Transitional supersonic axisymmetric wakes are investigated by conducting various numerical experiments. The main objective is to identify hydrodynamic instability mechanisms in the flow at $M\,{=}\,2.46$ for several Reynolds numbers, and to relate these to coherent structures that are found from various visualization techniques. The premise for this approach is the assumption that flow instabilities lead to the formation of coherent structures. Three high-order accurate compressible codes were developed in cylindrical coordinates for this work: a spatial Navier–Stokes (N-S) code to conduct direct numerical simulations (DNS), a linearized N-S code for linear stability investigations using axisymmetric basic states, and a temporal N-S code for performing local stability analyses. The ability of numerical simulations to exclude physical effects deliberately is exploited. This includes intentionally eliminating certain azimuthal/helical modes by employing DNS for various circumferential domain sizes. With this approach, the impact of structures associated with certain modes on the global wake-behaviour can be scrutinized. Complementary spatial and temporal calculations are carried out to investigate whether instabilities are of local or global nature. Circumstantial evidence is presented that absolutely unstable global modes within the recirculation region co-exist with convectively unstable shear-layer modes. The flow is found to be absolutely unstable with respect to modes $k\,{>}\,0$ for $Re_D\,{>}\,5000$ and with respect to the axisymmetric mode $k\,{=}\,0$ for $Re_D\,{>}\,100\,000$. It is concluded that azimuthal modes $k\,{=}\,2$ and $k\,{=}\,4$ are the dominant modes in the trailing wake, producing a ‘four-lobe’ wake pattern. Two possible mechanisms responsible for the generation of longitudinal structures within the recirculation region are suggested.

This paper provides a detailed study of turbulent statistics and scale-by-scale budgets in turbulent Rayleigh–Bénard convection. It aims at testing the applicability of Kolmogorov and Bolgiano theories in the case of turbulent convection and at improving the understanding of the underlying inertial-range scalings, for which a general agreement is still lacking. Particular emphasis is laid on anisotropic and inhomogeneous effects, which are often observed in turbulent convection between two differentially heated plates. For this purpose, the SO(3) decomposition of structure functions and a method of description of inhomogeneities are used to derive inhomogeneous and anisotropic generalizations of Kolmogorov and Yaglom equations applying to Rayleigh–Bénard convection, which can be extended easily to other types of anisotropic and/or inhomogeneous flows. The various contributions to these equations are computed in and off the central plane of a convection cell using data produced by a direct numerical simulation of turbulent Boussinesq convection at $\hbox{\it Ra}\,{=}\,10^6$ and $\hbox{\it Pr}\,{=}\,1$ with aspect ratio $A\,{=}\,5$. The analysis of the isotropic part of the Kolmogorov equation demonstrates that the shape of the third-order velocity structure function is significantly influenced by buoyancy forcing and large-scale inhomogeneities, while the isotropic part of the mixed third-order structure function $\langle(\Delta\theta)^2\Delta\vec{u}\rangle$ appearing in the Yaglom equation exhibits a clear scaling exponent 1 in a small range of scales. The magnitudes of the various low $\ell$ degree anisotropic components of the equations are also estimated and are shown to be comparable to their isotropic counterparts at moderate to large scales. The analysis of anisotropies notably reveals that computing reduced structure functions (structure functions computed at fixed depth for correlation vectors $\boldsymbol{r}$ lying in specific planes only) in order to reveal scaling exponents predicted by isotropic theories is misleading in the case of fully three-dimensional turbulence in the bulk of a convection cell, since such quantities involve linear combinations of different $\ell$ components which are not negligible in the flow. This observation also indicates that using single-point measurements together with the Taylor hypothesis in the particular direction of a mean flow to test the predictions of asymptotic dimensional isotropic theories of turbulence or to calculate intermittency corrections to these theories may lead to significant bias for mildly anisotropic three-dimensional flows. A qualitative analysis is finally used to show that the influence of buoyancy forcing at scales smaller than the Bolgiano scale is likely to remain important up to $\hbox{\it Ra}\,{=}\,10^9$, thus preventing Kolmogorov scalings from showing up in convective flows at lower Rayleigh numbers.

