The interaction of barotropic tidal currents and baroclinic geostrophic eddies is considered theoretically and numerically to determine whether energy can be transferred to an internal wave field by this process. The eddy field evolves independently of the tide, suggesting that it acts catalytically in facilitating energy transfer from the barotropic tide to the internal wave field, without exchanging energy with the other flow components. The interaction is identically zero and no waves are generated when the barotropic tidal current is horizontally uniform. Optimal internal wave generation occurs when the scales of tide and eddy fields satisfy resonant conditions. The most efficient generation is found if the tidal current horizontal scale is comparable to that of the eddies, with a weak maximum when the scales differ by a factor of two. Thus, this process is not an effective mechanism for internal wave excitation in the deep ocean, where tidal current scales are much larger than those of eddies, but it may provide an additional source of internal waves in coastal areas where horizontal modulation of the tide by topography can be significant.

We study the stationary, ideal flow on a free fluid surface with a prescribed shape. It is demonstrated that the flow is governed by a self-contained set of equations for the surface flow field without any reference to the bulk flow. To write down these equations for arbitrary surfaces, we apply a covariant formulation using Riemannian geometry and we show how to include surface tension and velocity-dependent forces such as the Coriolis force. We write down explicitly the equations for cases where the surface elevation can be written as function of either Cartesian or polar coordinates in the plane, and we obtain solutions for the important case of rotational symmetry and the perturbed flow when this symmetry is slightly broken. To understand the general character and solubility of the equations, we introduce the associated dynamical system describing the motion along the streamlines. The existence of orbits with transversal intersections, as well as quasi-periodic and chaotic solutions, show that not all boundary value problems are well-posed. In the particular case of unforced motion the streamlines are geodesic curves and in this case the existence of a non-trivial surface velocity field requires that the surface can be foliated by a family of non-intersecting geodesic curves.

Microtextured surfaces are commonly used to study complex hydrodynamic phenomena such as spreading and splashing of liquid droplets. However, although surface topography is known to modify near-surface flow, there is no theory able to quantitatively predict the dramatic changes in dynamics of liquid spreading and splashing. Here, we investigate experimentally water bells formed on micropatterned surfaces in order to characterize the hydrodynamics of inertia-dominated flows through regular porous layers. Water bells are self-suspended catenary-shaped liquid films created when a jet impinges on a horizontal disc called an impactor. We show that the presence of micrometre-sized posts regularly arranged on the impactor results in a decrease of the water bell radius and the loss of axisymmetry as open water bells adopt polygonal shapes. We introduce a simple model that captures the main features of the inertia-dominated flow and reveals the role of the hydrodynamic interactions between neighbouring posts. In addition to their applications for tunable jet atomization, these polygonal sheets provide a paradigmatic system for understanding inertia-dominated flow in porous media.

The recently understood relationship between high-Reynolds-number vortex–wave interaction theory and computationally generated self-sustaining processes provides a possible route to an understanding of some of the underlying structures of fully turbulent flows. Here vortex–wave interaction (VWI) theory is used in the long streamwise wavelength limit to continue the development found at order-one wavelengths by Hall & Sherwin (J. Fluid Mech., vol. 661, 2010, pp. 178–205). The asymptotic description given reduces the Navier–Stokes equations to the so-called boundary-region equations, for which we find equilibrium states describing the change in the VWI as the wavelength of the wave increases from $O(h)$ to $O(Rh)$, where $R$ is the Reynolds number and $2h$ is the depth of the channel. The reduced equations do not include the streamwise pressure gradient of the perturbation or the effect of streamwise diffusion of the wave–vortex states. The solutions we calculate have an asymptotic error proportional to ${R}^{- 2} $ when compared to the full Navier–Stokes equations. The results found correspond to the minimum drag configuration for VWI states and might therefore be of relevance to the control of turbulent flows. The key feature of the new states discussed here is the thickening of the critical layer structure associated with the wave part of the flow to completely fill the channel, so that the roll part of the flow is driven throughout the flow rather than as in Hall & Sherwin as a stress discontinuity across the critical layer. We identify a critical streamwise wavenumber scaling, which, when approached, causes the flow to localize and take on similarities with computationally generated or experimentally observed turbulent spots. In effect, the identification of this critical wavenumber for a given value of the assumed high Reynolds number fixes a minimum box length necessary for the emergence of localized structures. Whereas nonlinear equilibrium states of the Navier–Stokes equations are thought to form a backbone on which turbulent flows hang, our results suggest that the localized states found here might play a related role for turbulent spots.

