We obtain exact solutions of the linearized perturbation equations for the convective motions in a Boussinesq fluid contained in an infinite rectangular channel and heated from below. The top and bottom are assumed to be perfect heat conductors. The sides can either be conducting or insulating. We assume that the sidewalls are rigid, but allow the top and bottom to be free so that we can separate variables.

We find that the preferred modes of convection closely resemble transverse ‘finite rolls’ [as predicted by Davis (1967) for convection in a box] for channels with height to width ratios outside the range 0·1 to 1. Inside this range they show noticeable departures from roll form. We prove, however, that, except when the sidewalls are relaxed to infinity, ‘finite rolls’ are never exact solutions of the linearized equations, even though in most cases they are good approximations.

We also find that bringing the sidewalls closer together inhibits convection and, generally, produces thinner cells at the onset of convection.

This paper presents experimental and theoretical results on the internal waves emitted by a bluff body moving horizontally in a linearly stratified fluid. Three different bluff bodies (a sphere, a spheroid and a cylinder) have been used in order to study the effect of the shape of the bluff body, although most of the results are obtained for the sphere. Two types of internal waves have been observed experimentally: large wavelength lee waves generated by the bluff body itself and small wavelength coherent wake waves generated by the turbulent wake. First, the lee waves are separated from the wake waves by averaging the experimental measurements in the frame moving with the bluff body. The velocity amplitude of the lee waves scales as the inverse of the Froude number $F=2U_{B}/(ND)$ for $F>2$ (where $U_{B}$ is the towing velocity, $D$ the diameter and $N$ the buoyancy frequency). This scaling proves that the internal waves are related to the drag of the bluff body which is due to the separation of the flow behind the bluff body. This separation is usually not taken into account in the classical models which assume that the flow is dipolar. The drag can be modelled as a point force in the Navier–Stokes equations, which gives a correct prediction of the structure and the amplitude of the lee waves. Second, the wake waves have been separated from the lee waves by averaging the velocity fields in the frame moving at the phase velocity of the waves. The phase velocity and the wavelength scale as $F^{-2/3}$ and $F^{1/3}$ respectively which correspond to the velocity and distance between same sign vortices of the von Kármán vortex street. A simplified model is derived for the internal waves emitted by the double row of moving point vortices of the von Kármán street. The amplitude of the wake waves is measured experimentally and seems to depend on the Reynolds number.

Numerical simulations are presented of shear flow past two-dimensional droplets adhering to a wall, at moderate Reynolds numbers. The results were obtained using a level-set method to track the interface, with measures to eliminate any errors in the conservation of mass of droplets. First, the case of droplets whose contact lines are pinned is considered. Data are presented for the critical value of the dimensionless shear rate (Weber number, $\hbox{\it We}$), beyond which no steady state is found, as a function of Reynolds number, $\hbox{\it Re}$. $\hbox{\it We}$ and $\hbox{\it Re}$ are based on the initial height of the droplet and shear rate; the range of Reynolds numbers simulated is $\hbox{\it Re} \leq 25$. It is shown that, as $\hbox{\it Re}$ is increased, the critical value $\hbox{\it We}_c$ changes from $\hbox{\it We}_c\propto \hbox{\it Re}$ to $\hbox{\it We}_c\approx$ const., and that the deformation of droplets at $\hbox{\it We}$ just above $\hbox{\it We}_c$ changes fundamentally from a gradual to a sudden dislodgement. In the second part of the paper, drops are considered whose contact lines are allowed to move. The contact-line singularity is removed by using a Navier-slip boundary condition. It is shown that macroscale contact angles can be defined that are primarily functions of the capillary number based on the contact-line speed, instead of the value of $\hbox{\it We}$ of the shear flow. It is shown that a Cox–Voinov-type expression can be used to describe the motion of the downstream contact line. A qualitatively different relation is tested for the motion of the upstream contact line. In a third part of this paper, results are presented for droplets moving on a wall with position-dependent sliplength or contact-angle hysteresis window, in an effort to stabilize or destabilize the drop.

