bluuebunny/crossref_metadata_2025 huggingface dataset:
On narrow V-like ship wakes
Authors: Gu Daifang , Phillips O. M.
Published: April 2006

This paper is concerned with the generation of short gravity waves and their radiation from the outer edge of the turbulent boundary layer and wake of a ship. They arise primarily near the ship's stern. The wave spectrum in the direction of wavenumber vector at an angle (90° – δ) to the ship's track is:
\[\Phi_{\delta}(\omega) = \Psi\left(\frac{UT_d}{2l},\frac{U\sin\delta}{c_g},\frac{R}{UT_d}\right)\frac{1}{k_0R}\frac{2l\omega^2}{g^2}\gamma\left(0,\frac{\pi}{l};0,\omega \right),\]
where Ψ is dimensionless and a function of three dimensionless parameters. γ is the spectrum of the oscillating motion at the boundary, U the ship speed, Td the decay timescale of the oscillating motion, 2l the lengthscale of the eddies, and R the distance away from the boundary along the wavenumber vector. Generally, Φδ has large values near δ = 0 and small values at large δ; it behaves as 1/R at distances not far from the ship, then may vary slower than 1/R at intermediate distances, and finally behaves as 1/R again at distances far from the ship. These are consistent with the pattern found in SAR images of narrow V-like ship wakes. The method developed here is also applicable to various problems of surface wave generation by turbulence in water.
Shear flow over a translationally symmetric cylindrical bubble pinned on a slot in a plane wall
Authors: Feng James Q. , Basaran Osman A.
Published: April 2006

Steady states of a translationally-symmetric cylindrical bubble protruding from a slot in a solid wall into a liquid undergoing a simple shear flow are investigated. Deformations of and the flow past the bubble are determined by solving the nonlinear free-boundary problem comprised of the two-dimensional Navier–Stokes system by the Galerkin/finite element method. Under conditions of creeping flow, the results of finite element computations are shown to agree well with asymptotic results. When the Reynolds number Re is finite, flow separates from the free surface and a recirculating eddy forms behind the bubble. The length of the separated eddy measured in the flow direction increases with Re, whereas its width is confined to within the region that lies between the supporting solid surface and the separation point at the free surface. By tracking solution branches in parameter space with an arc-length continuation method, curves of bubble deformation versus Reynolds number are found to exhibit turning points when Re reaches a critical value Rec. Therefore, along a family of bubble shapes, solutions do not exist when Re > Rec. The locations of turning points and the structure of flow fields are found to be governed virtually by a single parameter, We = Ca Re, where We and Ca are Weber and capillary numbers. Two markedly different modes of bubble deformation are identified at finite Re. One is dominant when Re is small and is tantamount to a plain skewing or tilting of the bubble in the downstream direction; the other becomes more pronounced when Re is large and corresponds to a pure upward stretching of the bubble tip.
Low-Reynolds-number effects on near-wall turbulence
Authors: Antonia R. A. , Kim J.
Published: April 2006

Direct numerical simulations of a fully developed turbulent channel flow for two relatively small values of the Reynolds number are used to examine its influence on various turbulence quantities in the near-wall region. The limiting wall behaviour of these quantities indicates important increases in the r.m.s. value of the wall pressure fluctuations and its derivatives, the r.m.s. streamwise vorticity and in the average energy dissipation rate and the Reynolds shear stress. If the normalization is based on the wall shear stress and the kinematic viscosity, these changes are shown to be consistent with an increase in strength – but not the average diameter or average location – of the quasi-streamwise vortices in the buffer region. Evidence of this strengthening is provided by the increased sum of the stretching terms for the meansquare streamwise vorticity. It is also shown that a normalization based on Kolmogorov velocity and lengthscales, defined at the wall, is more appropriate in the near-wall region than scaling on the wall shear stress and kinematic viscosity.
Solitary wave, soliton and shelf evolution over variable depth
Authors: Johnson R. S.
Published: April 2006

The familiar problem of the propagation of surface waves over variable depth is reconsidered. The surface wave is taken to be a slowly evolving nonlinear wave (governed by the Korteweg–de Vries equation) and the depth is also assumed to be slowly varying; the fluid is stationary in its undisturbed state. Two cases are addressed: the first is where the scale of the depth variation is longer than that on which the wave evolves, and the second is where it is shorter (but still long). The first case corresponds to that discussed by a number of previous authors, and is the problem which has been approached through the perturbation of the inverse scattering transform method, a route not followed here. Our more direct methods reveal a new element in the solution: a perturbation of the primary wave, initiated by the depth change, which arises at the same order as the left-going shelf. The resulting leading-order mass balance is described, with more detail than hitherto (made possible by the use of a special depth variation). The second case is briefly presented using the same approach, and some important similarities are noted.
A numerical study of vortex ring formation at the edge of a circular tube
Authors: Nitsche Monika , Krasny Robert
Published: April 2006

