This paper examines how increasing the value of the Reynolds number $Re$ affects the ability of spanwise-forcing techniques to yield turbulent skin-friction drag reduction. The considered forcing is based on the streamwise-travelling waves of spanwise-wall velocity (Quadrio et al., J. Fluid Mech., vol. 627, 2009, pp. 161–178). The study builds upon an extensive drag-reduction database created via direct numerical simulation of a turbulent channel flow for two fivefold separated values of $Re$, namely $Re_{\unicode[STIX]{x1D70F}}=200$ and $Re_{\unicode[STIX]{x1D70F}}=1000$. The sheer size of the database, which for the first time systematically addresses the amplitude of the forcing, allows a comprehensive view of the drag-reducing characteristics of the travelling waves, and enables a detailed description of the changes occurring when $Re$ increases. The effect of using a viscous scaling based on the friction velocity of either the non-controlled flow or the drag-reduced flow is described. In analogy with other wall-based drag-reduction techniques, like riblets for example, the performance of the travelling waves is well described by a vertical shift of the logarithmic portion of the mean streamwise velocity profile. Except when $Re$ is very low, this shift remains constant with $Re$, at odds with the percentage reduction of the friction coefficient, which is known to present a mild, logarithmic decline. Our new data agree with the available literature, which is however mostly based on low-$Re$ information and hence predicts a quick drop of maximum drag reduction with $Re$. The present study supports a more optimistic scenario, where for an airplane at flight Reynolds numbers a drag reduction of nearly 30 % would still be possible thanks to the travelling waves.

This investigation addresses the dynamics of annular viscoelastic films flowing down a flexible tube. The fluid viscoelasticity is assumed to be weak in order to obtain approximate explicit expressions for the stresses. Based on Shkadov’s integral boundary layer method (Fluid Dyn., vol. 2(1), 1967, pp. 29–34), a set of nonlinear evolution equations is derived that is valid for flows with moderate Reynolds numbers. The linear stability property of the system is examined by using normal-mode analysis, which is verified by comparing the results with those resulting from the linearization of the full Navier–Stokes equations. The results indicate that the fluid viscoelasticity plays an unstable role in the stability of the annular film flow. The tube flexibility, which includes wall damping and wall tension, plays a dual role. A bifurcation analysis is performed, and the families of steady travelling waves are catalogued. It is found that the stiffness of the tube tends to stimulate the interfacial capillary ripples. The fluid viscoelasticity acts to strengthen the dispersion of the interfacial waves but weakens the interfacial capillary ripples. The spatio-temporal evolutions of the system are also solved numerically. When the tube radius is small enough, tube closure can be observed due to the Plateau–Rayleigh instability. The fluid viscoelasticity acts to promote tube closure while the tube radius is relatively small. However, it plays a role in postponing the closure of the tube with a large radius.

This work is focused on the modelling of the shear and normal stresses in fibre suspensions that are subjected to a simple shear flow in the presence of short-range lubrication forces, van der Waals and electrostatic forces, as well as solid friction forces between fibres. All of these forces are weighed by the contact probability. The theory is developed for attractive fibres with van der Waals interaction dominating over electrostatic repulsion. The model predicts a simple Bingham law for both the shear stress and the first normal stress difference, with the apparent shear and normal yield stresses proportional to the second and the third power of the particle volume fraction respectively. The model is applied to the experimental data of Rakatekar et al. (Adv. Mater., vol. 21, 2009, pp. 874–878) and Natale et al. (AIChE J., vol. 60, 2014, pp. 1476–1487) on suspensions of carbon nanotubes dispersed in a Newtonian epoxy resin. It reproduces well the quadratic dependence of the apparent yield stress on the particle volume fraction $(\unicode[STIX]{x1D70E}_{Y}\propto \unicode[STIX]{x1D719}^{2})$ for average particle aspect ratios of $r=160$ and 1200, while it underpredicts the power-law exponent for $r=80$ (always predicting $\unicode[STIX]{x1D719}^{2}$ behaviour instead of $\unicode[STIX]{x1D719}^{3.2}$).

