We present a theoretical asymptotic solution for high-speed transient flow through microporous media in this work by addressing the inertia effect in the high-pressure-difference pulse-decay process. The capillaric model is adopted, in which a bundle of straight circular tubes with a high length–radius ratio is used to represent the internal flow paths of microporous media so that the flow is described by a simplified incompressible Navier–Stokes equation based on the mean density, capturing the major characteristics of mass flow rate. By order-of-magnitude analysis and asymptotic perturbation, the inertial solution with its dimensionless criterion for the high-pressure-difference pulse-decay process is derived. To be compared with experimental data, the theoretical solution involves all three related effects, including the inertia effect, the slippage effect and the compressibility effect. A self-built experimental platform is therefore established to measure the permeability of microporous media by both pulse-decay and steady-state methods to validate the theoretical solution. The results indicate that the relative difference between two methods is less than 30 % even for permeability at as low as $48.2$ nD $(10^{-21}\,{\rm m}^2)$, and the present theoretical solution can accurately capture the inertia effect in the high-pressure-difference pulse-decay process, which significantly accelerates the measurements for ultra-low-permeability samples.

The linear temporal and absolute/convective stability characteristics of a thermal plume generated along a heated vertical cylinder are investigated theoretically under the Boussinesq approximation. Special focus is given to the uniform-wall-buoyancy-flux case whereby the cylinder surface sustains the same linear temperature gradient as the environment. A competition between the axisymmetric and helical modes is a remarkable feature of the instability, distinguishing these ‘annular plumes’ from free plumes/jets for which the helical mode is generally dominant. It is found that higher surface curvature stabilises the temporal axisymmetric mode significantly, but only has moderate effects on the helical mode. The most temporally unstable perturbation mode switches from a helical into an axisymmetric mode when the Prandtl number increases beyond a critical value. Both the roles of shear and buoyancy during the destabilisation are identified through an energy analysis which indicates that, while the shear work is usually a major source of perturbation energy, the buoyancy work manifests for long-wave axisymmetric perturbation modes, and for thin cylinders and high Prandtl numbers. For the specific temperature configuration considered herein, an annular plume is always convectively unstable whereas decreasing the cylinder radius from the planar limiting case first decreases and then increases the tendency of the flow towards being absolutely unstable. The helical mode is especially susceptible to being absolutely unstable on very thin cylinders. Several conditions for the onset of cellular thermal convection and plume detrainment are proposed based on our results and a hypothesis which connects the absolute instability to the detrainment phenomenon.

We compare a higher-order asymptotic construction for balance in geophysical flows with the method of ‘optimal balance’, a purely numerical approach to separating inertia–gravity waves from vortical modes. Both methods augment the linear geostrophic mode with dependent inertia–gravity wave mode contributions, the so-called slaved modes, such that the resulting approximately balanced states are characterized by very small residual wave emission during subsequent time evolution. In our benchmark setting – the single-layer rotating shallow water equations in the quasi-geostrophic regime – the performance of both methods is comparable across a range of Rossby numbers and for different initial conditions. Cross-balancing, i.e. balancing the model with one method and diagnosing the imbalance with the other, suggests that both methods find approximately the same balanced state. Our results also reinforce results from previous studies suggesting that spontaneous wave emission from balanced flow is very small. We further compare two numerical implementations of each of the methods: one pseudospectral, and the other a finite difference scheme on the standard C-grid. We find that a state that is balanced relative to one numerical scheme is poorly balanced for the other, independent of the method that was used for balancing. This shows that the notion of balance in the discrete case is fundamentally tied to a particular scheme.

