bluuebunny/crossref_metadata_2025 huggingface dataset:
Self-similar spiral instabilities in elastic flows between a cone and a plate
Authors: Mckinley Gareth H. , Öztekin Alparslan , Byars Jeffrey A. , Brown Robert A.
Published: April 2006

Experimental observations and linear stability analysis are used to quantitatively describe a purely elastic flow instability in the inertialess motion of a viscoelastic fluid confined between a rotating cone and a stationary circular disk. Beyond a critical value of the dimensionless rotation rate, or Deborah number, the spatially homogeneous azimuthal base flow that is stable in the limit of small Reynolds numbers and small cone angles becomes unstable with respect to non-axisymmetric disturbances in the form of spiral vortices that extend throughout the fluid sample. Digital video-imaging measurements of the spatial and temporal dynamics of the instability in a highly elastic, constant-viscosity fluid show that the resulting secondary flow is composed of logarithmically spaced spiral roll cells that extend across the disk in the self-similar form of aBernoulli Spiral.

Linear stability analyses are reported for the quasi-linear Oldroyd-B constitutive equation and the nonlinear dumbbell model proposed by Chilcott & Rallison. Introduction of a radial coordinate transformation yields an accurate description of the logarithmic spiral instabilities observed experimentally, and substitution into the linearized disturbance equations leads to a separable eigenvalue problem. Experiments and calculations for two different elastic fluids and for a range of cone angles and Deborah numbers are presented to systematically explore the effects of geometric and rheological variations on the spiral instability. Excellent quantitative agreement is obtained between the predicted and measured wavenumber, wave speed and spiral mode of the elastic instability. The Oldroyd-B model correctly predicts the non-axisymmetric form of the spiral instability; however, incorporation of a shear-rate-dependent first normal stress difference via the nonlinear Chilcott–Rallison model is shown to be essential in describing the variation of the stability boundaries with increasing shear rate.
On convection and mixing driven by sedimentation
Authors: Cardoso Silvana S. S. , Woods Andrew W.
Published: April 2006

The sedimentation of small particles from a suspension and the
concomitant release of light interstitial fluid may constitute a buoyancy
source for the development of convective motions. When the dense suspension
is emplaced beneath a stratified fluid an intermediate convecting layer
between the sedimenting front and the density gradient above gradually grows
in depth by erosion of the overlying stratified fluid. Novel laboratory
experiments involving sedimentation below a two-layer stratified region show
that turbulent mixing and entrainment across the top density interface is
significant for a broad range of the Richardson number. A simple theoretical
model predicting the rate of erosion of the stratification above the
convecting layer agrees well with these experiments. The model is then
extended to include the case of an overlying continuous density gradient and
compared successfully with both new experimental data and the original data
of Kerr (1991). Owing to the effects of dispersion of grain sizes, small
particles in the convecting fluid may lower the efficiency of the
interfacial mixing by the turbulent eddies.

Our model calculations suggest that turbulent mixing and entrainment
driven by sedimentation may be significant in the atmospheric and oceanic
contexts, in both of which stratification is weak. Such mixing may also
occur in molten magma chambers following the sedimentation of suspended
crystals, and in this case it may suppress large-scale overturning
events.
Direct computation of the sound from a compressible co-rotating vortex
pair
Authors: Mitchell Brian E. , Lele Sanjiva K. , Moin Parviz
Published: April 2006

The far-field sound generated by compressible co-rotating vortices is
computed by direct computation of the unsteady compressible Navier–Stokes
equations on a computational domain that extends to two acoustic wavelengths
in all directions. The vortices undergo a period of co-rotation followed by
a sudden merger. The directly computed far-field sound is compared to the
prediction of the acoustic analogy due to Möhring (1978, 1979), a modified
form of the analogy developed by Lighthill (1952), and an acoustic analogy
derived by Powell (1964). All three predictions are in excellent agreement
with the simulation. Results of far-field pressure fluctuations from an
acoustically non-compact, co-rotating vortex pair are also presented. In
this case, the vortex sound theory over-predicts the sound by 65% in
accordance with the analysis of Yates (1978).
On the viscous decay rates of inertial waves in a rotating circular
cylinder
Authors: Kerswell R. R. , Barenghi C. F.
Published: April 2006

