bluuebunny/crossref_metadata_2025 huggingface dataset:
Time-dependent isolated anomalies in zonal flows
Authors: Helfrich Karl R. , Pedlosky Joseph
Published: April 2006

A theory is developed for time-dependent coherent structures in a marginally stable atmospheric zonal flow. The coherent structures have the form of solitary waves travelling in the zonal direction. Analytical solutions are found for stationary solitary waves but these are shown to be always unstable. The instability manifests itself either as a fission of the structure subsequently emitting two oppositely directed travelling solitary waves or as an implosion in which the structure becomes increasingly more narrow and intense. Which of the two occurs depends sensitively on initial conditions. These solitary waves are stable in head-on collisions only if their joint zonally integrated amplitude is less than a critical value; otherwise, the implosion instability occurs. General initial conditions can give rise to solitary waves which either split, implode, or break down to form a train of nonlinear wave packets. A scenario for the birth and decay of isolated disturbances is given, utilizing the slow parametric transit of the marginal stability curve of the background zonal flow.
Direct numerical simulation of isotropic turbulence interacting with a weak shock wave
Authors: Lee Sangsan , Lele Sanjiva K. , Moin Parviz
Published: April 2006

Interaction of isotropic quasi-incompressible turbulence with a weak shock wave was studied by direct numerical simulations. The effects of the fluctuation Mach number Mt of the upstream turbulence and the shock strength M21 — 1 on the turbulence statistics were investigated. The ranges investigated were 0.0567 ≤ Mt ≤ 0.110 and 1.05 ≤ M1 ≤ 1.20. A linear analysis of the interaction of isotropic turbulence with a normal shock wave was adopted for comparisons with the simulations.

Both numerical simulations and the linear analysis of the interaction show that turbulence is enhanced during the interaction with a shock wave. Turbulent kinetic energy and transverse vorticity components are amplified, and turbulent lengthscales are decreased. The predictions of the linear analysis compare favourably with simulation results for flows with M2t < a(M21 — 1) with a ≈ 0.1, which suggests that the amplification mechanism is primarily linear. Simulations also showed a rapid evolution of turbulent kinetic energy just downstream of the shock, a behaviour not reproduced by the linear analysis. Investigation of the budget of the turbulent kinetic energy transport equation shows that this behaviour can be attributed to the pressure transport term.

Shock waves were found to be distorted by the upstream turbulence, but still had a well-defined shock front for M2t < a(M21— 1) with a ≈ 0.1). In this regime, the statistics of shock front distortions compare favourably with the linear analysis predictions. For flows with M2t > a(M21— 1 with a ≈ 0.1, shock waves no longer had well-defined fronts: shock wave thickness and strength varied widely along the transverse directions. Multiple compression peaks were found along the mean streamlines at locations where the local shock thickness had increased significantly.
Flow past a circular cylinder on a β-plane
Authors: Johnson E. R. , Page M. A.
Published: April 2006

This paper gives analytical and numerical solutions for both westward and eastward flows past obstacles on a β-plane. The flows are considered in the quasi-geostrophic limit where nonlinearity and viscosity allow deviations from purely geostrophic flow. Asymptotic solutions for the layer structure in almost-inviscid flow are given for westward flow past both circular and more elongated cylindrical obstacles. Structures are given for all strengths of nonlinearity from purely linear flow through to strongly nonlinear flows where viscosity is negligible and potential vorticity conserved. These structures are supported by accurate numerical computations. Results on detraining nonlinear western boundary layers and corner regions in Page & Johnson (1991) are used to present the full structure for eastward flow past an obstacle with a Huff rear face, completing previous analysis in Page & Johnson (1990) of eastward flow past obstacles without rear stagnation points. Viscous separation is discussed and analytical structures proposed for separated flows. These lead to predictions for the size of separated regions that reproduce the behaviour observed in experiments and numerical computations on β-plane flows.
On longitudinal vortices in curved channel flow
Authors: Bottaro Alessandro
Published: April 2006

The laminar flow in a curved channel is studied numerically to analyse the initial formation, development and interaction phenomena of an array of centrifugally induced longitudinal vortices arranged across the span of the channel. Simulations employing streamwise periodic boundary conditions (temporal model) as well as inlet-outlet conditions (spatial model) are carried out. In the temporal approach the interactions (pairing of vortices and growth of new vortex pairs) of fully developed vortex pairs are time-dependent, whereas in the spatial approach these events are inherently steady and concern vortices not in their fully developed state. The initial spatial development of the vortices is in excellent agreement with results of a linear stability analysis up to fairly large disturbance amplitudes. In the nonlinear regime a good agreement with experimental results has also been found. The receptivity of the flow is very important in a convectively unstable situation such as the present one and different behaviour is found at fixed Reynolds number (equal to 2.43 times the critical value for the onset of Dean vortices): the flow can be either steady or undergo a continuous sequence of merging and splitting events, depending on the inlet conditions. In the latter situation decorrelated patterns of low- and high-speed streaks are produced in streamwise-spanwise planes and they bear several similarities to near-wall coherent structures of turbulent boundary layers.
The interaction of waves with a row of circular cylinders
Authors: Linton C. M. , Evans D. V.
Published: April 2006