An axisymmetric vortex-sheet model is applied to simulate an experiment of Didden (1979) in which a moving piston forces fluid from a circular tube, leading to the formation of a vortex ring. Comparison between simulation and experiment indicates that the model captures the basic features of the ring formation process. The computed results support the experimental finding that the ring trajectory and the circulation shedding rate do not behave as predicted by similarity theory for starting flow past a sharp edge. The factors responsible for the discrepancy between theory and observation are discussed.
Large-scale Langmuir circulation and double-diffusive convection: evolution equations and flow transitions
Authors: Cox Stephen M. , Leibovich Sidney
Published: April 2006

Two-dimensional Langmuir circulation in a layer of stably stratified water and the mathematically analogous problem of double-diffusive convection are studied with mixed boundary conditions. When the Biot numbers that occur in the mechanical boundary conditions are small and the destabilizing factors are large enough, the system will be unstable to perturbations of large horizontal length. The instability may be either direct or oscillatory depending on the control parameters. Evolution equations are derived here to account for both cases and for the transition between them. These evolution equations are not limited to small disturbances of the nonconvective basic velocity and temperature fields, provided the spatial variations in the horizontal remain small. The direct bifurcation may be supercritical or subcritical, while in the case of oscillatory motions, stable finite-amplitude travelling waves emerge. At the transition, travelling waves, standing waves, and modulated travelling waves all are stable in sub-regimes.
Periodic steady vortices in a stagnation-point flow
Authors: Kerr Oliver S. , Dold J. W.
Published: April 2006

A stagnation point flow of the form U = (0, Ay, — Az) is unstable to three-dimensional disturbances. It has been shown that the vorticity components of such a disurbance that are perpendicular to the direction of the diverging flow will decay, and that the parallel component of vorticity can grow. We augment these findings by showing that fully nonlinear steady-state deviations from this flow exist that consist of a periodic distribution of counter-rotating vortices whose axes lie parallel to the direction of the diverging flow. These solutions have two independent parameters: the dimensionless strength of the converging flow, and the intensity of the vortices. We examine the structure of these vortices in the asymptotic limits of large strain rate of the converging flow, and of large amplitude of the vortices.
On the mechanism of shear flow instabilities
Authors: Baines Peter G. , Mitsudera Humio
Published: April 2006

In homogeneous and density-stratified inviscid shear flows, the mechanism for instability that is most commonly invoked and discussed is Kelvin–Helmholtz instability, as it occurs for a simple velocity discontinuity. There is a second mechanism, the wave-interaction mechanism, which is much more general, and is the subject of this paper. This mechanism depends on two free waves that propagate in opposite directions in a stratified shear flow, and which may become stationary relative to each other because of the shear. If this occurs, and their relative phase is suitably chosen, the velocity field of each wave increases the displacement of the other, and so the disturbance grows.

We show that this mechanism is responsible for instability in a general class of symmetric but otherwise arbitrary velocity and density profiles, provided that the Richardson number Ri < ¼ in a central region of arbitrarily small thickness. A critical layer exists in this central region for the growing disturbance, but its role in the instability process is incidental. When Ri > ¼ everywhere, the flow is stable because the free waves described above are absorbed by the critical layer, and hence are heavily damped. The necessary criteria of Rayleigh and Fjortoft for instability in homogeneous fluid are seen to provide a suitable geometry for two interacting waves. Some specific examples are given, including a succinct explanation of Holmboe waves.
Thermocapillary convection in a cylinder with a strong non-uniform axisymmetric magnetic field
Authors: Khine Yu Yu , Walker John S.
Published: April 2006

This paper treats a surface-tension-driven liquid-metal flow in a cylinder with a steady externally applied non-uniform axisymmetric magnetic field. The top boundary consists of an annular free surface around a solid disk, modelling the Czochralski growth of silicon crystals. A radial temperature gradient produces a decrease of the surface tension from the disk edge to the vertical cylinder wall. The magnetic flux density is sufficiently large that inertial effects and convective heat transfer are negligible. First we present large-Hartmann-number asymptotic solutions for magnetic fields with either a non-zero or a zero axial component at the free surface. The asymptotic solutions indicate that a purely radial magnetic field at the free surface represents a singular limit of more general magnetic fields. Secondly we present numerical solutions for arbitrary values of the Hartmann number, and we treat the evolution of the thermocapillary convection as the axial magnetic field at the free surface is changed continuously from the full field strength to zero.
Nonlinear long-wave stability of superposed fluids in an inclined channel
Authors: Tilley B. S. , Davis S. H. , Bankoff S. G.
Published: April 2006

