Turbulence is a complex nonlinear and multi-scale phenomenon. Although the fundamental underlying equations, the Navier–Stokes equations, have been known for two centuries, it remains extremely challenging to extract from them the statistical properties of turbulence. Therefore, for practical purposes, a sustained effort has been devoted to obtaining an effective description of turbulence, that we may call turbulence modelling, or statistical theory of turbulence. In this respect, the renormalisation group (RG) appears as a tool of choice, since it is precisely designed to provide effective theories from fundamental equations by performing in a systematic way the average over fluctuations. However, for Navier–Stokes turbulence, a suitable framework for the RG, allowing in particular for non-perturbative approximations, has been missing, which has thwarted RG applications for a long time. This framework is provided by the modern formulation of the RG called the functional renormalisation group (FRG). The use of the FRG has enabled important progress in the theoretical understanding of homogeneous and isotropic turbulence. The major one is the rigorous derivation, from the Navier–Stokes equations, of an analytical expression for any Eulerian multi-point multi-time correlation function, which is exact in the limit of large wavenumbers. We propose in this JFM Perspectives article a survey of the FRG method for turbulence. We provide a basic introduction to the FRG and emphasise how the field-theoretical framework allows one to systematically and profoundly exploit the symmetries. We stress that the FRG enables one to describe fully developed turbulence forced at large scales, which was not accessible by perturbative means. We show that it yields the energy spectrum and second-order structure function with accurate estimates of the related constants, and also the behaviour of the spectrum in the near-dissipative range. Finally, we expound the derivation of the spatio-temporal behaviour of $n$-point correlation functions, and largely illustrate these results through the analysis of data from experiments and direct numerical simulations.

Pattern-forming with externally imposed symmetry is ubiquitous in nature but little studied. We present experimental studies of pattern formation and selection by spatial periodic forcing in rapidly rotating convection. When periodic topographic structures are constructed on the heated boundary, they modulate the local temperature and velocity fields. Symmetric convection patterns in the form of regular vortex lattices are observed near the onset of convection, when the periodicity of the external forcing is set close to the intrinsic vortex spacing. We show that the new patterns arise as a dynamical process of imperfect bifurcation which is well described by a Ginzburg–Landau-like model. We explore the phase diagram of buoyancy strength and periodicity of external forcing to find the optimal experimental settings for which the vortex patterns best match that of the external forcing.

We investigate the modulation of turbulence caused by the presence of finite-size dispersed particles. Bluff (isotropic) spheres versus slender (anisotropic) fibres are considered to understand the influence of the shape of the objects on altering the carrier flow. While at a fixed mass fraction – but different Stokes number – both objects provide a similar bulk effect characterized by a large-scale energy depletion, a scale-by-scale analysis of the energy transfer reveals that the alteration of the whole spectrum is intrinsically different. For bluff objects, the classical energy cascade shrinks in its extension but is unaltered in the energy content and its typical features, while for slender ones we find an alternative energy flux which is essentially mediated by the fluid–solid coupling.

It has been reported that a fully localized turbulent band in channel flow becomes sustained when the Reynolds number is above a threshold. Here we show evidence that turbulent bands are of a transient nature instead. When the band length is controlled to be fixed, the lifetime of turbulent bands appears to be stochastic and exponentially distributed, a sign of a memoryless transient nature. Besides increasing with the Reynolds number, the mean lifetime also strongly increases with the band length. Given that the band length always changes over time in real channel flow, this size dependence may translate into a time dependence, which needs to be taken into account when clarifying the relationship between channel flow transition and the directed percolation universality class.

The shape of a sessile drop on a horizontal substrate depends upon the Bond number $Bo$ and the contact angle $\alpha$. Inspired by puddle approximations at large $Bo$ (Quéré, Rep. Prog. Phys., vol. 68, 2005, p. 2495), we address here the limit of small contact angles at fixed drop volume and arbitrary $Bo$. It readily leads to a pancake shape approximation, where the drop height and radius scale as $\alpha$ and $\alpha ^{-1/2}$, respectively, with capillary forces being appreciable only near the edge. The pancake approximation breaks down for $Bo=\textrm {ord}(\alpha ^{2/3})$. In that distinguished limit, capillary and gravitational forces are comparable throughout, and the drop height and radius scale as $\alpha ^{2/3}$ and $\alpha ^{-1/3}$, respectively. For $Bo\ll \alpha ^{2/3}$ these scalings remain, with the drop shape turning into a spherical cap. The asymptotic results are compared with a numerical solution of the exact problem.

