A reduced-order analysis based on optimized dynamic mode decomposition (DMD) is performed on the turbulent wake of a generic axisymmetric space launcher configuration computed via a zonal large-eddy simulation at the free stream Mach number $Ma_{\infty }=0.8$ and the Reynolds number based on the main body diameter $Re_{D}=6\times 10^{5}$ to investigate the buffet phenomenon. The transonic wake is characterized by an unsteady recirculation region occurring around the nozzle due to the separation of the turbulent boundary layer at the main body shoulder and subsequent dynamic interaction of the unstable free-shear layer with the nozzle surface. This results in strongly periodic and antisymmetric wall pressure fluctuations, for which three distinct frequency ranges are identified using conventional spectral analysis, i.e. $Sr_{D}\approx 0.1$, $Sr_{D}\approx 0.2$ and $Sr_{D}\approx 0.35$. For the spatially integrated side (buffet) loads on the nozzle, the second range is found to be energetically most dominant. To clarify the origin of the detected wake dynamics, the underlying spatio-temporal coherent modes are extracted using DMD. Subsequent analysis of the reduced-order modelled flow field based on the identified DMD modes reveals that at $Sr_{D}\approx 0.1$ a longitudinal cross-pumping motion of the separation bubble takes place, caused by a harmonic antisymmetric oscillation of the main recirculation vortex in the streamwise direction. At $Sr_{D}\approx 0.2$, a cross-flapping motion of the shear layer is determined, triggered by antisymmetric vortex shedding which is in phase with the cross-pumping motion such that it occurs at twice the frequency value. The last range of $Sr_{D}\approx 0.35$ is attributed to a swinging motion of the shear layer caused by a higher harmonic of the vortex shedding mode. Conclusively, the controversial aspect of the true three-dimensional shape of the antisymmetric mode at $Sr_{D}\approx 0.2$ that dominates the buffet phenomenon is scrutinized. Inclined elongated closed-loop vortices are identified that are shed in alternating sequence from azimuthally opposite positions in a longitudinal plane of symmetry that changes its momentary orientation irregularly, maintaining an axisymmetric time-averaged field and spatially isotropic buffet loads.

Unsteady flows in highly curved geometries are of interest in many engineering applications and also in physiological flows. In this study, we use flow visualization and computational fluid dynamics to study unsteady flows in a highly curved tube ($\unicode[STIX]{x1D6FD}=0.3$) with square cross-section; here, $\unicode[STIX]{x1D6FD}$ is the ratio of the half edge length to the radius of curvature of the tube. To explore the combined effects of curvature and pulsatility, we use a single flow pulse of duration $T$ and peak area averaged axial velocity $U_{p(max)}$, which are independently varied to investigate a range of Dean and Womersley numbers. This range includes cases corresponding to flows in the ascending aorta. We observe radially inward moving secondary flows which have the structure of wall jets on the straight walls; their subsequent collision on the inner wall leads to a re-entrant radially outward moving jet. The wall jet arises due to an imbalance between the centrifugal force and the radial pressure gradient. During the deceleration phase, the low-axial-momentum fluid accumulated in the jet reverses direction and leads to flow separation near the inner wall. We use boundary layer equations to derive scales, which have not been reported earlier, for the secondary flow velocities, the wall shear stress components and the distance ($\hat{P}$) traversed by the secondary flow structures in the transverse plane. We show that $\hat{P}$ predicts the movement of vortical structures until collision. In the limit $\unicode[STIX]{x1D6FD}\rightarrow 0$, the Reynolds number based on this secondary flow velocity scale asymptotes to the secondary streaming Reynolds number proposed by Lyne (J. Fluid Mech., vol. 45 (01), 1971, pp. 13–31) in loosely curved pipes. The magnitude of the secondary flow velocity is high and ${\sim}40\,\%$ of $U_{p(max)}$ for physiological flow conditions. We show that the flow separation on the inner wall has origins in the secondary flow, which was reported in a few earlier studies, and is not due to the axial pressure gradient in the tube as proposed earlier. The wall shear stress components, hypothesized to be important in arterial mechanobiology, may be estimated using our scaling relations for geometries with different curvatures and varying pulsatilities.

