The subgrid-scale (SGS) Prandtl number ($Pr$) is an important parameter in large-eddy simulation. Prior models often assume that the ‘$-5/3$’ inertial subrange scaling applies to the wavenumber range from 0 to $k_{\unicode[STIX]{x1D6E5}}$ (the wavenumber corresponding to the filter scale $\unicode[STIX]{x1D6E5}$) and yield a $Pr$ that is stability-independent and scale-invariant, which is inconsistent with experimental data and the results of dynamic models. In this study, the SGS Prandtl number is revisited by solving the co-spectral budgets of momentum and heat fluxes in an idealized but thermally stratified atmospheric surface layer. The SGS Prandtl number from the co-spectral budget model shows a strong dependence on the atmospheric stability and increases (decreases) as the atmosphere becomes stable (unstable), which is in good agreement with recent field experimental data. The dependence of $Pr$ on the filter scale is also captured by the co-spectral budget model: as the filter scale becomes smaller, the SGS Prandtl number decreases. Finally, the value of SGS Prandtl number under neutral conditions is shown to be caused by the dissimilarity between momentum and heat in the pressure decorrelation term and the flux transfer term. When the dissimilarity exists only in the flux transfer term, the fact that under neutral conditions the SGS Prandtl number is usually smaller than the turbulent Prandtl number for Reynolds-averaged Navier–Stokes simulations is an indication of a stronger spectral transfer coefficient for heat than for momentum. The model proposed in this study can be readily implemented.

Vortex reconnection, as the topological change of vortex lines or surfaces, is a critical process in transitional flows, but is challenging to accurately characterize, particularly in shear flows. We apply the vortex-surface field (VSF), whose isosurface is the vortex surface consisting of vortex lines, to study vortex reconnection in the Klebanoff-type temporal transition in channel flow. The VSF evolution can capture the reconnection of the hairpin-like vortical structures evolving from the initial vortex sheets in opposite halves of the channel. The incipient vortex reconnection is characterized by the vanishing minimum distance between a pair of vortex surfaces and the reduction of vorticity flux through the region enclosed by the wall and the VSF isoline of the channel half-height on the spanwise symmetric plane. We find that the surge of the wall-friction coefficient begins at the identified reconnection time. From the Biot–Savart law, the rapid reconnection of vortex lines can induce a velocity opposed to the mean flow, which partially blocks the flow near the central region and generally accelerates the near-wall fluid motion in the flow with constant mass flux. Therefore, the vortex reconnection appears to play an important role in the sudden increase of wall friction in transitional channel flows.

Understanding the origin of large-scale structures in high-Reynolds-number wall turbulence has been a central issue over a number of years. Recently, Rawat et al. (J. Fluid Mech., vol. 782, 2015, pp. 515–540) have computed invariant solutions for the large-scale structures in turbulent Couette flow at $Re_{\unicode[STIX]{x1D70F}}\simeq 128$ using an overdamped large-eddy simulation with the Smagorinsky model to account for the effect of the surrounding small-scale motions. Here, we extend this approach to Reynolds numbers an order of magnitude higher in turbulent channel flow, towards the regime where the large-scale structures in the form of very-large-scale motions (long streaky motions) and large-scale motions (short vortical structures) emerge energetically. We demonstrate that a set of invariant solutions can be computed from simulations of the self-sustaining large-scale structures in the minimal unit (domain of size $L_{x}=3.0h$ streamwise and $L_{z}=1.5h$ spanwise) with midplane reflection symmetry at least up to $Re_{\unicode[STIX]{x1D70F}}\simeq 1000$. By approximating the surrounding small scales with an artificially elevated Smagorinsky constant, a set of equilibrium states are found, labelled upper- and lower-branch according to their associated drag. It is shown that the upper-branch equilibrium state is a reasonable proxy for the spatial structure and the turbulent statistics of the self-sustaining large-scale structures.

The concept of edge states is investigated in the asymptotic suction boundary layer in relation to the receptivity process to noisy perturbations and the nucleation of turbulent spots. Edge tracking is first performed numerically, without imposing any discrete symmetry, in a large computational domain allowing for full spatial localisation of the perturbation velocity. The edge state is a three-dimensional localised structure recurrently characterised by a single low-speed streak that experiences erratic bursts and planar shifts. This recurrent streaky structure is then compared with predecessors of individual spot nucleation events, triggered by non-localised initial noise. The present results suggest a nonlinear picture, rooted in dynamical systems theory, of the nucleation process of turbulent spots in boundary-layer flows, in which the localised edge state plays the role of state-space mediator.

