For the particular application on which we focus, we show that, while the maximal flux achievable depends on the membrane thickness and permeance, the optimal membrane configuration is always in one of two set-ups: centred and diagonal across the full domain; or angled and in the corner of the domain.

The formation of orifice-generated vortex rings, at a Reynolds number of $5300$ and for a tube-to-orifice diameter ratio of $2.0$, is experimentally investigated for stroke-to-diameter ratios of $0.5(0.5)5.0$. A significant increase is observed in the production of the total invariants of the motion, namely the circulation $\varGamma$, the hydrodynamic impulse $I$ and the kinetic energy $E$, compared with the equivalent nozzle-generated vortex rings. The formation number, as defined by Gharib et al. (J. Fluid Mech., vol. 360, 1998, pp. 121–140), is found to be approximately $2.0$. By measuring the kinematics and the invariants of the ring for increasing stroke ratios, a limiting process in the ring formation is observed, which allows us to define the critical parameters and time scales in the vortex formation process. In particular, it was shown that the ring circulation, impulse, and energy do not reach their asymptotic state at the same non-dimensional time and stroke ratio, hence these two terms cannot be used interchangeably. The stroke ratio required to produce a ring with maximum energy is defined as the ‘optimal stroke ratio’, which is found to be around $4$. The non-dimensional time at which the ring reaches this state, termed the ‘optimal formation time’, is found to be approximately 6–7. The non-dimensional vortex ring numbers $\alpha =E/\rho ^{1/2}\varGamma ^{3/2}I^{1/2}$, $\beta =\varGamma /\rho ^{-1/3}I^{1/3}U^{2/3}$ and , are measured to be $0.33$, $1.8$ and $1.9$, respectively, consistent with previous experimental, numerical and analytical work, suggesting these numbers to be universal for all isolated vortex rings.

The behaviour of the diffusion front of a magnetic fluid in contact with a miscible non-magnetic fluid in a normal magnetic field is studied. It was found that the magnetic field is a cause of the diffusion front bending and its movement accompanied by intense advective flows. These flows lead to the fast growth of the wavy shape of the diffusion front and formation of the peaks. This phenomenon is studied both numerically and experimentally. The reasons for the instability of the diffusion front in a magnetic field are discussed. The influence of the parameters of the problem (Schmidt number, magnetic Rayleigh number, magnetic field, the thickness of the layer, diffusion front width, etc.) on the instability parameters is studied both numerically and experimentally. It is shown that the studied instability differs from Rosensweig's instability.

The closure of a human lung airway is modelled as a pipe coated internally with a liquid that takes into account the viscoelastic properties of mucus. For a thick-enough coating, the Plateau–Rayleigh instability blocks the airway by the creation of a liquid plug, and the preclosure phase is dominated by the Newtonian behaviour of the liquid. Our previous study with a Newtonian-liquid model demonstrated that the bifrontal plug growth consequent to airway closure induces a high level of stress and stress gradients on the airway wall, which is large enough to damage the epithelial cells, causing sublethal or lethal responses. In this study, we explore the effect of the viscoelastic properties of mucus by means of the Oldroyd-B and FENE-CR model. Viscoelasticity is shown to be very relevant in the postcoalescence process, introducing a second peak of the wall shear stresses. This second peak is related to an elastic instability due to the presence of the polymeric extra stresses. For high-enough Weissenberg and Laplace numbers, this second shear stress peak is as severe as the first one. Consequently, a second lethal or sublethal response of the epithelial cells is induced.