We investigate theoretically the steady incompressible viscoelastic flow in a rigid axisymmetric cylindrical pipe with varying cross-section. We use the Oldroyd-B viscoelastic constitutive equation to model the fluid viscoelasticity. First, we derive exact general formulae: for the total average pressure-drop as a function of the wall shear rate and the viscoelastic axial normal extra-stress; for the viscoelastic extra-stress tensor and the Trouton ratio as functions of the fluid velocity on the axis of symmetry; and for the viscoelastic extra-stress tensor along the wall in terms of the shear rate at the wall. Then we exploit the classic lubrication approximation, valid for small values of the square of the aspect ratio of the pipe, to simplify the original governing equations. The final equations are solved analytically using a regular perturbation scheme in terms of the Deborah number, De, up to eighth order in De. For a hyperbolically shaped pipe, we reveal that the reduced pressure-drop and the Trouton ratio can be recast in terms of a modified Deborah number, Dem, and the polymer viscosity ratio, η, only. Furthermore, we enhance the convergence and accuracy of the eighth-order solutions by deriving transformed analytical formulae using Padé diagonal approximants. The results show the decrease of the pressure drop and the enhancement of the Trouton ratio with increasing Dem and/or increasing η. Comparison of the transformed solutions with numerical simulations of the lubrication equations using pseudospectral methods shows excellent agreement between the results, even for high values of Dem and all values of η, revealing the robustness, validity and efficiency of the theoretical methods and techniques developed in this work. Last, it is shown that the exact solution for the Trouton ratio gives a well-defined and finite solution for any value of Dem and reveals the reason for the failure of the corresponding high-order perturbation series for Dem > 1/2.

Turbulent entrainment at the turbulent/non-turbulent interface (TNTI) plays an important role in understanding the turbulent diffusion. While entrainment in fully developed canonical turbulent flows has been extensively studied, the evolution of entrainment in spatially developing flows remains poorly understood. In this work, characteristics of entrainment and the effect of vortices on entrainment of the shear layer separated from a wall-mounted fence are studied by the experiment in a water channel. The shedding vortex experiences a series of stages, including generation, growth, deformation and breakdown into smaller vortices. With the development of the flow, entrainment varies correspondingly. The prograde vortex near the TNTI is found to suppress entrainment but have little effect on the detrainment process, while the retrograde vortex promotes entrainment and suppresses detrainment as well. Consequently, the local entrainment velocity is decreased by the prograde vortex and increased more significantly by the retrograde vortex. Along the streamwise direction, the time-mean entrainment velocity is smallest where the prograde vortex is strongest in the vortex deformation stage. However, the largest time-mean entrainment velocity is located where the enstrophy gradient near the TNTI is greatest after reattachment, rather than where the retrograde vortex is strongest shortly after the breakdown of the shedding vortex, because the scarcity of retrograde vortices in the vicinity of the TNTI makes their long-time cumulative contribution not as significant as their local enhancement. The present study reveals how entrainment evolves in the separated and reattaching flow, and improves our understanding of the effect of vortices on entrainment.

In convergent geometry, the effect of convergence and compression on the Rayleigh–Taylor instability (RTI) and Richtmyer–Meshkov instability (RMI) modifies the growth rate and behaviour of the instabilities. In order to better understand how compression/expansion caused by axial strain rates (i.e. strain rates normal to the interface) change the instability dynamics, axial strain rates are applied to RMI in planar geometry, isolated from the effects of convergence. Potential flow theory for the linear regime shows the growth rate of the instability is modified to include the background velocity difference of the instability's width. Resolved two-dimensional simulations of single-mode RMI showed the potential flow model is accurate whilst the amplitude is small compared with the wavelength. The application of strain rate to an RMI-induced mixing layer was investigated using three-dimensional implicit large eddy simulations (ILES) of the quarter-scale $\theta$-group case by Thornber et al. (Phys. Fluids, vol. 29, 2017, 105107). Whilst the background strain rate contributed to the mixing layer's growth, it was to a smaller extent than expected. The shear production of axial turbulent kinetic energy from the strain rate modified the rate of bulk entrainment, affecting the mixing layer's growth and mixedness, such that the strained simulations no longer attained the same self-similar state. The capability of the buoyancy-drag model by Youngs & Thornber (Physica D, vol. 410, 2020, 132517) to predict the mixing layer width was investigated, using a model calibrated to the unstrained case. New terms were introduced into the buoyancy-drag model, which correspond to the shear production of turbulent kinetic energy.

