The interaction of an oblique shock wave and a laminar boundary layer developing over a flat plate is investigated by means of numerical simulation and global linear-stability analysis. Under the selected flow conditions (free-stream Mach numbers, Reynolds numbers and shock-wave angles), the incoming boundary layer undergoes separation due to the adverse pressure gradient. For a wide range of flow parameters, the oblique shock wave/boundary-layer interaction (OSWBLI) is seen to be globally stable. We show that the onset of two-dimensional large-scale structures is generated by selective noise amplification that is described for each frequency, in a linear framework, by wave-packet trains composed of several global modes. A detailed analysis of both the eigenspectrum and eigenfunctions gives some insight into the relationship between spatial scales (shape and localization) and frequencies. In particular, OSWBLI exhibits a universal behaviour. The lowest frequencies correspond to structures mainly located near the separated shock that emit radiation in the form of Mach waves and are scaled by the interaction length. The medium frequencies are associated with structures mainly localized in the shear layer and are scaled by the displacement thickness at the impact. The linear process by which OSWBLI selects frequencies is analysed by means of the global resolvent. It shows that unsteadiness are mainly associated with instabilities arising from the shear layer. For the lower frequency range, there is no particular selectivity in a linear framework. Two-dimensional numerical simulations show that the linear behaviour is modified for moderate forcing amplitudes by nonlinear mechanisms leading to a significant amplification of low frequencies. Finally, based on the present results, we draw some hypotheses concerning the onset of unsteadiness observed in shock wave/turbulent boundary-layer interactions.

We study the viscous spatial linear stability characteristics of the time-averaged flow in turbulent subsonic jets issuing from serrated (chevroned) nozzles, and compare them to analogous round jet results. Linear parabolized stability equations (PSE) are used in the calculations to account for the non-parallel base flow. By exploiting the symmetries of the mean flow due to the regular arrangement of serrations, we obtain a series of coupled two-dimensional PSE problems from the original three-dimensional problem. This reduces the solution cost and manifests the symmetries of the stability modes. In the parallel-flow linear stability theory (LST) calculations that are performed near the nozzle to initiate the PSE, we find that the serrated nozzle reduces the growth rates of the most unstable eigenmodes of the jet, but their phase speeds are approximately similar. We obtain encouraging validation of our linear PSE instability wave results vis-à-vis near-field hydrodynamic pressure data acquired on a phased microphone array in experiments, after filtering the latter with proper orthogonal decomposition (POD) to extract the energetically dominant coherent part. Additionally, a large-eddy simulation database of the same serrated jet is investigated, and its POD-filtered pressure field is found to compare favourably with the corresponding PSE solution within the jet plume. We conclude that the coherent hydrodynamic pressure fluctuations of jets from both round and serrated nozzles are reasonably consistent with the linear instability modes of the turbulent mean flow.

Internal hydraulic jumps in flows with upstream shear are investigated using two-layer shock-joining theories and numerical solutions of the Navier–Stokes equations. The role of upstream shear has not previously been thoroughly investigated, although it is important in many oceanographic situations, including exchange flows. The full solution spaces of several two-layer theories, distinguished by how dissipation is distributed between the layers, with upstream shear are found, and the physically allowable solution space is identified. These two-layer theories are then evaluated using more realistic numerical simulations that have continuous density and velocity profiles and permit turbulence and mixing. Two-dimensional numerical simulations show that none of the two-layer theories reliably predicts the relation between jump height and speed over the full range of allowable solutions. The numerical simulations also show that different qualitative types of jumps can occur, including undular bores, energy-conserving conjugate state transitions, smooth-front jumps with trailing turbulence and overturning turbulent jumps. Simulation results are used to investigate mixing, which increases with jump height and upstream shear. A few three-dimensional simulations results were undertaken and are in quantitative agreement with the two-dimensional simulations.