In the literature, there are two different asymptotic results for the
viscous decay rates of inertial modes in a rotating circular cylinder. In
the absence of a viscous corner solution, either result can only be an
estimate of the true decay rate. In this note, we numerically calculate the
viscous decay rates for some experimentally excited inertial modes (Malkus
1989; Malkus & Waleffe 1991) in order to (i) assist in the
interpretation of these experiments and (ii) to assess the usefulness of the
two asymptotic estimates available. Our results indicate that the asymptotic
estimate due to Kudlick (1966) is more accurate and that the asymptotic
regime in which this estimate is useful (accurate to within 10%) can be
smaller than is commonly thought.
A model for large-scale structures in turbulent shear flows
Authors: Poje Andrew C. , Lumley J. L.
Published: April 2006

A procedure based on energy stability arguments is presented as a method
for extracting large-scale, coherent structures from fully turbulent shear
flows. By means of two distinct averaging operators, the instantaneous flow
field is decomposed into three components: a spatial mean, coherent field
and random background fluctuations. The evolution equations for the coherent
velocity, derived from the Navier–Stokes equations, are examined to
determine the mode that maximizes the growth rate of volume-averaged
coherent kinetic energy. Using a simple closure scheme to model the effects
of the background turbulence, we find that the spatial form of the maximum
energy growth modes compares well with the shape of the empirical
eigenfunctions given by the proper orthogonal decomposition. The discrepancy
between the eigenspectrum of the stability problem and the empirical
eigenspectrum is explained by examining the role of the mean velocity field.
A simple dynamic model which captures the energy exchange mechanisms between
the different scales of motion is proposed. Analysis of this model shows
that the modes which attain the maximum amplitude of coherent energy density
in the model correspond to the empirical modes which possess the largest
percentage of turbulent kinetic energy. The proposed method provides a means
for extracting coherent structures which are similar to those produced by
the proper orthogonal decomposition but which requires only modest
statistical input.
The effects of a vertical contraction on turbulence dynamics in a stably
stratified fluid
Authors: Thoroddsen S. T. , Van Atta C.W.
Published: April 2006

We have experimentally studied the effects of mean strain on the
evolution of stably stratified turbulence. Grid-generated turbulence
($Re_{\lambda \leqslant
25}$) in a stable linear mean background density
gradient was passed through a two-dimensional contraction, contracting the
stream only in the vertical direction. This induces an increase in
stratification strength, which reduces the largest vertical overturning
scales allowed by buoyancy forces. The mean strain through the contraction
causes, on the other hand, stretching of streamwise vortices tending to
increase the fluctuation levels of the transverse velocity components. This
competition between buoyancy and vortex stretching dominates the turbulence
dynamics inside and downstream of the contraction. Comparison between
non-stratified and stratified experiments shows that the stratification
significantly reduces the vertical velocity fluctuations. The vertical heat
flux is initially enhanced through the contraction. Then, farther downstream
the flux quickly reverses, leading to very strong restratification
coinciding with an increase in the vertical velocity fluctuations. The
vertical heat flux collapses much more rapidly than in the stratified case
without an upstream contraction and the restratification intensity is also
much stronger, showing values of normalized flux as strong as −0.55.
Velocity spectra show that the revival of vertical velocity fluctuations,
due to the strong restratification, starts at the very largest scales but is
then subsequently transferred to smaller scales. The distance from the
turbulence-generating grid to the entrance of the contraction is an
important parameter which was varied in the experiments. The larger this
distance, the larger the integral length scale can grow, approaching the
limit set by buoyancy, before entering the contraction. The evolution of the
various turbulence length scales is described. Two-point measurements of
velocity and temperature transverse integral scales were also performed
inside the contraction. The emergence of ‘zombie’ turbulence, for large
buoyancy times, is in good quantitative agreement with the numerical
simulations of Gerz & Yamazaki (1993) for stratification number larger
than 1.
Marangoni effects of adsorption—desorption controlled surfactants on the leading end of an infinitely long bubble in a capillary
Authors: Stebe K. J. , Barthès-Biesel D.
Published: April 2006

