Axisymmetric spherical Couette flow between two concentric differentially rotating spheres is computed numerically as an initial-value problem. The time-independent spherical Couette flows with zero, one and two Taylor vortices computed in our simulations are found to be reflection-symmetric about the equator despite the fact that our pseudospectral numerical method did not impose these properties. Our solutions are examined for self-consistency, compared with other numerical calculations, and tested against laboratory experiments. At present, the most precise laboratory measurements are those that measure Taylor-vortex size as a function of Reynolds number, and our agreement with these results is within a few per cent. We analyse our flows by plotting their meridional circulations, azimuthal angular velocities, and energy spectra. At Reynolds numbers just less than the critical value for the onset of Taylor vortices, we find that pinches develop in the flow in which the meridional velocity redistributes the angular momentum. Taylor vortices are easily differentiated from pinches because the fluid in a Taylor vortex is isolated from the rest of the fluid by a streamline that extends from the inner to the outer sphere, whereas the fluid in a pinch mixes with the rest of the flow.

We examine the transitions among the steady-state axisymmetric spherical Couette flows with zero, one, and two vortices per hemisphere. The steady flows are reflection symmetric with respect to the equator, but some of the transitions that we find break this symmetry. This is the first study to reproduce numerically the transitions to the one-vortex flow from the zero- and two-vortex flows. In our study, we use a numerical initial-value code to: (i) compute the bifurcation diagrams of the steady (stable and unstable) states, (ii) solve for the most unstable or least stable linear eigenmode and eigenvalue of a steady state, (iii) calculate the velocity field as a function of time during both the linear and nonlinear stages of the transitions, and (iv) determine the energy transfer mechanism into and out of the antireflection-symmetric components of the flow during the transitions.

Our study provides the explanation of the laboratory observation that some transitions occur only when the inner sphere of the Couette-flow apparatus is accelerated or decelerated quickly, whereas other transitions occur only when the acceleration is slow.

In this paper we consider theoretical and experimental aspects of axisymmetric, swirling flow which is generated in a column of liquid metal by a rotating magnetic field. Two cases are discussed, one in which there is no axial variation in the stirring force, and one where the body force is restricted to a relatively short length of the column. The latter case is of considerable practical interest in continuous casting.

One-dimensional stirring, where the swirl is independent of z and θ, is well understood. The magnetic body force is balanced by shear, all inertial forces being zero (except for the centripetal acceleration). However, in two-dimensional axisymmetric stirring, the axial variation in swirl drives a strong secondary poloidal flow. The principal local force balance is between the magnetic torque and inertia. The body force spins up the fluid as it passes through the forced region and the secondary flow sweeps this angular momentum into the unforced region. Consequently, the size and distribution of the swirl is controlled by the secondary flow.

The role of wall friction is considered and shown to control the length of the recirculating eddy. An approximate solution of the inviscid equations of motion, based on the angular momentum integral, is derived for the flow in the forced region. This is compared with the results of numerical experiments.

The analysis predicts that the swirl velocity scales on {B(σ/ρω)½}ωR, has a maximum at the bottom of the driven region, and penetrates an axial distance of the order ℝR away from the forced region. (For turbulent flow the Reynolds number ℝ must be based on an effective eddy viscosity.) All these features were reproduced experimentally.

The channel model of Stocker & Hutter (1986, 1987) is used to construct topographic wave solutions in a rectangular basin on the f-plane with variable but symmetric bathymetry. We show that in a narrow period band three types of eigenmodes can be discerned which exhibit local, midscale and global structure, respectively. Wave motion can be trapped either at the long sides of the elongated basin (channel mode) or at the ends of it (bay mode) or alternatively, a basinwide phase rotation is observed (Ball mode). The new bay modes are explained as resonances of topographic wave reflection in a semi-infinite channel. The influence of the variation of the aspect ratio of the rectangle and the topography parameter on the wave periods is also investigated.

Turbulent flows subjected to rotation display vortices parallel to the rotation axis and exhibiting a long timescale compared to the turbulent turnover time and the rotation period. A similar flow pattern is observed arising from the thermal instability in a rotating fluid. We demonstrate the analogy between turbulence and thermal convection in a rotating fluid. A basic quasi-geostrophic turbulent flow is considered which is forced at the bottom of the layer by a stochastic component of velocity parallel to the rotation axis. The turbulent basic state has no mean flow and the gradient along the rotation axis of the turbulent kinetic energy −∂z〈ω2〉 is analogous to the mean temperature profile in thermal convection. The linear perturbation equations of this basic turbulent state are given, where the thermal diffusion equation is replaced by the turbulent kinetic energy equation. Using a simple closure of this equation the model demonstrates the occurrence of an instability when the Reynolds number exceeds a critical value. Marginal stability curves are deduced by numerical integration of the perturbation equations. The results show order-of-magnitude agreement with laboratory experiments.