The two-dimensional acoustics problem of the scattering of an obliquely incident plane wave by a row of equally-spaced circular cylinders is solved using multipole expansions. The method is superior to existing techniques available for this problem as it allows the far-field behaviour of the solution to be evaluated in a straightforward manner, and extensive results for the reflection and transmission coefficients are given. The problem described above has a direct analogue in the theory of water waves and this is also discussed.
Wave packets, resonant interactions and soliton formation in inlet pipe flow
Authors: Savenkov Igor V.
Published: April 2006

The development of disturbances (two-dimensional non-linear and three-dimensional linear) in the entrance region of a circular pipe is studied in the limit of Reynolds numberR→ ∞ in the framework of triple-deck theory. It is found that lower-branch axisymmetric disturbances can interact in the resonant manner. Numerical calculations show that a two-dimensional nonlinear wave packet grows much more rapidly than that in the boundary layer on a flat plate, producing a spike-like solution which seems to become singular at a finite time. Large-sized, short-scaled disturbances are also studied. In this case the development of axisymmetric disturbances is governed by single one-dimensional equation in the form of the Korteweg-de Vries and Benjamin-Ono equations in the long- and short-wave limits respectively. The nonlinear interactions of these disturbances lead to the formation of solitons which can run both upstream and downstream. Linear three-dimensional wave packets are also calculated.
On the dispersion relation for trapped internal waves
Authors: Barber B. C.
Published: April 2006

An analysis is constructed in order to estimate the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface. The main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves. It is demonstrated that, in general 1/cp= 1/CpO+k/ωmax+ ∈(k) wherecpis the wave phase speed for a particular mode,CpOis the phase speed atk= 0, ωmaxis the maximum possible wave angular frequency and ωmax≤NmaxwhereNmaxis the maximum buoyancy frequency. Also, ∈(0) = 0, ∈(k) =o(k) forklarge, and is bounded for finitek.In particular, when ∈(k) can be neglected, the dispersion relation for a lowest mode wave is approximately 1/cp≈ (∫∞0N2(y)ydy)-½+k/ωmax. The eigenvalue problem is analysed for a class of buoyancy frequency squared functionsN2(x) which is taken to be a class of realvalued functions of a real variablexwhere O ≤x∞ such thatN2(x) =O(e-βx) asx→ ∞ and 1/β is an arbitrary length scale. It is demonstrated thatN2(x) can be represented by a power series in e-βx. The eigenfunction equation is constructed for such a function and it is shown that there are two cases of the equation which have solutions in terms of known functions (Bessel functions and confluent hypergeometric functions). For these two cases it is shown that ∈(k) can be neglected and that, in addition, ωmax=Nmax. More generally, it is demonstrated that whenk→ ∞ it is possible to approximate the equation uniformly in such a way that it can be compared with the confluent hypergeometric equation. The eigenvalues are then, approximately, zeros of the Whittaker functions. The main result which follows from this approach is that ifN2(x) isO(e-βx) asx→ ∞ and has a maximum valueN2maxthen a sufficient condition for 1/cp∼k/Nmaxto hold for largekfor the lowest mode is thatN2(t)/tis convex for O ≤t≤ 1 wheret= e-βx.
Adaptation of the near-surface wind to the development of sand transport
Authors: Mcewan Ian K. , Willetts Brian B.
Published: April 2006

A model of wind-blown sand transport is described with particular emphasis on the feedback between the grain cloud and the near-surface wind. The results from this model are used to develop Owen's (1964) hypothesis that ‘the grain layer behaves, so far as the flow outside it is concerned, as increased aerodynamic roughness whose height is proportional to the thickness of the layer’. The hypothesis is developed to show the influence this dynamic roughness has on the turbulent boundary layer above the saltation layer. Two processes are identified which influence the path of the system towards equilibrium. The first is the feedback between the near-surface wind and the grain cloud in which the quantity of sand transported is limited by the carrying capacity of the wind. The second is due to the temporal development of an internal boundary layer in response to the additional roughness imposed on the flow above the grain layer by the grain cloud. A similarity is noted between the temporal response of a turbulent boundary layer to sand transport and the spatial response of a turbulent boundary layer downstream of a step increase in surface roughness. Finally it is noted that the work may have important implications for transport rate prediction in unsteady winds.
Experimental investigation of wave boundary layers with a sudden change in roughness
Authors: Fredsøe J. , Sumer B. M. , Laursen T. S. , Pedersen C.
Published: April 2006