Data reconstruction of rotating turbulent snapshots is investigated utilizing data-driven tools. This problem is crucial for numerous geophysical applications and fundamental aspects, given the concurrent effects of direct and inverse energy cascades. Additionally, benchmarking of various reconstruction techniques is essential to assess the trade-off between quantitative supremacy, implementation complexity and explicability. In this study, we use linear and nonlinear tools based on the proper orthogonal decomposition (POD) and generative adversarial network (GAN) for reconstructing rotating turbulence snapshots with spatial damages (inpainting). We focus on accurately reproducing both statistical properties and instantaneous velocity fields. Different gap sizes and gap geometries are investigated in order to assess the importance of coherency and multi-scale properties of the missing information. Surprisingly enough, concerning point-wise reconstruction, the nonlinear GAN does not outperform one of the linear POD techniques. On the other hand, the supremacy of the GAN approach is shown when the statistical multi-scale properties are compared. Similarly, extreme events in the gap region are better predicted when using GAN. The balance between point-wise error and statistical properties is controlled by the adversarial ratio, which determines the relative importance of the generator and the discriminator in the GAN training.

In this paper, we present a systematic study of the nonlinear evolution of the travelling Mack modes in a Mach 3 supersonic boundary layer over a rotating cone with a $7^{\circ }$ half-apex angle using the nonlinear parabolic stability equation (NPSE). To quantify the effect of cone rotation, six cases with different rotation rates are considered, and from the same streamwise position, a pair of oblique Mack modes with the same frequency but opposite circumferential wavenumbers are introduced as the initial perturbations for NPSE calculations. As the angular rotation rate $\varOmega$ increases such that $\bar \varOmega$ (defined as the ratio of the rotation speed of the cone to the streamwise velocity at the boundary-layer edge) varies from 0 to $O(1)$, three distinguished nonlinear regimes appear, namely the oblique-mode breakdown, the generalised fundamental resonance and the centrifugal-instability-induced transition. For each regime, the mechanisms for the amplifications of the streak mode and the harmonic travelling waves are explained in detail, and the dominant role of the streak mode in triggering the breakdown of the laminar flow is particularly highlighted. Additionally, from the linear stability theory, the dominant travelling mode undergoes the greatest amplification for a moderate $\varOmega$, which, according to the $e^N$ transition-prediction method, indicates premature transition to turbulence. However, this is in contrast to the NPSE results, in which a delay of the transition onset is observed for a moderate $\varOmega$. Such a disagreement is attributed to the different nonlinear regimes appearing for different rotation rates. Therefore, the traditional transition-prediction method based on the linear instability should be carefully employed if multiple nonlinear regimes may appear.

This paper presents a comprehensive study of flow-induced vibrations of a D-section prism with various angles of attack $\alpha$ ($= 0^{\circ }\unicode{x2013}180^{\circ }$) and reduced velocity $U^*$ (= 2–20) via direct numerical simulations at a Reynolds number ${Re} = 100$. The prism is allowed to vibrate in both streamwise and transverse directions. Based on the characteristics of vibration amplitudes and frequencies, the responses are classified into nine different regimes: typical VIV regime ($\alpha = 0^{\circ }\unicode{x2013}30^{\circ }$), hysteretic VIV regime ($\alpha = 35^{\circ }\unicode{x2013}45^{\circ }$), extended VIV regime ($\alpha = 50^{\circ }\unicode{x2013}55^{\circ }$), first transition response regime ($\alpha = 60^{\circ }\unicode{x2013}65^{\circ }$), dual galloping regime ($\alpha = 70^{\circ }$), combined VIV and galloping regime ($\alpha = 75^{\circ }\unicode{x2013}80^{\circ }$), narrowed VIV regime ($\alpha = 85^{\circ }\unicode{x2013}145^{\circ }$), second transition response regime ($\alpha = 150^{\circ }\unicode{x2013}160^{\circ }$) and transverse-only galloping regime (${\alpha = 165^{\circ }\unicode{x2013}180^{\circ }}$). In the typical and narrowed VIV regimes, the vibration frequencies linearly increase with increasing $U^*$. In the hysteretic and extended VIV regimes, the vibration amplitudes are large in a wider range of $U^*$ as a result of the closeness of the vortex shedding frequency to the natural frequency of the prism because of the shear layer reattachment and separation point movement. In the two galloping regimes, the transverse amplitude keeps increasing with $U^*$ while the streamwise amplitude stays small or monotonically increases with increasing $U^*$. In the combined VIV and galloping regime, the vibration amplitude is relatively small in the VIV region while drastically increasing with increasing $U^*$ in the galloping region. In the transition response regimes, the vibration frequencies are galloping-like but the divergent amplitude cannot persist at high $U^*$. Furthermore, a wake mode map in the examined parametric space is offered. Particular attention is paid to physical mechanisms for hysteresis, dual galloping and flow intermittency. Finally, we probe the dependence of the responses on Reynolds numbers, mass ratios and degrees of freedom, and analyse the roles of the shear layer reattachment and separation point movement in the appearance of multiple responses.