We use the results of a direct numerical simulation (DNS) with a particle-tracking model to investigate three-dimensional transport by breaking internal gravity waves on slopes. Onshore transport occurs within an upslope surge of dense fluid after breaking. Offshore transport occurs due to an intrusion of mixed fluid that propagates offshore and resembles an intermediate nepheloid layer (INL). Entrainment of particles into the INL is related to irreversible mixing of the density field during wave breaking. Maximum onshore and offshore transport are calculated as a function of initial particle position, and can be of the order of the initial wave length scale for particles initialized within the breaking region. An effective cross-shore dispersion coefficient is also calculated, and is roughly three orders of magnitude larger than the molecular diffusivity within the breaking region. Particles are transported laterally due to turbulence that develops during wave breaking, and this lateral spreading is quantified with a lateral turbulent diffusivity. Lateral turbulent diffusivity values calculated using particles are elevated by more than one order of magnitude above the molecular diffusivity, and are shown to agree well with turbulent diffusivities estimated using a generic length scale turbulence closure model. Based on a favourable comparison of DNS results with those of a similar two-dimensional case, we use two-dimensional simulations to extend our cross-shore transport results to additional wave amplitude and bathymetric slope conditions.

We conduct a series of large-eddy simulations (LES) to examine the mean flow behaviour within the roughness layer of turbulent boundary layer flow over rough surfaces. We consider several configurations consisting of arrays of rectangular-prism roughness elements with various spacings, aspect ratios and height distributions. The results provide clear evidence for exponential behaviour of the mean flow with respect to the wall normal distance. Such behaviour has been proposed before (see, e.g., Cionco, 1966 Tech. Rep. DTIC document), and is represented as $U(z)/U_{h}=\exp [a(z/h-1)]$, where $U(z)$ is the spatially/temporally averaged fluid velocity, $z$ is the wall normal distance, $h$ represents the height of the roughness elements and $U_{h}$ is the velocity at $z=h$. The attenuation factor $a$ depends on the density of the roughness element distribution and details of the roughness distribution on the wall. Once established, the generic velocity profile shape is used to formulate a fully analytical model for the effective drag exerted by turbulent flow on a surface covered with arrays of rectangular-prism roughness elements. The approach is based on the von Karman–Pohlhausen integral method, in which a shape function is assumed for the mean velocity profile and its parameters are determined based on momentum conservation and other fundamental constraints. In order to determine the attenuation parameter $a$, wake interactions among surface roughness elements are accounted for by using the concept of flow sheltering. The model transitions smoothly between ‘$k$’ and ‘$d$’ type roughness conditions depending on the surface coverage density and the detailed geometry of roughness elements. Comparisons between model predictions and experimental/numerical data from the existing literature as well as LES data from this study are presented. It is shown that the analytical model provides good predictions of mean velocity and drag forces for the cases considered, thus raising the hope that analytical roughness modelling based on surface geometry is possible, at least for cases when the location of flow separation over surface elements can be easily predicted, as in the case of wall-attached rectangular-prism roughness elements.

A model for predicting the behaviour of a compressible flow laden with shocks interacting with granular material has been developed and tested. The model consists of two sets of coupled Euler equations, one for the gas phase and the other for the granular phase. Drag, convective, heat transfer and non-conservative terms couple the two sets of governing equations. Intergranular stress acting on the grains is modelled using granular kinetic theory in dilute regimes where particle collisions are dominant and frictional–collisional pressure in dense regions where layers of granular material slide over one another. The two-phase granular–gaseous model, as a result, is valid from dilute to densely packed granular regimes. The solution of these nonlinearly coupled Euler equations is challenging due to the presence of the non-conservative nozzling and work terms. A numerical technique, based on Godunov’s method, was designed for solving these equations. This method takes advantage of particle incompressibility to simplify the nozzling terms. It also uses the observation that a Riemann problem is valid in the region where gas can flow between particles and can be used to provide a physically accurate approximation of the non-conservative terms. The model and solution method are verified by comparisons to test problems involving granular shocks and two-phase shock-tube problems, and they are validated against experimental measurements of shock and dense particle-curtain interactions and transmitted oblique granular shocks.