Natural convection in an infinite horizontal layer subject to periodic heating along the lower wall has been investigated using a combination of numerical and asymptotic techniques. The heating maintains the same mean temperatures at both walls while producing sinusoidal temperature variations along one horizontal direction, with its spatial distribution characterized by the wavenumber $\alpha $ and the amplitude expressed in terms of a Rayleigh number $R{a}_{p} $. The primary response of the system takes the form of stationary convection consisting of rolls with the axis orthogonal to the heating wave vector and structure determined by the particular values of $R{a}_{p} $ and $\alpha $. It is shown that for sufficiently large $\alpha $ convection is limited to a thin layer adjacent to the lower wall with a uniform conduction zone emerging above it; the temperature in this zone becomes independent of the heating pattern and varies in the vertical direction only. Linear stability of the above system has been considered and conditions leading to the emergence of secondary convection have been identified. Secondary convection gives rise to either longitudinal rolls, transverse rolls or oblique rolls at the onset, depending on $\alpha $. The longitudinal rolls are parallel to the primary rolls and the transverse rolls are orthogonal to the primary rolls, and both result in striped patterns. The oblique rolls lead to the formation of convection cells with aspect ratio dictated by their inclination angle and formation of rhombic patterns. Two mechanisms of instability have been identified. In the case of $\alpha = O(1)$, parametric resonance dominates and leads to a pattern of instability that is locked in with the pattern of heating according to the relation ${\delta }_{cr} = \alpha / 2$, where ${\delta }_{cr} $ denotes the component of the critical disturbance wave vector parallel to the heating wave vector. The second mechanism, the Rayleigh–Bénard (RB) mechanism, dominates for large $\alpha $, where the instability is driven by the uniform mean vertical temperature gradient created by the primary convection, with the critical disturbance wave vector ${\delta }_{cr} \rightarrow 1. 56$ for $\alpha \rightarrow \infty $ and the fluid response becoming similar to that found in the case of a uniformly heated wall. Competition between these mechanisms gives rise to non-commensurable states in the case of longitudinal rolls and the appearance of soliton lattices, to the formation of distorted transverse rolls, and to the appearance of the wave vector component in the direction perpendicular to the forcing direction. A rapid stabilization is observed when the heating wavenumber is reduced below $\alpha \approx 2. 2$ and no instability is found when $\alpha \lt 1. 6$ in the range of $R{a}_{p} $ considered. It is shown that $\alpha $ plays the role of an effective pattern control parameter and its judicious selection provides a means for the creation of a wide range of flow responses.

The growth and collapse of a vapour bubble inside a microtube is studied both experimentally and theoretically. The length of the bubble, and the velocity and acceleration of its interface, are obtained from a high-speed image recording (typically 1.25 × 105 fps) for various energy inputs and two tube diameters. To understand the underlying dynamics of the system, two theoretical models are compared with experiment. A model based on a discontinuous time dependence of the vapour pressure inside the bubble is at variance with the data. It proves necessary to account in greater detail for the time dependence of the vapour pressure. A new model is proposed for this purpose which includes heat transfer in addition to inertia and viscous friction. Both the data and the model show that the vapour pressure decreases with time continuously instead of abruptly. The length, velocity and acceleration from the numerical simulations are found to be in good agreement with experimental data. Both the experiments and simulations clearly indicate that thermal effects play an important role throughout the whole growth and collapse process.

We present an extensive compilation of direct numerical simulation (DNS) data for Poiseuille and Couette flows, from the laminar into the fully turbulent regime, with the goal of highlighting similarities and differences. The data suggest that, for a given bulk velocity, Couette flow yields less resistance than Poiseuille flow and greater turbulence kinetic energy, which may be beneficial for more efficient diffusion, thus suggesting the effectiveness of fluid transport devices based on moving belts as opposed to classical ducts. Both flows exhibit similar trends for the wall-parallel velocity variances, which increase logarithmically with the Reynolds number. The shear stress and the wall-normal stress tend to saturate faster in Couette flow, which can thus be regarded as a limit to which Poiseuille flow tends, in the limit of high Reynolds number. Excess production over dissipation is found in the outer part of Poiseuille and Couette flow, which is responsible for non-local transfer of energy. However, the structure of the core flow seems to attain an asymptotic state which consists of a parabolic and linear mean velocity profile, respectively, and it seems unlikely that substantial changes to this scenario will occur at Reynolds numbers reachable by DNS in the foreseeable future.

In order to investigate the turbulent drag reduction phenomenon and understand its mechanism, direct numerical simulation (DNS) was carried out on decaying homogeneous isotropic turbulence (DHIT) with and without polymer additives. We explored the polymer effect on DHIT from the energetic viewpoint, i.e. the decay of the total turbulent kinetic energy and energy distribution at each scale in Fourier space and from the phenomenological viewpoint, i.e. the alterations of vortex structures, the enstrophy and the strain. It was obtained that in DHIT with polymer additives the decay of the turbulent kinetic energy is faster than that in the Newtonian fluid case and a modification of the turbulent kinetic energy transfer process for the Newtonian fluid flow is observed due to the release of the polymer elastic energy into flow structures at certain small scales. Besides, we deduced the transport equations of the enstrophy and the strain, respectively, for DHIT with polymer additives. Based on the analyses of these transport equations, it was found that polymer additives depress both the enstrophy and the strain in DHIT as compared to the Newtonian fluid case, indicating the inhibition effect on small-scale vortex structures and turbulence intensity by polymers.

