Topological transition and helicity conversion of vortex torus knots and links are studied using direct numerical simulations of the incompressible Navier–Stokes equations. We find three topological transitional routes (viz. merging, reconnection and transition to turbulence) in the evolution of vortex knots and links over a range of torus aspect ratios and winding numbers. The topological transition depends not only on the initial topology but also on the initial geometry of knots/links. For small torus aspect ratios, the initially knotted or linked vortex tube rapidly merges into a vortex ring with a complete helicity conversion from the writhe and link components to the twist. For large torus aspect ratios, the vortex knot or link is untied into upper and lower coiled loops via the first vortex reconnection, with a helicity fluctuation including loss of writhe and link, and generation of twist. Then, the relatively unstable lower loop can undergo a secondary reconnection to split into multiple small vortices with a similar helicity fluctuation. Surprisingly, for moderate torus aspect ratios, the incomplete reconnection of tangled vortex loops together with strong vortex interactions triggers transition to turbulence, in which the topological helicity decomposition fails due to the breakdown of vortex core lines.

The vortical structures over a thin rectangular wing with a very low aspect ratio of 0.277 were investigated in a wind tunnel at an effective Reynolds number of $3 \times 10^6$. When applying pitch-up motion pivoted at mid-chord, the maximum lift angle was increased with an increase in the pitch rate, but the maximum lift coefficient was reduced. The pitching motion also caused delay in the vortical development over the wing, which was increased with an increase in the pitch rate. The delay in the leading-edge vortex development due to the pitching motion was nearly identical to that in the tip vortex development, indicating that the dynamics of the leading-edge vortex was strongly influenced by the tip vortex. This was confirmed by particle image velocimetry measurements, which demonstrated that the tip vortex over a very low aspect-ratio wing induced strong downwash to influence the development of the leading-edge vortex during the pitching motion, which led to a delay in flow separation.

A dynamically consistent scaling of mean skin friction in zero-pressure-gradient turbulent boundary layers and fully developed pipe and channel flows, is derived. Theoretical arguments are based on transfer of kinetic energy from mean flow to large eddies of turbulence. A single new velocity scale $M/\nu$ is shown to be dynamically relevant for scaling skin friction in all flows; $M$ is the planar kinematic momentum rate of the shear flow and $\nu$ is fluid kinematic viscosity. An asymptotic $-1/2$ power scaling law (in $M$–$\nu$ scaling) is shown to be universally applicable. It is observed that the semi-empirical finite-$Re$ skin friction model, resulting from the asymptotic scaling law, applies well to individual flows, but fails to describe all flows in a universal fashion. This non-universality could be due to the differences in flow boundary conditions at finite Reynolds numbers and flow geometry, that affect the outer-layer structures in these flows. It is argued that these differences may be simply absorbed by considering differences in the shapes of mean velocity profiles amongst these flows. An empirical correction to $M$–$\nu$ scaling is proposed based on Clauser's shape factor $G$. It is demonstrated that data from all flows in this new, semi-empirical $M$–$\nu$–$G$ scaling collapse remarkably well onto a single universal curve. The corresponding universal finite-$Re$ model in $M$–$\nu$–$G$ scaling is shown to describe this curve to an excellent accuracy. These results underscore the importance of a dynamically consistent approach towards revealing universality of skin friction scaling in wall turbulence.

Dynamics of two-dimensional flow past a rigid flat plate with a trailing closed flexible filament acting as a deformable afterbody are investigated numerically by an immersed boundary-lattice Boltzmann method for the fluid flow and a finite element method for the filament motion. The effects of Reynolds number ($Re$) and length ratio ($Lr$) on the flow patterns and dynamics of the rigid-flexible coupling system are studied. Based on our numerical results, five typical state modes have been identified in $Lr\unicode{x2013}Re$ plane in terms of the filament shape and corresponding dynamics, i.e. static deformation, micro-vibration, multi-frequency flapping, periodic flapping and chaotic flapping modes, respectively. Benefiting from the passive flow control by using the flexible filament as a deformable afterbody, the coupled system may enjoy a significant drag reduction (up to $22\,\%$) compared with bare plate scenarios ($Lr=1$). Maximum drag reduction achieved at $L_{c,{min}} \in [1.8, 2]$ is often accompanied by the onset of the system state transition. The flow characteristic and its relation to the change in hydrodynamic drag are further explored in order to reveal the underlying mechanisms of the counterintuitive dynamical behaviour of the coupled system. The scaling laws for the form drag and the friction drag, which arise from the pressure and viscous effects, respectively, are proposed to estimate the overall drag acting on the system. The results obtained in the present study may shed some light on understanding the dynamical behaviour of rigid-flexible coupling systems.

