The structure of viscous and thermal boundary layers at the heated and cooled plates in turbulent thermally driven flows are of fundamental importance for heat transfer and its dependence on the thermal forcing (the Rayleigh number in non-dimensional form). The paper by Shi, Emran & Schumacher (J. Fluid Mech., this issue, vol. 706, 2012, pp. 5–33) stresses the deviations of the boundary layer vertical profiles from the Prandtl–Blasius–Pohlhausen theory. Recent papers showing very similar results, in contrast, focus more on the similarities.

The structure of the boundary layers in turbulent Rayleigh–Bénard convection is studied by means of three-dimensional direct numerical simulations. We consider convection in a cylindrical cell at aspect ratio one for Rayleigh numbers of and at fixed Prandtl number . Similar to the experimental results in the same setup and for the same Prandtl number, the structure of the laminar boundary layers of the velocity and temperature fields is found to deviate from the prediction of Prandtl–Blasius–Pohlhausen theory. Deviations decrease when a dynamical rescaling of the data with an instantaneously defined boundary layer thickness is performed and the analysis plane is aligned with the instantaneous direction of the large-scale circulation in the closed cell. Our numerical results demonstrate that important assumptions of existing classical laminar boundary layer theories for forced and natural convection are violated, such as the strict two-dimensionality of the dynamics or the steadiness of the fluid motion. The boundary layer dynamics consists of two essential local dynamical building blocks, a plume detachment and a post-plume phase. The former is associated with larger variations of the instantaneous thickness of velocity and temperature boundary layer and a fully three-dimensional local flow. The post-plume dynamics is connected with the large-scale circulation in the cell that penetrates the boundary region from above. The mean turbulence profiles taken in localized sections of the boundary layer for each dynamical phase are also compared with solutions of perturbation expansions of the boundary layer equations of forced or natural convection towards mixed convection. Our analysis of both boundary layers shows that the near-wall dynamics combines elements of forced Blasius-type and natural convection.

We validate a new law for the aspect ratio of vortices in a rotating, stratified flow, where and are the vertical half-height and horizontal length scale of the vortices. The aspect ratio depends not only on the Coriolis parameter and buoyancy (or Brunt–Väisälä) frequency of the background flow, but also on the buoyancy frequency within the vortex and on the Rossby number of the vortex, such that . This law for is obeyed precisely by the exact equilibrium solution of the inviscid Boussinesq equations that we show to be a useful model of our laboratory vortices. The law is valid for both cyclones and anticyclones. Our anticyclones are generated by injecting fluid into a rotating tank filled with linearly stratified salt water. In one set of experiments, the vortices viscously decay while obeying our law for , which decreases over time. In a second set of experiments, the vortices are sustained by a slow continuous injection. They evolve more slowly and have larger while still obeying our law for . The law for is not only validated by our experiments, but is also shown to be consistent with observations of the aspect ratios of Atlantic meddies and Jupiter’s Great Red Spot and Oval BA. The relationship for is derived and examined numerically in a companion paper by Hassanzadeh, Marcus & Le Gal (J. Fluid Mech., vol. 706, 2012, pp. 46–57).

We derive a relationship for the vortex aspect ratio (vertical half-thickness over horizontal length scale) for steady and slowly evolving vortices in rotating stratified fluids, as a function of the Brunt–Väisälä frequencies within the vortex and in the background fluid outside the vortex , the Coriolis parameter and the Rossby number of the vortex: . This relation is valid for cyclones and anticyclones in either the cyclostrophic or geostrophic regimes; it works with vortices in Boussinesq fluids or ideal gases, and the background density gradient need not be uniform. Our relation for has many consequences for equilibrium vortices in rotating stratified flows. For example, cyclones must have ; weak anticyclones (with ) must have ; and strong anticyclones must have . We verify our relation for with numerical simulations of the three-dimensional Boussinesq equations for a wide variety of vortices, including: vortices that are initially in (dissipationless) equilibrium and then evolve due to an imposed weak viscous dissipation or density radiation; anticyclones created by the geostrophic adjustment of a patch of locally mixed density; cyclones created by fluid suction from a small localized region; vortices created from the remnants of the violent breakups of columnar vortices; and weakly non-axisymmetric vortices. The values of the aspect ratios of our numerically computed vortices validate our relationship for , and generally they differ significantly from the values obtained from the much-cited conjecture that in quasi-geostrophic vortices.