We consider the two-layer flow of immiscible, viscous, incompressible fluids in an inclined channel. We use long-wave theory to obtain a strongly nonlinear evolution equation which describes the motion of the interface. This equation includes the physical effects of viscosity stratification, density stratification, and shear. A weakly nonlinear analysis of this equation yields a Kuramoto–Sivashinsky equation, which possesses a quadratic nonlinearity. However, certain physical situations exist in two-layer flow for which modifications of the Kuramoto–Sivashinsky equation are physically pertinent. In particular, the presence of the second layer can mediate the wave-steepening instability found in single-phase falling films, requiring the inclusion of a cubic nonlinearity in the weakly nonlinear analysis. The introduction of the cubic nonlinearity destroys the symmetry-breaking bifurcations of the Kuramoto–Sivashinsky equation, and new isolated solution branches emerge as the strength of the cubic nonlinearity increases. Bistability between these new solutions and those associated with the Kuramoto–Sivashinsky equation is found, as well as the formation of a hysteresis loop from smaller-amplitude travelling waves to larger-amplitude travelling waves. The physical implications of these dynamics to the phenomenon of laminar flooding in a channel are discussed.
Hypersonic aerodynamics on thin bodies with interaction and upstream influence
Authors: Khorrami A. Farid , Smith Frank T.
Published: April 2006

In the fundamental configuration studied here, a steady hypersonic free stream flows over a thin sharp aligned airfoil or flat plate with a leading-edge shock wave, and the flow field in the shock layer (containing a viscous and an inviscid layer) is steady laminar and two-dimensional, for a perfect gas without real and high-temperature gas effects. The viscous and inviscid layers are analysed and computed simultaneously in the region from the leading edge to the trailing edge, including the upstream-influence effect present, to determine the interactive flow throughout the shock layer and the positions of the shock wave and the boundary-layer edge, where matching is required. Further theoretical analysis of the shock layer helps to explain the computational results, including the nonlinear breakdown possible when forward marching against enhanced upstream influence, for example as the wall enthalpy increases towards its insulated value. Then the viscous layer is computed by sweeping methods, for higher values of wall enthalpies, to prevent this nonlinear breakdown for airfoils including the flat plate. Thin airfoils in hypersonic viscous flow are treated, for higher values of the wall enthalpies and with the upstream-influence effect, as are hypersonic inviscid flows, by modifying the computational methods used for the flat plate. Also, the behaviour of the upstream influence for bodies of relatively large thickness, and under wall velocity slip and enthalpy jump for flat plates, is discussed briefly from a theoretical point of view.

Subsequent to the present work, computations based on the Navier–Stokes and on the parabolized Navier–Stokes equations have yielded excellent and good agreement respectively with the present predictions for large Mach and Reynolds numbers.
The onset of transverse recirculations during flow of gases in horizontal ducts with differentially heated lower walls
Authors: Ingle N. K. , Mountziaris T. J.
Published: April 2006

A computational study has been performed to identify the onset of transverse buoyancy-driven recirculations during laminar flow of hydrogen and nitrogen in horizontal ducts with cool upper walls, and lower walls consisting of three sections: a cool upstream section, a heated middle section and a cool downstream section. The motivation for this work stems from the need to identify operating conditions maximizing the thickness uniformity, the interface abruptness and the precursor utilization during growth of thin films and multi-layer structures of semiconductors by metalorganic chemical vapour deposition (MOCVD). A mathematical model describing the flow and heat transfer along the vertical midplane of MOCVD reactors with the above geometry has been developed and computer simulations were performed for a variety of operating conditions using the Galerkin finite-element method. At atmospheric pressure and low inlet velocities, transverse recirculations form near the upstream and downstream edges of the heated section. These can be suppressed either by increasing the inlet velocity of the gas, so that forced convection dominates natural convection, or by decreasing the operating pressure to reduce the effects of buoyancy. The onset of transverse recirculations has been determined for Grashof (Gr) and Reynolds (Re) numbers covering the following ranges: 10−3 < Re < 100 and 1 < Gr < 106, with Gr and Re computed using fluid properties at the inlet conditions. The computations indicate that, for abrupt temperature changes along the lower wall (worst-case scenario), transverse recirculations are always absent if the following criteria are satisfied:
\[(Gr/Re) < 100\quad {\rm for}\quad 10^{-3} < Re \leqslant 4\quad {\rm and}\quad (Gr/Re^2) < 25\quad {\rm for}\quad 4 \leqslant Re < 100.\]

The predicted critical values of Re, which correspond to the onset of transverse recirculations, agree well with reported experimental observations. The above criteria can be used for optimal design and operation of horizontal MOCVD reactors and may also be useful for heat transfer studies in horizontal ducts with differentially heated lower walls.
Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2. Couette and Poiseuille flows
Authors: Feng J. , Hu H. H. , Joseph D. D.
Published: April 2006

