Research Article
A stronginteraction theory for the motion of arbitrary threedimensional clusters of spherical particles at low Reynolds number
 Qaizar Hassonjee, Peter Ganatos, Robert Pfeffer

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 21 April 2006, pp. 137

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This paper contains an ‘exact’ solution for the hydrodynamic interaction of a threedimensional finite cluster at arbitrarily sized spherical particles at low Reynolds number. The theory developed is the most general solution to the problem of an assemblage of spheres in a threedimensional unbounded media. The boundarycollocation truncatedseries solution technique of Ganatos, Pfeffer & Weinbaum (1978) for treating planar symmetric Stokes flow problems has been extensively modified to treat the nonsymmetric multibody problem. The orthogonality properties of the eigenfunctions in the azimuthal direction are used to satisfy the noslip boundary conditions exactly on entire rings on the surface of each particle rather than just at discrete points.
Detailed comparisons with the exact bipolar solutions for two spheres show the present theory to be accurate to five significant figures in predicting the translational and angular velocity components of the particles at all orientations for interparticle gap widths as close as 0.1 particle diameter. Convergence of the results to the exact solution is rapid and systematic even for unequalsized spheres (a1/a2 = 2). Solutions are presented for several interesting and intriguing configurations involving three or more spherical particles settling freely under gravity in an unbounded fluid or in the presence of other rigidly held particles. Advantage of symmetry about the origin is taken for symmetric configurations to reduce the collocation matrix size by a factor of 64. Solutions for the force and torque on threedimensional clusters of up to 64 particles have been obtained, demonstrating the multiparticle interaction effects that arise which would not be present if only pair interactions of the particles were considered. The method has the advantage of yielding a rather simple expression for the fluid velocity field which is of significance in the treatment of convective heat and mass transport problems in multiparticle systems.
Hydromagnetic screw dynamo
 Alexander Ruzmaikin, Dmitry Sokoloff, Anvar Shukurov

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 21 April 2006, pp. 3956

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We solve the problem of magnetic field generation by a laminar flow of conducting fluid with helical (screwlike) streamlines for large magnetic Reynolds numbers, Rm. Asymptotic solutions are obtained with help of the singular perturbation theory. The generated field concentrates within cylindrical layers whose position, the magnetic field configuration and the growth rate are determined by the distribution of the angular, Ω, and longitudinal, Vz, velocities along the radius. The growth rate is proportional to Rm−½. When Ω and Vz are identically distributed along the radius, the asymptotic forms are of the WKB type; for different distributions, singularlayer asymptotics of the Prandtl type arise. The solutions are qualitatively different from those obtained for solidbody screw motion. The generation threshold strongly depends on the velocity profiles.
Numerical calculations of the primaryflow exchange process in the Taylor problem
 K. A. Cliffe

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 21 April 2006, pp. 5779

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Numerical methods are used to study the way in which the number of cells present in the Taylor experiment changes as the length of the comparatively short annulus varies. The structure of the solution surface is determined by following paths of singular points in a finiteelement discretization of the axisymmetric Navier–Stokes equations. The numerical results are compared with the experiments of Benjamin (1978b), Mullin (1982) and Mullin et al. (1982). The calculations are in agreement with the qualitative theory of Benjamin (1978a) and Schaeffer (1980) except that in the interaction involving four and sixcell flows, the numerical calculations indicate that the sixcell flow can become unstable owing to perturbations that are antisymmetric about the midplane.
Modifications to the wake of a wire across Poiseuille flow due to a unipolar space charge
 F. M. J. Mccluskey, P. Atten

