The incompressible two-dimensional Euler equations on a sphere constitute a fundamental model in hydrodynamics. The long-time behaviour of solutions is largely unknown; statistical mechanics predicts a steady vorticity configuration, but detailed numerical results in the literature contradict this theory, yielding instead persistent unsteadiness. Such numerical results were obtained using artificial hyperviscosity to account for the cascade of enstrophy into smaller scales. Hyperviscosity, however, destroys the underlying geometry of the phase flow (such as conservation of Casimir functions), and therefore might affect the qualitative long-time behaviour. Here, we develop an efficient numerical method for long-time simulations that preserve the geometric features of the exact flow, in particular conservation of Casimirs. Long-time simulations on a non-rotating sphere then reveal three possible outcomes for generic initial conditions: the formation of either 2, 3 or 4 coherent vortex structures. These numerical results contradict the statistical mechanics theory and show that previous numerical results, suggesting 4 coherent vortex structures as the generic behaviour, display only a special case. Through integrability theory for point vortex dynamics on the sphere we present a theoretical model which describes the mechanism by which the three observed regimes appear. We show that there is a correlation between a first integral $\unicode[STIX]{x1D6FE}$ (the ratio of total angular momentum and the square root of enstrophy) and the long-time behaviour: $\unicode[STIX]{x1D6FE}$ small, intermediate and large yields most likely 4, 3 or 2 coherent vortex formations. Our findings thus suggest that the likely long-time behaviour can be predicted from the first integral $\unicode[STIX]{x1D6FE}$.

The surface texture of materials plays a critical role in wettability, turbulence and transport phenomena. In order to design surfaces for these applications, it is desirable to characterise non-smooth and porous materials by their ability to exchange mass and momentum with flowing fluids. While the underlying physics of the tangential (slip) velocity at a fluid–solid interface is well understood, the importance and treatment of normal (transpiration) velocity and normal stress is unclear. We show that, when the slip velocity varies at an interface above the texture, a non-zero transpiration velocity arises from mass conservation. The ability of a given surface texture to accommodate a normal velocity of this kind is quantified by a transpiration length. We further demonstrate that normal momentum transfer gives rise to a pressure jump. For a porous material, the pressure jump can be characterised by so-called resistance coefficients. By solving five Stokes problems, the introduced measures of slip, transpiration and resistance can be determined for any anisotropic non-smooth surface consisting of regularly repeating geometric patterns. The proposed conditions are a subset of the effective boundary conditions derived from formal multi-scale expansion. We validate and demonstrate the physical significance of the effective conditions on two canonical problems – a lid-driven cavity and a turbulent channel flow, both with non-smooth bottom surfaces.

Inclined turbulent thermal convection in liquid sodium is studied at large Rayleigh numbers $Ra\gtrsim 10^{7}$ based on the results of both experimental measurements and high-resolution numerical simulations. For a direct comparison, the considered system parameters are set to be similar: $Ra=1.67\times 10^{7}$ in the direct numerical simulations (DNS), $Ra=1.5\times 10^{7}$ in the large-eddy simulations and $Ra=1.42\times 10^{7}$ in the experiments, while the Prandtl number of liquid sodium is very small ($Pr\approx 0.009$). The cylindrical convection cell has an aspect ratio of one; one circular surface is heated, while the other one is cooled. Additionally, the cylinder is inclined with respect to gravity and the inclination angle varies from $\unicode[STIX]{x1D6FD}=0^{\circ }$, which corresponds to Rayleigh–Bénard convection (RBC), to $\unicode[STIX]{x1D6FD}=90^{\circ }$, as in a vertical convection (VC) set-up. Our study demonstrates quantitative agreement of the experimental and numerical results, in particular with respect to the global heat and momentum transport, temperature and velocity profiles, as well as the dynamics of the large-scale circulation (LSC). The DNS reveal that the twisting and sloshing of the LSC at small inclination angles periodically affects the instantaneous heat transport (up to $\pm 44\,\%$ of the mean heat transport). The twisted LSC is associated with a weak heat transport, while the sloshing mode that brings together the hot and cold streams of the LSC is associated with a strong heat transport. The experiments show that the heat transport scales as $Nu\sim Ra^{0.22}$ in both limiting cases (RBC and VC) for Rayleigh numbers around $Ra\approx 10^{7}$, while any inclination of the cell, $0<\unicode[STIX]{x1D6FD}\leqslant 90^{\circ }$, leads to an increase of $Nu$.

