bluuebunny/crossref_metadata_2025 huggingface dataset:
The structure and control of a turbulent reattaching flow
Authors: Sigurdson L. W.
Published: April 2006

An experimental study was made of the effect of a periodic velocity perturbation on the separation bubble downstream of the sharp-edged blunt face of a circular cylinder aligned coaxially with the free stream. Velocity fluctuations were produced with an acoustic driver located within the cylinder and a small circumferential gap located immediately downstream of the fixed separation line to allow communication with the external flow. The flow could be considerably modified when forced at frequencies lower than the initial Kelvin-Helmholtz frequencies of the free shear layer, and with associated vortex wavelengths comparable to the bubble height. Reattachment length, bubble height, pressure at separation, and average pressure on the face were all reduced. The effects on the large-scale structures were studied using flow photographs obtained by the smoke-wire technique. The forcing increased the entrainment near the leading edge. It was concluded that the final vortex of the shear layer before reattachment is an important element of the flow structure. There are two different instabilities involved: the Kelvin-Helmholtz instability of the free shear layer and the ‘shedding’-type instability of the entire bubble. The former consists of an interaction of the shear layer vorticity with itself, the latter with its images that result because of the presence of a wall. In order to determine the optimum forcing frequency, a method of frequency scaling is proposed which correlates data for a variety of bubbles and supports an analogy with Kármán vortex shedding.
Direct numerical simulation of controlled transition in a flat-plate boundary layer
Authors: Rist U. , Fasel H.
Published: April 2006

The three-dimensional development of controlled transition in a flat-plate boundary layer is investigated by direct numerical simulation (DNS) using the complete Navier-Stokes equations. The numerical investigations are based on the so-called spatial model, thus allowing realistic simulations of spatially developing transition phenomena as observed in laboratory experiments. For solving the Navier-Stokes equations, an efficient and accurate numerical method was developed employing fourth-order finite differences in the downstream and wall-normal directions and treating the spanwise direction pseudo-spectrally. The present paper focuses on direct simulations of the wind-tunnel experiments by Kachanovet al.(1984, 1985) of fundamental breakdown in controlled transition. The numerical results agreed very well with the experimental measurements up to the second spike stage, in spite of relatively coarse spanwise resolution. Detailed analysis of the numerical data allowed identification of the essential breakdown mechanisms. In particular, from our numerical data, we could identify the dominant shear layers and vortical structures that are associated with this breakdown process.
On the kinematics of short waves in the presence of surface flows of larger scales
Authors: Ramamonjiarisoa A.
Published: April 2006

When the resonance condition is satisfied, i.e. that the local group velocity of short surface waves matches the local velocity associated with a larger-scale surface flow, it is known that the short waves are reflected or trapped by the flow. A typical example is the case of short surface waves propagating on long surface waves. By direct numerical resolution of the kinematic equations, some aspects of the reflection or trapping are first examined. Next, the effects of a second long wave on the trajectory of the short waves are considered. It is found that the trajectory is strongly distorted in general. Reflection still occurs, having a larger effect on the variation of the shortwave wavenumber than when only a single long wave is present. The entrapment becomes more sporadic. At short time intervals, a forced Mathieu equation is found to govern the short-wave development. This leads to a discussion on a more general physical context.
Oscillatory internal shear layers in rotating and precessing flows
Authors: Hollerbach R. , Kerswell R. R.
Published: April 2006

We present a direct numerical solution of a particular inertial oscillation, the so-called 'spin-over’ mode, in spherical geometry. This mode is particularly relevant to the fluid flow within a precessing oblate spheroid. We demonstrate that the oscillatory Ekman layer breaks down at ±30° latitude, and that this breakdown spawns internal shear layers. We show that the structure of these shear layers is different for a full sphere and a spherical shell, as noted in the preceding paper (Kerswell 1995). Despite the existence of these shear layers, however, the numerical decay rates agree to within 1 % with the asymptotic decay rates, which neglect any possible shear layers. Finally, we consider the nonlinear mean flow profiles driven by this mode, and demonstrate that our numerical results agree reasonably well with experimental results.
A wall-wake model for the turbulence structure of boundary layers. Part 1. Extension of the attached eddy hypothesis
Authors: Perry A. E. , Marušić Ivan
Published: April 2006

The attached eddy hypothesis developed for zero pressure gradient boundary layers and for pipe flow is extended here to boundary layers with arbitrary streamwise pressure gradients, both favourable and adverse. It is found that in order to obtain the correct quantitative results for all components of the Reynolds stresses, two basic types of eddy structure geometries are required. The first type, called type-A, is interpreted to give a ‘wall structure’ and the second, referred to as type-B, gives a ‘wake structure’. This is in analogy with the conventional mean velocity formulation of Coles where the velocity is decomposed into a law of the wall and a law of the wake.

