We investigated experimentally the settling behaviour of vertically aligned spherical particles within various quiescent media at different release frequencies. The particles had a diameter of d = 4$d = 4$ mm and density of \rho _s = 2200$\rho _s = 2200$ kg m^{-3}$^{-3}$, and were released near the free surface of water, ethanol, a G60 water–glycerine mixture (60 % glycerine by weight) and oil media at frequencies of f_P = 4$f_P = 4$, 6 and 8 Hz, thereby allowing study of Galileo numbers, Ga \in [16, 976]$Ga \in [16, 976]$. Particle tracking velocimetry quantified the motion of nearly 800 particles in a 600 mm high tank, and particle image velocimetry examined flow patterns around the particles. Results revealed that the centre of mass of the particle trajectories exhibited preferential in-plane motions, with significant lateral dispersion and large Ga$Ga$ in water and ethanol, and nearly vertical paths with low Ga$Ga$ in the G60 mixture and oil media. Varying degrees of particle separation resulted in higher terminal velocities than for a single particle. Hence, particle drag decreased in all cases, with the oil medium showing the highest drag reduction under the closest particle separation, reaching up to nearly 70 % of that for the single particle. The vertical and lateral pair dispersions, R^2_z$R^2_z$ and R^2_L$R^2_L$, exhibited ballistic scaling, with dependences on the initial separation, r_0$r_0$, and the type of medium. With large Ga$Ga$, R^2_z$R^2_z$ displayed a ballistic regime followed by a slower rate, whereas with small Ga$Ga$, R^2_z$R^2_z$ maintained a consistent ballistic regime throughout settling. Finally, normalized R^2_z$R^2_z$ demonstrated distinct scaling (exponent 2/3 and 1) dependent on the normalized initial separation and Ga$Ga$.

Quasistatic magnetoconvection of a fluid with low Prandtl number ({\textit {Pr}}=0.025${\textit {Pr}}=0.025$) with a vertical magnetic field is considered in a unit-aspect-ratio box with no-slip boundaries. At high relative magnetic field strengths, given by the Hartmann number {\textit {Ha}}${\textit {Ha}}$, the onset of convection is known to result from a sidewall instability giving rise to the wall-mode regime. Here, we carry out three-dimensional direct numerical simulations of unprecedented length to map out the parameter space at {\textit {Ha}} = 200, 500, 1000${\textit {Ha}} = 200, 500, 1000$, varying the Rayleigh number ({\textit {Ra}}${\textit {Ra}}$) over the range 6\times 10^5 \lesssim {\textit {Ra}} \lesssim 5\times 10^8$6\times 10^5 \lesssim {\textit {Ra}} \lesssim 5\times 10^8$. We track the development of stable equilibria produced by this primary instability, identifying bifurcations leading to limit cycles and eventually to chaotic dynamics. At {\textit {Ha}}=200${\textit {Ha}}=200$, the steady wall-mode solution undergoes a symmetry-breaking bifurcation producing a state that features a coexistence between wall modes and a large-scale roll in the centre of the domain, which persists to higher {\textit {Ra}}${\textit {Ra}}$. However, under a stronger magnetic field at {\textit {Ha}}=1000${\textit {Ha}}=1000$, the steady wall-mode solution undergoes a Hopf bifurcation producing a limit cycle which further develops to solutions that shadow an orbit homoclinic to a saddle point. Upon a further increase in {\textit {Ra}}${\textit {Ra}}$, the system undergoes a subsequent symmetry break producing a coexistence between wall modes and a large-scale roll, although the large-scale roll exists only for a small range of {\textit {Ra}}${\textit {Ra}}$, and chaotic dynamics primarily arise from a mixture of chaotic wall-mode dynamics and arrays of cellular structures.

The efficiency of water electrolysis is significantly impacted by the generation of micro- and nanobubbles on the electrodes. Here molecular dynamics simulations are used to investigate the dynamics of single electrolytic nanobubbles on nanoelectrodes. The simulations reveal that, depending on the value of current, nucleated nanobubbles either grow to an equilibrium state or grow unlimitedly and then detach. To account for these findings, the stability theory for surface nanobubbles is generalized by incorporating the electrolytic gas influx at the nanobubble's contact line and adopting a real gas law, leading to accurate predictions for the numerically observed transient growth and stationary states of the nanobubbles. With this theory, the minimum current for bubble detachment can also be derived analytically. In the detachment regime, the radius of the nanobubble first increases with time (t) as R\propto t^{1/2}$R\propto t^{1/2}$ and then as R\propto t^{1/3}$R\propto t^{1/3}$, up to bubble detachment.

