A study has been made of how initially planar shocks in air propagate around 90° bends in channels of nearly rectangular cross-section. In shallow bends for which the radius of curvature R is much greater than the radius r of the channel, the shock recovers from a highly curved profile at the start of the bend to regain planarity towards the end of the bend. This occurs on account of the acceleration of the triple point across the channel following its interaction with the expansion waves generated at the convex wall. In sharp bends the shock profiles retain their pronounced curvature for some distance downstream of the bend.

At the start of a shallow bend (R/r ≈ 6) the shock at the concave wall, initial Mach number M0, accelerates to Mw = 1.15M0 and remains at this value until towards the end of the bend it begins to attenuate. At the convex wall, shocks of M0 > 1.7 attenuate to Mw = 0.7M0 and propagate at this value for some distance around the bend. In the early stages of a sharper bend (R/r ≈ 3) the shock at the concave wall strengthens to Mw = 1.3M0, remaining at this value for some distance downstream of the bend. At the convex wall the shock decelerates to 0.6M0.

Whitham's (1974) ray theory is shown to predict with reasonable accuracy the Mach numbers of the wall shocks at both surfaces for both bends tested and the range of incident shock velocities used, 1.2 < M0 < 3. The agreement between the theory and experimental results is particularly close for stronger shocks propagating along the inner bend. Predictions from 3-shock theory (Courant & Friedrichs 1948) of the Mach number at the outer wall are consistently higher than those from Whitham's analysis. In turn, the latter tends to slightly overestimate the strength of the wall shock.

A model is developed, based on an extension of Whitham's analysis, and is shown to predict the length of the Mach stem produced by shocks of M0 > 2 over the initial stages of the bend.

The search for a statistical stability criterion characterizing steady-state turbulent shear flows ‘has led us to the study of a function proportional to the ratio of the fluctuation dissipation rate and the dissipation rate of the mean. The first Euler–Lagrange equations for an upper bound of this function have optimal solutions with the observed scaling laws and an asymptotic velocity defect for turbulent channel flow. As in the case of maximum transport, the optimal solution for the model equations is a discrete spectrum of streamwise vortices. It is shown how these solutions can be brought closer to the realized flow with additional constraints on the smallest vortex.

We have conducted a detailed analysis of scaling for longitudinal and transverse velocity structure functions in a turbulent free shear flow. The free shear flow is generated via a mixing layer under varying conditions of upstream flow disturbances. Two velocity components are simultaneously measured with a pair of cross-wires at two spanwise locations, with varying positions of the second cross-wire, which allows us to study the statistics of two longitudinal and four transverse velocity increments. Spectra, probability density functions of the velocity increments, and scaling exponents are measured and discussed in relation to flow structures such as streamwise and spanwise vortices. Scaling exponents of the velocity structure functions are interpreted in the phenomenological framework of the hierarchical structure (HS) model of She & Leveque (Phys. Rev. Lett. vol. 72, 1994, p. 336). One HS parameter ($\beta$) specifying similarity between weak and strong vortices is shown to be universal for all structure functions, and another HS parameter ($\gamma$) related to the singularity index of the so-called most intermittent structures shows strong dependence on flow structures. The strongest intermittency occurs in the form of streamwise vortices. The results confirm that coherent small-scale flow structures are responsible for intermittency effects and anomalous scaling, and a complete set of measurements of longitudinal and transverse velocity variations are required to derive flow structural information.

A linear stability analysis of vapour–liquid counter flow in porous media is carried out. For the vapour-dominated basic state the development in time of both pressure and saturation disturbances is studied. The pressure field is shown to be asymptotically stable for all choices of thermal boundary conditions, excluding the insulating–insulating boundary condition for which it is neutrally stable. The saturation field is proven to be Lyapunov stable: the saturation disturbance remains bounded by an infinitesimal number at all times. For both vapour- and liquid-dominated basic states the direction of propagation of small saturation disturbances is determined. These results explain the formation of two-layer geothermal structures and why alternative structures cannot develop within homogeneous reservoirs.

