Plumes generated from a point buoyant source are relevant to hydrothermal vents in lakes and oceans on and beyond Earth. They play a crucial role in determining heat and material transport and thereby local biospheres. In this study, we investigate the development of rotating point plumes in an unstratified environment using both theory and numerical simulations. We find that in a sufficiently large domain, point plumes cease to rise beyond a penetration height $h_{{f}}$, at which buoyancy flux from the heat source is leaked laterally to the ambient fluid. The height $h_{{f}}$ is found to scale with the rotational length scale $h_{ \!{ f}}\sim L_{ \!\textit{ rot}}^p\equiv ({F_0}/{f^3})^{{1}/{4}},$ where $F_0$ is the source buoyancy flux, and $f=2\varOmega$ is the Coriolis parameter ($\varOmega$ is the rotation rate). In a limited domain, the plume may reach the top boundary or merge with neighbouring plumes. Whether rotational effects dominate depends on how $L_{\textit{rot}}^{p}$ compares to the height of the domain $H$ and the distance between the plumes $L$. Four parameter regimes can therefore be identified, and are explored here through numerical simulation. Our study advances the understanding of hydrothermal plumes and heat/material transport, with applications ranging from subsurface lakes to oceans in icy worlds such as Snowball Earth, Europa and Enceladus.

Vertically bounded, horizontally propagating internal waves may become unstable through triad resonant instability, in which two sibling waves in background noise draw energy from a parent internal tide. If the background stratification is uniform, then the condition for pure resonance between the parent and sibling wave frequencies and horizontal and vertical wavenumbers can be found semi-analytically from the roots of a polynomial expression. In non-uniform stratification, determining the frequencies and horizontal wavenumbers for which resonance occurs is less straightforward. We develop a theory for near-resonant excitation of a pair of sibling waves from a low-mode internal wave in which the proximity to pure resonance is characterised by the discrepancy between the forced sibling wave frequencies and the natural frequency of these modes. Knowing this discrepancy can be used methodically to determine pure resonance conditions. This inviscid theory is compared with numerical simulations of effectively inviscid waves. For comparison with laboratory experiments, the theory is adapted to include viscous effects both in the bulk of the fluid and at the side walls of the tank. We find that our theoretical predictions for frequencies and wavenumbers of the fastest growing sibling waves are generally consistent between theory, simulations and experiments, though theory overpredicts the growth rate observed in experiments. In all cases, the growth rate of sibling waves decreases with decreasing parent wave frequency, becoming negligibly small in experiments if the parent wave has frequency less than $\approx 0.7$ of the buoyancy frequency at the surface.

Direct numerical simulations (DNS) are performed to investigate the dependence of the Prandtl number ($\textit{Pr}$) and radius ratio ($\eta =r_{i}/r_{o}$) on the asymmetry of the mean temperature radial profiles in turbulent Rayleigh–Bénard convection (RBC) within spherical shells. Unlike planar RBC, the temperature drop, and the thermal and viscous boundary layer thicknesses, at the inner and outer boundaries are not identical in spherical shells. These differences in the boundary layer properties in spherical RBC contribute to the observed asymmetry in the radial profiles of temperature and velocity. The asymmetry originates from the differences in curvature and gravity at the two boundaries, and in addition, is influenced by $\textit{Pr}$. To investigate the $\eta$ and $\textit{Pr}$ dependence of these asymmetries, we perform simulations of Oberbeck–Boussinesq convection for $\eta = 0.2,0.6$ and $0.1 \leqslant Pr \leqslant 50$, and for a range of Rayleigh numbers ($Ra$) varying between $5 \times 10^{6}$ and $5 \times 10^{7}$. The Prandtl numbers that we choose cover a broad range of geophysical relevance, from low-$\textit{Pr}$ regimes ($\textit{Pr}=0.1$) representative of gas giants such as Jupiter and Saturn, to high-$\textit{Pr}$ regimes characteristic of organic flows used in the convection experiments ($\textit{Pr}=50$). A centrally condensed mass, with the gravity profile $g \sim 1/r^{2}$, is employed in this study. Our results show that the asymmetry at smaller $\eta$ exhibits a stronger $\textit{Pr}$ dependence than at larger $\eta$. Various assumptions for quantifying this asymmetry are evaluated, revealing that different assumptions are valid in different $\textit{Pr}$ regimes. It is shown that the assumption of the equal characteristic plume separation at the inner and outer boundaries, as well as the assumption of the identical thermal fluctuation scales between the two boundary layers, is valid only for $0.2 \lesssim Pr \lesssim 1$. In contrast, assumptions based on the equivalency of the local thermal boundary layer Rayleigh numbers and laminar natural-convective boundary layers are validated at $\textit{Pr}=50$ for the explored parameter space. Furthermore, new assumptions based on the statistical analysis of the inter-plume islands are proposed for $\textit{Pr}=0.1$ and $50$, and these are validated against the DNS data. These findings provide insights into the $(Pr,\eta)$ dependence of asymmetry in spherical RBC, and offer a framework for studying similar systems in geophysical and astrophysical contexts.

