The two-dimensional inviscid transonic flow about a circular cylinder is investigated. To do this, the Euler equations are integrated numerically with a time-dependent technique. The integration is based on an high-resolution finite volume upwind method.

Time scales are introduced and the flow at very short, short and large times is studied. Attention is focused on the behaviour of the numerical solution at large times, after the breakdown of symmetry and the onset of an oscillating solution have occurred. This solution is known to be periodic at Mach number between 0.5 and 0.6.

At higher speed, however, a richer behaviour is observed. As the Mach number is increased from 0.6 to 0.98 the numerical solution undergoes two transitions. Through a first one the periodical, regular flow enters a chaotic, turbulent regime. Through the second transition the chaotic flow comes back to an almost stationary state. The flow in the chaotic and in the almost stationary regimes is investigated. A numerical conjecture for the behaviour of the solution at large times is advanced.

The characteristics of dense gravity currents in coastal regions, where free-surface gravity waves are dominant, have yet to be studied in the laboratory. This paper provides a first insight into the dynamics of dense saline gravity currents moving beneath regular progressive free-surface water waves. The gravity currents were generated by releasing a finite volume of saline into a large wave tank with an established periodic wave field. After the initial collapse, the gravity currents propagated horizontally with two fronts, one propagating in the wave direction and the other against the wave direction. The fronts of the gravity currents oscillated with an amplitude and phase that correlated with the orbital velocities within a region close to the bed. To leading order, the overall length of the gravity current was found to be weakly affected by the wave action and the dynamics of the current could be approximated by simply considering the buoyancy of the released fluid. Other characteristics such as the position of the gravity current centre and the shape of the two leading profiles were found to be significantly affected by the wave action. The centre was displaced at constant speed dependent on the second-order wave-induced mean Lagrangian velocity. For long waves, the centre was advected downstream in the direction of wave propagation owing to the dominance of Stokes drift. For short waves, the gravity current centre moved upstream against the wave direction, as under these wave conditions Stokes drift is negligible at the bed. An asymmetry in the shape of the upstream and downstream current heads was observed, with the gravity current front moving against the waves being much thicker and the front steeper, similar to the case of a current moving in a stream.

Although models of the pressure–strain term explain many features of nearly uniform homogeneous shear flows, a discrepancy remains (Leslie 1980). It is suggested that the discrepancy is caused by use of an empirical expression for the fluctuation part of the pressure–strain term, the part usually denoted by ϕij,1. The discrepancy is eliminated by replacement of the empirical ϕij,1 with a recent theoretical expression. Relatedly, the Launder, Reece & Rodi (1975) model for the mean-field part ϕij,2 is shown to be a good approximation for both a strongly and weakly sheared flow. This model of ϕij,2 when combined with the theoretical ϕij,1 is found to provide an explanation for experiments of both Champagne, Harris & Corrsin (1970) and Harris, Graham & Corrsin (1977). Full correction requires that deviations from local isotropy be accounted for. Special emphasis is given to a theoretical demonstration that the pressure–strain term does not cause a retun to isotropy but, rather, it resists large anisotropy – a weaker effect.

A horizontal channel flow of two immiscible fluid layers with different densities, viscosities and thicknesses, subject to vertical gravitational forces and with an insoluble surfactant monolayer present at the interface, is investigated. The base Couette flow is driven by the uniform horizontal motion of the channel walls. Linear and nonlinear stages of the (inertialess) surfactant and gravity dependent long-wave instability are studied using the lubrication approximation, which leads to a system of coupled nonlinear evolution equations for the interface and surfactant disturbances. The (inertialess) instability is a combined result of the surfactant action characterized by the Marangoni number $Ma$ and the gravitational effect corresponding to the Bond number $Bo$ that ranges from $-\infty$ to $\infty$. The other parameters are the top-to-bottom thickness ratio $n$, which is restricted to $n\geqslant 1$ by a reference frame choice, the top-to-bottom viscosity ratio $m$ and the base shear rate $s$. The linear stability is determined by an eigenvalue problem for the normal modes, where the complex eigenvalues (determining growth rates and phase velocities) and eigenfunctions (the amplitudes of disturbances of the interface, surfactant, velocities and pressures) are found analytically by using the smallness of the wavenumber. For each wavenumber, there are two active normal modes, called the surfactant and the robust modes. The robust mode is unstable when $Bo/Ma$ falls below a certain value dependent on $m$ and $n$. The surfactant branch has instability for $m<1$, and any $Bo$, although the range of unstable wavenumbers decreases as the stabilizing effect of gravity represented by $Bo$ increases. Thus, for certain parametric ranges, even arbitrarily strong gravity cannot completely stabilize the flow. The correlations of vorticity-thickness phase differences with instability, present when gravitational effects are neglected, are found to break down when gravity is important. The physical mechanisms of instability for the two modes are explained with vorticity playing no role in them. This is in marked contrast to the dynamical role of vorticity in the mechanism of the well-known Yih instability due to effects of inertia, and is contrary to some earlier literature. Unlike the semi-infinite case that we previously studied, a small-amplitude saturation of the surfactant instability is possible in the absence of gravity. For certain $(m,n)$-ranges, the interface deflection is governed by a decoupled Kuramoto–Sivashinsky equation, which provides a source term for a linear convection–diffusion equation governing the surfactant concentration. When the diffusion term is negligible, this surfactant equation has an analytic solution which is consistent with the full numerics. Just like the interface, the surfactant wave is chaotic, but the ratio of the two waves turns out to be constant.