If the above mean velocity formulation is accepted, then in principle, once the eddy geometries are fixed for the two eddy types, all Reynolds stresses and associated spectra contributed from the attached eddies can be computed without any further empirical constants. This is done by using the momentum equation and certain convolution integrals developed here based on the attached eddy hypothesis. The theory is developed using data from equilibrium and quasi-equilibrium flows. In Part 2 the authors’ non-equilibrium data are used.
A numerical study of flow past a rotating circular cylinder using a hybrid vortex scheme
Authors: Chew Y. T. , Cheng M. , Luo S. C.
Published: April 2006

The vortex shedding and wake development of a two-dimensional viscous incompressible flow generated by a circular cylinder which begins its rotation and translation impulsively in a stationary fluid is investigated by a hybrid vortex scheme at a Reynolds number of 1000. The rotational to translational speed ratio α varies from 0 to 6. The method used to calculate the flow can be considered as a combination of the diffusion-vortex method and the vortex-in-cell method. More specifically, the full flow field is divided into two regions: near the body surface the diffusion-vortex method is used to solve the Navier–Stokes equations, while the vortex-in-cell method is used in the exterior inviscid domain. Being more efficient, the present computation scheme is capable of extending the computation to a much larger dimensionless time than those reported in the literature.

The time-dependent pressure, shear stress and velocity distributions, the Strouhal number of vortex shedding as well as the mean lift, drag, moment and power coefficients are determined together with the streamline and vorticity flow patterns. When comparison is possible, the present computations are found to compare favourably with published experimental and numerical results. The present results seem to indicate the existence of a critical α value of about 2 when a closed streamline circulating around the cylinder begins to appear. Below this critical α, Kármán vortex shedding exists, separation points can be found, the mean lift and drag coefficients and Strouhal number increase almost linearly with α. Above α ≈ 2, the region enclosed by the dividing closed streamline grows in size, Kármán vortex shedding ceases, the flow structure, pressure and shear stress distributions around the cylinder tend towards self-similarity with increase α, and lift and drag coefficients approach asymptotic values. The optimum lift to drag ratio occurs at α ≈ 2. The present investigation confirms Prandtl's postulation of the presence of limiting lift force at high α, and thus the usefulness of the Magnus effect in lift generation is limited.

The results show that the present method can be used to calculate not only the global characteristics of the separated flow, but also the precise evolution with time of the fine structure of the flow field.
Liquid-metal flow in a system of electrically coupled U-bends in a strong uniform magnetic field
Authors: Molokov Sergei , Stieglitz Robert
Published: April 2006

Liquid-metal magnetohydrodynamic flow in a system of electrically coupled U-bends in a strong uniform magnetic field is studied. The ducts composing the bends are electrically conducting and have rectangular cross-sections. It has been anticipated that very strong global electric currents are induced in the system, which modify the flow pattern and produce a very high pressure drop compared to the flow in a single U-bend. A detailed asymptotic analysis of flow for high values of the Harmann number (in fusion blanket applications of the order of 103−104) shows that circulation of global currents results in several types of peculiar flow patterns. In ducts parallel to the magnetic field a combination of helical and recirculatory flow types may be present and vary from one bend to another. The magnitude of the recirculatory motion may become very high depending on the flow-rate distribution between the bends in the system. The recirculatory flow may account for about 50% of the flow in all bends. In addition there are equal and opposite jets at the walls parallel to the magnetic field, which are common to any two bends. The pressure drop due to three-dimensional effects linearly increases with the number of bends in a system and may significantly affect the total pressure drop. To suppress this and some other unwelcome tendencies either the ducts perpendicular to the magnetic field should be electrically separated, or the flow direction in the neighbouring ducts should be made opposite, so that leakage currents cancel each other.
Generalization of the Forchheimer-extended Darcy flow model to the tensor premeability case via a variational principle
Authors: Knupp P. M. , Lage J. L.
Published: April 2006

A convex variational principle is used to obtain a generalization of the empirical nonlinear one-dimensional Forchheimer-extended Darcy flow equation to the multidimensional and anisotropic (tensor permeability) case. A modified permeability that is a function of flow velocity (or pressure gradient) is introduced in order to transform the nonlinear flow equation into a pseudo-linear form. Imposing an incompressibility condition on this pseudo-linear equation leads to a flow equation in Euler–Lagrange form which is used to build the corresponding variational principle. It is demonstrated that the variational principle is based on minimizing the power (time rate of doing work) required by the fluid to flow at a certain velocity under a prescribed pressure gradient. A consistent generalization of the Forchheimer equation to the tensor case then follows from the variational principle. The existence and uniqueness of solutions to the nonlinear flow equations might also be demonstrated using the variational principle on a case by case basis, once appropriate boundary conditions are chosen.
Lateral capillary interaction between particles protruding from a spherical liquid layer
Authors: Kralchevsky P. A. , Paunov V. N. , Nagayama Kuniaki
Published: April 2006