The spreading of a compound drop on a partially wetting solid substrate is numerically simulated using a diffuse-interface method. Compared with a simple drop, the spreading of a compound drop exhibits much more complex behaviour. Depending on the core–shell size ratio and the substrate wettability, various flow regimes are identified in which the interfacial morphology evolves in distinct ways. A phase diagram is constructed in the parameter space of the core–shell size ratio and the wetting angle. For relatively small inner drops, the outer interface does not rupture during the spreading and the inner drop either remains suspended and encapsulated or attaches onto the substrate. Otherwise, the compound drop spontaneously breaks up and releases the inner drop into the ambient fluid. Several breakup scenarios are observed depending on the location of the initial rupture. In some regimes, the wetting of the substrate by one fluid can entrap secondary drops of the other, which can either attach to the substrate or stay suspended. The viscosity ratio mainly affects the spreading rate and plays a minor role in the morphology evolution.

Large-scale patterns, which are well-known from the spiral defect chaos (SDC) regime of thermal convection at Rayleigh numbers $\mathit{Ra}<10^{4}$, continue to exist in three-dimensional numerical simulations of turbulent Rayleigh–Bénard convection in extended cylindrical cells with an aspect ratio ${\it\Gamma}=50$ and $\mathit{Ra}>10^{5}$. They are revealed when the turbulent fields are averaged in time and turbulent fluctuations are thus removed. We apply the Boussinesq closure to estimate turbulent viscosities and diffusivities, respectively. The resulting turbulent Rayleigh number $\mathit{Ra}_{\ast }$, that describes the convection of the mean patterns, is indeed in the SDC range. The turbulent Prandtl numbers are smaller than one, with $0.2\leqslant \mathit{Pr}_{\ast }\leqslant 0.4$ for Prandtl numbers $0.7\leqslant \mathit{Pr}\leqslant 10$. Finally, we demonstrate that these mean flow patterns are robust to an additional finite-amplitude sidewall forcing when the level of turbulent fluctuations in the flow is sufficiently high.

This study investigates the initial transient hydrodynamic evolution of highly under-expanded slit and round jets. A closed-form analytic similarity solution is derived for the temporal evolution of temperature, pressure and density at the jet head for vanishing diffusive fluxes, generalizing a previous model of Chekmarev using Chernyi's boundary-layer method for hypersonic flows. Two-dimensional numerical simulations were also performed to investigate the flow field during the initial stages over distances of ~ 1000 orifice radii. The parameters used in the simulations correspond to the release of pressurized hydrogen gas into ambient air, with pressure ratios varying between approximately 100 and 1000. The simulations confirm the similarity laws derived theoretically and indicate that the head of the jet is laminar at early stages, while complex acoustic instabilities are established at the sides of the jet, involving shock interactions within the vortex rings, in good agreement with previous experimental findings. Very good agreement is found between the present model, the numerical simulations and previous experimental results obtained for both slit and round jets during the transient establishment of the jet. Criteria for Rayleigh–Taylor instability of the decelerating density gradients at the jet head are also derived, as well as the formulation of a model addressing the ignition of unsteady expanding diffusive layers formed during the sudden release of reactive gases.

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.

In this paper we consider a suspension of elastic solid particles in a viscous liquid. The particles are assumed to be neo-Hookean and can undergo finite elastic deformations. A polarization technique, originally developed for analogous problems in linear elasticity, is used to establish a theory for describing the finite-strain, time-dependent response of an ellipsoidal elastic particle in a viscous fluid flow under Stokes flow conditions. A set of coupled, nonlinear, first-order ODEs is obtained for the evolution of the uniform stress fields in the particle, as well as for the shape and orientation of the particle, which can in turn be used to characterize the rheology of a dilute suspension of elastic particles in a shear flow. When applied to a suspension of cylindrical particles with initially circular cross-section, the theory confirms the existence of steady-state solutions, which can be given simple analytical expressions. The two-dimensional, steady-state solutions for the particle shape and orientation, as well as for the effective viscosity and normal stress differences in the suspension, are in excellent agreement with direct numerical simulations of multiple-particle dispersions in a shear flow obtained by using an arbitrary Lagrangian–Eulerian (ALE) finite element method (FEM) solver. The corresponding solutions for the evolution of the microstructure and the rheological properties of suspensions of initially spherical (three-dimensional) particles in a simple shear flow are also obtained, and compared with the results of Roscoe (J. Fluid Mech., vol. 28, 1967, pp. 273–293) in the steady-state regime. Interestingly, the results show that sufficiently soft elastic particles can be used to reduce the effective viscosity of the suspension (relative to that of the pure fluid).

