Turbulent Taylor-Couette flow displays traces of axisymmetric Taylor vortices even at high Reynolds numbers. With this motivation, Feldmann & Avila (2025) carry out long-time numerical simulations of axisymmetric high-Reynolds-number Taylor-Couette flow. They find that the Taylor vortices, using the only degree of freedom that remains available to them, carry out Brownian motion in the axial direction, with a diffusion constant that diverges as the number of rolls is reduced below a critical value.

Vertical thermal convection system exhibits weak turbulence and spatio-temporally chaotic behaviour. In this system, we report seven equilibria and 26 periodic orbits, all new and linearly unstable. These orbits, together with four previously studied in Zheng et al. (2024b) bring the number of periodic orbit branches computed so far to 30, all solutions to the fully non-linear three-dimensional Navier–Stokes equations. These new invariant solutions capture intricate spatio-temporal flow patterns including straight, oblique, wavy, skewed and distorted convection rolls, as well as bursts and defects in rolls. These interesting and important fluid mechanical processes in a small flow unit are shown to appear locally and instantaneously in a chaotic simulation in a large domain. Most of the solution branches show rich spatial and/or spatio-temporal symmetries. The bifurcation-theoretic organisation of these solutions is discussed; the bifurcation scenarios include Hopf, pitchfork, saddle–node, period-doubling, period-halving, global homoclinic and heteroclinic bifurcations, as well as isolas. These orbits are shown to be able to reconstruct statistically the core part of the attractor, and these results may pave the way to quantitatively describing transitional fluid turbulence using periodic orbit theory.

Parametric oscillations of an interface separating two fluid phases create nonlinear surface waves, called Faraday waves, which organise into simple patterns, such as squares and hexagons, as well as complex structures, such as double hexagonal and superlattice patterns. In this work, we study the influence of surfactant-induced Marangoni stresses on the formation and transition of Faraday-wave patterns. We use a control parameter, $B$ , that assesses the relative importance of Marangoni stresses as compared with the surface-wave dynamics. Our results show that the threshold acceleration required to destabilise a surfactant-covered interface through vibration increases with increasing $B$ . For a surfactant-free interface, a square-wave pattern is observed. As $B$ is incremented, we report transitions from squares to asymmetric squares, weakly wavy stripes and ultimately to ridges and hills. These hills are a consequence of the bidirectional Marangoni stresses at the neck of the ridges. The mechanisms underlying the pattern transitions and the formation of exotic ridges and hills are discussed.

Geophysical flows are characterized by rapid rotation. Simulating these flows requires small timesteps to achieve stability and accuracy. Numerical stability can be greatly improved by the implicit integration of the terms that are most responsible for destabilizing the numerical scheme. We have implemented an implicit treatment of the Coriolis force in a rotating spherical shell driven by a radial thermal gradient. We modified the resulting timestepping code to carry out steady-state solving via Newton’s method, which has no timestepping error. The implicit terms have the effect of preconditioning the linear systems, which can then be rapidly solved by a matrix-free Krylov method. We computed the branches of rotating waves with azimuthal wavenumbers ranging from 4 to 12. As the Ekman number (the non-dimensionalized inverse rotation rate) decreases, the flows are increasingly axially independent and localized near the inner cylinder, in keeping with well-known theoretical predictions and previous experimental and numerical results. The advantage of the implicit over the explicit treatment also increases dramatically with decreasing Ek, reducing the cost of computation by as much as a factor of 20 for Ekman numbers of order of 10−5. We carried out continuation for both the Rayleigh and Ekman numbers and obtained interesting branches in which the drift velocity remained unchanged between pairs of saddle–node bifurcations.

Experiments and numerical simulations have shown that turbulence in transitional wall-bounded shear flows such as plane Couette and Poiseuille flow frequently takes the form of long oblique bands, if the domains are sufficiently large to accommodate them. At their upper Reynolds-number threshold, laminar regions carve out gaps in otherwise uniform turbulence, thereby forming regular oblique turbulent-laminar patterns with a large spatial wavelength. At the lower threshold, isolated turbulent bands sparsely populate otherwise laminar domains and complete laminarization takes place via their disappearance characterized by the 2D directed percolation scenario.