The lateral capillary interaction between two particles immersed in a spherical thin liquid film is investigated. The interfacial shape, the lateral capillary force and the interparticle energy are calculated by using the numerical solution of the linearized Laplace equation of capillarity. Orthogonal bipolar coordinates on a sphere (inducing biconical coordinates in space) are introduced as a helpful instrument for solving this problem and other problems of similar geometry. We consider two types of boundary conditions at the particle surfaces: fixed contact angle and fixed contact line. We established that for particles of fixed contact angle the capillary interaction energy depends monotonically on the interparticle distance whereas for particles of fixed contact line the interaction energy exhibits a maximum. The numerical results show that in both cases the capillary interaction is much larger than the thermal energy kT and can induce aggregation and ordering of submicrometre particles. These theoretical findings can be important for understanding the properties of Pickering emulsions (stabilized by particles) and liposomes or biomembranes containing incorporated membrane proteins.
Linear pressure waves in fogs
Authors: Duraiswami R. , Prosperetti A.
Published: April 2006

The modelling of small-amplitude pressure waves in dilute single- or multi-component fogs by means of averaged equations is considered. The problem is cast in a singular-perturbation framework in which the suspended droplets are the singularities. This point of view simplifies the local problem in the vicinity of the droplets. Matching in the overlap region provides the coupling with the averaged fields. Among the advantages of the method is the fact that the leading-order effects are clearly identified. In particular it is shown that, for low-amplitude waves and far below the fluid's critical point, phase change effects only start to be important when the vapour mean free path becomes comparable with the drop radius and dominate for yet smaller drops.

This present method for the derivation of effective equations appears to be of general applicability to a variety of multi-phase situations and is illustrated in detail.
Interfacial instabilities in plane Poiseuille flow of two stratified viscoelastic fluids with heat transfer. Part 1. Evolution equation and stability analysis
Authors: Joo Sang W.
Published: April 2006

An evolution equation is derived that describes the nonlinear development of the interface between two viscoelastic fluids flowing, under the action of imposed pressure gradient and gravity, in a vertical channel. The channel walls are kept at different temperatures, resulting in heat transfer across the layers. The equation, based on the lubrication approximation, models the effects of stratifications in density, viscosity, elasticity, shear thinning, and thermal conductivity. It also describes the capillary and thermocapillary effects, as well as the sensitivity of viscosities to temperature. Linear-stability analysis is performed based on the evolution equation to understand the competing effects of viscous, elastic, and Marangoni instabilities. Particular attention is paid to the active control of the interfacial instabilities through the thermocapillarity.
A numerical study of the wave-induced solute transport above a rippled bed
Authors: Shum K. T.
Published: April 2006

The role of wave-induced separated flow in solute transport above a rippled bed is studied from numerical solutions to the two-dimensional Navier–Strokes equations and the advection-diffusion equation. A horizontal ambient flow that varies sinusoidally in time is imposed far above the bed, and a constant concentration difference between the upper and lower boundaries of computation is assumed. The computed flow field is the sum of an oscillatory rectilinear flow and a vortical flow which is periodic both in time and in the horizontal. Poincaré sections of this flow suggest chaotic mixing. Vertical lines of fluid particles above the crest and above the trough deform into whorls and tendrils, respectively, in just one wave period. Horizontal lines near the bottom deform into Smale horseshoe patterns. The combination of high shear and vortex-induced normal velocity close to the sediment surface results in large net displacements of fluid particles in a period. The resulting advective transport normal to the bed can be higher than molecular diffusion from well within the viscous boundary layer up to a few ripple heights above the bed. When this flow field is applied to the transport equation of a passive scalar, two distinct features – regular temporal oscillations in concentration and a linear time-averaged vertical concentration profile – are found immediately above the bed. These features have also been observed previously in field measurements on oxygen concentration. Advective transport is shown to be dominant even in the region where the time-averaged concentration profile is linear, a region where vertical solute transport has often been estimated using diffusion-type models in many field studies.
Geometric focusing of internal waves
Authors: Maas Leo R. M. , Lam Frans-Peter A.
Published: April 2006

The spatial structure of the streamfunction field of free, linear internal waves in a two-dimensional basin is governed by the canonical, second-order, hyperbolic equation on a closed domain. Its solution can be determined explicitly for some simple shapes of the basin. It consists of an algorithm by which ‘webs’ of uniquely related characteristics can be constructed and the prescription of one (independent) value of a field variable, related to the streamfunction, on each of these webs. The geometric construction of the webs can be viewed as an alternative version of a billiard game in which the angle of reflection equals that of incidence with respect to the vertical (rather than to the normal). Typically, internal waves are observed to be globally attracted (‘focused’) to a limiting set of characteristics. This attracting set can be classified by the number of reflections it has with the surface (its period in the terminology of dynamical systems). This period of the attractor is a fractal function of the normalized period of the internal waves: large regions of smooth, low-period attractors are seeded with regions with high-period attractors. Occasionally, all internal wave rays fold exactly back upon themselves, a ‘resonance’: focusing is absent and a smooth pattern, familiar from the cellular pattern in a rectangular domain, is obtained. These correspond to the well-known seiching modes of a basin. An analytic set of seiching modes has also been found for a semi-elliptic basin. A necessary condition for seiching to occur is formulated.
A laminar roughness boundary condition
Authors: Tuck E. O. , Kouzoubov A.
Published: April 2006