This paper reports the results of a two-dimensional finite element simulation of the motion of a circular particle in a Couette and a Poiseuille flow. The size of the particle and the Reynolds number are large enough to include fully nonlinear inertial effects and wall effects. Both neutrally buoyant and non-neutrally buoyant particles are studied, and the results are compared with pertinent experimental data and perturbation theories. A neutrally buoyant particle is shown to migrate to the centreline in a Couette flow, and exhibits the Segré-Silberberg effect in a Poiseuille flow. Non-neutrally buoyant particles have more complicated patterns of migration, depending upon the density difference between the fluid and the particle. The driving forces of the migration have been identified as a wall repulsion due to lubrication, an inertial lift related to shear slip, a lift due to particle rotation and, in the case of Poiseuille flow, a lift caused by the velocity profile curvature. These forces are analysed by examining the distributions of pressure and shear stress on the particle. The stagnation pressure on the particle surface are particularly important in determining the direction of migration.
Dissolution or growth of soluble spherical oscillating bubbles
Authors: Fyrillas Marios M. , Szeri Andrew J.
Published: April 2006

A new theoretical formulation is presented for mass transport across the dynamic interface associated with a spherical bubble undergoing volume oscillations. As a consequence of the changing internal pressure of the bubble that accompanies volume oscillations, the concentration of the dissolved gas in the liquid at the interface undergoes large-amplitude oscillations. The convection-diffusion equations governing transport of dissolved gas in the liquid are written in Lagrangian coordinates to account for the moving domain. The Henry's law boundary condition is split into a constant and an oscillating part, yielding the smooth and the oscillatory problems respectively. The solution of the oscillatory problem is valid everywhere in the liquid but differs from zero only in a thin layer of the liquid in the neighbourhood of the bubble surface. The solution to the smooth problem is also valid everywhere in the liquid; it evolves via convection-enhanced diffusion on a slow timescale controlled by the Péclet number, assumed to be large. Both the oscillatory and smooth problems are treated by singular perturbation methods: the oscillatory problem by boundary-layer analysis, and the smooth problem by the method of multiple scales in time. Using this new formulation, expressions are developed for the concentration field outside a bubble undergoing arbitrary nonlinear periodic volume oscillations. In addition, the rate of growth or dissolution of the bubble is determined and compared with available experimental results. Finally, a new technique is described for computing periodically driven nonlinear bubble oscillations that depend on one or more physical parameters. This work extends a large body of previous work on rectified diffusion that has been restricted to the assumptions of infinitesimal bubble oscillations or of threshold conditions, or both. The new formulation represents the first self-consistent, analytical treatment of the depletion layer that accompanies nonlinear oscillating bubbles that grow via rectified diffusion.
On Markov modelling of turbulence
Authors: Pedrizzetti Gianni , Novikov Evgeny A.
Published: April 2006

We consider Lagrangian stochastic modelling of the relative motion of two fluid particles in the inertial range of a turbulent flow. Eulerian analysis of such modelling corresponds to an equation for the Eulerian probability distribution of velocity-vector increments which introduces a hierarchy of constraints for making the model consistent with results from the theory of locally isotropic turbulence. A nonlinear Markov process is presented, which is able to satisfy exactly, in the statistical sense, incompressibility, the exact results on the third-order structure function, and the experimental second-order statistics. The corresponding equation for the Eulerian probability density of velocity-vector increments is solved numerically. Numerical results show non-Gaussian statistics of the one-dimensional Lagrangian probability distributions, and a complex shape of the three-dimensional Eulerian probability density function. The latter is then compared with existing experimental data.
The influence of a mean magnetic field on three-dimensional magnetohydrodynamic turbulence
Authors: Oughton Sean , Priest Eric R. , Matthaeus William H.
Published: April 2006

Building on results from two-dimensional magnetohydrodynamic (MHD) turbulence (Shebalin, Matthaeus & Montgomery 1983), the development of anisotropic states from initially isotropic ones is investigated numerically for fully three-dimensional incompressible MHD turbulence. It is found that when an external d.c. magnetic field (B0) is imposed on viscous and resistive MHD systems, excitations are preferentially transferred to modes with wavevectors perpendicular to B0). The anisotropy increases with increasing mechanical and magnetic Reynolds numbers, and also with increasing wavenumber. The tendency of B0 to inhibit development of turbulence is also examined.
Gravity-induced coalescence of drops at arbitrary Péclet numbers
Authors: Zinchenko Alexander Z. , Davis Robert H.
Published: April 2006