In recent years it has been established that vortex–wave interaction theory forms an asymptotic framework to describe high Reynolds number coherent structures in shear flows. Comparisons between the asymptotic approach and finite Reynolds number computations of equilibrium states from the full Navier–Stokes equations have suggested that the asymptotic approach is extremely accurate even at quite low Reynolds numbers. However, unlike the situation with an approach based on solving the full Navier–Stokes equations numerically, the vortex–wave interaction approach has not yet been developed to study the instability of the structures it describes. In this work, a comprehensive study of the different instabilities of vortex–wave interaction states is given and it is shown that there are three different time scales on which instabilities can develop. The most dangerous type is a rapidly growing Rayleigh instability of the streak part of the flow. The least dangerous type is a slow mode operating on the diffusion time scale of the roll–streak part of the flow. The third mode of instability, which we will refer to as the edge mode of instability, occurs on a time scale midway between those of the other two modes. The existence of the latter mode explains why some exact coherent structures can act as edge states between the laminar and turbulent attractors. These stability results are compared to results from Navier–Stokes calculations.

Direct numerical simulations are conducted for double diffusive convection (DDC) bounded by two parallel plates. The Prandtl numbers, i.e. the ratios between the viscosity and the molecular diffusivities of scalars, are similar to the values of seawater. The DDC flow is driven by an unstable salinity difference (here across the two plates) and stabilized at the same time by a temperature difference. For these conditions the flow can be in the finger regime. We develop scaling laws for three key response parameters of the system: the non-dimensional salinity flux $\mathit{Nu}_{S}$ mainly depends on the salinity Rayleigh number $\mathit{Ra}_{S}$, which measures the strength of the salinity difference and exhibits a very weak dependence on the density ratio $\unicode[STIX]{x1D6EC}$, which is the ratio of the buoyancy forces induced by two scalar differences. The non-dimensional flow velocity $Re$ and the non-dimensional heat flux $\mathit{Nu}_{T}$ are dependent on both $\mathit{Ra}_{S}$ and $\unicode[STIX]{x1D6EC}$. However, the rescaled Reynolds number $Re\unicode[STIX]{x1D6EC}^{\unicode[STIX]{x1D6FC}_{u}^{eff}}$ and the rescaled convective heat flux $(\mathit{Nu}_{T}-1)\unicode[STIX]{x1D6EC}^{\unicode[STIX]{x1D6FC}_{T}^{eff}}$ depend only on $\mathit{Ra}_{S}$. The two exponents are dependent on the fluid properties and are determined from the numerical results as $\unicode[STIX]{x1D6FC}_{u}^{eff}=0.25\pm 0.02$ and $\unicode[STIX]{x1D6FC}_{T}^{eff}=0.75\pm 0.03$. Moreover, the behaviours of $\mathit{Nu}_{S}$ and $Re\unicode[STIX]{x1D6EC}^{\unicode[STIX]{x1D6FC}_{u}^{eff}}$ agree with the predictions of the Grossmann–Lohse theory which was originally developed for the Rayleigh–Bénard flow. The non-dimensional salt-finger width and the thickness of the velocity boundary layers, after being rescaled by $\unicode[STIX]{x1D6EC}^{\unicode[STIX]{x1D6FC}_{u}^{eff}/2}$, collapse and obey a similar power-law scaling relation with $\mathit{Ra}_{S}$. When $\mathit{Ra}_{S}$ is large enough, salt fingers do not extend from one plate to the other and horizontal zonal flows emerge in the bulk region. We then show that the current scaling strategy can be successfully applied to the experimental results of a heat–copper–ion system (Hage & Tilgner, Phys. Fluids, vol. 22, 2010, 076603). The fluid has different properties and the exponent $\unicode[STIX]{x1D6FC}_{u}^{eff}$ takes a different value $0.54\pm 0.10$.

