Models of flowing complex fluids such as polymer solutions often use a conformation tensor that reflects the state of the fluid microstructure. In polymer solutions, this quantity measures the orientation and stretching of the molecules, and reflects the fact that the squared length of a polymer molecule must be positive. By exploiting results from differential geometry and continuum mechanics, Hameduddin et al. (J. Fluid Mech., vol. 842, 2018, pp. 395–427) introduce a new approach for analysing the conformation tensor that respects this positivity constraint. With this approach, they present computational results for turbulent flow of a polymer solution that exhibits turbulent drag reduction, showing that the new measures of polymer stretching afforded by their approach lend insights not available in traditional methods.

The control of streamwise vortices in high Reynolds number boundary layer flows often aims at reducing the vortex energy as a means of mitigating the growth of secondary instabilities, which eventually delays the transition from laminar to turbulent flow. In this paper, we aim at utilizing such an energy reduction strategy using optimal control theory to limit the growth of Görtler vortices developing in an incompressible laminar boundary layer flow over a concave wall, and excited by a row of roughness elements with spanwise separation of the same order of magnitude as the boundary layer thickness. Commensurate with control theory formalism, we transform a constrained optimization problem into an unconstrained one by applying the method of Lagrange multipliers. A high Reynolds number asymptotic framework is utilized, wherein the Navier–Stokes equations are reduced to the boundary region equations, in which wall deformations enter the problem through an appropriate Prandtl transformation. In the optimal control strategy, the wall displacement or the wall transpiration velocity serves as the control variable, while the cost functional is defined in terms of the wall shear stress. Our numerical results indicate, among other things, that the optimal control algorithm is very effective in reducing the amplitude of the Görtler vortices, especially for the control based on wall displacement.

Wave loading on marine structures is the major external force to be considered in the design of such structures. The accurate prediction of the nonlinear high-order components of the wave loading has been an unresolved challenging problem. In this paper, the nonlinear harmonic components of hydrodynamic forces on a bottom-mounted vertical cylinder are investigated experimentally. A large number of experiments were conducted in the Danish Hydraulic Institute shallow water wave basin on the cylinder, both on a flat bed and a sloping bed, as part of a European collaborative research project. High-quality data sets for focused wave groups have been collected for a wide range of wave conditions. The high-order harmonic force components are separated by applying the ‘phase-inversion’ method to the measured force time histories for a crest focused wave group and the same wave group inverted. This separation method is found to work well even for locally violent nearly-breaking waves formed from bidirectional wave pairs. It is also found that the $n$th-harmonic force scales with the $n$th power of the envelope of both the linear undisturbed free-surface elevation and the linear force component in both time variation and amplitude. This allows estimation of the higher-order harmonic shapes and time histories from knowledge of the linear component alone. The experiments also show that the harmonic structure of the wave loading on the cylinder is virtually unaltered by the introduction of a sloping bed, depending only on the local wave properties at the cylinder. Furthermore, our new experimental results reveal that for certain wave cases the linear loading is actually less than 40 % of the total wave loading and the high-order harmonics contribute more than 60 % of the loading. The significance of this striking new result is that it reveals the importance of high-order nonlinear wave loading on offshore structures and means that such loading should be considered in their design.

A three-dimensional wavelet multi-resolution analysis of direct numerical simulations of a turbulent premixed flame is performed in order to investigate the spatially localized spectral transfer of kinetic energy across scales in the vicinity of the flame front. A formulation is developed that addresses the compressible spectral dynamics of the kinetic energy in wavelet space. The wavelet basis enables the examination of local energy spectra, along with inter-scale and subfilter-scale (SFS) cumulative energy fluxes across a scale cutoff, all quantities being available either unconditioned or conditioned on the local instantaneous value of the progress variable across the flame brush. The results include the quantification of mean spectral values and associated spatial variabilities. The energy spectra undergo, in most locations in the flame brush, a precipitous drop that starts at scales of the same order as the characteristic flame scale and continues to smaller scales, even though the corresponding decrease of the mean spectra is much more gradual. The mean convective inter-scale flux indicates that convection increases the energy of small scales, although it does so in a non-conservative manner due to the high aspect ratio of the grid, which limits the maximum scale level that can be used in the wavelet transform, and to the non-periodic boundary conditions, which exchange energy through surface forces, as explicitly elucidated by the formulation. The mean pressure-gradient inter-scale flux extracts energy from intermediate scales of the same order as the characteristic flame scale, and injects energy in the smaller and larger scales. The local SFS-cumulative contribution of the convective and pressure-gradient mechanisms of energy transfer across a given cutoff scale imposed by a wavelet filter is analysed. The local SFS-cumulative energy flux is such that the subfilter scales upstream from the flame always receive energy on average. Conversely, within the flame brush, energy is drained on average from the subfilter scales by convective and pressure-gradient effects most intensely when the filter cutoff is larger than the characteristic flame scale.