Using a multiple-scale analysis, Walker et al. (J. Fluid Mech., vol. 944, 2022, R2) obtain the long-time behaviour of a shape-changing swimmer in a Poiseuille flow. They show that the behaviour falls into one of three categories: endless tumbling at increasing distance from the midline of the flow; preserved initial behaviour of the swimmer; or convergence to upstream rheotaxis, where the swimmer is situated at the midline of the flow. Furthermore, a single swimmer-dependent constant is identified that determines which of the three behaviours is realised.

We report on theoretical and experimental studies describing the buoyancy-driven ascent of a Taylor long drop in a circular vertical pipe where the descending fluid is Newtonian, and the ascending fluid is non-Newtonian yield shear thinning and described by the three-parameter Herschel–Bulkley model, including the Ostwald–de Waele model as a special case for zero yield. Results for the Ellis model are included to provide a more realistic description of purely shear-thinning behaviour. In all cases, lubrication theory allows us to obtain the velocity profiles and the corresponding integral variables in closed form, for lock-exchange flow with a zero net flow rate. The energy balance allows us to derive the asymptotic radius of the inner current, corresponding to a stable node of the differential equation describing the time evolution of the core radius. We carried out a series of experiments measuring the rheological properties of the fluids, the speed and the radius of the ascending long drop. For some tests, we measured the velocity profile with the ultrasound velocimetry technique. The measured radius of the ascending current compares fairly well with the asymptotic radius as derived through the energy balance, and the measured ascent speed shows a good agreement with the theoretical model. The measured velocity profiles also agree with their theoretical counterparts. We have also developed dynamic similarity conditions to establish whether laboratory physical models, limited by the availability of real fluids with defined rheological characteristics, can be representative of real phenomena on a large scale, such as exchanges in volcanic conduits. Appendix B contains scaling rules for the approximated dynamic similarity of the physical process analysed; these rules serve as a guide for the design of experiments reproducing real phenomena.

An experiment was performed using stereo particle image velocimetry (SPIV) in the Laboratoire de Mécanique des Fluides de Lille boundary layer facility to determine all the derivative moments needed to estimate the average dissipation rate of the turbulence kinetic energy $\epsilon = 2 \nu \langle {\mathsf{s}}_{ij}{\mathsf{s}}_{ij} \rangle$, where ${\mathsf{s}}_{ij}$ is the fluctuating strain rate and $\langle ~\rangle$ denotes ensemble averages. Also measured were all the moments of the full average deformation rate tensor, as well as all of the first, second and third fluctuating velocity moments except those involving pressure. The Reynolds number was $Re_\theta = 7634$ or $Re_\tau = 2598$. The present paper gives the measured average dissipation, $\epsilon$ and the derivative moments comprising it. The results are compared with the earlier measurements of Balint, Wallace & Vukolavcevic (J. Fluid Mech., vol. 228, 1991, pp. 53–86) and Honkan & Andreopoulos (J. Fluid Mech., vol. 350, 1997, pp. 29–96) at lower Reynolds numbers and to new results from a plane channel flow DNS at comparable Reynolds number. Of special interest is the prediction by George & Castillo (Appl. Mech. Rev., vol. 50, 1997, pp. 689–729) and Wosnik, Castillo & George (J. Fluid Mech., vol. 421, 2000, pp. 115–145) that $\epsilon ^+ \propto {x_2^+}^{-1}$ for streamwise homogeneous flows and a nearly indistinguishable power law, $\epsilon \propto {x_2^+}^{\gamma -1}$, for boundary layers. In spite of the modest Reynolds number, the predictions seem to be correct. Then the statistical character of the velocity derivatives is examined in detail, and a particular problem is identified with the breakdown of local homogeneity inside $x_2^+ = 100$. A more general alternative for partially homogeneous turbulence flows is offered which is consistent with the observations. With the help of DNS, the spatial characteristics of the dissipation very near the wall are also examined in detail.