A modified slip boundary condition is obtained to represent the effects of small roughness-like perturbations to an otherwise-plane fixed wall which is acting as a boundary to steady laminar flow of a viscous fluid. In its simplest form, for low local Reynolds number and small roughness slope, this boundary condition involves a constant apparent backflow at the mean surface or, equivalently, represents a shift of the apparent plane boundary toward the flow domain. Extensions of the theory are also made to include finite local Reynolds number and finite roughness slope.
Surfactant effects on the dynamics of a thin liquid sheet
Authors: Luca Luigi De , Meola Carosena
Published: April 2006

The dynamics of a free-surface slender two-dimensional stream (liquid sheet) issuing from a nozzle in the gravitational field in still air, under the effect of surface-active agents, are analysed experimentally. The particular test section geometry (the liquid is forced to assume a bidimensional form between two vertical guides and a horizontal plate placed at a certain variable distance from the nozzle exit section) employed in this study gives rise to various flow regimes depending on the governing parameters: liquid flow rate, sheet height, surface pressure, gravity. Two basic phenomena are observed: thinning of the sheet (with recirculating motion inside it) and sheet-threadlines transition. For a certain surfactant (bulk) concentration, there exists a minimum critical flow rate value for which the sheet is seen to thin starting at both of the sheet bottom corners. A ridge, usually referred to as a Reynolds ridge in the literature, separates the sheet from the thin-film regions. The thin films exhibit recirculating flows (caused by the onset of surfactant-induced surface-pressure-driven convection in the gravitational field) and extend to the entire rectangular interface as the flow rate is reduced. At zero flow rate the thinned sheet resembles a plane vertical soap film showing a recirculating cellular structure. These phenomena are linked to the presence of surface-active material adsorbed at the liquid-air interface and occur when the sheet height exceeds a critical value. Otherwise, at a critical flow rate value the liquid sheet breaks up into an array of (more or less regularly distributed) discrete threadlines (vertical jets), whose spacing depends on the surface tension of the test liquid.
Stability of spatially developing boundary layers in pressure gradients
Authors: Govindarajan Rama , Narasimha R.
Published: April 2006

A new formulation of the stability of boundary-layer flows in pressure gradients is presented, taking into account the spatial development of the flow and utilizing a special coordinate transformation. The formulation assumes that disturbance wavelength and eigenfunction vary downstream no more rapidly than the boundary-layer thickness, and includes all terms nominally of orderR−1in the boundary-layer Reynolds numberR. In Blasius flow, the present approach is consistent with that of Bertolottiet al.(1992) toO(R−1) but simpler (i.e. has fewer terms), and may best be seen as providing a parametric differential equation which can be solved without having to march in space. The computed neutral boundaries depend strongly on distance from the surface, but the one corresponding to the inner maximum of the streamwise velocity perturbation happens to be close to the parallel flow (Orr-Sommerfeld) boundary. For this quantity, solutions for the Falkner-Skan flows show the effects of spatial growth to be striking only in the presence of strong adverse pressure gradients. As a rational analysis toO(R−1) demands inclusion of higher-order corrections on the mean flow, an illustrative calculation of one such correction, due to the displacement effect of the boundary layer, is made, and shown to have a significant destabilizing influence on the stability boundary in strong adverse pressure gradients. The effect of non-parallelism on the growth of relatively high frequencies can be significant at low Reynolds numbers, but is marginal in other cases. As an extension of the present approach, a method of dealing with non-similar flows is also presented and illustrated.

However, inherent in the transformation underlying the present approach is a lower-order non-parallel theory, which is obtained by dropping all terms of nominal orderR−1except those required for obtaining the lowest-order solution in the critical and wall layers. It is shown that a reduced Orr-Sommerfeld equation (in transformed coordinates) already contains the major effects of non-parallelism.
Bifurcation of swirl in liquid cones
Authors: Shtern Vladimir , Barrero Antonio
Published: April 2006

