The use of magnetic fields during the growth of semiconductor crystals from the melt in a Czochralski (CZ) crystal puller has shown promise in controlling the heat and mass transport to the growth interface. The magnetic field suppresses turbulence and thermal convection in the melt in which large thermal gradients are present, thus improving the quality of the crystal. In this paper, analytical solutions are presented for the isothermal melt motion and electric current density driven by the differential rotation of the crystal and crucible about their common vertical axis. There is an applied, non-uniform, axisymmetric magnetic field with only radial and axial components which are independent of the azimuthal coordinate. The melt motion with a uniform axial magnetic field represents a singular limit of the flow considered here: as the radial magnetic field component goes to zero, the radial and axial (meridional) velocity components decrease in magnitude by a factor of M-1, where M is the large Hartmann number. The uniform axial field is a singular limit because the centrifugal acceleration due to the azimuthal velocity is exactly perpendicular to the magnetic field. Since the radial isothermal motion near the growth interface controls the radial distributions of dopants and impurities in the crystals, a non-uniform axisymmetric magnetic field is better than the uniform axial field. In addition, the axisymmetric field avoids the detrimental deviations from axisymmetric heat and mass transport associated with a uniform transverse (horizontal) magnetic field.

Two classes of shaped fields are considered, with only one class leading to the presence of the large meridional flow driven by differential rotation. The small electrical conductivity of the crystal plays an important role in determining the behaviour of the melt's angular velocity, which is constant along each magnetic field line. Results for two simple field configurations are presented in order to illustrate the effect of the field configuration on the nature of the meridional circulation and the potential for flow tailoring with the shaped field.

Bubble plumes in a linearly stratified ambient fluid are studied. Four well-defined flow regions were observed: an upward-moving bubble core, an inner plume consisting of a mixture of bubbles and relatively dense fluid, an annular downdraught and beyond that a horizontal intrusion flow. Depending on the gas flow rate with respect to the stratification, three types of intrusions were documented. At large gas flow rates a single intrusion was observed. As the gas flow rate was decreased, the buoyancy flux was insufficient to carry the lower fluid to the surface and a stack of intrusions were formed. At very low gas flow rates the intrusions became unsteady. The transition between these three regimes was observed to occur at critical values of the parameters N3H4/(QBg), QBg/ (4πα2u3sH), and H/HT, where N is the buoyancy frequency, H is the water depth, HT is equal to H + HA, HA being the atmospheric pressure head, QB is the gas flow rate at the bottom, g the acceleration due to gravity, α the entrainment coefficient and us the differential between the bubble and the average water velocity commonly called the slip velocity. The height between intrusions was found to scale with the Ozmidov length (QBg/N3)¼, the plunge point entrainment with the inner plume volume flux ($(Q_0 g)^{\frac{3}{4}} N^{-\frac{5}{4}}$ and the radial distance to the plunge point with (Q0g/N3)¾, where Q0 is the gas flow rate at the free surface.

These results were used to construct a double annular plume model which was used to investigate the efficiency of conversion of the input bubble energy to potential energy of the stratification; the efficiency was found to first increase, reach a maximum, then decrease with decreasing gas flow rate. This agreed well with the results from the laboratory experiments.

A calculation is made of the steady profile adopted by a touching pair of vortex regions with equal and opposite vorticity in a bounded uniform stream. A family of possible solutions is deduced, depending upon the magnitude of a (non-dimensionalized) vorticity parameter. A similar calculation is carried out incorporating a flat plate normal to the stream at the upstream end of the vortex configuration. The requirement of tangential separation at the plate tip selects a unique value of the vorticity. It is found that, as the width of the plate is reduced in relation to that of the channel, the vortex profile asymptotically approaches one member of the above mentioned family. The asymptotic form of the flow in the vicinity of the plate is deduced for this case and compared with a previous calculation.

A description of the incipient bending of a round incompressible jet issuing into a weak crossflow is presented. Axial vorticity is shown to appear from the early stages of the jet evolution owing to the distortion and reorientation of the azimuthal vorticity, and it eventually dominates the flow around the jet and determines the shape of its cross-section.

The crossflow is considered weak enough for the distortion of the jet to occur downstream of its development region, where diffusion already influences the whole cross-section and the jet can be modelled as a point source of momentum. Axial pressure gradient and axial diffusion are negligible under these conditions, since the jet is a slender structure.

