We study decaying turbulence in the one-dimensional (1-D) Burgers equation (Burgulence) and 3-D Navier–Stokes (NS) turbulence. We first investigate the decay in time $t$ of the energy $E(t)$ in Burgulence, for a fractional Brownian initial potential, with Hurst exponent $H$, and demonstrate rigorously a self-similar time decay of $E(t)$, previously determined heuristically. This is a consequence of the non-trivial boundedness of the energy for any positive time. We define a spatially forgetful oblivious fractional Brownian motion (OFBM), with Hurst exponent $H$, and prove that Burgulence, with an OFBM as initial potential $\varphi _0(x)$, is not only intermittent, but it also displays a, hitherto unanticipated, large-scale bifractality or multifractality; the latter occurs if we combine OFBMs, with a distribution of $H\hbox{-}$values. This is the first rigorous proof of genuine multifractality for turbulence in a nonlinear hydrodynamical partial differential equation. We then present direct numerical simulations (DNSs) of freely decaying turbulence, capturing some aspects of this multifractality. For Burgulence, we investigate such decay for two cases: (a) $\varphi _0(x)$ a multifractal random walk that crosses over to a fractional Brownian motion beyond a cross-over scale $\mathcal{L}$, tuned to go from small- to large-scale multifractality; (b) initial energy spectra $E_0(k)$, with wavenumber $k$, having one or more power-law regions, which lead, respectively, to self-similar and non-self-similar energy decay. Our analogous DNSs of the 3-D NS equations also uncover self-similar and non-self-similar energy decay. Challenges confronting the detection of genuine large-scale multifractality, in numerical and experimental studies of NS and Magnetohydrodynamics turbulence, are highlighted.

Direct numerical simulation (DNS) of a Mach 4.9 zero-pressure-gradient turbulent boundary layer spatially developing over a cooled flat plate at wall-to-recovery temperature $T_w/T_r = 0.60$ is performed. Very long, streamwise contiguous domains are used in the DNS to achieve a wide continuous range of ‘useful’ friction Reynolds numbers of $1000 \lesssim {Re}_\tau \lesssim 2500$. The DNS datasets have been analysed to assess state-of-the-art compressibility scaling relations and turbulence modelling assumptions. The DNS data show a notable distinction in Reynolds number dependence between thermal and velocity fields. Although Reynolds stress and the budgets of turbulent kinetic energy have reached Reynolds number independence in the inner layer under semi-local scaling by ${Re}_\tau \simeq 1000$, the budget terms for temperature variance and turbulent heat flux retain a clear Reynolds number dependence near the wall over a broader range up to ${Re}_\tau \simeq 1900$. Such a stronger dependence of the thermal field on the Reynolds number may lead to inaccuracy in turbulence models that are calibrated on the basis of low-Reynolds-number data. Spectral and structural analysis suggests a more significant reduction in the prevalence of alternating positive and negative structures and an increase in the streamwise uniformity of streaks in the wall heat flux $q_w$ than in the wall shear stress $\tau _w$ when the Reynolds number increases.

This paper presents numerical results for Rayleigh–Bénard convection with suspended particles at Rayleigh numbers $Ra=10^7$ and $10^8$, and unit Prandtl number. Accounting for their finite size makes it possible to investigate in detail the mechanism by which the particles, which are 10 % heavier than the fluid, get resuspended after settling, thus maintaining a two-phase circulating flow. It is shown that an essential component of this mechanism is the formation of particle accumulations, or ‘dunes’, on the bottom of the Rayleigh–Bénard cell. Ascending plumes become localised on these dunes. Particles are dragged up the dune slopes, and when they reach the top, are entrained into the rising plumes. Direct resuspension of particles from the cell bottom, if it happens at all, is very rare. For $Ra=10^7$, aspect ratios (width/height) $\Gamma =1,2,4$ are considered. It is found that in these and in the other cases simulated, at steady state, a single dune evolves, the largest linear dimension of which is comparable to the cell size. A remarkable consequence is that even at the low volume fraction considered here, 3.27 %, the particles are able to structure the flow and to determine the size and position of the largest ascending plumes. Their effect on the Nusselt number, however, remains small. This and other results are explained on the basis of the ratio of the cell-bottom viscous boundary-layer thickness to the particle diameter.