We investigate experimentally the turbulent flow through a two-dimensional contraction. Using a water tunnel with an active grid we generate turbulence at Taylor microscale Reynolds number $Re_{\unicode[STIX]{x1D706}}\sim 250$ which is advected through a 2.5 : 1 contraction. Volumetric and time-resolved tomographic particle image velocimetry and shake-the-box velocity measurements are used to characterize the evolution of coherent vortical structures at three streamwise locations upstream of and within the contraction. We confirm the conceptual picture of coherent large-scale vortices being stretched and aligned with the mean rate of strain. This alignment of the vortices with the tunnel centreline is stronger compared to the alignment of vorticity with the large-scale strain observed in numerical simulations of homogeneous turbulence. We judge this by the peak probability magnitudes of these alignments. This result is robust and independent of the grid-rotation protocols. On the other hand, while the pointwise vorticity vector also, to a lesser extent, aligns with the mean strain, it principally remains aligned with the intermediate eigenvector of the local instantaneous strain-rate tensor, as is known in other turbulent flows. These results persist when the distance from the grid to the entrance of the contraction is doubled, showing that modest transverse inhomogeneities do not significantly affect these vortical-orientation results.

The physical mechanism governing the onset of transonic shock buffet on swept wings remains elusive, with no unequivocal description forthcoming despite over half a century of research. This paper elucidates the fundamental flow physics on a civil aircraft wing using an extensive experimental database from a transonic wind tunnel facility. The analysis covers a wide range of flow conditions at a Reynolds number of around $3.6\times 10^{6}$. Data at pre-buffet conditions and beyond onset are assessed for Mach numbers between 0.70 and 0.84. Critically, unsteady surface pressure data of high spatial and temporal resolution acquired by dynamic pressure-sensitive paint is analysed, in addition to conventional data from pressure transducers and a root strain gauge. We identify two distinct phenomena in shock buffet conditions. First, we highlight a low-frequency shock unsteadiness for Strouhal numbers between 0.05 and 0.15, based on mean aerodynamic chord and reference free stream velocity. This has a characteristic wavelength of approximately 0.8 semi-span lengths (equivalent to three mean aerodynamic chords). Such shock unsteadiness is already observed at low-incidence conditions, below the buffet onset defined by traditional indicators. This has the effect of propagating disturbances predominantly in the inboard direction, depending on localised separation, with a dimensionless convection speed of approximately 0.26 for a Strouhal number of 0.09. Second, we describe a broadband higher-frequency behaviour for Strouhal numbers between 0.2 and 0.5 with a wavelength of 0.2 to 0.3 semi-span lengths (0.6 to 1.2 mean aerodynamic chords). This outboard propagation is confined to the tip region, similar to previously reported buffet cells believed to constitute the shock buffet instability on conventional swept wings. Interestingly, a dimensionless outboard convection speed of approximately 0.26, coinciding with the low-frequency shock unsteadiness, is found to be nearly independent of frequency. We characterise these coexisting phenomena by use of signal processing tools and modal analysis of the dynamic pressure-sensitive paint data, specifically proper orthogonal and dynamic mode decomposition. The results are scrutinised within the context of a broader research effort, including numerical simulation, and viewed alongside other experiments. We anticipate our findings will help to clarify experimental and numerical observations in edge-of-the-envelope conditions and to ultimately inform buffet-control strategies.