The mechanism of reconfiguration of broad leaves subjected to wind loading is investigated. Circular plastic sheets cut along a radius are immersed in a water flow. They roll up into cones when held at their centres. The opening angle of the cone and the drag force exerted on the sheet are measured as a function of the flow velocity and of the sheet bending rigidity. The cone becomes sharper when the velocity increases or when the sheet stiffness decreases; the reconfiguration leads to a decrease in the drag coefficient. Scaling laws are derived from the mechanical equilibrium of the sheets – the balance between form drag and elastic forces – and the experimental data collapse onto master curves. Two models for the pressure field yield theoretical curves in semi-quantitative agreement with the experiments.

It is shown for the first time that the gradient diffusion hypothesis often adopted for thermal dispersion heat flux in heat transfer within porous media can be derived from a transport equation for the thermal dispersion heat flux based on the Navier–Stokes and energy equations. The transport equation valid for both thermal equilibrium and non-equilibrium cases is mathematically modelled so that all unknown spatial correlation terms, associated with redistribution and dissipation of the dispersion heat flux, are expressed in terms of determinable variables. The unknown coefficients are determined analytically by considering of macroscopically unidirectional flow through a tube as treated by Taylor. Taylor's expression for the dispersion has been generated from the transport equation. Both laminar and turbulent flow cases are investigated to obtain two distinct limiting expressions for low- and high-Péclet-number regimes. The results obtained for the Taylor diffusion problem are translated to the case of heat and fluid flow in a packed bed, to obtain the corresponding expressions for the axial dispersion coefficient in a packed bed. The resulting expression for the high-Péclet-number case agrees well with the empirical formula, validating of the present transport analysis.

When the individual particles in an otherwise quiescent suspension of freely suspended spherical particles are acted upon by external couples, the resulting suspension-scale fluid motion is characterized by a non-symmetric state of stress. Viewed at the interstitial scale (i.e. microscopic scale), this coupling between translational and rotational particle motions is a manifestation of particle–particle hydrodynamic interactions and vanishes with the volume fraction $\phi$ of suspended spheres. The antisymmetric portion of the stress is quantified by the suspension-scale vortex viscosity $\mu_v$, different from the suspension's shear viscosity $\mu$. Numerical boundary element method (BEM) simulations of such force-free suspensions of spheres uniformly dispersed in incompressible Newtonian liquids of viscosity $\mu_0$ are performed for circumstances in which external couples (of any specified suspension-scale position-dependence) are applied individually to each of the suspended particles in order to cause them to rotate in otherwise quiescent fluids. In the absence of external forces acting on either the spheres or boundaries, such rotations indirectly, through interparticle coupling, cause translational motions of the individual spheres which, owing to the no-slip boundary condition, drag neighbouring fluid along with them. In turn, this combined particle–interstitial fluid movement is manifested as a suspension-scale velocity field, generated exclusively by the action of external couples. Use of this scheme to create suspension-scale particle-phase spin fields $\mbox{\boldmath \Omega}$ and concomitant velocity fields $\bm v$ enables both the vortex and shear viscosities of suspensions to be determined as functions of $\phi$ in disordered systems. This scheme is shown, inter alia, to confirm the constitutive equation, ${\textsfbi T}^{\hspace*{1pt}a} \,{=}\, 2\mu_v\bm \mbox{\boldmath \varepsilon}\,{\bm\cdot}\,[(1/2) {\bm \nabla}\,{\times}\, \bm v - \mbox{\boldmath \Omega}],$ proposed in the continuum mechanics literature for the linear relation between the antisymmetric stress ${\textsfbi T}^{\hspace*{1pt}a}$ and the disparity existing between the particle-phase spin rate $\mbox{\boldmath \Omega}$ and half the suspension's vorticity, ${\bm \nabla} \,{\times}\, \bm v$ (with the third-rank pseudotensor $\bm \mbox{\boldmath \varepsilon}$ the permutation triadic). Our dynamically based BEM simulations confirm the previous computations of the Prosperetti et al. group for the dependence of the vortex viscosity upon the solids volume fraction in concentrated disordered suspensions, obtained by a rather different simulation scheme. Moreover, our dynamically based rheological calculations are confirmed by our semi-independent, energetically based, calculations that equate the rates of working (equivalently, the energy dissipation rates) at the respective interstitial and suspension scales. As an incidental by-product, the same BEM simulation results also verify the suspension-scale Newtonian constitutive equation, ${\textsfbi T}^{\hspace*{1pt}s}\,{=}\, \mu[{\bm \nabla}\bm {v} + ({\bm \nabla}\bm {v})^{\dag}]$, as well as the functional dependence of the shear viscosity $\mu$ upon $\phi$ found in the literature.