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 21 April 2006, pp. 81104

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A wake behind a wire in a developed Poiseuille flow is examined with and without ionic injection into the liquid by the wire. Mean electric fields of up to 50 kV/cm between the wire and two plate electrodes on either side were used to bring about this injection. The resulting Coulomb force can modify the wake flow in two ways. When this force is weak, the injected ions are transported downstream in a thin charged wake, the only effect being that the deficit velocity is compensated over shorter distances. Once the Coulomb force is strong, there are two charged and turbulent plumelike structures going from the wire to the plates. These are perpendicular to the plates for zero forced flow and are pushed downstream to smaller angles as the forced flow is increased. The wake in this case is not present. Different experimental laws are given to characterize the different regimes and the transition between them.
Ordered motion in the turbulent boundary layer over wind waves
 H. Kawamura, Y. Toba

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 21 April 2006, pp. 105138

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The turbulent boundary layer over young wind waves (C/u* ∼ 1, where C is the phase speed of wind waves and u* is the friction velocity) has been investigated in a laboratory tank. Ordered motions have been found, and their structures studied in detail. Visualization of the outer boundary layer (0.4δ–1δ, where δ is the boundarylayer thickness) by paraffin mist has demonstrated the existence of a train of largescale ordered motions having a horizontal lengthscale that corresponds to the wavelength of the underlying wind waves. Hotwire measurements combined with the visualization have shown that the passage of the outer boundarylayer bulge is related to the occurrence of a lowspeed air mass, usually accompanied by an upward velocity to produce large Reynolds stress. In the vicinity of the wave surface (0–0.15δ), flow separation occurs over these wind waves. Instantaneous velocity shear measurements, using two hot wires 0.15 cm apart vertically, have detected a highshear layer at the edge of the separation bubbles. This highshear layer, the potential site for generating much turbulence, reattaches on the windward side of the preceding wind waves. A pressure rise and a shearstress spike, expected near the reattachment region, could be the mechanisms for supplying energy to the wind waves.
The bursting phenomena over wind waves have been examined in detail in the logarithmic boundary layer (0.15δ–0.3δ). The bursting phenomena are a major mechanism for producing Reynolds stress and have a specific relationship with the phase of the wind wave. To explain the bursting phenomena, two mechanisms (not present in the boundary layer over a flat plate) are proposed, involving airflow separation and the largescale ordered motions, respectively. The two mechanisms are a ‘big burst’ related to the discharge of a whole separation bubble, and a ‘small burst’ which is the upward bursting of a lowspeed air mass from the unstable separated shear layer into the ordered motions passing over a separation bubble.
Turbulence reduction by screens
 Johan Groth, Arne V. Johansson

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 21 April 2006, pp. 139155

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Turbulence suppression by use of screens was studied in a small wind tunnel especially designed and built for the purpose. Wide ranges of mesh sizes and wirediameter Reynolds numbers were covered in the present investigation, enabling the study of sub and supercritical screens under the same, wellcontrolled, flow conditions. For the latter type smallscale fluctuations, produced by the screen itself, interact with the incoming turbulence. In the immediate vicinity of the screen the turbulence was found to be highly anisotropic and the intensities were higher than on the upstream side. Downstream of a short initial decay region, where the intensities decrease rapidly, the return to isotropy was found to be much slower than for the unmanipulated turbulence. The latter was generated by a square rod grid, and was shown to become practically isotropic beyond a distance of roughly 20 mesh widths from the grid. The role of the turbulence scales for the overall reduction effectiveness, and for the optimization of screen combinations for application in lowturbulence wind tunnels was studied.
Considerations on the moving contactline singularity, with application to frictional drag on a slender drop
 P. A. Durbin

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 21 April 2006, pp. 157169

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It has previously been shown that the noslip boundary condition leads to a singularity at a moving contact line and that this forces one to admit some form of slip. Present considerations on the energetics of slip due to shear stress lead to a yield stress boundary condition. A model for the distortion of the liquid state near solid boundaries gives a physical basis for this boundary condition. The yield stress condition is illustrated by an analysis of a slender drop rolling down an incline. That analysis provides a formula for the frictional drag resisting the drop movement. With the present boundary condition the length of the slip region becomes a property of the fluid flow.
On direct methods in waterwave theory
 Jonathan J. Shields, William C. Webster