The evolution of a horizontal shear layer in the presence of a horizontal density gradient is explored by three-dimensional numerical simulations. These flows exhibit characteristics of both free shear flows and gravity currents, but have complex dynamics due to strong interactions between the turbulent features of each. Vertical vortices produced by horizontal shear are tilted and stretched by the gravitational adjustment, rapidly enhancing vorticity. Shear intensification at frontal convergences produces high-wavenumber vertical vorticity and the slumping of the density interface produces horizontal Kelvin–Helmholtz vortices typical of a gravity current. The interaction between these instabilities promotes a rapid transition to three-dimensional turbulence. The flow development depends on the relative time scales of shear instability and gravitational adjustment, described by a parameter $\gamma $ (where the limits $\gamma \rightarrow \infty $ and $\gamma \rightarrow 0$ represent a pure gravity current and a pure mixing layer, respectively). The growth rate of three-dimensional instability and the mixing increase for smaller $\gamma $. When $\gamma $ is sufficiently small, there are two distinct regimes: an early period of during which the interface grows rapidly, followed by horizontal diffusive growth. Numerical results are consistent with field observations of tidal separation flows in the Haro Strait (Farmer, Pawlowicz & Jiang, Dyn. Atmos. Oceans., vol. 36, 2002, pp. 43–58), including the magnitude of downwelling vertical currents, horizontal scales of surface vortex features and mixing rate.

The aerodynamics of hovering flight in a hawkmoth (Manduca sexta) are examined using a computational modelling approach which combines a low-fidelity blade-element model with a high-fidelity Navier–Stokes-based flow solver. The focus of the study is on understanding the optimality of the hawkmoth-inpired wingstrokes with respect to lift generation and power consumption. The approach employs a tight coupling between the computational models and experiments; the Navier–Stokes model is validated against experiments, and the blade-element model is calibrated with the data from the Navier–Stokes modelling. In the first part of the study, blade-element and Navier–Stokes modelling are used concurrently to assess the predictive capabilities of the blade-element model. Comparisons between the two modelling approaches also shed insights into specific flow features and mechanisms that are lacking in the lower-fidelity model. Subsequently, we use blade-element modelling to explore a large kinematic parameter space of the flapping wing, and Navier–Stokes modelling is used to assess the performance of the wing-stroke identified as optimal by the blade-element parameter survey. This multi-fidelity optimization study indicates that even within a parameter space constrained by the animal’s natural flapping amplitude and frequency, it is relatively easy to synthesize a wing stroke that exceeds the aerodynamic performance of the hawkmoth wing stroke. Within the prescribed constraints, the optimal wing stroke closely approximates the condition of normal hover, and the implications of these findings on hawkmoth flight capabilities as well as on the issue of biomimetic versus bioinspired design of flapping wing micro-aerial vehicles, are discussed.