Competing models employing anti-parallel vortex collision in search of a finite-time singularity of Euler's equation have arisen recently. Both the vortex sheet model proposed by Brenner et al. (Phys. Rev. Fluids, vol. 1, 2016, 084503) and the ‘tent’ model proposed by Moffatt & Kimura (J. Fluid Mech., vol. 861, 2019, pp. 930–967) consider a vortex monopole exposed to a strain flow to model the evolution of interacting anti-parallel vortices, a fundamental element in the turbulent cascade. Herein we employ contour dynamics to explore the inviscid evolution of a vortex dipole subjected to an external strain flow with and without axial stretching. We find that for any strain-to-vorticity ratio $\mathcal {E}$, the constituent vortices compress indefinitely, with weaker strain flows causing flattening to occur more slowly. At low $\mathcal {E}$, the vortex dipole forms the well-documented head–tail structure, whereas increasing $\mathcal {E}$ results in the dipole compressing into a pair of vortex sheets with no appreciable head structure. Axial stretching effectively lowers $\mathcal {E}$ dynamically throughout the evolution, thus delaying the transition from the head–tail regime to the vortex sheet regime to higher strain-to-vorticity ratios. Findings from this study offer a bridge between the two cascade models, with the particular mechanism arising depending on $\mathcal {E}$. It also suggests limits for the ‘tent’ model for a finite-time singularity, wherein the curvature-induced strain flow must be very weak in comparison with the vorticity density-driven mutual attraction such that the convective time scale of the evolution exceeds the core flattening time scale.

Turbulent wakes are often characterized by dominant coherent structures over disparate scales. Dynamics of their behaviour can be attributed to interscale energy dynamics and triadic interactions. We develop a methodology to quantify the dynamics of kinetic energy of specific scales. Coherent motions are characterized by the triple decomposition and used to define mean, coherent and random velocity. Specific scales of coherent structures are identified through dynamic mode decomposition, whereby the total coherent velocity is separated into a set of velocities classified by frequency. The coherent kinetic energy of a specific scale is defined by a frequency triad of scale-specific velocities. Equations for the balance of scale-specific coherent kinetic energy are derived to interpret interscale dynamics. The methodology is demonstrated on three wake flows: (i) ${Re}=175$ flow over a cylinder; (ii) a direct numerical simulation of ${Re}=3900$ flow over a cylinder; and (iii) a large-eddy simulation of a utility-scale wind turbine. The cylinder wake cases show that energy transfer starts with vortex shedding and redistributes energy through resonance of higher harmonics. The scale-specific coherent kinetic energy balance quantifies the distribution of transport and transfer among coherent, mean and random scales. The coherent kinetic energy in the rotor scales and the hub vortex scale in the wind turbine interact to produce new scales. The analysis reveals that vortices at the blade root interact with the hub vortex formed behind the nacelle, which has implications for the proliferation of scales in the downwind near wake.