Supersonic internal flows often exhibit multiple reflected shocks within a limited distance. These shocks can interact with each other in a complex manner due to the characteristics of the shock wave–turbulent boundary layer interaction (STBLI), including flow distortion and the relaxing boundary layer. This study aims to characterise this type of interaction and to clarify its fluid physics. A separated STBLI zone was established either upstream or downstream, and another weaker STBLI was established in the opposing position to serve as a perturbation. Time-resolved measurements were employed to characterise the mean separation and unsteadiness as the two regions approached each other, as well as their relationship. The experimental results indicated that the STBLI could affect the separation and reattachment of the other STBLI through either the decelerated or relaxing boundary layer. Despite a small deflection angle, the incident shock can amplify the low-frequency oscillations in the downstream STBLI region. Additionally, the interaction in the downstream region can be influenced by both low- and high-frequency oscillations associated with the upstream STBLI through a relaxing boundary layer. Despite the limited correlation observed between the low-frequency fluctuations in the downstream region and the boundary layer flow not far upstream, there still exists some degree of correlation between the low-frequency shock motions even when they are widely separated. Both the ‘upstream mechanism’ and ‘downstream mechanism’ have been observed, and the significance of low-frequency dynamics in the separated flow, relative to that of the upstream flow, is closely associated with interaction intensity.

In transonic flow conditions, buffeting associated with finite-amplitude lift fluctuations can limit the operational envelope of an aircraft. For both airfoils and wings, these oscillations have been linked to global flow instabilities that arise from a Hopf bifurcation. We employ a combination of numerical simulations and global stability analysis to investigate the near-critical behaviour of the oscillatory buffet-onset instability on airfoils. The flow is governed by the unsteady Reynolds-averaged Navier–Stokes equations, with a basic state provided by a steady-state solution. In the weakly nonlinear formulation, the disturbance amplitude is described by the Landau equation. The linear growth rate can be determined from either the simulations or the stability analysis, and the Landau constant is derived from simulations resulting in finite-amplitude equilibrium states. The results show that the Landau constant is nearly independent of Mach number and angle of attack for a given airfoil. Using the Landau constant derived from a small number of simulations, the stability analysis can be employed to efficiently capture the essential finite-amplitude behaviour needed to estimate the buffet-onset boundary. The stability analysis is shown to capture the envelope of lift oscillations during a continuous pitch of an airfoil, from pre-buffet through post-buffet lift levels.

We present a thin-film viscoplastic fluid model of a mountain range, where a uniform fluid layer is deformed by a vertical backstop moving at constant speed. This represents a simplification of the geometry in subduction zones, where an overlying tectonic plate scrapes sediments off the underthrusting plate, thereby forming an accretionary wedge. By using a viscoplastic rheology, we aim to generalise Newtonian models that capture the effective viscous behaviour of rock at large length scales. The model system is characterised by the dimensionless Bingham number, which is the ratio of the yield stress to characteristic shear stress. At low and high Bingham numbers and at early and late times the system is found to be asymptotically self-similar, which we confirm by solving the governing equations numerically. In addition, we test the high-Bingham-number results experimentally using ultrasound gel as the working fluid. The experiments reproduce many features of the theoretically predicted behaviour. The size of the observed wedge grows in the manner predicted, and the fluid surface profile is found to collapse to a universal shape. The viscoplastic fluid wedge exhibits features of both viscous continuum and Coulomb models of accretionary wedges, suggesting that a viscoplastic rheology may provide quantitative insights into the dynamics of real accretionary wedges found in convergent tectonic settings.

Quantifying transport by strongly stratified turbulence in low Prandtl number ($Pr$) fluids is critically important for the development of better models for the structure and evolution of stellar and planetary interiors. Motivated by recent numerical simulations showing strongly anisotropic flows suggestive of a scale-separated dynamics, we perform a multiscale asymptotic analysis of the governing equations. We find that, in all cases, the resulting slow–fast systems take a quasilinear form. Our analysis also reveals the existence of several distinct dynamical regimes depending on the emergent buoyancy Reynolds and Péclet numbers, $Re_b = \alpha ^2 Re$ and $Pe_b = Pr Re_b$, respectively, where $\alpha$ is the aspect ratio of the large-scale turbulent flow structures, and $Re$ is the outer-scale Reynolds number. Scaling relationships relating the aspect ratio, the characteristic vertical velocity and the strength of the stratification (measured by the Froude number $Fr$) naturally emerge from the analysis. When $Pe_b \ll \alpha$, the dynamics at all scales is dominated by buoyancy diffusion, and our results recover the scaling laws empirically obtained from direct numerical simulations by Cope et al. (J. Fluid Mech., vol. 903, 2020, A1). For $Pe_b \ge O(1)$, diffusion is negligible (or at least subdominant) at all scales and our results are consistent with those of Chini et al. (J. Fluid Mech., vol. 933, 2022) for strongly stratified geophysical turbulence at $Pr =O(1)$. Finally, we have identified a new regime for $\alpha \ll Pe_b \ll 1$, in which slow, large scales are diffusive while fast, small scales are not. We conclude by presenting a map of parameter space that clearly indicates the transitions between isotropic turbulence, non-diffusive stratified turbulence, diffusive stratified turbulence and viscously dominated flows, and by proposing parameterisations of the buoyancy flux, mixing efficiency and turbulent diffusion coefficient for each regime.