The impact of turbulent fluctuations on the forces exerted by a fluid on a towed spherical particle is investigated by means of high-resolution direct numerical simulations. The measurements are carried out using a novel scheme to integrate the two-way coupling between the particle and the incompressible surrounding fluid flow maintained in a high-Reynolds-number turbulent regime. The main idea consists of combining a Fourier pseudo-spectral method for the fluid with an immersed-boundary technique to impose the no-slip boundary condition on the surface of the particle. This scheme is shown to converge as the power $3/ 2$ of the spatial resolution. This behaviour is explained by the ${L}_{2} $ convergence of the Fourier representation of a velocity field displaying discontinuities of its derivative. Benchmarking of the code is performed by measuring the drag and lift coefficients and the torque-free rotation rate of a spherical particle in various configurations of an upstream-laminar carrier flow. Such studies show a good agreement with experimental and numerical measurements from other groups. A study of the turbulent wake downstream of the sphere is also reported. The mean velocity deficit is shown to behave as the inverse of the distance from the particle, as predicted from classical similarity analysis. This law is reinterpreted in terms of the principle of ‘permanence of large eddies’ that relates infrared asymptotic self-similarity to the law of decay of energy in homogeneous turbulence. The developed method is then used to attack the problem of an upstream flow that is in a developed turbulent regime. It is shown that the average drag force increases as a function of the turbulent intensity and the particle Reynolds number. This increase is significantly larger than predicted by standard drag correlations based on laminar upstream flows. It is found that the relevant parameter is the ratio of the viscous boundary layer thickness to the dissipation scale of the ambient turbulent flow. The drag enhancement can be motivated by the modification of the mean velocity and pressure profile around the sphere by small-scale turbulent fluctuations. It is demonstrated that the variance of the drag force fluctuations can be modelled by means of standard drag correlations. Temporal correlations of the drag and lift forces are also presented.

The objective of this study is to investigate the motion of an ellipsoidal capsule in a simple shear flow when its revolution axis is initially placed off the shear plane. We consider prolate capsules with an aspect ratio of two or three enclosed by a membrane, which is either strain-hardening or strain-softening. We seek the equilibrium motion of the capsule as we increase the capillary number $\mathit{Ca}$, which measures the ratio between the viscous and elastic forces. The three-dimensional fluid–structure interaction problem is solved numerically by coupling a boundary integral method (for the internal and external flows) with a finite element method (for the wall deformation). For any initial inclination with the flow vorticity axis, a given capsule converges towards a unique equilibrium configuration which depends on $\mathit{Ca}$. At low capillary number, the stable equilibrium motion is the rolling regime: the capsule aligns its long axis with the vorticity axis, while the membrane tank-treads. As $\mathit{Ca}$ increases, the capsule takes a complex wobbling motion at equilibrium, precessing around the vorticity axis. As $\mathit{Ca}$ is further increased, the capsule long axis oscillates about the shear plane, while the membrane rotates around a capsule cross-section that also oscillates over time (oscillating–swinging regime). The amplitude of the oscillations about the shear plane decreases as $\mathit{Ca}$ increases and the capsule finally takes a swinging motion in the shear plane. It is found that the transitions from rolling to wobbling and from wobbling to oscillating–swinging depend on the mean energy stored in the membrane.