The interface defining a biological cell is a thin membrane, which acts as a leaky capacitor. We investigate the influence of capacitance and conductivity on the stability of a planar membrane subjected to a DC electric field. We develop a zero-thickness model of the membrane, in which the bilayer finite thickness is effectively accounted for by membrane electro-mechanical properties such as bending modulus, capacitance and conductance. The linear stability analysis shows that membrane conductance and asymmetry in the embedding electrolyte solutions destabilize the interface. However, the capacitive charging acts to stabilize the system under conditions where an ordinary fluid–fluid interface is unstable.

We study formation and properties of new coherent structures: ageostrophic modons in the two-layer rotating shallow water model. The ageostrophic modons are obtained by ‘ageostrophic adjustment’ of the exact modon solutions of the two-layer quasi-geostrophic equations with the free surface, which are used to initialize the full two-layer shallow water model. Numerical simulations are performed using a well-balanced high-resolution finite volume numerical scheme. For large enough Rossby numbers, the initial configurations undergo ageostrophic adjustment towards asymmetric ageostrophic quasi-stationary coherent dipoles. This process is accompanied by substantial emission of inertia–gravity waves. The resulting dipole is shown to be robust and survives frontal collisions. It contains captured inertia–gravity waves and, for higher Rossby numbers and weak stratification, carries a (baroclinic) hydraulic jump at its axis. For stronger stratifications and high enough Rossby numbers, ‘rider’ coherent structures appear as a result of adjustment, with a monopole in one layer and a dipole in another. Other ageostrophic coherent structures, such as two-layer tripoles and two-layer modons with nonlinear scatter plot, result from the collisions of ageostrophic modons. They are shown to be long-living and robust, and to capture waves.

In many applications it is advantageous to construct effective slip boundary conditions, which could fully characterize flow over patterned surfaces. Here we focus on laminar shear flows over smooth anisotropic surfaces with arbitrary scalar slip , varying in only one direction. We derive general expressions for eigenvalues of the effective slip-length tensor, and show that the transverse component is equal to half of the longitudinal one, with a two times larger local slip, . A remarkable corollary of this relation is that the flow along any direction of the one-dimensional surface can be easily determined, once the longitudinal component of the effective slip tensor is found from the known spatially non-uniform scalar slip.

Strongly turbulent Taylor–Couette flow with independently rotating inner and outer cylinders with a radius ratio of is experimentally studied. From global torque measurements, we analyse the dimensionless angular velocity flux as a function of the Taylor number and the angular velocity ratio in the large-Taylor-number regime and well off the inviscid stability borders (Rayleigh lines) for co-rotation and for counter-rotation. We analyse the data with the common power-law ansatz for the dimensionless angular velocity transport flux , with an amplitude and an exponent . The data are consistent with one effective exponent for all , but we discuss a possible dependence in the co- and weakly counter-rotating regimes. The amplitude of the angular velocity flux is measured to be maximal at slight counter-rotation, namely at an angular velocity ratio of , i.e. along the line . This value is theoretically interpreted as the result of a competition between the destabilizing inner cylinder rotation and the stabilizing but shear-enhancing outer cylinder counter-rotation. With the help of laser Doppler anemometry, we provide angular velocity profiles and in particular identify the radial position of the neutral line, defined by for fixed height . For these large values, the ratio , which is close to , is distinguished by a zero angular velocity gradient in the bulk. While for moderate counter-rotation , the neutral line still remains close to the outer cylinder and the probability distribution function of the bulk angular velocity is observed to be monomodal. For stronger counter-rotation the neutral line is pushed inwards towards the inner cylinder; in this regime the probability distribution function of the bulk angular velocity becomes bimodal, reflecting intermittent bursts of turbulent structures beyond the neutral line into the outer flow domain, which otherwise is stabilized by the counter-rotating outer cylinder. Finally, a hypothesis is offered allowing a unifying view and consistent interpretation for all these various results.

