Research Article
Plume generation in natural thermal convection at high Rayleigh and Prandtl numbers
 C. LITHGOWBERTELLONI, M. A. RICHARDS, C. P. CONRAD, R. W. GRIFFITHS

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 22 June 2001, pp. 121

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We study natural thermal convection of a fluid (corn syrup) with a large Prandtl number (103–107) and temperaturedependent viscosity. The experimental tank (1 × 1 × 0.3m) is heated from below with insulating top and side boundaries, so that the fluid experiences secular heating as experiments proceed. This setup allows a focused study of thermal plumes from the bottom boundary layer over a range of Rayleigh numbers relevant to convective plumes in the deep interior of the Earth's mantle. The effective value of Ra, based on the viscosity of the fluid at the interior temperature, varies from 105 at the beginning to almost 108 toward the end of the experiments. Thermals (plumes) from the lower boundary layer are trailed by continuous conduits with long residence times. Plumes dominate flow in the tank, although there is a weaker largescale circulation induced by material cooling at the imperfectly insulating top and sidewalls. At large Ra convection is extremely timedependent and exhibits episodic bursts of plumes, separated by periods of quiescence. This bursting behaviour probably results from the inability of the structure of the thermal boundary layer and its instabilities to keep pace with the rate of secular change in the value of Ra. The frequency of plumes increases and their size decreases with increasing Ra, and we characterize these changes via in situ thermocouple measurements, shadowgraph videos, and videos of liquid crystal films recorded during several experiments. A scaling analysis predicts observed changes in plume head and tail radii with increasing Ra. Since inertial effects are largely absent no transition to ‘hard’ thermal turbulence is observed, in contrast to a previous conclusion from numerical calculations at similar Rayleigh numbers. We suggest that bursting behaviour similar to that observed may occur in the Earth's mantle as it undergoes secular cooling on the billionyear time scale.
Modelling Rayleigh–Taylor instability of a sedimenting suspension of several thousand circular particles in a direct numerical simulation
 T. W. PAN, D. D. JOSEPH, R. GLOWINSKI

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 22 June 2001, pp. 2337

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In this paper we study the sedimentation of several thousand circular particles in two dimensions using the method of distributed Lagrange multipliers for solid–liquid flow. The simulation gives rise to fingering which resembles Rayleigh–Taylor instabilities. The waves have a welldefined wavelength and growth rate which can be modelled as a conventional Rayleigh–Taylor instability of heavy fluid above light. The heavy fluid is modelled as a composite solid–liquid fluid with an effective composite density and viscosity. Surface tension cannot enter this problem and the characteristic shortwave instability is regularized by the viscosity of the solid–liquid dispersion. The dynamics of the Rayleigh–Taylor instability are studied using viscous potential flow, generalizing work of Joseph, Belanger & Beavers (1999) to a rectangular domain bounded by solid walls; an exact solution is obtained.
Spatiotemporal behaviour in an enclosed corotating disk pair
 ANTHONY RANDRIAMAMPIANINA, ROLAND SCHIESTEL, MICHAEL WILSON

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 22 June 2001, pp. 3964

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We present a numerical investigation of the flow between corotating disks with a stationary outer casing – the enclosed corotating disk pair configuration. It is known that in such a geometry, axisymmetric and threedimensional flow regimes develop depending on the value of the rotation rate. The threedimensional flow is always unsteady flowing to its wavy structure in the radialtangential plane. Axisymmetric regimes exhibit first a pitchfork bifurcation, characterized by a symmetry breaking with respect to the interdisk midplane, before a Hopf bifurcation is established. The regime diagrams for these bifurcations are given in the (Re, G)plane, where Re(= Ωb2/ν) is the rotational Reynolds number and G(= s/(b−a)) is the gap ratio. For values of G smaller than a critical limit Gc ∼ 0.26, there exists a range of rotation rates where the motion becomes timedependent before bifurcating to a steady symmetry breaking regime. It is shown that for G [ges ] Gc the transition to unsteady threedimensional flow occurs after the pitchfork bifurcation, and the flow structure is characterized by a shiftandreflect symmetry. The transition to threedimensional flow is consistent with experimental observations made by Abrahamson et al. (1989) where multiple solutions develop (known as the intransitivity phenomenon) with the presence of quasiperiodic behaviour resulting from successive vortex pairings. On the other hand, for smaller values of gap ratio, the threedimensional flow shows a symmetry breaking. Finally, it is found that the variation of torque coefficient as a function of the rotation rate is the same for both the axisymmetric and threedimensional solutions.
Annular and spiral patterns in flows between rotating and stationary discs
 E. SERRE, E. CRESPO DEL ARCO, P. BONTOUX