Sufficiently near the origin, a finite-length entraining wake is identified on the lee side of the jet, which gradually merges with the main core. At the same time, the cross-section begins to acquire a characteristic elongated shape, with the jet concentrating in a thin layer. Farther downstream the axial vorticity of the jet rearranges into a couple of large locally two-dimensional contrarotating vortices standing against the wind under the action of their own induced velocity, and a smaller vortex sheet coinciding with the distorted jet.

The ignition of the supersonic boundary layer flow of a combustible mixture over a flat plate is studied through both direct numerical integration and activation energy asymptotics. Since ignition can be induced through either internally generated viscous heating or heat transfer from a hot wall, analyses are conducted for both an adiabatic wall and an isothermal wall whose temperature can be either higher or lower than the maximum frozen temperature in the flow. The analyses provide a description of the flow structure under various ignition situations, especially the extent of flow non-similarity and the interaction between the inner reaction region and the outer frozen regions. Explicit expressions for the ignition distance are obtained for all ignition situations, and the corresponding effects of the physical parameters on the ignition delay are also assessed. Specifically, it is demonstrated that, for low free-stream Mach number M∞, the ignition distance increases linearly with M∞ because of the decreased residence time, and for high M∞ it decreases exponentially with M∞ because of viscous heating. Results from the asymptotic analyses are found to compare well with those obtained from the direct numerical integration.

A unified theory is developed which describes nonlinear evolution of surface gravity waves propagating over an uneven bottom in the case of two-dimensional incompressible and inviscid fluid of arbitrary depth. Under the assumptions that the bottom of the fluid has a slowly varying profile and the wave steepness is small, a system of approximate nonlinear evolution equations (NEEs) for the surface elevation and the horizontal component of surface velocity is derived on the basis of a systematic perturbation method with respect to the steepness parameter. A single NEE for the surface elevation is also presented. These equations are expressed in terms of original coordinate variables and therefore they have a direct relevance to physical systems. Since the formalism does not rely on the often used assumptions of shallow water and long waves, the NEEs obtained are uniformly valid from shallow water to deep water and have wide applications in various wave phenomena of physical and engineering importance. The shallow- and deep-water limits of the equations are discussed and the results are compared with existing theories. It is found that our theory includes as specific cases almost all approximate theories known at present.

We present optical shadowgraph flow visualization and heat transport measurements of Rayleigh–Bénard convection with rotation about a vertical axis. The fluid, water with Prandtl number 6.4, is confined in a cylindrical convection cell with radius-to-height ratio Γ = 1. For dimensionless rotation rates 150 < Ω < 8800, the onset of convection occurs at critical Rayleigh numbers Rc(Ω) much less than those predicted by linear stability analysis for a laterally infinite system and qualitatively consistent with finite-aspect-ratio, linear-stability calculations of Buell & Catton (1983). As in the calculations, the forward bifurcation at onset is to states of localized flow near the lateral walls with azimuthal periodicity of 3 < m < 8. These states precess in the rotating frame, contrary to the assumptions of Buell & Catton (1983) but in quantitative agreement with recent calculations of Goldstein et al. (1992), with a frequency that is finite at onset but goes to zero as Ω goes to zero. At Ω = 2145 we find primary and secondary stability boundaries for states with m = 4, 5, 6, and 7. Further, we show that at higher Rayleigh number, there is a transition to a vortex state where the vortices form with the symmetry of the existing azimuthal periodicity of the sidewall state. Aperiodic, time-dependent heat transport begins for Rayleigh numbers at or slightly above the first appearance of vortices. Visualization of the formation and interactions of thermal vortices is presented, and the behaviour of the Nusselt number at high Rayleigh numbers is discussed.

The mass transfer mechanism across a sheared air–water interface without bubble entrainment due to wave breaking was experimentally investigated in terms of the turbulence structure of the organized motions in the interfacial region in a wind-wave tank. The transfer velocity of the carbon dioxide (CO2) on the water side was measured through reaeration experiments of CO2, and the fluid velocities in the air and water flows were measured using both a hot-wire anemometer and a laser-Doppler velocimeter. The results show that the mass transfer across a sheared air–water interface is more intensively promoted in wind shear, compared to an unsheared interface. However, the effect of the wind shear on the mass transfer tends to saturate in the high-shear region in the present wind-wave tank, where the increasing rate of mass transfer velocity with the wind shear decreases rapidly. The effect of the wind shear on the mass transfer can be well explained on the basis of the turbulence structure near the air–water interface. That is, surface-renewal eddies are induced on the water side through the high wind shear on the air–water interface by the strong organized motion generated in the air flow above the interface, and the renewal eddies control the mass transfer across a sheared interface. The mass transfer velocity is correlated with the frequency of the appearance of the surface-renewal eddies, as it is in open-channel flows with unsheared interfaces, and it increases approximately in proportion to the root of the surface-renewal frequency. The surface-renewal frequency increases with increasing the wind shear, but for high shear the rate of increase slows. This results in the saturated effect of the wind shear on the mass transfer in the high-shear region in the present wind-wave tank. The mass transfer velocity can be well estimated by the surface-renewal eddy-cell model based on the concept of the time fraction when the surface renewal occurs.