We consider freely decaying, anisotropic, statistically axisymmetric, Saffman turbulence in which , where is the energy spectrum and the wavenumber. We note that such turbulence possesses two statistical invariants which are related to the form of the spectral tensor at small . These are and , where the subscripts and indicate quantities parallel and perpendicular to the axis of symmetry. Since and , and being integral scales, self-similarity of the large scales (when it applies) demands and . This, in turn, requires that is constant, contrary to the popular belief that freely decaying turbulence should exhibit a ‘return to isotropy’. Numerical simulations performed in large periodic domains, with different types and levels of initial anisotropy, confirm that and are indeed invariants and that, in the fully developed state, . Somewhat surprisingly, the same simulations also show that is more or less constant in the fully developed state. Simple theoretical arguments are given which suggest that, when and are both constant, the integral scales should evolve as and , irrespective of the level of anisotropy and of the presence of helicity. These decay laws, first proposed by Saffman (Phys. Fluids, vol. 10, 1967, p. 1349), are verified by the numerical simulations.

In this paper we consider a fluid whose viscosity depends on both the mean normal stress and the shear rate flowing down an inclined plane. Such flows have relevance to geophysical flows. In order to make the problem amenable to analysis, we consider a generalization of the lubrication approximation for the flows of such fluids based on the development of the generalization of the Reynolds equation for such flows. This allows us to obtain analytical solutions to the problem of propagation of waves in a fluid flowing down an inclined plane. We find that the dependence of the viscosity on the pressure can increase the breaking time by an order of magnitude or more than that for the classical Newtonian fluid. In the viscous regime, we find both upslope and downslope travelling wave solutions, and these solutions are quantitatively and qualitatively different from the classical Newtonian solutions.

In this article, a biglobal stability approach is used in conjunction with direct numerical simulation (DNS) to identify the instability mode coupling that may be responsible for triggering large thrust oscillations in segmented solid rocket motors (SRMs). These motors are idealized as long porous cylinders in which a Taylor–Culick type of motion may be engendered. In addition to the analytically available steady-state solution, a computed mean flow is obtained that is capable of securing all of the boundary conditions in this problem, most notably, the no-slip requirement at the chamber headwall. Two sets of unsteady simulations are performed, static and dynamic, in which the injection velocity at the chamber sidewall is either held fixed or permitted to vary with time. In these runs, both DNS and biglobal stability solutions converge in predicting the same modal dependence on the size of the domain. We find that increasing the chamber length gives rise to less stable eigenmodes. We also realize that introducing an eigenmode whose frequency is sufficiently spaced from the acoustic modes leads to a conventional linear evolution of disturbances that can be accurately predicted by the biglobal stability framework. While undergoing spatial amplification in the streamwise direction, these disturbances will tend to decay as time elapses so long as their temporal growth rate remains negative. By seeding the computations with the real part of a specific eigenfunction, the DNS outcome reproduces not only the imaginary part of the disturbance, but also the circular frequency and temporal growth rate associated with its eigenmode. For radial fluctuations in which the vorticoacoustic wave contribution is negligible in relation to the hydrodynamic stability part, excellent agreement between DNS and biglobal stability predictions is ubiquitously achieved. For axial fluctuations, however, the DNS velocity will match the corresponding stability eigenfunction only when properly augmented by the vorticoacoustic solution for axially travelling waves associated with the Taylor–Culick profile. This analytical approximation of the vorticoacoustic mode is found to be quite accurate, especially when modified using a viscous dissipation function that captures the decaying envelope of the inviscid acoustic wave amplitude. In contrast, pursuant to both static and dynamic test cases, we find that when the frequency of the introduced eigenmode falls close to (or crosses over) an acoustic mode, a nonlinear mechanism is triggered that leads to the emergence of a secondary eigenmode. Unlike the original eigenmode, the latter materializes naturally in the computed flow without being artificially seeded. This natural occurrence may be ascribed to a nonlinear modal interplay in the form of internal, eigenmode-to-eigenmode coupling instead of an external, eigenmode pairing with acoustic modes. As a result of these interactions, large amplitude oscillations are induced.