We investigate the nonlinear dynamics of turbulent shear flows, with and without rotation, in the context of a simple but physically motivated closure of the equation governing the evolution of the Reynolds stress tensor. We show that the model naturally accounts for some familiar phenomena in parallel shear flows, such as the subcritical transition to turbulence at a finite Reynolds number and the occurrence of a universal velocity profile close to a wall at large Reynolds number. For rotating shear flows we find that, depending on the Rayleigh discriminant of the system, the model predicts either linear instability or nonlinear instability or complete stability as the Reynolds number is increased to large values. We investigate the properties of Couette–Taylor flows for varying inner and outer cylinder rotation rates and identify the region of linear instability (similar to Taylor's), as well as regions of finite-amplitude instability qualitatively compatible with recent experiments. We also discuss quantitative predictions of the model in comparison with a range of experimental torque measurements. Finally, we consider the relevance of this work to the question of the hydrodynamic stability of astrophysical accretion disks.

A boundary-layer scale analysis is presented for steady, zonally symmetric flow in a Cartesian channel of rectangular cross-section, subject to uniform internal heating, and cooling at the lateral boundaries, using an approach based on that of Hignett, Ibbetson & Killworth for a related system. Six main flow regimes are identified, depending chiefly upon the magnitude of the parameter ℙ defined as the square of the ratio of the (non-rotating) thermal-boundary-layer thickness scale to that of the Ekman layers adjacent to the horizontal boundaries. For ${\mathbb P} \ll {\mathbb A}^{\frac{1}{6}}\epsilon^{\frac{1}{2}}(\gg 1$, where [Aopf ] is the Rayleigh number and ε the channel aspect ratio), the flow consists of an advectively dominated interior, characterized by a balance between vertical advection and internal heat generation, diffusively dominated thermal boundary layers adjacent to the sidewalls, and horizontal, viscously dominated Ekman layers (for non-zero rotation rate). If ℙ [Lt ] 1, the flow is only weakly modified by rotation, but as ℙ increases through unity, rotation tends to inhibit heat transfer and thickens the thermal boundary layers. Provided ℙ [Gt ] ε2σ −2, (where σ is the Prandtl number), the zonal flow is predominantly geostrophic, though not given by the conventional thermal-wind scale (based on the total thermal contrast ΔT) unless ℙ [Gt ] 1.

The results of the scale analysis are compared with laboratory measurements and numerical simulations of steady flow in a rotating, cylindrical annulus subject to (radially non-uniform) internal heating and sidewall cooling. Over the range of parameters accessible in the laboratory, the azimuthal velocity scale and thermal contrast were found to vary with rotation and heating rates in the way predicted from the scale analysis for the Cartesian system. Above a certain critical value of ℙ (for the geometry used here ℙt ≈ 1), the baroclinic wave regime was found to occur, corresponding to where rotational constraints first begin to influence significantly the heat transfer of the axisymmetric flow. The numerical simulations are compared with the laboratory measurements, and used to extend the ranges of rotation rate and aspect ratio over which the scale analysis could be verified. Good agreement was found for the dependence of globally averaged flow parameters on ℙ, and the dynamical characteristics of each regime were further verified using explicit calculations of the balance of terms in the basic equations from the numerical model.

Further applications of the scaling technique to other, related systems are also discussed, together with a consideration of its generalization to systems of geophysical interest.

In this paper the propagation of long waves (tides) into a canal is studied. The canal is assumed to be rotating at constant angular velocity and the depth of the fluid is uniform.

The rate of rotation can have a considerable effect on the amplitude of the unattenuated modes in the channel. This is due partly to the modification of the known solution when the rotation is zero, and partly to the fact that even if there is only a single semi-infinite barrier unattenuated waves of a special type (Kelvin waves) may be propagated in the rotating system into what is normally called the ‘shadow’ region behind the barrier (Crease 1956).

The purpose of this theoretical investigation is to seek a partial explanation of the behaviour of tides and storm surges in the North Sea. For this reason two models of this region are discussed. In Example I the model is of two parallel semi-infinite barriers in the path of a plane progressive wave, and in Example II there are two parallel barriers, one semi-infinite and the other infinite.