Turbidity currents (TCs) are a common kind of particle-laden flow in underwater natural environments. This work employs a Eulerian–Lagrangian model to investigate the dynamic regimes of lock-exchange TC in a moderate flow Reynolds number range (${Re} = 1716-3836$) as well as the formation and evolution mechanisms of interfacial Kelvin–Helmholtz (KH) billows composed of a fluid–particle mixture. The results demonstrate that a fluid streak with high stretching at the interface, which twists and takes on a braided structure, is the key to the onset of KH instability. An increase in ${\textit{Re}}$ results in a higher interfacial fluid velocity gradient that intensifies the shear instability, and an increase in the convergent fluid force acting on the particles. This provides an explanation for the significant increases both in quantity and strength of KH vortices as ${\textit{Re}}$ rises. The enhanced KH vortices contribute to particle suspension and streamwise transport at larger ${\textit{Re}}$, leading to an extension in the duration of the slumping stage, which exhibits a constant forward velocity regime. The spatially continuous braided structure in the vorticity sheet region is responsible for the intriguing merging phenomenon of interfacial vortices. Furthermore, TC kinetic energy increases with the increasing ${\textit{Re}}$, and the system dissipation rate decreases in the early and middle stages of the TC. This behaviour may be correlated to the reducing shear between the TC and ambient fluid by interfacial KH billows. Regarding the turbulent kinetic energy dissipation of interfacial vortices, normal strain predominates in the middle stage, while shear deformation is most prevalent in the early and later stages.

Flow through a square-duct at a moderate Reynolds number is investigated. We first employ an edge-tracking procedure in the $\pi$-rotationally symmetric sub-space of state space and identify a streamwise-localised invariant solution for square-duct flow, which is a steady travelling wave with mirror symmetries across bisectors of the duct walls. The identified invariant solution features four vortices placed in pairs at opposite duct walls and exhibits significant streamwise localisation making it the first reported localised solution in the square-duct flow. Additionally, this solution remains very close to the laminar attractor in the sense of the velocity perturbation energy and the corresponding hydraulic losses. Stability analysis of this solution demonstrates that the identified state is an edge state in the $\pi$-rotationally symmetric sub-space but not in the full space. Next, a long-time turbulence behaviour and its relevance to the symmetric streamwise-localised invariant solution are discussed. We focus on the characteristics of the averaged flow and the recurring patterns of eight- or four-vortex states, typical for the square-duct flow and related to Prandtl’s secondary flows of the second type. Through heuristic arguments, we illustrate that turbulent flow exhibits relatively quiescent interludes of increased symmetry of the velocity field across wall bisectors. We show that those periods correlate to episodes where, statistically, a four-vortex flow configuration emerges from the otherwise eight-vortex state, which is also associated with decreased symmetry of the flow field. Our results suggest that the four-vortex state appearing in the relatively quiescent periods in the flow time history, accompanied by flow field symmetrisation and the onset of streamwise localisation of turbulent flow, bears a striking similarity to the found symmetric streamwise-localised invariant solution.

This study investigates the fluid mechanisms underlying the interaction between ventilated shoulder and tail cavities under vertical launching conditions. It is found that expansion and contraction coexist within the tail cavity. When the expansion rate exceeds the contraction rate, the volume of the tail cavity increases; conversely, it decreases. Through this process, the cavity undergoes cyclic pulsation during its vertical evolution, including expansion, over-expansion, contraction and over-contraction. Before the shoulder cavity extends to the position of the tail cavity, wall confinement restricts the tail cavity from expanding towards the vehicle’s lateral wall. After the encounter between the shoulder and tail cavities, the re-entrant flow at the end of the shoulder cavity induces the tail cavity to overcome wall confinement and expand towards the lateral wall, initiating their fusion. As a result, a supercavity forms and attaches to the surface of the vehicle. Moreover, after the fusion, the pressure driving mode at the vehicle’s bottom wall shifts from the tail cavity pulsation to the re-entrant flow. In addition, an increase in the ventilation rate induces progressive expansion of the shoulder cavity’s radial dimension, and accelerates its downstream propagation. The fusion mode between the shoulder and tail cavities transitions from progressive fusion to coverage fusion.