Features of the turbulent cascade are investigated for various datasets from three different turbulent flows, namely free jets as well as wake flows of a regular grid and a cylinder. The analysis is focused on the question as to whether fully developed turbulent flows show universal small-scale features. Two approaches are used to answer this question. First, two-point statistics, namely structure functions of longitudinal velocity increments, and, second, joint multiscale statistics of these velocity increments are analysed. The joint multiscale characterisation encompasses the whole cascade in one joint probability density function. On the basis of the datasets, evidence of the Markov property for the turbulent cascade is shown, which corresponds to a three-point closure that reduces the joint multiscale statistics to simple conditional probability density functions (cPDFs). The cPDFs are described by the Fokker–Planck equation in scale and its Kramers–Moyal coefficients (KMCs). The KMCs are obtained by a self-consistent optimisation procedure from the measured data and result in a Fokker–Planck equation for each dataset. Knowledge of these stochastic cascade equations enables one to make use of the concepts of non-equilibrium thermodynamics and thus to determine the entropy production along individual cascade trajectories. In addition to this new concept, it is shown that the local entropy production is nearly perfectly balanced for all datasets by the integral fluctuation theorem (IFT). Thus, the validity of the IFT can be taken as a new law of the turbulent cascade and at the same time independently confirms that the physics of the turbulent cascade is a memoryless Markov process in scale. The IFT is taken as a new tool to prove the optimal functional form of the Fokker–Planck equations and subsequently to investigate the question of universality of small-scale turbulence in the datasets. The results of our analysis show that the turbulent cascade contains universal and non-universal features. We identify small-scale intermittency as a universality breaking feature. We conclude that specific turbulent flows have their own particular multiscale cascades, in other words, their own stochastic fingerprints.

This paper presents an experimental study on shock-wave/turbulent-boundary-layer interaction unsteadiness and delves specifically into the shear layer’s role. A range of axisymmetric step-induced interactions is investigated and the scale of separation is altered by over an order of magnitude – mass in the recirculation by two orders – while subjected to constant separation-shock strength. The effect of the separated shear layer on interaction unsteadiness is thus isolated and its kinematics are characterised. Results point at a mechanism whereby the depletion of separated flow is dictated by the state of the large eddy structures at their departure from the bubble. Low-frequency pulsations are found to adjust in response and sustain a reconciling view of an entrainment–recharge process, with both an inherent effect of the upstream boundary layer on shear layer inception and an increase in the mass locally acquired by eddies as they develop downstream.

The present work focuses on the flame–wall interaction (FWI) based on direct numerical simulations (DNS) of a head-on premixed flame quenching configuration at the statistically stationary state. The effects of FWI on the turbulent flame temperature, wall heat flux, flame dynamics and flow structures were investigated. In turbulent head-on quenching, particularly for high turbulence intensity, the distorted flames generally consist of the head-on flame part and the entrained flame part. The flame properties are jointly influenced by turbulence, heat generation from chemical reactions and heat loss to the cold wall boundary. For the present FWI configuration, as the wall is approached, the ‘influence zone’ can be identified as the region within which the flame temperature, scalar gradient and flame dilatation start to decrease, whereas the wall heat flux tends to increase. As the distance to the wall drops below the flame-quenching distance, approximately where the wall heat flux reaches its maximum value, chemical reactions become negligibly weak inside the ‘quenching zone’. A simplified counter-flow model is also proposed. With the reasonably proposed relation between the flame speed and the flame temperature, the model solutions match well with the DNS results, both qualitatively and quantitatively. Moreover, near-wall statistics of some important flame properties, including the flame dilatation, reaction progress variable gradient, tangential strain rate and curvature were analysed in detail under different wall boundary conditions.