Experiments and numerical simulations have shown that turbulence in transitional wall-bounded shear flows such as plane Couette and Poiseuille flow frequently takes the form of long oblique bands, if the domains are sufficiently large to accommodate them. At their upper Reynolds-number threshold, laminar regions carve out gaps in otherwise uniform turbulence, thereby forming regular oblique turbulent-laminar patterns with a large spatial wavelength. At the lower threshold, isolated turbulent bands sparsely populate otherwise laminar domains and complete laminarization takes place via their disappearance characterized by the 2D directed percolation scenario.

Vertical thermal convection is a non-equilibrium system in which both buoyancy and shear forces play a role in driving the convective flow. Beyond the onset of convection, the driven dissipative system exhibits chaotic dynamics and turbulence. In a three-dimensional domain extended in both the vertical and the transverse dimensions, Gao et al. (Phys. Rev. E, vol. 97, 2018, 053107) have observed a variety of convection patterns which are not described by linear stability analysis. We investigate the fully nonlinear dynamics of vertical convection using a dynamical-systems approach based on the Oberbeck-Boussinesq equations. We compute the invariant solutions of these equations and the bifurcations that are responsible for the creation and termination of various branches. We map out a sequence of local bifurcations from the laminar base state, including simultaneous bifurcations involving patterned steady states with different symmetries. This atypical phenomenon of multiple branches simultaneously bifurcating from a single parent branch is explained by the role of D-4 symmetry. In addition, two global bifurcations are identified: first, a homoclinic cycle from modulated transverse rolls and second, a heteroclinic cycle linking two symmetry-related diamond-roll patterns. These are confirmed by phase space projections as well as the functional form of the divergence of the period close to the bifurcation points. The heteroclinic orbit is shown to be robust and to result from a 1:2 mode interaction. The intricacy of this bifurcation diagram highlights the essential role played by dynamical systems theory and computation in hydrodynamic configurations.

Directed percolation (DP), a universality class of continuous phase transitions, has recently been established as a possible route to turbulence in subcritical wall-bounded flows. In canonical straight pipe or planar flows, the transition occurs via discrete large-scale turbulent structures, known as puffs in pipe flow or bands in planar flows, which either self-replicate or laminarize. However, these processes might not be universal to all subcritical shear flows. Here, we design a numerical experiment that eliminates discrete structures in plane Couette flow and show that it follows a different, simpler transition scenario: turbulence proliferates via expanding fronts and decays via spontaneous creation of laminar zones. We map this phase transition onto a stochastic one-variable system. The level of turbulent fluctuations dictates whether moving-front transition is discontinuous, or continuous and within the DP universality class, with profound implications for other hydrodynamic systems.

In a vertical channel driven by an imposed horizontal temperature gradient, numerical simulations have previously shown steady, time-periodic and chaotic dynamics. We explore the observed dynamics by constructing invariant solutions of the three-dimensional Oberbeck-Boussinesq equations, characterizing the stability of these equilibria and periodic orbits, and following the bifurcation structure of the solution branches under parametric continuation in Rayleigh number. We find that in a narrow vertically-periodic domain of aspect ratio ten, the flow is dominated by the competition between three and four co-rotating rolls. We demonstrate that branches of three and four-roll equilibria are connected and can be understood in terms of their discrete symmetries. Specifically, the D4 symmetry of the four-roll branch dictates the existence of qualitatively different intermediate branches that themselves connect to the three-roll branch in a transcritical bifurcation due to D3 symmetry. The physical appearance, disappearance, merging and splitting of rolls along the connecting branch provide a physical and phenomenological illustration of the equivariant theory of D3-D4 mode interaction. We observe other manifestations of the competition between three and four rolls, in which the symmetry in time or in the transverse direction is broken, leading to limit cycles or wavy rolls, respectively. Our work highlights the interest of combining numerical simulations, bifurcation theory, and group theory, in order to understand the transitions between and origin of flow patterns.