The ready availability of full-field velocity measurements in present-day experiments has kindled interest in using such data for force estimation, especially in situations where direct measurements are difficult. Among the methods proposed, a formulation based on impulse is attractive, for both practical and physical reasons. However, evaluation of the impulse requires a complete description of the vorticity field, and this is particularly hard to achieve in the important region close to a body surface. This paper presents a solution to the problem. The incomplete experimental-vorticity field is augmented by a vortex sheet on the body, with strength determined by the no-slip boundary condition. The impulse is then found from the sum of vortex-sheet and experimental contributions. Components of physical interest can straightforwardly be recognised; for example, the classical ‘added mass’ associated with fluid inertia is represented by an explicit term in the formulation for the vortex sheet. The method is implemented in the context of two-dimensional flat-plate flow, and tested on velocity-field data from a translating wing experiment. The results show that the vortex-sheet contribution is significant for the test data set. Furthermore, when it is included, good agreement with force-balance measurements is found. It is thus recommended that any impulse-based force calculation should correct for (likely) data incompleteness in this way.

Geophysical granular flows, such as avalanches, debris flows, lahars and pyroclastic flows, are always strongly influenced by the basal topography that they flow over. In particular, localised bumps or obstacles can generate rapid changes in the flow thickness and velocity, or shock waves, which dissipate significant amounts of energy. Understanding how a granular material is affected by the underlying topography is therefore crucial for hazard mitigation purposes, for example to improve the design of deflecting or catching dams for snow avalanches. Moreover, the interactions with solid boundaries can also have important applications in industrial processes. In this paper, small-scale experiments are performed to investigate the flow of a granular avalanche over a two-dimensional smooth symmetrical bump. The experiments show that, depending on the initial conditions, two different steady-state regimes can be observed: either the formation of a detached jet downstream of the bump, or a shock upstream of it. The transition between the two cases can be controlled by adding varying amounts of erodible particles in front of the obstacle. A depth-averaged terrain-following avalanche theory that is formulated in curvilinear coordinates is used to model the system. The results show good agreement with the experiments for both regimes. For the case of a shock, time-dependent numerical simulations of the full system show the evolution to the equilibrium state, as well as the deposition of particles upstream of the bump when the inflow ceases. The terrain-following theory is compared to a standard depth-averaged avalanche model in an aligned Cartesian coordinate system. For this very sensitive problem, it is shown that the steady-shock regime is captured significantly better by the terrain-following avalanche model, and that the standard theory is unable to predict the take-off point of the jet. To retain the practical simplicity of using Cartesian coordinates, but have the improved predictive power of the terrain-following model, a coordinate mapping is used to transform the terrain-following equations from curvilinear to Cartesian coordinates. The terrain-following model, in Cartesian coordinates, makes identical predictions to the original curvilinear formulation, but is much simpler to implement.

Barchan dunes are crescent-shaped formations of sand that can dominate both desert and subaqueous landscapes when the supply of sand is scarce. Because of the complexity and scale of the underlying phenomena, the mechanisms governing the entire process from the genesis to the long-term evolution of barchan fields still remain to be better understood. Herein, we attempt to present a description of this process in a subaqueous environment by employing a large-eddy simulation approach that couples turbulent flow and sand-bed morphodynamics. We show that the seeds of the emergent structure in barchan fields are random turbulent flow motions near the initially flat bed. We also provide high-resolution insights into phenomena such as barchan migration and merging, and show how transverse sand waves are formed and migrate over the barchan horns. Furthermore, the transverse sand waves over the barchan horns are shown to be the seeds of the newly born barchans at the end points of the two horns of a barchan through the process known as calving. To show this, we examine the celerity, wavelength and amplitude of the transverse sand waves over the barchan as they approach the end of its horn. The celerity and wavelength of these transverse sand waves are shown to be the defining factors in determining the frequency of the calving process. The amplitude of the newly born barchans (through calving) is also shown to be associated with the amplitude of the transverse waves near the end of the horn. The simulation data also show that the wavelength of the newly born barchans (the distance between individual dunes) is closely related to that of the transverse sand waves over their maternal barchan. Finally, we use the simulation results to discuss past conclusions derived from theory, conventional models and field observations.