In this study we experimentally investigate bubbly drag reduction in a highly turbulent flow of water with dispersed air at $5.0\times 10^{5}\leqslant Re\leqslant 1.7\times 10^{6}$ over a non-wetting surface containing micro-scale roughness. To do so, the Taylor–Couette geometry is used, allowing for both accurate global drag and local flow measurements. The inner cylinder – coated with a rough, hydrophobic material – is rotating, whereas the smooth outer cylinder is kept stationary. The crucial control parameter is the air volume fraction $\unicode[STIX]{x1D6FC}$ present in the working fluid. For small volume fractions ($\unicode[STIX]{x1D6FC}<4\,\%$), we observe that the surface roughness from the coating increases the drag. For large volume fractions of air ($\unicode[STIX]{x1D6FC}\geqslant 4\,\%$), the drag decreases compared to the case with both the inner and outer cylinders uncoated, i.e. smooth and hydrophilic, using the same volume fraction of air. This suggests that two competing mechanisms are at play: on the one hand, the roughness invokes an extension of the log layer – resulting in an increase in drag – and, on the other hand, there is a drag-reducing mechanism of the hydrophobic surface interacting with the bubbly liquid. The balance between these two effects determines whether there is overall drag reduction or drag enhancement. For further increased bubble concentration $\unicode[STIX]{x1D6FC}=6\,\%$ we find a saturation of the drag reduction effect. Our study gives guidelines for industrial applications of bubbly drag reduction in hydrophobic wall-bounded turbulent flows.

Experiments to study the effect of edge undulation on vortex formation have been conducted on impulsively accelerated plates. Abstractions of propulsors found in nature are produced by imprinting undulatory features with varying wavelengths onto the circumferential vortex-forming edge of circular plates. The effects of the small-scale disturbances introduced by these modifications are accessed by means of force measurements and time-resolved particle image velocimetry. Investigations of four different geometries at two different Reynolds numbers reveal an insensitivity of the flow towards length scales smaller than or similar to the thickness of the feeding shear layer. However, the instabilities in the shear layer and the coherence of the vortex wake are influenced when the wavelength of the undulation exceeds the shear-layer thickness by a significant margin. This results in a force augmentation due to enhanced entrainment into the turbulent vortex core, and thus an associated faster vortex growth rate. Yet, contrary to prior expectations, the time of vortex pinch-off remains constant for all edge modifications. The cause–effect relationship behind the stability of the vortex wake is further investigated. While for small edge undulations a turbulent transition of the vortex core results in vortex pinch-off, for larger edge undulations the turbulent vortex core is found to be fed constantly with additional circulation from the shear layer.

We present a direct comparison between interface-resolved and one-way-coupled point-particle direct numerical simulations (DNS) of gravity-free turbulent channel flow laden with small inertial particles, with high particle-to-fluid density ratio and diameter of approximately three viscous units. The most dilute flow considered, solid volume fraction $O(10^{-5})$, shows the particle feedback on the flow to be negligible, whereas differences with respect to the unladen case, notably a drag increase of approximately 10 %, are found for a volume fraction $O(10^{-4})$. This is attributed to a dense layer of particles at the wall, caused by turbophoresis, flowing with large particle-to-fluid apparent slip velocity. The most dilute case is therefore taken as the benchmark for assessing the validity of a widely used point-particle model, where the particle dynamics results only from inertial and nonlinear drag forces. In the bulk of the channel, the first- and second-order moments of the particle velocity from the point-particle DNS agree well with those from the interface-resolved DNS. Close to the wall, however, most of the statistics show major qualitative differences. We show that this difference originates from the strong shear-induced lift force acting on the particles in the near-wall region. This mechanism is well captured by the lift force model due to Saffman (J. Fluid Mech., vol. 22 (2), 1965, pp. 385–400), while other widely used, more elaborate, approaches aiming at extending the lift model for a wider range of particle Reynolds numbers can actually underpredict the magnitude of the near-wall particle velocity fluctuations for the cases analysed here.