Self-similar solutions to the compressible Euler equations with nonlinear conduction are considered as particular instances of unsteady radiative deflagration – or ‘ablation’ – waves with the goal of characterizing the actual hydrodynamic properties that such flows may present. The chosen family of solutions, corresponding to the ablation of an initially quiescent perfectly cold and homogeneous semi-infinite slab of inviscid compressible gas under the action of increasing external pressures and radiation fluxes, is well suited to the description of the early ablation of a target by gas-filled cavity X-rays in experiments of high energy density physics. These solutions are presently computed by means of a highly accurate numerical method for the radiative conduction model of a fully ionized plasma under the approximation of a non-isothermal leading shock wave. The resulting set of solutions is unique for its high fidelity description of the flows down to their finest scales and its extensive exploration of external pressure and radiative flux ranges. Two different dimensionless formulations of the equations of motion are put forth, yielding two classifications of these solutions which are used for carrying out a quantitative hydrodynamic analysis of the corresponding flows. Based on the main flow characteristic lengths and on standard characteristic numbers (Mach, Péclet, stratification and Froude numbers), this analysis points out the compressibility and inhomogeneity of the present ablative waves. This compressibility is further analysed to be too high, whether in terms of flow speed or stratification, for the low Mach number approximation, often used in hydrodynamic stability analyses of ablation fronts in inertial confinement fusion (ICF), to be relevant for describing these waves, and more specifically those with fast expansions which are of interest in ICF. Temperature stratification is also shown to induce, through the nonlinear conductivity, supersonic upstream propagation of heat-flux waves, besides a modified propagation of quasi-isothermal acoustic waves, in the flow conduction regions. This description significantly departs from the commonly admitted depiction of a quasi-isothermal conduction region where wave propagation is exclusively ascribed to isothermal acoustics and temperature fluctuations are only diffused.

The dynamics of two-dimensional (2-D) ideal fluid vortices is studied experimentally in the presence of an irrotational strain flow. Laboratory experiments are conducted using strongly magnetized pure electron plasmas, a technique which is made possible by the isomorphism between the drift–Poisson equations describing plasma dynamics transverse to the field and the 2-D Euler equations describing an ideal fluid. The electron plasma system provides an excellent opportunity to study the dynamics of a 2-D Euler fluid due to weak dissipation and weak 3-D effects, simple diagnosis and precise control. The plasma confinement apparatus used here was designed specifically to study vortex dynamics under the influence of external flow by applying boundary conditions in two dimensions. Additionally, vortex-in-cell simulations are carried out to complement the experimental results and to extend the parameter range of the studies. It is shown that the global dynamics of a quasi-flat vorticity profile is in good quantitative agreement with the theory of a piecewise-constant elliptical patch of vorticity, including the equilibria, dynamical orbits and stability properties. Deviations from the elliptical patch theory are observed for non-flat vorticity profiles; they include inviscid damping of the orbits and modified stability limits. The dependence of these phenomena on the flatness of the initial profile is discussed. The relationship of these results to other theoretical, numerical and experimental studies is also discussed.

Direct numerical simulation of a turbulent boundary layer (TBL) subjected to a moderate adverse pressure gradient (APG, $\unicode[STIX]{x1D6FD}=1.45$) is performed to explore the contribution of large scales to the skin friction, where $\unicode[STIX]{x1D6FD}$ is the Clauser pressure gradient parameter. The Reynolds number based on the momentum thickness develops from $Re_{\unicode[STIX]{x1D703}}\approx 110$ to 6000 with an equilibrium region in $Re_{\unicode[STIX]{x1D703}}=4000$–5500. The spanwise wavelength ($\unicode[STIX]{x1D706}_{z}$) spectra of the streamwise and spanwise velocity fluctuations show that the large-scale energy is significantly enhanced throughout the boundary layer. We quantify the superposition and amplitude modulation effects of these enhanced large scales on the skin friction coefficient ($C_{f}$) by employing two approaches: (i) spanwise co-spectra of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$; (ii) conditionally averaged $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$. The velocity–vorticity correlations $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ are related to the advective transport and the vortex stretching, respectively. Although $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ negatively contributes to $C_{f}$, the positive contribution of the large scales ($\unicode[STIX]{x1D706}_{z}>0.5\unicode[STIX]{x1D6FF}$) is observed in the co-spectra of weighted $\langle v\unicode[STIX]{x1D714}_{z}\rangle$. For the co-spectra of weighted $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$, we observe an outer peak at $\unicode[STIX]{x1D706}_{z}\approx 0.75\unicode[STIX]{x1D6FF}$ and the superposition of the large scales in the buffer region, leading to the enhancement of $C_{f}$. The magnitude of $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ depends on the large-scale streamwise velocity fluctuations ($u_{L}$). In particular, the negative-$u_{L}$ events amplify $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ in the outer region, and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ is enhanced by the positive-$u_{L}$ events. As a result, the skin friction induced by $\langle v\unicode[STIX]{x1D714}_{z}\rangle$ and $\langle -w\unicode[STIX]{x1D714}_{y}\rangle$ increases in the present APG TBL.