We show that rotation appears owing to bifurcation in primarily pure meridional steady motion of viscous incompressible fluid. This manifestation of the laminar axisymmetric 'swirl dynamo’ occurs in flows inside liquid conical menisci with the cone half-angle θc < 90°. The liquid flows towards the cone apex near the surface and moves away along the axis driven by (i) surface shear stresses (typical for electrosprays) or (ii) by body Lorentz forces (e.g. in the process of cathode eruption). When the motion intensity increases and passes a critical value, new swirling regimes appear resulting from the supercritical pitchfork bifurcation. This agrees with recent observations of swirl in Taylor cones. We find that when the swirl Reynolds number Γc reaches a threshold value, flow separation occurs and the meridional motion becomes two-cellular with inflows near both the surface and the axis, and an outflow near the cone θ = θs, 0 < θs < θc. In the limit of high Γc, the angular thickness of the near-surface cell tends to zero. In case (i) the swirl is concentrated near the surface while the motion inside the inner cell becomes purely meridional with the radial velocity being uniform. We also study the two-phase flow of a liquid inside and a gas outside the meniscus. Flow separation occurs in both media and then swirl is concentrated near the interface. In case (ii) we reveal another interesting effect: a cascade of flow separations near the axis. As the driving forces increase, meridional motion becomes multi-cellular although very slow in comparison with swirl. To cover all ranges of parameters we combine numerical calculations and asymptotic analyses.
Mixed-convection laminar film condensation on a semi-infinite vertical plate
Authors: Shu Jian-Jun , Wilks Graham
Published: April 2006

The flow of a uniform stream of pure saturated vapour past a cold, semi-infinite vertical plate is examined. The formulation incorporates the limits of both pure forcedconvection and pure body-force-convection laminar film condensation. Detailed asymptotic and exact numerical solutions are obtained and comparisons drawn with approximate methods and experimental results reported in the literature.
Symmetry, sidewalls, and the transition to chaos in baroclinic systems
Authors: Mundt Michael D. , Hart John E. , Ohlsen Daniel R.
Published: April 2006

A high-resolution, quasi-geostrophic numerical model is utilized to examine two-layer baroclinic flow in a cylinder. Particular attention is given to the role of horizontal shear of the basic state induced by viscosity near the cylinder wall, and to the desymmetrization brought about by the cylindrical geometry, in the transition to baroclinic chaos. Solutions are computed for both f-plane and β-plane situations, and the results are compared to previous laboratory experiments. Agreement in the former case is found to be good, although the onset of chaos occurs at slightly lower forcing in the laboratory when its basic flow is prograde, and at higher forcing amplitude when the experimental basic azimuthal currents are retrograde. This suggests that the modest discrepancies may be attributable to computationally neglected ageostrophic effects in the interior fluid and Ekman boundary layers. When β ≠ 0, the numerical and laboratory results are in excellent agreement. The computational simulations indicate that the viscous sidewall boundary layer plays a pivotal role in the dynamics. Moreover, in contrast to previous studies performed in a periodic, rectilinear channel, the route to chaos is largely temporal and involves relatively few spatial modes. The reduction in symmetries upon going from f-plane channel to either f-plane or β-plane cylinder models leads to fewer secondary instabilities and fewer spatial modes that are active in the dynamics.
The interaction of an isotropic field of acoustic waves with a shock wave
Authors: Mahesh Krishnan , Lee Sangsan , Lele Sanjiva K. , Moin Parviz
Published: April 2006

Moore's (1954) inviscid linear analysis of the interaction of a shock wave with a plane acoustic wave is evaluated by comparison to computation. The analysis is then extended to study the interaction of an isotropic field of acoustic waves with a normal shock wave. The evolution of fluctuating kinetic energy, sound level and thermodynamic fluctuations across the shock wave are examined in detail.

The interaction of acoustic fluctuations with the shock is notably different from that of vortical fluctuations. The kinetic energy of the acoustic fluctuationsdecreasesacross the shock wave for Mach numbers between 1.25 and 1.8. For Mach numbers exceeding 3, the kinetic energy amplifies by levels that significantly exceed those found in the interaction of vortical fluctuations with the shock. Upon interacting with the shock wave, the acoustic waves generate vortical fluctuations whose contribution to the far-field kinetic energy increases with increasing Mach number. The level of sound increases across the shock wave. The rise in the sound pressure level across the shock varies from 5 to 20 dB for Mach number varying from 1.5 to 5. The fluctuations behind the shock wave are nearly isentropic for Mach number less than 1.5, beyond which the generation of entropy fluctuations becomes significant.
The spreading of variable-viscosity axisymmetric radial gravity currents: applications to the emplacement of Venusian ‘pancake’ domes
Authors: Sakimoto S. E. H. , Zuber M. T.
Published: April 2006