The ratio of the observed height of the semi-diurnal M2 tide in the North Sea to the height of the incident tide from the Atlantic lies between the results predicted by the two models and is in closer agreement with the model of Example I.

Settling characteristics of bidisperse suspensions containing light and heavy particles in inclined channels have been studied both experimentally and theoretically. The suspension is relatively dilute with a total volume fraction of no more than 0.16. In this dilute range the flow-visualization experiments indicate the formation of distinct zones with clear interfaces between them. There is no evidence of lateral segregation of particles. The convection currents formed near an inclined boundary are transferred to different zones depending on the relative concentration of the two species. Based on the flow-visualization experiments, the well known Ponder-Nakamura-Kuroda (PNK) model has been adopted to predict the settling characteristics of bidisperse suspensions.

We have investigated the modulated and unmodulated travelling azimuthal waves appearing on the toroidal Taylor–Görtler (TG) vortices in a fluid contained between two concentric spheres with the inner sphere rotating. For smaller-clearance cases, toroidal TG vortices appear at the equator, just as in the flow between two concentric cylinders with the inner cylinder rotating. When the Reynolds number of the flow increases quasi-statically, spiral TG vortices appear in addition to toroidal TG vortices, and no modulation occurs, even if the Reynolds number further increases quasi-statically. However, when the Reynolds number is increased from zero to a particular value with a specific acceleration of the inner sphere, modulated wavy toroidal TG vortices appear. We found that the necessary condition for occurrence of modulation is the prevention of spiral TG vortices. Using simultaneous flow-visualization and spectral techniques, and measuring the fluctuation of sinks and sources of vortex boundaries, we obtained the frequency f1 of travelling azimuthal waves passing a fixed point in the laboratory and the modulation frequencies f2 and f′2 of these waves, as determined by an observer in the laboratory and an observer fixed in a reference frame that rotates in phase with the travelling azimuthal waves, respectively. The relationship among the characteristic frequencies, f1, f2 and f′2, obtained by modal analysis and the experimental results, is (f′2 + kf1/m)/f2 = − 1, where k and m are a modulation parameter and the wavenumber of travelling azimuthal waves, respectively.

Some new exact relations are derived between integral properties of a progressive irrotational gravity wave of finite amplitude in deep water. In particular it is shown that the Eulerian-mean angular momentum $\overline{A}_{\rm E}$ is directly proportional to the Lagrangian T−V, through the relation \[ \overline{A}_{\rm E} = 2c(T-V)/g, \] where c is the phase speed and g denotes the acceleration due to gravity. Moreover, for waves of constant length, the differential relation \[ {\rm d}\overline{A}_{\rm E} = 2(3T-V)\,{\rm d}c/g \] also holds.

In a wave of limiting steepness it was shown previously that the level of action ya is very nearly equal to the crest level ymax. This is further discussed, and is shown to be probably a numerical coincidence.

Transport of dense fluid by an inclined gravity current can control the vertical density structure of the receiving basin in many natural and industrial settings. A case familiar to many is a lake fed by river water that is dense relative to the lake water. In laboratory experiments, we pulsed dye into the basin inflow to visualise the transport pathway of the inflow fluid through the basin. We also measured the evolving density profile as the basin filled. The experiments confirmed previous observations that when the turbulent gravity current travelled through ambient fluid of uniform density, only entrainment into the dense current occurred. When the gravity current travelled through the stratified part of the ambient fluid, however, the outer layers of the gravity current outflowed from the current by a peeling detrainment mechanism and moved directly into the ambient fluid over a large range of depths. The prevailing model of a filling box flow assumes that a persistently entraining gravity current entrains fluid from the basin as the current descends to the deepest point in the basin. This model, however, is inconsistent with the transport pathway observed in visualisations and poorly matches the stratifications measured in basin experiments. The main contribution of the present work is to extend the prevailing filling box model by incorporating the observed peeling detrainment. The analytical expressions given by the peeling detrainment model match the experimental observations of the density profiles more closely than the persistently entraining model. Incorporating peeling detrainment into multiprocess models of geophysical systems, such as lakes, will lead to models that better describe inflow behaviour.

In this paper we examine the effect of thermal expansion on the stability of a planar unstrained diffusion flame and provide a comprehensive characterization of diffusive-thermal instabilities while realistically accounting for density variations. The possible patterns that are likely to be observed as a result of differential and preferential diffusion are identified for a whole range of parameters including the distinct Lewis numbers associated with the fuel and oxidizer, the initial mixture strength and the flow conditions. Although we find that thermal expansion has a marked influence on flame instability, it does not play a crucial role as it does in premixed combustion. It primarily affects the parameter regime associated with the onset of the instabilities and the growth rate of the unstable modes. Perhaps the most surprising result is that its has a different influence on the various modes of instability – a destabilizing influence on the formation of cellular structures and a stabilizing influence on the onset of oscillations.