Marginal stability arguments are used to describe the rotation number dependence of torque in Taylor–Couette (TC) flow for radius ratios $\unicode[STIX]{x1D702}\geqslant 0.9$ and shear Reynolds number $\mathit{Re}_{S}=2\times 10^{4}$. With an approximate representation of the mean profile by piecewise linear functions, characterised by the boundary-layer thicknesses at the inner and outer cylinder and the angular momentum in the centre, profiles and torques are extracted from the requirement that the boundary layers represent marginally stable TC subsystems and that the torque at the inner and outer cylinder coincide. This model then explains the broad shoulder in the torque as a function of rotation number near $R_{\unicode[STIX]{x1D6FA}}\approx 0.2$. For rotation numbers $R_{\unicode[STIX]{x1D6FA}}<0.07$ the TC stability conditions predict boundary layers in which the shear Reynolds numbers are very large. Assuming that the TC instability is bypassed by some shear instability, a second narrower maximum in torque appears, in very good agreement with numerical simulations. The results show that marginal stability theory, despite its shortcomings in other cases, can explain quantitatively the non-monotonic torque variation with rotation number for both the broad maximum as well as the narrow maximum.

Using direct numerical simulations (DNS), we explore local and volume-integrated measures of turbulence and mixing in breaking internal waves on slopes. We consider eight breaking wave cases with a range of normalized pycnocline thicknesses $k\unicode[STIX]{x1D6FF}$, where $k$ is the horizontal wavenumber and $\unicode[STIX]{x1D6FF}$ is the pycnocline thickness, but with similar incoming wave properties. The energetics of wave breaking is quantified in terms of local turbulent dissipation and irreversible mixing using the method of Scotti & White (J. Fluid Mech., vol. 740, 2014, pp. 114–135). Local turbulent mixing efficiencies are calculated using the irreversible flux Richardson number $R_{f}^{\ast }$ and are found to be a function of the turbulent Froude number $Fr_{k}$. Volume-integrated measures of the turbulent mixing efficiency during wave breaking are also made, and are found to be functions of $k\unicode[STIX]{x1D6FF}$. The bulk turbulent mixing efficiency ranges from 0.25 to 0.37 and is maximized when $k\unicode[STIX]{x1D6FF}\approx 1$. In order to connect local and bulk mixing efficiency measures, the variation in the bulk turbulent mixing efficiency with $k\unicode[STIX]{x1D6FF}$ is related to the turbulent Froude number at which the maximum total mixing occurs over the course of the breaking event, $Fr_{k}^{max}$. We find that physically, $Fr_{k}^{max}$ is controlled by the vertical length scale of billows at the interface during wave breaking.

The rotational kinematics of inertial cylinders in homogeneous isotropic turbulence is investigated via laboratory experiments. The effects of particle size and shape on rotation statistics are measured for near-neutrally buoyant particles whose sizes are within the inertial subrange of turbulence. To examine the effects of particle size, three right-circular cylinders (aspect ratio $\unicode[STIX]{x1D706}=1$) are considered, with size $d_{eq}=16\unicode[STIX]{x1D702}$, $27\unicode[STIX]{x1D702}$ and $67\unicode[STIX]{x1D702}$. Here, $d_{eq}$ is the diameter of a sphere whose volume is equal to that of the particle and $\unicode[STIX]{x1D702}$ is the Kolmogorov length scale. Results show that the variance of the particle rotation rate follows a $-4/3$ power-law scaling with respect to $d_{eq}$. To examine the effect of particle shape, two cylinders with identical volumes and different aspect ratios ($\unicode[STIX]{x1D706}=1$ and $\unicode[STIX]{x1D706}=4$) are measured. Their motion also scales with $d_{eq}$ regardless of shape. Simultaneous measurements of orientation and rotation for $\unicode[STIX]{x1D706}=4$ particles allows a decomposition of rotation along the primary axes of each particle. This analysis shows that there is no preference for rotation about a particle’s symmetry axis, unlike the preference displayed by sub-Kolmogorov-scale particles in previous studies.