A small eddy viscosity or mass diffusivity that varies with height has been found to have unexpected effects on the Kelvin–Helmholtz (KH) instability of a stably stratified shear layer near the neutral stability boundary. In particular, varying viscosity can increase the growth rate of the instability in contrast to the effect of uniform viscosity. Here, these results are extended to parameter ranges relevant in many geophysical and engineering contexts. We find that linearization of the viscous terms based on the assumption of weak viscosity/diffusivity is valid for non-dimensional values (inverse Reynolds number) up to ${\sim}10^{-2}$. Decreasing the Richardson number far below its critical value $1/4$ can change, or even reverse, the effects of eddy viscosity and diffusivity. A primary goal is to explain the unexpected destabilization by viscosity. Varying viscosity affects vorticity (and other fluid properties) in a manner identical to advection with an advecting velocity equal to minus the gradient of viscosity. Destabilization occurs when this viscous ‘advection’ reinforces the vorticity distribution of a growing mode.

An analysis is presented of the lateral solidification of a semi-infinite mushy region influenced by vertical, buoyancy-driven convection of the residual, interstitial melt. We consider a parameter regime in which the flow is steady on the time scale of the transient evolution of the mushy region. Our idealized model predicts patterns of macrosegregation consistent with earlier experimental studies and sheds light on the mechanisms involved.

The flow response of a rapidly rotating fluid-filled cube to low-amplitude librational forcing is investigated numerically. Librational forcing is the harmonic modulation of the mean rotation rate. The rotating cube supports inertial waves which may be excited by libration frequencies less than twice the rotation frequency. The response is comprised of two main components: resonant excitation of the inviscid inertial eigenmodes of the cube, and internal shear layers whose orientation is governed by the inviscid dispersion relation. The internal shear layers are driven by the fluxes in the forced boundary layers on walls orthogonal to the rotation axis and originate at the edges where these walls meet the walls parallel to the rotation axis, and are hence called edge beams. The relative contributions to the response from these components is obscured if the mean rotation period is not small enough compared to the viscous dissipation time, i.e. if the Ekman number is too large. We conduct simulations of the Navier–Stokes equations with no-slip boundary conditions using parameter values corresponding to a recent set of laboratory experiments, and reproduce the experimental observations and measurements. Then, we reduce the Ekman number by one and a half orders of magnitude, allowing for a better identification and quantification of the contributions to the response from the eigenmodes and the edge beams.

We present the results of an experimental investigation of the effect of a magnetic field on the stability of convection in a liquid metal. A rectangular container of gallium is subjected to a horizontal temperature gradient and a uniform magnetic field is applied separately in three directions. The magnetic field suppresses the oscillation most effectively when it is applied in the vertical direction and is least efficient when applied in the direction of the temperature gradient. The critical temperature difference required for the onset of oscillations is found to scale exponentially with the magnitude of the magnetic field for all three orientations. Comparisons are made with available theory and qualitative differences are discussed.

A non-Rayleigh mechanism of instability in horizontally homogeneous hydrodynamic systems consisting of two iimmiscible fluids and heated uniformly from above, i.e. ‘anticonvection’, was first investigated theoretically by Welander (1964) and more recently (and independently) by Gershuni & Zhukhovitsky (1980). This paper discusses the possibility of ‘anticonvection’ occurring in a water–air system, taking into account evaporation, stratification with respect to moisture, thermocapillary effects and the presence of a surface heat source at the water–air interface. For this purpose, a linear problem of stability is solved, in which the Rayleigh number analogues in each of the fluids are its eigenvalues in one case and the increments of disturbances in the other. It has been shown that taking into account evaporation and stratification with respect to moisture in the air reveals a new feedback in the system resulting in the disappearance of one of the previously known areas of anticonvection and the formation of new areas of instability. The mechanism of the oscillatory regime of the loss of stability in the system under study was found and considered in detail. Increments and wavelengths of the growing modes are calculated, and the possibility of experimental investigation of moist anticonvection in laboratory and field conditions is discussed.

This work treats the vertically stratified system of two homogeneous fluid layers confined between horizontal infinite rotating disks which rotate steadily about a common vertical axis. Allowance is made for uniform injection of fluid at either disk. With appropriate restrictions on disk rotational speeds and injection rates a flat interface is possible, and the problem admits similarity solutions to the Navier-Stokes equations of the Kármán-Bödewadt-Batchelor variety. This type of flow allows for a uniformly accessible surface of interphase mass and heat transfer at the two-fluid interface, and, with that as the primary motivation, the present work provides exploratory numerical solutions of the above equations for both corotation and counter-rotation combined with injection.