This study investigates the effects of thermal buoyancy on the ascent or descent dynamics and path instabilities of a finite-size sphere through direct numerical simulations with the immersed boundary method. By parametrically varying the density ratio $(\rho _r)$, Richardson number $({\textit{Ri}})$ and Galileo number $(\textit{Ga})$, four distinct motion regimes are identified: stable vertical, zigzagging, spiralling and chaotic regimes. These regimes emerge from the competition between particle inertial, gravitational forces and fluid thermal-buoyant forces. Compared with isothermal cases, particles with positive Richardson numbers exhibit accelerated motion due to thermal buoyancy. The critical Reynolds numbers ${\textit{Re}}_{p,cr}$ for their path instability are significantly reduced by amplifying wake recirculation zones and triggering vortex shedding. This destabilization mechanism is markedly more pronounced for light particles $(\rho _r \lt 1)$ than heavy particles $(\rho _r \gt 1)$. The present results reveal that the dynamics of heated light particles $(\rho _r=0.5, {\textit{Ri}}\gt 0)$ are governed by the codependent interplay of thermal-buoyancy intensity (${\textit{Ri}}$) and gravitational force (${\textit{Ga}}$), which collectively dictate velocity modulation and path instability patterns. Notably, thermal buoyancy elevates particle Reynolds numbers $({\textit{Re}}_p)$ while could reduce Nusselt numbers, arising from competing mechanisms between intensified convective transport and impaired conductive heat transfer – particularly pronounced for low ${\textit{Ga}}$ particles. These findings bridge the gap between fundamental fluid mechanics and thermal engineering, offering insights to optimize thermal management in particle-laden flows systems, such as industrial heat exchangers and fluidized bed reactors, where thermohydrodynamic coupling effect plays a key role in the performance.

This study experimentally investigates wake recovery mechanisms behind a floating wind turbine subjected to imposed fore-aft (surge) and side-to-side (sway) motions. Wind tunnel experiments with varying free-stream turbulence intensities ($\textit{TI}_{\infty } \in [1.1, 5.8]\,\%$) are presented. Rotor motion induces large-scale coherent structures – pulsating for surge and meandering for sway – whose development critically depends on the energy ratio between the incoming turbulence and the platform motion. The results provide direct evidence supporting the role of these structures in enhancing wake recovery, as previously speculated by Messmer, Peinke & Hölling (J. Fluid Mech., vol. 984, 2024, A66). These periodic structures significantly increase Reynolds shear stress gradients, particularly in the streamwise–lateral direction, which are key drivers of wake recovery. However, their influence diminishes with increasing $\textit{TI}_{\infty }$: higher background turbulence weakens the coherent flow patterns, reducing their contribution to recovery. Beyond a threshold turbulence level – determined by the energy, frequency and direction of motion – rotor-induced structures no longer contribute meaningfully to recovery, which becomes primarily driven by the free-stream turbulence. Finally, we show that the meandering structures generated by sway motion are more resilient in turbulent backgrounds than the pulsating modes from surge, making sway more effective for promoting enhanced wake recovery.