We report high-spatial-resolution measurements of the thermal boundary layer (BL) properties in turbulent thermal convection. The experiment was made near the lower conducting plate of a water-filled rectangular convection cell of height 0.76 m, with a Prandtl number $\mathit{Pr}= 4. 3$ and over the Rayleigh-number range $2\times 1{0}^{10} \lt \mathit{Ra}\lt 7\times 1{0}^{11} $. Time series of the local temperature at various vertical distance $z$ from the plate were measured. Statistical properties of the profiles of the temperature, i.e. the mean temperature $\langle T\rangle $, fluctuating temperature root mean square (r.m.s.) ${\sigma }_{T} $, temperature skewness ${S}_{T} $, and flatness ${F}_{T} $, and those of the temperature time derivative, i.e. the r.m.s. ${ \sigma }_{T}^{\prime } $, skewness ${ S}_{T}^{\prime } $ and flatness ${ F}_{T}^{\prime } $ of the derivative, are studied. It is found that most of these quantities exhibit some degree of invariability with $\mathit{Ra}$, especially for the regime inside the thermal BL. When comparing with the mean temperature profiles, the profiles of the second moment of temperature seem to possess a higher level of universality. It is shown that the distance ${\delta }_{\sigma } $ from the plate to the maximal temperature r.m.s. position provides a natural length scale for the characterization of the thermal BL, as the statistical properties of the temperature field, such as its r.m.s., skewness and flatness, are all sharply different below and above this length scale, i.e. below ${\delta }_{\sigma } $, ${\sigma }_{T} $ increases linearly with the vertical distance $z$ from the plate and ${S}_{T} $ is close to zero and ${F}_{T} $ is close to three and both quantities remains nearly constant, whereas above ${\delta }_{\sigma } $ the decay of ${\sigma }_{T} $ obeys a logarithmic behaviour and ${S}_{T} $ and ${F}_{T} $ both exhibit a hill-like structure. It is also found that near the plate $\langle T\rangle $, ${\sigma }_{T} $ and ${ \sigma }_{T}^{\prime } $ all increase linearly with $z$. Our observations further reveal that such linear dependence occurs within a self-similar region of the thermal BL, where the temperature probability density functions can be scaled onto a single distribution that differs slightly from the Gaussian distribution. The $\mathit{Ra}$-dependencies of various thermal BL properties are also studied and our results yield ${\delta }_{th} / H= (6. 85\pm 0. 70){\mathit{Ra}}^{- 0. 33\pm 0. 03} $, ${\delta }_{\sigma } / H= (2. 86\pm 0. 30){\mathit{Ra}}^{- 0. 31\pm 0. 03} $ and ${ \delta }_{\sigma }^{\prime } / H= (25\pm 3){\mathit{Ra}}^{- 0. 38\pm 0. 05} $, where $H$ is the height of the cell, ${\delta }_{th} $ and ${ \delta }_{\sigma }^{\prime } $ are the BL thicknesses determined respectively from the profiles of $\langle T\rangle $ and ${ \sigma }_{T}^{\prime } $.

When a few bubbles are entrained in a travelling vortex ring, it has been shown that, even at extremely low volume loadings, their presence can significantly affect the structure of the vortex core (Sridhar & Katz, J. Fluid Mech., vol. 397, 1999, pp. 171–202). A typical Euler–Lagrange point-particle model with two-way coupling for this dilute system, wherein the bubbles are assumed subgrid and momentum point sources are used to model their effect on the flow, is shown to be unable to capture accurately the experimental trends of bubble settling location, bubble escape and vortex distortion for a range of bubble parameters and vortex strengths. The bubbles experience significant amounts of drag, lift, added mass, pressure and gravity forces. However, these forces are in balance with each other as the bubbles reach a mean settling location away from the vortex core. The reaction force on the fluid due to the net summation of these forces alone is thus very small and is unable to affect the vortex core. By accounting for fluid volume displacement due to bubble motion, experimental trends on vortex distortion and bubble settling location are captured accurately. The fluid displacement effects are studied by computing various contributions to an effective volume displacement force and are found to be important even at low volume loadings. As the bubble size and hence bubble Reynolds number increase, the bubbles settle further away from the vortex centre and have strong potential for vortex distortion. The net volume displacement force depends on the radial pressure force, the radial settling location of the bubble, as well as the vortex Reynolds number. The resultant of the volume displacement force is found to be roughly at $4{5}^{\circ } $ with the vortex travel direction, resulting in wakes directed towards the vortex centre. Finally, a simple modification to the standard point-particle two-way coupling approach is developed wherein the interphase reaction source terms are consistently altered to account for the fluid displacement effects and reactions due to bubble accelerations.