The leading end of an infinitely long gas bubble which displaces a viscous surfactant solution in a capillary tube is studied. The surfactant is present at elevated concentration and has sorption controlled mass transfer. The displaced fluid wets the wall, forming a continuous liquid film between the bubble and the capillary wall. Both the thickness of this film and the additional pressure required to aspirate the bubble depend upon the Marangoni stresses caused by non-uniform surfactant adsorption along the interface. The equations governing this flow are solved at asymptotically small capillary number for the case where the balances of momentum and mass transfer are coupled to leading order. As the Marangoni effect is increased over several orders of magnitude, the additional pressure and the wetting-layer thickness increase above the stress-free interface values found by Bretherton (1961) and approach an upper bound of 42/3 times the Bretherton values. Accompanying changes in the surface tension, Marangoni stress and surface velocity profiles as this upper bound is approached are described. Finally, surface viscosities that are intrinsic to the interfacial region are incorporated in the analysis. When small departures from surface equilibrium states are considered, the terms representing surface viscous effects have the same functional form as the Marangoni stresses and result in thicker films and higher additional pressures.
Free-surface flow generated by a small submerged circular cylinder starting from rest
Authors: Tyvand Peder A. , Miloh Touvia
Published: April 2006

The impulsively starting motion of a small circular cylinder submerged horizontally below a free surface is studied analytically using a small-time expansion. The cylinder is considered small if the ratio between its radius and initial submergence depth is much smaller than one. The surface elevation is calculated up to third order. The hydrodynamic force on the small cylinder is also discussed. Certain inconsistencies in our small-cylinder approximation (assuming locally uniform flow around the cylinder) are found in the force prediction. The present work is an accompanying paper to Tyvand & Miloh (1995), where the same problem is studied for arbitrary radius versus submergence depth.
Dynamic simulation of the motion of capsules in pipelines
Authors: Feng J. , Huang P. Y. , Joseph D. D.
Published: April 2006

In this paper we report results of two-dimensional simulations of the motion of elliptic capsules carried by a Poiseuille flow in a channel. The numerical method allows computation of the capsule motion and the fluid flow around the capsule, and accurate evaluation of the lift force and torque. Results show that the motion of a capsule which is heavier than the carrying fluid may be decomposed into three stages: initial lift-off, transient oscillations and steady flying. The behaviour of the capsule during initial lift-off and steady flying is analysed by studying the pressure and shear stress distributions on the capsule. The dominant mechanism for the lift force and torque is lubrication or inertia or a combination of the two under different conditions. The lift-off velocity for the ellipse in two dimensions is compared with experimental values for cylindrical capsules in pipes. Finally, the mechanisms of lift for capsules are applied to flying core flows, and it is argued that inertial forces are responsible for levitating heavy crude oil cores lubricated by water in a horizontal pipeline.
A dynamic localization model for large-eddy simulation of turbulent flows
Authors: Ghosal Sandip , Lund Thomas S. , Moin Parviz , Akselvoll Knut
Published: April 2006

In a previous paper, Germano, et al. (1991) proposed a method for computing coefficients of subgrid-scale eddy viscosity models as a function of space and time. This procedure has the distinct advantage of being self-calibrating and requires no a priori specification of model coefficients or the use of wall damping functions. However, the original formulation contained some mathematical inconsistencies that limited the utility of the model. In particular, the applicability of the model was restricted to flows that are statistically homogeneous in at least one direction. These inconsistencies and limitations are discussed and a new formulation that rectifies them is proposed. The new formulation leads to an integral equation whose solution yields the model coefficient as a function of position and time. The method can be applied to general inhomogeneous flows and does not suffer from the mathematical inconsistencies inherent in the previous formulation. The model has been tested in isotropic turbulence and in the flow over a backward-facing step.
Chaotic mode competition in the shape oscillations of pulsating bubbles
Authors: Zardi D. , Seminara G.
Published: April 2006