Long-surface-wave instability in dense granular flows down inclined planes is analysed using recently proposed three-dimensional constitutive equations. A full linear stability analysis of the local governing equations is performed and compared to previous experimental results obtained with glass beads. We show that the proposed rheology is able to capture all the features of the instability quantitatively. In particular, it predicts well the behaviour and scaling for the cutoff frequency of the instability observed in the experiments. This result gives strong support for the three-dimensional rheology proposed and suggests new terms in the Saint-Venant equations used to describe free-surface granular flows.

We consider the deformation and breakup of a non-Newtonian slender drop in a Newtonian liquid, subject to an axisymmetric extensional flow, and the influence of inertia in the continuous phase. The non-Newtonian fluid inside the drop is described by the simple power-law model and the unsteady deformation of the drop is represented by a single partial differential equation. The steady-state problem is governed by four parameters: the capillary number; the viscosity ratio; the external Reynolds number; and the exponent characterizing the power-law model for the non-Newtonian drop. For Newtonian drops, as inertia increases, drop breakup is facilitated. However, for shear thinning drops, the influence of increasing inertia results first in preventing and then in facilitating drop breakup. Multiple stationary solutions were also found and a stability analysis has been performed in order to distinguish between stable and unstable stationary states.

The three-dimensional governing macroscopic equations of the flow field which is developed when an elasto-plastic highly deformable open-cell porous medium whose pores are uniformly filled with liquid and gas is struck head-on by a planar shock wave, are developed using a multiphase approach. The one-dimensional version of these equations is solved numerically using an arbitrary Lagrangian Eulerian (ALE) based numerical code. The numerical predictions are compared qualitatively to experimental results from various sources and good agreements are obtained. This study complements our earlier studies in which we solved, using an ALE-based numerical code, the one-dimensional governing equations of the flow field which is developed when an elasto-plastic flexible open-cell porous medium, capable of undergoing extremely large deformations, whose pores are saturated with gas only, is struck head-on by a planar shock wave.

The linear stability of a rotating flow in an elliptically deformed cylindrical shell with an imposed radial temperature contrast is studied using local and global approaches. We demonstrate that (i) a stabilizing temperature profile can either increase or decrease the growth rate of the elliptical instability depending on the selected mode and on the strength of the radial buoyancy force; (ii) when the temperature profile is destabilizing, the elliptical instability coexists with two-dimensional convective instabilities at relatively small values of the Rayleigh number, the fastest growing mode depending on the relative values of the Rayleigh number and of the eccentricity; (iii) the elliptical instability totally disappears for larger values of the Rayleigh number. We argue that thermal effects have to be taken into account when looking for the occurrence and influence of inertial instabilities in geophysical and astrophysical systems, especially in planetary cores.

In this paper we present ray-tracing results on the interaction of inertia–gravity waves with the velocity field of a vortex in a rotating stratified fluid. We consider rays that interact with a Rankine-type vortex with a Gaussian vertical distribution of vertical vorticity. The rays are traced, solving the WKB equations in cylindrical coordinates for vortices with different aspect ratios. The interactions are governed by the value of $\hbox{\it Fr} R/\lambda$ where $\hbox{\it Fr}$ is the vortex Froude number, $R$ its radius, and $\lambda$ the incident wavelength. The Froude number is defined as $ {\hbox{\it Fr}}\,{=}\,U_{max}/(NR)$ with $U_{max}$ the maximum azimuthal velocity and $N$ the buoyancy frequency. When $\hbox{\it Fr} R/\lambda\,{>}\,1$, part of the incident wave field strongly decreases in wavelength while its energy is trapped. The vortex aspect ratio, $H/R$, determines which part of this incident wave field is trapped, and where its energy accumulates in the vortex. Increasing values of $\hbox{\it Fr} R/\lambda$ are shown to be associated with a narrowing of the trapping region and an increase of the energy amplification of trapped rays. In the inviscid approximation, the infinite energy amplification predicted for unidirectional flows is retrieved in the limit $\hbox{\it Fr} R/\lambda \,{\rightarrow}\, \infty$. When viscous damping is taken into account, the maximal amplification of the wave energy becomes a function of $\hbox{\it Fr} R/\lambda$ and a Reynolds number, $Re_{wave}\,{=}\,\sqrt{U_L^2+U_H^2}/\nu k^2$, with $U_L$ and $U_H$ typical values of the shear in, respectively, the radial and vertical directions; the kinematic viscosity is $\nu$, and the wavenumber $k$, for the incident waves. In a sequel paper, we compare WKB simulations with experimental results.