This work experimentally explores the alignment of the vorticity vector and the strain-rate tensor eigenvectors at locations of extreme upscale and downscale energy transfer. We show that the turbulent von Kármán flow displays vorticity–strain alignment behaviour across a large range of Reynolds numbers, which is very similar to previous studies on homogeneous, isotropic turbulence. We observe that this behaviour is amplified for the largest downscale energy transfer events, which tend to be associated with sheet-like geometries. These events are also shown to have characteristics previously associated with high flow field nonlinearity and singularities. In contrast, the largest upscale energy transfer events display very different structures which showcase a strong preference for vortex compression. Notably, in both cases we find that these trends are strengthened as the probed scales approach the Kolmogorov scale. We then show further evidence for the argument that strain self-amplification is the most salient feature in characterising the cascade direction. Finally, we identify possible invariant behaviour for the largest energy transfer events, even at scales near the Kolmogorov scale.

The presence of trapped air on a solid surface can alter the direction of the liquid jets induced by cavitation bubbles, which prevents or reduces erosion. In this study, we numerically investigate mutual interaction between air trapped in a pocket on a wall and a nearby bubble in water, as well as the resultant hydrodynamic loading. Both the depth and radius of the cylindrical pocket are similar to the maximum bubble radius. The pressure imposed on the inner wall of the air pocket is assessed for various values of the air pocket size and the stand-off parameter. The deformation of the air pocket and the bubble is analysed in each of three sequential stages. During the bubble expansion stage, a shock wave reflects at the water–air interface of the pocket, and the wall inside the compressed pocket is protected from the shock wave. As the bubble jet induced during bubble contraction tends to move away from the air pocket, other liquid jets formed at the water–air interface, namely central and lateral pocket jets, can directly collide with the inner wall of the pocket after the bubble collapses. These collisions exert significant pressure on the wall under certain conditions. The formation of the central pocket jet originates from the strong fluctuation of the water–air interface by the expanding and contracting bubble. The development of the lateral pocket is related to changes in the potential energy of the air under its second contraction.

We use direct numerical simulations to examine the onset of stratified turbulence triggered by the zigzag instability recently identified in columnar Taylor–Green vortices (Guo et al. 2024, J. Fluid Mech., vol. 997, A34) and its role in layer formation within the flow. The study focuses on Froude numbers $0.125 \leqslant \textit{Fr} \leqslant 2.0$ and Reynolds numbers ${\textit{Re}}$ ranging from 800 to 3200. The breakdown of the freely evolving vortex array is driven by local density overturns, combining shear and convective mechanisms initiated by the primary zigzag instability. Our results show a linear relationship between the peak buoyancy Reynolds number ${{\textit{Re}}}_b^{\star }$, driven by the zigzag instability, and ${\textit{Re}}\, {\textit{Fr}}^2$. When the flow does not exhibit local shear or convective instability, the value of ${{\textit{Re}}}_b^{\star }$ falls below unity. Both density and momentum layers arise from the zigzag instability: horizontal velocity layers are strong and persistent, while density layers are weaker and more transient. The vertical scale of the mean shear layers increases with ${\textit{Fr}}$ for ${\textit{Fr}} \leqslant 1$, shows weak dependence on ${\textit{Re}}$, and agrees well with the length scale associated with the fastest-growing linear mode of the zigzag instability. Further analysis in the sorted buoyancy coordinate highlights the role of density overturns caused by the zigzag instability in forming buoyancy layers during the transition to turbulence.

Over-expansion flow can generate asymmetric shock wave interactions, which lead to significant lateral forces on a nozzle. However, there is still a lack of a suitable theory to explain the phenomenon of asymmetry. The current work carefully investigates the configurations of shock wave interactions in a planar nozzle, and proposes a theoretical method to analyse the asymmetry of over-expansion flows. First, various possible flow patterns of over-expansion flows are discussed, including regular and Mach reflections. Second, the free interaction theory and the minimum entropy production principle are used to analyse the boundary layer flow and main shock wave interactions, establish the relationship between the separation shock strength and separation position, and predict asymmetric configurations. Finally, experiments are conducted to validate the theoretical method, and similar experiments from other studies are discussed to demonstrate the effectiveness of the proposed method. Results demonstrate that the direction of asymmetric over-expansion flow is random, and the separated flow strives to adopt a pattern with minimal total pressure loss. Asymmetric interaction is a mechanism through which the flow can achieve a more efficient thermodynamic balance by minimising entropy production.