The stability criterion for almost cylindrical dielectric liquid bridges subjected to axial electric fields in the presence of residual axial gravity is obtained. In its absence, a perfectly cylindrical equilibrium solution is allowed for all values of the relevant parameters, which are the slenderness of the liquid bridge, the electrical Bond number and the relative permittivity between the outer and inner media. This basic solution is unstable beyond a critical slenderness which varies with the electrical parameters (González et al. 1989). The destabilization takes place axisymmetrically. The inclusion of the gravitational Bond number as a new, small parameter may be treated by means of the Lyapunov-Schmidt Method, a well-known projection technique that gives the local bifurcation diagram relating the admissible equilibrium amplitudes for the liquid bridge and the aforementioned parameters. As in the absence of applied electric field, the gravitational Bond number breaks the pitchfork diagram into two isolated branches of axisymmetric equilibrium solutions. The stable one has a turning point whose location determines the new stability criterion. Quantitative results are presented after solving the resulting set of linear recursive problems by means of the method of lines.

Finite-amplitude bifurcation diagrams of axisymmetric liquid bridges anchored between two plane parallel electrodes subjected to a potential difference and in the presence of an axial gravity field are found by solving simultaneously the Laplace equation for the electric potential and the Young–Laplace equation for the interface by means of the Galerkin/finite element method. Results show the strong stabilizing effect of the electric field, which plays a role somewhat similar to the inverse of the slenderness. It is also shown that the electric field may determine whether the breaking of the liquid bridge leads to two equal or unequal drops. Finally, the sensitivity of liquid bridges to an axial gravity in the presence of the electric field is studied.

Experiments were performed in order to elucidate the effects of hydrodynamic interactions between two drops on their gravity-induced relative motion. The relative trajectories of two drops, their relative velocities, and the travel time for them to flow around each other were measured for different initial horizontal separations. Two size ratios and two viscosity ratios were investigated. Hydrodynamic interactions significantly reduce the relative velocity of two nearby drops and cause them to flow around each other with curved trajectories, resulting in a longer duration of the close encounter, compared with that for two non-interacting drops. These effects increase with decreasing drop separation, decreasing size ratio, and increasing viscosity ratio. Experimental results are in good agreement with theoretical predictions, except when the drops become sufficiently close that interface deformation occurs.

Wave-drift damping results from low-frequency, oscillatory- motions of a floating body, in the presence of an incident wave field. Previous works have analysed this effect in a quasi-steady manner, based on the rate of change of the added resistance in waves, with respect to a small steady forward velocity. In this paper the wave-drift damping coefficient is derived more directly, from a perturbation analysis where the low-frequency body oscillations are superposed on the diffraction field. Unlike the case of body oscillations in calm water, where the damping due to wave radiation is asymptotically small for low frequencies, the superposition of oscillatory motions on the diffraction field results in an order-one damping coefficient. All three degrees of freedom are considered in the horizontal plane. The resulting matrix of damping coefficients is derived from pressure integration on the body, and transformed in special cases to a far-field control surface.

An air jet emanating from a square conduit having an equivalent diameter of 4.34 cm and a centreline velocity of 4 m/s is forced using four resonantly driven piezoelectric actuators placed along the sides of the square exit. Excitation is effected via amplitude modulation of the resonant carrier waveform. The flow is normally receptive to time–harmonic excitation at the modulating frequency but not at the resonant frequency of the actuators. When the excitation amplitude is high enough, the excitation waveform is demodulated by a nonlinear process that is connected with the formation and coalescence of nominally spanwise vortices in the forced segments of the jet shear layer. As a result, the modulating and carrier wave trains undergo spatial amplification and attenuation, respectively, downstream of the exit plane. Strong instabilities of the jet column are excited when the jet is forced at phase relationships between actuators that correspond (to lowest order) to the azimuthal modes m = 0, ±1, ±2, and −1 of an axisymmetric flow. The streamwise velocity component is measured phase locked to the modulating signal in planes normal to the mean flow. Resonantly driving the actuators with different carrier amplitudes results in a distorted mean flow having a featureless spectrum that can be tailored to provide favourable conditions for the introduction and propagation of desirable low-frequency disturbances.