This study deals with turbulent oscillatory boundary-layer flows over a plane bed with a sudden spatial change in roughness. Two kinds of ‘change in the roughness’ were investigated: in one, the roughness changed from a smooth-wall roughness to a roughness equal to 4.8 mm, and in the other, it changed from a roughness equal to 0.35 mm to the same roughness as in the previous experiment (4.8 mm). The free-stream flow was a purely oscillating flow with sinusoidal velocity variation. Mean flow and turbulence properties were measured. The Reynolds number was 6 × 106 for the major part of the experiments, with a maximum velocity of approximately 2 m/s and the stroke of the motion about 6 m. The response of the boundary layer to the sudden change in roughness was found to occur over a transitional length of the flow. The bed shear stress over this transitional length attains a peak value over the bed section with the larger roughness. It was found that the amplification in the bed shear stress due to this peak could be up to 2.5 times its asymptotic value. Also, it was found that the turbulence is quantitatively different in the two half periods; a much stronger turbulence is experienced in the half period where the flow is towards the less-rough section. The present experiments further showed that a constant streaming occurs near the bed in the neighbourhood of the junction between the two bed sections. This streaming is directed towards the section with the larger roughness.
Streaming flow induced by an oscillating cascade of circular cylinders
Authors: Yan B. , Ingham D. B. , Morton B. R.
Published: April 2006

The fluid flow induced by a cascade of circular cylinders which oscillates harmonically in an unbounded, incompressible, viscous fluid which is otherwise at rest is investigated both numerically and experimentally. Attention in this paper is mainly concentrated on the induced steady streaming flow which occurs when the ratio of the amplitude of the oscillation of the cascade to the size of the cylinder, ε, is very small. The leading-order flow is then governed by the steady Navier-Stokes equations. In order to solve these equations numerically we first generate numerically a grid system using the boundary element method and then use a finite-difference scheme on the newly generated rectangular grid system. Numerical results show that for small values of the streaming Reynolds number Rs there are four recirculating flows of equal strength around each circular cylinder of the cascade. At large values of Rs symmetry breaks down and numerical solutions are found for asymmetrical flows. Numerically, a critical value of Rs, Rso say, is identified such that the flow is symmetrical when Rs < Rso and asymmetrical when Rs > Rso and these results are in reasonable agreement with experimental results, which are also presented in this paper.
Impingement of an unsteady two-phase jet on unheated and heated flat plates
Authors: Özdemir İ. Bedii , Whitelaw J. H.
Published: April 2006

This paper is concerned with an experimental investigation of the oblique impingement of an unsteady, axisymmetric two-phase jet on heated surfaces. Size and velocity were measured simultaneously with a phase-Doppler velocimeter, and the spatial distributions over the wall jet were found to be correlated with the interfacial activities as inferred from vertical velocity measurements in the vicinity of the wall. These results are discussed together with size measurements by a laser-diffraction technique to quantify the effect of the approach conditions of the inflowing jet droplet field and wall temperature in relation to mechanisms of secondary atomization.

Two mechanisms of secondary atomization were identified; the first did not involve direct wall contact and was due to the strain acting on the droplets by the continuous phase within the impingement region and was enhanced by thermal effects from the wall to cause breakup. The approaching velocity of the inflowing droplets to the plate was important to this process so that higher velocities increased the rate of strain within the impingement region and, consequently, the wall temperature promoting the secondary atomization shifted towards lower values. The second mechanism required direct wall contact and involved atomization of the film deposited on the wall by the impingement of the inflowing two-phase jet. With the penetration of high-speed inflowing droplets into the film, liquid mass was raised into the two-phase medium due to splashes from the film so that a new size class with larger droplets was generated. The resulting large droplets tended to stay close to the wall within the impingement region with small vertical velocities

In between the injections, the suspended droplet field above the film oscillated normal to the plate as a cloud so that the impact of large droplets on the film resulted in coalescence with the film and the ejection of smaller numbers of small droplets. The unsteady wall jet flow, caused by the arrival of the spray at the plate, swept the vertically oscillating droplet cloud radially outwards so that the resulting radial transport caused the dynamics of the unsteady film to be correlated with the size characteristics of the unsteady wall jet. Based on this phenomenological description, a radial droplet transport equation is derived.