Addition of polymers modifies a turbulent flow in a manner that depends non-trivially on the interplay of fluid inertia, quantified by the Reynolds number $\textit{Re}$, and the elasticity of the dissolved polymers, given by the Deborah number $\textit{De}$. We use direct numerical simulations to study polymeric flows at different $\textit{Re}$ and $\textit{De}$ numbers, and uncover various features of their dynamics. Polymeric flows exhibit a non-unique scaling of the energy spectrum that is a function of $\textit{Re}$ and $\textit{De}$, owing to different dominant contributions to the total energy flux across scales, with the weakening of fluid nonlinearity with decreasing $\textit{Re}$ also leading to the reduction of the polymeric scaling range. This behaviour is also manifested in the real space scaling of structure functions. We also shed light on how the addition of polymers results in slowing down the fluid nonlinear cascade resulting in a depleted flux, as velocity fluctuations with less energy persist for longer times in polymeric flows, especially at intermediate $\textit{Re}$ numbers. These velocity fluctuations exhibit intermittent, large deviations similar to that in a Newtonian flow at large $\textit{Re}$, but differ more and more as $\textit{Re}$ becomes smaller. This observation is further supported by the statistics of fluid energy dissipation in polymeric flows, whose distributions collapse on to the Newtonian at large $\textit{Re}$, but increasingly differ from it as $\textit{Re}$ decreases. We also show that polymer dissipation is significantly less intermittent compared with fluid dissipation, and even less so when elasticity becomes large. Polymers, on an average, dissipate more energy when they are stretched more, which happens in extensional regions of the flow. However, owing to vortex stretching, regions with large rotation rates also correlate with large polymer extensions, albeit to a relatively less degree than extensional regions.

This study examines the dynamics of vortical interactions and their implications for mitigating thermoacoustic instability in a turbulent combustor. The regions of intense vortical interactions are identified as vortical communities in the network space of weighted directed vortical networks constructed from two-dimensional experimental velocity data. One can expect vortical interactions in the combustor to be strongest near the moment of vortex shedding, as the shed vortices gradually weaken due to dissipation while convecting downstream. However, we show that, during the state of thermoacoustic instability, there is a non-trivial consistent phase lag of approximately $52^\circ$ between the shedding of the coherent structures from the backward-facing step and the time instant when the vortical interactions attain their local maximum value. We explain this phase lag by investigating the correlation between acoustic pressure fluctuations, spatio-temporal dynamics of coherent structures and vortical interactions in the reaction field of the combustor. We also show the aperiodic variation of vortical interactions during the states of combustion noise and aperiodic epochs of intermittency. Furthermore, the spatio-temporal evolution of pairs of vortical communities with the maximum inter-community interactions provides insight into explaining the critical regions detected in the reaction field during the states of intermittency and thermoacoustic instability, also identified in previous studies. We further show that the most efficient suppression of thermoacoustic instability via air microjet injection is achieved when steady air jets are introduced to disrupt the maximum inter-community interactions present during the state of thermoacoustic instability.

Flow-induced compaction of soft, elastically deformable porous media occurs in numerous industrial processes. A theoretical study of this problem, and its interplay with gravitational and mechanical compaction, is presented here in a one-dimensional configuration. First, it is shown that soft media can be categorised into two ‘types’, based on their compaction behaviour in the limit of large applied fluid pressure drop. This behaviour is controlled by the constitutive laws for effective pressure and permeability, which encode the rheology of the solid matrix, and can be linked to the well-known poroelastic diffusivity. Next, the interaction of gravitational and flow-induced compaction is explored, with the resultant asymmetry between upward and downward flow leading to distinct compaction behaviour. In particular, flow against gravity – upwards – must first relieve gravitational stresses before any bulk compaction of the medium can occur, so upward flow may result in compaction of some regions and decompaction of others, such that the overall depth remains fixed. Finally, the impact of a fixed mechanical load on the sample is considered: again, it is shown that flow must ‘undo’ this external load before any bulk compaction of the whole medium can occur in either flow direction. The interplay of these different compaction mechanisms is explored, and qualitative differences in these behaviours based on the ‘type’ of the medium are identified.