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 22 June 2001, pp. 65100

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Different instabilities of the boundary layer flows that appear in the cavity between stationary and rotating discs are investigated using threedimensional direct numerical simulations. The influence of curvature and confinement is studied using two geometrical configurations: (i) a cylindrical cavity including the rotation axis and (ii) an annular cavity radially confined by a shaft and a shroud. The numerical computations are based on a pseudospectral Chebyshev–Fourier method for solving the incompressible Navier–Stokes equations written in primitive variables. The high level accuracy of the spectral methods is imperative for the investigation of such instability structures. The basic flow is steady and of the Batchelor type. At a critical rotation rate, stationary axisymmetric and/or threedimensional structures appear in the Bödewadt and Ekman layers while at higher rotation rates a second transition to unsteady flow is observed. All features of the transitions are documented. A comparison of the wavenumbers, frequencies, and phase velocities of the instabilities with available theoretical and experimental results shows that both type II (or A) and type I (or B) instabilities appear, depending on flow and geometric control parameters. Interesting patterns exhibiting the coexistence of circular and spiral waves are found under certain conditions.
Nonlinear stability of a fluidloaded elastic plate with mean flow
 N. PEAKE

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 22 June 2001, pp. 101118

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It has been known for some time that the unsteady interaction between a simple elastic plate and a mean flow has a number of interesting features, which include, but are not limited to, the existence of negativeenergy waves (NEWs) which are destabilized by the introduction of dashpot dissipation, and convective instabilities associated with the flow–surface interaction. In this paper we consider the nonlinear evolution of these two types of waves in uniform mean flow. It is shown that the NEW can become saturated at weakly nonlinear amplitude. For general parameter values this saturation can be achieved for wavenumber k corresponding to lowfrequency oscillations, but in the realistic case in which the coefficient of the nonlinear tension term (in our normalization proportional to the square of the solid–fluid density ratio) is large, saturation is achieved for all k in the NEW range. In both cases the nonlinearities act so as increase the restorative stiffness in the plate, the oscillation frequency of the dashpots driving the NEW instability decreases, and the system approaches a state of static deflection (in agreement with the results of the numerical simulations of Lucey et al. 1997). With regard to the marginal convective instability, we show that the wavetrain evolution is described by the defocusing form of the nonlinear Schrödinger (NLS) equation, suggesting (at least for wave trains with compact support) that in the longtime limit the marginal convective instability decays to zero. In contrast, expansion about a range of other points on the neutral curve yields the focusing form of the NLS equation, allowing the existence of isolated soliton solutions, whose amplitude is shown to be potentially significant for realistic parameter values. Moreover, when slow spanwise modulation is included, it turns out that even the marginal convective instability can exhibit solitarywave behaviour for modulation directions lying outside broad wedges about the flow direction.
Propagation of a curved weak shock
 A. MONICA, PHOOLAN PRASAD

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 22 June 2001, pp. 119151

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Propagation of a curved shock is governed by a system of shock ray equations which is coupled to an infinite system of transport equations along these rays. For a twodimensional weak shock, it has been suggested that this system can be approximated by a hyperbolic system of four partial differential equations in a ray coordinate system, which consists of two independent variables (ζ, t) where the curves t = constant give successive positions of the shock and ζ = constant give rays. The equations show that shock rays not only stretch longitudinally due to finite amplitude on a shock front but also turn due to a nonuniform distribution of the shock strength on it. These changes finally lead to a modification of the amplitude of the shock strength. Since discontinuities in the form of kinks appear on the shock, it is necessary to study the problem by using the correct conservation form of these equations. We use such a system of equations in conservation form to construct a totalvariationbounded finite difference scheme. The numerical solution captures converging shock fronts with a pair of kinks on them – the shock front emerges without the usual folds in the caustic region. The shock strength, even when the shock passes through the caustic region, remains so small that the smallamplitude theory remains valid. The shock strength ultimately decays with a welldefined geometrical shape of the shock front – a pair of kinks which separate a central disc from a pair of wings on the two sides. We also study the ultimate shape and decay of shocks of initially periodic shapes and plane shocks with a dent and a bulge.
Diffuseinterface modelling of thermocapillary flow instabilities in a HeleShaw cell
 M. VERSCHUEREN, F. N. VAN DE VOSSE, H. E. H. MEIJER