Optimal transitional mechanisms are analysed for an incompressible shear layer developing over a short, pressure gradient-induced laminar separation bubble (LSB) with peak reversed flow of 2 %. Although the bubble remains globally stable, the shear layer destabilises due to the amplification of external time- and spanwise-periodic disturbances. Using linear resolvent analysis, we demonstrate that the pressure gradient modifies boundary layer receptivity, shifting from Tollmien–Schlichting (T-S) waves and streaks in a zero-pressure-gradient environment to Kelvin–Helmholtz (K-H) and centrifugal instabilities in the presence of the LSB. To characterise the nonlinear evolution of these disturbances, we employ the harmonic-balanced Navier–Stokes (N-S) framework, solving the N-S equations in spectral space with a finite number of Fourier harmonics. Additionally, adjoint optimisation is incorporated to identify forcing disturbances that maximise the mean skin friction drag, conveniently chosen as the cost function for the optimisation problem since it is commonly observed to increase in the transitional stage. Compared with attached boundary layers, this transition scenario exhibits both similarities and differences. While oblique T-S instability is replaced by oblique K-H instability, both induce streamwise rotational forcing through the quadratic nonlinearity of the N-S equations. However, in separated boundary layers, centrifugal instability first generates strong streamwise vortices due to multiple centrifugal resolvent modes, which then develop into streaks via lift-up. Finally, we show that the progressive distortion and disintegration of K-H rollers, driven by streamwise vortices, lead to the breakdown of large coherent structures.

The evolution of settling fine particle clouds in transition or rarefied flow regimes is a fundamental yet insufficiently understood problem in fluid mechanics. Here, we address this challenge numerically using a kinematic model, and approximate the hydrodynamic interaction between particles by superposing velocity disturbances from rarefied gas flows past individual particles. The effect of electrostatic interactions among charged particles is also studied. As an application, we simulate the sedimentation of small dust clouds under Martian conditions, focusing on the 10$\,\unicode{x03BC}$m diameter fraction of ‘settled dust’. Our results show that under Martian conditions, dust clouds develop elongated tails during sedimentation, with up to 25 % of particles leaking from the bulk over a 10 minute period. Unlike Earth-based scenarios, the clouds do not break apart owing to the weaker hydrodynamic interactions in Mars’ thin atmosphere. By examining the interplay between hydrodynamic and electrostatic interactions, which influence particle leakage in opposite ways, we demonstrate that larger dust clouds are also likely to evolve with sustained tail formation. Fully suppressing particle leakage would require particle charges well above $10^4e$, levels unlikely to occur under typical Martian conditions. New analytical expressions are derived for the cloud settling velocity and tail evolution, providing theoretical insights and a foundation for future studies on particle dynamics in transition/rarefied environments.

Droplet impacts with rough surfaces described by Fourier series are investigated assuming gas cushioning is negligible. For impacts leading to a contiguous contact patch, a mixed boundary value problem for the displacement potential is formulated by extending models of inertially dominated droplet impacts with a flat plate. For large times after impact, the contact line evolution for impacts with periodic rough substrates is found to tend to the contact line evolution obtained for a droplet impact with a flat plate vertically positioned at the average height of the rough substrate. For symmetric impacts with even substrate geometries represented by Fourier cosine series, the contact line evolution is given by a Schlömilch series in which the coefficients are related to the coefficients of the corresponding Fourier series. A method for determining whether secondary impacts occur for particular geometries is described and regime diagrams, which show the boundary of the region of substrate parameters associated with single contiguous impacts, are obtained. The loads associated with droplet impacts with periodic rough substrates are calculated and compared with the loads associated with impacts with a flat plate. As the height of the roughness increases, the load associated with an impact with a rough substrate may initially differ significantly from the flat-plate case, although the load on a flat plate is recovered in the limit of large time. The implications of the results for more general droplet impacts with roughness are discussed from both a theoretical and experimental standpoint.