In order to improve the physical understanding of the development of turbulent structures, the compressible evolution equations for the first three invariants $P$, $Q$ and $R$ of the velocity gradient tensor have been derived. The mean evolution of characteristic turbulent structure types in the $QR$-space were studied and compared at different wall-normal locations of a compressible turbulent boundary layer. The evolution of these structure types is fundamental to the physics that needs to be captured by turbulence models. Significant variations of the mean evolution are found across the boundary layer. The key features of the changes of the mean trajectories in the invariant phase space are highlighted and the consequences of the changes are discussed. Further, the individual elements of the overall evolution are studied separately to identify the causes that lead to the evolution varying with the distance to the wall. Significant impact of the wall-normal location on the coupling between the pressure-Hessian tensor and the velocity gradient tensor was found. The highlighted features are crucial for the development of more universal future turbulence models.

Stratified shear flows occur in many geophysical contexts, from oceanic overflows and river estuaries to wind-driven thermocline layers. We explore a turbulent wall-bounded shear flow of lighter miscible fluid into a quiescent fluid of higher density with a range of Richardson numbers $0.05\lesssim Ri\lesssim 1$. In order to find a stability parameter that allows close comparison with linear theory and with idealized experiments and numerics, we investigate different definitions of $Ri$. We find that a gradient Richardson number defined on fluid interface sections where there is no overturning at or adjacent to the maximum density gradient position provides an excellent stability parameter, which captures the Miles–Howard linear stability criterion. For small $Ri$ the flow exhibits robust Kelvin–Helmholtz instability, whereas for larger $Ri$ interfacial overturning is more intermittent with less frequent Kelvin–Helmholtz events and emerging Holmboe wave instability consistent with a thicker velocity layer compared with the density layer. We compute the perturbed fraction of interface as a quantitative measure of the flow intermittency, which is approximately 1 for the smallest $Ri$ but decreases rapidly as $Ri$ increases, consistent with linear theory. For the perturbed regions, we use the Thorpe scale to characterize the overturning properties of these flows. The probability distribution of the non-zero Thorpe length yields a universal exponential form, suggesting that much of the overturning results from increasingly intermittent Kelvin–Helmholtz instability events. The distribution of turbulent kinetic energy, conditioned on the intermittency fraction, has a similar form, suggesting an explanation for the universal scaling collapse of the Thorpe length distribution.

Pipe bifurcations are common flow configurations in both natural and man-made systems. This study follows our previous report (Chen et al., Phys. Fluids, vol. 27, 2015, 034107) by describing three aspects of flows through junction angles of $70^{\circ }$, $90^{\circ }$ and $110^{\circ }$, with a square cross-section. First, the inflow creates tightly spiralling vortices in four quadrants of the junction. For sufficiently large Reynolds number $Re$, these vortices undergo behaviour resembling steady near-axisymmetric breakdown. With increasing $Re$, the flow through the $90^{\circ }$ junction remains steady and stable until the first Hopf bifurcation. Beyond the Hopf bifurcation, the vortices undergo a helical instability. The $70^{\circ }$ and $110^{\circ }$ junctions, however, first exhibit pitchfork bifurcations leading to asymmetric solutions. Second, the direct eigenmodes of the linearised flow are large in vortices in the outlet pipes, whereas the adjoint eigenmodes primarily reside in a small region in the inlet and the junction, near the front and back walls. Third, the sensitivities of the eigenvalues to spatially localised feedback and base flow modifications are greatest in and near the junction vortices. We highlight the regions of high growth rate and frequency sensitivity, as well as regions where the production and transport of perturbations by modifications of the base flow contribute most to the base flow sensitivity. The flow separation at the corners of the junction does not coincide with the eigenmodes or sensitivity regions.