Dispersion of microswimmers is widespread in environmental and biomedical applications. In the category of continuum modelling, the present study investigates the dispersion of microswimmers in a confined unidirectional flow under a diffuse reflection boundary condition, instead of the specular reflection and the Robin boundary conditions prevailing in existing studies. By the moment analysis based on the Smoluchowski equation, the asymptotic and transient solutions are directly obtained, as validated against random walk simulations, to illustrate the effects of mean flow velocity, swimming velocity and gyrotaxis on the migration and distribution patterns of elongated microswimmers. Under the diffuse reflection boundary condition, microswimmers are found more likely to exhibit M-shaped low-shear trapping and even pronounced centreline aggregation, and elongated shape affects depletion at the centreline. Along the flow direction, they readily form unimodal distributions oriented downstream, resulting in prominent downstream migration. Near the centreline, the migration is almost entirely downstream, while upstream and vertical migrations are confined near the boundaries. When the mean flow velocity and swimming velocity are comparable, the system undergoes a temporal transition from M-shaped low-shear trapping to M-shaped high-shear trapping and ultimately to centreline aggregation. The downstream migration continuously strengthens over time, while the upstream first strengthens and then weakens. Moreover, the coupling between swimming-induced diffusion and convective dispersion leads to non-monotonic, fluctuating trends in both drift velocity and dispersivity over time. These results contribute to a deeper understanding of the underlying mechanisms governing the locomotion and control of natural and synthetic microswimmers.

The evolution of the mixing layer in rotation-driven Rayleigh–Taylor (RT) turbulence is investigated theoretically and numerically. It is found that the evolution of the turbulent mixing layer in rotation-driven RT turbulence is self-similar, but the width of the mixing layer does not follow the classical quadratic growth observed in planar RT turbulence induced by constant external acceleration. Based on the approach used in cylindrical RT turbulence without rotation (Zhao et al. 2021, Phys. Rev. E, vol. 104, 055104), a theoretical model is established to predict the growth of mixing widths in rotation-driven RT turbulence, and the model’s excellent agreement with direct numerical simulations (DNS) serves to validate its reliability. The model proposes a rescaled time that allows for the unification of the evolutions of the mixing layers in rotation-driven RT turbulence with various Atwood numbers and rotation numbers. It is further identified that the growth law described by the model of rotation-driven RT turbulence can be recovered to quadratic growth when the effects of geometrical curvature, radial inhomogeneity of the centrifugal force, and Coriolis force become negligible. Moreover, based on the DNS results, we find that turbulent mixing layers in rotation-driven RT turbulence cover a wide range of length scales. The strong rotation at the same Atwood number enhances the generation of fine-scale structures but is not conducive to overall fluid mixing within the mixing layer.

A series of new laboratory experiments explore the transient flow in an enclosed space of depth $H$, which is subject to an upward displacement ventilation flux, $Q_V$, and which contains a localised heat source of buoyancy flux $F_s$, when the buoyancy of the ventilation air changes by $\Delta g'$. Initially, the plume, produced by the heat source, entrains the ventilation air, leading to a two-layer stratification which depends on the dimensionless strength of convection, $\mu \propto F_s^{1/3}H^{5/3}/Q_V$. When the buoyancy of the ventilation air decreases, $\Delta g' \lt 0$, a new layer of relatively dense fluid grows next to the floor. The fluid entrained by the plume from this new layer causes the plume to intrude between the original upper and lower layers. For a sufficiently large decrease in buoyancy, $|\Delta g' Q_V /F_s| \gt 1$, then as the new lower layer grows, the plume eventually becomes negatively buoyant relative to the original lower layer and intrudes between the new lowest layer and the original lower layer. When the buoyancy of the air supply increases, $\Delta g'\gt 0$, it mixes with the fluid in the original lower layer. If the increase in buoyancy is sufficient, $\Delta g' Q_V/F_s\gt 1$, then the new supply air eventually also mixes with the original upper layer. In each case, a new two-layer stratification becomes re-established. We propose new models for the evolution of the transient flow, assuming that the buoyancy profile can be approximated by a staircase of well-mixed layers. These layers are emptied or filled through the action of the plume and ventilation. We find that the model predictions are consistent with our new experiments in each of the four regimes. We conclude by discussing the implications of these transient flows for thermal comfort and the mixing of contaminants into the occupied lower region of the space.