We study experimentally the flow of light granular material along the free surface of a liquid of greater density. Despite a rich set of related geophysical and environmental phenomena, such as the spreading of calved ice, volcanic ash, debris and industrial wastes, there are few previous studies on this topic. We conduct a series of lock-release experiments of buoyant spherical beads into a rectangular tank initially filled with either fresh or salt water, and record the time evolution of the interface shape and the front location of the current of beads. We find that following the release of the lock the front location obeys a power-law behaviour during an intermediate time period before the nose of beads reaches a maximum runout distance within a finite time. We investigate the dependence of the scaling exponent and runout distance on the total amount of beads, the initial lock length, and the properties of the liquid that fills the tank in the experiments. Scaling arguments are provided to collapse the experimental data into universal curves, which can be used to describe the front dynamics of buoyant granular flows with different size and buoyancy effects and initial lock aspect ratios.

Particles subjected to flow are known to acquire electrostatic charges through repeated contacts with each other and with other surfaces. These charges alter gas–particle flow behaviour at different scales. In this work, we present a continuum framework for analysing the interplay between tribocharging and the flow of a monodisperse assembly of particles characterized by a single effective work function. Specifically, we have derived the continuum, kinetic theory transport equations for gas–particle flow and local-averaged charge on particles directly from the Boltzmann equation. We also derive the auxiliary conditions to capture tribocharging at bounding conducting walls. The resulting two-fluid model with tribocharging and boundary conditions has then been validated against results from discrete element simulations that have been specially designed to probe specific terms in the models.

We perform an experimental and theoretical study of the wave pattern generated by the leg strokes of water striders during a propulsion cycle. Using the synthetic schlieren method, we are able to measure the dynamic response of the free surface accurately. In order to match experimental conditions, we extend Bühler’s theory of impulsive forcing (J. Fluid Mech., vol. 573, 2007, pp. 211–236) to finite depth. We demonstrate the improved ability of this approach to reproduce the experimental findings, once the observed continuous forcing and hence non-zero temporal and spatial extent of the leg strokes is also taken into account.

We investigate the merger of two co-rotating geophysical vortices at finite Rossby and Froude number. The initial conditions consist of two uniform potential vorticity vortices in near-equilibrium and in a nearly ‘balanced’ state (i.e. with negligible emission of inertia–gravity wave radiation). We determine the critical merger distance between the two vortices. This distance is found to increase with the magnitude of the Rossby number: intense cyclones or intense anticyclones are able to merge from further apart compared to weaker cyclones and anticyclones. Note that the Froude number is proportional to the Rossby number for the near-equilibrium initial conditions considered. The critical merging distance also depends on the sign of the potential vorticity anomaly, which is positive for ‘cyclones’ and negative for ‘anticyclones’. We show that ageostrophic motions occurring at finite Rossby number tend to draw cyclones together but draw anticyclones apart. On the other hand, we show that anticyclones tend to deform more, in particular when subject to vertical shear (as when the vortices are vertically offset). These two effects compete. Overall, nearly aligned cyclones tend to merge from further apart than their anticyclonic counterparts, while vertically offset anticyclones merge from further apart than cyclones.