The reflection on a wavy wall of a planar shock propagating at Mach number 1.5 in air is simulated in a two-dimensional geometry by solving the fully compressible Navier–Stokes equations. A high-order spectral difference numerical discretization is used over an unstructured mesh composed of quadrilateral elements. The shock discontinuity is handled numerically through a specific treatment, which is limited in space to a small portion of the computational cell through which the shock is travelling. In the conditions under investigation, the reflection on the wavy wall leads to a weak and smooth deformation of the shock front without singularities just after reflection. Long-living triple points (Mach stems) are spontaneously formed on the reflected shock at a finite distance from the wavy wall. They then propagate on the front in both directions and collide regularly, forming a periodic cellular pattern quite similar to that of a cellular detonation. Transverse waves, issued from the triple points, are generated in the shocked gas. As a result of their mutual interaction, a complex and strongly unsteady flow is produced in the shocked gas. The topology of the instantaneous streamline patterns is characterized by short-lived critical points that appear intermittently. Due to the compressible character of the unsteady two-dimensional flow, the topology of critical points which can be observed is more diverse than would be expected for incompressible two-dimensional flows. Some of them take the form of short-lived sources or sinks. The mechanism of formation and the dynamics of the triple points, as well as the instantaneous streamline patterns, are analysed in the present paper. The results are useful for deciphering the cellular structure of unstable detonations.

Substantial discrepancy in the conditions for attainment of different closure modes of a ventilated supercavity has existed widely in the published literature. In this study, supercavity closure is investigated with an objective to understand the physical mechanisms determining closure formation and transition between different closure modes and to reconcile the observations from prior studies under various flow settings. The experiments are conducted in a closed-wall recirculating water tunnel to image ventilated supercavity closure using high speed and high-resolution photography and simultaneously measure pressure inside the cavity. The flow conditions are varied systematically to cover a broad range of water velocity, ventilation flow rate and cavitator size, which correspond to different Froude numbers, air entrainment coefficients and blockage ratios, respectively. In addition to the classical closure modes reported in the literature (e.g. re-entrant jet, twin vortex, quad vortex, etc.), the study has revealed a number of new closure modes that occur during the transition between classical modes, or under very specific flow conditions. Closure maps are constructed to depict the flow regimes, i.e. the range of Froude number and air entrainment coefficient, for various closure modes at different blockage ratios. From the closure map at each blockage ratio, a critical ventilation flow rate, below which the supercavity collapses into foamy cavity upon reduction of Froude number, is identified. The air entrainment coefficients corresponding to such critical ventilation rate are found to be independent of blockage ratio. It has been observed that in the process of generating a supercavity by increasing ventilation flow rate, the cavitation number gradually reduces to a minimum value and stays fixed upon further increments in the ventilation rate. Once a supercavity is formed, the ventilation rate can be decreased to a much lower value with no change in cavitation number while still maintaining a supercavity. This process is accompanied by a change in closure modes, which generally goes from twin vortex, to quad vortex, and then to re-entrant jet. In addition, the blockage effect is shown to play an important role in promoting the occurrence of twin-vortex closure modes. Subsequently, a physical framework governing the variation of different closure modes is proposed, and is used to explain mode transition upon the change of flow conditions, including the blockage effect. This framework is further extended to shed light on the occurrence of closure modes for ventilated supercavitation experiments across different types of flow facilities, the natural supercavity closure and the pulsating supercavity reported in the literature. Finally, in combination with a recent numerical study, our research discusses the role of the internal flow physics on the observed features during supercavity formation and closure-mode transition, paving the way for future investigations in this direction.

The influence of insoluble surfactants on dynamic wetting failure during displacement of Newtonian fluids in a rectangular channel is studied in this work. A hydrodynamic model for steady Stokes flows of dilute surfactant solutions is developed and evaluated using three approaches: (i) a one-dimensional (1D) lubrication-type approach, (ii) a novel hybrid of a 1D description of the receding phase and a 2D description of the advancing phase, and (iii) an asymptotic theory of Cox (J. Fluid Mech., vol. 168, 1986b, pp. 195–220). Steady-state solution families in the form of macroscopic contact angles as a function of the capillary number are determined and limit points are identified. When air is the receding fluid, Marangoni stresses are found to increase the receding-phase pressure gradients near the contact line by thinning the air film without significantly changing the capillary-pressure gradients there. As a consequence, the limit points shift to lower capillary numbers and the onset of wetting failure is promoted. The model predictions are then used to interpret decades-old experimental observations concerning the influence of surfactants on air entrainment (Burley & Kennedy, Chem. Engng Sci., vol. 31, 1976, pp. 901–911). In addition to being a computationally efficient alternative for the rectangular geometries considered here, the hybrid modelling approach developed in this paper could also be applied to more complicated geometries where a thin air layer is present near a contact line.