A possible mechanism for the occurrence of the phenomenon of erratic drift of bubbles in liquids subjected to acoustic waves was proposed by Benjamin & Ellis (1990) who showed that nonlinear interactions between adjacent perturbation modes expressed in terms of spherical harmonics of any order may lead to the excitation of mode 1 which is equivalent to a displacement of the bubble centroid. We show that indeed such a mechanism can give rise to a chaotic process at least under the conditions experimentally investigated by Benjamin & Ellis (1990). In fact we examine the case in which the angular frequency ω of the incident wave is sufficiently close to both the natural frequency of mode n + 1 (ωn + 1) and twice the natural frequency of mode n (2ωn) thus exciting simultaneously a subharmonic mode n and a synchronous mode n + 1. The value of n is set equal to 3 in accordance with Benjamin & Ellis' (1990) observation. A classical multiple scale analysis allows us to follow the development of these perturbations in the weakly nonlinear regime to find an autonomous system of quadratically coupled nonlinear differential equations governing the evolution of the amplitudes of the perturbations on a slow time scale. As obtained by Gu & Sethna (1987) for the Faraday resonance problem, we find both regular and chaotic solutions of the above system. Chaos is found to develop for large enough values of the amplitude of the acoustic excitation within some region in the parameter space and is reached through a period-doubling sequence displaying the typical characteristics of Feigenbaum scenario.
Stokes problems for moving half-planes
Authors: Zeng Y. , Weinbaum S.
Published: April 2006

New exact solutions of the Navier–Stokes equations are obtained for the unbounded and bounded oscillatory and impulsive tangential edgewise motion of touching half-infinite plates in their own plane. In contrast to Stokes classical solutions for the harmonic and impulsive motion of an infinite plane wall, where the solutions are separable or have a simple similarity form, the present solutions have a two-dimensional structure in the near region of the contact between the half-infinite plates. Nevertheless, it is possible to obtain relatively simple closed-form solutions for the flow field in each case by defining new variables which greatly simplify the r- and θ-dependence of the solutions in the vicinity of the contact region. These solutions for flow in a half-infinite space are then extended to bounded flows in a channel using an image superposition technique. The impulsive motion has application to the motion near geophysical faults, whereas the oscillatory motion has arisen in the design of a novel oscillating half-plate flow chamber for examining the effect of fluid shear stress on cultured cell monolayers.
Existence and stability of stationary vortices in a uniform shear flow
Authors: Nycander J.
Published: April 2006

Isolated vortices in a background flow of constant shear are studied. The flow is governed by the two-dimensional Euler equation. An infinite family of integral invariants, the Casimirs, constrain the flow to an isovortical surface. An isovortical surface consists of all flows that can be obtained by some incompressible deformation of a given vorticity field. It is proved that on every isovortical surface satisfying appropriate conditions there exists a stationary solution, stable to all exponentially growing disturbances, which represents a localized vortex that is elongated in the direction of the external flow. The most important condition is that the vorticity anomaly q in the vortex has the same sign as the external shear. The validity of the proof also requires that q is non-zero only in a finite region, and that 0 < qmin ≤ q ≤ qmax ≤ ∞ in this region (assuming the external shear to be positive).
Cavitation in the rotational structures of a turbulent wake
Authors: Belahadji B. , Franc J. P. , Michel J. M.
Published: April 2006

Experiments show that cavitation, if moderately developed, makes three kinds of vortical coherent structures visible inside the turbulent wake of a two-dimensional obstacle: Bénard–Kármán vortices, streamwise three-dimensional vortices and finally the vortices which appear on the borders of the very near wake. The latter, which are called here near-wake vortices, result by successive pairing in the first ones and there is some indication that they are also the origin of streamwise vortices. Cavitation is not a passive agent of visualization, as can be established on the basis of fundamental arguments, and it reacts with the flow as soon as it appears; when it is developed, it breaks the connection between the elongation rate and the vorticity rate of the vortex filaments. Then the subsequent evolution of a cavitating vortex and its final implosion are rather complicated. Despite its active character, cavitation in rotational structures, if properly interpreted, can give information of interest on the basic non-cavitating turbulent flow. By adapting a simple model due to Kermeen & Parkin (1957) and Arndt (1976), and counting near-wake vortices, it is possible to accurately predict the conditions of cavitation inception: consideration of coherent rotational structures is probably the best approach to explain, in an almost deterministic way, the large difference between the absolute value of the mean pressure coefficient at the obstacle base and the incipient cavitation number.
Experimental observations of edge waves
Authors: Buchan S. J. , Pritchard W. G.
Published: April 2006