We examine the injection of fluid of one viscosity and density into a horizontal permeable aquifer initially saturated with a second fluid of different viscosity and density. The novel feature of the analysis is that we allow the permeability to vary vertically across the aquifer. This leads to recognition that the interface may evolve as either a rarefaction wave that spreads at a rate proportional to $t$, a shock-like front of fixed length or a mixture of shock-like regions and rarefaction-wave-type regions. The classical solutions in which there is no viscosity ratio between the fluids and in which the formation has constant permeability lead to an interface that spreads laterally at a rate proportional to $t^{1/2}$. However, these solutions are unstable to cross-layer variations in the permeability owing to the vertical shear which develops in the flow, causing the structure of the interface to evolve to the rarefaction wave or shock-like structure. In the case that the viscosities of the two fluids are different, it is possible that the solution involves a mixture of shock-like and rarefaction-type structures as a function of the distance above the lower boundary. Using the theory of characteristics, we develop a regime diagram to delineate the different situations. We consider the implications of such heterogeneity for the prediction of front locations during $\text{CO}_{2}$ sequestration. If we neglect the permeability fluctuations, the model always predicts rarefaction-type solutions, while even modest changes in the permeability across a layer can introduce shocks. This difference may be very significant since it leads to the $\text{CO}_{2}$ plume occupying a greater fraction of the pore space between the injector and the leading edge of the $\text{CO}_{2}$ front in a layer of the same mean permeability. This has important implications for estimates of the fraction of the pore space that the $\text{CO}_{2}$ may access.

The present experimental investigation characterises the dynamic response and wake structure of a sinusoidally rotating circular cylinder with a low mass ratio (defined as the ratio of the total oscillating mass to the displaced fluid mass) undergoing cross-stream flow-induced vibration (FIV). The study covers a wide parameter space spanning the forcing rotary oscillation frequency ratio $0\leqslant f_{r}^{\ast }\leqslant 4.5$ and the forcing rotation speed ratio $0\leqslant \unicode[STIX]{x1D6FC}_{r}^{\ast }\leqslant 2.0$, at reduced velocities associated with the vortex-induced vibration (VIV) upper and lower amplitude response branches. Here, $f_{r}^{\ast }=f_{r}/f_{nw}$ and $\unicode[STIX]{x1D6FC}_{r}^{\ast }=\unicode[STIX]{x1D6FA}_{o}D/(2U)$, where $f_{r}$ is the forcing rotary oscillation frequency, $f_{nw}$ is the natural frequency of the system in quiescent fluid (water), $\unicode[STIX]{x1D6FA}_{o}$ is the peak angular rotation speed, $D$ is the cylinder diameter and $U$ is the free-stream velocity; the reduced velocity is defined by $U^{\ast }=U/(\,f_{nw}D)$. The fluid–structure system was modelled using a low-friction air-bearing system in conjunction with a free-surface recirculating water channel, with axial rotary motion provided by a microstepping motor. The cylinder was allowed to vibrate with only one degree of freedom transverse to the oncoming free-stream flow. It was found that in specific ranges of $f_{r}^{\ast }$, the body vibration frequency may deviate from that seen in the non-rotating case and lock onto the forcing rotary oscillation frequency or its one-third subharmonic. The former is referred to as the ‘rotary lock-on’ (RLO) region and the latter as the ‘tertiary lock-on’ (TLO) region. Significant increases in the vibration amplitude and suppression of VIV could both be observed in different parts of the RLO and TLO regions. The peak amplitude response in the case of $U^{\ast }=5.5$ (upper branch) was observed to be $1.2D$, an increase of approximately $50\,\%$ over the non-rotating case, while in the case of $U^{\ast }=8.0$ (lower branch), the peak amplitude response was $2.2D$, a remarkable increase of $270\,\%$ over the non-rotating case. Notably, the results showed that the amplitude responses at moderate Reynolds numbers ($Re=UD/\unicode[STIX]{x1D708}=2060$ and $2940$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid) in the present study showed significant differences from those of a previous low-Reynolds-number ($Re=350$) numerical study at similar reduced velocities by Du & Sun (Phys. Fluids, vol. 14 (8), 2015, pp. 2767–2777). Remarkably, in an additional study examining the cylinder vibration as a function of $U^{\ast }$ while the fixed forcing rotary oscillation parameters were kept constant at $(f_{r}^{\ast },\unicode[STIX]{x1D6FC}_{r}^{\ast })=(1.0,1.0)$, the cylinder experienced substantially larger oscillations than in the non-rotating case, and a rotation-induced galloping response was observed for $U^{\ast }>12$, where the amplitude increased monotonically to reach approximately $3.0D$ at the highest reduced velocity ($U^{\ast }=20$) tested. Furthermore, new wake modes were identified in the RLO and TLO regions using particle image velocimetry measurements at selected points in the $f_{r}^{\ast }-\unicode[STIX]{x1D6FC}_{r}^{\ast }$ parameter space.