We examine the abrupt increase in the rise velocity of an isolated bubble in a viscoelastic fluid occurring at a critical value of its volume, under creeping flow conditions. This ‘velocity discontinuity’, in most experiments involving shear-thinning fluids, has been somehow associated with the change of the shape of the bubble to an inverted teardrop with a tip at its pole and/or the formation of the ‘negative wake’ structure behind it. The interconnection of these phenomena is not fully understood yet, making the mechanism of the ‘velocity jump’ unclear. By means of steady-state analysis, we study the impact of the increase of bubble volume on its steady rise velocity and, with the aid of pseudo arclength continuation, we are able to predict the stationary solutions, even lying in the discontinuous area in the diagrams of velocity versus bubble volume. The critical area of missing experimental results is attributed to a hysteresis loop. The use of a boundary-fitted finite element mesh and the open-boundary condition are essential for, respectively, the correct prediction of the sharply deformed bubble shapes caused by the large extensional stresses at the rear pole of the bubble and the accurate application of boundary conditions far from the bubble. The change of shape of the rear pole into a tip favours the formation of an intense shear layer, which facilitates the bubble translation. At a critical volume, the shear strain developed at the front region of the bubble sharply decreases the shear viscosity. This change results in a decrease of the resistance to fluid displacement, allowing the developed shear stresses to act more effectively on bubble motion. These coupled effects are the reason for the abrupt increase of the rise velocity. The flow field for stationary solutions after the velocity jump changes drastically and intense recirculation downstream of the bubble is developed. Our predictions are in quantitative agreement with published experimental results by Pilz & Brenn (J. Non-Newtonian Fluid Mech., vol. 145, 2007, pp. 124–138) on the velocity jump in fluids with well-characterized rheology. Additionally, we predict shapes of larger bubbles when both inertia and elasticity are present and obtain qualitative agreement with experiments by Astarita & Apuzzo (AIChE J., vol. 11, 1965, pp. 815–820).

Two-dimensional numerical simulations are performed to examine internal wave reflection at a sloping boundary. Owing to reflection, the reflected wave amplitude and wavenumber increase. At low values of the incoming wave amplitude, the reflected wave beam is linear and its properties agree well with linear inviscid theory. Linear theory overestimates the reflected wave Froude number, $Fr_{r}$, for higher values of incoming wave amplitude. Nonlinearity sets in with increasing value of incoming wave Froude number, $Fr_{i}$, leading to parametric subharmonic instability (PSI) of the reflected wave beam: two subharmonics emerge from the reflection region with frequencies $0.33{\it\Omega}$ and $0.67{\it\Omega}$ and wavenumbers that add up to those of the reflected wave. The amplification of Froude number due to reflection must be sufficiently large for PSI to occur implying that the off-criticality in wave angle cannot be too large. The simulations also show that, all other parameters being fixed, a threshold in beam amplitude is required for the onset of PSI in the reflected beam, consistent with results from a previous weakly-nonlinear asymptotic theory for a freely propagating finite-width beam. Growth rates of subharmonic modes at moderate reflected wave amplitude are in reasonable agreement with that theory. However, for $Fr_{r}>0.5$, small scale fluctuations becomes prominent and the subharmonic energy growth rates saturate in the simulations in contrast to the theoretical prediction. Increasing the incoming beam thickness (number of carrier wavelengths) increases the strength of PSI. Keeping the incoming Froude number constant and increasing the incoming Reynolds number by a factor of 50 does not have an effect on the unequal division of frequencies among the subharmonic modes that is found in the simulations.