Boundary layers of Novec649, a low-global-warming potential fluid of interest for low-grade heat recovery, are investigated numerically by means of linear stability theory, direct numerical simulation (DNS) and large-eddy simulations (LES). This organic vapour is of interest in organic Rankine cycle (ORC) turbines and realistic thermodynamic conditions are selected. Under these conditions, the vapour behaves as a dense gas and, due to its high molecular complexity, real-gas effects occur. In addition, the fluid exhibits large and highly variable heat capacities and density- as well as temperature-dependent transport properties. More specifically we report the first direct and LES of transitional and turbulent boundary layers of Novec649 at high-subsonic conditions $M=0.9$. A controlled transition is performed by using oblique modes determined by linear stability theory extended to dense gases. An oblique-type transition is obtained as in low-speed air flows, where sinuous streaks develop by the lift-up mechanism and break down into turbulence. In the turbulent state, the profiles of dynamic flow properties (velocities, turbulent intensities, turbulent kinetic energy budgets) are little affected by the gas properties and remain very close to incompressible DNS, despite the high-subsonic flow speed. The fluctuations levels for thermodynamic properties have been quantified with respect to air flows. Notwithstanding a drastic reduction, genuine compressibility effects are present. For example, the fluctuating Mach number and the acoustic mode are characteristic of high-speed flows. The influence of forcing frequency and amplitude on the established turbulent state has been investigated using LES. An analysis of integral quantities shows a slow relaxation towards a canonical equilibrium turbulent state for all cases due to the high Reynolds numbers typical of dense gas flows. Overall the present DNS constitutes a valuable reference not only for forthcoming experiments but also for future studies of free-stream transition and loss mechanisms in ORC turbines.

This study aims to extract and characterize structures in fully developed pipe flow at a friction Reynolds number of $\textit {Re}_\tau = 12\,400$. To do so, we employ data-driven wavelet decomposition (DDWD) (Floryan & Graham, Proc. Natl Acad. Sci. USA, vol. 118, 2021, e2021299118), a method that combines features of proper orthogonal decomposition and wavelet analysis in order to extract energetic and spatially localized structures from data. We apply DDWD to streamwise velocity signals measured separately via a thermal anemometer at 40 wall-normal positions. The resulting localized velocity structures, which we interpret as being reflective of underlying eddies, are self-similar across streamwise extents of 40 wall units to one pipe radius, and across wall-normal positions from $y^+=350$ to $y/R=1$. Notably, the structures are similar in shape to Meyer wavelets. Projections of the data onto the DDWD wavelet subspaces are found to be self-similar as well, but in Fourier space; the bounds of self-similarity are the same as before, except streamwise self-similarity starts at a larger length scale of $450$ wall units. The evidence of self-similarity provided in this study lends further support to Townsend's attached eddy hypothesis, although we note that the self-similar structures are detected beyond the log layer and extend to large length scales.

Direct numerical simulations are performed in rotating turbulence for different regimes at various Rossby and inertial Reynolds numbers ($\textit {Re}_I$). A new algorithm, adapted from stratified turbulence (Lam et al., J. Fluid Mech., vol. 923, 2021, A31) to rotating turbulence, permits to separate the three-dimensional velocity field into three parts: inertial waves (IWs), eddies and a geostrophic mode (GM). It uses the space–time properties of waves and their advection by the GM to filter the IWs from the rest of the motion. We obtain balance equations for the separate energies of waves, eddies and the GM. Their mutual interactions are evaluated and analysed via Sankey diagrams that provide a global picture of energy exchanges. When the flow is forced at large scale, it mainly feeds the wave part and the multiple interactions lead to energy dissipation in eddy and GM motion. We also show that, in addition to the wave/wave interaction that feeds the GM, corresponding to different mechanisms described in the literature, other non-documented interactions feed it, as the eddy/wave interaction or the eddy/eddy interaction at moderate $\textit {Re}_I$. We propose a scale-by-scale analysis of the transfer to the GM: we show that transfers from wave or eddy occur at large scale, that they either inject or remove energy, and that this occurs with or without direct cascade depending on the kind of interaction, wave/wave, eddy/wave or eddy/eddy. The self-interaction of the GM is an inverse cascade for its horizontal component, shaping it into a very large-scale flow.

