Direct numerical simulations (DNS) of a supercritical temporal mixing layer are conducted for the purpose of exploring the characteristics of high-pressure transitional mixing behaviour. The conservation equations are formulated according to fluctuation-dissipation (FD) theory, which is consistent with non-equilibrium thermodynamics and converges to kinetic theory in the low-pressure limit. According to FD theory, complementing the low-pressure typical transport properties (viscosity, diffusivity and thermal conductivity), the thermal diffusion factor is an additional transport property which may play an increasingly important role with increasing pressure. The Peng–Robinson equation of state with appropriate mixing rules is coupled to the dynamic conservation equations to obtain a closed system. The boundary conditions are periodic in the streamwise and spanwise directions, and of non-reflecting outflow type in the cross-stream direction. Due to the strong density stratification, the layer is considerably more difficult to entrain than equivalent gaseous or droplet-laden layers, and exhibits regions of high density gradient magnitude that become very convoluted at the transitional state. Conditional averages demonstrate that these regions contain predominantly the higher-density, entrained fluid, with small amounts of the lighter, entraining fluid, and that in these regions the mixing is hindered by the thermodynamic properties of the fluids. During the entire evolution of the layer, the dissipation is overwhelmingly due to species mass flux followed by heat flux effects with minimal viscous contribution, and there is a considerable amount of backscatter in the flow. Most of the species mass flux dissipation is due to the molecular diffusion term with significant contributions from the cross-term proportional to molecular and thermal diffusion. These results indicate that turbulence models for supercritical fluids should primarily focus on duplicating the species mass flux rather than the typical momentum flux, which constitutes the governing dissipation in atmospheric mixing layers. Examination of the passive-scalar probability density functions (PDFs) indicates that neither the Gaussian, nor the beta PDFs are able to approximate the evolution of the DNS-extracted PDF from its inception through transition. Furthermore, the temperature–species PDFs are well correlated, meaning that their joint PDF is not properly approximated by the product of their marginal PDFs; this indicates that the traditional reactive flow modelling based on replacing the joint PDF representing the reaction rate by the product of the marginal PDFs is not appropriate. Finally, the subgrid-scale temperature–species PDFs are also well correlated, and the species PDF exhibits important departures from the Gaussian. These results suggest that classic PDFs used in atmospheric pressure flows would not capture the physics of this supercritical mixing layer, either in an assumed PDF model at the larger scale, or at the subgrid scale.

In the present work second-order Stokes theory has been extended to the case of a generally shaped bottom profile connecting two half-strips of constant (but possibly different) depths, initiating a method for generalizing the Stokes hierarchy of second- and higher-order wave theory, without the assumption of spatial periodicity. In modelling the wave–bottom interaction three partial problems arise: the first order, the unsteady second order and the steady second order. The three problems are solved by using appropriate extensions of the consistent coupled-mode theory developed by the present authors for the linearized problem. Apart from the Stokes small-amplitude expansibility assumption, no additional asymptotic assumptions have been introduced. Thus, bottom slope and curvature may be arbitrary, provided that the resulting wave dynamics is Stokes-compatible. Accordingly, the present theory can be used for the study of various wave phenomena (propagation, reflection, diffraction) arising from the interaction of weakly nonlinear waves with a general bottom topography, in intermediate water depth. An interesting phenomenon, that is also very naturally resolved, is the net mass flux induced by the depth variation, which is consistently calculated by means of the steady second-order potential. The present method has been validated against experimental results and fully nonlinear numerical solutions. It has been found that it correctly predicts the second-order harmonic generation, the amplitude nonlinearity, and the amplitude variation due to non-resonant first-and-second harmonic interaction, up to the point where the energy transfer to the third and higher harmonics can no longer be neglected. Under the restriction of weak nonlinearity, the present model can be extended to treat obliquely incident waves and the resulting second-order refraction patterns, and to study bichromatic and/or bidirectional wave–wave interactions, with application to the transformation of second-order random seas in variable bathymetry regions.