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 21 April 2006, pp. 171199

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Model equations for threedimensional, inviscid flow between two arbitrary, timevarying material surfaces are derived using a ‘direct’ or variational approach due to Kantorovich. This approach results in a hierarchy of approximate theories, each of a higher level of spatial approximation and complexity. It can be shown that the equations are equivalent in substance to ‘the theory of directed fluid sheets’ of Green & Naghdi (1974, 1976).
The theory can be used to study the propagation of long waves in water of finite depth and, as such, competes with theories derived using the classical Rayleigh–Boussinesq perturbation methods. In order to demonstrate that there is an advantage to the present approach, we compare predictions for steady, twodimensional waves over a horizontal bottom. Numerical solutions indicate that the direct theory converges more rapidly than the perturbation theories. Also, the equations of the higherorder direct theories contain singularities related to waves of limiting height, and indeed such waves can be predicted with relative accuracy. Finally, the range of applicability of the direct theory is far greater: waves as short as three times the water depth can be modelled. This is essentially a deepwater condition, well beyond the range of convergence of the Rayleigh–Boussinesq approach.
On the stability of laminar boundarylayer flow over a flat plate with a compliant surface
 P. K. Sen, D. S. Arora

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 21 April 2006, pp. 201240

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The problem has been examined using a kinematic model for wall pliability, wherein a kinematic postulation of the wall boundary conditions is made. A form of the normalized walldisplacement and its phase are used as additional parameters in an extended eigenvalue problem. Using this technique the entire gamut of possibilities regarding stability of flow past (normally) pliable walls can be examined, yet without recourse to any specific material properties for the wall. Rather, the results based on the kinematic model can be used to backcalculate the material properties corresponding to any chosen model for the dynamics of the wall. A sample back calculation is discussed herein for the Benjamin–Landahl wall model, and based on this some predictions are made regarding both stabilization of the flow and physical realizability of modes. It is believed that the kinematic model will prove useful in further understanding of the problem, and in the design of stabilizing coatings.
The results show that there are three important ‘mode classes’ (distinct from ‘modes’), namely the Tollmien–Schlichting (TS), resonant (R) and Kelvin–Helmholtz (KH). Whereas the TS and R mode classes broadly agree with modes bearing similar names as found by earlier workers, the present KH mode class is difficult to classify based on earlier work. Moreover, there are also important transitional mode classes in the regions of bifurcations of the regular mode classes.
Two important concepts evolve in connection with the TS and R mode classes, namely the existence of ‘stable pockets’ for the former and ‘unstable pockets’ for the latter. It is also confirmed herein that there are conflicting requirements on the damping d to stabilize TS and R modes. Considering these points it has been suggested that TS and R modes be avoided by keeping soft surfaces as compliant coatings. However, this in turn leads to instabilities from one of the transitional mode classes. It is also seen that a soft surface that is also marginally active (i.e. having a small negative value of d) could render even better stabilization.
The force exerted on a body in inviscid unsteady nonuniform rotational flow
 T. R. Auton, J. C. R. Hunt, M. Prud'Homme