The correlation suggests that the secondary atomization with direct wall contact is the dominant process for the generation of a new size class within the wall flow and initiates the mutual interaction between the unsteady film and wall jet droplet field.
Effects of surface cooling on the spreading of lava flows and domes
Authors: Griffiths Ross W. , Fink Jonathan H.
Published: April 2006

Scaling analyses describe the evolution of an extrusion of viscous or plastic fluid in the presence of surface cooling and solidification, under the assumption that the flows consist of two components: an isothermal interior and a surface crust. The ‘crust’ is a complex thermal and rheological boundary layer which we model using a viscous, plastic or brittle rheology. These models are thought to be relevant to some types of lava flows and address the effects of cooling on their morphology and rate of advance of the flow front. They show that effects of crust strength will dominate over both viscous and yield stresses in the interior when the ratio of crust thickness to flow length exceeds the ratio of effective yield stress of the crust to basal shear stress exerted on the bulk of the flow, a condition that appears likely to be met by many lava flows and small outgrowths on large flows. Similarity solutions are compared with measurements on the spreading of extrusions of wax beneath cold water in the laboratory, where the extruded liquid is viscous but develops a solid crust. Crust strength provides the dominant retarding force for the wax flows in cases where surface solidification occurs rapidly compared with lateral advection. These conditions give flows topped by sheets of solid that buckle or rift apart, or extrusions that enlarge by small bulbous outgrowths (analogous to ‘pillows’ on submarine lavas and ‘toes’ on some sub-aerial basalt flows). A comparison of the models with data for the growth of lava domes in the craters of Mount St Helens and Soufrière of St Vincent volcanoes reveals that spreading of those domes was not controlled by stresses in the flow interior. Instead the data are consistent with a balance between gravity and yield stresses in a thin crustal layer over most of the growth period.
Thermocapillary migration of a bidisperse suspension of bubbles
Authors: Wang Y. , Mauri R. , Acrivos A.
Published: April 2006

We consider the thermocapillary motion of a well-mixed suspension of non-conducting spherical bubbles of negligible viscosity in a viscous conducting liquid under conditions of vanishingly small Reynolds and Marangoni numbers. Recently, Acrivos, Jeffrey & Saville (1990) showed that when all the bubbles are of identical size, the ensemble-averaged migration velocity $\bar{U}_{1}$ of a test bubble of radius a1 within the suspension equals $U^{(0)}_1[1-\frac{3}{2}c_1+O(c^2_1)]$, where c1 is the volume fraction of the bubbles and U1(0) is the thermocapillary velocity of single bubble given by Young, Goldstein & Block (1959). Here we extend this result to a bi-disperse suspension containing bubbles of radii a1 and a2 ≡ λa1 in which case $\bar{U}_1 = U^{(0)}_1[1-\frac{3}{2}c_1 - S(\lambda)c_2+\ldots]$, where c1 and c2 are the corresponding volume fractions of the two sets of bubbles. Values for S(Λ) are presented for some typical size ratios Λ, and asymptotic expressions for S(Λ) are derived for Λ → 0 and for Λ → ∞.
The motion of particles in the Hele-Shaw cell
Authors: Pozrikidis C.
Published: April 2006

The force and torque on a particle that translates, rotates, or is held stationary in an incident flow within a channel with parallel-sided walls, are considered in the limit of Stokes flow. Assuming that the particle has an axisymmetric shape with axis perpendicular to the channel walls, the problem is formulated in terms of a boundary integral equation that is capable of describing arbitrary three-dimensional Stokes flow in an axisymmetric domain. The method involves: (a) representing the flow in terms of a single-layer potential that is defined over the physical boundaries of the flow as well as other external surfaces, (b) decomposing the polar cylindrical components of the velocity, boundary surface force, and single-layer potential in complex Fourier series, and (c) collecting same-order Fourier coefficients to obtain a system of one-dimensional Fredholm integral equations of the first kind for the coefficients of the surface force over the traces of the natural boundaries of the flow in an azimuthal plane. In the particular case where the polar cylindrical components of the boundary velocity exhibit a first harmonic dependence on the azimuthal angle, we obtain a reduced system of three real integral equations. A numerical method of solution that is based on a standard boundary element-collocation procedure is developed and tested. For channel flow, the effect of domain truncation on the nature of the far flow is investigated with reference to plane Hagen–Poiseuille flow past a cylindrical post. Numerical results are presented for the force and torque exerted on a family of oblate spheroids located above a single plane wall or within a parallel-sided channel. The effect of particle shape on the structure of the flow is illustrated, and some novel features of the motion are discussed. The numerical computations reveal the range of accuracy of previous asymptotic solutions for small or tightly fitting spherical particles.
Baroclinic instability of Kirchhoff's elliptic vortex
Authors: Miyazaki Takeshi , Hanazaki Hideshi
Published: April 2006

The linear instability of Kirchhoff's elliptic vortex in a vertically stratified rotating fluid is investigated using the quasi-geostrophic, f-plane approximation. Any elliptic vortex is shown to be unstable to baroclinic disturbances of azimuthal wavenumber m = 1 (bending mode) and m = 2 (elliptical deformation). The axial wavenumber of the unstable bending mode approaches Λc = 1.7046 in the limit of small ellipticity, indicating that it is a short-wave baroclinic instability. The instability occurs when the bending wave rotates around the vortex axis with angular velocity identical to the rotation rate of the undisturbed elliptic vortex. On the other hand, the wavenumber of the elliptical deformation mode approaches zero in the same limit, showing that it is a long-wave sideband instability.
Internal wave generation in uniformly stratified fluids. Part 2. Moving point sources
Authors: Voisin Bruno
Published: April 2006

The Green's function method is applied to the generation of internal gravity waves by a moving point mass source. Arbitrary motion of a source of arbitrary time dependence is treated using the impulsive Green's function, while ‘classical’ approaches of uniform motion of a steady or oscillatory source are recovered using the monochromatic Green's function. Waves have locally the structure of impulsive waves, emitted at the retarded time tr, and having propagated with the group velocity; at each position and time an implicit equation defines tr, in terms of which the waves are written. A source both oscillating and moving generates two systems of waves, with respectively positive and negative frequencies, and when oscillations vanish these systems merge into one.

