This study presents a buoyancy-driven stability analysis in a three-dimensional inclined porous medium with a capillary transition zone that is formed between a non-wetting and an underlying wetting phase. In this two-phase, two-component, partially miscible system, a solute from a non-wetting phase diffuses into a porous layer saturated with a wetting-phase fluid, creating a dense diffusive boundary layer beneath an established capillary transition zone. Transient concentration and gravity-driven velocity fields are derived for the wetting phase while the saturation field remains fixed. Linear stability analysis with the quasi-steady-state approximation is employed to determine the onset of solutal convective instability for buoyancy-dominant, in-transition and capillary-dominant systems. The analysis of the problem leads to a differential eigenvalue problem composed of a system of three complex-valued equations that are numerically solved to determine the critical times, critical wavenumbers and neutral stability curves as a function of inclination angle for different Bond numbers. The layer inclination is shown to play an essential role in the stability of the problem, where the gravity-driven flow removes solute concentrations in the diffusive boundary layer. The results indicate that the horizontal porous layer exhibits the fastest onset of instability, and longitudinal rolls are always more unstable than oblique and transverse rolls. The inclination angle has a more substantial impact on stabilizing the diffusive boundary layer in the buoyancy-dominant than in the capillary-dominant systems. Furthermore, for both buoyancy-dominant and capillary-dominant systems, the critical times and wavenumbers vary exponentially with inclination angle ≤ 60° and follow the Stirling model.

We present direct numerical simulations of developing turbulent channel flows subjected to thermal expansion or contraction downstream of a heated or cooled wall. Using different constitutive relations for viscosity we analyse the response of variable property flows to streamwise acceleration/deceleration by separating the effect of streamwise acceleration/deceleration from the effect of wall-normal property variations. We demonstrate that, beyond a certain streamwise location, the flow can be considered in a state of ‘quasi-equilibrium’ regarding semilocally scaled variables. As such, we claim that the development of turbulent quantities due to streamwise acceleration/deceleration is localized to the region of impulsive heating/cooling, while changes in turbulence occurring farther downstream can be attributed solely to property variations. This finding allows us to study turbulence modulation in accelerating/decelerating flows using the semilocal scaling framework. By investigating the energy redistribution among the turbulent velocity fluctuations, we conclude that a change in bulk streamwise velocity has a non-local effect which originates from the change in mean shear and modifies the energy pathways through velocity-pressure-gradient correlations. On the other hand, the wall-normal property gradients have a local effect and act through the modification of the viscous dissipation. We show that it is possible to superimpose and compare the two different effects when using the semilocal scaling framework.

We study turbulent flows in a smooth straight pipe of circular cross-section up to friction Reynolds number $({\textit {Re}}_{\tau }) \approx 6000$ using direct numerical simulation (DNS) of the Navier–Stokes equations. The DNS results highlight systematic deviations from Prandtl friction law, amounting to approximately $2\,\%$, which would extrapolate to approximately $4\,\%$ at extreme Reynolds numbers. Data fitting of the DNS friction coefficient yields an estimated von Kármán constant $k \approx 0.387$, which nicely fits the mean velocity profile, and which supports universality of canonical wall-bounded flows. The same constant also applies to the pipe centreline velocity, thus providing support for the claim that the asymptotic state of pipe flow at extreme Reynolds numbers should be plug flow. At the Reynolds numbers under scrutiny, no evidence for saturation of the logarithmic growth of the inner peak of the axial velocity variance is found. Although no outer peak of the velocity variance directly emerges in our DNS, we provide strong evidence that it should appear at ${\textit {Re}}_{\tau } \gtrsim 10^4$, as a result of turbulence production exceeding dissipation over a large part of the outer wall layer, thus invalidating the classical equilibrium hypothesis.