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 22 June 2001, pp. 153166

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In this paper we present the results of a diffuseinterface model for thermocapillary or Marangoni flow in a HeleShaw cell. We use a Galerkintype spectral element discretization, based on Gauss–Lobatto quadrature, for numerical implementation of the governing equations resulting from the diffuseinterface model. The results are compared to classical results for a linear and circular fixed interface. It is found that the diffuseinterface solution converges to the classical solution in the sharpinterface limit. The results are sufficiently accurate if the interfacial thickness is only small compared to the size of the thermocapillary boundary layer, even if the interfacial thickness used is much larger than the real interfacial thickness. We also consider freely movable interfaces with a temperature gradient perpendicular to the interface. It will be shown that this situation can lead to a destabilizing Marangoni convection.
The constantV vortex
 ALAN J. FALLER

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 22 June 2001, pp. 167180

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It has been found that the generation of swirl by a continuous rotary oscillation of a rightcircular cylinder partially filled with water can leave a vortex with a radially constant tangential velocity, V, i.e. ∂V/∂r = 0, excepting a small central core and the sidewall boundary layer. This vortex maintains ∂V/∂r = 0 during viscous decay by the turbulent bottom boundary layer, a fact that suggests that ∂V/∂r = 0 is a stable condition for a decaying vortex.
Theory shows that such a profile of V and its steady decay is possible only if the radial transport per unit length in the turbulent Bödewadt boundary layer is TB,t = AVr/2 where A ≈ 0.072 is a dimensionless constant found from the experiment. This model of turbulent transport is extended to a case with ∂V/∂r ≠ 0 by an analysis of vortex decay in an experiment started from solid rotation. For this case an additional term proportional to ∂V/∂r is added to the transport equation.
The degeneration of largescale interfacial gravity waves in lakes
 D. A. HORN, J. IMBERGER, G. N. IVEY

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 22 June 2001, pp. 181207

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Mechanisms for the degeneration of largescale interfacial gravity waves are identified for lakes in which the effects of the Earth's rotation can be neglected. By assuming a simple twolayer model and comparing the timescales over which each of these degeneration mechanisms act, regimes are defined in which particular processes are expected to dominate. The boundaries of these regimes are expressed in terms of two lengthscale ratios: the ratio of the amplitude of the initial wave to the depth of the thermocline, and the ratio of the depth of the thermocline to the overall depth of the lake. Comparison of the predictions of this timescale analysis with the results from both laboratory experiments and field observations confirms its applicability. The results suggest that, for small to medium sized lakes subject to a relatively uniform windstress, an important mechanism for the degeneration of largescale internal waves is the generation of solitons by nonlinear steepening. Since solitons are likely to break at the sloping boundaries, leading to localized turbulent mixing and enhanced dissipation, the transfer of energy from an initial basinscale seiche to shorter solitons has important implications for the lake ecology.
Shear layers driven by turbulent plumes
 A. B. D. WONG, R. W. GRIFFITHS, G. O. HUGHES

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 22 June 2001, pp. 209241

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A turbulent plume from a continuous source of buoyancy in a long tank is shown to generate a series of quasisteady counterflowing horizontal shear layers throughout the tank. Both the horizontal flow velocity and the depth of the shear layers are observed to decrease with distance above/below the plume outflow. The shear layers are supported by the stable density stratification produced by the plume and are superimposed on the vertical advection and entrainment inflow that make up the socalled ‘filling box’ circulation. Thus, at some depths, the surrounding water flows away from the plume instead of being entrained, although we see no evidence of ‘detrainment’ of dense plume water. Given the stratification produced by the plume at large times, the timescale for the velocity structure to adjust to changes in forcing is proportional to the time for long internal gravity waves to travel the length of the tank. The shear layers are interpreted in terms of internal normal modes that are excited by, and which in turn determine, the horizontal plume outflow. The sixth and seventh baroclinic modes typically dominate because at the level of the plume outflow their phase speed is approximately equal and opposite to the vertical advection in the ‘filling box’. Also, the approximate balance between phase speed and advection is found to hold throughout the tank, resulting in the observed quasisteady flow structure. Viscosity causes the horizontal velocity in the shear layers to decrease with distance above/below the plume outflow, and is thought to be responsible for a lowfrequency oscillation in the flow structure that is observed during experiments.
Instability of a gravitymodulated fluid layer with surface tension variation
 J. RAYMOND LEE SKARDA