Measurements of the mean and fluctuating velocities have been obtained with pressure and hot-wire probes in the attached boundary layers and wakes of two airfoil models at a low Mach number. The first model is a conventional airfoil at zero incidence and the second an advanced supercritical airfoil at an angle of attack of 4°. The mean-flow and Reynolds-stress data and related quantities are presented with emphasis on the trailing-edge region. The results indicate that the flow around the conventional airfoil is a minor perturbation of a symmetric flat-plate flow with small wake curvature and weak viscous–inviscid interaction. The flow around the supercritical airfoil is in considerable contrast with strong streamwise pressure gradients, non-negligible normal pressure gradients, and large surface and streamline curvatures of the trailing-edge flow. The near wake is strongly curved and intense mixing occurs between the retarded upper-surface boundary layer and strongly accelerated lower-surface boundary layer.

We modify a recent theory of Longuet-Higgins (1989a, b) to study the resonant interaction between an isotropic mode and one or two distort ional modes of an oscillating bubble in water when the isotropic mode is forced by ambient sound. Gravity and buoyant rise are ignored. The energy exchange between modes is strong enough so that both (or all three) can attain comparable amplitudes after a long time. We show that for two-mode interactions the mode-coupling equations are similar to those studied in other physical contexts such as nonlinear optics, coupled oscillators and standing waves in a basin. Instability around fixed points is examined for various bubble radii, phase mismatch, and detuning of the external forcing. Numerical evidences of chaotic bubble oscillations and sound radiation are discussed. It is found that in a certain parameter domain, Hopf bifurcations are possible, and chaos is reached via a period-doubling sequence. However, when there are three interacting modes, each of the two distortion modes interacts with the breathing mode directly and the route to chaos is via a quasi-periodic 2-torus. Possible relevance of this theory to the observed erratic drifting of a bubble is discussed.

In this study, we consider the effects of linear and nonlinear instability waves on the near-field dynamics and aeroacoustics of two-dimensional laminar compressible mixing layers. Through a combination of direct computations, linear and nonlinear stability calculations, we demonstrate the significant role of nonlinear mechanisms in accurately describing the behaviour of instability waves. In turn, these processes have a major impact on sound generation mechanisms such as Mach wave radiation and vortex pairing sound. Our simulations show that the mean flow correction, which is required in order to accurately describe the dynamics of large-scale vortical structures, is intrinsically tied to the nonlinear modal interactions and accurate prediction of saturation amplitudes of instability waves. In addition, nonlinear interactions are largely responsible for the excitation and development of higher harmonics in the flow which contribute to the acoustic radiation. Two flow regimes are considered: In supersonic shear layers, where the far-field sound is determined by the instability wave solution at sufficiently high Mach numbers, it is shown that these nonlinear effects directly impact the Mach wave radiation. In subsonic shear layers, correctly capturing the near-field vortical structures and the interactions of the subharmonic and fundamental modes become critical due to the vortex pairing sound generation process. In this regime, a method is proposed to combine the instability wave solution with the Lilley–Goldstein acoustic analogy in order to predict far-field sound.

The existence of steady, one-dimensional, finite-amplitude waves in a cold collison-free plasma is investigated for the case in which the plasma and the magnetic field are uniform far ahead of the wave. It is supposed that the magnetic field at infinity is inclined at an angle β to the direction normal to the plane of the waves (0 [les ] ½π). It is shown that the problem is equivalent to determining the orbits of a particle in a uniformly rotating field of force.

Two types of waves are found to exist for β ≠ 0: solitary waves and quasishocks. In a quasi-shock, the plasma and magnetic field oscillate irregularly behind the wave front about mean values, which are different from the values ahead of the wave. The width of the wave front is of the order of the ion gyroradius. The quasi-shocks resemble oblique magnetohydrodynamic shock-waves. For the case β = 0, only solitary waves exist, which have already been described (Saffman 1961).