The wake behind a circularly capped bubble rising in fluids of different viscosity has been experimentally investigated using aspirin powder for flow visualization and high-speed photography synchronized with pressure-time measurements to measure the pressure field. The bubble plus its primary wake with a cusped tail is observed to contain symmetric pressure minima within the primary wake. Adjacent to the boundary wake is a free shear layer which contains large-scale vortices generated near the bubble rim that remain essentially stationary to an observer in the laboratory reference frame. The change in wake geometry and the transition to an ellipsoidal bubble shape as fluid viscosity increases is documented.

The airfoil shape of the boundary of the circularly capped bubble and its closed primary wake is modelled using a Joukowski transformation in which the Joukowski constant is adjusted to match the experimental and potential-flow pressures along the bubble cap. The model successfully predicts the frontal pressure field, and the wake size and shape. The Davies & Taylor bubble-cap boundary condition is also verified.

An explicit representation of an analytical solution to the problem of decay of a plane shock wave of arbitrary strength is proposed. The solution satisfies the basic equations exactly. The approximation lies in the (approximate) satisfaction of two of the Rankine-Hugoniot conditions. The error incurred is shown to be very small even for strong shocks. This solution analyses the interaction of a shock of arbitrary strength with a centred simple wave overtaking it, and describes a complete history of decay with a remarkable accuracy even for strong shocks. For a weak shock, the limiting law of motion obtained from the solution is shown to be in complete agreement with the Friedrichs theory. The propagation law of the non-uniform shock wave is determined, and the equations for shock and particle paths in the (x, t)-plane are obtained. The analytic solution presented here is uniformly valid for the entire flow field behind the decaying shock wave.

We consider the steady self-propagation with respect to the fluid at infinity of two equal symmetrically shaped vortices in a compressible fluid. Each vortex core is modelled by a region of stagnant constant-pressure fluid bounded by closed constant-pressure, constant-speed streamlines of unknown shape. The external flow is assumed to be irrotational inviscid isentropic flow of a perfect gas. The flow is therefore shock free but may be locally supersonic. The nonlinear free-boundary problem for the vortex-pair flow is formulated in the hodograph plane of compressible-flow theory, and a numerical solution method based on finite differences is described. Specific results are presented for a range of parameters which control the flow, namely the Mach number of the pair translational motion and the fluid speed on each vortex bounding streamline. Perturbation-theory predictions are developed, valid for vortices of small core radius when the pair Mach number is much less than unity. These are in good agreement with the hodograph-plane calculations. The numerical and the perturbation-theory results together confirm the recently discovered (Barsony-Nagy, Er-El & Yungster 1987) existence of continuous shock-free transonic compressible flows with embedded vortices. For the vortex-pair geometry studied, solution branches corresponding to physically acceptable flows that could be calculated using the present hodograph-plane numerical method were found to be terminated when either the flow on the streamline of symmetry separating the vortiqes tends to become superonic or when limiting lines appear in the hodograph plane giving a locally multivalued mapping to the physical plane.

Heat-transport measurements are reported for thermal convection in a rectangular box of aspect’ ratio 10 x 5. Results are presented for Rayleigh numbers up to 35Rc, Prandtl numbers between 2 and 20, and wavenumbers between 0.6 and 1.0kc, where Rc and kc are the critical Rayleigh number and wavenumber for the onset of convection in a layer of infinite lateral extent. The measurements are in good agreement with a phenomenological model which combines the calculations of Nusselt number, as a function of Rayleigh number and roll wavenumber for two-dimensional convection in an infinite layer, with a nonlinear amplitude-equation model developed to account for sidewell attenuation. The appearance of bimodal convection increases the heat transport above that expected for simple parallel-roll convection.