The theory of solitary topographic Rossby waves (modons) in a uniformly rotating two-layer ocean over a constant slope is developed. The modon is described by an exact, form-preserving, uniformly translating, horizontally localized, nonlinear solution to the inviscid quasi-geostrophic equations. Baroclinic topographic modons are found to translate steadily along contours of constant depth in both directions: either with negative speed (within the range of the phase velocities of linear topographic waves) or with positive speed (outside the range of the phase velocities of linear topographic waves). The lack of resonant wave radiation in the first case is due to the orthogonality of the flow field in the modon exterior to the linear topographic wave field propagating with the modon translation speed, that is impossible for barotropic modons. Another important property of a baroclinic topographic modon is that its integral angular momentum must be zero only in the bottom layer; the total angular momentum can be non-zero unlike for the beta-plane modons over flat bottom. This feature allows modon solutions superimposed by intense monopolar vortices in the surface layer to exist. Explicit analytical solutions for the baroclinic topographic modons with piecewise linear dependence of the potential vorticity on the streamfunction are presented and analysed.

We consider a homogeneous fluid of viscosity confined within an oblate spheroidal cavity, , with eccentricity . The spheroidal container rotates rapidly with an angular velocity , which is fixed in an inertial frame and defines a small Ekman number , and undergoes weak latitudinal libration with frequency and amplitude , where is the Poincaré number quantifying the strength of Poincaré force resulting from latitudinal libration. We investigate, via both asymptotic and numerical analysis, fluid motion in the spheroidal cavity driven by latitudinal libration. When , an asymptotic solution for and in oblate spheroidal coordinates satisfying the no-slip boundary condition is derived for a spheroidal cavity of arbitrary eccentricity without making any prior assumptions about the spatial–temporal structure of the librating flow. In this case, the librationally driven flow is non-axisymmetric with amplitude , and the role of the viscous boundary layer is primarily passive such that the flow satisfies the no-slip boundary condition. When , the librationally driven flow is also non-axisymmetric but latitudinal libration resonates with a spheroidal inertial mode that is in the form of an azimuthally travelling wave in the retrograde direction. The amplitude of the flow becomes at and the role of the viscous boundary layer becomes active in determining the key property of the flow. An asymptotic solution for describing the librationally resonant flow is also derived for an oblate spheroidal cavity of arbitrary eccentricity. Three-dimensional direct numerical simulation in an oblate spheroidal cavity is performed to demonstrate that, in both the non-resonant and resonant cases, a satisfactory agreement is achieved between the asymptotic solution and numerical simulation at .

In this paper, we address the linear stability analysis of a confined potential vortex with a free surface. This particular flow has been recently used by Tophøj et al. (Phys. Rev. Lett., vol. 110(19), 2013, article 194502) as a model for the swirling flow of fluid in an open cylindrical container, driven by rotating the bottom plate (the rotating bottom experiment) to explain the so-called rotating polygons instability (Vatistas J. Fluid Mech., vol. 217, 1990, pp. 241–248; Jansson et al., Phys. Rev. Lett., vol. 96, 2006, article 174502) in terms of surface wave interactions leading to resonance. Global linear stability results are complemented by a Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) analysis in the shallow-water limit as well as new experimental observations. It is found that global stability results predict additional resonances that cannot be captured by the simple wave coupling model presented in Tophøj et al. (2013). Both the main resonances (thought to be at the root of the rotating polygons) and these secondary resonances are interpreted in terms of over-reflection phenomena by the WKBJ analysis. Finally, we provide experimental evidence for a secondary resonance supporting the numerical and theoretical analysis presented. These different methods and observations allow to support the unstable wave coupling mechanism as the physical process at the origin of the polygonal patterns observed in free-surface rotating flows.

We theoretically investigate the lift force on spherical nanoparticles in a shear flow of a dilute gas, wherein the non-rigid-body collision between the particle and the gas molecules is considered. The analytical formula of the lift force is derived based on the gas kinetic theory. In the limit of rigid-body collision, the formula is consistent with the theoretical results in the literature (Liu & Bogy, Phys. Fluids, vol. 20, 2008, 107102), which predicts that the lift force is in the opposite direction to the fluid velocity gradient (negative lift force). However, by taking into account gas–particle intermolecular interactions, the direction of the lift force on the nanoparticle is found to be dependent on temperature, i.e. both positive and negative lift forces exist in a certain temperature range. An explanation for the direction change of the lift force is given based on the analysis of the scattering angle under non-rigid-body particle–molecule collisions.