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 21 April 2006, pp. 241257

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A general expression is derived for the fluid force on a body of simple shape moving with a velocity v through inviscid fluid in which there is an unsteady nonuniform rotational velocity field u0(x,t) in two or three dimensions. It is assumed that the radius is small compared with the scale over which the strain rate changes, though for the sphere it is also assumed that the changes in the ambient velocity field over the scale of the sphere are small compared with the velocity of the body relative to the flow. Given these approximations it is shown that the effects of the rate of change of the vorticity of the ambient flow is of second order and can be neglected. However the rate of change of the irrotational straining motion is included in the analysis. It is shown that the inertial forces derived by many authors for irrotational flow can be simply added to a generalization of the lift force derived by Auton (1987) in a companion paper. It is shown how this lift force is made up of a rotational and an inertial or addedmass component. For threedimensional bluff bodies the latter is generally larger (by a factor of three for a sphere), and can be simply calculated from the addedmass coefficient. For illustration, the general expression is used to derive formulae for (i) the motion of a spherical bubble in a steady nonuniform flow to contrast with the motion in an unsteady flow, and (ii) the motion of rigid volumes of neutral density across an inviscid shear flow. These results show how addedmass (and lift) forces lead to different motions for a sphere and a cylinder. The general expression is useful in twophase flow calculations, and for indicating the forces and motions of ‘lumps of fluid’ in turbulent flows.
The effect of small surface perturbations on the pulsatile boundary layer on a semiinfinite flat plate
 P. W. Duck

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 21 April 2006, pp. 259293

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The laminar pulsatile flow over a semiinfinite flat plate, on which is located a small (steady) surface distortion is investigated; tripledeck theory provides the basis for the study. The problem is of direct relevance to the externally imposed acoustic excitation of boundary layers. The investigation is primarily numerical and involves the solution of the nonlinear, unsteady boundarylayer equations which arise from the lower deck. The numerical method involves the use of finite differencing in the transverse direction, CrankNicolson marching in time, and Fourier transforms in the streamwise direction, and as such is an extension of the spectral method of Burggraf & Duck (1982). Supersonic and incompressible flows are studied. A number of the computations presented suggest that the small surface distortion can excite a largewavenumber, rapidly growing instability, leading to a breakdown of the solution, with the wall shear at a point seeming to increase without bound as a finite time is approached. Rayleigh modes for the basic (undisturbed) velocity profile are computed and there is some correlation between the existence and magnitude of the growth rate of these unstable modes, and the occurrence of the apparent singularity. Streamline plots indicate that this phenomenon is linked to the formation of closed (or ‘catseye’) eddies in the main body of the boundary layer, away from the wall. TollmienSchlichting instabilities are clearly seen in the case of incompressible flows.
Nonlinear spatial evolution of an externally excited instability wave in a free shear layer
 M. E. Goldstein, Lennart S. Hultgren

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 21 April 2006, pp. 295330

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We consider a disturbance that evolves from a strictly linear finitegrowthrate instability wave, with nonlinear effects first becoming important in the critical layer. The local Reynolds number is assumed to be just small enough so that the spatialevolution, nonlinearconvection, and viscousdiffusion terms are of the same order of magnitude in the interactive criticallayer vorticity equation. The numerical results show that viscous effects eventually become important even when the viscosity is very small due to continually decreasing scales generated by the nonlinear effects. The vorticity distribution diffuses into a more regular pattern visavis the inviscid case, and the instabilitywave growth ultimately becomes algebraic. This leads to a new dominant balance between linear and nonlinearconvection terms and an equilibrium critical layer of the Benney & Bergeron (1969) type begins to emerge, but the detailed flow field, which has variable vorticity within the cat'seye boundary, turns out to be somewhat different from theirs. The solution to this rescaled problem is compared with the numerical results and is then used to infer the scaling for the next stage of evolution of the flow. The instabilitywave growth is simultaneously affected by meanflow divergence and nonlinear criticallayer effects in this latter stage of development and is eventually converted to decay. The neutral stability point is the same as in the corresponding linear case, however.
Finiteamplitude evolution of twolayer geostrophic vortices
 Karl R. Helfrich, Uwe Send