The result of a linear stability calculation of solitary waves which propagate steadily along the free surface of a liquid layer of constant depth is examined numerically by employing a time-stepping scheme based on a boundary-integral method. The initial’ growth rate that is found for sufficiently small perturbations agrees well with the growth rate expected from the linear stability calculation. In calculating the later ‘nonlinear’ stage of the instability, it is found that two distinct types of long-time evolution are possible. These depend only on the sign of the unstable normal-mode perturbation that is superimposed initially on the steady wave. The growth of the perturbation ultimately leads to breaking for one sign. Unexpectedly, for the opposite sign, there is a monotonic decrease in the total height of the wave. In this latter case there is a smooth evolution to a stable solitary wave of lesser amplitude but very nearly the same energy.

Intended as a contribution towards understanding the multiple processes entailed in the development of coastal sand bars due to wave action, this theoretical and experimental study deals with the Bragg reflection of long-crested surface waves in a water channel whose bed is corrugated sinusoidally. The present findings complement and in a few respects improve upon those in previous investigations, particularly Davies & Heathershaw (1984).

In §2 a linearized theory is presented, being directed to the elucidation of experimental situations where monochromatic waves propagate into a channel with a limited stretch of corrugations on its bed and an imperfectly absorbing beach at its far end. Allowance is made fully for dispersive effects (§2.2) and approximately for small frictional effects (§2.3). Points of interpretation (§2.4) include accounts of degenerate but non-trivial solutions that apply at frequencies terminating the stopping band, wherein the spatial wavefield has an exponential envelope. The experimental results presented in §4 derive from measurements of the wavefield over a stretch of 24 corrugations, at various frequencies both inside and outside the stopping band. Quantitative comparisons (§4.2 and 4.3) demonstrate close agreements with the theory.

A jet is a stream of one fluid entering another at high speed. In the simplest classical model of jet flow, the geometry is two-dimensional, gravity and viscosity are ignored, the moving fluid is a liquid, and the stationary fluid is a gas whose influence is assumed negligible. The description of this idealized flow can be reduced to a problem of complex analysis, but, except for very simple nozzle geometries, that problem cannot be solved analytically. This paper presents an efficient procedure for solving the jet problem numerically in the case of an arbitrary polygonal nozzle.

The radial dynamics of a spherical bubble in a compressible liquid is studied by means of a rigorous singular-perturbation method to second order in the bubble-wall Mach number. The results of Part 1 (Prosperetti & Lezzi, 1986) are recovered at orders zero and one. At second order the ordinary inner and outer structure of the solution proves inadequate to correctly describe the fields and it is necessary to introduce an intermediate region the characteristic length of which is the geometric mean of the inner and outer lengthscales. The degree of indeterminacy for the radial equation of motion found at first order is significantly increased by going to second order. As in Part 1 we examine several of the possible forms of this equation by comparison with results obtained from the numerical integration of the complete partial-differential-equation formulation. Expressions and results for the pressure and velocity fields in the liquid are also reported.

We consider the problem of global stability of the rigid rotation of two fluids. The realized interfacial configurations minimize a potential. We derive the most general form of the potential in which the working of the contact line may be expressed as a potential. The resulting variational problem for the interfacial potential is solved when the contact-line conditions are prescribed and for coating flows in which the interface makes a tangent contact with the wetted rod. In the former case, good agreement with experiments is obtained except near lines of contact. This shows that a spinning rod interfacial tensiometer is viable. In the latter case of coating flow, we get good agreement with experiments when the effects of gravity are not too large. The problem of bifurcation of coating flow is discussed qualitatively and some experimental results are given. We show how bifurcating sequences fit well into our qualitative description of the solution which must minimize the interfacial potential as the angular velocity is increased. The last bifurcations lead to pendant drops on a rotating ‘ceiling’ under the influence of centripetal forces which replace gravity. The dynamics of rollers of oil in water, or part in water and part in air, are explained in terms of the wavelength dependence of rotating drops.

The effect of vertical throughflow on the onset of convection in a fluid layer, between permeable horizontal boundaries, when heated uniformly from below, is re-examined analytically. It is shown that when the Péclet number Q is large in magnitude, the critical Rayleigh number Rc is proportional to Qn, where n = 0, 1, 2, 3 or 4, with a coefficient depending on the Prandtl number P, according to the types of boundaries. When the upper and lower boundaries are of different types, the effect of a small amount of throughflow in one direction is to decrease Rc. This is so when the throughflow is away from the more restrictive boundary. Contributions arise from the curvature of the basic temperature profile, and from the vertical transport of perturbation velocity and perturbation temperature. The decrease in Rc is small if P ∼ 1 but can be of significant size if P [Lt ] 1 or P [Gt ] 1.