Four different applications of spectral proper orthogonal decomposition (SPOD) are demonstrated on large-eddy simulation data of a turbulent jet. These are: low-rank reconstruction, denoising, frequency–time analysis and prewhitening. We demonstrate SPOD-based flow-field reconstruction using direct inversion of the SPOD algorithm (frequency-domain approach) and propose an alternative approach based on projection of the time series data onto the modes (time-domain approach). We further present a SPOD-based denoising strategy that is based on hard thresholding of the SPOD eigenvalues. The proposed strategy achieves significant noise reduction while facilitating drastic data compression. In contrast to standard methods of frequency–time analysis such as wavelet transform, a proposed SPOD-based approach yields a spectrogram that characterises the temporal evolution of spatially coherent flow structures. A convolution-based strategy is proposed to compute the time-continuous expansion coefficients. When applied to the turbulent jet data, SPOD-based frequency–time analysis reveals that the intermittent occurrence of large-scale coherent structures is directly associated with high-energy events. This work suggests that the time-domain approach is preferable for low-rank reconstruction of individual snapshots, and the frequency-domain approach for denoising and frequency–time analysis.

A Lagrangian perspective has yielded many new insights in our quest to reveal the intricacies of turbulent flows. Much of this progress has been possible by following the trajectories of idealised, inertialess objects (tracers) traversing through the flow. Their spins and tumbles provide a glimpse into the underlying local velocity gradients of the turbulent field. While it is known that the spinning and tumbling rates of anisotropic particles are modified in turbulence – compared with those in a random flow field – a quantitative explanation for this has remained elusive. Now, Pujara et al. (J. Fluid Mech., vol. 922, 2021, R6) have made an attempt to predict the split between spinning and tumbling rates by accessing the particle's alignment with the local vorticity. Their analysis of filtered turbulent fields reveals a Lagrangian scale invariance, whereby key quantities relating to the particle's rotational statistics are preserved from the dissipative to the integral scale.

We consider flow along a finite-length collapsible channel driven by a fixed upstream flux, where a section of one wall of a planar rigid channel is replaced by a plane-strain elastic beam subject to uniform external pressure. A modified constitutive law is used to ensure that the elastic beam is energetically conservative. We apply the finite element method to solve the fully nonlinear steady and unsteady systems. In line with previous studies, we show that the system always has at least one static solution and that there is a narrow region of the parameter space where the system simultaneously exhibits two stable static configurations: an (inflated) upper branch and a (collapsed) lower branch, connected by a pair of limit point bifurcations to an unstable intermediate branch. Both upper and lower static configurations can each become unstable to self-excited oscillations, initiating either side of the region with multiple static states. As the Reynolds number increases along the upper branch the oscillatory limit cycle persists into the region with multiple steady states, where interaction with the intermediate static branch suggests a nearby homoclinic orbit. These oscillations approach zero amplitude at the upper branch limit point, resulting in a stable tongue between the upper and lower branch oscillations. Furthermore, this new formulation allows us to calculate a detailed energy budget over a period of oscillation, where we show that both upper and lower branch instabilities require an increase in the work done by the upstream pressure to overcome the increased dissipation.

Recently, the authors considered a thin steady developed viscous free-surface flow passing the sharp trailing edge of a horizontally aligned flat plate under surface tension and the weak action of gravity, acting vertically, in the asymptotic slender-layer limit (J. Fluid Mech., vol. 850, 2018, pp. 924–953). We revisit the capillarity-driven short-scale viscous–inviscid interaction, on account of the inherent upstream influence, immediately downstream of the edge and scrutinise flow detachment on all smaller scales. We adhere to the assumption of a Froude number so large that choking at the plate edge is insignificant but envisage the variation of the relevant Weber number of $O(1)$. The main focus, tackled essentially analytically, is the continuation of the structure of the flow towards scales much smaller than the interactive ones and where it no longer can be treated as slender. As a remarkable phenomenon, this analysis predicts harmonic capillary ripples of Rayleigh type, prevalent on the free surface upstream of the trailing edge. They exhibit an increase of both the wavelength and amplitude as the characteristic Weber number decreases. Finally, the theory clarifies the actual detachment process, within a rational description of flow separation. At this stage, the wetting properties of the fluid and the microscopically wedge-shaped edge, viewed as infinitely thin on the larger scales, come into play. As this geometry typically models the exit of a spout, the predicted wetting of the wedge is related to what in the literature is referred to as the teapot effect.