A literature review suggests that the flows past simply connected bodies with aspect ratio close to unity and symmetries aligned with the flow follow a consistent sequence of regimes (steady, periodic, quasiperiodic) as the Reynolds number increases. However, evidence is fragmented, and studies are rarely conducted using comparable numerical or experimental set-ups. This paper investigates the wake dynamics of two canonical bluff bodies with distinct symmetries: a cube (discrete) and a sphere (continuous). Employing three-dimensional (3-D) global linear stability analysis and nonlinear simulations within a unified numerical framework, we identify the bifurcation sequence driving these regime transitions. The sequence: a pitchfork bifurcation breaks spatial symmetry; a Hopf bifurcation introduces temporal periodicity ($St_1$); a Neimark–Sacker bifurcation destabilises the periodic orbit, leading to quasiperiodic dynamics with two incommensurate frequencies ($St_1, St_2$). A Newton–Krylov method computes the unstable steady and periodic base flows without imposing symmetry constraints. Linear stability reveals similarities between the cube and sphere in the spatial structure of the leading eigenvectors and in the eigenvalue trajectories approaching instability. This study provides the first confirmation of a Neimark–Sacker bifurcation to quasiperiodicity in these 3-D wakes, using Floquet stability analysis of computed unstable periodic orbits and their Floquet modes. The quasiperiodic regime is described in space and time by the Floquet modes’ effects on the base flow and a spectrum dominated by the two incommensurate frequencies and tones arising from nonlinear interactions. Although demonstrated for a cube and a sphere, this bifurcation sequence, leading from steady state to quasiperiodic dynamics, suggests broader applicability beyond these geometries.

We study convection in a volumetrically heated fluid which is cooled from both plates and is under rotation through the use of direct numerical simulations. The onset of convection matches similar systems and predictions from asymptotic analysis. At low rotation rates, the fluid becomes more organised, enhancing heat transport and increasing boundary layer asymmetry, whereas high rotation rates suppress convection. Velocity and temperature statistics reveal that the top unstably stratified boundary layer exhibits behaviour consistent with other rotating convective systems, while the bottom boundary shows a unique interaction between unstable stratification and Ekman boundary layers. Additional flow statistics such as energy dissipation are analysed to rationalise the flow behaviour.

This study investigates the onset of linear instabilities and their later nonlinear interactions in the shear layer of an initially laminar jet using high-fidelity simulations. We present a quantitative analysis of the vortex-pairing phenomenon by computing the spatial growth rates and energy budget of the dominant frequencies. Compared with a turbulent jet, the hydrodynamic instabilities and vortex pairing are enhanced in an initially laminar jet. Using local linear theory, we identify the fundamental as the frequency with the largest spatial growth rate, and its exponential growth causes the shear layer to roll up into vortices. Visualisations and conditional $x$–$t$ plots reveal that fundamental vortices pair to form subharmonic vortices, which then merge to produce second subharmonic vortices. The energy transfer during this process is evaluated using the spectral turbulent kinetic energy equation, focusing on dominant coherent structures identified through spectral proper orthogonal decomposition. Spectral production and nonlinear transfer terms show that the fundamental frequency gains energy solely from the mean flow, while subharmonics gain energy both linearly from the mean flow and nonlinearly through backscatter from the fundamental frequency. Our results confirm Monkewitz’s theoretical model of a resonance mechanism between the fundamental and subharmonic, which supplies energy to the subharmonic. We highlight the energetic versus dynamical importance of tonal frequencies. The second subharmonic corresponds to the largest spectral peak, while the fundamental, though the fourth largest spectral peak, is dynamically dominant, as it determines all other spectral peaks and supplies energy to the subharmonics through a reverse energy cascade.