The behaviour of a dispersion of infinitely polarizable slender rods in an electric field is described using theory and numerical simulations. The polarization of the rods results in the formation of dipolar charge clouds around the particle surfaces, which in turn drive nonlinear electrophoretic flows under the action of the applied field. This induced-charge electrophoresis causes no net migration for uncharged particles with fore–aft symmetry, but can lead to rotations and to relative motions as a result of hydrodynamic interactions. A slender-body formulation is derived that accounts for induced-charge electrophoresis based on a thin double layer approximation, and shows that the effects of the electric field on a single rod can be modelled by a linear slip velocity along the rod axis, which causes particle alignment and drives a stresslet flow in the surrounding fluid. Based on this slender-body model, the hydrodynamic interactions between a pair of aligned rods are studied, and we identify domains of attraction and repulsion, which suggest that particle pairing may occur. An efficient method is implemented for the simulation of dispersions of many Brownian rods undergoing induced-charge electrophoresis, that accounts for far-field hydrodynamic interactions up to the stresslet term, as well as near-field lubrication and contact forces. Simulations with negligible Brownian motion show that particle pairing indeed occurs in the suspension, as demonstrated by sharp peaks in the pair distribution function. The superposition of all the electrophoretic flows driven on the rod surfaces is observed to result in a diffusive motion at long times, and hydrodynamic dispersion coefficients are calculated. Results are also presented for colloidal suspensions, in which Brownian fluctuations are found to hinder particle pairing and alignment. Orientation distributions are obtained for various electric field strengths, and are compared to an analytical solution of the Fokker–Planck equation for the orientation probabilities in the limit of infinite dilution.

We present a spectral theory of the low-frequency response of a harbour to short and random incident waves. Assuming the incident sea to be stationary and Gaussian, nonlinear extensions are made for the response spectrum. Advantage is taken of the typical wind-wave spectrum which is dominated by high-frequency components. After showing that nonlinearity is needed only up to the second order in wave steepness, we extend the mild-slope approximation for constructing the transfer functions. Numerical examples are presented for a square harbour and constant depth. Discounting friction losses, the effects of different entrances are compared.

When a viscoelastic fluid blob is stretched out into a thin horizontal filament, it sags and falls gradually under its own weight, forming a catenary-like structure that evolves dynamically. If the ends are brought together rapidly after stretching, the falling filament tends to straighten by rising. These two effects are strongly influenced by the elasticity of the fluid and yield qualitatively different behaviours from the case of a purely viscous filament analysed previously (Teichman & Mahadevan, J. Fluid Mech. vol. 478, 2003, p. 71). Starting from the bulk equations for the motion of a viscoelastic fluid, we derive a simplified equation for the dynamics of a viscoelastic filament and analyse this equation in some simple settings to explain our observations.

This paper concerns the stability of the von Kármán swirling flow between coaxial disks. A linear stability analysis shows that for moderate Reynolds numbers ($Re\le50$) and for any rotation ratio $s\in[-1,1[$ there is a radial location $r_{pc}$ from which the self-similar von Kármán solutions become unstable to axisymmetric disturbances. When the disks are moderately counter-rotating ($s\in[-0.56,0[$), two different disturbances (types I and II) appear at the same critical radius. A spatio-temporal analysis shows that, at a very short distance from this critical radius, the first disturbance (type I) becomes absolutely unstable whereas the second (type II) remains convectively unstable. Outside this range of aspect ratios, all the disturbances examined are found to be absolutely unstable. The flow between two coaxial rotating disks enclosed in a stationary sidewall is then numerically investigated. For sufficently large aspect ratios, the cavity flow is found to be globally unstable for axisymmetric disturbances similar to that calculated with the self-similar solutions. The flow in cavities with aspect ratios smaller than $R\,{\approx}\,10.3$ (and $Re\,{\le}\,50$) is not destabilized by these axisymmetric disturbances. An experimental investigation conducted for a cavity with aspect ratio $R\,{=}\,15$ confirms the numerical results. Axisymmetric disturbances similar to those calculated for the same cavity are detected and three-dimensional modes can also be observed near the sidewall.