The Magellan images of Venus have revealed a number of intriguing volcanic features, including the steep-sided or ‘pancake’ domes. These volcanic domes or flows have morphologies that suggest formation by a single continuous emplacement of lava with a higher viscosity than that of the surrounding basaltic plains. Numerous investigators have suggested that such high viscosity is due to high silica content, leading to the conclusion that the domes are evidence of evolved magmatic products on Venus. However, viscosity depends on crystallinity as well as on silica content: high viscosity could therefore also be due to a cooler (and therefore higher crystal content) lava. Models of dome emplacement which include both cooling and composition factors are thus necessary in order to determine the ranges of crystallinity and silica content which might lead to the observed gross dome morphologies. Accordingly, in this study domes are modelled as radial viscous gravity currents with an assumed cooling-induced viscosity increase to include both effects. Analytical and numerical results indicate that pancake dome formation is feasible with compositions ranging from basaltic to rhyolitic. Therefore, observations of gross dome morphology alone are insufficient for determining composition and the domes do not necessarily represent strong evidence for evolved magmatism on Venus.
Parasitic capillary waves: a direct calculation
Authors: Longuet-Higgins Michael S.
Published: April 2006

As in a previous theory (Longuet-Higgins 1963) parasitic capillary waves are considered as a perturbation due to the local action of surface tension forces on an otherwise pure progressive gravity wave. Here the theory is improved by: (i) making use of our more accurate knowledge of the profile of a steep Stokes wave; (ii) taking account of the influence of gravity on the capillary waves themselves, through the effective gravitational acceleration g* for short waves riding on longer waves.

Nonlinearity in the capillary waves themselves is not included, and certain other approximations are made. Nevertheless, the theory is shown to be in essential agreement with experiments by Cox (1958), Ebuchi, Kawamura & Toba (1987) and Perlin, Lin & Ting (1993).

A principal result is that for gravity waves of a given length L > 5 cm there is a critical steepness parameter (AK)c at which the surface velocity (in a frame of reference moving with the phase-speed) equals the minimum (local) speed of capillary-gravity waves. On subcritical gravity waves, with steepness AK < (AK)c, capillary waves may be generated at all points of the wave surface. On supercritical waves, with AK > (AK)c, capillary waves can only be generated in the wave troughs; they are trapped between two caustics near the crests. Generally, the amplitude of the parasitic capillaries is greatest on gravity waves of near critical (but not maximum) steepness.
Hydrodynamic stability of a sheared liquid film
Authors: Miesen Rob , Boersma Bendiks Jan
Published: April 2006

We study the hydrodynamic stability of a thin layer of liquid that is sheared by a gas. First, the interface conditions for the free surface approximation of the problem are discussed. We then study the stability of the flow to disturbances with phase speeds smaller than the maximum velocity in the liquid film, i.e. the internal mode, extending previous results and resolving some apparent contradictions.

The dynamic effect of the gas is studied by dropping the free surface approximation and solving the Orr-Sommerfeld equation for the gas together with that for the liquid. The effect on the stability of the liquid film is very large, which is explained by the fact that the imaginary part of the wave speed (which determines the stability of the film) is very small. Consequently the free surface approximation is, in general, not correct.

We then study the dependence of the critical Reynolds number on the Weber number, on the curvature of the liquid velocity profile and on the properties of the gas. With the gas included, a second mode of instability is found which has a phase velocity that is, in general, larger than the maximum liquid velocity and corresponds to capillary-gravity waves. We compare results with experiments from the literature; good agreement is found. Finally, a suggestion on the relevance of this study to the generation of ‘roll waves’, which are important from a practical point of view, is given.
The inviscid transonic flow about a cylinder
Authors: Botta Nicola
Published: April 2006

The two-dimensional inviscid transonic flow about a circular cylinder is investigated. To do this, the Euler equations are integrated numerically with a time-dependent technique. The integration is based on an high-resolution finite volume upwind method.

Time scales are introduced and the flow at very short, short and large times is studied. Attention is focused on the behaviour of the numerical solution at large times, after the breakdown of symmetry and the onset of an oscillating solution have occurred. This solution is known to be periodic at Mach number between 0.5 and 0.6.

At higher speed, however, a richer behaviour is observed. As the Mach number is increased from 0.6 to 0.98 the numerical solution undergoes two transitions. Through a first one the periodical, regular flow enters a chaotic, turbulent regime. Through the second transition the chaotic flow comes back to an almost stationary state. The flow in the chaotic and in the almost stationary regimes is investigated. A numerical conjecture for the behaviour of the solution at large times is advanced.
The nonlinear excitation of synchronous edge waves by a monochromatic wave normally approaching a plane beach
Authors: Blondeaux P. , Vittori G.
Published: April 2006

The possible excitation of synchronous edge waves by a monocromatic wave normally approaching a plane beach is studied. Use is made of the full three-dimensional water wave theory but only beach angles β = π/(2N), where N is an integer, are considered. A weakly nonlinear stability analysis is used to investigate the interaction of subharmonic, synchronous and 3/2-frequency edge waves with the incoming wave field. It is shown that values of β exist for which an energy transfer from the incoming wave to the synchronous edge waves takes place through the intermediary of the subharmonic components.
Three-dimensional convection driven by centrifugal buoyancy
Authors: Auer M. , Busse F. H. , Clever R. M.
Published: April 2006