Point-particle direct numerical simulations have been employed to quantify the turbulence modulation and particle responses in a turbulent particle-laden jet in the two-way coupled regime with an inlet Reynolds number based on bulk velocity and jet diameter $({D_j})$ of ~10 000. The investigation focuses on three cases with inlet bulk Stokes numbers of 0.3, 1.4 and 11.2. Special care is taken to account for the particle–gas slip velocity and non-uniform particle concentrations at the nozzle outlet, enabling a reasonable prediction of particle velocity and concentration fields. Turbulence modulation is quantified by the variation of the gas-phase turbulent kinetic energy (TKE). The presence of the particle phase is found to damp the gas-phase TKE in the near-field region within $5{D_j}$ from the inlet but subsequently increases the TKE in the intermediate region of (5–20)Dj. An analysis of the gas-phase TKE transport equation reveals that the direct impact of the particle phase is to dissipate TKE via the particle-induced source term. However, the finite inertia of the particle phase affects the gas-phase velocity gradients, which indirectly affects the TKE production and dissipation, leading to the observed TKE attenuation and enhancement. Particle response to the gas-phase flow is quantified. Particles are found to exhibit notably stronger response to the gas-phase axial velocity than to the radial velocity. A new dimensionless figure is presented that collapses both the axial and radial components of the particle response as a function of the local Stokes number based on their respective integral length scales.

We investigate the instabilities and transition mechanisms of Boussinesq stratified boundary layers on sloping boundaries when subjected to oscillatory body forcing parallel to the slope. We examine idealized forms of boundary layers on hydraulically smooth abyssal slopes in tranquil mid- to low-latitude regions, where low-wavenumber internal tides gently heave isopycnals up and down adiabatic slopes in the absence of mean flows, high-wavenumber internal tides, shelf breaks, resonant tide–bathymetry interactions (critical slopes) and other phenomena associated with turbulence ‘hot spots’. In non-rotating low-Reynolds-number flow, increased stratification on the downslope phase has a relaminarizing effect, while on the upslope phase we find transition-to-turbulence pathways arise from shear production triggered by gravitational instabilities. When rotation is significant (low slope Burger numbers) we find that boundary layer turbulence is sustained throughout the oscillation period, resembling stratified Stokes–Ekman layer turbulence. Simulation results suggest that oscillating boundary layers on smooth slopes at low Reynolds number ($\textit {Re}\leqslant 840$), unity Prandtl number and slope Burger numbers greater than unity do not cause significant irreversible turbulent buoyancy flux (mixing), and that flat-bottom dissipation rate models derived from the tide amplitude are accurate within an order of magnitude.

Couette–Poiseuille (CP) flow in the presence of longitudinal grooves is studied by means of numerical analysis. The flow is actuated by movement of the flat wall and pressure imposed in the opposite direction. The stationary wall features longitudinal grooves that modify the flow, change hydrodynamic drag on the driving wall and cause onset of hydrodynamic instability in the form of travelling waves with a consequent supercritical bifurcation, already at moderate ranges of the Reynolds number. We show that by manipulating this system it is possible to significantly decrease phase speed of the unstable wave and to effectively decouple time scales of wave propagation and amplification with a potential to significantly reduce the distance required for the onset of nonlinear effects. Current analysis begins with concise characterization of stationary, laminar CP flow and the effects of applying a selected corrugation pattern, followed by determination of conditions leading to the onset of instabilities. In the second part we illustrate selected nonlinear solutions obtained for low, supercritical values of the Reynolds numbers and due to the amplification of unstable travelling waves of possibly low phase velocities. This work is concluded with a short discussion of a linear evolution of a wave packet consisting of a superposition of a number of unstable waves and initiated by a localized pulse. This part illustrates that in addition to the reduction of the phase velocity of a single, unstable mode, imposition of the Couette component also reduces group velocity of a wave packet.

Linearized flow past a submerged obstacle with an elastic sheet resting on the flow surface are studied in the limit that the bending length is small compared to the obstacle depth, in two and three dimensions. Gravitational effects are included in the two-dimensional geometry, but absent in the three-dimensional geometry; the Froude number is chosen so that gravitational and elastic restoring forces are comparable in size. In each of these problems, the waves are exponentially small in the asymptotic limit, and can be computed using exponential asymptotic methods. In the two-dimensional problem, flow past a submerged step is considered. It is found that the relative strengths of the gravitational and elastic restoring forces produce two distinct classes of elastic sheet behaviour. In one parameter regime, constant-amplitude elastic waves and gravity waves extend indefinitely upstream and downstream from the obstacle. In the other parameter regime, all waves decay exponentially away from the obstacle. The equivalent nonlinear two-dimensional geometry is then studied; this asymptotic analysis predicts the existence of a third intermediate regime in which waves persist indefinitely in only one direction, depending on whether the submerged step rises or falls. In the three-dimensional geometry, it is predicted that the elastic waves extend ahead of the submerged source, decaying algebraically in space. The form of these elastic waves is computed, and validated by comparison with numerical computations of the elastic sheet behaviour.