The mechanism of trailing vortex wandering has long been debated and is often attributed to either wind-tunnel effects or an instability. Using particle image velocimetry data obtained in the wake of a NACA0012 airfoil, we remove the effect of wandering from the measured velocity field and, through a triple decomposition, recover the coherent wandering motion. Based on this wandering motion, the most energetic structures are computed using the proper orthogonal decomposition (POD) and exhibit a helical mode with an azimuthal wavenumber of $|m|=1$ whose kinetic energy grows monotonically in the downstream direction. To investigate the nature of the vortex wandering, we perform a spatial stability analysis of a matched Batchelor vortex. The primary stability mode is found to be marginally stable and nearly identical in both size and structure to the leading POD mode. The strikingly similar structure, coupled with the measured energy growth, supports the proposition that the vortex wandering is the result of an instability. We conclude that the cause of the wandering is the non-zero radial velocity of the $|m|=1$ mode on the vortex centreline, which acts to transversely displace the trailing vortex, as observed in experiments. However, the marginal nature of the stability mode prevents a definitive conclusion regarding the specific type of instability.

We present results of numerical simulations of leading-edge acoustic receptivity for acoustic waves impinging on the leading edge of a finite-thickness flat plate. We use both compressible and incompressible flow solvers fitted with high-order high-accuracy numerical methods and independent methods of estimating the receptivity coefficient. The results show that the level of acoustic receptivity in the existing literature appears to be one order of magnitude too high. Our review of previous numerical simulations and experiments clearly identifies some contradictory trends. In the limit of an infinitely thin flat plate, our results are consistent with asymptotic theory and numerical simulations.

Leading-edge vortices (LEVs) can form and remain attached to a rotating wing indefinitely, but the mechanisms of stable attachment are not well understood. Taking for granted that such stable structures do form, a practical question arises: what is the trajectory of the LEV core? Noting that span-wise flow exists within the LEV core, it is apparent that a mean streamline aligned with the axis of the stable LEV must exist. The present work uses the Navier–Stokes equations along such a steady, axial streamline in order to consider the accelerations that act in the streamline-normal direction to affect its local curvature. With some simplifying assumptions, a coupled system of ordinary differential equations is derived that describes the trajectory of an axial streamline through the vortex core. The model is compared to previous work, and is found to predict the trajectory of the LEV core well at span-wise locations inboard of the midspan. This result suggests that Coriolis acceleration is responsible for limiting the span-wise extent of a stable LEV by tilting it into the wake within several chord lengths from the centre of rotation. The downwash due to the tip vortex also appears to play a role, as the only significant differences between model-predicted LEV trajectories and previous results are in the plate-normal direction.

We consider dynamo action driven by three-dimensional rotating anelastic convection in a spherical shell. Motivated by the behaviour of the solar dynamo, we examine the interaction of hydromagnetic modes with different symmetries and demonstrate how complicated interactions between convection, differential rotation and magnetic fields may lead to modulation of the basic cycle. For some parameters, type 1 modulation occurs by the transfer of energy between modes of different symmetries with little change in the overall amplitude; for other parameters, the modulation is of type 2, where the amplitude is significantly affected (leading to grand minima in activity) without significant changes in symmetry. Most importantly, we identify the presence of ‘supermodulation’ in the solutions, where the activity switches chaotically between type 1 and type 2 modulation; this is believed to be an important process in solar activity.

Internal phoretic flows due to the interactions of solid boundaries with local chemical gradients may be created using chemical patterning. Alternatively, we demonstrate here that internal flows might also be induced by geometric asymmetries of chemically homogeneous surfaces. We characterise the circulatory flow created in a cavity enclosed between two eccentric cylindrical walls of uniform chemical activity. Local gradients of the diffusing solute induce a slip flow along the surface of the cylinders, leading to a circulatory bulk flow pattern which can be solved analytically in the diffusive limit. The flow strength can be controlled by adjusting the relative positions of the cylinders, and an optimal configuration is identified. These results provide a model system for tunable phoretic pumps.