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 22 June 2001, pp. 243271

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Gravity modulation of an unbounded fluid layer with surface tension variations along its free surface is investigated. The stability of such systems is often characterized in terms of the wavenumber, α and the Marangoni number, Ma. In (α, Ma) parameter space, modulation has a destabilizing effect on the unmodulated neutral stability curve for large Prandtl number, Pr, and small modulation frequency, Ω, while a stabilizing effect is observed for small Pr and large Ω. As Ω → ∞ the modulated neutral stability curves approach the unmodulated neutral stability curve. At certain values of Pr and Ω, multiple minima are observed and the neutral stability curves become highly distorted. Closed regions of subharmonic instability are also observed. In (1/Ω, g1Ra)space, where g1 is the relative modulation amplitude, and Ra is the Rayleigh number, alternating regions of synchronous and subharmonic instability separated by thin stable regions are observed. However, fundamental differences between the stability boundaries occur when comparing the modulated Marangoni–Bénard and Rayleigh–Bénard problems. Modulation amplitudes at which instability tongues occur are strongly influenced by Pr, while the fundamental instability region is weakly affected by Pr. For large modulation frequency and small amplitude, empirical relations are derived to determine modulation effects. A oneterm Galerkin approximation was also used to reduce the modulated Marangoni–Bénard problem to a Mathieu equation, allowing qualitative stability behaviour to be deduced from existing tables or charts, such as Strutt diagrams. In addition, this reduces the parameter dependence of the problem from seven transport parameters to three Mathieu parameters, analogous to parameter reductions of previous modulated Rayleigh–Bénard studies. Simple stability criteria, valid for small parameter values (amplitude and damping coefficients), were obtained from the oneterm equations using classical method of averaging results.
On the vortical structure in a plane impinging jet
 J. SAKAKIBARA, K. HISHIDA, W. R. C. PHILLIPS

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 22 June 2001, pp. 273300

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The vortical structure of a plane impinging jet is considered. The jet was locked both in phase and laterally in space, and time series digital particle image velocimetry measurements were made both of the jet exiting the nozzle and as it impinged on a perpendicular wall. Isovorticity and isoλ2 surfaces coupled with critical point theory were used to identify and clarify structure. The flow near the nozzle was much as observed in mixing layers, where the shear layer evolves into spanwise rollers, only here the rollers occurred symmetrically about the jet midplane. Accordingly the rollers were seen to depict spanwise perturbations with the wavelength of flutes at the nozzle edge and were connected, on the same side of the jet, with streamwise ‘successive ribs’ of the same wavelength. This wavelength was 0.71 of the distance between rollers and, contrary to some experiments in mixing layers, did not double when the rollers paired. Structures not reported previously but evident here with isovorticity, λ2 and critical point theory are ‘cross ribs’, which extend from the downstream side of each roller to its counterpart across the symmetry plane; their spanwise periodic spacing exceeds that of successive ribs by a factor of three. Cross ribs stretch because of the diverging flow as the rollers approach the wall and move apart, causing the vorticity within them to intensify. This process continues until the cross ribs reach the wall and merge with ‘wall ribs’. Wall ribs remain near the wall throughout the cycle and are composed of vorticity of the same sign as the cross ribs, but the intensity level of the vorticity within them is cyclic. Details of the expansion of fluid elements, evaluated from the rate of strain tensor, revealed that both cross and successive ribs align with the principal axis and that the vorticity comprising them is continuously amplified by stretching. It is further shown, by appeal to the production terms of the phaseaveraged vorticity equation, that wall ribs are sustained by merging and stretching rather than reorientation of vorticity. Moreover production of vorticity is a maximum when cross and wall ribs merge and is greatest near the symmetry plane of the jet. The demise of successive ribs on the other hand occurs away from the symmetry plane and would appear to be less important dynamically than cross ribs merging with wall ribs.
Interaction of water waves with threedimensional periodic topography
 R. PORTER, D. PORTER