A computational investigation of variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells is presented. As a first step, two-dimensional base states are obtained by means of simulations of the Stokes equations, which are nonlinear due to the dependence of the viscosity on the local concentration. Here, the vertical position of the displacement front is seen to reach a quasisteady equilibrium value, reflecting a balance between viscous and gravitational forces. These base states allow for two instability modes: first, there is the familiar tip instability driven by the unfavourable viscosity contrast of the displacement, which is modulated by the presence of density variations in the gravitational field; second, a gravitational instability occurs at the unstably stratified horizontal interface along the side of the finger. Both of these instability modes are investigated by means of a linear stability analysis. The gravitational mode along the side of the finger is characterized by a wavelength of about one half to one full gap width. It becomes more unstable as the gravity parameter increases, even though the interface is shifted closer to the wall. The growth rate is largest far behind the finger tip, where the interface is both thicker, and located closer to the wall, than near the finger tip. The competing influences of interface thickness and wall proximity are clarified by means of a parametric stability analysis. The tip instability mode represents a gravity-modulated version of the neutrally buoyant mode. The analysis shows that in the presence of density stratification its growth rate increases, while the dominant wavelength decreases. This overall destabilizing effect of gravity is due to the additional terms appearing in the stability equations, which outweigh the stabilizing effects of gravity onto the base state.

Direct numerical simulations of the variable density and viscosity Navier–Stokes equations are employed, in order to explore three-dimensional effects within miscible displacements in horizontal Hele-Shaw cells. These simulations identify a number of mechanisms concerning the interaction of viscous fingering with a spanwise Rayleigh–Taylor instability. The dominant wavelength of the Rayleigh–Taylor instability along the upper, gravitationally unstable side of the interface generally is shorter than that of the fingering instability. This results in the formation of plumes of the more viscous resident fluid not only in between neighbouring viscous fingers, but also along the centre of fingers, thereby destroying their shoulders and splitting them longitudinally. The streamwise vorticity dipoles forming as a result of the spanwise Rayleigh–Taylor instability place viscous resident fluid in between regions of less viscous, injected fluid, thereby resulting in the formation of gapwise vorticity via the traditional, gap-averaged viscous fingering mechanism. This leads to a strong spatial correlation of both vorticity components. For stronger density contrasts, the streamwise vorticity component increases, while the gapwise component is reduced, thus indicating a transition from viscously dominated to gravitationally dominated displacements. Gap-averaged, time-dependent concentration profiles show that variable density displacement fronts propagate more slowly than their constant density counterparts. This indicates that the gravitational mixing results in a more complete expulsion of the resident fluid from the Hele-Shaw cell. This observation may be of interest in the context of enhanced oil recovery or carbon sequestration applications.

Shock–vortex interaction is analysed for strong gaseous shock waves and a ratio of specific heats close to unity. A nonlinear wave equation for the wrinkles of the shock front is obtained for weak vortices. The solution breaks down after a finite time and the slope of the front develops jump discontinuities, indicating the formation of Mach stems. Shock–turbulence interactions are also briefly discussed.

Helically invariant reductions due to a reduced set of independent variables $(t, r, \xi )$ with $\xi = az+ b\varphi $ emerging from a cylindrical coordinate system of viscous and inviscid time-dependent fluid flow equations, with all three velocity components generally non-zero, are considered in primitive variables and in the vorticity formulation. Full sets of equations are derived. Local conservation laws of helically invariant systems are systematically sought through the direct construction method. Various new sets of conservation laws for both inviscid and viscous flows, including families that involve arbitrary functions, are derived. For both Euler and Navier–Stokes flows, infinite sets of vorticity-related conservation laws are derived. In particular, for Euler flows, we obtain a family of conserved quantities that generalize helicity. The special case of two-component flows, with zero velocity component in the invariant direction, is additionally considered, and special conserved quantities that hold for such flows are computed. In particular, it is shown that the well-known infinite set of generalized enstrophy conservation laws that holds for plane flows also holds for the general two-component helically invariant flows and for axisymmetric two-component flows.