We study the formation of fine radial jets during the impact of a compound drop on a smooth solid surface. The disperse-phase droplets are heavier than the outer continuous phase of the main drop and sink to the bottom of the drop before it is released from the nozzle. The droplets often arrange into a regular pattern around the axis of symmetry. This configuration produces narrow high-speed jets aligned with every internal droplet. These radial jets form during the early impulsive phase of the impact, by local focusing of the outer liquid, which is forced into the narrowing wedge under each internal droplet. The pressure-driven flow forces a thin sheet under and around each droplet, which levitates and separates from the solid surface. Subsequently, surface tension re-forms this horizontal sheet into a cylindrical jet, which is typically as narrow as ${\sim}35~\unicode[STIX]{x03BC}\text{m}$, while smaller droplets can produce even thinner jets. We systematically change the number of inner droplets and the properties of the main drop to identify the jetting threshold. The jet speed and thickness are minimally affected by the viscosity of the outer liquid, suggesting pure inertial focusing. The jets emerge at around eight times the drop impact velocity. Jetting stops when the density of the inner droplets approaches that of the continuous phase. The interior droplets are often greatly deformed and broken up into satellites by the outer viscous stretching, through capillary pinch-off or tip streaming.

For the problem of horizontal convection the Nusselt number based on entropy production is bounded from above by $C\,Ra^{1/3}$ as the horizontal convective Rayleigh number $Ra\rightarrow \infty$ for some constant $C$ (Siggers et al., J. Fluid Mech., vol. 517, 2004, pp. 55–70). We re-examine the variational arguments leading to this ‘ultimate regime’ by using the Wentzel–Kramers–Brillouin method to solve the variational problem in the $Ra\rightarrow \infty$ limit and exhibiting solutions that achieve the ultimate $Ra^{1/3}$ scaling. As expected, the optimizing flows have a boundary layer of thickness ${\sim}Ra^{-1/3}$ pressed against the non-uniformly heated surface; but the variational solutions also have rapid oscillatory variation with wavelength ${\sim}Ra^{-1/3}$ along the wall. As a result of the exact solution of the variational problem, the constant $C$ is smaller than the previous estimate by a factor of $2.5$ for no-slip and $1.6$ for no-stress boundary conditions. This modest reduction in $C$ indicates that the inequalities used by Siggers et al. (J. Fluid Mech., vol. 517, 2004, pp. 55–70) are surprisingly accurate.

Hot liquid metal drops impacting onto a cold substrate solidify during their subsequent spreading. Here we experimentally study the influence of solidification on the outcome of an impact event. Liquid tin drops are impacted onto sapphire substrates of varying temperature. The impact is visualised both from the side and from below, which provides a unique view on the solidification process. During spreading, an intriguing pattern of radial ligaments rapidly solidifies from the centre of the drop. This pattern determines the late-time morphology of the splat. A quantitative analysis of the drop spreading and ligament formation is supported by scaling arguments. Finally, a phase diagram for drop bouncing, deposition and splashing as a function of substrate temperature and impact velocity is provided.

This work deals with the closed-loop control of streaky structures induced by free-stream turbulence (FST), at the levels of 3.0 % and 3.5 %, in a zero-pressure-gradient transitional boundary layer, by means of localized sensors and actuators. A linear quadratic Gaussian regulator is considered along with a system identification technique to build reduced-order models for control. Three actuators are developed with different spatial supports, corresponding to a baseline shape with only vertical forcing, and to two other shapes obtained by different optimization procedures. A computationally efficient method is derived to obtain an actuator that aims to induce the exact structures that are inside the boundary layer, given in terms of their first spectral proper orthogonal decomposition (SPOD) mode, and an actuator that maximizes the energy of induced downstream structures. All three actuators lead to significant delays in the transition to turbulence and were shown to be robust to mild variations in the FST levels. Integrated total drag reductions observed were up to 21 % and 19 % for turbulence intensity levels of 3.0 % and 3.5 %, respectively, depending on the considered actuator. Differences are understood in terms of the SPOD of actuation and FST-induced fields along with the causality of the control scheme when a cancellation of disturbances is considered along the wall-normal direction. The actuator optimized to generate the leading downstream SPOD mode, representing the streaks in the open-loop flow, leads to the highest transition delay, which can be understood due to its capability of closely cancelling structures in the boundary layer.