We present the results of large-eddy simulations (LES) of turbulent thermal convection generated by surface cooling in a finite-depth stably stratified horizontal layer with an isothermal bottom surface. The flow is a simplified model of turbulent convection occurring in the warm shallow ocean during adverse weather events. Simulations are performed in a 6 × 6 × 1 aspect ratio computational domain using the pseudo-spectral Fourier method in the horizontal plane and finite-difference discretization on a high-resolution clustered grid in the vertical direction. A moderate value of the Reynolds number and two different values of the Richardson number corresponding to a weak initial stratification are considered. A version of the dynamic model is applied as a subgrid-scale (SGS) closure. Its performance is evaluated based on comparison with the results of direct numerical simulations (DNS) and simulations using the Smagorinsky model. Comprehensive study of the spatial structure and statistical properties of the developed turbulent state shows some similarity to Rayleigh–Bénard convection and other types of turbulent thermal convection in horizontal layers, but also reveals distinctive features such as the dominance of a large-scale pattern of descending plumes and strong turbulent fluctuations near the surface.

Investigations of the Richtmyer–Meshkov instability carried out in shock tubes have traditionally used membranes to separate the two gases. The use of membranes, in addition to introducing other experimental difficulties, impedes the use of advanced visualization techniques such as planar laser-induced fluorescence (PLIF). Jones & Jacobs (1997) recently developed a new technique by which a perturbed, membrane-free gas–gas interface can be created in a shock tube. The gases enter the shock tube from opposite ends and exit through two small slots on opposite sides of the test section, forming a stagnation point flow at the interface location. A gentle rocking motion of the shock tube then provides the initial perturbation in the form of a standing wave. The original investigation using this technique utilized dense fog seeding for visualization, which allowed large-scale effects to be observed, but was incapable of resolving smaller-scale features. PLIF visualization is used in the present study to investigate the instability generated by two incident shock strengths (Ms = 1.11 and 1.21), yielding very clear digital images of the flow. Early-time growth rate measurements obtained from these experiments are found to be in excellent agreement with incompressible linear stability theory (appropriately adjusted for a diffuse interface). Very good agreement is also found between the late-time amplitude measurements and the nonlinear models of Zhang & Sohn (1997) and Sadot et al. (1998). Comparison of images from the Ms = 1.11 and 1.21 sequences reveals a significant increase in the amount of turbulent mixing in the higher-Mach-number experiments, suggesting that a mixing transition has occurred.

The effect of viscosity on jet formation for bubbles collapsing near solid boundaries is studied numerically. A numerical technique is presented which allows the Navier–Stokes equations with free-surface boundary conditions to be solved accurately and efficiently. Good agreement is obtained between experimental data and numerical simulations for the collapse of large bubbles. However, the bubble rebound in our simulation is larger than that observed in laboratory experiments. This leads us to conclude that compressible and thermal effects should be taken into account to obtain a correct model of the rebound. A parametric study of the effect of viscosity on jet impact velocity is undertaken. The jet impact velocity is found to decrease as viscosity increases and above a certain threshold jet impact is impossible. We study how this critical Reynolds number depends on the initial radius and the initial distance from the wall. A simple scaling law is found to link this critical Reynolds number to the other non-dimensional parameters of the problem.

We explain the emergence and robustness of intense jets in highly turbulent planetary atmospheres, like that on Jupiter, by a general statistical mechanics approach to potential vorticity patches. The idea is that potential vorticity mixing leads to the formation of a steady organized coarse-grained flow, corresponding to the statistical equilibrium state. Our starting point is the quasi-geostrophic 1-1/2 layer model, and we consider the relevant limit of a small Rossby radius of deformation. Then narrow jets are obtained, in the sense that they scale like the radius of deformation. These jets can be either zonal, or closed into a ring bounding a vortex. Taking into account the beta-effect and a sublayer deep shear flow, we predict organization of the turbulent atmospheric layer into an oval-shaped vortex within a background shear. Such an isolated vortex is centred over an extremum of the equivalent topography, combining the interfacial geostrophic tilt due to the deep shear flow and the planetary beta-effect (the resulting effective beta-effect is locally quadratic). This prediction is in agreement with an analysis of wind data in major Jovian vortices (Great Red Spot and Oval BC).