The refraction of surface gravity waves by currents leads to spatial modulations in the wave field and, in particular, in the significant wave height. We examine this phenomenon in the case of waves scattered by a localised current feature, assuming (i) the smallness of the ratio between current velocity and wave group speed, and (ii) a swell-like, highly directional wave spectrum. We apply matched asymptotics to the equation governing the conservation of wave action in the four-dimensional position–wavenumber space. The resulting explicit formulas show that the modulations in wave action and significant wave height past the localised current are controlled by the vorticity of the current integrated along the primary direction of the swell. We assess the asymptotic predictions against numerical simulations using WAVEWATCH III for a Gaussian vortex. We also consider vortex dipoles to demonstrate the possibility of ‘vortex cloaking’ whereby certain currents have (asymptotically) no impact on the significant wave height. We discuss the role of the ratio of the two small parameters characterising assumptions (i) and (ii) above, and show that caustics are significant only for unrealistically large values of this ratio, corresponding to unrealistically narrow directional spectra.

The dynamics of turbulent flows is chaotic and difficult to predict. This makes the design of accurate reduced-order models challenging. The overarching objective of this paper is to propose a nonlinear decomposition of the turbulent state to predict the flow based on a reduced-order representation of the dynamics. We divide the turbulent flow into a spatial problem and a temporal problem. First, we compute the latent space, which is the manifold onto which the turbulent dynamics live. The latent space is found by a series of nonlinear filtering operations, which are performed by a convolutional autoencoder (CAE). The CAE provides the decomposition in space. Second, we predict the time evolution of the turbulent state in the latent space, which is performed by an echo state network (ESN). The ESN provides the evolution in time. Third, by combining the CAE and the ESN, we obtain an autonomous dynamical system: the CAE-ESN. This is the reduced-order model of the turbulent flow. We test the CAE-ESN on the two-dimensional Kolmogorov flow and the three-dimensional minimal flow unit. We show that the CAE-ESN: (i) finds a latent-space representation of the turbulent flow that has {\lesssim }1\,\%${\lesssim }1\,\%$ of the degrees of freedom than the physical space; (ii) time-accurately and statistically predicts the flow at different Reynolds numbers; and (iii) takes {\lesssim }1\,\%${\lesssim }1\,\%$ computational time to predict the flow with respect to solving the governing equations. This work opens possibilities for nonlinear decomposition and reduced-order modelling of turbulent flows from data.

Trip-resolved large-eddy simulations of the DARPA SUBOFF are performed to investigate the development of turbulent boundary layers (TBLs) in model-scale studies. The primary consideration of the study is the extent to which the details of tripping affect statistics in large-eddy simulations of complex geometries, which are presently limited to moderate Reynolds number TBLs. Two trip wire configurations are considered, along with a simple numerical trip (wall-normal blowing), which serves as an exemplar of artificial computational tripping methods often used in practice. When the trip wire height exceeds the laminar boundary layer thickness, shedding from the trip wire initiates transition, and the near field is characterized by an elevation of the wall-normal Reynolds stress and a modification of the turbulence anisotropy and mean momentum balance. This trip wire also induces a large jump in the boundary layer thickness, which affects the way in which the TBL responds to the pressure gradients and streamwise curvature of the hull. The trip-induced turbulence decays along the edge of the TBL as a wake component that sits on top of the underlying TBL structure, which dictates the evolution of the momentum and displacement thicknesses. In contrast, for a trip wire height shorter than the laminar boundary layer thickness, transition is initiated at the reattachment point of the trip-induced recirculation bubble, and the artificial trip reasonably replicates the resolved trip wire behaviour relatively shortly downstream of the trip location. For each case, the inner layer collapses rapidly in terms of the mean profile, Reynolds stresses and mean momentum balance, which is followed by the collapse of the Reynolds stresses in coordinates normalized by the local momentum thickness, and finally against the 99 % thickness. By this point, the lasting impact of the trip is the offset in boundary layer thickness due to the trip itself, which becomes a diminishing fraction of the total boundary layer thickness as the TBL grows. The importance of tripping the model appendages is also highlighted due to their lower Reynolds numbers and susceptibility to laminar separations.