A linearized theory is given for the case of nearly rigid rotation, with explicit analytical results for the large-Reynolds-number boundary-layer limit. Also, we offer a theoretical discussion of the inviscid limit for arbitrary rotation and injection rates. Based on the type of Euler-cell solutions identified in previous work, we derive the remarkably simple formula \[ \rho_1\omega_1^2\cot^2\frac{\omega_1d_1}{V_1} = \rho_2\omega^2_2\cot^2\frac{\omega_2d_2}{V_2} \] connecting densities ρ, depths d, rotation speeds ω and injection velocities V.

Sample calculations and comparisons are given for property ratios typical of water-kerosene layers. In this case, the linearized theory works exceedingly well for corotation with small injectional Rossby numbers V/ωd. The simple inviscid theory cited above shows excellent agreement with the numerical computations for Reynolds numbers greater than 500 and for Rossby numbers > 1/π, corresponding to strong blowing in the inviscid regime. The larger-wavelength inviscid cell structure appears to provide the kind of stagnation-flow pattern essential to the application envisaged.

Multi-particle dispersion is studied using direct numerical simulations of temporally evolving mixing layers and planar jets for tetrahedra consisting of four fluid particles which are seeded in the turbulent regions or in the non-turbulent regions near the turbulent/non-turbulent interface (TNTI). The modified Richardson law for decaying turbulence is observed for particle pairs. The size dependence of the mean and relative motions of the entrained tetrahedra indicates that the characteristic length scale of the entrained lumps of fluid is approximately 10 times the Kolmogorov microscale. When the tetrahedra move within the TNTI layer they are flattened and elongated by vortex stretching at a deformation rate that is characterized by the Kolmogorov time scale. The shape evolutions of the tetrahedra show that in free-shear flows, thin-slab structures of advected scalars are generated within the TNTI layers.

We analyse the linear stability of stably stratified fluids whose density depends on two scalar fields where one of the scalar fields is unstably stratified and involves a settling velocity. Such conditions may be found, for example, in flows involving the transport of sediment in addition to heat or salt. A linear stability analysis for constant-gradient base states demonstrates that the settling velocity generates a phase shift between the perturbation fields of the two scalars, which gives rise to a novel, settling-driven instability mode. This instability mechanism favours the growth of waves that are inclined with respect to the horizontal. It is active for all density and diffusivity ratios, including for cases in which the two scalars diffuse at identical rates. If the scalars have unequal diffusivities, it competes with the elevator mode waves of the classical double-diffusive instability. We present detailed linear stability results as a function of the governing dimensionless parameters, including for lateral gradients of the base state density fields that result in predominantly horizontal intrusion instabilities. Highly resolved direct numerical simulation results serve to illustrate the nonlinear competition of the various instabilities for such flows in different parameter regimes.

A theory of a jet in a rotating, viscous fluid is developed. It is suggested that it may be related to a jet observed in an experiment with a rotating spherical shell of liquid and, in addition, may partly explain the existence of the subsurface equatorial current in the Pacific Ocean. The fundamental physical idea is that of a balance of vorticity brought into the jet by advection and diffused by friction. The necessary approximations are borrowed from boundary-layer theory. The linear case is solved completely and inferences about the non-linear jet are obtained by dimensional reasoning.

The resonant excitation of weakly nonlinear, standing edge waves of frequency ω and longshore wavenumber k by a normally incident, non-breaking gravity wave of frequency 2ω and shoreline amplitude a to on a bottom that descends smoothly from a shoreline depth of zero and slope σ to an offshore depth h∞ is calculated for ka [Lt ] σ [Lt ] kh∞ [Lt ] 1. The analysis generalizes hose of Guza & Bowen (1976) and Rockliff (1978), which assume uniform slope and perfect reflection, and culminates in a pair of evolution equations for the slowly varying, quadrature amplitudes (or, equivalently, amplitude and phase) of the edge wave. Weak, linear damping (which implies imperfect reflection) is incorporated, and the resulting fixed points and bifurcation points of the evolution equations are determined. It is shown that the solution for prescribed initial conditions must tend to one of the stable fixed points, which correspond to an edge-wave amplitude of either zero or O((σa/k)½), depending on whether the damping exceeds or is inferior to a certain critical value. The restriction σ [Lt ] kh∞ is relaxed for the special depth profile h/h∞ = 1 −exp (−σx/h∞), for which the inviscid, shallow-water equations admit exact solutions. These solutions serve to validate the asymptotic (σ/kh↓O) approximations for arbitrary depth profiles.