The collision efficiency in a dilute suspension of sedimenting drops is considered, with allowance for particle Brownian motion and van der Waals attractive force. The drops are assumed to be of the same density, but they differ in size. Drop deformation and fluid inertia are neglected. Owing to small particle volume fraction, the analysis is restricted to binary interactions and includes the solution of the full quasi-steady Fokker—Planck equation for the pair-distribution function. Unlike previous studies on drop or solid particle collisions, a numerical solution is presented for arbitrary Péclet numbers, Pe, thus covering the whole range of particle size in typical hydrosols. Our technique is mainly based on an analytical continuation into the plane of complex Péclet number and a special conformal mapping, to represent the solution as a convergent power series for all real Péclet numbers. This efficient algorithm is shown to apply to a variety of convection—diffusion problems. The pair-distribution function is expanded into Legendre polynomials, and a finite-difference scheme with respect to particle separation is used. Two-drop mobility functions for hydrodynamic interactions are provided from exact bispherical coordinate solutions and near-field asymptotics. The collision efficiency is calculated for wide ranges of the size ratio, the drop-to-medium viscosity ratio, and the Péclet number, both with and without interdroplet forces. Solid spheres are considered as a limiting case; attractive van der Waals forces are required for non-zero collision rates in this case. For Pe [Gt ] 1, the correction to the asymptotic limit Pe → ∞ is O(Pe−1/2). For Pe [Lt ] 1, the first two terms in an asymptotic expansion for the collision efficiency are C/Pe + ½C2, where the constant C is determined from the Brownian solution in the limit Pe → 0. The numerical results are in excellent agreement with these limits. For intermediate Pe, the numerical results show that Brownian motion is important for Pe ≤ O(102). For Pe = 10, the trajectory analysis for Pe → ∞ may underestimate the collision rate by a factor of two. A simpler, approximate solution based on neglecting the transversal diffusion is also considered and compared to the exact solution. The agreement is within 2–3% for all conditions investigated. The effect of van der Waals attractions on the collision efficiency is studied for a wide range of droplet sizes. Except for very high drop-to-medium viscosity ratios, the effect is relatively small, especially when electromagnetic retardation is accounted for.
Joint statistics of a passive scalar and its dissipation in turbulent flows
Authors: Anselmet F. , Djeridi H. , Fulachier L.
Published: April 2006

The statistical relationship between a passive scalar and its dissipation is important for both a basic understanding of turbulence small-scale properties and for various aspects of turbulent combustion modelling. This problem is studied in two different flows through spectral analysis as well as probability density functions using temperature as a passive scalar. Particular attention is paid to the experimental determination of the three squared derivatives involved in the temperature dissipation. As a first step, it is found that basic properties such as the correlation coefficient between temperature and its dissipation are strongly related to the asymmetry of the scalar fluctuations, so that the usually assumed statistical independence between these variables is not justified. These trends are the same for the two flows investigated here, a boundary layer and a jet. This connection appears to be related to fluctuations of small amplitude for both quantities which are associated with relatively low frequencies lying between the integral scale and the Taylor microscale. In regions where the temperature skewness factor is nearly zero, the correlation coefficient is also very small, and several tests show that the assumption of independence is then fully justified. Thus, the main parameter influencing joint statistics of temperature and its dissipation is the asymmetric feature of temperature fluctuations, but the asymmetry of the longitudinal temperature derivative, which results from the flow boundary conditions and is usually felt through the presence of the so-called temperature ramps, is also involved. Even though the magnitude of the derivative skewness factor is almost uniformly distributed in both flows, the secondary effect becomes the dominant one in flow regions where the influence of the temperature asymmetry is relatively weak.
Melting driven by vigorous compositional convection
Authors: Kerr Ross C.
Published: April 2006

The melting of a solid in contact with a hot fluid is quantified for the case in which a difference between the densities of the fluid and of the melted solid is able to drive vigorous compositional convection. A scaling analysis is first used to obtain a theoretical expression for the melting rate that is valid for a certain range of Stefan numbers. This expression is then compared with melting velocities measured in laboratory experiments in which ice and wax are melted when they are overlain or underlain by hot aqueous solutions. The melting velocities are consistent with the theoretical expression, and are found to depend on the heats of solution that are released when the melted solids mix with the solutions. The experiments also indicate that, for vigorous convection to occur during the melting of a floor, the unstable compositional buoyancy needs to be at least twice the stabilizing thermal buoyancy.

An important geological situation in which melting occurs is when large volumes of basaltic magma are intruded into the Earth's continental crust. The theoretical and experimental results are used and extended to examine quantitatively the melting of the floor and walls of the magma chamber, and of crustal blocks that fall into the chamber.
A stationary potential-flow approximation for a breaking-wave crest
Authors: Jenkins Alastair D.
Published: April 2006

The flow in a breaking-wave crest is represented by a complex velocity potential on a Riemann surface, satisfying the Bernoulli condition on two free boundaries. The flow is assumed to be stationary in the reference frame which moves with the wave crest, and at large distances approximates Stokes corner flow in the main part of the fluid and a parabolic descending flow in the jet. The interaction of the jet with the rest of the fluid is neglected.

The solution is obtained by means of a conformal transformation from a bounded, teardrop-shaped domain, using a Faber polynomial expansion. The Bernoulli condition is applied at a number of discrete points on the boundaries, and the resulting nonlinear equations for the expansion coefficients are solved iteratively. The resulting surface form is similar to that obtained by laboratory experiments and time-dependent numerical simulations of waves up to the point of breaking, with a stagnation point at the top of the crest, an overturning loop with major axis$\ap 8g^{-\frac{1}{3}}\Psi^{\frac{2}{3}}$, and a maximum acceleration of ≈ 5.4g, wheregis the gravitational acceleration and ψ is the flux in the jet.
Biflagellate gyrotaxis in a shear flow
Authors: Jones M. S. , Baron L. Le , Pedley T. J.
Published: April 2006