Convection driven by centrifugal buoyancy in a cylindrical fluid annulus cooled from the inside, heated from the outside, and rotating about its axis is described. While at high values of the dimensionless rotation parameter τ convection rolls aligned with the axis are preferred, three-dimensional patterns of convection are introduced at low values of τ through either the cross-roll instability or a subharmonic varicose instability. The instabilities are studied in terms of simple analytical relationships as well as through numerical methods based on the Galerkin scheme. Analytical expressions for the steady three-dimensional patterns induced by the instabilities are also derived.
Effects of interaction between Marangoni and double-diffusive instabilities
Authors: Tanny J. , Chen C. C. , Chen C. F.
Published: April 2006

The effect of surface tension on the onset of convection in horizontal double-diffusive layer was studied both experimentally and by linear stability analysis. The experiments were conducted in a rectangular tank with base dimension of 25×13 cm and 5 cm in height. A stable solute (NaCl) stratification was first established in the tank, and then a vertical temperature gradient was imposed. Vertical temperature and concentration profiles were measured using a thermocouple and a conductivity probe and the flow patterns were visualized by a schlieren system. Two types of experiments were carried out which illustrate the effect of surface tension on the onset of convection. In the rigid–rigid experiments, when the critical thermal Rayleigh number, RT, is reached, large double-diffusive plumes were seen simultaneously to rise from the heated bottom and descend from the cooled top. In the rigid–free experiments, owing to surface tension effects, the first instability onset was of the Marangoni type. Well-organized small plumes were seen to emerge and persist close to the top free surface at a relatively small RTM (where subscript M denotes ‘Marangoni’). At larger RTt > RTM (where subscript t denotes ‘top’) these plumes evolved into larger double-diffusive plumes. The onset of double-diffusive instability at the bottom region occurred at a still higher RTb > RTt (where subscript b denotes ‘bottom‘). A series of stability experiments was conducted for a layer with an initial top concentration of 2 wt% and different concentration gradients. The stability map shows that in the rigid–free case the early Marangoni instability in the top region reduces significantly the critical RT for the onset of double-diffusive convection. Compared with the rigid–rigid case, the critical RT in the top region is reduced by about 60% and in the bottom region by about 30%. The results of the linear stability analysis, which takes into account both surface tension and double-diffusive effects, are in general agreement with the experiments. The analysis is then applied to study the stability characteristics of such a layer as gravity is reduced to microgravity level. Results show that even at 10 −4g0, where g0 is the gravity at sea level, the double-diffusive effect is of equal importance to the Marangoni effect.
Transverse instability of plane wavetrains in gas-fluidized beds
Authors: Göz M. F.
Published: April 2006

Using a two-fluid model of gas-fluidized beds, it is shown that periodic plane voidage waves travelling against gravity are unstable to perturbations with large transverse wavelength. This secondary instability sets in at arbitrarily small amplitudes of the plane wave and correspondingly small transverse wavenumbers of the two-dimensional perturbation. More precisely, if the bed is wide enough to accommodate sufficiently long horizontal waves, then the plane wave becomes unstable as soon as its amplitude has grown to the order of the square of the transverse wavenumber. The instability can be stationary or oscillatory in nature and has its origin in the interaction between the plane wave and four least-stable modes with small transverse wavenumber. Two of them represent a pair of bubble-like ‘mixed modes‘; the other two are initially, i. e. at the onset of the primary wave, pure transverse modes, one consisting only of an initially pure vertical velocity perturbation of the state of uniform fluidization. Depending on a relation between the eigenvalues of the least-stable modes at the primary bifurcation point, either one of these can be the dominant mode, which becomes (most) unstable along the growing vertically travelling plane wave. While the transverse modes gain longitudinal structure during this process, the mixed modes obtain a vertical component of the vertically averaged velocity as well, so that it appears that the secondary instability described here is a variant of Batchelor & Nitsche's (1991) ‘overturning’ instability found recently for unbounded stratified fluids, see also Batchelor (1993).
The structure of highly sheared turbulence
Authors: Souza F. A. de , Nguyen V. D. , Tavoularis S.
Published: April 2006

Uniformly sheared flows have been generated in a high-speed wind tunnel at shear rates higher than previously achieved, in an effort to approach those in the inner turbulent boundary layer. As at lower shear rates, the turbulence structure was found to attain a self-similar state with approximately constant anisotropies and exponential kinetic energy growth. The normal Reynolds stress anisotropies showed no systematic dependence upon the mean shear within the examined range; however, the shear stress anisotropy was significantly lower than the low-shear values, in conformity with boundary layer measurements and direct numerical simulations of homogeneous shear flow.
On improving a one-layer ocean model with thermodynamics
Authors: Ripa P.
Published: April 2006

A popular method used to incorporate thermodynamic processes in a shallow water model (e.g. one used to study the upper layer of the ocean) is to allow for density variations in time and horizontal position, but keep all dynamical fields as depth independent. This is achieved by replacing the horizontal pressure gradient by its vertical average. These models have limitations, for instance they cannot represent the ‘thermal wind’ balance (between the horizontal density gradient and the vertical shear of the velocity) which dominates at low frequencies. A new model is now proposed which uses velocity and density fields varying linearly with depth, with coefficients that are functions of horizontal position and time. This model can explicitly represent the thermal wind balance, but its use is not restricted to low- frequency dynamics.