An experimental investigation of two-dimensional dispersively focused laboratory breaking waves is presented. We describe the bandwidth effect on breaking wave energetics, including spectral energy evolution, characteristic group velocity, energy dissipation and its rate, and breaking strength parameter, $b$. To evaluate the role of bandwidth, three definitions of wave group steepness are adopted where $S_s$ and $S_n$ are bandwidth-dependent and $S_p$ remains constant when bandwidth is changed. Our data show two regimes of spectral energy evolution in breaking wave groups, with both regimes bandwidth-dependent: energy dissipation and gain occur at $f > 0.95f_p$ ($f_p$ is the peak frequency) and $f < 0.95f_p$, respectively. The characteristic group velocity, which is used in energy dissipation calculations, increases by up to 7 % after wave breaking, being larger for higher bandwidth breaking waves. An unambiguous bandwidth dependence is found between $S_p$ and both the fractional and absolute wave energy dissipation. Wave groups of larger bandwidth break at a lower value of $S_p$ and consequently lose relatively more energy. The energy dissipation rate depends on the breaking duration which itself is bandwidth dependent. Consequently, no clear bandwidth effect is observed in energy dissipation rate when compared with either $S_p$ or $S_s$. However, there is a systematic bandwidth dependence in the variation of $b$ when parameterised in terms of $S_p$, with their relationship becoming increasingly nonlinear as bandwidth increases. When parameterised with $S_s$, $b$ shows a markedly reduced bandwidth dependence. Finally, the numerical breaking onset and relationship between $b$ and $S_s$ in the numerical study of Derakhti & Kirby (J. Fluid Mech., vol. 790, 2016, pp. 553–581) is validated experimentally.

In this work we investigate the effect of vertical confinement and inertia on the flow past thin ellipses in a Hele-Shaw cell (with centre line velocity $U_c$ and height 2$h$) with different aspect ratios for symmetrical flows and at an angle of attack, using asymptotic methods and numerical simulations. A Stokes region is identified at the ellipse vertices which results in flow different to flow past bluff bodies. Comparison with asymptotic analysis indicates close agreement over the ‘flat’ portion of the ellipse, for $\delta =(b/a)=0.05$, where $a$ and $b$ are the semi-major and -minor ellipse axes, respectively. Two flow conditions are investigated for ellipses at an angle of attack of 10$^\circ$ for a fixed $\delta =0.05$. Firstly, for $\varLambda =(U_ca/\nu )(h/a)^2 \ll 1$, the effect of increasing the vertical confinement of the Hele-Shaw cell results in the rear stagnation point (RSP) moving from close to the potential-flow prediction when $\epsilon =h/a$ is very small to the two-dimensional Stokes-flow prediction when $\epsilon$ is large. Secondly, for a fixed $\epsilon \ll 1$, when inertia is increased past $\varLambda ={O}(\epsilon )$ the RSP moves towards the trailing edge and is located there for $\varLambda ={O}(1)$. Under these conditions an attached exponentially decaying shear layer or ‘viscous tail’ is formed. A modified Bernoulli equation of the depth-averaged flow, together with the Kutta–Joukowski theorem is used to predict the drag and lift coefficients on the ellipse, which include a linear and a nonlinear contribution, corresponding to a Hele-Shaw and circulation component, respectively. Close agreement is found up to $\varLambda ={O}(1)$.

Membrane viscosity is known to play a central role in the transient dynamics of isolated viscoelastic capsules by decreasing their deformation, inducing shape oscillations and reducing the loading time, that is, the time required to reach the steady-state deformation. However, for dense suspensions of capsules, our understanding of the influence of the membrane viscosity is minimal. In this work, we perform a systematic numerical investigation based on coupled immersed boundary–lattice Boltzmann (IB-LB) simulations of viscoelastic spherical capsule suspensions in the non-inertial regime. We show the effect of the membrane viscosity on the transient dynamics as a function of volume fraction and capillary number. Our results indicate that the influence of membrane viscosity on both deformation and loading time strongly depends on the volume fraction in a non-trivial manner: dense suspensions with large surface viscosity are more resistant to deformation but attain loading times that are characteristic of capsules with no surface viscosity, thus opening the possibility to obtain richer combinations of mechanical features.