In this work a cephalopod-like deformable body that fills an internal cavity with fluid and expels it to propel an escape manoeuvre, while undergoing a drastic external shape change through shrinking, is shown to employ viscous as well as mainly inviscid hydrodynamic mechanisms to power an impressively fast start. First, we show that recovery of added-mass energy enables a shrinking rocket in a dense inviscid flow to achieve greater escape speed than an identical rocket in a vacuum. Next, we extend the shrinking body results of Weymouth & Triantafyllou (J. Fluid Mech., vol. 702, 2012, pp. 470–487) to three-dimensional bodies and show that three hydrodynamic mechanisms must be combined to achieve rapid escape performance in a viscous fluid: added-mass energy recovery; flow separation elimination; and an optimized energy storage and recovery. In particular, we show that the mechanism of separation elimination achieved through rapid body shrinking, coordinated with the mechanism of recovering the initially imparted added-mass energy, is critical to achieving a high escape speed. Hence a flexible, collapsing body can be vastly superior to a rigid-shell jet-propelled body.

A characteristic feature of the onset of turbulence in shear flows is the appearance of an ‘edge’, a codimension-one invariant manifold that separates ‘lower’ orbits, which decay directly to the laminar state, from ‘upper’ orbits, which decay more slowly and less directly. The object of this paper is to elucidate the structure of the edge that makes this behaviour possible. To this end we consider a succession of low-dimensional models. In doing this we isolate geometric features that are robust under increase of dimension and are therefore candidates for explaining analogous features in higher dimension. We find that the edge, which is the stable manifold of a ‘lower-branch’ state, winds endlessly around an ‘upper-branch’ state in such a way that upper orbits are able to circumnavigate the edge and return to the laminar state.

Square and triangular shape actuator panels mounted on the wall of a wind tunnel beneath an air flow have been impulsively rotated with an angular velocity between 3 and $26~\mathrm{rad} ~{\mathrm{s} }^{- 1} $. A custom-designed balance was used to measure the time-dependent lift and drag forces during the deployment of the actuator, the position of which was monitored by a digital encoder. The measured forces have been compensated for inertia effects which are significant. The results indicated that all lift and drag force coefficients during the transient deployment are different than the corresponding coefficients under stationary conditions at the same deployment angle. It was found that these dynamic effects are augmented with increasing velocity ratio $\mathit{Str}$. The square actuator was found to have better aerodynamic performance than the triangular ones. Additional experiments within different boundary layers reveal that the generated unsteady forces on the moving panels are affected by the characteristics of the incoming boundary layers. The results showed that the thinner the boundary layer is the higher the forces are. Time-resolved flow visualization studies indicated that during the deployment of the panel the upstream turbulent boundary layer structures and the free stream fluid are decelerated and squeezed in the longitudinal direction as they approach the moving plate. A very thin and highly sheared wall layer develops over the moving panel, it generates a substantial amount of vorticity and it subsequently separates from the three edges of the panel to form a large-scale ring-like vortical structure which is responsible for the transient augmentation of the aerodynamic forces. This structure consists of wrapped around separated shear layers which contain pockets of compressed eddies and free stream fluid originated in the upstream incoming boundary layer and free stream. A horseshoe vortex starts to form over the moving plate and during the final stages of deployment it has been moved upstream while the incoming boundary layer turbulent structures are pushed and diverted upwards.

When a liquid metal flows around a truncated cylinder in the presence of a magnetic field which is parallel to the axis of the cylinder, a stagnant region develops above the cylinder. We call this region a Ludford column. The Ludford column represents the magnetohydrodynamics (MHD) analogue of the well-known Taylor columns in rotating flows. Whereas Taylor columns can be easily visualized using dye, the visualization of Ludford columns has remained elusive up to now because liquid metals are opaque. We demonstrate that this fundamental limitation of experimental MHD can be overcome by using a superconducting 5 T magnet. This facility permits us to perform MHD experiments in which the opaque liquid metals are replaced with a transparent electrolyte while maintaining the key MHD effects. We report results of a series of flow experiments in which an aqueous solution of sulphuric acid flows around a bar with square cross-section (which for simplicity shall be referred to as a cylinder). We vary the Reynolds number in the range $5\lt Re\lt 100$ and the Hartmann number in the range $0\lt Ha\lt 14$. The experimental procedure involves flow visualizations using tracer particles as well as velocity measurements using particle image velocimetry (PIV). Our experiments provide direct access to the Ludford column for the first time and reveal the spatial structure of this basic feature of MHD flows.