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 22 June 2001, pp. 301335

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The scattering and trapping of water waves by threedimensional submerged topography, infinite and periodic in one horizontal coordinate and of finite extent in the other, is considered under the assumptions of linearized theory. The mildslope approximation is used to reduce the governing boundary value problem to one involving a form of the Helmholtz equation in which the coefficient depends on the topography and is therefore spatially varying.
Two problems are considered: the scattering by the topography of parallelcrested obliquely incident waves and the propagation of trapping modes along the periodic topography. Both problems are formulated in terms of ‘domain’ integral equations which are solved numerically.
Trapped waves are found to exist over any periodic topography which is ‘sufficiently’ elevated above the unperturbed bed level. In particular, every periodic topography wholly elevated above that level supports trapped waves. Fundamental differences are shown to exist between these trapped waves and the analogous Rayleigh–Bloch waves which exist on periodic gratings in acoustic theory.
Results computed for the scattering problem show that, remarkably, there exist zeros of transmission at discrete wavenumbers for any periodic bed elevation and for all incident wave angles. One implication of this property is that total reflection of an incident wave of a particular frequency will occur in a channel with a single symmetric elevation on the bed. The zeros of transmission in the scattering problem are shown to be related to the presence of a ‘nearly trapped’ mode in the corresponding homogeneous problem.
The scattering of waves by multiple rows of periodic topography is also considered and it is shown how Bragg resonance – wellestablished in scattering of waves by twodimensional ripple beds – occurs in modes other than the input mode.
Weakly nonlinear stability of viscous flow past a flexible surface
 V. SHANKAR, V. KUMARAN

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 22 June 2001, pp. 337354

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The weakly nonlinear stability of viscous fluid flow past a flexible surface is analysed in the limit of zero Reynolds number. The system consists of a Couette flow of a Newtonian fluid past a viscoelastic medium of nondimensional thickness H (the ratio of wall thickness to the fluid thickness), and viscosity ratio μr (ratio of the viscosities of wall and fluid media). The wall medium is bounded by the fluid at one surface and two different types of boundary conditions are considered at the other surface of the wall medium – for ‘grafted’ gels zero displacement conditions are applied while for ‘adsorbed’ gels the displacement normal to the surface is zero but the surface is permitted to move in the lateral direction. The linear stability analysis reveals that for grafted gels the most unstable modes have α ∼ O(1), while for adsorbed gels the most unstable modes have α → 0, where α is the wavenumber of the perturbations. The results from the weakly nonlinear analysis indicate that the nature of the bifurcation at the linear instability is qualitatively very different for grafted and absorbed gels. The bifurcation is always subcritical for the case of flow past grafted gels. It is found, however, that relatively weak but finiteamplitude disturbances do not significantly reduce the critical velocity required to destabilize the flow from the critical velocity predicted by the linear stability theory. For the case of adsorbed gels, it is found that a supercritical equilibrium state could exist in the limit of small wavenumber for a wide range of parameters μr and H, while the bifurcation becomes subcritical at larger values of the wavenumber and there is a transition from supercritical to subcritical bifurcation as the wavenumber is increased.
Transitions and instabilities of flow in a symmetric channel with a suddenly expanded and contracted part
 J. MIZUSHIMA, Y. SHIOTANI

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 22 June 2001, pp. 355369

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Transitions and instabilities of twodimensional flow in a symmetric channel with a suddenly expanded and contracted part are investigated numerically by three different methods, i.e. the time marching method for dynamical equations, the SOR iterative method and the finiteelement method for steadystate equations. Linear and weakly nonlinear stability theories are applied to the flow. The transitions are confirmed experimentally by flow visualizations. It is known that the flow is steady and symmetric at low Reynolds numbers, becomes asymmetric at a critical Reynolds number, regains the symmetry at another critical Reynolds number and becomes oscillatory at very large Reynolds numbers. Multiple stable steadystate solutions are found in some cases, which lead to a hysteresis. The critical conditions for the existence of the multiple stable steadystate solutions are determined numerically and compared with the results of the linear and weakly nonlinear stability analyses. An exchange of modes for oscillatory instabilities is found to occur in the flow as the aspect ratio, the ratio of the length of the expanded part to its width, is varied, and its relation with the impinging freeshearlayer instability (IFLSI) is discussed.
Stability of inviscid shear flow over a flexible boundary
 JOHN MILES

