The linear, weakly nonlinear and strongly nonlinear evolution of unstable waves in a geostrophic shear layer is examined. In all cases, the growth of initially small-amplitude waves in the periodic domain causes the shear layer to break up into a series of eddies or pools. Pooling tends to be associated with waves having a significant varicose structure. Although the linear instability sets the scale for the pooling, the wave growth and evolution at moderate and large amplitudes is due entirely to nonlinear dynamics. Weakly nonlinear theory provides a catastrophic time ts at which the wave amplitude is predicted to become infinite. This time gives a reasonable estimate of the time observed for pools to detach in numerical experiments with marginally unstable and rapidly growing waves.

Flow-induced morphological instability of a mushy layer Journal of Fluid Mechanics, vol. 391 (1999), pp. 337–357 By J. A. NEUFELD J. S. WETTLAUFER D. L. FELTHAMAND M. G. WORSTER

The development with time of the impulsively started laminar flow of a viscous fluid away from a stagnation point is investigated. A series expansion in time is formulated for the shear stress and displacement thickness. This series expansion is obtained from a numerical solution of the full Navier–Stokes equations, and 44 terms are computed for the shear-stress series. The series is analysed and series-improvement techniques are employed to improve its convergence properties. The final series that results converges even for infinite time, and acceptable agreement with the Proudman & Johnson calculations of shear stress for steady-state flow at a stagnation point is obtained. Only 17 terms in the displacement-thickness series are reported, owing to numerical difficulties which are considerably more of an obstacle than in the shear-stress calculation. However, it is observed that the displacement thickness grows exponentially with time. Acceptable agreement with calculations of Proudman & Johnson is obtained for small time. For dimensionless time greater than 2.5, it is concluded that not enough terms are known to extrapolate the displacement-thickness series further.

The transient adjustment process of a compressible fluid in a rapidly rotating pipe is studied. The system Ekman number E is small, and the assumptions of small Mach number and the heavy-gas limit (γ = 1.0) are invoked. Fluid motion is generated by imposing a step-change perturbation in the temperature at the pipe wall Tw. Comprehensive analytical solutions are obtained by deploying the matched asymptotic technique with proper timescales O(E−1/2) and O(E−1). These analytical solutions are shown to be consistent with corresponding full numerical solutions. The detailed profiles of major variables are delineated, and evolution of velocity and temperature fields is portrayed. At moderate times, the entire flow field can be divided into two regions. In the inner inviscid region, thermo-acoustic compression takes place, and the process is isothermal–isentropic with the angular momentum being conserved. In the outer viscous region, diffusion of angular momentum occurs. The principal dynamic mechanisms are discussed, and physical rationalizations are offered. The essential differences between the responses of a compressible and an incompressible fluid are highlighted.

The issue of stability of the analytically obtained flow is addressed by undertaking a formal stability analysis. It is illustrated that, within the range of parameters of present concern, the flow is stable when ε ∼ O(E).

We consider the motion of small-amplitude surface gravity waves over variable bathymetry. Although the governing equations of motion are linear, for general bathymetric variations they are non-separable and cannot be solved exactly. For slowly varying bathymetry, however, approximate solutions based on geometric (ray) techniques may be used. The ray equations are a set of coupled nonlinear ordinary differential equations with Hamiltonian form. It is argued that for general bathymetric variations, solutions to these equations - ray trajectories - should exhibit chaotic motion, i.e. extreme sensitivity to initial and environmental conditions. These ideas are illustrated using a simple model of bottom bathymetry, h(x,y) = h0(1 + εcos (2πx/L) cos (2πy/L)). The expectation of chaotic ray trajectories is confirmed via the construction of Poincaré sections and the calculation of Lyapunov exponents. The complexity of chaotic geometric wavefields is illustrated by considering the temporal evolution of (mostly) chaotic wavecrests. Some practical implications of chaotic ray trajectories are discussed.

We study unstable waves in gas–liquid two-layer channel flows driven by a pressure gradient, under stable stratification, not assumed to be set in motion impulsively. The basis of the study is direct numerical simulation (DNS) of the two-phase Navier–Stokes equations in two and three dimensions for moderately large Reynolds numbers, accompanied by a theoretical description of the dynamics in the linear regime (Orr–Sommerfeld–Squire equations). The results are compared and contrasted across a range of density ratios $r=\unicode[STIX]{x1D70C}_{liquid}/\unicode[STIX]{x1D70C}_{gas}$. Linear theory indicates that the growth rate of small-amplitude interfacial disturbances generally decreases with increasing $r$; at the same time, the cutoff wavenumbers in both streamwise and spanwise directions increase, leading to an ever-increasing range of unstable wavenumbers, albeit with diminished growth rates. The analysis also demonstrates that the most dangerous mode is two-dimensional in all cases considered. The results of a comparison between the DNS and linear theory demonstrate a consistency between the two approaches: as such, the route to a three-dimensional flow pattern is direct in these cases, i.e. through the strong influence of the linear instability. We also characterize the nonlinear behaviour of the system, and we establish that the disturbance vorticity field in two-dimensional systems is consistent with a mechanism proposed previously by Hinch (J. Fluid Mech., vol. 144, 1984, p. 463) for weakly inertial flows. A flow-pattern map constructed from two-dimensional numerical simulations is used to describe the various flow regimes observed as a function of density ratio, Reynolds number and Weber number. Corresponding simulations in three dimensions confirm that the flow-pattern map can be used to infer the fate of the interface there also, and show strong three-dimensionality in cases that exhibit violent behaviour in two dimensions, or otherwise the development of behaviour that is nearly two-dimensional behaviour possibly with the formation of a capillary ridge. The three-dimensional vorticity field is also analysed, thereby demonstrating how streamwise vorticity arises from the growth of otherwise two-dimensional modes.