We study a fifty-year-old problem of fast acoustic streaming, that is, the generation of moderate or large hydrodynamic Reynolds number (\textit {Re}$\textit {Re}$) acoustic streaming (or steady flow) by the convection of momentum in an acoustic wave (or another periodic flow), while the latter is simultaneously altered by the former. The intrinsic disparity of length and time scales makes a brute-force solution of the full Navier–Stokes and continuity equations a formidable problem. Circumventing this difficulty, we split the problem into a time-averaged system of equations for the steady flow component and a dynamic system of equations for its quasi-periodic flow counterpart. The latter system of equations is obtained by subtracting the time-averaged Navier–Stokes equation from its original dynamic form, and is rendered a nonlinear wave equation using the continuity equation and an adiabatic connection between density and pressure. The resulting equations are compatible with the theory by Eckart for small \textit {Re}$\textit {Re}$ flow, and capture large-\textit {Re}$\textit {Re}$ effects. Scaling analysis and a case study show that acoustic streaming is weak and does not contribute to the acoustic wave close to the wave source, relevant to many microfluidic systems. At small \textit {Re}$\textit {Re}$, the streaming magnitude is proportional to an inverse Strouhal number, a small quantity in experiments. Moderate and large \textit {Re}$\textit {Re}$ render the streaming magnitude comparable to the pre-attenuating periodic flow (or particle velocity of the wave) at approximately a wave attenuation length away from the wave source or further; the wave is altered by the streaming that it generates, and the streaming dominates the flow far from the wave source.

When a saturated brine layer is cooled from above, both a convective temperature front as well as a front of sedimenting salt crystals can form. We employ direct numerical simulations to investigate the evolution and interaction of these two density fronts. Depending on the ratio of the temperature front velocity and the crystal settling velocity, which is governed by a dimensionless parameter in the form of a Rayleigh number, we find that either two separate fronts exist for all times, two initially separate fronts combine into a single front after some time or a single front exists at all times. We furthermore propose approximate scaling laws for the propagation of the thermal and crystal fronts in each regime and compare them with the simulation data, with generally good agreement.

Flux conditions are semi-analytical expressions that can be used to determine the flux at the grounding line of marine ice sheets. In the glaciology literature, such flux conditions have initially been established for the Weertman and Coulomb friction laws. However, the extension to more general and complex friction laws, such as the Budd friction law, for which basal friction depends on both the basal velocity and the effective pressure, is a topic of recent research. Several studies have also shown that hybrid friction laws, which consider a transition between a power-law friction far from the grounding line and a plastic behaviour close to it, were good candidates for improved modelling of marine ice sheets. In this article, we show that the flux conditions previously derived for the Weertman and Coulomb friction laws can be generalised to flux conditions for the Budd friction law with two different effective-pressure models. In doing so, we build a bridge between the results obtained for these two friction laws. We provide a justification for the existence and uniqueness of a solution to the boundary-layer problem based on asymptotic developments. We also generalise our results to hybrid friction laws, based on a parametrisation of the flux condition. Finally, we discuss the validity of the assumptions made during the derivation, and we provide additional explicit expressions for the flux that stay valid when the bedrock slopes are important or when the friction coefficients are relatively small.

The turbulent wake past a square-back Ahmed body in close proximity to the ground experiences random side-to-side switching between two asymmetric positions, a phenomenon known as bi-modality. It has been observed to be sensitive to the dynamics of the upstream boundary layers formed along the body surfaces. Close to the body fore end, these separate and reattach, with hairpin vortices emanating from the reattachment points and growing along the surfaces before breaking down upstream of the base. This study uses wall-resolved large eddy simulations to investigate the effect of using suction to suppress these upstream boundary layer separations on the wake bi-modality. It is seen that, in the unforced flow (in the absence of suction), the smaller top and side surface vortices resulting from breakdown interact as they convect downstream. Steady suction is confirmed to suppress the boundary layer separations on the different body surfaces. When the boundary layer separations on the two side (vertical) surfaces are suppressed, it is found that horizontal bi-modality is completely inhibited with weak vertical asymmetry preserved. The interaction of the small top/side surface vortices is interrupted, damping boundary layer fluctuations just upstream of the base. Applying suction on different combinations of side/top/bottom boundary layer separations is found to have different effects on the underbody flow and the wake vertical balance, with bi-modality suppression dependent on side surface suction. This confirms that bi-modality is triggered, at least in part, by boundary layer disturbances on the surfaces perpendicular to the switching direction.