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 21 April 2006, pp. 331348

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The finiteamplitude evolution of circular twolayer quasigeostrophic vortices with piecewise uniform potential vorticity in each layer (also termed ‘heton’ clouds by Hogg & Stommel 1985a and Pedlosky 1985) is studied using the contour dynamics method. The numerical investigations are preceded by a linear stability analysis which shows the stabilizing influence of deepening the lower layer. Net barotropic flow may be either stabilizing or destabilizing. The contour dynamics calculations for baroclinic vortices show that supercritical (i.e. linearly unstable) conditions may lead to explosive break up of the vortex via the generation of continuous hetons at the cloud boundary. The number of vortex pairs is equal to the azimuthal mode number of the initial disturbance. An additional weakly supercritical regime in which amplitude vacillation occurs, but not explosive growth, is identified. Vortices with net barotropic circulation behave similarly except that the layer with vorticity opposite to the barotropic circulation will break up first. Strong barotropic circulation can inhibit the development of hetons. The stronger layer may eject thin filaments, but remain mostly intact. Calculations for initial conditions composed of several unstable modes show that the linearly most unstable mode dominates at finite amplitude.
On the instability of geostrophic vortices
 Glenn R. Flierl

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 21 April 2006, pp. 349388

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The instabilities of barotropic and baroclinic, quasigeostrophic, fplane, circular vortices are found using a linearized contour dynamics model. We model the vortex using a circular region of horizontally uniform potential vorticity surrounded by an annulus of uniform, but different, potential vorticity. We concentrate mostly upon isolated vortices with no circulation in the basic state outside the outer radius b. In addition to linear analyses, we also consider weakly nonlinear waves. The amplitude equation has a cubic nonlinearity and, depending upon the sign of the coefficient of the cubic term, may give nonlinear stabilization or nonlinear enhancement of the growth. Barotropic isolated eddies are unstable when the outer annulus is narrow enough; on the other hand, if the scale of the whole vortex is sufficiently small compared to the radius of deformation of a baroclinic mode, the break up may be preferentially to a depthvarying disturbance corresponding to a twisting and tilting of the vortex. As the vortex becomes more baroclinic, we find that largescale vortices show an elliptical mode baroclinic instability as well which is relatively insensitive to the scale of the outer annulus. When the baroclinic currents in the basic state dominate, the twisting mode disappears, and we see only the instabilities associated with either strong enough shear in the annular region or sufficiently large vortices compared with the deformation radius. The finite amplitude results show that the baroclinic instability mode for large enough vortices is nonlinearly stabilized while in most cases, the other two kinds of instability are nonlinearly destabilized.
Substructures in a turbulent spot
 R. Sankaran, M. Sokolov, R. A. Antonia

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 21 April 2006, pp. 389414

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Substructures within a turbulent spot which develops in a slightly heated laminar boundary layer have been identified using arrays of cold wires aligned in either a streamwise direction or in a direction normal to the wall. At any given streamwise distance from the spot origin, histograms of the number of detected substructures exhibit a peak, defining the most probable spot or the spot with the most likely number of substructures. The number of substructures in the most probable spot increases with streamwise distance but all substructures are convected at approximately the same velocity for any given distance from the wall. This velocity is approximately equal to that of the leading edge of the spot and increases slightly with distance from the wall. The increase in the number of substructures accounts for the streamwise growth of the spot. A simple relation is derived for determining the number of substructures at a particular streamwise station and a geometrical construction is proposed for identifying the origin of a new substructure. There is sufficient evidence for suggesting that the new substructures are formed near the trailing edge of the spot. The convection velocity, inclination and lengthscales of the substructures compare favourably with the corresponding characteristics of hairpin vortices.
Gravity–capillary rings generated by water drops
 Bernard Le Méhauté

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 21 April 2006, pp. 415427

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A theory for water waves created by the impact of small objects such as raindrops on an initially quiescent body of water is established. Capillary and dissipative viscous effects are taken into account in addition to gravity. It is shown that the prevailing waves are in a mixed capillary–gravity regime around a wavenumber km which corresponds to the minimum value of the group velocity. The waves are described as function of time and distance by the linear superposition of two transient wave components, a ‘subkm’ (k < km) component and a ‘superkm’ (k > km) component. The superkm components prevail at a short distance from the drop, whereas only the subkm ones remain at a larger distance. The relative time history of the wavetrain is independent of the size of the drop, and its amplitude is proportional to the drop momentum when it hits the free surface. The wave pattern is composed of a multiplicity of rings of amplitude increasing towards the drop location and is terminated by a trailing wave with an exponential decay. The number of rings increases with time and distance.
Critical layers in accelerating twolayer flows
 Donald B. Altman