This paper investigates the dynamics of velocity gradients for a spatially developing flow generated by a single square element of a fractal square grid at low inlet Reynolds number through direct numerical simulation. This square grid-element is also the fundamental block of a classical grid. The flow along the grid-element centreline is initially irrotational and becomes turbulent further downstream due to the lateral excursions of vortical turbulent wakes from the grid-element bars. We study the generation and evolution of the symmetric and anti-symmetric parts of the velocity gradient tensor for this spatially developing flow using the transport equations of mean strain product and mean enstrophy respectively. The choice of low inlet Reynolds number allows for fine spatial resolution and long simulations, both of which are conducive in balancing the budget equations of the above quantities. The budget analysis is carried out along the grid-element centreline and the bar centreline. The former is observed to consist of two subregions: one in the immediate lee of the grid-element which is dominated by irrotational strain, and one further downstream where both strain and vorticity coexist. In the demarcation area between these two subregions, where the turbulence is inhomogeneous and developing, the energy spectrum exhibits the best $-5/3$ power-law slope. This is the same location where the experiments at much higher inlet Reynolds number show a well-defined $-5/3$ spectrum over more than a decade of frequencies. Yet, the $Q{-}R$ diagram, where $Q$ and $R$ are the second and third invariants of the velocity gradient tensor, remains undeveloped in the near-grid-element region, and both the intermediate and extensive strain-rate eigenvectors align with the vorticity vector. Along the grid-element centreline, the strain is the first velocity gradient quantity generated by the action of pressure Hessian. This strain is then transported downstream by fluctuations and strain self-amplification is activated a little later. Further downstream, vorticity from the bar wakes is brought towards the grid-element centreline, and, through the interaction with strain, leads to the production of enstrophy. The strain-rate tensor has a statistically axial stretching form in the production region, but a statistically biaxial stretching form in the decay region. The usual signatures of velocity gradients such as the shape of $Q{-}R$ diagrams and the alignment of vorticity vector with the intermediate eigenvector are detected only in the decay region even though the local Reynolds number (based on the Taylor length scale) is only between 30 and 40.

Several recent studies have demonstrated how large-scale vortices may arise spontaneously in rotating planar convection. Here, we examine the dynamo properties of such flows in rotating Boussinesq convection. For moderate values of the magnetic Reynolds number ($100\lesssim Rm\lesssim 550$, with $Rm$ based on the box depth and the convective velocity), a large-scale (i.e. system-size) magnetic field is generated. The amplitude of the magnetic energy oscillates in time, nearly out of phase with the oscillating amplitude of the large-scale vortex. The large-scale vortex is disrupted once the magnetic field reaches a critical strength, showing that these oscillations are of magnetic origin. The dynamo mechanism relies on those components of the flow that have length scales lying between that of the large-scale vortex and the typical convective cell size; smaller-scale flows are not required. The large-scale vortex plays a crucial role in the magnetic induction despite being essentially two-dimensional; we thus refer to this dynamo as a large-scale-vortex dynamo. For larger magnetic Reynolds numbers, the dynamo is small scale, with a magnetic energy spectrum that peaks at the scale of the convective cells. In this case, the small-scale magnetic field continuously suppresses the large-scale vortex by disrupting the correlations between the convective velocities that allow it to form. The suppression of the large-scale vortex at high $Rm$ therefore probably limits the relevance of the large-scale-vortex dynamo to astrophysical objects with moderate values of $Rm$, such as planets. In this context, the ability of the large-scale-vortex dynamo to operate at low magnetic Prandtl numbers is of great interest.

We present an extension to anisotropic flows of the recently developed Helmholtz and wave–vortex decomposition method for one-dimensional spectra measured along ship or aircraft tracks in Bühler et al. (J. Fluid Mech., vol. 756, 2014, pp. 1007–1026). Here, anisotropy refers to the statistical properties of the underlying flow field, which in the original method was assumed to be homogeneous and isotropic in the horizontal plane. Now, the flow is allowed to have a simple kind of horizontal anisotropy that is chosen in a self-consistent manner and can be deduced from the one-dimensional power spectra of the horizontal velocity fields and their cross-correlation. The key result is that an exact and robust Helmholtz decomposition of the horizontal kinetic energy spectrum can be achieved in this anisotropic flow setting, which then also allows the subsequent wave–vortex decomposition step. The anisotropic method is as easy to use as its isotropic counterpart and it robustly converges back to it if the observed anisotropy tends to zero. As a by-product of our analysis we also found a simple test for statistical correlation between rotational and divergent flow components. The new method is developed theoretically and tested with encouraging results on challenging synthetic data as well as on ocean data from the Gulf Stream.