The main purpose of this paper is to provide some carefully documented experimental observations of the subharmonic generation of edge waves over a plane beach by waves normally incident on the beach from the distant ocean. In order to establish experimentally the details of the subharmonic instability mechanism, it is important first to determine the properties of the primary wave field whose stability is to be investigated. Thus, a detailed appraisal has been made of the wave field established in the tank and over the beach in the absence of edge waves. These data have been particularly useful in defining the amplitudes of the incident- and reflected-wave fields at the toe of the beach, the magnitudes of which are central to the triggering of the edge-wave instability. Details are presented of the edge-wave field over the beach, as well as marginal stability curves and initial growth rates of the instability, the latter two of which are compared with theoretical estimates obtained from extant theories of the instability mechanism. Some experiments are also described in which edge waves were established as modes forced by topographical imperfections at the beach.
On the neutral curve of the flat-plate boundary layer: comparison between experiment, Orr–Sommerfeld theory and asymptotic theory
Authors: Healey J. J.
Published: April 2006

The neutral stability curve for the flat-plate boundary layer has been calculated using the Orr–Sommerfeld equation and compared to those obtained using upper- and lower-branch scalings. The Orr–Sommerfeld results agree well with the lower-branch scaling at Reynolds numbers relevant to experiment, but agree well with the upper-branch scaling only for Rδ > 105. It is shown that the critical layer only emerges from the viscous wall layer when Rδ > 105. This suggests that for Rδ < 105, when the critical layer lies within the viscous wall layer, the disturbance has a triple-deck structure, even for the upper branch of the neutral curve (which can be reached if the phase jump across the critical layer is retained).

The transition from a triple-deck to a five-deck structure with increasing Reynolds number on the upper branch occurs relatively abruptly and can be associated with a square-root branch point in the Tietjens function. Essentially, the lower- and upper-branch scalings pertain to two different modes, the first possessing a triple-deck structure, the second a five-deck structure. The modes are connected at the branch point, and the neutral curves of each mode join to give a single curve close to this branch point. The asymptotic expansions for the upper- and lower-branch neutral curves depend upon the analyticity of the dispersion relationship, and so the proximity of the branch point indicates where these expansions will be liable to inaccuracies. This explains the poor neutral-curve predictions made by five-deck analyses at the Reynolds numbers where transition occurs.
Unsteady flow interactions between an advected cylindrical vortex tube and a spherical particle
Authors: Kim Inchul , Elghobashi Said , Sirignano William A.
Published: April 2006

The unsteady, three-dimensional, incompressible, viscous flow interactions between a vortical (initially cylindrical) structure advected by a uniform free stream and a spherical particle held fixed in space is investigated numerically for a range of particle Reynolds numbers 20 ≤ Re ≤ 100. The counter-clockwise rotating vortex tube is initially located ten sphere radii upstream from the sphere centre. The finite-difference computations yield the flow properties and the temporal distributions of lift, drag, and moment coefficients of the sphere. Initially, the lift force is positive owing to the upwash on the sphere, then becomes negative owing to the downwash as the vortex tube passes the sphere. Varying the size of the vortex core (σ) shows that the r.m.s. lift coefficient is linearly proportional to the circulation of the vortex tube at small values of σ. At large values of σ, the r.m.s. lift coefficient is linearly proportional to the maximum fluctuation velocity (vmax) induced by the vortex tube but independent of σ. For intermediate values of σ, the r.m.s. lift coefficient depends on both σ and vmax (or equivalently both σ and the circulation). We observe some interesting flow phenomena in the near wake as a function of time owing to the passage of the vortex tube.
Simplified equations for the interaction of nearly parallel vortex filaments
Authors: Klein Rupert , Majda Andrew J. , Damodaran Kumaran
Published: April 2006