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 22 June 2001, pp. 371378

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The stability of an inviscid flow that comprises a thin shear layer and a uniform outer flow over a flexible boundary is investigated. It is shown that the flow is temporally unstable for all wavenumbers. This instability is either Kelvin–Helmholtzlike or induced by the phase shift across the critical layer. The threshold of absolute instability is determined in the form F = F∗(1 + Cεn) for ε [Lt ] 1, where F (a Froude number) and ε are, respectively, dimensionless measures of the flow speed and the shearlayer thickness, F∗ is the limiting value of F for a uniform flow, C < 0 and n = 1 in the absence (as for a brokenline velocity profile) of a phase shift across the critical layer, and C > 0 and n = 2/3 in the presence of such a phase shift. Explicit results are determined for an elastic plate (and, in an Appendix, for a membrane) with a brokenline, parabolic, or Blasius boundarylayer profile. The predicted threshold for the brokenline profile agrees with Lingwood & Peake's (1999) result for ε [Lt ] 1, but that for the Blasius profile contradicts their conclusion that the threshold for ε ↓ 0 is a ‘singular and unattainable limit’.
Equations relating structure functions of all orders
 REGINALD J. HILL

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 22 June 2001, pp. 379388

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Exact equations are given that relate velocity structure functions of arbitrary order with other statistics. ‘Exact’ means that no approximations are used except that the Navier–Stokes equation and incompressibility condition are assumed to be accurate. The exact equations are used to determine the structure function equations of all orders for locally homogeneous but anisotropic turbulence as well as for the locally isotropic case. The uses of these equations for investigating the approach to local homogeneity as well as to local isotropy and the balance of the equations and identification of scaling ranges are discussed. The implications for scaling exponents and investigation of intermittency are briefly discussed.
Scaledependent droplet clustering in turbulent clouds
 A. B. KOSTINSKI, R. A. SHAW

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 22 June 2001, pp. 389398

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The current understanding of fundamental processes in atmospheric clouds, such as nucleation, droplet growth, and the onset of precipitation (collision–coalescence), is based on the assumption that droplets in undiluted clouds are distributed in space in a perfectly random manner, i.e. droplet positions are independently distributed with uniform probability. We have analysed data from a homogeneous cloud core to test this assumption and gain an understanding of the nature of droplet transport. This is done by examining onedimensional cuts through clouds, using a theory originally developed for xray scattering by liquids, and obtaining statistics of droplet spacing. The data reveal droplet clustering even in cumulus cloud cores free of entrained ambient air. By relating the variance of droplet counts to the integral of the pair correlation function, we detect a systematic, scaledependent clustering signature. The extracted signal evolves from sub to superPoissonian as the length scale increases. The subPoisson tail observed below mmscales is a result of finite droplet size and instrument resolution. Drawing upon an analogy with the hardsphere potential from the theory of liquids, this subPoisson part of the signal can be effectively removed. The remaining part displays unambiguous clustering at mm and cmscales. Failure to detect this phenomenon until now is a result of the previously unappreciated cumulative nature, or ‘memory,’ of the common measures of droplet clustering.
The supercritical bore produced by a highspeed ship in a channel
 T. P. GOURLAY

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 22 June 2001, pp. 399409

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An experimental investigation is made into the various flow regimes accompanying a ship travelling in a channel at supercritical speeds. The phenomena of smooth solitons, broken solitons, bores, and steady supercritical flow are observed. We look at the conditions under which each phenomenon exists, and the depthbased Froude numbers at which the transitions occur. Special emphasis is placed on ship bores, and we put forward a simple theoretical model for predicting the form of the bores as well as the transition to steady supercritical flow.
Addendum
Schedule of International Conferences on Fluid Mechanics

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 22 June 2001, pp. 412413

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