The generation of indirect combustion noise by compositional inhomogeneities is examined theoretically. For this, the compact-nozzle theory of Marble & Candel (J. Sound Vib., vol. 55 (2), 1977, pp. 225–243) is extended to a multi-component gas mixture, and the chemical potential function is introduced as an additional acoustic source mechanism. Transfer functions for subcritical and supercritical nozzle flows are derived, and the contribution of compositional noise is compared to entropy noise and direct noise by considering an idealized nozzle downstream of the combustor exit. It is shown that compositional noise is dependent on the local mixture composition and can exceed entropy noise for fuel-lean conditions and supercritical nozzle flows. This suggests that the compositional indirect noise requires potential consideration with the implementation of low-emission combustors.

Impulsively started, low-aspect-ratio elliptical flat plates have been investigated experimentally to understand the vortex pinch-off dynamics at transitional and fully turbulent Reynolds numbers. The range of Reynolds numbers investigated is representative of those observed in animals that employ rowing and paddling modes of drag-based propulsion and manoeuvring. Elliptical flat plates with five aspect ratios ranging from one to two have been considered, as abstractions of propulsor planforms found in nature. It has been shown that Reynolds-number scaling is primarily determined by plate aspect ratio in terms of both drag forces and vortex pinch-off. Due to vortex-ring growth time scales that are longer than those associated with the development of flow instabilities, the scaling of drag is Reynolds-number-dependent for the aspect-ratio-one flat plate. With increasing aspect ratio, the Reynolds-number dependency decreases as a result of the shorter growth time scales associated with high-aspect-ratio elliptical vortex rings. Large drag peaks are observed during early-stage vortex growth for the higher-aspect-ratio flat plates. The collapse of these peaks with Reynolds number provides insight into the evolutionary convergence process of propulsor planforms used in drag-based swimming modes over diverse scales towards aspect ratios greater than one.

TriGlobal linear instability analysis and direct numerical simulations have been performed to unravel the mechanisms ultimately responsible for transition of steady laminar flow over a long rectangular finite-span open cavity with dimensions $L$ : $D$ : $W$$=$ 6 : 1 : 2 to unsteadiness. The steady laminar three-dimensional flow loses stability at $\mathit{Re}_{D,cr}\approx 1080$ as a consequence of linear amplification of a travelling eigenmode that is qualitatively analogous to the shear-layer mode known from analyses of flow in spanwise-periodic cavities, but has a three-dimensional structure which is strongly influenced by the cavity lateral walls. Differences in the eigenspectrum of the present and the spanwise homogeneous flow configuration are documented. Topological changes exerted on the steady laminar flow by linear amplification of the unstable shear-layer mode are reminiscent of observations in experiments at an order of magnitude higher Reynolds number.

A series of direct numerical simulations were performed of Taylor–Couette (TC) flow, the flow between two coaxial cylinders, with the outer cylinder rotating and the inner one fixed. Three cases were considered, where the Reynolds number of the outer cylinder was $Re_{o}=5.5\times 10^{4}$, $Re_{o}=1.1\times 10^{5}$ and $Re_{o}=2.2\times 10^{5}$. The ratio of radii ${\it\eta}=r_{i}/r_{o}$ was fixed to ${\it\eta}=0.909$ to mitigate the effects of curvature. Axially periodic boundary conditions were used, with the aspect ratio of vertical periodicity ${\it\Gamma}$ fixed to ${\it\Gamma}=2.09$. Being linearly stable, TC flow with outer cylinder rotation is known to have very different behaviour than TC flow with pure inner cylinder rotation. Here, we find that the flow nonetheless becomes turbulent, but the torque required to drive the cylinders and level of velocity fluctuations was found to be smaller than those for pure inner cylinder rotation at comparable Reynolds numbers. The mean angular momentum profiles showed a large gradient in the bulk, instead of the constant angular momentum profiles of pure inner cylinder rotation. The near-wall mean and fluctuation velocity profiles were found to coincide only very close to the wall, showing large deviations from both pure inner cylinder rotation profiles and the classic von Karman law of the wall elsewhere. Finally, transport of angular velocity was found to occur mainly through intermittent bursts, and not through wall-attached large-scale structures as is the case for pure inner cylinder rotation.