We investigate exact nonlinear waves on surfaces locally approximating the rotating sphere for two-dimensional inviscid incompressible flow. Our first system corresponds to a $\beta$-plane approximation at the equator, and the second to a $\gamma$ approximation, with the latter describing flow near the poles. We find exact wave solutions in the Lagrangian reference frame that cannot be written down in closed form in the Eulerian reference frame. The wave particle trajectories, contours of potential vorticity and Lagrangian mean velocity take relatively simple forms. The waves possess a non-trivial Lagrangian mean flow that depends on the amplitude of the waves and on a particle label that characterizes values of constant potential vorticity. The mean flow arises due to potential vorticity conservation on fluid particles. Solutions over the entire space are generated by assuming that the flow far from the origin is zonal and there is a region of uniform potential vorticity between this zonal flow and the waves. In the $\gamma$ approximation, a class of waves is found that, based on analogous solutions on the plane, we call Ptolemaic vortex waves. The mean flow of some of these waves, which we can describe in highly nonlinear scenarios due to the exact nature of the solutions, resembles polar jet streams. Several illustrative solutions are used as initial conditions in the fully spherical rotating Navier–Stokes equations, where integration is performed via the numerical scheme presented in Salmon & Pizzo (Atmosphere, vol. 14, issue 4, 2023, 747). The potential vorticity contours found from these numerical experiments vary between stable permanent progressive form and fully turbulent flows generated by wave breaking.

A novel selective mode decomposition, proper orthogonal decomposition and dynamic mode decomposition methods are used to analyse large-eddy simulation data of the flow field about a NACA0012 airfoil at low Reynolds numbers of $5\times 10^4$ and $9\times 10^4$, and at near-stall conditions. The objective of the analysis is to investigate the structure of the laminar separation bubble (LSB) and its associated low-frequency flow oscillation (LFO). It is shown that the flow field can be decomposed into three dominant flow modes: two low-frequency modes (LFO-Mode-1 and LFO-Mode-2) that govern an interplay of a triad of vortices and sustain the LFO phenomenon, and a high-frequency oscillating (HFO) mode featuring travelling Kelvin–Helmholtz waves along the wake of the airfoil. The structure and dynamics of the LSB depend on the energy content of these three dominant flow modes. At angles of attack lower than the stall angle of attack and above the angle of a full stall, the flow is dominated by the HFO mode. At angles of attack above the stall angle of attack the LFO-Mode-2 overtakes the HFO mode, triggers instability in the LSB and initiates the LFO phenomenon. Previous studies peg the structure, stability and bursting conditions of the separation bubble to local flow parameters. However, the amplitude of these local flow parameters is dependent on the energy content of the three dominant flow modes. Thus, the present work proposes a more robust bursting criterion that is based on global eigenmodes.

We investigate the dynamics of heavy inertial particles in a flow field due to an isolated, non-axisymmetric vortex. For our study, we consider a canonical elliptical vortex – the Kirchhoff vortex and its strained variant, the Kida vortex. Contrary to the anticipated centrifugal dispersion of inertial particles, which is typical in open vortical flows, we observe the clustering of particles around co-rotating attractors near the Kirchhoff vortex due to its non-axisymmetric nature. We analyse the inertia-modified stability characteristics of the fixed points, highlighting how some of the fixed points migrate in physical space, collide and then annihilate with increasing particle inertia. The introduction of external straining, the Kida vortex being an example, introduces chaotic tracer transport. Using a Melnikov analysis, we show that particle inertia and external straining can compete, where chaotic transport can be suppressed beyond a critical value of particle inertia.

As wind farms continue to grow in size, mesoscale effects such as blockage and gravity waves become increasingly important. Allaerts & Meyers (J. Fluid Mech., vol. 862, 2019, pp. 990–1028) proposed an atmospheric perturbation model (APM) that can simulate the interaction of wind farms and the atmospheric boundary layer while keeping computational costs low. The model resolves the mesoscale flow, and couples to a wake model to estimate the turbine inflow velocities at the microscale. This study presents a new way of coupling the mesoscale APM to a wake model, based on matching the velocity between the models throughout the farm. This method performs well, but requires good estimates of the turbine-level velocity fields by the wake model. Additionally, we investigate the mesoscale effects of a large wind farm, and find that aside from the turbine forces and increased turbulence levels, the dispersive stresses due to subgrid flow heterogeneity also play an important role at the entrance of the farm, and contribute to the global blockage effect. By using the wake model coupling, we can explicitly incorporate these stresses in the model. The resulting APM is validated using 27 prior large-eddy simulations of a large wind farm under different atmospheric conditions. The APM and large-eddy simulation results are compared on both mesoscale and turbine scale, and on turbine power output. The APM captures the overall effects that gravity waves have on wind farm power production, and significantly outperforms standard wake models.