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.

A new arbitrary Lagrangian–Eulerian (ALE) formulation for Navier–Stokes flow on self-evolving surfaces is presented. It is based on a general curvilinear surface parameterisation that describes the motion of the ALE frame. Its in-plane part becomes fully arbitrary, while its out-of-plane part follows the material motion of the surface. This allows for the description of flows on deforming surfaces using only surface meshes. The unknown fields are the fluid density or pressure, the fluid velocity and the surface motion, where the latter two share the same normal velocity. The corresponding field equations are the continuity equation or area-incompressibility constraint, the surface Navier–Stokes equations and suitable surface mesh equations. Particularly advantageous are mesh equations based on membrane elasticity. The presentation focuses on the coupled set of strong and weak form equations, and presents several manufactured steady and transient solutions. These solutions are used together with numerical simulations to illustrate and discuss the properties of the proposed new ALE formulation. They also serve as basis for the development and verification of corresponding computational methods. The new formulation allows for a detailed study of fluidic membranes such as soap films, capillary menisci and lipid bilayers.

Discontinuous shear-thickening (DST) fluids exhibit unique instability properties in a wide range of flow conditions. We present numerical simulations of a scalar model for DST fluids in a planar simple shear using the smoothed particle hydrodynamics approach. The model reproduces the spatially homogeneous instability mechanism based on the competition between the inertial and microstructural time scales, with good congruence to the theoretical predictions. Spatial inhomogeneities arising from a stress-splitting instability are rationalised within the context of local components of the microstructure evolution. Using this effect, the addition of non-locality in the model is found to produce an alternative mechanism of temporal instabilities, driven by the inhomogeneous pattern formation. The reported arrangement of the microstructure is generally in agreement with the experimental data on gradient pattern formation in DST. Simulations in a parameter space representative of realistic DST materials resulted in aperiodic oscillations in measured shear rate and stress, driven by formation of gap-spanning frictional structures.

Underwater capillary tubes fill rapidly with the surrounding liquid. Capillary and hydrostatic pressures push the liquid into the tube, causing the air to exit as bubbles at the other end. We study the natural filling process of a vertical capillary tube immersed in water during several bubble formation events. A theoretical model is proposed that captures the dynamics of the meniscus inside the capillary tube as it fills with water. We find good agreement with the experimental data that describe this special case of spontaneous flow using a dynamic contact angle model based on molecular kinetic theory.

We investigate the influence of the Reynolds number on the spatial development of an incompressible planar jet. The study relies on direct numerical simulations (DNS) at inlet Reynolds numbers between $500 \leqslant Re \leqslant 13\,500$, being the widest range and the largest values considered so far in DNS. At the lowest $Re$, the flow is transitional and characterised by large quasi-two-dimensional vortices; at the largest $Re$, the flow reaches a fully turbulent regime with a well-developed self-similar region. We provide a complete description of the flow, from the instabilities in the laminar near-inlet region, to the self-similar regime in the turbulent far field. At the inlet, the leading destabilisation mode is sinusoidal/asymmetric at low Reynolds number and varicose/symmetric at large Reynolds number, with both modes coexisting at intermediate $Re$. In the far field, the mean and fluctuating statistics converge to self-similar profiles only for $Re\geqslant 4500$; the flow anisotropy, the budget of the Reynolds stresses and the energy spectra are addressed. The spreading of the jet is quantified via the turbulent–non-turbulent interface (TNTI). We find that the thickness of the turbulent region, and the shape and fractal dimension of the TNTI become $Re$-independent for $Re \geqslant 4500$. Comparisons with previous numerical and experimental works are provided whenever available.

A liquid film flowing down a fibre becomes unstable, leading to the formation of droplets that travel downstream. The droplet spacing and speed depend on the flow rate for a given nozzle and fibre radii. We show that fibre morphology further modifies the droplet spacing. In particular, we study the effect of the size of the beads in a granular chain on the evolution of the film thickness. We show that, when the size of the bead exceeds a critical value, the selection mechanism for instability modes is modified from regularly spaced droplets to coarsening by droplet merging. Droplet formation for flow over a single bead on the fibre is modified successively over subsequent beads in the downstream. Further, we show that if the perturbation in the flow produced by the bead is introduced as a velocity perturbation at the nozzle inlet, the formation of droplets on the fibre is qualitatively similar to that for the bead.