A submerged body subject to a sudden shape change experiences large forces due to the variation of added-mass energy. While this phenomenon has been studied for single actuation events, application to sustained propulsion requires the study of periodic shape change. We do so in this work by investigating a spring–mass oscillator submerged in quiescent fluid subject to periodic changes in its volume. We develop an analytical model to investigate the relationship between added-mass variation and viscous damping, and demonstrate its range of application with fully coupled fluid–solid Navier–Stokes simulations at large Stokes number. Our results demonstrate that the recovery of added-mass kinetic energy can be used to completely cancel the viscous damping of the fluid, driving the onset of sustained oscillations with amplitudes as large as four times the average body radius $r_{0}$. A quasi-linear relationship is found to link the terminal amplitude of the oscillations $X$ to the extent of size change $a$, with $X/a$ peaking at values from 4 to 4.75 depending on the details of the shape-change kinematics. In addition, it is found that pumping in the frequency range of $1-a/2r_{0}<{\it\omega}^{2}/{\it\omega}_{n}^{2}<1+a/2r_{0}$, with ${\it\omega}/{\it\omega}_{n}$ being the ratio between frequency of actuation and natural frequency, is required for sustained oscillations. The results of this analysis shed light on the role of added-mass recovery in the context of shape-changing bodies and biologically inspired underwater vehicles.

The flux Richardson number $R_{f}$ (often referred to as the mixing efficiency) is a widely used parameter in stably stratified turbulence which is intended to provide a measure of the amount of turbulent kinetic energy $k$ that is irreversibly converted to background potential energy (which is by definition the minimum potential energy that a stratified fluid can attain that is not available for conversion back to kinetic energy) due to turbulent mixing. The flux Richardson number is traditionally defined as the ratio of the buoyancy flux $B$ to the production rate of turbulent kinetic energy $P$. An alternative generalized definition for $R_{f}$ was proposed by Ivey & Imberger (J. Phys. Oceanogr., vol. 21, 1991, pp. 650–658), where the non-local transport terms as well as unsteady contributions were included as additional sources to the production rate of $k$. While this definition precludes the need to assume that turbulence is statistically stationary, it does not properly account for countergradient fluxes that are typically present in more strongly stratified flows. Hence, a third definition that more rigorously accounts for only the irreversible conversions of energy has been defined, where only the irreversible fluxes of buoyancy and production (i.e. the dissipation rates of $k$ and turbulent potential energy ($E_{PE}^{\prime }$)) are used. For stationary homogeneous shear flows, all of the three definitions produce identical results. However, because stationary and/or homogeneous flows are typically not found in realistic geophysical situations, clarification of the differences/similarities between these various definitions of $R_{f}$ is imperative. This is especially true given the critical role $R_{f}$ plays in inferring turbulent momentum and heat fluxes using indirect methods, as is commonly done in oceanography, and for turbulence closure parameterizations. To this end, a careful analysis of two existing direct numerical simulation (DNS) datasets of stably stratified homogeneous shear and channel flows was undertaken in the present study to compare and contrast these various definitions. We find that all three definitions are approximately equivalent when the gradient Richardson number $Ri_{g}\leqslant 1/4$. Here, $Ri_{g}=N^{2}/S^{2}$, where $N$ is the buoyancy frequency and $S$ is the mean shear rate, provides a measure of the stability of the flow. However, when $Ri_{g}>1/4$, significant differences are noticeable between the various definitions. In addition, the irreversible formulation of $R_{f}$ based on the dissipation rates of $k$ and $E_{PE}^{\prime }$ is the only definition that is free from oscillations at higher gradient Richardson numbers. Both the traditional definition and the generalized definition of $R_{f}$ exhibit significant oscillations due to the persistence of linear internal wave motions and countergradient fluxes that result in reversible exchanges between $k$ and $E_{PE}^{\prime }$. Finally, we present a simple parameterization for the irreversible flux Richardson number $R_{f}^{\ast }$ based on $Ri_{g}$ that produces excellent agreement with the DNS results for $R_{f}^{\ast }$.