The disturbance flow due to the motion of a small sphere parallel to the streamlines of an unbounded linear shear flow is evaluated at small Reynolds number using the method of matched expansions. Decaying laws are obtained for all velocity components in a far inviscid region and viscous wakes. The z component (in the direction of the shear-rate gradient) of the disturbance velocity is cylindrically symmetric in the inviscid region. It decays with the distance r from the sphere like r−5/3, while the y component (in the direction of vorticity) decays like r−4/3. The widths of two viscous wakes, located upstream and downstream of the sphere, grow with the longitudinal coordinate x as yw ~ zw ~ |x|1/3. The maximum x and z components of the velocity are located in the wake cores; they scale like |x|−2/3 and |x|−1 respectively. For two particles interacting through their outer regions, the migration velocity of each particle is the sum of the velocity of an isolated particle and of a disturbance velocity induced by the other one. Particles placed in the normal or transversal directions repel each other. When each particle is located in a wake of the other one, they may either attract or repel each other.

The assumptions of the shock-expansion method are re-examined. Although the shock-expansion method and Whitham's rule are used to treat two different classes of problems, certain similarities between these two methods are noted. It is suggested that a single procedure leads to solutions for the entire flow field for both classes of problems.

The gravitational settling of a homogeneous suspension of Brownian particles on an inclined plate is considered. The hindered settling towards the wall and the viscous, buoyancy-driven bulk motion of the sediment are considered assuming steady conditions and accounting for the effects of Brownian diffusion, shear-induced diffusion and migration of particles due to a gradient in shear stress. Generally, the results show the development of a sediment boundary layer where the settling towards the wall is balanced by Brownian diffusion at the beginning of the plate and by shear-induced diffusion further downstream. Compared to previous results in the literature, the present theory allows steady-state solutions for extended values of the plate inclination and particle volume fraction above the sediment; upon reconsidering the case with non-Brownian particles, a new similarity solution, with a stable shock in particle density, is developed.

To bridge quasi-geostrophic dynamics and its discrete representation by a series of piecewise constant potential vorticity (PV), the dispersion relation for the Rossby wave in the single-layer β-plane is compared with that for the normal mode of edge waves straddling an infinite series of PV discontinuities (‘PV staircase’). It is shown that the edge waves over evenly spaced, uniform-height PV steps converge to the Rossby wave on the β-plane as Δ → 0, L → 0, Δ/L = βeff (Δ, L and βeff are the step size, step separation and the effective β, respectively), whereas they reduce to the single-step edge wave in the short-wave limit. For sufficiently small step separations, the difference in the phase velocities of the edge wave and the Rossby wave scales as O(L2). Two effects of increasing L on the zonal propagation are identified: (i) increased phase and group velocities in the short-wave limit due to an increased zonal wind at the PV steps and (ii) decreased phase and group velocities in the long-wave limit due to a decreased effective meridional tilt of the mode. The reduced tilt also severely limits the meridional group propagation. The relationship between the edge wave mode and the finite-difference approximation to the Rossby wave is also discussed.

Long-term marginal stability of a new family of isolated oceanic vortices is analysed. Sign reversal of the radial gradient of the potential vorticity anomaly, as implied by the isolation requirement, leads to vortex unsteadiness but does not break the coherence of the vortex, which remains marginally stable even for high absolute Rossby numbers $Ro\simeq 0.8$. The marginally stable vortices are characterized by a zero amount of potential vorticity anomaly on every isopycnal. The marginally stable final state is an unsteady vortex whose inner one-signed potential vorticity anomaly experiences revolution, rotation, precession and nutation.

It is shown that the advection–diffusion equation for contaminant dispersion in water of variable depth admits of an inverse approach in which the stochastic mean concentration at a fixed sensitive location is calculated for arbitrary discharge sites. A ray method is used to obtain simple approximate solutions for the inverse problem when the sensitive location is situated at the shoreline. This makes explicit the considerable improvements in shoreline pollution levels which can be gained by siting effluent outfalls further away from the shore.