The wake characteristics of bluff-body-stabilized flames are a strong function of the density ratio across the flame and the relative offset between the flame and shear layer. This paper describes systematic experimental measurements and stability calculations of the dependence of the flow field characteristics and flame sheet dynamics upon flame density ratio, , over the Reynolds number range of 1000–3300. We show that two fundamentally different flame/flow behaviours are observed at high and low values: a stable, noise-driven fixed point and limit-cycle oscillations, respectively. These results are interpreted as a transition from convective to global instability and are captured well by stability calculations that used the measured velocity and density profiles as inputs. However, in this high-Reynolds-number flow, the measurements show that no abrupt bifurcation in flow/flame behaviour occurs at a given value. Rather, the flow field is highly intermittent in a transitional range, with the relative fraction of the two different flow/flame behaviours monotonically varying with . This intermittent behaviour is a result of parametric excitation of the global mode growth rate in the vicinity of a supercritical Hopf bifurcation. It is shown that this parametric excitation is due to random fluctuations in relative locations of the flame and shear layer.

Addition of particles or droplets to turbulent liquid flows or addition of droplets to turbulent gas flows can lead to modulation of turbulence characteristics. Corresponding observations have been reported for very small particle or droplet volume loadings ${\Phi }_{v} $ and therefore may be important when simulating such flows. In this work, a modelling framework that accounts for preferential concentration and reproduces isotropic and anisotropic turbulence attenuation effects is presented. The framework is outlined for both Reynolds-averaged Navier–Stokes (RANS) and joint probability density function (p.d.f.) methods. Validations are performed involving a range of particle and flow-field parameters and are based on the direct numerical simulation (DNS) study of Boivin, Simonin & Squires (J. Fluid Mech., vol. 375, 1998, pp. 235–263) dealing with heavy particles suspended in homogeneous isotropic turbulence (Stokes number $\mathit{St}= O(1{\unicode{x2013}} 10)$, particle/fluid density ratio ${\rho }_{p} / \rho = 2000$, ${\Phi }_{v} = O(1{0}^{- 4} )$) and the experimental investigation of Poelma, Westerweel & Ooms (J. Fluid Mech., vol. 589, 2007, pp. 315–351) involving light particles ($\mathit{St}= O(0. 1)$, ${\rho }_{p} / \rho \gtrsim 1$, ${\Phi }_{v} = O(1{0}^{- 3} )$) settling in grid turbulence. The development in this work is restricted to volume loadings where particle or droplet collisions are negligible.

We study the dispersal of a plume of incompressible buoyant fluid injected into a confined inclined aquifer in which there is a background flow. We assume that, to prevent pressure buildup in the system, there is an outflow from the aquifer, with flux equal to the injection flux, through a producing well. Using the method of characteristics, we identify that the trajectory of the plume of injected fluid depends on the magnitudes of both the injection flux and the background aquifer flux relative to the buoyancy-driven exchange flow of injected and original fluid within the aquifer , on the direction of the background aquifer flow, and on whether the producing well lies upslope or downslope from the injecting well. We find the values of the controlling parameters and for which all injected fluid flows up-dip, for which the injected fluid partitions into a component moving up-dip and a component moving down-dip, and for which all injected fluid flows down-dip. A key learning from the analysis is that there may be very different plume trajectories when a buoyant fluid is injected into a confined, inclined aquifer, and prediction of the trajectory depends on knowledge of the background flow as well as the injection rate and location of the producing wells. In the process of sequestration, this range of initial plume geometries can inform analysis of longer-term geological storage and assessment of the risk of activating different possible leakage pathways to the surface.