The hydrodynamic stability of a rapidly elongating, viscous liquid jet such as obtained in shaped charges is presented. The flow field depends on three characteristic timescales associated with the growth of perturbations (due essentially to the effect of the surface tension), the elongation of the jet, and the inward diffusion of vorticity from the free surface, respectively. The latter process introduces a time lag resulting in the current values of the free-surface perturbation and its time derivative being a function of their past history. Solutions of the integro-differential equation for the evolution of disturbances exhibit a novel dual role played by the viscosity: besides the traditional damping effect it is associated with a destabilizing mechanism in the elongating jet. The wavelength of maximum instability is also a function of time elapsed since the jet formation, longer wavelengths becoming dominant at later stages. Understanding of these instability processes can help in both promoting and delaying instability as required by specific applications.

The splitting of a single pressure discontinuity into a propagating two-wave system is studied for the case of saturated-liquid expansion (liquid-evaporation wave splitting) and vapour compression (vapour-condensation wave splitting). Experimental results from the Max-Planck-Institut für Strömungsforschung and from Rensselaer Polytechnic Institute show that splitting occurs in test fluids of large molar heat capacity, such as iso-octane (Cv0/R ≈ 37). Each of the two forms of splitting results in a single-phase forerunner wave carrying a pressure discontinuity followed by a phase-change wave, also with a pressure discontinuity. The thermodynamic state between the forerunner wave and the phase-change wave is metastable (supersaturated liquid or vapour). The waves are quantitatively described by systems of adiabats, e.g. shock adiabats. It appears that nucleation processes are predominantly homogeneous.

In vapour-compression shock-wave splitting, a combined wave (liquefaction shock) splits into discrete forerunner and condensation waves at a triple point, the intersection of a liquefaction shockfront, forerunner shock and condensation discontinuity: such a point occurs just at critical supersaturation (i.e. the Wilson-line state), where condensation is spontaneous and immediate. For shock waves that produce a metastable state of subcritical supersaturation, condensation is delayed, that is, the condensation discontinuity propagates more slowly; for a split-shock system, the condensation discontinuity propagates subsonically. The pressure amplitude of a real split-shock system is much larger than that predicted by an equilibrium model.

In liquid-evaporation wave splitting, the forerunner wave is an acoustic expansion wave and the second wave an evaporation wave with a propagation velocity approximately determined by the Chapman-Jouguet condition for deflagration. Such evaporation wavefronts are increasingly distinct as the temperature approaches the critical-point value. The evaporation rates across the wavefront are comparable to those found in vapour explosions.

Settling of a dilute, monodisperse suspension between two inclined, narrowly spaced parallel plates is considered. Effects of sediment motion are accounted for. Lubrication theory and a simplified model for the particle motion lead to a system of two coupled nonlinear hyperbolic equations for the evolution of the two interfaces between clear fluid, suspension and sediment. Two problems are solved: batch settling and the filling of a channel that initially contains clear fluid. In the batch-settling case, the sediment has no major qualitative effect on the motion. In the filling problem, however, effects of sediment are important.

A simple, direct method is presented for the calculation of mobility functions for the translational and rotational velocities and stresslets of two equal-sized spheres in unbounded low-Reynolds-number flow when the ambient velocity field is a superposition of a uniform stream, a vorticity and a rate-of-strain field. Our numerical procedure furnishes accurate values for touching spheres and coefficients for the near-field asymptotic expansions. The singular behaviour of the mobility functions is clarified. These results have been used to determine accurately the coefficient of the O(c2) term in the expression for the bulk stress in a suspension of spheres (6.95c2 instead of 7.6c2).

The sources of error associated with the use of particle-tracking techniques in the measurement of velocity and vorticity fields in moderately three-dimensional turbulent flows are analysed. The two dominant sources of error are the visualisation error, resulting from the limited resolution of the optical data acquisition system, and the sampling error, due to limited particle concentration. Their relative importance is discussed.

The performance of the interpolation methods used to translate the measurements from the positions of the particles to an arbitrary point is discussed, and a non-parametric algorithm is given to estimate the errors that arise, using only the available data. The smoothing of the results to produce flow maps of a given statistical significance is also discussed. Finally, the method is validated using simultaneous laser-Doppler velocity measurements.

The system is applied to measurements of the near wake of a circular cylinder. Velocity and vorticity maps are provided which throw light on the process by which the large eddies form and relax to their final equilibrium configurations.