In this paper we consider the axisymmetric flow of a rotating stratified fluid into a point sink. Linear analysis of the initial value problem of flow of a linearly stratified fluid into a point sink that is suddenly switched on shows that a spatially variable selective withdrawal layer is established through the outward propagation of inertial shear waves. The amplitude of these waves decays with distance from the sink; the e-folding scale of a given mode is equal to the Rossby radius of that mode. As a consequence, the flow reaches an asymptotic state, dependent on viscosity and species diffusion, in which the withdrawal-layer structure only exists for distances less than the Rossby radius based on the wave speed of the lowest mode, R1. If the Prandtl number, Pr, is large, then the withdrawal layer slowly re-forms in a time that is O(δ2iκ-1), such that it extends out much farther to a distance that is O(R1Prδ2iδ-2e) rather than O(R1).

Because there is no azimuthal pressure gradient to balance the Coriolis force associated with the radial, sinkward flow, a strong swirling flow develops. Using scaling arguments, we conclude that this swirl causes the withdrawal-layer thickness to grow like $(ft)^{\frac{1}{3}}$, such that eventually there is no withdrawal layer anywhere in the flow domain. Scaling arguments also suggest that this thickening takes place in finite-size basins.

These analyses of swirl-induced thickening and diffusive thinning can be combined to yield a classification scheme that shows how different types of flows are possible depending on the relative sizes of a parameter J, which we define as fQ(Nhv)-1, E (the Ekman number fh2v-1), and Pr.

This work involves active control of fundamental two- and three-dimensional amplified modes in an axisymmetric jet by introducing localized acoustic disturbances produced by an azimuthal array of miniature speakers placed close to the jet lip on the exit face. The independent control of each speaker output allowed different azimuthal amplitude and phase distributions of periodic inputs. The types of inputs used in this study consisted of conditions to force helical mode pairs with the same frequency and equal but opposite azimuthal wavenumbers, m = ±1, separately, or with axisymmetric (m = 0) modes. Three forcing conditions were studied in detail. The first consisted of a weakly amplified helical mode pair which was essentially ‘superposed’ with the natural jet instability modes. This provided a reference to the second case which consisted of the same helical mode pairs along with an axisymmetric mode at the harmonic streamwise wavenumber. This combination led to the resonant growth of the otherwise weakly (linear) amplified subharmonic helical modes. A weakly nonlinear three-wave amplitude evolution equation with a coupling coefficient derived from the data was found to model the enhanced growth of the subharmonic helical modes well. The third case consisted of forcing only m = ±1 helical modes at a frequency which was close to the most amplified. This was compared to the results of Corke et al. (1991) who forced an axisymmetric mode at the same frequency and found it to lead to the enhanced growth of near-subharmonic modes, as well as numerous sum and difference modes. The helical modes had effects identical to the previous work and confirmed the resonant amplification of a near-subharmonic mode. The amplitude development was also well represented by the nonlinear amplitude equation, including the dependence of the streamwise amplification rate on the azimuthal change in the fundamental-mode initial amplitude. However, the coupling coefficient in this case was approximately one-third that with exact fundamental-subharmonic resonance. Finally we offer some explanation for the selection of the different mode frequencies in this case.

The boundary layer develops along a flat plate with a Reynolds number high enough to sustain turbulence and allow accurate experimental measurements, but low enough to allow a direct numerical simulation. A favourable pressure gradient just downstream of the trip (experiment) or inflow boundary (simulation) helps the turbulence to mature without unduly increasing the Reynolds number. The pressure gradient then reverses, and the β-parameter rises from −0.3 to +2. The wall-pressure distribution and Reynolds number of the simulation are matched to those of the experiment, as are the gross characteristics of the boundary layer at the inflow. This information would be sufficient to calculate the flow by another method. Extensive automation of the experiment allows a large measurement grid with long samples and frequent calibration of the hot wires. The simulation relies on the recent ‘fringe method’ with its numerical advantages and good inflow quality. After an inflow transient good agreement is observed; the differences, of up to 13%, are discussed. Moderate deviations from the law of the wall are found in the velocity profiles of the simulation. They are fully correlated with the pressure gradient, are in fair quantitative agreement with experimental results of Nagano, Tagawa & Tsuji. and are roughly the opposite of uncorrected mixing-length-model predictions. Large deviations from wall scaling are observed for other quantities, notably for the turbulence dissipation rate. The a1 structure parameter drops mildly in the upper layer with adverse pressure gradient.