New simplified asymptotic equations for the interaction of nearly parallel vortex filaments are derived and analysed here. The simplified equations retain the important physical effects of linearized local self-induction and nonlinear potential vortex interaction among different vortices but neglect other non-local effects of self-stretching and mutual induction. These equations are derived systematically in a suitable distinguished asymptotic limit from the Navier–Stokes equations. The general Hamiltonian formalism and conserved quantities for the simplified equations are developed here. Properties of these asymptotic equations for the important special case involving nearly parallel pairs of interacting filaments are developed in detail. In particular, strong evidence is presented that for any filament pair with a negative circulation ratio, there is finite-time collapse in a self-similar fashion independent of the perturbation but with a structure depending on the circulation ratio. On the other hand, strong evidence is presented that no finite-time collapse is possible for perturbations of a filament pair with a positive circulation ratio. The present theory is also compared and contrasted with earlier linear and nonlinear stability analyses for pairs of filaments.
Eddy viscosity of three-dimensional flow
Authors: Wirth A. , Gama S. , Frisch U.
Published: April 2006

Detailed theoretical and numerical results are presented for the eddy viscosity of three-dimensional forced spatially periodic incompressible flow.

As shown by Dubrulle & Frisch (1991), the eddy viscosity, which is in general a fourth-order anisotropic tensor, is expressible in terms of the solution of auxiliary problems. These are, essentially, three-dimensional linearized Navier–Stokes equations which must be solved numerically.

The dynamics of weak large-scale perturbations of wavevector k is determined by the eigenvalues – called here ‘eddy viscosities’ – of a two by two matrix, obtained by contracting the eddy viscosity tensor with two k-vectors and projecting onto the plane transverse to k to ensure incompressibility. As a consequence, eddy viscosities in three dimensions, but not in two, can become complex. It is shown that this is ruled out for flow with cubic symmetry, the eddy viscosities of which may, however, become negative.

An instance is the equilateral ABC-flow (A = B = C = 1). When the wavevector k is in any of the three coordinate planes, at least one of the eddy viscosities becomes negative for R = 1/v > Rc [bsime ] 1.92. This leads to a large-scale instability occurring for a value of the Reynolds number about seven times smaller than instabilities having the same spatial periodicity as the basic flow.
An experimental study of the onset of parametrically pumped surface waves in viscous fluids
Authors: Bechhoefer John , Ego Valerie , Manneville Sebastien , Johnson Brad
Published: April 2006

We measure the threshold accelerations necessary to excite surface waves in a vertically vibrated fluid container (the Faraday instability). Under the proper conditions, the thresholds and onset wavelengths agree with recent theoretical predictions for a laterally infinite, finite-depth container filled with a viscous fluid. Experimentally, we show that by using a viscous, non-polar fluid, the finite-size effects of sidewalls and the effects of surface contamination can be made negligible. We also show that finite-size corrections are of order h/L, where h is the fluid depth and L the container size. Based on these measurements, one can more easily interpret certain unexpected observations from previous experimental studies of the Faraday instability.
Nonlinear energy transfer to short gravity waves in the presence of long waves
Authors: Ölmez H. S. , Milgram J. H.
Published: April 2006

Existing theories for calculating the energy transfer rates to gravity waves due to resonant nonlinear interactions among wave components whose lengths are long in comparison to wave elevations have been verified experimentally and are well accepted. There is uncertainty, however, about prediction of energy transfer rates within a set of waves having short to moderate lengths when these are present simultaneously with a long wave whose amplitude is not small in comparison to the short wavelengths. Here we implement both a direct numerical method that avoids small-amplitude approximations and a spectral method which includes perturbations of high order. These are applied to an interacting set of short- to intermediate-length waves with and without the presence of a large long wave. The same cases are also studied experimentally. Experimentally and numerical results are in reasonable agreement with the finding that the long wave does influence the energy transfer rates. The physical reason for this is identified and the implications for computations of energy transfer to short waves in a wave spectrum are discussed.
The motion of axisymmetric dipolar particles in homogeneous shear flow
Authors: Almog Y. , Frankel I.
Published: April 2006

The respective effects of an external field and departure of suspended particles from spherical shape on the rotary motion of axisymmetric dipolar particles placed in a homogeneous shear flow are studied. The analysis shows that, owing to cumulative effects, even a weak external field or a small deviation from spherical shape can significantly modify the resulting motion relative to that found in the corresponding classical problems of torque-free particles or dipolar spheres in homogeneous shear. Thus, unlike the latter problems, there are in the present problem cases when all particles approach a single limit cycle; in other cases multiple stable equilibria simultaneously coexist and the orientation space is appropriately divided into corresponding domains of attraction; in some situations possessing appropriate symmetry properties, particles may, depending upon their respective initial orientations, either move along a family of periodic orbits or else converge to a stable equilibrium orientation.
Wavenumber transport: scattering of small-scale internal waves by large-scale wavepackets
Authors: Bruhwiler David L. , Kaper Tasso J.
Published: April 2006