This experimental study investigated the control induced by a spanwise surface wire on a rigid circular cylinder undergoing vortex-induced vibration (VIV) under the conditions of low mass damping in the lower synchronization branch. Being motivated by the idea of VIV-based energy harvesting from ocean and river flows, this elastically mounted cylinder was immersed in a water channel, leaving a free end at its bottom spanwise end, while the free water surface bounded its top. The cylinder was constrained to vibrate in the cross-stream direction. The wire diameter was 6.25 % of the cylinder diameter. Experimental research was conducted by attaching this large-scale wire along the span of the cylinder at various angular positions ranging from 0° to 180° (with respect to the most upstream point of the cylinder) at a fixed Reynolds number of 104 (based on the cylinder diameter). Simultaneous to measuring the trajectory of the cylinder motion via a laser distance sensor, the instantaneous velocity field in the near wake of the cylinder was obtained using particle image velocimetry. Several VIV response categories were identified depending on the angular position of the wire, which led to the classification of distinct angular ranges for the wire application. Associated with the structural vibrations in these categories, different vortex-formation modes induced by the wire were revealed. For specific wire positions, decreases of up to 98 % and increases of up to 102 % were identified in the oscillation amplitude of the cylinder compared with the amplitude of the clean cylinder under similar conditions.

A straight cylindrical duct is considered containing an axial mean flow that is uniform everywhere except within a boundary layer near the wall, which need not be thin. Within this boundary layer the mean flow varies parabolically. The linearized Euler equations are Fourier transformed to give the Pridmore-Brown equation, for which the Green's function is constructed using Frobenius series. The critical layer gives a non-modal contribution from the continuous spectrum branch cut, and dominates the downstream pressure perturbation in certain cases, particularly for thicker boundary layers. The continuous spectrum branch cut is also found to stabilize what are otherwise convectively unstable modes by hiding them behind the branch cut. Overall, the contribution from the critical layer is found to give a neutrally stable non-modal wave when the source is located within the sheared flow region, and to decay algebraically along the duct as $O(x^{-{5}/{2}})$ for a source located with the uniform flow region. The Frobenius expansion, in addition to being numerically accurate close to the critical layer where other numerical methods lose accuracy, is also able to locate modal poles hidden behind the branch cut, which other methods are unable to find; this includes the stabilized hydrodynamic instability. Matlab code is provided to compute the Green's function.

Spectral proper orthogonal decomposition (SPOD) is an increasingly popular modal analysis method in the field of fluid dynamics due to its specific properties: a linear system forced with white noise should have SPOD modes identical to response modes from resolvent analysis. The SPOD, coupled with the Welch method for spectral estimation, may require long time-resolved datasets. In this work, a linearised Ginzburg–Landau model is considered in order to study the method's convergence. Spectral proper orthogonal decomposition modes of the white-noise forced equation are computed and compared with corresponding response resolvent modes. The quantified error is shown to be related to the time length of Welch blocks (spectral window size) normalised by a convective time. Subsequently, an algorithm based on a temporal data shift is devised to further improve SPOD convergence and is applied to the Ginzburg–Landau system. Next, its efficacy is demonstrated in a numerical database of a boundary layer subject to bypass transition. The proposed approach achieves substantial improvement in mode convergence with smaller spectral window sizes with respect to the standard method. Furthermore, SPOD modes display growing wall-normal and spanwise velocity components along the streamwise direction, a feature which had not yet been observed and is also predicted by a global resolvent calculation. The shifting algorithm for the SPOD opens the possibility for using the method on datasets with time series of moderate duration, often produced by large simulations.