Three particular cases are considered: the uniform horizontal and vertical motions of a steady source, and the uniform horizontal motion of an oscillatory source. Waves spread downstream of the steady source. For the oscillatory source they can extend both upstream and downstream, depending on the ratio of the source frequency to the buoyancy frequency, and are contained inside conical wavefronts, parts of which are caustics. For horizontal motion, moreover, the steady analysis (based on the monochromatic Green's function) reveals the presence of two insignificant contributions overlooked by the unsteady analysis (based on the impulsive Green's function), but which for an extended source may become of the same order as the main contribution. Among those is an upstream columnar disturbance associated with blocking.
Evolution equations for counterpropagating edge waves
Authors: Pierce R. D. , Knobloch E.
Published: April 2006

Asymptotically exact evolution equations for counterpropagating shallow-water edge waves are derived. The structure of the equations depends only on the symmetries of the problem and on the fact that the group velocity of the edge waves is of order one. As a result the equations take the form of parametrically forced Davey–Stewartson equations with mean-field coupling. The calculations extend existing work on parametric excitation of edge waves by normally incident waves to arbitrary beach profiles with asymptotically constant depth, and include coupling to wave-generated mean longshore currents. Dissipation arises generically from radiation damping, but we also consider heuristically the effects of linear boundary-layer damping. Spatially modulated waves do not couple to the parametric forcing due to the non-locality of the evolution equations and are damped. Thus only spatially uniform wavetrains are expected as stable solutions. If linear dissipation is included the parametric coupling selects standing waves, but in the undamped case travelling wave states are possible. Both classes of solutions are examined for modulational instabilities, and stability conditions for the generic evolution equations are presented. However, modulational instability is found to be excluded in the shallow-water formulation through the effects of the mean flow. Explicit numerical results for two experimentally relevant beach profiles, exponentially decaying and piecewise linear, are presented.
Transient and asymptotic stability of granular shear flow
Authors: Schmid P. J. , Kytömaa H. K.
Published: April 2006

The linear stability of granular material in an unbounded uniform shear flow is considered. Linearized equations of motion derived from kinetic theories are used to arrive at a linear initial-value problem for the perturbation quantities. Two cases are investigated: (a) wavelike disturbances with time constant wavenumber vector, and (b) disturbances that will change their wave structure in time owing to a shear-induced tilting of the wavenumber vector. In both cases, the stability analysis is based on the solution operator whose norm represents the maximum possible amplification of initial perturbations. Significant transient growth is observed which has its origin in the non-normality of the involved linear operator. For case (a), regions of asymptotic instability are found in the two-dimensional wavenumber plane, whereas case (b) is found to be asymptotically stable for all physically meaningful parameter combinations. Transient linear stability phenomena may provide a viable and fast mechanism to trigger finite-amplitude effects, and therefore constitute an important part of pattern formation in rapid particulate flows.
Solitary waves on a vorticity layer
Authors: Higuera F. J. , Jiménez J.
Published: April 2006

Contour dynamics methods are used to determine the shapes and speeds of planar, steadily propagating, solitary waves on a two-dimensional layer of uniform vorticity adjacent to a free-slip plane wall in an, otherwise irrotational, unbounded incompressible fluid, as well as of axisymmetric solitary waves propagating on a tube of azimuthal vorticity proportional to the distance to the symmetry axis. A continuous family of solutions of the Euler equations is found in each case. In the planar case they range from small-amplitude solitons of the Benjamin–Ono equation to large-amplitude waves that tend to one member of the touching pair of counter-rotating vortices of Pierrehumbert (1980), but this convergence is slow in two small regions near the tips of the waves, for which an asymptotic analysis is presented. In the axisymmetric case, the small-amplitude waves obey a Korteweg–de Vries equation with small logarithmic corrections, and the large-amplitude waves tend to Hill's spherical vortex.
The viscous interaction between sound waves and the trailing edge of a supersonic splitter plate
Authors: Peake N.
Published: April 2006