Pressure-driven laminar and turbulent flow in a horizontal partially filled pipe was investigated using stereoscopic particle imaging velocimetry (S-PIV) in the cross-stream plane. Laminar flow velocity measurements are in excellent agreement with a recent theoretical solution in the literature. For turbulent flow, the flow depth was varied independently of a nominally constant Reynolds number (based on hydraulic diameter, $D_{H}$; bulk velocity, $U_{b}$ and kinematic viscosity $\unicode[STIX]{x1D708}$) of $Re_{H}=U_{b}D_{H}/\unicode[STIX]{x1D708}\approx 30\,000\pm 5\,\%$. When running partially full, the inferred friction factor is no longer a simple function of Reynolds number, but also depends on the Froude number $Fr=U_{b}/\sqrt{gD_{m}}$ where $g$ is gravitational acceleration and $D_{m}$ is hydraulic mean depth. S-PIV measurements in turbulent flow reveal the presence of secondary currents which causes the maximum streamwise velocity to occur below the free surface consistent with results reported in the literature for rectangular cross-section open channel flows. Unlike square duct and rectangular open channel flow the mean secondary motion observed here manifests only as a single pair of vortices mirrored about the vertical bisector and these rollers, which fill the half-width of the pipe, remain at a constant distance from the free surface even with decreasing flow depth for the range of depths tested. Spatial distributions of streamwise Reynolds normal stress and turbulent kinetic energy exhibit preferential arrangement rather than having the same profile around the azimuth of the pipe as in a full pipe flow. Instantaneous fields reveal the signatures of elements of canonical wall-bounded turbulent flows near the pipe wall such as large-scale and very-large-scale motions and associated hairpin packets whilst near the free surface, the signatures of free surface turbulence in the absence of imposed mean shear such as ‘upwellings’, ‘downdrafts’ and ‘whirlpools’ are present. Two-point spatio-temporal correlations of streamwise velocity fluctuation suggest that the large-scale coherent motions present in full pipe flow persist in partially filled pipes but are compressed and distorted by the presence of the free surface and mean secondary motion.

Finite-amplitude manifestations of stratified shear flow instabilities and their spatio-temporal coherent structures are believed to play an important role in turbulent geophysical flows. Such shear flows commonly have layers separated by sharp density interfaces, and are therefore susceptible to the so-called Holmboe instability, and its finite-amplitude manifestation, the Holmboe wave. In this paper, we describe and elucidate the origin of an apparently previously unreported long-lived coherent structure in a sustained stratified shear flow generated in the laboratory by exchange flow through an inclined square duct connecting two reservoirs filled with fluids of different densities. Using a novel measurement technique allowing for time-resolved, near-instantaneous measurements of the three-component velocity and density fields simultaneously over a three-dimensional volume, we describe the three-dimensional geometry and spatio-temporal dynamics of this structure. We identify it as a finite-amplitude, nonlinear, asymmetric confined Holmboe wave (CHW), and highlight the importance of its spanwise (lateral) confinement by the duct boundaries. We pay particular attention to the spanwise vorticity, which exhibits a travelling, near-periodic structure of sheared, distorted, prolate spheroids with a wide ‘body’ and a narrower ‘head’. Using temporal linear stability analysis on the two-dimensional streamwise-averaged experimental flow, we solve for three-dimensional perturbations having two-dimensional, cross-sectionally confined eigenfunctions and a streamwise normal mode. We show that the dispersion relation and the three-dimensional spatial structure of the fastest-growing confined Holmboe instability are in good agreement with those of the observed confined Holmboe wave. We also compare those results with a classical linear analysis of two-dimensional perturbations (i.e. with no spanwise dependence) on a one-dimensional base flow. We conclude that the lateral confinement is an important ingredient of the confined Holmboe instability, which gives rise to the CHW, with implications for many inherently confined geophysical flows such as in valleys, estuaries, straits or deep ocean trenches. Our results suggest that the CHW is an example of an experimentally observed, inherently nonlinear, robust, long-lived coherent structure which has developed from a linear instability. We conjecture that the CHW is a promising candidate for a class of exact coherent states underpinning the dynamics of more disordered, yet continually forced stratified shear flows.