The stability of an equilibrium system of two drops suspended from circular holes in a horizontal plate is examined. The drop surfaces are the disconnected axisymmetric surfaces pinned to the edges of the holes. The holes lie in the same horizontal plane and the two drops are connected by a liquid layer that lies above the plate. The total liquid volume is constant. For identical pendant drops pinned to holes of equal radii, axisymmetric perturbations are always the most dangerous. The stability region for two identical drops differs considerably from that for a solitary pendant drop. A bifurcation analysis shows that the loss of stability leads to a continuous transition from a critical system of identical drops to a stable system of axisymmetric non-identical drops. With increasing total protruded liquid volume this system of non-identical drops reaches its own collective stability limit (to axisymmetric perturbations) which gives rise to dripping or streaming from the holes. Critical volumes and heights for non-identical drops have been calculated as functions of the dimensionless hole radius (associated with the Bond number). For unequal hole radii, there are three intervals of the larger dimensionless hole radius, $R_{1}^{0}$, with qualitatively different bifurcation patterns which in turn can depend on the smaller dimensionless hole radius, $R_2 ^0$. Loss of stability may occur when the drop suspended from the larger hole reaches its stability limit (to non-axisymmetric perturbations) as a solitary drop or when the system reaches the collective stability limit (to axisymmetric perturbations). Typical situations are illustrated for selected values of $R_1 ^0$, and then the basic characteristics of the stability for a dense set of $R_1 ^0$ are presented.

Continuous mode transition is an instance of the bypass route to boundary-layer turbulence. The stages that precede breakdown are explained in terms of continuous Orr–Sommerfeld and Squire spectra. In that context, the role of pressure gradient is less evident than it is in natural transition. Its role is investigated using linear theory and numerical simulations. Both approaches demonstrate that adverse pressure gradients enhance the coupling of low-frequency vortical disturbances to the boundary-layer shear. The result is stronger boundary-layer perturbation jets – or Klebanoff distortions. The correlation between the intensity of the perturbation jets and transition location is tested by direct numerical simulations of pairwise continuous mode interactions; such interactions can reproduce the entire transition process. The results confirm that stronger perturbation jets are more unstable, and hence provoke early transition in adverse pressure gradient. This is also consistent with the experimental observation that transition becomes independent of pressure gradient at high turbulent intensities. Under such conditions, boundary-layer streaks are highly unstable and transition is achieved swiftly, independent of the mean gradient in pressure.

In this paper, we present a new passive control device for form-drag reduction in flow over a two-dimensional bluff body with a blunt trailing edge. The device consists of small tabs attached to the upper and lower trailing edges of a bluff body to effectively perturb a two-dimensional wake. Both a wind-tunnel experiment and large-eddy simulation are carried out to examine its drag-reduction performance. Extensive parametric studies are performed experimentally by varying the height and width of the tab and the spanwise spacing between the adjacent tabs at three Reynolds numbers of $\hbox{\it Re}\,{=}\,u_\infty h/\nu\,{=}\,20\,000$, 40 000 and 80 000, where $u_\infty$ is the free-stream velocity and $h$ is the body height. For a wide parameter range, the base pressure increases (i.e. drag reduces) at all three Reynolds numbers. Furthermore, a significant increase in the base pressure by more than 30% is obtained for the optimum tab configuration. Numerical simulations are performed at much lower Reynolds numbers of $\hbox{\it Re}\,{=}\,320$ and 4200 to investigate the mechanism responsible for the base-pressure increase by the tab. Results from the velocity measurement and numerical simulations show that the tab introduces the spanwise mismatch in the vortex-shedding process, resulting in a substantial reduction of the vortical strength in the wake and significant increases in the vortex formation length and wake width.