When a laser beam of high intensity interacts with a dense material, an ablation front appears in the high-temperature plasma resulting from the interaction. Such a front can be used to accelerate and compress the dense material. The dynamics of the ablation front is strongly coupled to that of the absorption front where the laser energy is absorbed. The present paper determines analytical solutions of the front internal structure in the fully compressible case.

A theoretical investigation of similarity solutions for interactive laminar boundary layers is presented. The questions of uniqueness and of the appearance of homogeneous eigensolutions are discussed. The similarity solutions yielding the asymptotic behaviour of the nonlinear triple-deck equations in the far field can be used either to improve the development of computational schemes or to check the accuracy of numerical results. A special similarity solution governed by a modified Falkner-Skan boundary-value problem determines the shape of a wall generating the largest possible deflection of a laminar boundary layer in supersonic flow if separation is to be avoided. Increasing the controlling parameter of this special pressure distribution (for both supersonic and subsonic flows) beyond a cutoff value leads to a global breakdown of the interacting laminar-boundary-layer approach which cannot be removed or avoided.

Measurements of the rapidly changing gaseous composition in engines at low speed can be made via narrow tubes which convey the gases to monitoring equipment in a less hostile environment. This paper quantifies the extent to which the tube smooths out any changes in concentration. Exact (and approximate) formulae are derived for the temporal variance as weighted double (and single) integrals of the steady flow properties along the tube. Such is the non-uniformity that typically the region near the engine contributes 100 times as much to the spreading as does the region near the monitoring equipment. The advantages of keeping the sampling tubes short and heated are made explicit.

An inviscid fluid in steady two-dimensional motion in a region bounded by a closed streamline must have constant vorticity. We solve here for some such flows where the boundary is in part free, the fluid velocity magnitude being constant on the free boundary. A trivial example of such a flow is a circular cylinder of fluid rotating about its axis as if rigid, for which the whole circular boundary is free, irrespective of its radius. We now ask what happens to that flow when it comes into contact with solid boundaries. There is no steady flow when the contact is with a finite segment of a single plane wall, but a unique solution exists when the rotating fluid mass is in contact with some concave boundaries. Computed results are obtained for vortices lying inside a parabolic dish, or in a corner between two planes.

Freely decaying turbulent flows in a stably stratified fluid are simulated with a pseudo-spectral numerical code solving the fully nonlinear Navier–Stokes equations under the Boussinesq approximation with periodic boundary conditions. The flow is decomposed into a turbulent field and a horizontal mean flow ū(z, t) defined as the average of the horizontal velocity component in a horizontal plane at height z and time t. Similarly, the density field is decomposed into a turbulent field and a (stable) mean density profile ρ(z, t) defined as the average of the density field in a horizontal plane at height z and time t. Attention is paid to the effect of the turbulent velocity field on an initial z-periodic horizontal mean flow (Simulation A) or an initial z-periodic perturbation of the mean density profile (Simulation B). Both A and B are performed under conditions of moderate and strong stratification and are compared to the non-stratified simulations.

Simulation A shows that the turbulence–mean flow interaction is strongly affected by the buoyancy forces. In the absence of a stratification, the mean flow perturbation decays rapidly due to the turbulent diffusion of momentum. When a moderate stratification is applied, the mean flow perturbation decays much more slowly whereas it oscillates and grows with time when the stratification is strong. These results are interpreted by defining a time-dependent eddy viscosity. Whereas the eddy viscosity coefficient has positive values in the non-stratified simulation, it is affected by the buoyancy forces and decreases after a period of order N−1. For large times, the eddy diffusivity oscillates and its time-averaged value over a few turnover timescales is positive but small when the stratification is moderate, and roughly zero when the stratification is strong. These results are interpreted by defining a time-dependent eddy viscosity. Whereas the eddy viscosity coefficient has positive values in the non-stratified simulation, it is affected by the buoyancy forces and decreases after a period of order N−1 in the stratified simulations (where N is the Brunt–Väisälä frequency associated with the background linear stratification). At large time, we find that the eddy viscosity remains roughly zero when the stratification is moderate, whereas it oscillates but remains persistently negative in the strongly stratified case, which causes the horizontal mean flow to accelerate.

We conclude that the presence of a stable stratification strongly affects the temporal behaviour of the mean quantities ū and ρ in turbulent flows and partly explains the formation of horizontal layers in stratified geofluids such as oceans and atmospheres.

For a reactive solute, with weak second-order recombination, an investigation is made of the near-source behaviour (where concentrations are high), and of the far field (where the recombination has an accumulative effect). Despite the loss of material and increased spread due to recombination, the far-field concentration distribution is shown to be nearly Gaussian. This permits a simplified (Gaussian) treatment of the chemical nonlinearity. Explicit solutions are given for the total amount of solute, variance and kurtosis for solutes with no first-order reactions.