A novel reduced-order model for time-varying nonlinear flows arising from a resolvent decomposition based on the time-mean flow is proposed. The inputs required for the model are the mean-flow field and a small set of velocity time-series data obtained at isolated measurement points, which are used to fix relevant frequencies, amplitudes and phases of a limited number of resolvent modes that, together with the mean flow, constitute the reduced-order model. The technique is applied to derive a model for the unsteady three-dimensional flow in a lid-driven cavity at a Reynolds number of 1200 that is based on the two-dimensional mean flow, three resolvent modes selected at the most active spanwise wavenumber, and either one or two velocity probe signals. The least-squares full-field error of the reconstructed velocity obtained using the model and two point velocity probes is of the order of 5 % of the lid velocity, and the dynamical behaviour of the reconstructed flow is qualitatively similar to that of the complete flow.

An analytical model is proposed for flow around a plane bubble rising steadily in a channel. The fluid surrounding the bubble is assumed to be inviscid, and the flow is assumed to be irrotational. The resulting potential flow is sought as the sum of a uniform stream and a source in the unit strip. Explicit expressions for the bubble shape and rise velocity, and for the streamlines, are given as functions of the dimensionless surface tension. The predictions are in good agreement with existing experimental and numerical results.

The mean velocity profile (MVP) in thermally stratified atmospheric surface layers (ASLs) deviates from the classic logarithmic form. A theoretical framework was recently proposed (Katul et al.Phys. Rev. Lett., vol. 107, 2011, 268502) to link the MVP to the spectrum of turbulence and was found to successfully predict the MVP for unstable stratification. However, the theory failed to reproduce the MVP in stable conditions (Salesky et al.Phys. Fluids, vol. 25, 2013, 105101), especially when ${\it\zeta}>0.2$ (where ${\it\zeta}$ is the atmospheric stability parameter). In the present study, it is demonstrated that this shortcoming is due to the failure to identify the appropriate length scale that characterizes the size of momentum transporting eddies in the stable ASL. Beyond ${\it\zeta}\approx 0.2$ (near where the original theory fails), the Ozmidov length scale becomes smaller than the distance from the wall $z$ and hence is a more stringent constraint for characterizing the size of turbulent eddies. An expression is derived to connect the Ozmidov length scale to the normalized MVP (${\it\phi}_{m}$), allowing ${\it\phi}_{m}$ to be solved numerically. It is found that the revised theory produces a prediction of ${\it\phi}_{m}$ in good agreement with the widely used empirical Businger–Dyer relation and two experimental datasets in the stable ASL. The results here demonstrate that the behaviour of ${\it\phi}_{m}$ in the stable ASL is closely linked to the size of momentum transporting eddies, which can be characterized by the Ozmidov scale under mildly to moderately stable conditions ($0.2<{\it\zeta}<1-2$).

The focus of this study is to analyse the tip region of a hydraulic fracture, for which a fluid flow inside the crack transitions from the laminar to the turbulent regime away from the tip. To tackle the problem, a phenomenological formula for flow in pipes has been adapted to describe flow in a fracture through the concept of a hydraulic diameter. The selected model is able to capture laminar, turbulent and transition regimes of the flow. The near-tip region of a hydraulic fracture is analysed by focusing on steady propagation of a semi-infinite hydraulic fracture with leak-off, for which the aforementioned phenomenological formula for the fluid flow is utilized. First, the distance from the tip within which a laminar solution applies is estimated. Then, expressions for asymptotic solutions that are associated with fully developed turbulent flow inside the semi-infinite hydraulic fracture are derived. Finally, the laminar zone size and the asymptotic solutions are compared with the numerical solution, where the latter captures all regimes of the fluid flow.

We report on a study, employing direct numerical simulations, of the turbulent/non-turbulent interface of a wake in a stably stratified fluid. It is found that thresholds for both enstrophy and potential enstrophy are needed to identify the interface. Using conditional averaging relative to the location of the interface, various quantities of interest are examined. The thickness of the interface is found to scale with the Kolmogorov scale. From an examination of the Ozmidov and Kolmogorov length scales as well as the buoyancy Reynolds number, it is found that the buoyancy Reynolds number decreases and becomes of order 1 near the interface, indicating the suppression of the turbulence there by the stable stratification. Finally the overall rate of loss of energy due to internal wave radiation is found to be comparable to the overall rate of loss due to turbulent kinetic energy dissipation.