Submerged vegetation plays a subtle role in exchanging the fluid mass and energy in the vegetated flow zone, where the swaying motions of flexible plants are the important source of turbulent kinetic energy production. Flume experiments were conducted to study the modes, characteristics and factors of swaying of individual submerged flexible plants. A modified plant model in a new form, representing the highly flexible vegetation with clustered leaves, was employed. A ‘rigid-like’ synchronous swaying mode and a ‘whip-like’ asynchronous flapping mode are found to appear alternately for the individual plants. The interaction between these modes depends on the resulting local flow structure affected by the plants. Compared with a plant in isolation with the same flow Reynolds number, the swaying motions of a plant within the vegetation patch are less frequent but more prone to the synchronous mode. The eigen frequency of the motions increases linearly with an increase in flow Reynolds number in the range of 2 × 104–5 × 104, but the normalised amplitude reaches a saturation at a high flow Reynolds number. Moreover, the in-line and spanwise motions have a 2 : 1 frequency ratio for an ‘8’ shaped trajectory on the horizontal plane and a 1 : 1 ratio for a ‘0’ shaped circular trajectory, or a combination of both.

We carry out direct numerical simulations of flow in a plane open channel at friction Reynolds number up to ${{Re}}_{\tau } \approx 6000$. We find solid evidence for the presence of universal large-scale organization in the outer layer, with eddies that are larger and stronger than in the closed channel flow. As a result, velocity fluctuations are found to be stronger than in closed channels, throughout the depth. The inner-layer peak of the streamwise velocity variance is observed to grow logarithmically, as in Townsend's attached-eddy model (Townsend, The Structure of Turbulent Shear Flow, 2nd edn, Cambridge University Press, 1976), but saturation of the growth cannot be discarded based on the present data. Although we do not observe a clear outer peak of the streamwise velocity variance, we present substantial evidence that such a peak should emerge at a Reynolds number barely higher than achieved herein. The most striking feature of the flow is the presence of an extended logarithmic layer, with associated Kármán constant asymptoting to $k \approx 0.375$, in line with observations made in shear-free Couette–Poiseuille flow (Coleman et al., Flow Turbul. Combust., vol. 99, issue 3, 2017, pp. 553–564). The virtual absence of a wake region and of corrective terms to the log law in the present flow leads us to conclude that deviations from the log law observed in internal flows are likely due to the effects of the opposing walls, rather than the presence of a driving pressure gradient.

Multigrain/polydispersity has a significant impact on turbidity current (TC). Despite the fact that several researches have looked into this effect, the impact of the fluid–particle interactions is not fully understood. Motivated by this, we employ the Eulerian–Lagrangian computational fluid dynamics–discrete element method model to investigate the dynamics of the bidisperse lock-exchange TC. Results show that, because the coarse particles will settle faster and stop moving forward, the two phases of bidisperse transport and fine component transport can be distinguished in the evolution of the bidisperse TC. During the bidisperse transport stage, the upper interface of each component is primarily determined by their own settling and transport characteristics and does not strongly depend on the relative fine particle volume fraction $\phi _F$. Fine particles are primarily responsible for the vortical structures near the upper interface of the TC head, and the increase of $\phi _F$ promotes their streamwise development. In comparison, fragmented vortical coherent structures are closely related to the presence of coarse particles, which can be seen in the lower layers. Bidisperse segregation alters the collision process between dispersed phases, which differs from monodisperse TC. The collisions and segregation-induced flow establish interconnections between the two dispersed phases. In the latter stage, the transport of fine particles is inhibited by both the lift force and the contact force produced by the collision with the deposited materials. As $\phi _F$ rises, the negative contact force weakens, and its change is essentially balanced by the rise in negative lift force.