The current work presents a realizable method to control streaky disturbances in boundary layer flows and delay transition to turbulence by means of active flow control. Numerical simulations of the nonlinear transitional regime in a Blasius boundary layer are performed where streaks are excited in the boundary layer by means of a high level of free-stream turbulence. The occurring disturbances are measured by means of localized wall-shear-stress sensors and damped out using near-wall actuators, which resemble ring plasma actuators. Each actuator is powered by a time-varying signal whose amplitude is computed by processing signals from the sensors. The processed signal is the result of two control laws: the linear quadratic Gaussian regulator (LQG) and the inverse feed-forward control technique (IFFC). The use of the first control method, LQG, requires a state-space representation of the system dynamics, so the flow is described by means of a linear time-invariant operator that captures only the most relevant information of the dynamics and results in a reduced-order model (ROM). The ROM is computed by means of the eigensystem realization algorithm (ERA), which is based on the impulse responses of the real system. Collecting such impulse responses may be unfeasible when considering free-stream turbulence because of the high dimensionality of the input forcing needed for a precise description of such a phenomenon. Here, a new method to identify the relevant system dynamics and generate the needed impulse responses is proposed, based on additional shear-stress measurements in an upstream location. Transfer functions between such measurements and other downstream sensors are obtained and allow the derivation of the ERA system, in a data-driven approach that would be realizable in experiments. Finally, in order to discuss the advantages of the LQG based on the ROM and analyse its performance, the implemented LQG is compared to the IFFC, which consists of wave cancellation. The work (i) presents a systematic and straightforward way to deal with high-dimensional disturbances in order to build ROMs for a feasible control technique, and (ii) shows that even when considering practical constraints, such as the type and size of actuators and sensors, it is possible to achieve at least as large delay of bypass transition as that obtained in more idealized cases found in the literature.

The characteristics of turbulent boundary layers (TBLs) subjected to adverse pressure gradients are analysed through well-resolved large-eddy simulations. The geometries under study are the NACA0012 and NACA4412 wing sections, at $0^{\circ }$ and $5^{\circ }$ angle of attack, respectively, both of them at a Reynolds number based on inflow velocity and chord length of $Re_{c}=400\,000$. The turbulence statistics show that adverse pressure gradients (APGs) have a significant effect on the mean velocity, velocity fluctuations and turbulent kinetic energy budget, and this effect is more prominent on the outer region of the boundary layer. Furthermore, the effect of flow history is assessed by means of an integrated Clauser pressure-gradient parameter $\overline{\unicode[STIX]{x1D6FD}}$ (Vinuesa et al., Flow Turbul. Combust., vol. 99, 2017, pp. 565–587), through the study of cases with matching local values of $\unicode[STIX]{x1D6FD}$ and the friction Reynolds number $Re_{\unicode[STIX]{x1D70F}}$ to isolate this effect. Our results show a noticeable effect of the flow history on the outer region, however the differences in the near-wall peak of the tangential velocity fluctuations appear to be mostly produced by the local APG magnitude. The one-dimensional power-spectral density shows energetic small scales in the outer region of APG TBLs, whereas these energetic scales do not appear in zero-pressure-gradient (ZPG) TBLs, suggesting that small scales near the wall are advected towards the outer layer by the APG. Moreover, the linear coherence spectra show that the spectral outer peak of high-Reynolds-number ZPG TBLs is highly correlated with the near-wall region (Baars et al., J. Fluid Mech., vol. 823, 2017, R2), unlike APG TBLs which do not show such a correlation. This result, together with the different two-dimensional spectra of APG and high-Reynolds-number ZPG TBLs, suggests different energisation mechanisms due to APG and increase in Reynolds number. To the authors’ knowledge, this is the first in-depth analysis of the TBL characteristics over wings, including detailed single-point statistics, spectra and coherence.