We performed large-eddy simulations of flow over a series of three-dimensional dunes at laboratory scale (Reynolds number based on the average channel depth and streamwise velocity was 18 900) using the Lagrangian dynamic eddy-viscosity subgrid-scale model. The bedform three-dimensionality was imposed by shifting a standard two-dimensional dune shape in the streamwise direction according to a sine wave. The statistics of the flow are discussed in 10 cases with in-phase and staggered crestlines, different deformation amplitudes and wavelengths. The results are validated qualitatively against experiments. The three-dimensional separation of flow at the crestline alters the distribution of wall pressure, which in turn may cause secondary flow across the stream, which directs low-momentum fluid, near the bed, toward the lobe (the most downstream point on the crestline) and high-momentum fluid, near the top surface, toward the saddle (the most upstream point on the crestline). The mean flow is characterized by a pair of counter-rotating streamwise vortices, with core radius of the order of the flow depth. However, for wavelengths smaller than the flow depth, the secondary flow exists only near the bed and the mean flow away from the bed resembles the two-dimensional case. Staggering the crestlines alters the secondary motion; the fastest flow occurs between the lobe and the saddle planes, and two pairs of streamwise vortices appear (a strong one, centred about the lobe, and a weaker one, coming from the previous dune, centred around the saddle). The distribution of the wall stress and the focal points of separation and attachment on the bed are discussed. The sensitivity of the average reattachment length, depends on the induced secondary flow, the streamwise and spanwise components of the channel resistance (the skin friction and the form drag), and the contribution of the form drag to the total resistance are also studied. Three-dimensionality of the bed increases the drag in the channel; the form drag contributes more than in the two-dimensional case to the resistance, except for the staggered-crest case. Turbulent-kinetic energy is increased in the separated shear layer by the introduction of three-dimensionality, but its value normalized by the plane-averaged wall stress is lower than in the corresponding two-dimensional dunes. The upward flow on the stoss side and higher deceleration of flow on the lee side over the lobe plane lift and broaden the separated shear layer, respectively, affecting the turbulent kinetic energy.

This paper describes analyses of the nonlinear dynamics of harmonically forced, turbulent premixed flames. A key objective of this work is to analyse the ensemble-averaged dynamics of the flame front position, $\langle \xi \rangle $, excited by harmonic forcing of amplitude $\varepsilon $, in the presence of stochastic flow fluctuations of amplitude $\mu $. Low-amplitude and/or near-field effects are quantified by a third-order perturbation analysis, while the more general case is analysed computationally by solving the three-dimensional level-set equation, extracting the instantaneous flame position, and ensemble averaging the results. We show that different mechanisms contribute to smoothing of flame wrinkles, manifested as progressive decay in the magnitude of $\langle \xi \rangle $. Near the flame holder, random phase jitter, associated with stochastic velocity fluctuations tangential to the flame, is dominant. Farther downstream, propagation of the ensemble-averaged front normal to itself at the time-averaged turbulent burning velocity, $ \overline{{S}_{T, eff} } $, leads to destruction of wrinkles, analogous to the laminar case, an effect that scales with $\mu $. A second, new result is the demonstration that the ensemble-averaged turbulent burning velocity, ${S}_{T, eff} (s, t)$, is modulated by the harmonic forcing, i.e. ${S}_{T, eff} (s, t)= \overline{{S}_{T, eff} (s)} + { S}_{T, eff}^{\prime } (s, t)$, where ${ S}_{T, eff}^{\prime } $ has an inverse dependence upon the instantaneous, ensemble averaged-flame curvature, an effect that scales with $\varepsilon $ and $\mu $. We show that this curvature dependence follows from basic application of Huygens propagation to flames with stochastic wrinkling superimposed upon base curvature. This effect also leads to smoothing of flame wrinkles and is analogous to stretch processes in positive-Markstein-length, laminar flames.