A flagellated, bottom-heavy micro-organism's swimming direction in a shear flow is determined from a balance between the gravitational and viscous torques (gyrotaxis). Hitherto, the cell has been assumed to be a spheroid and the flagella have been neglected. Here we use resistive-force theory to calculate both the magnitude and the direction of a biflagellate cell's swimming velocity and angular velocity relative to the fluid when there is an arbitrary linear flow far from the cell. We present an idealized model for the flagellar beat but, in calculating the velocity of the fluid relative to an element of a flagellum, the presence of the cell body is not neglected. Results are given for the case of a spherical cell body whose flagella beat in a vertical plane, when the ambient linear flow is in the same vertical plane. Results show that resistive-force theory can be used for organisms where the cell body has significant effect on the flow past the flagella and that the viscous torque on the flagella is a significant term in the torque balance equations. A model is presented for the calculation of a cell's velocity and angular velocity in a shear flow which is valid up to high magnitudes of rate of strain or vorticity. The main application of the results will be to modify a recent continuum model for suspensions of gyrotactic micro-organisms (Pedley & Kessler 1990).
Sound radiation from a cylindrical duct. Part 1. Ray structure of the duct modes and of the external field
Authors: Chapman C. J.
Published: April 2006

This paper determines the ray structure of a spinning acoustic mode propagating inside a semi-infinite circular cylindrical duct, and thereby determines the ray structure of the field radiated from the end of the duct. Inside the duct, but outside of a caustic cylindrical surface, the rays are piecewise linear helices; on striking the rim of the end-face of the duct, these rays produce ‘Keller cones’ of diffracted rays. The cones determine the structure of the radiated field: for example, no rays penetrate two cone-shaped far-field quiet zones centred on the duct axis; two rays pass through each point in a forward loud zone; and one ray passes through each point in a rearward loud zone. The two rays through each point in the forward loud zone interfere to produce an oscillatory directivity pattern. One quarter of the rays on each cone point back inside the duct and produce the reflected field. Thus the rim of the end-face of the duct acts as a ‘ring source’, in which the radiated and reflected fields have their origin. Every propagating duct mode determines a polar angle and an azimuthal angle; these are taken as parameters specifying the mode and are used to calculate the positions and angles of all the rays. The mathematical method on which the paper is based is Debye's approximation for the Bessel function which appears in the expression for the duct modes; the approximation shows also that the duct contains a region of smooth helical rays on which the field consists of inhomogeneous waves: this region is the inner cylinder, lying inside the annulus of piecewise linear helical rays. The results of the paper are very promising for the application of Keller's geometrical theory of diffraction to detailed calculations of the sound radiated from aeroengine ducts. An alternative description of the field, using Cargill's meridional rays, is summarized.
Resonant sloshing near a critical depth
Authors: Waterhouse D. D.
Published: April 2006

Oscillations of a tank at a near-resonant frequency have been shown to produce a response which changes from a ‘hard-spring’ to a ‘soft-spring’ response as the depth passes through a critical value. This paper investigates the transition region and it is shown, using a symbolic manipulator, that in fact the large-amplitude response is that of a soft spring on either side of this critical depth.
On the stability of flow in an elliptic pipe which is nearly circular
Authors: Davey A. , Salwen H.
Published: April 2006

In an earlier paper (Davey 1978) the first author investigated the linear stability of flow in a straight pipe whose cross-section was an ellipse, of small ellipticity e, by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. That paper contained some serious errors which we correct herein. We show analytically that for the most important mode n = 1, for which the circular problem has a double eigenvalue c0 as the ‘swirl’ can be in either direction, the ellipticity splits the double eigenvalue into two separate eigenvalues c0 ± e2c12, to leading order, when the cross-sectional area of the pipe is kept fixed. The imaginary part of c12 is non-zero and so the ellipticity always makes the flow less stable. This specific problem is generic to a much wider class of fluid dynamical problems which are made less stable when the symmetry group of the dynamical system is reduced from S1 to Z2.

In the Appendix, P. G. Drazin describes simply the qualitative structure of this problem, and other problems with the same symmetries, without technical detail.
Nonlinear spatial evolution of helical disturbances to an axial jet
Authors: Churilov S. M. , Shukhman I. G.
Published: April 2006

We investigate the weakly nonlinear spatial evolution of helical disturbances of an axisymmetrical jet which are the analogue of three-dimensional disturbances, such as a single oblique wave (the wave vector is directed at an angle to the main flow velocity) in plane-parallel flows. It is shown that when a supercriticality is large enough, the perturbation amplitude A grows in the streamwise direction (along z) explosively: A ∼ (z0 — z)−5/2, though more slowly than in the case of essentially three-dimensional disturbances in the form of a pair of oblique waves (A ∼ (z0 — z)−3; Goldstein & Choi 1989). The nonlinearity needed for such a growth, is due equally to the cylindricity of shear layer and to the spatial character of the evolution (in the temporal problem the ‘evolution’ contribution is absent). At a smaller supercriticality, the evolution equation has a non-local (integral in z) nonlinearity, unusual for the regime of a viscous critical layer. Scenarios of disturbance development for different levels of supercriticality are studied, with proper account taken of viscous broadening of the flow.
Low-dimensional description of free-shear-flow coherent structures and their dynamical behaviour
Authors: Rajaee Mojtaba , Karlsson Sture K. F. , Sirovich Lawrence
Published: April 2006