We revisit the classical problem of a liquid film falling along a vertical wall due to the action of gravity, i.e. the Kapitza paradigm (Kapitza, Zh. Eksp. Teor. Fiz., vol. 18, 1948, pp. 3–28). The free surface of such a flow is typically deformed into a train of solitary pulses that consists of large asymmetric wave humps preceded by small precursory ripples, designated as ‘capillary waves’. We set out to answer four fundamental questions. (i) By what mechanism do the precursory ripples form? (ii) How can they travel at the same celerity as the large-amplitude main humps? (iii) Why are they designated as ‘capillary waves’? (iv) What determines their wavelength and number and why do they attenuate in space? Asymptotic expansion as well as direct numerical simulations and calculations with a low-dimensional integral boundary-layer model have yielded the following conclusions. (i) Precursory ripples form due to an inertia-based mechanism at the foot of the leading front of the main humps, where the local free-surface curvature is large. (ii) The celerity of capillary waves is matched to that of the large humps due to the action of surface tension, which speeds up the former and slows down the latter. (iii) They are justly designated as ‘capillary waves’ because their wavelength is systematically shorter than the visco-capillary cutoff wavelength of the Kapitza instability. Due to a nonlinear effect, namely that their celerity decreases with decreasing amplitude, they nonetheless attain/maintain a finite amplitude because of being continuously compressed by the pursuing large humps. (iv) The number and degree of compression of capillary waves is governed by the amplitude of the main wave humps as well as the Kapitza number. Large-amplitude main humps travel fast and strongly compress the capillary waves in order for these to speed up sufficiently. Also, the more pronounced the first capillary wave becomes, the more (spatially attenuating) capillary waves are needed to allow a smooth transition to the back of the next main hump. These effects are amplified by decreasing the Kapitza number, whereby, at very small values, streamwise viscous diffusion increasingly attenuates the amplitude of the capillary waves.

We present a study of the dynamics of the free-surface shape of a flow in a cylinder driven by a rotating bottom. Near the critical Reynolds number of the laminar–turbulent transition of the boundary layer, the free surface exhibits irregular surface switching between axisymmetric and non-axisymmetric shapes, and the switching often occurs with a significant change in the free-surface height. Although such surface deformation is known to be caused by the flow structures, the detailed flow structures of a rotating fluid with a large surface deformation have yet to be analysed. We thus investigate the velocity distribution and surface shape dynamics and show that the flow field during the loss of its axisymmetry is similar to that predicted by the theory of Tophøj et al. (Phys. Rev. Lett., vol. 110, 2013, 194502). The slight difference observed by quantitative comparison is caused by the fact that the basic flow of our study contains a weak rigid-body rotation in addition to the potential flow assumed by the theory. Furthermore, the observed non-axisymmetric surface shape, which has an elliptic horizontal cross-section, is generally associated with a quadrupole vortex structure. It is also found that the relative position between the free surface and the flow structure changes before and after the detachment of the free surface from the bottom. The change just after the detachment is drastic and occurs via a transient dipole-like vortex structure.

In this paper we used simulation tools to study turbulent boundary-layer structures in the roughness sublayer. Of particular interest is the case of a neutrally-stratified atmospheric boundary layer in which the lower boundary is covered by a homogeneous plant canopy. The goal of this study was to formulate a consistent conceptual model for the creation and evolution of the dominant coherent structures associated with canopy roughness and how they link with features observed in the overlying inertial sublayer. First, coherent structures were examined using temporally developing flow where the full range of turbulent scales had not yet developed, which allowed for instantaneous visualizations. These visualizations were used to formulate a conceptual model, which was then further tested using composite-averaged structure realizations from fully-developed flow with a very large Reynolds number. This study concluded that quasi two-dimensional mixing-layer-like roller structures exist in the developed flow and give the largest contributions to mean Reynolds stresses near the canopy. This work fully acknowledges the presence of highly three-dimensional and localized vortex pairing processes. The primary argument is that, as in a mixing layer, the smaller three-dimensional vortex interactions do not destroy the larger two-dimensional structure. Because the flow has a very large Reynolds number, the roller-like structures are not well-defined vortices but rather are a conglomerate of a large range of smaller-scale vortex structures that create irregularities. Because of this, the larger-scale structure is more difficult to detect in correlation or conditional sampling analyses. The frequently reported ‘scalar microfronts’ and associated spikes in pressure occur in the slip-like region between adjacent rollers. As smaller vortices within roller structures stretch, they evolve to form arch- and hairpin-shaped structures. Blocking by the low-flux canopy creates vertical asymmetry, and tends to impede the vertical progression of head-down structures. Head-up hairpins are allowed to continually stretch upward into the overlying inertial sublayer, where they evolve into the hairpin structures commonly reported to populate wall-bounded flows. This process is thought to be modulated by boundary-layer-scale secondary instability, which enhances head-up hairpin formation along quasi-streamwise transects.