In a vertical channel driven by an imposed horizontal temperature gradient, numerical simulations (Gao et al., Phys. Rev. E, vol. 88, 2013, 023010; Phys. Rev. E, vol. 91, 2015, 013006; Phys. Rev. E, vol. 97, 2018, 053107) have previously shown steady, time-periodic and chaotic dynamics. We explore the observed dynamics by constructing invariant solutions of the three-dimensional Oberbeck-Boussinesq equations, characterizing the stability of these equilibria and periodic orbits, and following the bifurcation structure of the solution branches under parametric continuation in Rayleigh number. We find that in a narrow vertically periodic domain of aspect ratio 10, the flow is dominated by the competition between three and four co-rotating rolls. We demonstrate that branches of three- and four-roll equilibria are connected and can be understood in terms of their discrete symmetries. Specifically, the D(4 )symmetry of the four-roll branch dictates the existence of qualitatively different intermediate branches that themselves connect to the three-roll branch in a transcritical bifurcation due to D-3 symmetry. The physical appearance, disappearance, merging and splitting of rolls along the connecting branch provide a physical and phenomenological illustration of the equivariant theory of D-3-D-4 mode interaction. We observe other manifestations of the competition between three and four rolls, in which the symmetry in time or in the transverse direction is broken, leading to limit cycles or wavy rolls, respectively. Our work highlights the interest of combining numerical simulations, bifurcation theory and group theory, in order to understand the transitions between and origin of flow patterns

This paper is associated with a poster winner of a 2023 American Physical Society Division of Fluid Dynamics (DFD) Milton Van Dyke Award for the work presented at the DFD Gallery of Fluid Motion. The original poster is available online at the Gallery of Fluid Motion, https://doi.org/10.1103/APS.DFD.2023.GFM.P0030.

Convection in a vertical channel subjected to a horizontal temperature gradient is numerically investigated. Previous numerical simulations reveal a variety of behaviors: steady, time-periodic and chaotic. We extend previous numerical investigations by constructing stable and unstable branches of equilibria and periodic orbits of the underlying Oberbeck-Boussinesq equations by parametric continuation. In a narrow domain of vertical aspect ratio ten, the observed flow structures are dominated by the competition between three and four co-rotating rolls. We identify the invariant solution branches underlying the observed flow dynamics, characterize the bifurcations creating those branches and link the observed bifurcation scenarios to the symmetries of the primary branches. Specifically, we show that D4 symmetry dictates that several intermediate branches bifurcate simultaneously from the four-roll branch, while D3 symmetry requires that the intersection of these intermediate branches with the three-roll branch be transcritical. We observe other manifestations of the competition between three and four rolls, in which the symmetry in time or in the transverse direction is broken, leading to limit cycles or wavy rolls, respectively. Our work highlights the interest of combining numerical simulations, bifurcation theory, and group theory, in order to understand the transitions between and origin of flow patterns.

Low Reynolds number turbulence in wall-bounded shear flows \emph{en route} to laminar flow takes the form of oblique, spatially-intermittent turbulent structures. In plane Couette flow, these emerge from uniform turbulence via a spatiotemporal intermittent process in which localised quasi-laminar gaps randomly nucleate and disappear. For slightly lower Reynolds numbers, spatially periodic and approximately stationary turbulent-laminar patterns predominate. The statistics of quasi-laminar regions, including the distributions of space and time scales and their Reynolds number dependence, are analysed. A smooth, but marked transition is observed between uniform turbulence and flow with intermittent quasi-laminar gaps, whereas the transition from gaps to regular patterns is more gradual. Wavelength selection in these patterns is analysed via numerical simulations in oblique domains of various sizes. Via lifetime measurements in minimal domains, and a wavelet-based analysis of wavelength predominance in a large domain, we quantify the existence and non-linear stability of a pattern as a function of wavelength and Reynolds number. We report that the preferred wavelength maximises the energy and dissipation of the large-scale flow along laminar-turbulent interfaces. This optimal behaviour is primarily due to the advective nature of the large-scale flow, with turbulent fluctuations playing only a secondary role.