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 21 April 2006, pp. 429451

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A series of laboratory experiments on accelerating twolayer shear flows over topography is described. The mean flow reverses at the interface of the layers, forcing a critical layer to occur there. It is found that for a sufficiently thin interface, a slowly growing recirculating region, the ‘acceleration rotor’, develops on the interfacial wave at meanflow Richardson numbers of O(0.5). This, in turn, can induce a secondary dynamical shear instability on the trailing edge of the wave. A singlemode, linear, twolayer numerical model reproduces many features of the acceleration rotor if meanflow acceleration and bottom forcing are included. Velocity measurements are obtained from photographs using image processing software developed for the automated reading of particlestreak photographs. Typical results are shown.
The compressible turbulent shear layer: an experimental study
 Dimitri Papamoschou, Anatol Roshko

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 21 April 2006, pp. 453477

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The growth rate and turbulent structure of the compressible, plane shear layer are investigated experimentally in a novel facility. In this facility, it is possible to flow similar or dissimilar gases of different densities and to select different Mach numbers for each stream. Ten combinations of gases and Mach numbers are studied in which the freestream Mach numbers range from 0.2 to 4. Schlieren photography of 20ns exposure time reveals very low spreading rates and largescale structures. The growth of the turbulent region is defined by means of Pitotpressure profiles measured at several streamwise locations. A compressibilityeffect parameter is defined that correlates and unifies the experimental results. It is the Mach number in a coordinate system convecting with the velocity of the dominant waves and structures of the shear layer, called here the convective Mach number. It happens to have nearly the same value for each stream. In the current experiments, it ranges from 0 to 1.9. The correlations of the growth rate with convective Mach number fall approximately onto one curve when the growth rate is normalized by its incompressible value at the same velocity and density ratios. The normalized growth rate, which is unity for incompressible flow, decreases rapidly with increasing convective Mach number, reaching an asymptotic vaue of about 0.2 for supersonic convective Mach numbers.
Bifurcation phenomena in Taylor–Couette flow with buoyancy effects
 K. S. Ball, B. Farouk

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 21 April 2006, pp. 479501

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A numerical study has been conducted to determine the various modes of Taylor–Couette flow that exist between concentric vertical cylinders, as the aspect ratio Γ (height to gap width, H/d) and the Reynolds number Re (based on the inner cylinder speed) are varied. Furthermore, the effects of the introduction of buoyancy on the development of the flow are examined. This is accomplished by considering both cylinders to be isothermal, with the rotating inner cylinder at a higher temperature than the stationary outer cylinder. Results are presented for a wide range of the Grashof number Gr (based on the temperature difference ΔT across the annular gap). The structure of the Taylor vortices is observed to be distorted considerably with the buoyant flows, and the nature of the onset and subsequent development of the vortices is altered. The hysteresis between the different modes of cellular flow, characteristic of the bifurcation phenomena, is also substantially modified.
The reflection of a solitary wave by a vertical wall
 J. G. B. ByattSmith

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 21 April 2006, pp. 503521

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In this paper we consider the headon collision of two equal solitary waves this being equivalent, in the absence of viscosity to the reflection of one solitary wave by a vertical wall. The perturbation expansion of the Euler equations, which lead to the Boussinesq equation at lowest order, is recast to obtain two weakly coupled KdV equations. We show analytically that the amplitude of the solitary wave after reflection is reduced. This change in amplitude is shown to be fifth order in ε, the amplitude of the wave. It is also shown that the experimentally observed transient loss of amplitude can be explained by the presence of the thirdorder dispersive tail.