Algorithms for reconstructing and predicting nonlinear ocean wave fields from remote measurements are presented. Three types of synthetic observations are used to quantify the influence of remote measurement modulation mechanisms on the algorithms’ performance. First, the observations correspond to randomly distributed surface elevations. Then, they are related to a marine radar model – the second type takes the wave shadowing modulation into account whereas the third one also includes the tilt modulation. The observations are numerically generated based on unidirectional waves of various steepness values. Linear and weakly nonlinear prediction algorithms based on analytical models are considered, as well as a highly nonlinear algorithm relying on the high-order spectral (HOS) method. Reconstructing surfaces from shadowed observations is found to have an impact limited to the non-visible regions, while tilt modulation affects the reconstruction more generally due to the indirect, more complex extraction of wave information. It is shown that the accuracy of the surface reconstruction mainly depends on the correct modelling of the wave shape nonlinearities. Modelling the nonlinear correction of the dispersion relation, in particular the frequency-dependent wave phase effects in the case of irregular waves, substantially improves the prediction. The suitability of the algorithms for severe wave conditions in finite depth and using non-perfect observations is assessed through wave tank experiments. It shows that only the third-order HOS solution predicts the right amplitude and phase of an emerging extreme wave, emphasizing the relevance of the corresponding physical modelling.

Measurement data sets are presented for the turbulence intensity profile of three velocity components (u$u$, v$v$ and w$w$) and turbulent kinetic energy (TKE, k$k$) over a wide range of Reynolds numbers from Re_\tau =990$Re_\tau =990$ to 20 750 in a pipe flow. The turbulence intensity profiles of the u$u$- and w$w$-component show logarithmic behaviour, and that of the v$v$-component shows a constant region at high Reynolds numbers, Re_\tau >10\,000$Re_\tau >10\,000$. Furthermore, a logarithmic region is also observed in the TKE profile at y/R=0.055\unicode{x2013}0.25$y/R=0.055\unicode{x2013}0.25$. The Reynolds number dependences of peak values of u$u$-, w$w$-component and TKE fit to both a logarithmic law (Marusic et al., Phys. Rev. Fluids, vol. 2, 2017, 100502) and an asymptotic law (Chen and Sreenivasan, J. Fluid Mech., vol. 908, 2020, R3), within the uncertainty of measurement. The Reynolds number dependence of the bulk TKE k^+_{bulk}$k^+_{bulk}$, which is the total amount of TKE in the cross-sectional area of the pipe also fits to both laws. When the asymptotic law is applied to the k^+_{bulk}$k^+_{bulk}$, it asymptotically increases to the finite value k^+_{bulk}=11$k^+_{bulk}=11$ as the Reynolds number increases. The contribution ratio \langle u'^2\rangle /k$\langle u'^2\rangle /k$ reaches a plateau, and the value tends to be constant within 100< y^+<1000$100< y^+<1000$ at Re_\tau >10\ 000$Re_\tau >10\ 000$. Therefore, the local balance of each velocity component also indicates asymptotic behaviour. The contribution ratios are balanced in this region at high Reynolds numbers as \langle u'^2\rangle /k\simeq 1.25$\langle u'^2\rangle /k\simeq 1.25$, \langle w'^2\rangle /k\simeq 0.5$\langle w'^2\rangle /k\simeq 0.5$ and \langle v'^2\rangle /k \simeq 0.25$\langle v'^2\rangle /k \simeq 0.25$.

The gravity flow of a granular material between two vertical walls separated by a width 2W$2W$ is simulated using the discrete element method (DEM). Periodic boundary conditions are applied in the flow (vertical) and the other horizontal directions. The mass flow rate is controlled by specifying the average solids fraction \bar {\phi }$\bar {\phi }$, the ratio of the volume of the particles to the volume of the channel. A steady fully developed state can be achieved for a narrow range of \bar {\phi }$\bar {\phi }$, \bar {\phi }_{max} \geq \bar {\phi } \geq \bar {\phi }_{cr}$\bar {\phi }_{max} \geq \bar {\phi } \geq \bar {\phi }_{cr}$, and the material is in free fall for \bar {\phi } < \bar {\phi }_{min}$\bar {\phi } < \bar {\phi }_{min}$. For an intermediate range of \bar {\phi }$\bar {\phi }$ (\bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$\bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$), there are oscillations in the horizontal coordinate of the centre of mass, velocity components and stress. As \bar {\phi }$\bar {\phi }$ decreases in the range \bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$\bar {\phi }_{cr} > \bar {\phi } \geq \bar {\phi }_{min}$, the amplitude of the oscillations increases proportional to \sqrt {\bar {\phi }_{cr} - \bar {\phi }}$\sqrt {\bar {\phi }_{cr} - \bar {\phi }}$ and the frequency appears to tend to a non-zero value as \bar {\phi } \rightarrow \bar {\phi }_{cr}$\bar {\phi } \rightarrow \bar {\phi }_{cr}$, indicating a supercritical Hopf bifurcation. The relation between the dominant frequency and the higher harmonics of the position, velocity and stress fluctuations are explained using the momentum balance. It is found that dissipation in the inter-particle and particle–wall contacts is critical for the presence of an oscillatory state.