Three-dimensionalities in the wake of flow past a circular cylinder executing sinusoidal rotary oscillations about its axis is studied experimentally. The results of water tunnel experiments on a rotationally oscillating cylinder for Reynolds number of 250 with varying amplitude and forcing frequency are discussed. Qualitative studies using hydrogen bubble and laser-induced fluorescence flow visualisation techniques are performed. Observation made for oscillating amplitude, $\theta _{0} = {\rm \pi}/4$ and $\theta _{0}=3{\rm \pi} /4$, and a low normalised forcing frequency, $FR$, of 0.75 and 0.5, respectively, confirmed a mode having a spanwise non-dimensional wavelength of $\sim$1.8 which is also observed for a rotating cylinder. On increasing forcing frequency this mode disappears and a new mode with a bean-shaped structure and a much smaller spanwise normalised wavelength of $\sim$0.8 appears at an $FR$ of 1 and an oscillation amplitude of ${\rm \pi} /2$. This mode remains almost stable until a forcing frequency of $FR=1.4$. At higher forcing frequency, $FR=2.75$, and oscillation amplitude of $3{\rm \pi} /4$, a mode with cellular structure and a normalised spanwise wavelength of $\sim$1.6 is identified. The cells in this mode flatten up with increasing downstream distance and are shed alternately with respect to the adjacent cell. Certain combinations of forcing parameters resulted in a forced two-dimensionality of the wake. Quantitative studies using hot-wire measurements and particle image velocimetry confirm the presence of these modes and wake characteristics. Wake mode map in the forcing frequency and amplitude plane is presented showing regions of newly discovered modes and wake lock-on boundaries.

An analysis of laminar natural convection in inclined slots subjected to patterned heating has been performed. The imposed heating takes a simple form characterized by a single Fourier mode combined with uniform heating. It is shown that periodic heating applied at the lower plate produces no net flow when the slot is either horizontal or vertical, but a net upward flow is generated when the slot is tilted. Periodic heating applied at the upper plate produces net downward flow in the inclined situation. The addition of uniform heating promotes the upward flow while cooling has the opposite effect. There is a critical inclination angle at which the maximum net flow rate is greatest. Dynamic and thermal boundary layers are present when the wavenumber of the imposed heating is large. The use of heating at both plates, with the same wavenumber, leads to a flow dominated by the plate exposed to a more intense heating; when the two plates are heated equally no net flow is observed irrespective of the inclination angle. Changes of the relative positions of the two patterns can change the net flow rate by up to 50 %. The intensity of the flow increases with reduction of the Prandtl number. If the heating applied to the plates is of different wavelength, but of the same intensity, a wide range of behaviours of the flow system is possible. The details of this response are sensitive to the ratio of the two wavenumbers.

Hydraulic fractures propagating at depth are subjected to buoyant forces caused by the density contrast between fluid and solid. This paper is concerned with the analysis of the transition from an initially radial fracture towards an elongated buoyant growth – a critical topic for understanding the extent of vertical hydraulic fractures in the upper Earth crust. Using fully coupled numerical simulations and scaling arguments, we show that a single dimensionless number governs buoyant hydraulic fracture growth, namely the dimensionless viscosity of a radial hydraulic fracture at the time when buoyancy becomes of order 1. It quantifies whether the transition to buoyancy occurs when the growth of the radial hydraulic fracture is either still in the regime dominated by viscous flow dissipation or already in the regime where fracture energy dissipation dominates. A family of fracture shapes emerge at late time from finger-like (toughness regime) to inverted elongated cudgel-like (viscous regime). Three-dimensional toughness-dominated buoyant fractures exhibit a finger-like shape with a constant-volume toughness-dominated head and a viscous tail having a constant uniform horizontal breadth: there is no further horizontal growth past the onset of buoyancy. However, if the transition to buoyancy occurs while in the viscosity-dominated regime, both vertical and horizontal growths continue to match scaling arguments. As soon as the fracture toughness is not strictly zero, horizontal growth stops when the dimensionless horizontal toughness becomes of order 1. The horizontal breadth follows the predicted scaling.

We consider steady, fully-developed flows of deformable, inelastic grains driven by gravity between identical bumpy walls. Using constitutive relations from extended kinetic theory (EKT) for the erodible bed near the centreline and the collisional flow between the surfaces of the bed and the walls, we calculate the fields of mean velocity, fluctuation velocity and solid volume fraction across the chute. We consider both situations in which the solid volume fraction at and near the centreline is high enough to form a bed and when it is not. We compare results predicted by EKT with recent discrete element simulations results, and obtain very good agreement.

The classical Mott-Smith solution for one-dimensional normal shock wave structure is extended to the two-dimensional regular shock reflection problem. The solution for the non-equilibrium molecular velocity distribution function along the symmetry-plane streamline is obtained as a weighted sum of four Maxwellians. An analysis of applicability of the solution has been performed using the results of direct simulation Monte Carlo calculations for a range of incident shock wave intensities. Accuracy of the solution improves with increasing $Ma_n$, the Mach number normal to the shock front, so that the solution becomes rather accurate for strong shocks with $Ma_n>8$.