Direct numerical simulations of the flow around a NACA-0012 aerofoil are conducted, employing an immersed boundary method to represent flat-plate trailing-edge extensions both with and without serrations. Properties of the turbulent boundary layer convecting over the trailing edge are similar for both cases. For cases with serrations, the trailing-edge noise produced by the flow over the aerofoil is observed to decrease in amplitude, and the frequency interval over which the noise reduction occurs differs depending on the serration length. The directivity and spanwise coherence of the trailing-edge noise appears largely unaffected by the serrations. The hydrodynamic behaviour in the vicinity of the trailing-edge extensions is investigated. The streamwise discontinuity imparted upon the turbulent flow by the straight trailing edge can clearly be observed in statistical quantities, whereas for the serrated case no spanwise homogeneous discontinuities are observed. The trailing-edge serrations appear to break up the larger turbulent structures convecting into the wake, and to promote the development of horseshoe vortices originating at the serrations themselves.

In 1875, Lord Kelvin stated an energy-based argument for equilibrium and stability in conservative flows. The possibility of building an implementation of Kelvin’s argument, based on the construction of a simple bifurcation diagram, has been the subject of debate in the past. In this paper, we build on work from dynamical systems theory, and show that an essential requirement for constructing a meaningful bifurcation diagram is that families of solutions must be accessed through isovortical (i.e. vorticity-preserving), incompressible rearrangements. We show that, when this is the case, turning points in fluid impulse are linked to changes in the number of the positive-energy modes associated with the equilibria (and therefore in the number of modes likely to be linearly unstable). In addition, the shape of a velocity-impulse diagram, for a family of solutions, determines whether a positive-energy mode is lost or gained at the turning point. Further to this, we detect bifurcations to new solution families by calculating steady flows that have been made ‘imperfect’ through the introduction of asymmetries in the vorticity field. The resulting stability approach, which employs ‘imperfect velocity-impulse’ (IVI) diagrams, can be used to determine the number of positive-energy (likely unstable) modes for each equilibrium flow belonging to a family of steady states. As an illustration of our approach, we construct IVI diagrams for several two-dimensional flows, including elliptical vortices, opposite-signed vortex pairs (of both rotating and translating type), single and double vortex rows, as well as gravity waves. By also considering an example involving the Chaplygin–Lamb dipole, we illustrate how the stability of a specific flow may be determined, by embedding it within a properly constructed solution family. The stability data from our IVI diagrams agree precisely with results in the literature. To the best of our knowledge, for a few of the flows considered here, our work yields the first available stability boundaries. Further to this, for several of the flows that we examine, the IVI diagram methodology leads us to the discovery of new families of steady flows, which exhibit lower symmetry.

An jet in cross-flow (JICF) of air is studied using three-dimensional direct numerical simulation with and without chemical reaction in order to investigate the role of the complex JICF turbulent flow field in the mechanism of fast fuel-oxidant mixing and of aerodynamic flame stabilization in the near field of the jet nozzle. Focus is on delineating the flow/mixing/chemistry conditions that are necessary and/or sufficient to achieve flame anchoring that ultimately enables the formulation of more reliable and precise guidelines for design of fuel injection nozzles. A mixture averaged diffusion formulation that includes the effect of thermal diffusion is used along with a detailed chemical kinetics mechanism for hydrogen–air combustion. A new parametrization technique is used to describe the jet trajectory: solution of Laplace’s equation upon, and then within, an opportune scalar surface anchored by Dirichlet boundary conditions at the jet nozzle and plume exit from the domain provides a smoothly varying field along the jet path. The surface is selected to describe the scalar mixing and reaction associated with a transverse jet. The derived field, , is used as a condition to mark the position along the natural jet trajectory when analysing the variation of relevant flow, mixing and reaction quantities in the present direct numerical simulation (DNS) datasets. Results indicate the presence of a correlation between the flame base location in parameter space and a region of low velocity magnitude, high enstrophy, high mixing rate and high equivalence ratio (flame root region). Instantaneously, a variety of vortical structures, well known from the literature as important contributors to fuel-oxidant mixing, are observed in both inert and reactive cases with a considerable span in length scales. Moreover, instantaneous plots from reactive cases illustrate that the most upstream flame tongues propagate close to the trailing edge of the fuel jet potential core near the jet shear layer vortex shedding position. Some degree of asymmetry with respect to the domain mid-plane in the spanwise direction is observed in the averaged fields, both for the inert and reactive cases.