We study the spectral energy transfer due to wave–triad interactions in the Garrett–Munk spectrum of internal gravity waves based on a numerical evaluation of the collision integral in the wave kinetic equation. Our numerical evaluation builds on the reduction of the collision integral on the resonant manifold for a horizontally isotropic spectrum. We evaluate directly the downscale energy flux available for ocean mixing, whose value is in close agreement with the finescale parameterization. We further decompose the energy transfer into contributions from different mechanisms, including local interactions and three types of non-local interactions, namely parametric subharmonic instability, elastic scattering (ES) and induced diffusion (ID). Through analysis on the role of each mechanism, we resolve two long-standing paradoxes regarding the mechanism for forward cascade in frequency and zero ID flux for the GM76 spectrum. In addition, our analysis estimates the contribution of each mechanism to the energy transfer in each spectral direction, and reveals new understanding of the importance of local interactions and ES in the energy transfer.

An experimental study of long interfacial gravity waves was conducted in a closed wave tank containing two layers of viscous immiscible fluids. The study focuses on the development in time of the mean particle drift that occurs close to the interface where the two fluids meet. From a theoretical analysis by Weber & Christensen (Eur. J. Mech.-B/Fluids, vol. 77, 2019, pp. 162–170) it is predicted that the growing drift in this region is associated with the action of the virtual wave stress. This effect has not been explored experimentally before. Interfacial waves of different amplitudes and frequencies were produced by a D-shaped paddle. Particle tracking velocimetry was used to track the time development of the Lagrangian mean drift. The finite geometry of the wave tank causes a mean return flow that is resolved by mass transport considerations. The measurements clearly demonstrate the importance of the virtual wave stress mechanism in generating wave drift currents near the interface.

To draw concrete pictures of the hierarchy of multiscale coherent vortices in turbulence behind a cylinder and to reveal their sustaining mechanism, we conduct direct numerical simulations of the turbulence at the Reynolds number, which is defined by the uniform inflow velocity and the cylinder diameter, 5000$5000$. The turbulence consists of three kinds of hierarchies of coherent vortices in three distinct regions: namely, the downstream region, the recirculation region just behind the cylinder and the separate shear layers. By tracking the temporal evolution of multiscale vortices in each of these regions, we demonstrate that, in all the three regions, smaller coherent vortices tend to align in the direction perpendicular to larger ones. This implies that smaller vortices are stretched and amplified in the strain-rate fields around larger ones. We also show the relevance of this observation to the energy cascade. Smaller-scale vortices receive the kinetic energy in the regions where they are stretched by larger-scale vortices; and, at the same time, they tend to compress larger ones, thus reducing larger-scale kinetic energy.

In this paper, transient granular flows are examined both numerically and experimentally. Simulations are performed using the continuous three-dimensional (3-D) granular model introduced in Daviet & Bertails-Descoubes (ACM Trans. Graph., vol. 35, no. 4, 2016b, p. 102), which represents the granular medium as an inelastic and dilatable continuum subject to the Drucker–Prager yield criterion in the dense regime. One notable feature of this numerical model is to resolve such a non-smooth rheology without any regularisation. We show that this non-smooth model, which relies on a constant friction coefficient, is able to reproduce with high fidelity various experimental granular collapses over inclined erodible beds, provided the friction coefficient is set to the avalanche angle – and not to the stop angle, as generally done. In order to better characterise the range of validity of the fully plastic rheology in the context of transient frictional flows, we further revisit scaling laws relating the shape of the final collapse deposit to the initial column aspect ratio, and accurately recover established power-law dependences up to aspect ratios of the order of 10. The influence of sidewall friction is then examined through experimental and simulated collapses with varying channel widths. The analysis offers a comprehensive framework for estimating the effective flow thickness in relation to the channel width, thereby challenging previously held assumptions regarding its estimation in the literature. Finally, we discuss the possibility to extend the constant coefficient model with a hysteretic model in order to refine the predictions of the early-stage dynamics of the collapse. This illustrates the potential effects of such phenomenology on transient flows, paving the way to more elaborate analysis.