The essence of sub-critical transition of oscillatory boundary-layer flows is the non-modal growth of finite-amplitude disturbances. The current understanding of the mechanisms of the orderly and bypass transitions of oscillatory boundary-layer flows is limited. The present study adopts optimisation approaches to predict the maximum energy amplification of two- and three-dimensional perturbations in response to the optimal initial disturbance with or without external forcing. A series of direct numerical simulations are also performed to compare with the results obtained from the stability analyses. In particular, the optimal initial perturbation similar to a Tollmien–Schlichting (T–S) wave yields the largest transient growth under the combined effects of the Orr mechanism and inflectional point instability. With a considerable level of two-dimensional disturbance, the vortex tube nonlinearly develops from the T–S-like wave, and then either deforms into a $\varLambda$-vortex in the near-wall region or rolls up to the free shear region. The further burst of turbulence can follow the first pathway as K-type transition or the second one as vortex tube breakdown due to the elliptical instability. Additionally, non-modal growth can initiate the inception of streaky structures by favourable three-dimensional initial perturbations and/or forcing. The secondary instabilities responsible for the streak breakdown are classified as the varicose (symmetric) and sinuous (anti-symmetric) modes. Under a sufficiently high level of three-dimensional disturbance, the bypass transition is predominantly characterised by the formation of the sinuous mode and turbulent spots, which leads to the suppression of inflection point instability.

The interaction between forward-facing steps of several heights and a pre-existing critical stationary crossflow instability of a swept-wing boundary layer is analysed. Direct numerical simulations (DNS) are performed of the incompressible three-dimensional laminar base flow and the stationary distorted flow that arise from the interaction between an imposed primary stationary crossflow perturbation and the steps. These DNS are complemented with solutions of the linear and the nonlinear parabolised stability equations, used towards identifying the influence of linearity and non-parallelism near the step. A fully stationary solution of the Navier–Stokes equations is enforced numerically, in order to isolate the mechanisms pertaining to the interaction of the stationary disturbance with the step. Results provide insight into the salient modifications of the base laminar boundary layer due to the step, and the response of the incoming crossflow instability to these changes. The fundamental spanwise Fourier mode of the disturbance field gradually lifts up as it approaches the step and passes over it. The flow environment around the step is characterised by a sudden spanwise modulation of the base-flow streamlines. Additional stationary perturbation structures are induced at the step, which manifest in the form of spanwise-aligned velocity streaks near the wall. Shortly downstream of the step, the fundamental component of the crossflow perturbation maintains a rather constant amplification for the smallest steps studied. For the largest step, however, the fundamental crossflow perturbation is stabilised significantly shortly downstream of the largest step. This surprising result is ascribed to a modulation of the kinetic energy transfer between the base flow and the fundamental perturbation field, which is brought forward as a new step interaction mechanism. Possible non-modal growth effects at the step are discussed. Furthermore, the results from DNS indicate significant amplification of the high-order harmonic crossflow components downstream of the step.

A model is constructed to describe the flow field and arbitrary deformation of a drop or vesicle that contains and is embedded in an electrolyte solution, where the flow and deformation are caused by an applied electric field. The applied field produces an electrokinetic flow, which is set up on the charge-up time scale $\tau _{*c}=\lambda _{*} a_{*}/D_{*}$, where $\lambda _{*}$ is the Debye screening length, $a_{*}$ is the inclusion length scale and $D_{*}$ is an ion diffusion constant. The model is based on the Poisson–Nernst–Planck and Stokes equations. These are reduced or simplified by forming the limit of strong electrolytes, for which dissolved salts are completely ionised in solution, together with the limit of thin Debye layers. Debye layers of opposite polarity form on either side of the drop interface or vesicle membrane, together forming an electrical double layer. Two formulations of the model are given. One utilises an integral equation for the velocity field on the interface or membrane surface together with a pair of integral equations for the electrostatic potential on the outer faces of the double layer. The other utilises a form of the stress-balance boundary condition that incorporates the double layer structure into relations between the dependent variables on the layers’ outer faces. This constitutes an interfacial boundary condition that drives an otherwise unforced Stokes flow outside the double layer. For both formulations relations derived from the transport of ions in each Debye layer give additional boundary conditions for the potential and ion concentrations outside the double layer.

We investigate the mechanisms leading to acoustic whistling for a jet passing through a circular hole in a thick plate connecting two domains. Two generic situations are considered. In the first one, the upstream domain is a closed cavity while the downstream domain is open, leading to a class of conditionally unstable modes. In this case, the instability source lies in the recirculation region within the thickness of the plate, but coupling with a conveniently tuned resonator is needed to select the conditional instability range. In the second situation, the two regions, upstream and downstream of the hole, are considered as open, leading to a class of hydrodynamic modes where instability of the recirculation region is sufficient to generate self-oscillations without the need of any resonator. A matched asymptotic model, valid in the low Mach limit, is used to derive a global impedance of the system, combining the impedance of the hole and the modelled impedances of the upstream and downstream domains. It is shown that the knowledge of this global impedance along the real $\omega$-axis provides an instability criterion and a prediction of the eigenvalues of the full system. Validations against the solutions of the eigenvalue problem obtained from the linearized fully compressible formulation confirm the accuracy of the approach. Then, it is subsequently used to characterise the range of existence of instabilities as a function of the Reynolds number, the Mach number, the aspect ratio of the hole and (for the cavity configuration) the dimensionless volume of the cavity.