Volume, mass, buoyancy variance, energy and momentum are conserved in the new model. Furthermore, these integrals of motion have the same dependence on the dynamical fields as the exact (continuously stratified) case. The evolution of the three components of the absolute vorticity field are correctly represented. Conservation of density–potential vorticity is not fulfilled, though, owing to artificial removal of the vertical curvature of the velocity field.

The integrals of motion are used to construct a ‘free energy’ [Escr ]f, which is quadratic to the lowest order in the deviation from a steady state with (at most) a uniform velocity field. [Escr ]fis positive definite, and therefore the free evolution of the system cannot lead to an ‘explosion’ of the dynamical fields. (This is not the case if the velocity shear and/or the density vertical gradient is excluded in the model, which results in a non-negative definite free energy.)

In a model with one active layer, linear waves on top of a steady state with no currents are, to a very good approximation, those of the first two vertical modes of the continuously stratified model. These are the familiar geophysical gravity and vortical waves (e.g. Poincaré, Rossby, and coastal Kelvin waves at mid-latitudes, equatorial waves, etc.).

Finally, baroclinic instability is well represented in the new model. For long perturbations (wavelengths of the order of the deformation radius of the first mode) the agreement with more precise calculations is excellent. On the other hand, the comparison with the eigenvalues of Eady's problem (which corresponds to wavelengths of the order of the deformation radius of the second mode) shows differences of the order of 40%. Nevertheless, the new model does have a high-wavenumber cutoff, even though it is constrained to linear profiles in depth and therefore cannot reproduce the exponential trapping of Eady's problem eigensolutions.

In sum, the integrals of motion, vorticity dynamics, free waves and baroclinic instability results all give confidence in the new model. Its main novelty, however, lies in the ability to incorporate thermodynamic processes.
Stability of linear shear flows in shallow water
Authors: Knessl Charles , Keller Joseph B.
Published: April 2006

The stability or instability of various linear shear flows in shallow water is considered. The linearized equations for waves on the surface of each flow are solved exactly in terms of known special functions. For unbounded shear flows, the exact reflection and transmission coefficients R and T for waves incident on the flow, are found. They are shown to satisfy the relation |R|2= 1+ |T|2, which proves that over reflection occurs at all wavenumbers. For flow bounded by a rigid wall, R is found. The poles of R yield the eigenvalue equation from which the unstable mides can be found. For flow in a channel, with two rigid walls, the eigenvalue equation for the modes is obtained. The results are compared with previous numerical results.
On two approaches to the problem of instability of short-crested water waves
Authors: Badulin Sergei I. , Shrira Victor I. , Kharif Christian , Ioualalen Mansour
Published: April 2006

The work is concerned with the problem of the linear instability of symmetric short-crested water waves, the simplest three-dimensional wave pattern. Two complementary basic approaches were used. The first, previously developed by Ioualalen & Kharif (1993, 1994), is based on the application of the Galerkin method to the set of Euler equations linearized around essentially nonlinear basic states calculated using the Stokes-like series for the short-crested waves with great precision. An alternative analytical approach starts with the so-called Zakharov equation, i.e. an integro-differential equation for potential water waves derived by means of an asymptotic procedure in powers of wave steepness. Both approaches lead to the analysis of an eigenvalue problem of the type
{\rm det}|{\boldmath A}-\gamma{\boldmath B}|=0
where A and B are infinite square matrices. The first approach should deal with matrices of quite general form although the problem is tractable numerically. The use of the proper canonical variables in our second approach turns the matrix B into the unit one, while the matrix A gets a very specific ‘nearly diagonal’ structure with some additional (Hamiltonian) properties of symmetry. This enables us to formulate simple necessary and sufficient a priori criteria of instability and to find instability characteristics analytically through an asymptotic procedure avoiding a number of additional assumptions that other authors were forced to accept.

A comparison of the two approaches is carried out. Surprisingly, the analytical results were found to hold their validity for rather steep waves (up to steepness 0.4) for a wide range of wave patterns. We have generalized the classical Phillips concept of weakly nonlinear wave instabilities by describing the interaction between the elementary classes of instabilities and have provided an understanding of when this interaction is essential. The mechanisms of the relatively high stability of short-crested waves are revealed and explained in terms of the interaction between different classes of instabilities. A helpful interpretation of the problem in terms of an infinite chain of interacting linear oscillators was developed.