In this work, we have obtained explicit analytical formulae expressing the wave resistance of a two-dimensional body in terms of geometric parameters of nonlinear downstream waves. The formulae have been constructed in the form of high-order asymptotic expansions in powers of the wave amplitude with coefficients depending on the mean depth. To obtain these expansions, the second Stokes method has been used. The analysis represents the next step of the research carried out in Maklakov & Petrov (J. Fluid Mech., vol. 776, 2015, pp. 290–315), where the properties of the waves have been computed by a numerical method of integral equations. In the present work, we have derived a quadratic system of equations with respect to the coefficients of the second Stokes method and developed an effective computer algorithm for solving the system. Comparison with previous numerical results obtained by the method of integral equations has been made.

An experimental and theoretical study of laminar vortical flows driven by oscillating electromagnetic forces that act in orthogonal directions in a shallow electrolytic fluid layer is presented. Forces are generated by the interaction of the field of a dipolar permanent magnet and two imposed alternating electric currents perpendicular to each other with independent frequencies varying in the range of 10–30 mHz. Velocity fields of the time-dependent flow are obtained using particle image velocimetry, while particle tracking allows exploration of the Lagrangian trajectories and time maps. An approximate two-dimensional analytical solution is obtained for the laminar creeping regime so that Lagrangian trajectories are integrated explicitly. These trajectories resemble Lissajous figures with the usual property that, when the ratio of the frequencies of the imposed currents is rational, closed paths are found, while non-closed paths occur when this ratio is irrational. Deviations of this regime that account for slight increase of inertial effects are explored through a quasi-two-dimensional numerical simulation. In this case, non-closed paths are found even for rational frequency ratios. This case was observed in the experiment. Lagrangian trajectories calculated numerically show a qualitative agreement with experimental particle tracking. Furthermore, numerical time maps obtained for increasing inertial effects and rational frequency ratios reveal a chaotic behaviour. Some features of the Lagrangian trajectories are validated experimentally. In particular, topological properties of the calculated and observed time maps are in qualitative agreement. In a characteristic case, a partial time map calculated numerically is compared with the section acquired from the experimental tracking of one particle.

The dynamics of small-amplitude perturbations, as well as the regime of fully developed nonlinear propagating waves, is investigated for pulsatile channel flows. The time-periodic base flows are known analytically and completely determined by the Reynolds number $Re$ (based on the mean flow rate), the Womersley number $Wo$ (a dimensionless expression of the frequency) and the flow-rate waveform. This paper considers pulsatile flows with a single oscillating component and hence only three non-dimensional control parameters are present. Linear stability characteristics are obtained both by Floquet analyses and by linearized direct numerical simulations. In particular, the long-term growth or decay rates and the intracyclic modulation amplitudes are systematically computed. At large frequencies (mainly $Wo\geqslant 14$), increasing the amplitude of the oscillating component is found to have a stabilizing effect, while it is destabilizing at lower frequencies; strongest destabilization is found for $Wo\simeq 7$. Whether stable or unstable, perturbations may undergo large-amplitude intracyclic modulations; these intracyclic modulation amplitudes reach huge values at low pulsation frequencies. For linearly unstable configurations, the resulting saturated fully developed finite-amplitude solutions are computed by direct numerical simulations of the complete Navier–Stokes equations. Essentially two types of nonlinear dynamics have been identified: ‘cruising’ regimes for which nonlinearities are sustained throughout the entire pulsation cycle and which may be interpreted as modulated Tollmien–Schlichting waves, and ‘ballistic’ regimes that are propelled into a nonlinear phase before subsiding again to small amplitudes within every pulsation cycle. Cruising regimes are found to prevail for weak base-flow pulsation amplitudes, while ballistic regimes are selected at larger pulsation amplitudes; at larger pulsation frequencies, however, the ballistic regime may be bypassed due to the stabilizing effect of the base-flow pulsating component. By investigating extended regions of a multi-dimensional parameter space and considering both two-dimensional and three-dimensional perturbations, the linear and nonlinear dynamics are systematically explored and characterized.