Elasto-inertial turbulence (EIT) has been demonstrated to be able to sustain in two-dimensional (2-D) channel flow; however the systematic investigations on 2-D EIT remain scarce. To address this gap, this study conducts direct numerical simulations of 2-D EIT at a modest Reynolds number ($Re=2000$) to examine its statistical characteristics and dynamic mechanisms. Meanwhile, this paper explores the similarities and differences between 2-D EIT with the maximum drag reduction (MDR) state in three-dimensional (3-D) flow. We demonstrate that statistical characteristics of 2-D EIT follow distinct trends compared to those in viscoelastic drag-reducing turbulence as nonlinear elasticity increases. These differences can be attributed to two different underlying dynamical processes: the gradual suppression of inertial turbulence in 3-D flow, and the progressive enhancement of EIT in 2-D flow. Also, we present the role of pressure, energy budget and spectral characteristics of 2-D EIT, which show significant similarities to those in the MDR state, thus providing compelling evidence for the 2-D nature of EIT. More strikingly, we identify an anomalous Reynolds stress in 2-D EIT that contributes negatively to flow resistance, which differs from the extremely small but positive Reynolds stress observed in the MDR state. Although with small values of Reynolds stress, the correlation analysis indicates clearly moderate positive correlation between the streamwise and normalwise velocity fluctuations rather than their being uncorrelated. Moreover, quadrant analysis of velocity fluctuations reveals the predominance of motions in the first and third quadrants, which are closely associated with the typical polymer extension sheet-like structures.

We study the stationary, intermittent and nonlinear dynamics of nominally ideally expanded, natural and forced supersonic twin-rectangular turbulent jets using spectral modal decomposition. We decompose large-eddy simulation data into four reflectional symmetry components about the major and minor axes. In the natural jet, spectral proper orthogonal decomposition (SPOD) uncovers two resonant instabilities antisymmetric about the major axis. Known as screech tones, the more energetic of the two is a steady flapping instability, while the other is an intermittent double-flapping instability. We test the hypothesis that symmetry breaking can be leveraged for control design. Time-periodic forcing symmetric about the major and minor axes is implemented using a plasma actuation model, and succeeds in removing screech from a different symmetry component. We investigate the spectral peaks of the forced jet using an extension of bispectral mode decomposition (BMD), where the bispectrum is bounded by unity and which conditionally recovers the SPOD. We explain the appearance of harmonic peaks as three sets of triadic interactions between reflectional symmetries, forming an interconnected triad network. BMD modes of active triads distil coherent structures comprising multiple coupled instabilities, including Kelvin–Helmholtz, core and guided-jet modes (G-JM). Downstream-propagating core modes can be symmetric or antisymmetric about the major axis, whereas upstream-propagating G-JM responsible for screech closure (Edgington-Mitchell et al. J. Fluid Mech.945, 2022, p. A8) are antisymmetric only. The dependence of G-JM on symmetry hence translates from the azimuthal symmetry of the round jet to the dihedral group symmetry of the twin-rectangular jet, and explains why the twin jet exhibits antisymmetric but not symmetric screech modes.