Understanding particle drift in suspension flows is of the highest importance in numerous engineering applications where particles need to be separated and filtered out from the suspending fluid. Commonly known drift mechanisms such as the Magnus force, Saffman force and Segré–Silberberg effect all arise only due to inertia of the fluid, with similar effects on all non-spherical particle shapes. In this work, we present a new shape-selective lateral drift mechanism, arising from particle inertia rather than fluid inertia, for ellipsoidal particles in a parabolic velocity profile. We show that the new drift is caused by an intermittent tumbling rotational motion in the local shear flow together with translational inertia of the particle, while rotational inertia is negligible. We find that the drift is maximal when particle inertial forces are of approximately the same order of magnitude as viscous forces, and that both extremely light and extremely heavy particles have negligible drift. Furthermore, since tumbling motion is not a stable rotational state for inertial oblate spheroids (nor for spheres), this new drift only applies to prolate spheroids or tri-axial ellipsoids. Finally, the drift is compared with the effect of gravity acting in the directions parallel and normal to the flow. The new drift mechanism is stronger than gravitational effects as long as gravity is less than a critical value. The critical gravity is highest (i.e. the new drift mechanism dominates over gravitationally induced drift mechanisms) when gravity acts parallel to the flow and the particles are small.

We present direct numerical simulation of a mechanism for creating longitudinal vortices in pipe flow, compared with a model theory. By furnishing the pipe wall with a pattern of crossing waves, secondary flow in the form of streamwise vortex pairs is created. The mechanism, ‘CL1’, is kinematic and known from oceanography as a driver of Langmuir circulation. CL1 is strongest when the ‘wall wave’ vectors make an acute angle with the axis, $\varphi =10^{\circ }$–$20^{\circ }$, changes sign near $45^{\circ }$ and is weak and of opposite sign beyond this angle. A competing, dynamic mechanism driving secondary flow in the opposite sense is also observed, created by the azimuthally varying friction. Whereas at smaller angles ‘CL1’ prevails, the dynamic effect dominates when $\varphi \gtrsim 45^{\circ }$, reversing the flow. Curiously, the circulation strength is a faster-than-linearly increasing function of Reynolds number for small $\varphi$. We explore an analogy with Prandtl's secondary motion of the second kind in turbulence. A transport equation for average streamwise vorticity is derived, and we analyse it for three different crossing angles, $\varphi =18.6^{\circ }, 45^{\circ }$ and $60^{\circ }$. Mean-vorticity production is organised in a ring-like structure with the two rings contributing to rotating flow in opposite senses. For the larger $\varphi$, the inner ring decides the main swirling motion, whereas for $\varphi =18.6^{\circ }$, outer-ring production dominates. For the larger angles, the outer ring is mainly driven by advection of vorticity and the inner by deformation (stretching) whereas, for $\varphi =18.6^{\circ }$, both contribute approximately equally to production in the outer ring.

The modelling of natural convection in porous media is receiving increased interest due to its significance in environmental and engineering problems. State-of-the-art simulations are based on the classic macroscopic Darcy–Oberbeck–Boussinesq (DOB) equations, which are widely accepted to capture the underlying physics of convection in porous media provided the Darcy number, $Da$, is small. In this paper we analyse and extend the recent pore-resolved direct numerical simulations (DNS) of Gasow et al. (J. Fluid Mech, vol. 891, 2020, p. A25) and show that the macroscopic diffusion, which is neglected in DOB, is of the same order (with respect to $Da$) as the buoyancy force and the Darcy drag. Consequently, the macroscopic diffusion must be modelled even if the value of $Da$ is small. We propose a ‘two-length-scale diffusion’ model, in which the effect of the pore scale on the momentum transport is approximated with a macroscopic diffusion term. This term is determined by both the macroscopic length scale and the pore scale. It includes a transport coefficient that solely depends on the pore-scale geometry. Simulations of our model render a more accurate Sherwood number, root mean square (r.m.s.) of the mass concentration and r.m.s. of the velocity than simulations that employ the DOB equations. In particular, we find that the Sherwood number $Sh$ increases with decreasing porosity and with increasing Schmidt number $(Sc)$. In addition, for high values of $Ra$ and high porosities, $Sh$ scales nonlinearly. These trends agree with the DNS, but are not captured in the DOB simulations.