This work studies the three-dimensional flow dynamics around a rotating circular cylinder of finite length, whose axis is positioned perpendicular to the streamwise direction. Direct numerical simulations and global stability analyses are performed within a parameter range of Reynolds number Re=DU_\infty /\nu <500$Re=DU_\infty /\nu <500$ (based on cylinder diameter D$D$, uniform incoming flow velocity U_\infty$U_\infty$), length-to-diameter ratio {\small \text{AR}}=L/D\leq 2${\small \text{AR}}=L/D\leq 2$ and dimensionless rotation rate \alpha =D\varOmega /2U_\infty \leq 2$\alpha =D\varOmega /2U_\infty \leq 2$ (where \varOmega$\varOmega$ is rotation rate). By solving Navier–Stokes equations, we investigated the wake patterns and explored the phase diagrams of the lift and drag coefficients. For a cylinder with {\small \text{AR}}=1${\small \text{AR}}=1$, we found that when the rotation effect is weak (0\leq \alpha \lesssim 0.3$0\leq \alpha \lesssim 0.3$), the wake pattern is similar to the unsteady wake past the non-rotating finite-length cylinder, but with a new linear unstable mode competing to dominate the saturation state of the wake. The flow becomes stable for 0.3\lesssim \alpha \lesssim 0.9$0.3\lesssim \alpha \lesssim 0.9$ when Re<360$Re<360$. When the rotation effect is strong (\alpha \gtrsim 0.9$\alpha \gtrsim 0.9$), new low-frequency wake patterns with stronger oscillations emerge. Generally, the rotation effect first slightly decreases and then sharply increases the Re$Re$ threshold of the flow instability when \alpha$\alpha$ is relatively small, but significantly decreases the threshold at high \alpha$\alpha$ (0.9<\alpha \leq 2$0.9<\alpha \leq 2$). Furthermore, the stability analyses based on the time-averaged flows and on the steady solutions demonstrate the existence of multiple unstable modes undergoing Hopf bifurcation, greatly influenced by the rotation effect. The shapes of these global eigenmodes are presented and compared, as well as their structural sensitivity, visualising the flow region important for the disturbance development with rotation. This research contributes to our understanding of the complex bluff-body wake dynamics past this critical configuration.

We investigate self-excited vortex-acoustic lock-in in a bluff body combustor. The bluff body not only anchors the flame but also sheds vortices. Mutual interaction between vortex shedding and the combustor acoustic field leads to lock-in. A lower-order model for vortex shedding is coupled to the acoustic equations to obtain a set of discrete dynamical maps, which are then solved numerically. Lock-in is identified when a definite phase relationship between vortex shedding and acoustic field remains unaltered with time. The common frequency during (phase) lock-in is either close to the natural acoustic or vortex shedding frequencies, accordingly termed A- or V-lock-in, respectively. Instability and amplitude suppression occur with most parts of the A- and V-lock-in regions, respectively. Furthermore, we relate the lock-in and pre-lock-in regimes observed in our previous experiments (Singh & Mariappan, Combust. Sci. Technol., 2019, pp. 1–29) to A- and V-lock-in phenomena, respectively. Among the two lock-in phenomena, combustion instability is favoured by 1:1$1:1$ A-lock-in, whose boundaries in the parameter space can be obtained from a forced response of the vortex shedding process. Finally, we discuss the bifurcations leading to lock-in.

We use the Dyson–Wyld diagrammatic technique to analyse the infinite series for the correlation functions of the velocity in hydrodynamic turbulence. We demonstrate the fundamental role played by the triple correlator of the velocity in determining the entire statistics of the hydrodynamic turbulence. All higher-order correlation functions are expressed through the triple correlator. This is shown through the suggested triangular re-summation of the infinite diagrammatic series for multi-point correlation functions. The triangular re-summation is the next logical step after the Dyson–Wyld line re-summation for the Green's function and the double correlator. In particular, it allows us to explain why the inverse cascade of the two-dimensional hydrodynamic turbulence is close to Gaussian. Since the triple correlator dictates the flux of energy \varepsilon$\varepsilon$ through the scales, we support the Kolmogorov-1941 idea that \varepsilon$\varepsilon$ is one of the main characteristics of hydrodynamic turbulence.