Highly turbulent Taylor–Couette flow with spanwise-varying roughness is investigated experimentally and numerically (direct numerical simulations with an immersed boundary method) to determine the effects of the spacing and spanwise width $s$ of the spanwise-varying roughness on the total drag and on the flow structures. We apply sandgrain roughness, in the form of alternating rough and smooth bands to the inner cylinder. Numerically, the Taylor number is $O(10^{9})$ and the roughness width is varied in the range $0.47\leqslant \tilde{s}=s/d\leqslant 1.23$, where $d$ is the gap width. Experimentally, we explore $Ta=O(10^{12})$ and $0.61\leqslant \tilde{s}\leqslant 3.74$. For both approaches the radius ratio is fixed at $\unicode[STIX]{x1D702}=r_{i}/r_{o}=0.716$, with $r_{i}$ and $r_{o}$ the radius of the inner and outer cylinder respectively. We present how the global transport properties and the local flow structures depend on the boundary conditions set by the roughness spacing $\tilde{s}$. Both numerically and experimentally, we find a maximum in the angular momentum transport as a function of $\tilde{s}$. This can be attributed to the re-arrangement of the large-scale structures triggered by the presence of the rough stripes, leading to correspondingly large-scale turbulent vortices.

Data from direct numerical simulations (DNS) of disperse bubbly flow in an upward vertical channel are used to develop a new second-moment closure for bubble-induced turbulence (BIT) in the Euler–Euler framework. The closure is an extension of a BIT model originally proposed by Ma et al. (Phys. Rev. Fluids, vol. 2, 2017, 034301) for two-equation eddy-viscosity models and focuses on the core region of the channel, where the interfacial term and dissipation term are in balance. Particular attention in this study is given to the treatment of the pressure–strain term for bubbly flows and the form of the interfacial term to account for BIT. For the latter, the concept of an effective BIT source is proposed, which leads to a significant simplification of the modelling work for both the pressure–strain correlation and the interfacial term itself. The anisotropy of bubbly flow is analysed with the aid of the anisotropy-invariant map obtained from the DNS data, and a parameter governing this issue is established. The complete second-moment closure is tested against reference data for different bubbly channel flows and a case of a bubble column. The agreement achieved with the DNS data is very good and the performance of the new model is better than obtained with the standard procedure. Furthermore, the model is shown to be robust and to fulfil the requirements of realizability.

The transition to turbulence in pipe flow proceeds through several distinct stages, eventually producing aggressively expanding regions of fluctuations, ‘slugs’, surrounded by laminar flow. By examining mean-velocity profiles, fluctuating-velocity profiles and Reynolds stress profiles, the seminal study of Wygnanski & Champagne (J. Fluid Mech., vol. 59 (2), 1973, 281–335) concluded that the flow inside slugs is ‘identical’ to fully turbulent flow. Although this conclusion is widely accepted, upon closer examination of their analysis, we find that their data cannot be used to substantiate this conclusion. We resolve this conflict via new experiments and simulations wherein we pair slugs and fully turbulent flow at the same value of Reynolds number ($Re$). We conclude that the flow inside a slug is indeed indistinguishable from a fully turbulent flow but only when the two flows share the same value of $Re$. Our work highlights the rich $Re$-dependence of transitional pipe flows.

This paper presents a theoretical and experimental study of the long-standing fluid mechanics problem involving the temporal resolution of a large localised initial disturbance into a sequence of solitary waves. This problem is of fundamental importance in a range of applications, including tsunami and internal ocean wave modelling. This study is performed in the context of the viscous fluid conduit system – the driven, cylindrical, free interface between two miscible Stokes fluids with high viscosity contrast. Owing to buoyancy-induced nonlinear self-steepening balanced by stress-induced interfacial dispersion, the disturbance evolves into a slowly modulated wavetrain and further into a sequence of solitary waves. An extension of Whitham modulation theory, termed the solitary wave resolution method, is used to resolve the fission of an initial disturbance into solitary waves. The developed theory predicts the relationship between the initial disturbance’s profile, the number of emergent solitary waves and their amplitude distribution, quantifying an extension of the well-known soliton resolution conjecture from integrable systems to non-integrable systems that often provide a more accurate modelling of physical systems. The theoretical predictions for the fluid conduit system are confirmed both numerically and experimentally. The number of observed solitary waves is consistently within one to two waves of the prediction, and the amplitude distribution shows remarkable agreement. Universal properties of solitary wave fission in other fluid dynamics problems are identified.