The dynamics of spherical particles resting on a horizontal wall and set into motion by an oscillatory flow is investigated by means of a fully coupled model. Both a smooth wall and a rough wall, the latter being composed of resting particles with a random arrangement and with the same diameter as the moving particles, are considered. The fluid and particle motions are determined by means of direct numerical simulations of Navier–Stokes equations and Newton’s laws, respectively. The immersed boundary approach is used to force the no-slip condition on the surface of the particles. In particular, the process of formation of transverse sediment chains, within the boundary layer but orthogonal to the direction of fluid oscillations, is simulated in parameter ranges matching those of laboratory experiments investigating rolling-grain ripple formation. The numerical results agree with the experimental observations and show that the transverse sediment chains are generated by steady recirculating cells, generated by the interaction of the fluid and particle oscillations.

Nutrient uptake is one of the most important factors in cell growth. Despite the biological importance, little is known about the effect of cell–cell hydrodynamic interactions on nutrient uptake in a suspension of swimming micro-organisms. In this study, we numerically investigate the nutrient uptake in an infinite suspension of squirmers. In the dilute limit, our results are in good agreement with a previous study by Magar et al. (Q. J. Mech. Appl. Maths, vol. 56, 2003, pp. 65–91). When we increased the volume fraction of squirmers, the nutrient uptake of individual cells was enhanced by the hydrodynamic interactions. The average nutrient concentration in the suspension decayed exponentially as a function of time, and the relaxation time could be scaled using the Sherwood number, the Péclet number and the volume fraction of cells. We propose a fitting function for the Sherwood number, which is useful in predicting nutrient uptake in the non-dilute regime. Furthermore, we analyse the swimming energy consumed by individual cells. The results indicate that both the energetic cost and the nutrient uptake increased as the volume fraction of cells was increased, and that the uptake per unit energy was not significantly affected by the volume fraction. These findings are important in understanding the mass transport and metabolism of swimming micro-organisms in nature and for industrial applications.

Numerical models of the geodynamo are usually classified into two categories: dipolar modes, observed when the inertial term is small enough; and multipolar fluctuating dynamos, for stronger forcing. We show that a third dynamo branch corresponding to a dominant force balance between the Coriolis force and the Lorentz force can be produced numerically. This force balance is usually referred to as the strong-field limit. This solution coexists with the often described viscous branch. Direct numerical simulations exhibit a transition from a weak-field dynamo branch, in which viscous effects set the dominant length scale, and the strong-field branch, in which viscous and inertial effects are largely negligible. These results indicate that a distinguished limit needs to be sought to produce numerical models relevant to the geodynamo and that the usual approach of minimising the magnetic Prandtl number (ratio of the fluid kinematic viscosity to its magnetic diffusivity) at a given Ekman number is misleading.

The hydrodynamics of a flagellated micro-organism is investigated when swimming close to a planar free-slip surface by means of numerical solutions of the Stokes equations obtained via a boundary element method. Depending on the initial conditions, the swimmer can either escape from the free-slip surface or collide with the boundary. Interestingly, the micro-organism does not exhibit a stable orbit. Independently of escape or attraction to the interface, close to a free-slip surface, the swimmer follows a counter-clockwise trajectory, in agreement with experimental findings (Di Leonardo et al., Phys. Rev. Lett., vol. 106 (3), 2011, 038101). The hydrodynamics is indeed modified by the free surface. In fact, when the same swimmer moves close to a no-slip wall, a set of initial conditions exists which result in stable orbits. Moreover, when moving close to a free-slip or a no-slip boundary, the swimmer assumes a different orientation with respect to its trajectory. Taken together, these results contribute to shed light on the hydrodynamical behaviour of micro-organisms close to liquid–air interfaces which are relevant for the formation of interfacial biofilms of aerobic bacteria.