Mass transport in suspensions of swimming microorganisms is one of the most important factors for the colonisation and growth of microorganisms. Hydrodynamic interactions among swimming microorganisms play an important role in mass transport, especially in highly concentrated suspensions. To elucidate the influence of highly concentrated cells on mass transport, we numerically simulated mass transport in lattices of squirmers that were fixed in space and oriented in the same direction. The effects of different volume fractions, Péclet numbers ($Pe$) and lattice configurations on mass transport were quantified by tracking Lagrangian material points that move with background flow with Brownian diffusivity. Although the flow field became periodic in space and each streamline basically extended in one direction, the motion of tracer particles became diffusive over long durations due to Brownian motion and cross-flows. Flow-induced diffusion was anisotropic and significantly enhanced over Brownian diffusion in the longitudinal direction. We also investigated mass transport in random configurations of squirmers to reproduce more general conditions. Similar enhanced diffusion was also observed in the random configurations, indicating that the flow-induced diffusion appears regardless of the configurations. The present flow-induced diffusion did not follow $Pe$ dependency of the conventional Taylor dispersion due to the cross-flows. The time and velocity scales were proposed, which enabled us to predict the flow-induced diffusivity from the data of the flow field and Brownian diffusivity without solving the mass conservation equation. The findings reported here improve our understanding of the transport phenomena in packed suspensions of swimming microorganisms.

Direct numerical simulation of a compressible round jet is carried out at Mach number of 0.9 and Reynolds number of 3600 and the data are used to perform velocity gradient tensor (VGT) analysis for different regions of the spatially developing jet. For the developed portion of the jet, the classical teardrop shape is observed for the joint probability density function (p.d.f.) of Q and R (second and third invariants of the VGT). In the region just after the potential core, between $X = 10$ and 15 $r_0$ ($r_0$ is the jet inlet radius), an inclination towards the third quadrant is observed in the Q–R joint p.d.f. which represents the presence of tube-like structures. It is also shown that this inclination in the turbulent/non-turbulent (T/NT) boundary and interface towards the third quadrant is mainly a contribution of points that lie in regions with negative dilatation. Regions with weak expansion also show this third quadrant inclination to some extent. Points that lie in regions with relatively higher positive dilatation show no such inclination towards the third quadrant but are inclined towards the fourth quadrant which indicates the presence of sheet-like structures. Similarly for the domain segment $X = 15$ to 20 $r_0$, it is observed that points that lie in the regions with positive dilatation have a joint p.d.f. with an inclination towards fourth quadrant, which suggests the presence of sheet-like structures at the T/NT boundary and interface. Points that lie in regions with negative dilatation show the appearance of a third quadrant lobe.

Lean hydrogen/air premixed flame flashback in a turbulent boundary layer over a flat plate is investigated using three-dimensional direct numerical simulation with detailed chemical kinetics. The upstream propagation of the flame takes place in near-wall turbulence and the interaction between the flame and the approaching reactant flow is studied. It is found that backflow regions are always present immediately upstream of flame bulges that are convex towards the reactants, confirming earlier observations. A flashback speed, including the effects of flame displacement speed and flow velocity, is introduced to quantify the flame flashback behaviour. This analysis indicates that the flashback speed is overall positive and it is considerably affected by the presence of the backflow regions. A budget analysis of the pressure transport equation is performed to explain the presence of the backflow regions. It is suggested that the positive dilatation and thermal diffusion terms near the leading edge of flame bulges are the main reasons for the pressure increase, leading to an adverse pressure gradient. The effects of the flame-induced adverse pressure gradient on the structures of the turbulent boundary layer are also investigated. It is revealed that the near-wall mean velocity and skin-friction coefficient are reduced due to the adverse pressure gradient. The coherent vortical structures of the boundary layer turbulence are lifted by the adverse pressure gradient. The analysis of the Reynolds stress component showed that the ejection event is augmented by combustion while the sweep event is attenuated, which facilitates the occurrence of flame flashback.