We propose a fully spatio-temporal approach for identifying spatially varying modes of oscillation in fluid dynamics simulation output by means of multitaper frequency–wavenumber spectral analysis. One-dimensional spectrum estimation has proven to be a valuable tool in the analysis of turbulence data applied spatially to determine the rate of energy transport between spatial scales, or temporally to determine frequencies of oscillatory flows. It also allows for the quantitative comparison of flow characteristics between two scenarios using a standard basis. It has the limitation, however, that it neglects coupling between spatial and temporal structures. Two-dimensional frequency–wavenumber spectral analysis allows one to decompose waveforms into standing or travelling variety. The extended higher-dimensional multitaper method proposed here is shown to have improved statistical properties over conventional non-parametric spectral estimators, and is accompanied by confidence intervals which estimate their uncertainty. Multitaper frequency–wavenumber analysis is applied to a canonical benchmark problem, namely, a direct numerical simulation of von Kármán vortex shedding off a square wall-mounted cylinder with two inflow scenarios with matching momentum-thickness Reynolds numbers $Re_{\unicode[STIX]{x1D703}}\approx 1000$ at the obstacle. Frequency–wavenumber analysis of a two-dimensional section of these data reveals that although both the laminar and turbulent inflow scenarios show a turbulent $-5/3$ cascade in wavenumber ($\unicode[STIX]{x1D708}$) and frequency ($f$), the flow characteristics differ in that there is a significantly more prominent discrete harmonic oscillation near $(f,\unicode[STIX]{x1D708})=(0.2,0.21)$ in wavenumber and frequency in the laminar inflow scenario than the turbulent scenario. This frequency–wavenumber pair corresponds to a travelling wave with velocity near one near the centre path of the vortex street.

Trailing-edge serrations are add ons retrofitted to wind-turbine blades to mitigate turbulent boundary-layer trailing-edge noise. This manuscript studies the physical mechanisms behind the noise reduction by investigating the far-field noise and the hydrodynamic flow field. A conventional sawtooth and a combed-sawtooth trailing-edge serration are studied. Combed-sawtooth serrations are obtained by filling the empty space between the teeth with combs (i.e. solid filaments). Both serration geometries are retrofitted to a NACA 0018 aerofoil at zero degree angle of attack. Computations are carried out by solving the explicit, transient, compressible lattice Boltzmann equation, while the acoustic far field is obtained by means of the Ffowcs Williams and Hawkings analogy. The numerical results are validated against experiments. It is confirmed that the combed-sawtooth serrations reduce noise more than the conventional sawtooth ones for the low- and mid-frequency range. It is found that the presence of combs affects the intensity of the scattered noise but not the frequency range of noise reduction. For both configurations, the intensity of the surface pressure fluctuations decreases from the root to the tip, and noise sources are mainly located at the serrations root for the low- and mid-frequency range. The presence of the filaments generates a more uniform distribution of the noise sources along the edges with respect to the conventional serration. The installation of combs mitigates the interaction between the two sides of the aerofoil at the trailing edge and the generation of a turbulent wake in the empty space between teeth. As a result, the inward (i.e. from the serration edge to the centreline) and outward (i.e. from the serration centreline to the edge) flow motions, due to the presence of the teeth, are mitigated. It is found that the installation of serrations affects the surface pressure fluctuations integral parameters. Both the spanwise correlation length and convective velocity of the surface pressure fluctuations increase with respect to the baseline straight configuration. When both quantities are similar to the one obtained for the straight trailing edge, the effect of the slanted edge is negligible, thus corresponding to no noise reduction. It is concluded that the changes in sound radiation are mainly caused by destructive interference of the radiated sound waves for which a larger spanwise correlation length is beneficial. Finally, the difference between measurements and the literature is caused by an incorrect modelling of the spanwise correlation length, which shows a different decay rate with respect to the one obtained for a straight trailing edge.