A model for obtaining scaling laws for Rayleigh–Bénard convection (RBC) at high Rayleigh numbers in tall, slender cells (cells with low aspect ratio, $\varGamma = d/H \ll 1$) is presented. Traditional RBC ($\varGamma \gtrsim 1$) is characterised by large-eddy circulation scaling with the height of the cell, a near-isothermal core and almost all the thermal resistance provided at the horizontal walls. In slender RBC cells, on the other hand, away from the horizontal walls, tube-like convection with eddies scaling with the tube diameter and a linear temperature gradient driving the convective flow is present. The crux of our approach is to split the cell into two components: (i) ‘wall convection’ near the top and bottom horizontal walls and (ii) ‘tube convection (TC)’ in the central part away from the walls. By applying the scaling relations for both wall convection and TC, and treating the total thermal resistance as a sum of their contributions, unified scaling relations for Nusselt number, Reynolds number and mean vertical temperature gradient in slender RBC cells are developed. Our model is applicable for high enough Rayleigh numbers, such that convection both at the wall and in the tube are turbulent. Our model predictions compare well with the data from various studies in slender RBC cells where these conditions are satisfied. In particular, the effects of $\varGamma$ and Prandtl number are well captured. We propose a scaled aspect ratio using which we obtain ‘universal’ correlations for the heat flux and for the fractional temperature drop in the tube that include the effects of Rayleigh and Prandtl numbers. The profiles of suitably scaled horizontal and vertical velocity fluctuations, along with estimates for boundary layer thickness near the horizontal walls, and the radial distribution of the velocity fluctuations in the tube part are also presented.

We develop an optimal resolvent-based estimator and controller to predict and attenuate unsteady vortex-shedding fluctuations in the laminar wake of a NACA 0012 airfoil at an angle of attack of 6.5°, chord-based Reynolds number of 5000 and Mach number of 0.3. The resolvent-based estimation and control framework offers several advantages over standard methods. Under equivalent assumptions, the resolvent-based estimator and controller reproduce the Kalman filter and LQG controller, respectively, but at substantially lower computational cost using either an operator-based or data-driven implementation. Unlike these methods, the resolvent-based approach can naturally accommodate forcing terms (nonlinear terms from Navier–Stokes) with coloured-in-time statistics, significantly improving estimation accuracy and control efficacy. Causality is optimally enforced using a Wiener–Hopf formalism. We integrate these tools into a high-performance-computing-ready compressible flow solver and demonstrate their effectiveness for estimating and controlling velocity fluctuations in the wake of the airfoil immersed in clean and noisy free streams, the latter of which prevents the flow from falling into a periodic limit cycle. Using four shear–stress sensors on the surface of the airfoil, the resolvent-based estimator predicts a series of downstream targets with approximately $3\,\%$ and $30\,\%$ error for the clean and noisy free stream conditions, respectively. For the latter case, using four actuators on the airfoil surface, the resolvent-based controller reduces the turbulent kinetic energy in the wake by $98\,\%$.

Hypersonic transition studies on systems sustaining multimodal dynamics are critical to understanding aerothermal loading on flight-relevant configurations. The present work evaluates transition mechanisms in hypersonic boundary layers over a cone–cylinder–flare geometry, and its sensitivity to free stream disturbance amplitudes, using a global linear stability approach and direct numerical simulations (DNS). Under relatively quiet conditions, the flow field resembles the laminar solution, consisting of a large separation zone over the cylinder–flare junction. Linear analysis identifies multiple convective instabilities including, oblique first modes and two-dimensional second modes over the cone segment, and shear layer instabilities over the separation zone. This separation zone also supports a stationary global instability, producing streamwise streaks with an azimuthal wavenumber, $m=21$, which eventually drives transition as captured in the DNS. Conversely, at higher disturbance amplitudes, the largely attached boundary layer transitions through a bypass mechanism, involving intermodal interactions between low-frequency streaks, and first mode instabilities. The resulting upstream shift in transition onset leads to a significant rise in both steady and unsteady surface loading. Peak thermal loading under quiet conditions displays the signature of the linear global instability over the flare, whereas that under noisier environments is dominated by an imprint of unsteady Görtler vortices over the cylinder–flare junction.