A numerical study of doubly periodic deep-water short-crested wave instabilities, arising from various quartet resonant interactions, is conducted using a high-order Boussinesq-type model. The model is first verified through a series of simulations involving classical class I plane wave instabilities. These correctly lead to well-known (nearly symmetric) recurrence cycles below a previously established breaking threshold steepness, and to an asymmetric evolution (characterized by a permanent transfer of energy to the lower side-band) above this threshold, with dissipation from a smoothing filter promoting this behaviour in these cases. A series of class Ia short-crested wave instabilities, near the plane wave limit, are then considered, covering a wide range of incident wave steepness. A close match with theoretical growth rates is demonstrated near the inception. It is shown that the unstable evolution of these initially three-dimensional waves leads to an asymmetric evolution, even for weakly nonlinear cases presumably well below breaking. This is characterized by an energy transfer to the lower side-band, which is also accompanied by a similar transfer to more distant upper side-bands. At larger steepness, the evolution leads to a permanent downshift of both the mean and peak frequencies, driven in part by dissipation, effectively breaking the quasi-recurrence cycle. A single case involving a class Ib short-crested wave instability at relatively large steepness is also considered, which demonstrates a reasonably similar evolution. These simulations consider the simplest physical situations involving three-dimensional instabilities of genuinely three-dimensional progressive waves, revealing qualitative differences from classical two-dimensional descriptions. This study is therefore of fundamental importance in understanding the development of three-dimensional wave spectra.

The classical bulk model for isolated jets and plumes due to Morton, Taylor & Turner (Proc. R. Soc. Lond. A, vol. 234, 1956, p. 1) is generalized to allow for time-dependence in the various fluxes driving the flow. This new system models the spatio-temporal evolution of jets in a homogeneous ambient fluid and Boussinesq and non-Boussinesq plumes in stratified and unstratified ambient fluids.

Separable time-dependent similarity solutions for plumes and jets are found in an unstratified ambient fluid, and proved to be linearly stable to perturbations propagating at the velocity of the ascending plume fluid. These similarity solutions are characterized by having time-independent plume or jet radii, with appreciably smaller spreading angles ($\tan^{-1}(2\alpha/3)$) than either constant-source-buoyancy-flux pure plumes (with spreading angle $\tan^{-1}(6\alpha/5)$) or constant-source-momentum-flux pure jets (with spreading angle $\tan^{-1}(2\alpha)$), where $\alpha$ is the conventional entrainment coefficient. These new similarity solutions are closely related to the similarity solutions identified by Batchelor (Q. J. R. Met. Soc., vol. 80, 1954, p. 339) in a statically unstable ambient, in particular those associated with a linear increase in ambient density with height.

If the source buoyancy flux (for a rising plume) or source momentum flux (for a rising jet) is decreased generically from an initial to a final value, numerical solutions of the governing equations exhibit three qualitatively different regions of behaviour. The upper region, furthest from the source, remains largely unaffected by the change in buoyancy flux or momentum flux at the source. The lower region, closest to the source, is an effectively steady plume or jet based on the final (lower) buoyancy flux or momentum flux. The transitional region, in which the plume or jet adjusts between the states in the lower and upper regions, appears to converge very closely to the newly identified stable similarity solutions. Significantly, the predicted narrowing of the plume or jet is observed. The size of the narrowing region can be determined from the source conditions of the plume or jet. Minimum narrowing widths are considered with a view to predicting pinch-off into rising thermals or puffs.

Solutions to the equations of Morton et al. (Proc. R. Soc. Lond. A, vol. 234, 1956, p. 1) describing turbulent plumes and jets rising in uniformly stratified environments are identified for the first time.

The evolution of plumes and jets with sources whose driving flux decreases with time is considered in a stratified environment. Numerical calculations indicate that as the source buoyancy flux, for a Boussinesq plume (or source momentum flux, for a Boussinesq jet), is decreased, a transitional narrowing region with characteristic spreading angle $\tan^{-1}(2\alpha/3)$ is formed, where $\alpha$ is the well-known entrainment coefficient. The plume or jet dynamics are modelled well by a separable solution to the governing equations which predicts stalling in the plume at a critical stall time $t_s\,{=}\,\upi/N$ and stalling in the jet at a critical stall time $t_s\,{=}\,\upi/(2N)$, where $N$ is the buoyancy frequency of the ambient background stratification. This stall time is independent of the driving source conditions, a prediction which is verified by numerical solution of the underlying evolution equations.