Vortex reconnection under Biot–Savart evolution is investigated geometrically and numerically using a tent model consisting of vortex filaments initially in the form of two tilted hyperbolic branches; the vortices are antiparallel at their points of nearest approach. It is shown that the tips of these vortices approach each other, accelerating as they do so to form a finite-time singularity at the apex of the tent. The minimum separation of the vortices and the maximum velocity and axial strain rate exhibit nearly self-similar Leray scaling, but the exponents of the velocity and strain rate deviate slightly from their respective self-similar values of $-1/2$ and $-1$; this deviation is associated with the appearance of distinct minima of curvature leading to cusp structures at the tips. The writhe and twist of each vortex are both zero at all times up to the instant of reconnection. By way of validation of the model, the structure of the eigenvalues and eigenvectors of the rate-of-strain tensor is investigated: it is shown that the second eigenvalue $\unicode[STIX]{x1D706}_{2}$ has dipole structure around the vortex filaments. At the tips, it is observed that $\unicode[STIX]{x1D706}_{2}$ is positive and the corresponding eigenvector is tangent to the filament, implying persistent stretching of the vortex.

Two-layer exchange flows are observed in the channels/straits connecting two water bodies of different densities. This study examines the nature of the barotropic forcing and its effect on the interfacial waves in two-layer exchange flows over a smooth/break underwater sill. Experiments were conducted with different initial conditions, distinguishing the case of hydrostatic disequilibrium and the case of a global pressure-balanced state. The experiments demonstrate that the baroclinic exchange flow is dominated by the barotropic-forcing-induced oscillations. A simplified barotropic model is developed to predict the period of the barotropic oscillation with an excellent agreement with experimental measurements. Detailed velocity and flow-rate measurements also indicate the importance of the barotropic forcing in exchange flows. The effect of the superimposed barotropic forcing on the interfacial wave characteristics is also investigated. Large two-dimensional surge-like structures are observed during the experiments, whose generation is shown to be related to the flow-rate oscillations. The length scales of these structures is comparable with the total water depth and is shown to increase with increasing Reynolds numbers.

Accelerations exceeding 20g in surface waves have been observed both in experiments and in numerically computed flows with a free surface. The present paper describes a family of analytic solutions which display such behaviour. They are expressible in parametric form as z = F sinh ω + iG cosh ω + γω + iH, where F, G and H are functions of the time t only, and γ is linear in t. ω is a complex parameter which is real at the free surface. The functions F(t) and G(t) satisfy two nonlinear, coupled ODEs, which can be solved numerically. Typically the solutions pass through an ‘inertial shock’, or singularity in the time, where the displacements vary as t2/3, the velocities as t1/3 and the accelerations as t-4/3. In this class of solution the free surface develops a cusp as t → ∞. In a special case, F and G vary as t4/7 and the cusp is reached in finite time. Gravity is neglected, but plays a part in setting up the initial conditions for the highly accelerated flow.

In future papers it will be shown that more general solutions exist in which the acceleration is momentarily large but bounded.

We present new insight into the classical problem of a uniform flow, linearly stratified in density, past an isolated three-dimensional obstacle. We demonstrate how, for a low-Froude-number obstacle, simple linear theory with a linearized boundary condition is capable of providing excellent quantitative agreement with laboratory measurements of the perturbation to the density field. It has long been known that such a flow may be divided into two regions, an essentially horizontal flow around the base of the obstacle and a wave-generating flow over the top of the obstacle, but until now the experimental diagnostics have not been available to test quantitatively the predicted features. We show that recognition of a small slope that develops across the obstacle in the surface separating these two regions is vital to rationalize experimental measurements with theoretical predictions. Utilizing the principle of stationary phase and causality arguments to modify the relationship between wavenumbers in the lee waves, linearized theory provides a detailed match in both the wave amplitude and structure to our experimental observations. Our results demonstrate that the structure of the lee waves is extremely sensitive to departures from horizontal flow, a detail that is likely to be important for a broad range of geophysical manifestations of these waves.

We consider the time domain problem of a floating body in two dimensions, constrained to move in heave and pitch only, subject to the linear equations of water waves. We show that using the acceleration potential, we can write the equations of motion as an abstract wave equation. From this we derive a generalized eigenfunction solution in which the time domain problem is solved using the frequency-domain solutions. We present numerical results for two simple cases and compare our results with an alternative time domain method.