The present experimental study addresses the flow of a yield stress fluid with some elasticity (Carbopol gel) in a square duct. The behaviour of two fluids with lower and higher yield stress is investigated in terms of the friction factor and flow velocities at multiple Reynolds numbers $Re^{\ast }\in$ (1, 200) and, hence, Bingham numbers $Bi\in$ (0.01, 0.35). Taking advantage of the symmetry planes in a square duct, we reconstruct the entire 3-component velocity field from two-dimensional particle image velocimetry (PIV). A secondary flow consisting of eight vortices is observed to recirculate the fluid from the core towards the wall centre and from the corners back to the core. The extent and intensity of these vortices grows with increasing $Re^{\ast }$ or, alternately, as the plug size decreases. The second objective of this study is to explore the change in flow in the presence of particles. To this end, almost neutrally buoyant finite-size spherical particles with a duct height, $2H$, to particle diameter, $d_{p}$, ratio of 12 are used at two volume fractions $\unicode[STIX]{x1D719}=5$ and 10 %. Particle tracking velocimetry is used to measure the velocity of these refractive-index-matched spheres in the clear Carbopol gel, and PIV to extract the fluid velocity. Additionally, simple shadowgraphy is also used to qualitatively visualise the development of the particle distribution along the streamwise direction. The particle distribution pattern changes from being concentrated at the four corners, at low flow rates, to being focussed along a diffused ring between the centre and the corners, at high flow rates. The presence of particles induces streamwise and wall-normal velocity fluctuations in the fluid phase; however, the primary Reynolds shear stress is still very small compared to turbulent flows. The size of the plug in the particle-laden cases appears to be smaller than the corresponding single-phase cases. Similar to Newtonian fluids, the friction factor increases due to the presence of particles, almost independently of the suspending fluid matrix. Interestingly, predictions based on an increased effective suspension viscosity agrees quite well with the experimental friction factor for the concentrations used in this study.

This work examines the complex flow field in a bistable turbulent swirl flame, where the flame alternates irregularly between a lifted-off M-shape and an attached V-shape. The flow field consists of various types of intermittent dynamics due to flame-shape switching that occur on different time scales. In order to properly identify, separate and temporally resolve these dynamic components, a novel method of multiresolution proper orthogonal decomposition (MRPOD) is developed by interfacing the maximum overlap discrete wavelet packet transform (MODWPT) of multidimensional data series with the conventional snapshot POD. Specific care is taken to select the wavelet filter, decomposition level and reconstruction bandwidth to achieve variable spectral bandpasses and adequate temporal resolutions. When applied to data series from high-speed three-component velocity field measurements in the bistable swirl flame, MRPOD is capable of isolating frequency components that are usually lumped into a single POD mode, with enhanced spatial/temporal coherence for even weak and highly intermittent dynamics. Owing to the improved spectral purity, a series of previously unknown dynamics is uncovered alongside well-characterised instabilities such as the precessing vortex core (PVC) and thermoacoustic (TA) instabilities. In particular, a non-periodic switch mode is found to couple with the previously identified shift mode only during flame-shape transitions, with pronounced modifications on the backflow and near the inlet of the combustor, a region known to influence the growth rate of PVC. TA oscillations are seen to drive repetitive flame reattachment during M–V transitions that eventually settles into a V-flame. But sustained high TA amplitudes alone do not appear to necessarily presage the onset of such a transition. Additional higher-order harmonics of PVC and evidence of TA modulations on PVC dynamics are uncovered, which also exhibit bi-modal behaviours: while maintaining their characteristic frequencies, these instabilities could attain either a single or double helical structure and are only active during either V- or M-flame periods.