A new perspective on the analysis of turbulent boundary layers on streamlined bodies is provided by deriving the axisymmetric Reynolds-averaged Navier–Stokes equations in an orthogonal coordinate system aligned with streamlines, streamline-normal lines and the plane of symmetry. Wall-resolved large-eddy simulation using an unstructured overset method is performed to study flow about the axisymmetric DARPA SUBOFF hull at a Reynolds number of $Re_L = 1.1 \times 10^{6}$ based on the hull length and free-stream velocity. The streamline-normal coordinate is naturally normal to the wall at the hull surface and perpendicular to the free-stream velocity far from the body, which is critical for studying bodies with concave streamwise curvature. The momentum equations naturally reduce to the differential form of Bernoulli's equation and the $s$–$n$ Euler equation for curved streamlines outside of the boundary layer. In the curved laminar boundary layer at the front of the hull, the streamline momentum equation represents a balance of the streamwise advection, streamwise pressure gradient and viscous stress, while the streamline-normal equation is a balance between the streamline-normal pressure gradient and centripetal acceleration. In the turbulent boundary layer on the mid-hull, the curvature terms and streamwise pressure gradient are negligible and the results conform to traditional analysis of flat-plate boundary layers. In the thick stern boundary layer, the curvature and streamwise pressure gradient terms reappear to balance the turbulent and viscous stresses. This balance explains the characteristic variation of static pressure observed for thick boundary layers at the tails of axisymmetric bodies.

A rapidly growing bubble close to a free surface induces jetting: a central jet protruding outwards and a crown surrounding it at later stages. While the formation mechanism of the central jet is known and documented, that of the crown remains unsettled. We perform axisymmetric simulations of the problem using the free software program BASILISK, where a finite-volume compressible solver has been implemented, which uses a geometric volume-of-fluid (VoF) method for the tracking of the interface. We show that the mechanism of crown formation is a combination of a pressure distortion over the curved interface, inducing flow focusing, and of a flow reversal, caused by the second expansion of the toroidal bubble that drives the crown. The work culminates in a parametric study with the Weber number, the Reynolds number, the pressure ratio and the dimensionless bubble distance to the free surface as control parameters. Their effects on both the central jet and the crown are explored. For high Weber numbers, we observe the formation of weaker ‘secondary crowns’, highly correlated with the third oscillation cycle of the bubble.

We demonstrate a novel instability found within unconfined viscous bands/rims, or free-surface flows involving a longitudinal viscosity contrast. Such instabilities may be described as viscous banding instabilities, non-porous viscous fingering instabilities or unconfined viscous fingering instabilities of free-surface flows involving the intrusion of a less viscous fluid into a band of more viscous fluid. A consequence of this work is that viscous fingering instabilities, widely known to occur in porous media following the seminal work of Saffman & Taylor (Proc. R. Soc. Lond. A, vol. 245, 1958, pp. 312–329), also occur in non-porous environments. Although the mechanism of the viscous banding instability is characteristically different from that of the Saffman–Taylor instability, there are important similarities between the two. The main similarity is that a viscosity contrast leads to instability. A distinguishing feature is that confinement, such as the rigid walls of a Hele-Shaw cell, is not necessary for viscous banding instabilities to occur. More precisely, Saffman–Taylor instabilities are driven by a jump in dynamic pressure gradient, whereas viscous banding instabilities, or non-porous viscous fingering instabilities, are driven by a jump in hydrostatic pressure gradient, directly related to a slope discontinuity across the intrusion front. We examine the onset of instability within viscous bands down an inclined plane, determine conditions under which viscous banding instabilities occur and map out a range of behaviours in parameter space in terms of two dimensionless parameters: the viscosity ratio and the volume of fluid ahead of the intrusion front.