This paper is associated with a poster winner of a 2022 American Physical Society's Division of Fluid Dynamics (DFD) Gallery of Fluid Motion Award for work presented at the DFD Gallery of Fluid Motion. The original poster is available online at the Gallery of Fluid Motion, https://doi.org/10.1103/APS.DFD.2022.GFM.P0027

Low Reynolds number turbulence in wall-bounded shear flows en route to laminar flow takes the form of spatially intermittent turbulent structures. In plane shear flows, these appear as a regular pattern of alternating turbulent and quasi-laminar flow. Both the physical and the spectral energy balance of a turbulent-laminar pattern in plane Couette flow are computed and compared to those of uniform turbulence. In the patterned state, the mean flow is strongly modulated and is fuelled by two mechanisms: primarily, the nonlinear self-interaction of the mean flow (via mean advection), and secondly, the extraction of energy from turbulent fluctuations (via negative spectral production, associated with an energy transfer from small to large scales). Negative production at large scales is also found in the uniformly turbulent state. Important features of the energy budgets are surveyed as a function of Re through the transition between uniform turbulence and turbulent-laminar patterns.

Transitional localized turbulence in shear flows is known to either decay to an absorbing laminar state or to proliferate via splitting. The average passage times from one state to the other depend superexponentially on the Reynolds number and lead to a crossing Reynolds number above which proliferation is more likely than decay. In this paper, we apply a rare-event algorithm, Adaptative Multilevel Splitting, to the deterministic Navier-Stokes equations to study transition paths and estimate large passage times in channel flow more efficiently than direct simulations. We establish a connection with extreme value distributions and show that transition between states is mediated by a regime that is self-similar with the Reynolds number. The super-exponential variation of the passage times is linked to the Reynolds number dependence of the parameters of the extreme value distribution. Finally, motivated by instantons from Large Deviation theory, we show that decay or splitting events approach a most-probable pathway. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 2)'.

For the Faraday instability, by which standing waves appear on the free surface of a vertically vibrated fluid layer, the wavelength is controlled by the forcing frequency rather than by the fluid depth, making it easy to destabilize multiple wavelengths everywhere simultaneously. In the 1990s, this technique was used to produce fascinating new phenomena such as quasipatterns by Edwards Fauve and superlattices by Gollub, Pier Kudrolli. This in turn sparked a renaissance of interest in Faraday waves, leading to new mathematical theories and numerical simulations. We will discuss some of the exotic patterns found in recent numerical simulations, such as quasi-hexagons alternating with beaded stripes, a supersquare divided into four subsquares with synchronized diagonal blocks, Platonic solids alternating with their duals while drifting, and a twisted sheared secondary instability of square waves.

Generically, a local bifurcation only affects a single solution branch. However, branches that are quite different may nonetheless share certain eigenvectors and eigenvalues, leading to coincident bifurcations. For the fluidic pinball, two supercritical pitchfork bifurcations, of the equilibrium and the periodic solutions, occur at nearly the same Reynolds number. The mechanism of this kind of non-generic coincidence is modelled and explained.

A number of approximations have been proposed to estimate basic hydrodynamic quantities, in particular the frequency of a limit cycle. One of these, RZIF (for Real Zero Imaginary Frequency), calls for linearizing the governing equations about the mean flow and estimating the frequency as the imaginary part of the leading eigenvalue. A further reduction, the SCM (for Self-Consistent Model), approximates the mean flow as well, as resulting only from the nonlinear interaction of the leading eigenmode with itself. Both RZIF and SCM have proven dramatically successful for the archetypal case of the wake of a circular cylinder. Here, the SCM is applied to thermosolutal convection, for which a supercritical Hopf bifurcation gives rise to branches of standing waves and traveling waves. The SCM is solved by means of a full Newton method coupling the approximate mean flow and leading eigenmode. Although the RZIF property is verified for the traveling waves, the SCM reproduces the nonlinear frequency only very near the onset of the bifurcation and for another isolated parameter value. Thus, the nonlinear interaction arising from the leading mode is insufficient to reproduce the nonlinear mean field and frequency.