The snapshot form of the Karhunen-Loéve (K–L) expansion has been applied to twodimensional, two-component hot-wire data from the region of a weakly pertubed free shear layer that includes the first pairing process. Low-level external perturbation was provided by a loudspeaker driven by a computer-generated signal composed of two sine waves of equal amplitude at the frequencies of the naturally developing fundamental instability wave and its first subharmonic, separated by a controllable initial phase angle difference. It was found that a large fraction of the fluctuation energy is carried by the first few modes. A low-dimensional empirical eigenfunction space is obtained which describes the shear-flow coherent structures well. Galerkin projection of the Navier-Stokes equations onto this basis set of principal eigenfunction modes results in a low-order system of dynamical equations, and solution of this system of equations describes the dynamics of the coherent structures associated with eigenfunctions. Finally the simulation, as obtained from the system of dynamical equations, is shown to compare reasonably well with the experiments.
Self-similar gravity currents with variable inflow revisited: plane currents
Authors: Gratton Julio , Vigo Claudio
Published: April 2006

We use shallow-water theory to study the self-similar gravity currents that describe the intrusion of a heavy fluid below a lighter ambient fluid. We consider in detail the case of currents with planar symmetry produced by a source with variable inflow, such that the volume of the intruding fluid varies in time according to a power law of the type tα. The resistance of the ambient fluid is taken into account by a boundary condition of the von Kármán type, that depends on a parameter β that is a function of the density ratio of the fluids. The flow is characterized by β, α, and the Froude number [Fscr ]0 near the source. We find four kinds of self-similar solutions: subcritical continuous solutions (Type I), continuous solutions with a supercritical-subcritical transition (Type II), discontinuous solutions (Type III) that have a hydraulic jump, and discontinuous solutions having hydraulic jumps and a subcritical-supercritical transition (Type IV). The current is always subcritical near the front, but near the source it is subcritical ([Fscr ]0 < 1) for Type I currents, and supercritical ([Fscr ]0 > 1) for Types II, III, and IV. Type I solutions have already been found by other authors, but Type II, III, and IV currents are novel. We find the intervals of parameters for which these solutions exist, and discuss their properties. For constant-volume currents one obtains Type I solutions for any β that, when β > 2, have a ‘dry’ region near the origin. For steady inflow one finds Type I currents for O < β < ∞ and Type II and III Currents for and β, if [Fscr ]0 is sufficiently large.
Steady inviscid rotational flows with free surfaces
Authors: Vanden-Broeck J.-M. , Tuck E. O.
Published: April 2006

An inviscid fluid in steady two-dimensional motion in a region bounded by a closed streamline must have constant vorticity. We solve here for some such flows where the boundary is in part free, the fluid velocity magnitude being constant on the free boundary. A trivial example of such a flow is a circular cylinder of fluid rotating about its axis as if rigid, for which the whole circular boundary is free, irrespective of its radius. We now ask what happens to that flow when it comes into contact with solid boundaries. There is no steady flow when the contact is with a finite segment of a single plane wall, but a unique solution exists when the rotating fluid mass is in contact with some concave boundaries. Computed results are obtained for vortices lying inside a parabolic dish, or in a corner between two planes.
On the linear stability of compressible plane Couette flow
Authors: Duck Peter W. , Erlebacher Gordon , Hussaini M. Yousuff
Published: April 2006

The linear stability of compressible plane Couette flow is investigated. The appropriate basic velocity and temperature distributions are perturbed by a small-amplitude normal-mode disturbance. The full small-amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in some detail. It is found that instabilities can occur, although the corresponding growth rates are often quite small; the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wave speed of the disturbances approaches the velocity of either of the walls, and these regimes are also analysed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.
Multiple linear instability of layered stratified shear flow
Authors: Caulfield Colm-Cille P.
Published: April 2006

We develop a simple model for the behaviour of an inviscid stratified shear flow with a thin mixed layer of intermediate fluid. We find that the flow is simultaneously unstable to oscillatory disturbances that are a generalization of those discussed by Holmboe (1962), purely unstable modes analogous to those considered by Taylor (1931), and a new type of oscillatory disturbance at large wavelength. The relative significance of these different types of instability depends on the ratio R of the depth of the intermediate layer to the depth of the shear layer. For small values of R, the new type of oscillatory wave has both the largest growthrate for given bulk Richardson number Ri0, and is also primarily unstable to disturbances propagating at an angle to the mean flow, i.e. such modes violate the conditions of Squire's theorem (1933), and thus the assumption of initial two dimensionality of such flows is invalid. For intermediate values of R, the Holmboe-type modes and the Taylor-types modes may have wavelengths and phase speeds conducive to the formation of a resonant triad over a wide range of Ri0. Thus the presence of an intermediate layer in a stratified shear flow markedly changes its stability properties.
Numerical simulation of inelastic frictional spheres in simple shear flow
Authors: Lun C. K. K. , Bent A. A.
Published: April 2006