The motion of a viscous incompressible liquid between two parallel disks, moving towards each other or in opposite directions, is considered. The description of possible conditions of motion is based on the exact solution of the Navier–Stokes equations. Both stationary and transient cases have been considered. The stability of the motion is analysed for different initial perturbations. Different types of stability were found according to whether the disks moved towards or away from each other.

In this paper, we describe a theoretical asymptotic model for large-amplitude travelling solitary waves in an axially symmetric rotating flow of an inviscid incompressible fluid confined in an infinitely long circular tube. By considering the special, but important, case when the upstream flow is close to that of uniform axial flow and uniform rotation, we are able to construct analytical solutions which describe solitary waves with ‘bubbles’, that is, recirculation zones with reversed flow, located on the axis of the tube. Such waves have amplitudes which slightly exceed the critical amplitude, where there is incipient flow reversal. The effect of the recirculation zone is to introduce into the governing amplitude equation an extra nonlinear term, which is proportional to the square of the difference between the wave amplitude and the critical amplitude. We consider in detail a special, but representative, class of upstream inflow conditions. We find that although the structure of the recirculation zone is universal, the presence of such solitary waves is quite sensitive to the actual upstream axial and rotational velocity shear configurations. Our results are compared with previous theories and observations, and related to the well-known phenomenon of vortex breakdown.

The dynamics of buoyant gravity currents in a rotating reference frame is a classical problem relevant to geophysical applications such as river water entering the ocean. However, existing scaling theories are limited to currents propagating along a vertical wall, a situation almost never realized in the ocean. A scaling theory is proposed for the structure (width and depth), nose speed and flow field characteristics of buoyant gravity currents over a sloping bottom as functions of the gravity current transport Q, density anomaly g′, Coriolis frequency f, and bottom slope α. The nose propagation speed is cp ∼ cw/ (1 + cw/cα) and the width of the buoyant gravity current is Wp ∼ cw/ f(1 + cw/cα), where cw = (2Qg′ f)1/4 is the nose propagation speed in the vertical wall limit (steep bottom slope) and cα = αg/f is the nose propagation speed in the slope-controlled limit (small bottom slope). The key non-dimensional parameter is cw/cα, which indicates whether the bottom slope is steep enough to be considered a vertical wall (cw/cα → 0) or approaches the slope-controlled limit (cw/cα → ∞). The scaling theory compares well against a new set of laboratory experiments which span steep to gentle bottom slopes (cw/cα = 0.11–13.1). Additionally, previous laboratory and numerical model results are reanalysed and shown to support the proposed scaling theory.

The behaviour of an insulating particle suspended in a liquid metal and subject to the influence of locally uniform electric and magnetic fields (E, B) is considered. The electric field drives a current J which is perturbed by the presence of the particle, and the resulting Lorentz force drives a flow. It is assumed that both the Reynolds number and the Hartmann number based on particle size are small. If the particle is fixed, it experiences a force and couple that are each bilinear in J and B; if it is freely suspended, then it moves with translational velocity U and angular velocity Ω each similarly bilinear in J and B. The general form of these bilinear relationships is determined, with particular attention to three types of particle symmetry: (i) isotropy; (ii) axisymmetry; and (iii) orthotropy.

It is well-known that magnetohydrodynamic (MHD) flows behave differently from conventional fluid flows in two ways: the magnetic field makes the flow field anisotropic in the sense that it becomes independent of the coordinate parallel to the field; and the flow of liquid across the field lines induces an electric current, leading to ohmic damping. In this paper, an experimental study is presented of the long-time decay of an initially three-dimensional flow structure subject to a steady magnetic field, when the ratio of the electromagnetic Lorentz forces to the nonlinear inertial forces, quantified by the magnetic interaction parameter, N0, takes large as well as moderate values. This investigation is markedly different from previous studies on quasi-two-dimensional MHD flows in thin layers of conducting fluids, where only Hartmann layer friction held the key to the dissipation of the flow.