Shore protection by small seabed bars was once considered possible because seafloor undulations strongly reflect surface waves of twice the wavelength by the so-called Bragg resonance mechanism. The idea, however, proved ‘unreliable’ when it was realized that a patch of longshore seabed bars adjacent to a reflective shore could result in larger waves at the shoreline than for the case of a flat seabed. Here we propose to revamp the Bragg resonance mechanism as a means of coastal protection by considering oblique seabed bars that divert, rather than reflect, shore-normal incident waves to the shore-parallel direction. We show, via multiple-scale analysis supported by direct numerical simulations, that the creation of a large protected wake near the shoreline requires a bi-chromatic patch to deflect the incident waves to the shore-parallel direction. With two superposed sets of oblique seabed bars, the incident wave energy becomes efficiently deflected far to the sides, leaving a wake of decreased wave activity downstream of the patch. We demonstrate that the shore protection efficiency provided by this novel arrangement is not affected by reflection of leaked waves at the shoreline, and that it is relatively robust against small frequency detuning.

This article reports an analysis of the first detailed chemistry direct numerical simulation (DNS) of a high Karlovitz number laboratory premixed flame. The DNS results are first compared with those from laser-based diagnostics with good agreement. The subsequent analysis focuses on a detailed investigation of the flame area, its local thickness and their rates of change in isosurface following reference frames, quantities that are intimately connected. The net flame stretch is demonstrated to be a small residual of large competing terms: the positive tangential strain term and the negative curvature stretch term. The latter is found to be driven by flame speed–curvature correlations and dominated in net by low probability highly curved regions. Flame thickening is demonstrated to be substantial on average, while local regions of flame thinning are also observed. The rate of change of the flame thickness (as measured by the scalar gradient magnitude) is demonstrated, analogously to flame stretch, to be a competition between straining tending to increase gradients and flame speed variations in the normal direction tending to decrease them. The flame stretch and flame thickness analyses are connected by the observation that high positive tangential strain rate regions generally correspond with low curvature regions; these regions tend to be positively stretched in net and are relatively thinner compared with other regions. High curvature magnitude regions (both positive and negative) generally correspond with lower tangential strain; these regions are in net negatively stretched and thickened substantially.

Analysis of natural convection in a horizontal slot formed by two corrugated isothermal plates has been carried out. The analysis is limited to subcritical Rayleigh numbers $Ra$ where no secondary motion takes place in the absence of corrugations. The corrugations have a sinusoidal form characterized by the wavenumber, the upper and lower amplitudes and the phase difference. The most intense convection occurs for corrugation wavelengths comparable to the slot height; it increases proportionally to $Ra$ and proportionally to the corrugation height. Placement of corrugations on both plates may either significantly increase or decrease the convection depending on the phase difference between the upper and lower corrugations, with the strongest convection found for corrugations being in phase, i.e. a ‘wavy’ slot, and the weakest for corrugations being out of phase, i.e. a ‘converging–diverging’ slot. It is shown that the shear forces would always contribute to the corrugation build-up if erosion was allowed, while the role of pressure forces depends on the location of the corrugations as well as on the corrugation height and wavenumber, and the Rayleigh number. Placing corrugations on both plates results in the formation of a moment which attempts to change the relative position of the plates. There are two limiting positions, i.e. the ‘wavy’ slot and the ‘converging–diverging’ slot, with the latter being unstable. The system would end up in the ‘wavy’ slot configuration if relative movement of the two plates was allowed. The presence of corrugations affects the conductive heat flow and creates a convective heat flow. The conductive heat flow increases with the corrugation height as well as with the corrugation wavenumber; it is largest for short-wavelength corrugations. The convective heat flow is relevant only for wavenumbers of $O(1)$, it increases proportionally to $Ra^{3}$ and proportionally to the second power of the corrugation height. Convection is qualitatively similar for all Prandtl numbers $Pr$, with its intensity increasing for smaller $Pr$ and with the heat transfer augmentation increasing for larger $Pr$.