Analytical wind turbine wake models are widely used to predict the wake velocity deficit. In these models, the wake growth rate is a key parameter specified mainly with empirical formulations. In this study, a new physics-based model is proposed and validated to predict the wake expansion downstream of a turbine based on the incoming ambient turbulence and turbine operating conditions. The new model utilises Taylor diffusion theory, the Gaussian wake model, turbulent mixing layer theory and the analogy between wind turbine wake expansion and scalar diffusion. These components ensure that the model conserves mass and momentum in the far wake and accounts for the ambient turbulence and turbine-induced turbulence effects on the wake expansion. To account for the turbulence relevant scales that contribute to the wake expansion, the model uses the root-mean-square of the low-pass filtered radial velocity component. A simplified version that only requires the unfiltered velocity standard deviation and turbulence integral scale is also proposed. In addition, a new relation for the near-wake length is derived. The model performance is validated using large-eddy simulation data of a wind turbine wake under neutral atmospheric conditions with a wide range of incoming turbulence levels. The results show that the proposed model yields reasonable predictions of the wake width, maximum velocity deficit and near-wake length. In the case with a relatively low incoming streamwise turbulence intensity of 0.05, the ambient and turbine-induced terms in the model contribute almost equally to the wake width, rendering them both crucial for reasonable wake predictions.

We show that the Fukagata et al.'s (Phys. Fluids, vol. 14, no. 11, 2002, pp. 73–76) identity for free-stream boundary layers simplifies to the von Kármán momentum integral equation relating the skin-friction coefficient and the momentum thickness when the upper bound in the integrals used to obtain the identity is taken to be asymptotically large. If a finite upper bound is used, the terms of the identity depend spuriously on the bound itself. Differently from channel and pipe flows, the impact of the Reynolds stresses on the wall-shear stress cannot be quantified in the case of free-stream boundary layers because the Reynolds stresses disappear from the identity. The infinite number of alternative identities obtained by performing additional integrations on the streamwise momentum equation also all simplify to the von Kármán equation. Analogous identities are found for channel flows, where the relative influence of the physical terms on the wall-shear stress depends on the number of successive integrations, demonstrating that the laminar and turbulent contributions to the skin-friction coefficient are only distinguished in the original identity discovered by Fukagata et al. (Phys. Fluids, vol. 14, no. 11, 2002, pp. 73–76). In the limit of large number of integrations, these identities degenerate to the definition of skin-friction coefficient and a novel twofold-integration identity is found for channel and pipe flows. In addition, we decompose the skin-friction coefficient uniquely as the sum of the change of integral thicknesses with the streamwise direction, following the study of Renard & Deck (J. Fluid Mech., vol. 790, 2016, pp. 339–367). We utilize an energy thickness and an inertia thickness, which is composed of a thickness related to the mean-flow wall-normal convection and a thickness linked to the streamwise inhomogeneity of the mean streamwise velocity. The contributions of the different terms of the streamwise momentum equation to the friction drag are thus quantified by these integral thicknesses.

We perform large-scale numerical simulations to study Rayleigh–Darcy convection in three-dimensional fluid-saturated porous media up to Rayleigh–Darcy number $Ra=8\times 10^4$. At these large values of $Ra$, the flow is dominated by large columnar structures – called megaplumes – which span the entire height of the domain. Near the boundaries, the flow is hierarchically organized, with fine-scale structures interacting and nesting to form larger-scale structures called supercells. We observe that the correlation between the flow structure in the core of the domain and at the boundaries decreases only slightly for increasing $Ra$, and remains rather high even at the largest $Ra$ considered here. This confirms that supercells are the boundary footprint of megaplumes dominating the core of the domain. In agreement with available literature predictions, we show that the thickness of the thermal boundary layer scales very well with the Nusselt number as $\delta \sim 1/ Nu$. Measurements of the mean wavenumber – inverse of the mean length scale – in the core of the flow support the scaling $\bar {k} \sim Ra^{0.49}$, in very good agreement with theoretical and numerical predictions. Interestingly, the behaviour of the mean wavenumber near the boundaries scales as $\bar {k} \sim Ra^{0.81}$, which is distinguishably different from the presumed linear behaviour. We hypothesize that a linear behaviour can only be observed in the ultimate regime, which we argue to set in only at $Ra$ in excess of $5\times 10^5$, whereas a sublinear behaviour is recovered at more modest $Ra$. The present results are expected to help the development of long desired reliable models to predict the large- and fine-scale structure of Rayleigh–Darcy convection in the high-$Ra$ regime typically encountered in geophysical processes, such as for instance in geological carbon dioxide sequestration.