The initial ‘linear’ phase of decay of an MHD flow, characterized by dominant Lorentz forces and modelled extensively in the literature, has been observed for the first time in a laboratory experiment. Further, when N0 is large compared to unity, a distinct regime of decay of a vortex follows this linear phase. This interesting trend can be explained in terms of the behaviour of the ratio of the actual magnitudes of the Lorentz to the nonlinear inertial forces – the true interaction parameter – which decreases to a constant of order unity towards the end of the linear phase of decay, and remains invariant during a subsequent ‘nonlinear’ phase.

In the present study the existence of multiple three-dimensional double-diffusive flow patterns in a horizontal rectangular porous cavity of a square cross-section, having horizontal aspect ratios Ax = Ay = 2 is investigated numerically. Opposing vertical gradients of temperature and concentration are applied between the two horizontal walls of the cavity, where the solute gradient is destabilizing against a stabilizing temperature gradient. All vertical walls are considered to be impermeable and adiabatic. The Brinkman and Forchheimer terms are included in the momentum equations where the convective terms are retained. The effect of the buoyancy ratio, N, thermal Rayleigh number, RaT and Lewis number, Le, on the formation of multiple flow patterns is investigated over a wide range of parameters. Altogether 36 symmetric flow structures have been identified when each of the parameters N, RaT, and Le is varied independently, keeping the others as constants. The results of the calculations are presented in terms of the average Sherwood number curves consisting of different solution branches, where transitions between the branches are indicated. The flow patterns are classified according to their symmetry properties and the type of symmetries broken or preserved are identified during the bifurcation processes.

Motivated by recent experimental results obtained in a low-Prandtl-number fluid (Jaletzky 1999), we study theoretically the rotating cylindrical annulus model with rigid boundary conditions. A boundary layer theory is presented which allows a systematic study of the linear properties of the system in the asymptotic regime of very fast rotation rates. It shows that the Stewartson layers have a (de)stabilizing influence at (high) low Prandtl numbers. In the weakly nonlinear regime and for low Prandtl numbers, a strong retrograde mean flow develops at quadratic order. The Poiseuille part of this mean flow is determined by an equation obtained by averaging of the Navier–Stokes equation. It thus gives rise to a new global-coupling term in the envelope equation describing modulated waves, which can be used for other systems. The influence of this global-coupling term on the sideband instabilities of the waves is studied. In the strongly nonlinear regime, the waves restabilize against these instabilities at small rotation rates, but they are destabilized by a short-wavelength mode at larger rotation rates. We also find an inversion in the dependence of the amplitude on the Rayleigh number at low Prandtl numbers and intermediate rotation rates.

We study the effect of a time-periodic, lateral acceleration on the two-dimensional flow of a fluid with a free surface subject to surface tension, confined between two plane, parallel walls under conditions of zero gravity. We assume that the velocity of each contact line is a prescribed, single-valued function of the dynamic contact angle between fluid and solid at the wall. We begin by obtaining analytical solutions for the small-amplitude standing waves that evolve when this function is linear, the fluid is inviscid and the lateral acceleration is sufficiently small. This leads to damping of the motion, unless either the contact angles are fixed or the contact lines are pinned. In these cases, we include the effect on the flow of the wall boundary layers, which are the other major sources of damping. We then consider the weakly nonlinear solution of the inviscid problem when the contact angle is almost constant and the external forcing is close to resonance. This solution indicates the possibility of a hysteretic response to changes in the forcing frequency. Finally, we examine numerical solutions of the fully nonlinear, inviscid problem using a desingularized integral equation technique. We find that periodic solutions, chaotic solutions and solutions where the topology of the fluid changes, either through self-intersection or pinch off, are all possible.

The linear stability of the Stokes layer generated by an oscillating flat plate is investigated using Floquet theory. The results obtained include the behaviour of the growth rate of the disturbances, part of the corresponding neutral curve and the structure of neutrally stable disturbances. Previously unknown properties of the growth rate cause the neutral curve to have a complicated geometry: the majority of the marginal curve is defined by waves propagating relative to the basic flow and the curve is smooth in character, but for certain very narrow bands of wavenumbers it was found that stationary modes are the first to become unstable. This phenomenon has the consequence that the underlying smooth neutral curve is punctuated by thin finger-like features. The structure of the eigenfunctions showed that the neutrally stable disturbances tend to grow most rapidly just after the wall velocity passes through zero.