At all scales, porous materials stir interstitial fluids as they are advected, leading to complex (and chaotic) distributions of matter and energy. Of particular interest is whether porous media naturally induce chaotic advection in Darcy flows at the macroscale, as these stirring kinematics profoundly impact basic processes such as solute transport and mixing, colloid transport and deposition and chemical, geochemical and biological reactivity. While the prevalence of pore-scale chaotic advection has been established, and many studies report complex transport phenomena characteristic of chaotic advection in heterogeneous Darcy flow, it has also been shown that chaotic dynamics are prohibited in a large class of Darcy flows. In this study we rigorously establish that chaotic advection is inherent to steady three-dimensional (3-D) Darcy flow with anisotropic and heterogeneous hydraulic conductivity fields. These conductivity fields generate non-trivial braiding of streamlines, leading to both chaotic advection and (purely advective) transverse macro-dispersion. We establish that steady 3-D Darcy flow has the same topology as unsteady 2-D flow and use braid theory to establish a quantitative link between transverse dispersivity and Lyapunov exponent in heterogeneous Darcy flow. Our main results show that chaotic advection and transverse dispersion occur in both anisotropic weakly heterogeneous and in heterogeneous weakly anisotropic conductivity fields, and that the quantitative link between these phenomena persists across a broad range of conductivity fields. As the ubiquity of macroscopic chaotic advection has profound implications for the myriad processes hosted in porous media, these results call for re-evaluation of transport and reaction methods in these systems.

Invariant maps are a useful tool for turbulence modelling, and the rapid growth of machine learning-based turbulence modelling research has led to renewed interest in them. They allow different turbulent states to be visualised in an interpretable manner and provide a mathematical framework to analyse or enforce realisability. Current invariant maps, however, are limited in machine learning models by the need for costly coordinate transformations and eigendecomposition at each point in the flow field. This paper introduces a new polar invariant map based on an angle that parametrises the relationship of the principal anisotropic stresses, and a scalar that describes the anisotropy magnitude relative to a maximum value. The polar invariant map reframes realisability in terms of a limiting anisotropy magnitude, allowing for new and simplified approaches to enforcing realisability that do not require coordinate transformations or explicit eigendecomposition. Potential applications to machine learning-based turbulence modelling include post-processing corrections for realisability, realisability-informed training, turbulence models with adaptive coefficients and general tensor basis models. The relationships to other invariant maps are illustrated through examples of plane channel flow and square duct flow. Sample calculations are provided for a comparison with a typical barycentric map-based method for enforcing realisability, showing an average 62 % reduction in calculation time using the equivalent polar formulation. The results provide a foundation for new approaches to enforcing realisability constraints in Reynolds-averaged turbulence modelling.

This study connected flow structure and morphological changes in and around a rectangular vegetation patch. The emergent patch was constructed in an 8 cm sand bed. Two patch densities were tested, using a regular configuration of rigid dowels. Near the leading edge of the patch, enhanced turbulence levels produced sediment erosion. Some of the eroded sediment was carried into the patch, forming an interior deposition dune. The denser patch resulted in a smaller dune due to stronger lateral flow diversion and weaker interior streamwise velocity. After the leading-edge dune, in the fully developed region of the patch, vortices formed in the shear layers along the patch lateral edges. Elevated turbulence at the patch edge produced local erosion. For the dense patch, material eroded from the edge was transported into the patch to form a flow-parallel ridge, and there was no net sediment loss/gain by the patch. For the sparse patch, material eroded from the edge was transported away from the patch, resulting in a net loss of sediment from the patch. In the wake of both patches, deposition occurred near the wake edges and not at the wake centreline, which was attributed to the weak lateral transport associated with the weakness of the von Kármán vortex street. Specifically, the lateral transport length scale was less than half the width of the patch. The increasing bedform height within the wake progressively weakened and narrowed the von Kármán vortex street, illustrating an important feedback from morphological evolution to the flow structure. Despite significant local sediment redistribution, the patch did not induce channel-scale sediment transport.

We study the interaction between a pair of particles suspended in a uniform oscillatory flow. The time-averaged behaviour of particles under these conditions, which arises from an interplay of inertial and viscous forces, is explored through a theoretical framework relying on small oscillation amplitude. We approximate the oscillatory flow in terms of dual multipole expansions, with which we compute time-averaged interaction forces using the Lorentz reciprocal theorem. We then develop analytic approximations for the force in the limit where Stokes layers surrounding the particles do not overlap. Finally, we show how the same formalism can be generalised to the situation where the particles are free to oscillate and drift in response to the applied flow. The results are shown to be in agreement with existing numerical data for forces and particle velocities. The theory thus provides an efficient means to quantify nonlinear particle interactions in oscillatory flows.