Since its introduction in the late 19th century, symmetry breaking has been found to play a crucial role in physics. In particular, it appears as one key phenomenon controlling hydrodynamic and acoustic instabilities in problems with rotational symmetries. A previous paper investigated its desired potential application to the control of circumferential thermoacoustic modes in one annular cavity coupled with multiple flames (Bauerheim et al., J. Fluid Mech., vol. 760, 2014, pp. 431–465). The present paper focuses on a similar problem when symmetry breaking appears unintentionally, for example when uncertainties due to tolerances are taken into account. It yields a large uncertainty quantification (UQ) problem containing numerous uncertain parameters. To tackle this well-known ‘curse of dimensionality’, a novel UQ methodology is used. It relies on the active subspace approach to construct a reduced set of input variables. This strategy is applied on two annular cavities coupled by 19 flames to determine its modal risk factor, i.e. the probability of an azimuthal acoustic mode being unstable. Since each flame is modelled by two uncertain parameters, it leads to a large UQ problem involving 38 parameters. An acoustic network model is then derived, which yields a nonlinear dispersion relation for azimuthal modes. This nonlinear problem, subject to bifurcations, is solved quasi-analytically. Results show that the dimension of the probabilistic problem can be drastically reduced, from 38 uncertain parameters to only 3. Moreover, it is found that the three active variables are related to physical quantities, which unveils underlying phenomena controlling the stability of the two coupled cavities. The first active variable is associated with a coupling strength controlling the bifurcation of the system, while the two others correspond to a symmetry-breaking effect induced by the uncertainties. Thus, an additional destabilization effect appear caused by the non-uniform pattern of the uncertainty distribution, which breaks the initial rotating symmetry of the annular cavities. Finally, the active subspace is exploited by fitting the response surface with polynomials (linear, quadratic and cubic). By comparing accuracy and cost, results prove that 5 % error can be achieved with only 30 simulations on the reduced space, whereas 2000 are required on the complete initial space. It exemplifies that this novel UQ technique can accurately predict the risk factor of an annular configuration at low cost as well as unveil key parameters controlling the stability.

Recent studies have demonstrated that large- and very-large-scale motions in the logarithmic region of turbulent boundary layers ‘amplitude modulate’ dynamics of the near-wall region (Marusic et al., Science, vol. 329, 2010, pp. 193–196; Mathis et al., J. Fluid Mech., vol. 628, 2009a, pp. 311–337). These contributions prompted development of a predictive model for near-wall dynamics (Mathis et al., J. Fluid Mech., vol. 681, 2011, pp. 537–566) that has promising implications for large-eddy simulations of wall turbulence at high Reynolds numbers (owing to the presence of smaller scales as the wall is approached). Existing studies on the existence of amplitude modulation in wall-bounded turbulence have addressed smooth-wall flows, though high Reynolds number rough-wall flows are ubiquitous. Under such conditions, the production of element-scale vortices ablates the viscous wall region and a new near-wall layer emerges: the roughness sublayer. The roughness sublayer depth scales with aggregate roughness element height, $h$, and is typically $2h\sim 3h$. Above the roughness sublayer, Townsend’s hypothesis dictates that turbulence in the logarithmic layer is unaffected by the roughness sublayer (beyond its role in setting the friction velocity and thus inducing a deficit in the mean streamwise velocity known as the roughness function). Here, we present large-eddy simulation results of turbulent channel flow over rough walls. We follow the decoupling procedure outlined in Mathis et al. (J. Fluid Mech., vol. 628, 2009a, 311–337) and present evidence that outer-layer dynamics amplitude modulate the roughness sublayer. Below the roughness element height, we report enormous sensitivity to the streamwise–spanwise position at which flow statistics are measured, owing to spatial heterogeneities in the roughness sublayer imparted by roughness elements. For $y/h\gtrsim 1.5$ (i.e. above the cubes, but within the roughness sublayer), topography dependence rapidly declines.