A numerical program is developed to simulate an assembly of inelastic frictional spheres inside a control volume undergoing rapid shearing motion induced by the top and bottom moving periodic boundaries. A sticking–sliding collision model is used to emulate binary collisions of real particles. After the flow has reached a steady state, ensemble averages of macroscopic properties such as translational and rotational granular temperatures, and kinetic and collisional stresses at different solids concentrations are obtained. The present results are compared with previous theoretical, numerical and experimental works, and favourable agreement is found among them. The simulation results show that the stresses are anisotropic and decrease with decreasing coefficient of restitution and increasing friction coefficient. At high solids fraction, above about 0.5, there exists a critical concentration where the layering effects of particles, the formation of high-density microstructures and the increase in correlation of particle velocities are the major causes of abrupt changes in flow properties.
A determination of the effective viscosity for the Brinkman–Forchheimer flow model
Authors: Givler R. C. , Altobelli S. A.
Published: April 2006

The effective viscosity μe for the Brinkman–Forchheimer flow (BFF) model has been determined experimentally for steady flow through a wall-bounded porous medium. Nuclear magnetic resonance (NMR) techniques were used to measure non-invasively the ensemble-average velocity profile of water flowing through a tube filled with an open-cell rigid foam of high porosity (ϕ = 0.972). By comparing these data with the BFF model, for which all remaining parameters were measured independently, it was determined that μe = (7.5+3.4−2.4)μf, where μf was the viscosity of the fluid. The Reynolds number, based upon the square root of the permeability, was 17.
An example of active circulation control of the unsteady separated flow past a semi-infinite plate
Authors: Cortelezzi L. , Leonard A. , Doyle J. C.
Published: April 2006

Active circulation control of the two-dimensional unsteady separated flow past a semiinfinite plate with transverse motion is considered. The rolling-up of the separated shear layer is modelled by a point vortex whose time-dependent circulation is predicted by an unsteady Kutta condition. A suitable vortex shedding mechanism introduced. A control strategy able to maintain constant circulation when a vortex is present is derived. An exact solution for the nonlinear controller is then obtained. Dynamical systems analysis is used to explore the performance of the controlled system. The control strategy is applied to a class of flows and the results are discussed. A procedure to determine the position and the circulation of the vortex, knowing the velocity signature on the plate, is derived. Finally, a physical explanation of the control mechanism is presented.
Nonlinear interaction of shear flow with a free surface
Authors: Dimas Athanassios A. , Triantafyllou George S.
Published: April 2006

In this paper the nonlinear evolution of two-dimensional shear-flow instabilities near the ocean surface is studied. The approach is numerical, through direct simulation of the incompressible Euler equations subject to the dynamic and kinematic boundary conditions at the free surface. The problem is formulated using boundary-fitted coordinates, and for the numerical simulation a spectral spatial discretization method is used involving Fourier modes in the streamwise direction and Chebyshev polynomials along the depth. An explicit integration is performed in time using a splitting scheme. The initial state of the flow is assumed to be a known parallel shear flow with a flat free surface. A perturbation having the form of the fastest growing linear instability mode of the shear flow is then introduced, and its subsequent evolution is followed numerically. According to linear theory, a shear flow with a free surface has two linear instability modes, corresponding to different branches of the dispersion relation: Branch I, at low wavenumbers; and Branch II, at high wavenumbers for low Froude numbers, and low wavenumbers for high Froude numbers. Our simulations show that the two branches have a distinctly different nonlinear evolution.

Branch I: At low Froude numbers, Branch I instability waves develop strong oval-shaped vortices immediately below the ocean surface. The induced velocity field presents a very sharp shear near the crest of the free-surface elevation in the horizontal direction. As a result, the free-surface wave acquires steep slopes, while its amplitude remains very small, and eventually the computer code crashes suggesting that the wave will break.

Branch II: At low Froude numbers, Branch II instability waves develop weak vortices with dimensions considerably smaller than their distance from the ocean surface. The induced velocity field at the ocean surface varies smoothly in space, and the free-surface elevation takes the form of a propagating wave. At high Froude numbers, however, the growing rates of the Branch II instability waves increase, resulting in the formation of strong vortices. The free surface reaches a large amplitude, and strong vertical velocity shear develops at the free surface. The computer code eventually crashes suggesting that the wave will break. This behaviour of the ocean surface persists even in the infinite-Froude-number limit.

It is concluded that the free-surface manifestation of shear-flow instabilities acquires the form of a propagating water wave only if the induced velocity field at the ocean surface varies smoothly along the direction of propagation.