Lattice Boltzmann simulations were carried out to investigate the noise mitigation mechanisms of a 3-D printed porous trailing-edge insert, elucidating the link between noise reduction and material permeability. The porous insert is based on a unit cell resembling a lattice of diamond atoms. It replaces the last 20 % chord of a NACA 0018 at zero angle-of-attack. A partially blocked insert is considered by adding a solid partition between 84 % and 96 % of the aerofoil chord. The regular porous insert achieves a substantial noise reduction at low frequencies, although a slight noise increase is found at high frequencies. The partially blocked porous insert exhibits a lower noise reduction level, but the noise emission at mid-to-high frequency is slightly affected. The segment of the porous insert near the tip plays a dominant role in promoting noise mitigation, whereas the solid-porous junction contributes, in addition to the rough surface, towards the high-frequency excess noise. The current study demonstrates the existence of an entrance length associated with the porous material geometry, which is linked to the pressure release process that is responsible for promoting noise mitigation. This process is characterised by the aerodynamic interaction between pressure fluctuations across the porous medium, which is found at locations where the porous insert thickness is less than twice the entrance length. Present results also suggest that the noise attenuation level is related to both the chordwise extent of the porous insert and the streamwise turbulent length scale. The porous inserts also cause a slight drag increase compared to their solid counterpart.

Submesoscale fronts with large horizontal buoyancy gradients and $O(1)$ Rossby numbers are common in the upper ocean. These fronts are associated with large vertical transport and are hotspots for biological activity. Submesoscale fronts are susceptible to symmetric instability (SI) – a form of stratified inertial instability which can occur when the potential vorticity is of the opposite sign to the Coriolis parameter. Here, we use a weakly nonlinear stability analysis to study SI in an idealised frontal zone with a uniform horizontal buoyancy gradient in thermal wind balance. We find that the structure and energetics of SI strongly depend on the front strength, defined as the ratio of the horizontal buoyancy gradient to the square of the Coriolis frequency. Vertically bounded non-hydrostatic SI modes can grow by extracting potential or kinetic energy from the balanced front and the relative importance of these energy reservoirs depends on the front strength and vertical stratification. We describe two limiting behaviours as ‘slantwise convection’ and ‘slantwise inertial instability’ where the largest energy source is the buoyancy flux and geostrophic shear production, respectively. The growing linear SI modes eventually break down through a secondary shear instability, and in the process transport considerable geostrophic momentum. The resulting breakdown of thermal wind balance generates vertically sheared inertial oscillations and we estimate the amplitude of these oscillations from the stability analysis. We finally discuss broader implications of these results in the context of current parameterisations of SI.

In Part 1 (Wienkers, Thomas & Taylor, J. Fluid Mech., vol. 926, 2021, A6), we described the theory for linear growth and weakly nonlinear saturation of symmetric instability (SI) in the Eady model representing a broad frontal zone. There, we found that both the fraction of the balanced thermal wind mixed down by SI and the primary source of energy are strongly dependent on the front strength, defined as the ratio of the horizontal buoyancy gradient to the square of the Coriolis frequency. Strong fronts with steep isopycnals develop a flavour of SI we call ‘slantwise inertial instability’ by extracting kinetic energy from the background flow and rapidly mixing down the thermal wind profile. In contrast, weak fronts extract more potential energy from the background density profile, which results in ‘slantwise convection.’ Here, we extend the theory from Part 1 using nonlinear numerical simulations to focus on the adjustment of the front following saturation of SI. We find that the details of adjustment and amplitude of the induced inertial oscillations depend on the front strength. While weak fronts develop narrow frontlets and excite small-amplitude vertically sheared inertial oscillations, stronger fronts generate large inertial oscillations and produce bore-like gravity currents that propagate along the top and bottom boundaries. The turbulent dissipation rate in these strong fronts is large, highly intermittent and intensifies during periods of weak stratification. We describe each of these mechanisms and energy pathways as the front evolves towards the final adjusted state, and in particular focus on the effect of varying the dimensionless front strength.

We investigate the effect of helicity on the scale-similar structures of homogeneous isotropic and non-mirror-symmetric turbulence based on the Lagrangian renormalised approximation (LRA), which is a self-consistent closure theory proposed by Kaneda (J. Fluid Mech., vol. 107, 1981, pp. 131–145). In this study, we focus on the time scale representing the scale-similar range. For the LRA, the Lagrangian two-time velocity correlation and response function determine the representative time scale. The LRA predicts that both the Lagrangian two-time velocity correlation and response function equation do not explicitly depend on helicity. We assume the extended scale-similar spectra and time scale by considering the helicity dissipation rate. Considering the small-scale structures, the requirements for the energy and helicity fluxes and response function equation to be scale similar, yield the conventional inertial-range power laws and provide the energy and helicity spectra $\propto k^{-5/3}$ and the time scale $\propto \varepsilon ^{-1/3} k^{-2/3}$, where $\varepsilon$ and $k$ denote the energy dissipation rate and wavenumber, respectively. Notably, energy flux can be scale similar only when $k^H /k \ll 1$, where $k^H = \varepsilon ^H/\varepsilon$ and $\varepsilon ^H$ denotes the helicity dissipation rate. This condition makes the energy cascade process in the scale-similar range completely independent of helicity. We also investigate the localness of the interscale interaction in the energy and helicity cascades for the LRA. We demonstrate that the helicity cascade is slightly non-local in scales compared with the energy cascade. This study provides a foundation on the modelling of non-mirror-symmetric turbulent flows.