The onset of thermal convection in the presence of an oscillatory, non-planar shear flow is investigated on a linear basis. For the case of planar oscillations, the basic shear has no effect upon the value of the critical Rayleigh number but does act as a pattern selection mechanism. For the non-planar case, when there are two horizontal components of the basic velocity, the same result is true if the components are either in phase or directly out of phase. For the general case, however, stabilization occurs because convection rolls experience the stabilizing effects of shear regardless of their orientation. The results are obtained both by expansion in terms of the amplitude of the oscillating flow and in terms of its frequency, assuming the frequency to be small. The degree of stabilization increases with the Prandtl number. Pattern selection still occurs with non-planar oscillations.

A series of experiments were conducted in water to study mixing in the field of a single, two-dimensional vortex. The experimental configuration is that of a laminar line vortex initiated along a diffusion layer between two streams of different scalar concentrations. Measurements of passive scalars in inert and chemically reactive environments were made using a planar laser-induced fluorescence technique. A fast acid/base isothermal reaction was utilized to highlight the molecular mixing. The experimental results show that the mixing enhancement in the presence of a vortex is linearly dependent on the vortex strength and the time elapsed since vortex initiation. In particular, the mixedness, defined as the spatially integrated second moment of concentration field in the vortex, and the spatially averaged scalar dissipation are found to follow this dependence. This variation is mainly attributed to the contact area generation along the diffusion layers between the two streams as a result of inviscid deformations in the vortical flow field. The results presented pertain to mixing in liquids and in the limit of high Schmidt numbers.

We investigate the role of buoyancy force on the generation and decay of random motion in a homogeneously stratified fluid by means of direct numerical simulations (DNS) of the dynamic and thermodynamic equations. The simulations start from a fluid which is at rest but has appreciable temperature fluctuations. Therefore the flow initially evolves by extracting energy from the potential energy field. Three free parameters, the Reynolds number Re, the Prandtl number Pr and the stratification number St, characterize the flow. Among these numbers the stratification number, St = (lT0/T′0) (dTR/dz), is the most crucial one for the investigated problem. Here T′0 and lT0 are the initial r.m.s. temperature and the initial integral temperature lengthscale, respectively, and dTR/dz is the background stratification. St is a measure of the strength of background-temperature gradient compared to the initial mean fluctuating temperature gradient in the fluid.

A critical stratification number of order one is found to separate an oscillating, non-turbulent flow from flow states which exhibit features of turbulence. When St > 1, the statistics reveal a nearly linear and strongly anisotropic flow as typical for gravity waves but the flow-field variables behave randomly. When St < 1, i.e. when the initial gradient of fluctuating temperature exceeds the gradient of its background value, the available potential energy is sufficient to create nonlinear motions which resemble turbulence in many aspects. The properties of such a flow are a transient state of enhanced stirring and mixing, enhanced rates of dissipation of temperature fluctuations, and a quick return to isotropy.

The stratification number is an easily measurable parameter in field experiments in the ocean as well as in the atmosphere. Therefore St may be a useful indicator of whether a flow regime contains sufficient potential energy to create turbulence.

In this paper we study the inviscid instability of a skewed compressible mixing layer between streams of different velocity magnitude and direction. The mean flow is governed by the three-dimensional laminar boundary-layer equations and can be reduced to a sum of a uniform flow and a two-dimensional shear flow. In the stability analysis, the amplification direction is assumed to be normal to the homogeneous direction of the mean flow. The results show that skewing enhances the instability by a factor of three for the incompressible mixing layer with velocity ratio 0.5 and uniform temperature. Under compressible conditions, skewing still increases the maximum amplification rate for a medium convective Mach number, but the enhancement is smaller. A scaling of the skewing effect is introduced which quantitatively explains the linear stability behaviour. Similarly, a suitably defined convective Mach number explains the compressibility effect.