We consider the flow resulting from the interaction between the trailing edge of a supersonic splitter plate and sound waves incident on the trailing edge from upstream, as a model problem of relevance to understanding the unsteady flow in the vicinity of a supersonic jet nozzle. Morgan has previously shown that there is only one plausible solution to the outer potential-flow problem in this supersonic system, in which the vortex-sheet deflection close to the trailing edge varies linearly with the distance from the trailing edge, and that, in contrast to the subsonic version of the problem, it is not possible to construct an outer solution in which the vortex sheet leaves the plate smoothly (i.e. with zero gradient). In this paper our aim is to establish that this supersonic potential-theory solution is consistent with the equations governing the viscous flow close to the plate, and to provide a description of the nature of this inner flow, and we proceed by applying asymptotic analysis in the limit of large Reynolds number. For appropriate choices of the incident-wave amplitude and frequency, the canonical triple-deck structure at the trailing edge is realized, and the governing equations are then simplified by linearizing about the steady base flow in the lower deck; upstream of the trailing edge the unsteady flow is calculated analytically, whilst downstream a two-region parabolic scheme is employed. Our inner viscous flow is seen to match onto the outer potential-theory solution, and in particular we verify that the downstream evolution of the lower-deck flow as it emerges into the outer region corresponds exactly to the behaviour of the vortex sheet at the trailing edge in the outer flow. Once the consistency of the outer solution has been established, the dependence on the various flow parameters can be investigated, and we demonstrate in particular that significant unsteady shear-layer disturbances can be generated at the trailing edge over a wide range of values of the incidence angle, and that the amplitude of these disturbances decreases with increasing supersonic flow speed.
Three-dimensionalization of barotropic vortices on the f-plane
Authors: Smyth W. D. , Peltier W. R.
Published: April 2006

We examine the stability characteristics of a two-dimensional flow which consists initially of an inflexionally unstable shear layer on an f-plane. Under the action of the primary instability, the vorticity in the shear-layer initially coalesces into two Kelvin–Helmholtz vortices which subsequently merge to form a single coherent vortex. At a sequence of times during this process, we test the stability of the two-dimensional flow to fully three-dimensional perturbations. A somewhat novel approach is developed which removes inconsistencies in the secondary stability analyses which might otherwise arise owing to the time-dependence of the two-dimensional flow.

In the non-rotating case, and before the onset of pairing, we obtain a spectrum of unstable longitudinal modes which is similar to that obtained previously by Pierrehumbert & Widnall (1982) for the Stuart vortex, and by Klaassen & Peltier (1985, 1989, 1991) for more realistic flows. In addition, we demonstrate the existence of a new sequence of three-dimensional subharmonic (and therefore ‘helical’) instabilities. After pairing is complete, the secondary instability spectrum is essentially unaltered except for a doubling of length- and timescales that is consistent with the notion of spatial and temporal self-similarity. Once pairing begins, the spectrum quickly becomes dominated by the unstable modes of the emerging subharmonic Kelvin–Helmholtz vortex, and is therefore similar to that which is characteristic of the post-pairing regime. Also in the context of non-rotating flow, we demonstrate that the direct transfer of energy into the dissipative subrange via secondary instability is possible only if the background flow is stationary, since even slow time-dependence acts to decorrelate small-scale modes and thereby to impose a short-wave cutoff on the spectrum.

The stability of the merged vortex state is assessed for various values of the planetary vorticity f. Slow rotation may either stabilize or destabilize the columnar vortices, depending upon the sign of f, while fast rotation of either sign tends to be stabilizing. When f has opposite sign to the relative vorticity of the two-dimensional basic state, the flow becomes unstable to new mode of instability that has not been previously identified. Modes whose energy is concentrated in the vortex cores are shown to be associated, even at non-zero f, with Pierrehumbert's (1986) elliptical instability. Through detailed consideration of the vortex interaction mechanisms which drive instability, we are able to provide physical explanations for many aspects of the three-dimensionalization process.
Flow structure in the frequency-modulated wake of a cylinder
Authors: Nakano M. , Rockwell D.
Published: April 2006

A cylinder is subjected to frequency-modulated (FM) excitation and the structure of its wake is characterized in terms of the modulation frequency and the frequency deviation. It is possible to destabilize or restabilize the degree of organization of the vortical structures in the near wake and thereby substantially manipulate the spectral content, relative to the case of purely sinusoidal excitation. These processes of destabilization and restabilization are attainable by varying the frequency deviation while holding the modulation frequency constant or vice versa. A phase-locked periodicity of the nearwake response is attainable at the period of the modulation frequency, as well as at double its period. This phase-locked periodicity, or lack of it, is related to the degree of organization of the wake. The structure of the far wake is strongly dependent upon the nature of the near wake modification. Either coherent or destabilized wake structure can be induced in the far wake, at a given value of nominal excitation frequency, by employing appropriate FM excitation.
Turbulence characteristics of the boundary layer on a rotating disk
Authors: Littell Howard S. , Eaton John K.
Published: April 2006