We consider the scaling of the mass flux and entrainment velocity across the turbulent/non-turbulent interface (TNTI) in the far field of an axisymmetric jet at high Reynolds number. Time-resolved, simultaneous multi-scale particle image velocimetry (PIV) and planar laser-induced fluorescence (PLIF) are used to identify and track the TNTI, and directly measure the local entrainment velocity along it. Application of box-counting and spatial-filtering methods, with filter sizes $\unicode[STIX]{x1D6E5}$ spanning over two decades in length, show that the mean length of the TNTI exhibits a power-law behaviour with a fractal dimension $D\approx 0.31{-}0.33$. More importantly, we invoke a multi-scale methodology to confirm that the mean mass flux, which is equal to the product of the entrainment velocity and the surface area, remains constant across the range of filter sizes. The results, within experimental uncertainty, also show that the entrainment velocity along the TNTI exhibits a power-law behaviour with $\unicode[STIX]{x1D6E5}$, such that the entrainment velocity increases with increasing $\unicode[STIX]{x1D6E5}$. In fact, the mean entrainment velocity scales at a rate that balances the scaling of the TNTI length such that the mass flux remains independent of the coarse-grain filter size, as first suggested by Meneveau & Sreenivasan (Phys. Rev. A, vol. 41, no. 4, 1990, pp. 2246–2248). Hence, at the smallest scales the entrainment velocity is small but is balanced by the presence of a very large surface area, whilst at the largest scales the entrainment velocity is large but is balanced by a smaller (smoother) surface area.

A specific feature of three-dimensional bluff body wakes, flow bistability, is a subject of particular recent interest. This feature consists of a random flipping of the wake between two asymmetric configurations and is believed to contribute to the pressure drag of many bluff bodies. In this study we apply the modelling approach recently suggested for axisymmetric bodies by Rigas et al. (J. Fluid Mech., vol. 778, 2015, R2) to the reflectional symmetry-breaking modes of a rectilinear bluff body wake. We demonstrate the validity of the model and its Reynolds number independence through time-resolved base pressure measurements of the natural wake. Further, oscillating flaps are used to investigate the dynamics and time scales of the instability associated with the flipping process, demonstrating that they are largely independent of Reynolds number. The modelling approach is then used to design a feedback controller that uses the flaps to suppress the symmetry-breaking modes. The controller is successful, leading to a suppression of the bistability of the wake, with concomitant reductions in both lateral and streamwise forces. Importantly, the controller is found to be efficient, the actuator requiring only 24 % of the aerodynamic power saving. The controller therefore provides a key demonstration of efficient feedback control used to reduce the drag of a high-Reynolds-number three-dimensional bluff body. Furthermore, the results suggest that suppression of large-scale structures is a fundamentally efficient approach for bluff body drag reduction.

We consider the steady flow of a viscous compressible gas through an axisymmetric or two-dimensional porous medium whose properties in the direction of the flow are sufficiently slowly varying. The study is partly motivated by a number of different applications in the Earth sciences, including the release of magmatic volatiles from a magma chamber beneath an active volcano and the discharge of geothermal fluids. The results are also relevant to evaluating the consequences of an accidental release of carbon dioxide from a storage reservoir within the Earth, as might happen at a carbon capture and storage (CCS) site. We consider both slow, thermally equilibrated, flows and fast, adiabatic flows. Because the flow is compressible, it is the mass (and not the volume) flux which is conserved along the flow. We determine this constant mass flux and the velocity and pressure fields, both of which vary with position along the flow, as a function of all the physical parameters. We find that the resultant pressure gradient in the medium is largest at the far, low-pressure end of the conduit because the velocity is largest at that end due to the smallest density being associated with the smallest pressures. This means that the pressure in the permeable conduit is always larger than the linear pressure distribution which joins the given pressures at depth and at the surface, as would be the situation if the flow were incompressible. The detailed pressure distribution is shown to depend on the variation with depth of the quantity $\unicode[STIX]{x1D707}T/(ka^{2})$, where $\unicode[STIX]{x1D707}$ is the dynamic viscosity of the vapour, $T$ is the external temperature, $k$ the permeability and $\unicode[STIX]{x03C0}a^{2}$ the cross-sectional area of the conduit. The resultant mass flux is determined to be proportional to the mean along the flow of $\unicode[STIX]{x1D707}T/(ka^{2})$. We present two numerical illustrations of the results.

One type of thin vortex filament structure that has attracted interest in recent years is that which obeys self-similar scaling. Among various applications, these filaments have been used to model the motion of quantized vortex filaments in superfluid helium after reconnection events. While similarity solutions have been described analytically and numerically using the local induction approximation (LIA), they have not been studied (or even shown to exist) under the non-local Biot–Savart model. In this present paper, we show not only that self-similar vortex filament solutions exist for the non-local Biot–Savart model, but that such solutions are qualitatively similar to their LIA counterparts. This suggests that the various LIA solutions found previously should be valid physically (at least in the small amplitude regime), since they agree well with the more accurate Biot–Savart model.