Tropical cyclogenesis is studied in the context of idealized three-dimensional Boussinesq dynamics with perhaps the simplest possible model for bulk cloud physics. With low-altitude input of water vapour on realistic length and time scales, numerical simulations capture the formation of vortical hot towers. From measurements of water vapour, vertical velocity, vertical vorticity and rain, it is demonstrated that the structure, strength and lifetime of the hot towers are similar to results from models including more detailed cloud microphysics. The effects of low-altitude vertical shear are investigated by varying the initial zonal velocity profile. In the presence of weak low-level vertical shear, the hot towers retain the low-altitude monopole cyclonic structure characteristic of the zero-shear case (starting from zero velocity). Some initial velocity profiles with small vertical shear can have the effect of increasing cyclonic predominance of individual hot towers in a statistical sense, as measured by the skewness of vertical vorticity. Convergence of horizontal winds in the atmospheric boundary layer is mimicked by increasing the frequency of the moisture forcing in a horizontal subdomain. When the moisture forcing is turned off, and again for zero shear or weak low-level shear, merger of cyclonic activity results in the formation of a larger-scale cyclonic vortex. An effect of the shear is to limit the vertical extent of the resulting emergent moist vortex. For stronger low-altitude vertical shear, the individual hot towers have a low-altitude vorticity dipole rather than a cyclonic monopole. The dipoles are not conducive to the formation of larger-scale vortices, and thus sufficiently strong low-level shear prevents the vortical-hot-tower route to cyclogenesis. The results indicate that the simplest condensation and evaporation schemes are useful for exploratory numerical simulations aimed at better understanding of competing effects such as low-level moisture and vertical shear.

This paper is an extension of work on separation from a downstream moving wall by Ruban et al. (J. Fluid. Mech., vol. 678, 2011, pp. 124–155) and is in particular concerned with the boundary-layer separation in unsteady two-dimensional laminar supersonic flow. In a frame attached to the wall, the separation is assumed to be provoked by a shock wave impinging upon the boundary layer at a point that moves downstream with a non-dimensional speed which is assumed to be of order where is the Reynolds number. In the coordinate system of the shock however, the wall moves upstream. The strength of the shock and its speed are allowed to vary with time on a characteristic time scale that is large compared to . The ‘triple-deck’ model is used to describe the interaction process. The governing equations of the interaction problem can be derived from the Navier–Stokes equations in the limit . The numerical solutions are obtained using a combination of finite differences along the streamwise direction and Chebyshev collocation along the normal direction in conjunction with Newton linearization. In the present study with the wall moving upstream, the evidence is inconclusive regarding the so-called ‘Moore–Rott–Sears’ criterion being satisfied. Instead it is observed that the pressure rise from its initial value is very slow and that a recirculation region forms, the upstream part of which is wedge-shaped, as also observed in turbulent marginal separation for large values of angle of attack.

It has been observed experimentally that when a free shear layer is perturbed by a disturbance consisting of two waves with frequencies and , components with the combination frequencies ( and being integers) develop to a significant level thereby causing flow randomization. This spectral broadening process is investigated theoretically for the case where the frequency difference is small, so that the perturbation can be treated as a modulated wavetrain. A nonlinear evolution system governing the spectral dynamics is derived by using the non-equilibrium nonlinear critical layer approach. The formulation provides an appropriate mathematical description of the physical concepts of sideband instability and amplitude–phase modulation, which were suggested by experimentalists. Numerical solutions of the nonlinear evolution system indicate that the present theory captures measurements and observations rather well.