Measurements of the boundary layer on an effectively infinite rotating disk in a quiescent environment are described for Reynolds numbers up to Reδ2 = 6000. The mean flow properties were found to resemble a ‘typical’ three-dimensional crossflow, while some aspects of the turbulence measurements were significantly different from two-dimensional boundary layers that are turned. Notably, the ratio of the shear stress vector magnitude to the turbulent kinetic energy was found to be at a maximum near the wall, instead of being locally depressed as in a turned two-dimensional boundary layer. Also, the shear stress and the mean strain rate vectors were found to be more closely aligned than would be expected in a flow with this degree of crossflow. Two-point velocity correlation measurements exhibited strong asymmetries which are impossible in a two-dimensional boundary layer. Using conditional sampling, the velocity field surrounding strong Reynolds stress events was partially mapped. These data were studied in the light of the structural model of Robinson (1991), and a hypothesis describing the effect of cross-stream shear on Reynolds stress events is developed.
Rotating axial flow of a continuously separating mixture
Authors: Dahlkild A. A. , Amberg G.
Published: April 2006

We consider the continuous separation process of a monodispersed suspension flowing axially through a rotating circular cylinder. This stationary problem can be regarded as a basic flow case of rotating mixtures in conjunction with previous studies of time-dependent flows like spin-up and batch settling in a cylinder. The ‘mixture model’ for two-phase flow is used to formulate the problem, which is solved in the range of small Ekman and Rossby numbers by asymptotic analytical methods and by a numerical code. The gradual separation of the axially injected suspension is manifested as a stationary stratification of the mixture which induces a swirl component of the velocity, in analogy with the thermal wind in the Earth's atmosphere. The presence of the azimuthal motion and induced secondary flow due to Ekman-layer pumping clearly influences the character of the stratification. Analytical and numerical results are in excellent agreement.
Wind effects on the nonlinear evolution of slowly varying gravity—capillary waves
Authors: Hara Tetsu , Mei Chiang C.
Published: April 2006

A train of uniform two-dimensional gravity waves in deep water is known to be unstable to certain sideband disturbances. If the time of propagation is sufficiently long for the fourth-order terms to be important, the sidebands may grow at unequal rates, resulting in a downward shift of peak frequency. But this shift is only a temporary phase of a recurrent evolution process. Recent work by us (Hara & Mei 1991) has shown that wind and dissipation can help maintain this downshift at large time. In this paper we examine a similar two-dimensional problem for capillary–gravity waves. The basic flow in air and water is assumed to be steady, horizontally uniform and turbulent; the wave-induced flow in both media is assumed to be laminar. Evolution equations are deduced with wind and dissipation included in such a way that their influence is comparable to the asymmetric spectral evolution. After finding the initial growth rates of unstable sidebands, the nonlinear development of modulational instability is examined by integrating the evolution equations numerically. Computed results show that persistent downshift of frequency can happen for relatively long waves, but upshift occurs for very short waves.
Temporal stability of Jeffery–Hamel flow
Authors: Hamadiche Mahmoud , Scott Julian , Jeandel Denis
Published: April 2006

In this study of the temporal stability of Jeffery–Hamel flow, the critical Reynolds number based on the volume flux, Rc, and that based on the axial velocity, Rec, are computed. It is found that both critical Reynolds numbers decrease very rapidly when the half-angle of the channel, α, increases, such that the quantity αRc remains very nearly constant and αRecis a nearly linear function of α. For a short channel there can be more than one value of the critical Reynolds number. A fully nonlinear analysis, for Re close to the critical value, indicates that the loss of stability is supercritical. The resulting asymmetric oscillatory solutions show staggered arrays of vortices positioned along the channel.
The initial development of a jet caused by fluid, body and free-surface interaction. Part 1. A uniformly accelerating plate
Authors: King A. C. , Needham D. J.
Published: April 2006

The flow field induced by a vertical plate accelerating into a stationary fluid of finite depth with a free surface and a gravitational restoring force is investigated. This is a model problem for some technologically important design issues such as the bow splash of a ship moving at forward speed. Experimentally it is found that a thin jet forms on the plate and rises rapidly upwards. We investigate this jet in the small-time approximation and find an analytical solution for the flow field in which the jet emerges out of a thin region where the horizontal momentum of the main flow is converted by inertial effects into a rising jet.
Dynamical properties of vortical structures on the beta-plane
Authors: Sutyrin G. G. , Hesthaven J. S. , Lynov J. P. , Rasmussen J. Juul
Published: April 2006

The long-time evolution of monopolar and dipolar vortices influenced by the largescale gradient of the ambient potential vorticity (the β-effect) is studied by direct numerical solutions of the equivalent barotropic quasi-geostrophic equation. Translation and reorganization of vortical structures are shown to depend strongly on their intensity. Transport of trapped fluid by vortical structures is illustrated by calculating particle trajectories and by considering closed isolines of potential vorticity and the streamfunction in a co-moving reference frame.