We investigate the dynamics of drop impact on a thin liquid film at short times in order to identify the mechanisms of splash formation. Using numerical simulations and scaling analysis, we show that it depends both on the inertial dynamics of the liquid and the cushioning of the gas. Two asymptotic regimes are identified, characterized by a new dimensionless number $J$: when the gas cushioning is weak, the jet is formed after a sequence of bubbles are entrapped and the jet speed is mostly selected by the Reynolds number of the impact. On the other hand, when the air cushioning is important, the lubrication of the gas beneath the drop and the liquid film controls the dynamics, leading to a single bubble entrapment and a weaker jet velocity.

We derive an asymptotic model that describes the nonlinear coupled evolution of (i) near-inertial waves (NIWs), (ii) balanced quasi-geostrophic flow and (iii) near-inertial second harmonic waves with frequency near $2f_{0}$, where $f_{0}$ is the local inertial frequency. This ‘three-component’ model extends the two-component model derived by Xie & Vanneste (J. Fluid Mech., vol. 774, 2015, pp. 143–169) to include interactions between near-inertial and $2f_{0}$ waves. Both models possess two conservation laws which together imply that oceanic NIWs forced by winds, tides or flow over bathymetry can extract energy from quasi-geostrophic flows. A second and separate implication of the three-component model is that quasi-geostrophic flow catalyses a loss of NIW energy to freely propagating waves with near-$2f_{0}$ frequency that propagate rapidly to depth and transfer energy back to the NIW field at very small vertical scales. The upshot of near-$2f_{0}$ generation is a two-step mechanism whereby quasi-geostrophic flow catalyses a nonlinear transfer of near-inertial energy to the small scales of wave breaking and diapycnal mixing. A comparison of numerical solutions with both Boussinesq and three-component models for a two-dimensional initial value problem reveals strengths and weaknesses of the model while demonstrating the extraction of quasi-geostrophic energy and production of small vertical scales.

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 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.

We describe new experiments in which a bubble plume, produced from a point source of bubbles, rises through an ambient fluid composed of two layers of fluid of different density. In the lower layer, the speed of the plume exceeds the bubble rise speed and the motion is well described using classical theory of turbulent buoyant plumes. As the mixture enters the upper layer, it is either buoyant and rises to the top of the layer, or is negatively buoyant and forms a fountain. In our experiments, in which a fountain forms in the upper layer, the bubble rise speed exceeds the characteristic speed of this fountain, and a separated flow develops. The bubbles rise to the top of the system, while the lower layer fluid in the fountain rises a finite distance into the upper layer, entrains some of the upper layer fluid, and then collapses. This mixture of fluids then feeds a growing layer of density which is intermediate between the upper and lower layer. The height of rise of the fountain scales with the square of the Froude number of the fountain and the rate of entrainment of upper layer fluid into the fountain is directly proportional to the height of the fountain. This is analogous to the scaling for single-phase fountains with Froude numbers in the same range, $1<Fr<7$, but the constants of proportionality are smaller. We illustrate the relevance of the work for the design of mixing and aeration systems in freshwater reservoirs.

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.

The effect of differential diffusion of two passive scalars having Schmidt numbers of unity and 0.25, respectively, is investigated using direct numerical simulation of a temporally evolving jet. The objective of the research is twofold: (i) to compare the turbulent/non-turbulent (T/NT) interface position using the scalar criterion between the unity- and low-Schmidt-number scalar; and (ii) to determine the impact of the T/NT interface on differential diffusion. For the latter, the T/NT interface is detected using the vorticity criterion. To quantify the effect of differential diffusion, a normalised differential diffusion parameter is analysed, clearly showing the dominance of differential diffusion at the T/NT interface. A transport equation for the scalar differences is then evaluated, which shows that differential diffusion originates at the interface. Further, the separation between the passive scalars, arising due to differential diffusion, is studied using conventional and conditional statistics with respect to the interface distance. Since differential diffusion is known to be present at large and small scales, the connection between them is analysed using the scalar dissipation rate. Moreover, the physical mechanism responsible for the departure of the two scalars is analysed using the scalar gradient alignment, the ratio of the diffusive fluxes and by a transport equation for the scalar gradients.