We perform a sparse identification of nonlinear dynamics (SINDy) for low-dimensionalized complex flow phenomena. We first apply the SINDy with two regression methods, the thresholded least square algorithm and the adaptive least absolute shrinkage and selection operator which show reasonable ability with a wide range of sparsity constant in our preliminary tests, to a two-dimensional single cylinder wake at $Re_D=100$, its transient process and a wake of two-parallel cylinders, as examples of high-dimensional fluid data. To handle these high-dimensional data with SINDy whose library matrix is suitable for low-dimensional variable combinations, a convolutional neural network-based autoencoder (CNN-AE) is utilized. The CNN-AE is employed to map a high-dimensional dynamics into a low-dimensional latent space. The SINDy then seeks a governing equation of the mapped low-dimensional latent vector. Temporal evolution of high-dimensional dynamics can be provided by combining the predicted latent vector by SINDy with the CNN decoder which can remap the low-dimensional latent vector to the original dimension. The SINDy can provide a stable solution as the governing equation of the latent dynamics and the CNN-SINDy-based modelling can reproduce high-dimensional flow fields successfully, although more terms are required to represent the transient flow and the two-parallel cylinder wake than the periodic shedding. A nine-equation turbulent shear flow model is finally considered to examine the applicability of SINDy to turbulence, although without using CNN-AE. The present results suggest that the proposed scheme with an appropriate parameter choice enables us to analyse high-dimensional nonlinear dynamics with interpretable low-dimensional manifolds.

Turbulent channel flow simulation of a shear-thinning fluid is considered – see Arosemena et al. (J. Fluid Mech., vol. 908, 2021, p. A43) – and compared with a Newtonian base case to reveal the effects of the shear-dependent rheology on the near-wall structures. Analyses of different flow statistics revealed that, for the shear-thinning fluid case, the streamwise vortices appear to grow in size, depart from the wall and present a lessening in their intensity. Information regarding variations in the quasi-longitudinal vortices is also obtained from three-dimensional structures identified through a normalized $Q$-criterion. With shear-thinning rheology, it is shown that the structures are comprised of wall-attached and -detached families which are taller than for a Newtonian fluid. Also, for a given height, the structures appear to be longer, with approximately the same width and overall larger volume for the shear-thinning fluid case; albeit their fractal dimension remains the same when compared to the Newtonian base case. Moreover, it is observed that the number density of vortical structures decreases with shear-thinning fluid behaviour. These observations, in conjunction with the known changes to the longitudinal velocity structures which appear to be less streaky, more spanwise separated and thickened with shear-thinning rheology, strongly suggest that the near-wall self-sustaining process has been disrupted. As we move slightly away from the wall and with shear-thinning behaviour, the local increase in viscosity seems to lead to less energetic vortices whereas the streaks are provided with an additional source of energy due to fluctuations in viscosity.

In jet engines and gas turbines, the annular shape of the combustion chamber allows the appearance of self-oscillating azimuthal thermoacoustic modes. We report experimental evidence of a new type of modal dynamics characterised by periodic switching of the spinning direction and develop a theoretical model that fully reproduces this phenomenon and explains the underlying mechanisms. It is shown that tiny asymmetries of the geometry, the mean temperature field, the thermoacoustic response of the flames or the acoustic impedance of the walls, present in any real systems, can induce these heteroclinic orbits. The model also explains experimental observations showing a statistically dominant spinning direction despite the absence of swirling flow, or pairs of preferred nodal line directions.