The stability of the waves is considered, and it is concluded that the quasishocks are stable but that the solitary waves for β ≠ 0 are unstable.

The velocity sensitivity of a resistance-wire temperature sensor is expressed in terms of sensor parameters, and the resulting errors in temperature derivative moments in isotropic turbulence are evaluated. It is shown that velocity sensitivity of a degree completely negligible for most purposes causes severe contamination of the measured third moment. The contamination terms are shown to be production rates of the mean square temperature gradient and vorticity, respectively, and therefore create positive values of measured derivative skewness. The dominant contamination term is related to the temperature spectrum through the balance equation for the mean-square temperature gradient, and calculations based on an assumed spectral form show that under typical conditions the measured skewness is large. This mechanism could provide an alternative to anisotropy as an explanation of the positive skewnesses recently measured in the atmosphere.

We investigate theoretically and experimentally the stability of two interlaced helical vortices with respect to displacement perturbations having wavelengths that are large compared to the size of the vortex cores. First, existing theoretical results are recalled and applied to the present configuration. Various modes of unstable perturbations, involving different phase relationships between the two vortices, are identified and their growth rates are calculated. They lead to a local pairing of neighbouring helix loops, or to a global pairing with one helix expanding and the other one contracting. A relation is established between this instability and the three-dimensional pairing of arrays of straight parallel vortices, and a striking quantitative agreement concerning the growth rates and frequencies is found. This shows that the local pairing of vortices is the driving mechanism behind the instability of the helix system. Second, an experimental study designed to observe these instabilities in a real flow is presented. Two helical vortices are generated by a two-bladed rotor in a water channel and characterised through dye visualisations and particle image velocimetry measurements. Unstable displacement modes are triggered individually, either by varying the rotation frequency of the rotor, or by imposing a small rotor eccentricity. The observed unstable mode structure, and the corresponding growth rates obtained from advanced processing of visualisation sequences, are in good agreement with theoretical predictions. The nonlinear late stages of the instability are also documented experimentally. Whereas local pairing leads to strong deformations and subsequent breakup of the vortices, global pairing results in a leapfrogging phenomenon, which temporarily restores the initial double-helix geometry, in agreement with recent observations from numerical simulations.

In this paper we examine the steady, two-dimensional convective motion which occurs in a horizontal circular cylinder whose wall is non-uniformly heated. One observes several qualitatively different physical phenomena depending on the wall temperature distribution and the value of the Rayleigh number. The low-Rayleigh-number behaviour for the single convective cell heated from below is related to the classical Rayleigh stability problem. The critical Rayleigh number for the single circular cell is approximately five times the value for Rayleigh's multi-celluar configuration. The flow which exhibits a nearly parabolic velocity profile near the critical Rayleigh number, gradually changes to a rigidly-rotating-core behaviour as the Rayleigh number increases. The speed of core rotation is a function of the Prandtl number, whereas the boundary-layer thickness is primarily a function of the Rayleigh number. When the heating is from side to side, the solution shows that as the Rayleigh number increases the core motion is progressively arrested leaving a narrow circulating band of fluid adjacent to the wall. An oblique heating produces a hybrid phenomenon, a low-Rayleigh-number behaviour which is characteristic of the sideways heating case and a high-Rayleigh-number interior motion characteristic of the bottom heating case. To determine the core motion in the high-Rayleigh-number limit, Batchelor's work concerning the uniqueness of incompressible, exactly steady, closed streamline flows with small viscosity is extended to include flows with small thermal conductivity.

By experiments and supporting computations we investigate two methods of transport enhancement in two-dimensional open cellular flows with inertia. First, we introduce a spatial dependence in the velocity field by periodic modulation of the shape of the wall driving the flow; this perturbs the steady-state streamlines in the direction perpendicular to the main flow. Second, we introduce a time dependence through transient acceleration–deceleration of a flat wall driving the flow; surprisingly, even though the streamline portrait changes very little during the transient, there is still significant transport enhancement. The range of Reynolds and Reynolds–Strouhal numbers studied is 7.7[les ]Re[les ]46.5 and 0.52[les ]ReSr[les ]12.55 in the spatially dependent mode and 12[les ]Re[les ]93 and 0.26[les ]ReSr[les ]5.02 in the time-dependent mode. The transport is described theoretically via lobe dynamics. For both modifications, a curve with one maximum characterizes the various transport enhancement measures when plotted as a function of the forcing frequency. A qualitative analysis suggests that the exchange first increases linearly with the forcing frequency and then decreases as 1/Sr for large frequencies.