The initial behaviour of strong monopoles is found to be well described by a recent approximate theory for the evolution of azimuthal mode one, even for times longer than the linear Rossby wave period. In the long-time limit, strong monopoles transport particles mainly westward, although the meridional displacement is several times larger than the initial vortex size. The appearance of an annulus with opposite radial gradient of the potential vorticity around the vortex core is demonstrated. This annulus forms owing to the meridional vortex drift on the β-plane and results in reorganization of a strong monopolar vortex into a rotating tripole. A critical value of the vortex intensity is found, below which the tripolar structure does not appear even in the case of an initially shielded vortex. Weak monopolar vortices are able to trap particles and provide some west-meridional fluid transport, even in the case when they decay like a linear Rossby wave packet.

The evolution of initial f-plane dipoles on the β-plane is strongly dependent on the initial direction of propagation. Strong dipoles adjust to steadily propagating modon solutions either accelerating (westward case), decelerating (eastward case) or oscillating with a decaying amplitude (meridional case), thereby carrying trapped particles predominantly eastward. A steady state is not reached if the dipole intensity is below a critical value which depends on the initial direction of propagation. Weak dipoles either decay and shrink owing to Rossby wave radiation (westward case), gradually separate and split (eastward case), or disintegrate (meridional case) without longdistance fluid transport. Thus, on the β-plane monopoles provide mainly westward transport of trapped fluid, whereas dipoles provide mainly eastward transport. Only strong monopoles are found to provide significant meridional fluid transport.
Analogy between laminar flows in curved pipes and orthogonally rotating pipes
Authors: Ishigaki Hiroshi
Published: April 2006

The secondary flow of a viscous fluid, caused by the Coriolis force, through a straight pipe rotating about an axis perpendicular to the pipe axis is analogous to that of a fluid, caused by the centrifugal force, through a stationary curved pipe. The quantitative analogy between these two fully developed laminar flows will be demonstrated through similarity arguments, computational studies and the use of experimental data. Similarity considerations result in two analogous governing parameters for each flow, which include a new one for the rotating flow. When one of these analogous pairs of parameters of the two flows is large, it will be demonstrated that there are strong similarities between the two flows regarding friction factors, heat transfer rates, flow patterns and flow properties for the same values of the other pair of parameters.
Resonant wave interactions near a critical level in a stratified shear flow
Authors: Grimshaw R.
Published: April 2006

Resonant interactions between internal gravity waves propagating in a stratified shear flow are considered for the case when the background density and shear flow vary slowly with respect to the waves. In Grimshaw (1988) triad resonances were considered, and interaction equations derived for the case when the resonance conditions are met only on certain space-time surfaces, being resonance sites. Here this analysis is extended to include higher-order resonances, with the aim of studying resonant wave interactions near a critical level. It is shown that a secondary resonant interaction between two incoming waves, in which two harmonic components of one incoming wave interact with a single harmonic component of another incoming wave, produces a reflected wave. This result is shown to agree with the study of Brown & Stewartson (1980, 1982a, b) who obtained this same result by a different approach.
The transient behaviour of alloys solidified from below prior to the formation of chimneys
Authors: Worster M. Grae , Kerr Ross C.
Published: April 2006

We investigate interactions between interfacial disequilibrium and compositional convection during the freezing of an alloy from below to form a mushy layer. A theoretical model is developed in which a stagnant mushy layer underlies a melt that is convecting vigorously, driven by compositional gradients associated with undercooling at the mush-liquid interface. In a series of laboratory experiments, we measure the interfacial undercooling in aqueous solutions of ammonium chloride contaminated to varying degrees by copper sulphate. It has recently been found (Huppert & Hallworth 1993) that a small amount of copper sulphate added to a solution of ammonium chloride significantly inhibits the formation of chimneys in the mushy layer that forms when the solution is cooled below its liquidus. It is our thesis that this phenomenon can be explained in large part by the consequences of the interactions between compositional convection and interfacial undercooling that are investigated herein. The measured undercooling is a function of the rate of advance of the interface and is found to be a very strong, increasing function of the concentration of copper sulphate in solution. The theoretical model is evaluated using parameter values appropriate to the experimental system and it is found that the transient development of the mushy layer depends significantly on the level of interfacial disequilibrium. In particular, it is predicted that the time taken for the Rayleigh number associated with the mushy layer to reach any particular value increases enormously as the level of interfacial disequilibrium increases and that the Rayleigh number can have an upper bound that is less than the critical value needed for the onset of convection within the mushy layer. This suggests that the formation of chimneys in the mushy layer can be similarly delayed or prohibited, in agreement with the experimental findings of Huppert & Hallworth (1993). Additionally, the model predicts that under certain conditions the solid fraction can increase away from the cooled boundary leading to trapping of the interstitial liquid. The model also describes a mechanism for macrosegregation of alloys cooled and solidified from below.