In this work, we treat the problem of small-scale, small-amplitude, internal waves interacting nonlinearly with a vigorous, large-scale, undulating shear. The amplitude of the background shear can be arbitrarily large, with a general profile, but our analysis requires that the amplitude vary on a length scale longer than the wavelength of the undulations. In order to illustrate the method, we consider the ray-theoretic model due to Broutman & Young (1986) of high-frequency oceanic internal waves that trap and detrap in a near-inertial wavepacket as a prototype problem. The near-inertial wavepacket tends to transport the high-frequency test waves from larger to smaller wavenumber, and hence to higher frequency. We identify the essential physical mechanisms of this wavenumber transport, and we quantify it. We also show that, for an initial ensemble of test waves with frequencies between the inertial and buoyancy frequencies and in which the number of test waves per frequency interval is proportional to the inverse square of the frequency, a single nonlinear wave–wave interaction fundamentally alters this initial distribution. After the interaction, the slope on a log-log plot is nearly flat, whereas initially it was -2. Our analysis captures this change in slope. The main techniques employed are classical adiabatic invariance theory and adiabatic separatrix crossing theory.
Gravity-driven flows in porous layers
Authors: Huppert Herbert E. , Woods Andrew W.
Published: April 2006

The motion of instantaneous and maintained releases of buoyant fluid through shallow permeable layers of large horizontal extent is described by a nonlinear advection–diffusion equation. This equation admits similarity solutions which describe the release of one fluid into a horizontal porous layer initially saturated with a second immiscible fluid of different density. Asymptotically, a finite volume of fluid spreads as t1/3. On an inclined surface, in a layer of uniform permeability, a finite volume of fluid propagates steadily alongslope under gravity, and spreads diffusively owing to the gravitational acceleration normal to the boundary, as on a horizontal boundary. However, if the permeability varies in this cross-slope direction, then, in the moving frame, the spreading of the current eventually becomes dominated by the variation in speed with depth, and the current length increases as t1/2. Shocks develop either at the leading or trailing edge of the flows depending upon whether the permeability increases or decreases away from the sloping boundary. Finally we consider the transient and steady exchange of fluids of different densities between reservoirs connected by a shallow long porous channel. Similarity solutions in a steadily migrating frame describe the initial stages of the exchange process. In the final steady state, there is a continuum of possible solutions, which may include flow in either one or both layers of fluid. The maximal exchange flow between the reservoirs involves motion in one layer only. We confirm some of our analysis with analogue laboratory experiments using a Hele-Shaw cell.
The motion of long bubbles in polygonal capillaries. Part 2. Drag, fluid pressure and fluid flow
Authors: Wong Harris , Radke C. J. , Morris S.
Published: April 2006

This work determines the pressure–velocity relation of bubble flow in polygonal capillaries. The liquid pressure drop needed to drive a long bubble at a given velocity U is solved by an integral method. In this method, the pressure drop is shown to balance the drag of the bubble, which is determined by the films at the two ends of the bubble. Using the liquid-film results of Part 1 (Wong, Radke & Morris 1995), we find that the drag scales as Ca2/3 in the limit Ca → 0 (Ca μU/σ, where μ is the liquid viscosity and σ the surface tension). Thus, the pressure drop also scales as Ca2/3. The proportionality constant for six different polygonal capillaries is roughly the same and is about a third that for the circular capillary.

The liquid in a polygonal capillary flows by pushing the bubble (plug flow) and by bypassing the bubble through corner channels (corner flow). The resistance to the plug flow comes mainly from the drag of the bubble. Thus, the plug flow obeys the nonlinear pressure–velocity relation of the bubble. Corner flow, however, is chiefly unidirectional because the bubble is long. The ratio of plug to corner flow varies with liquid flow rate Q (made dimensionless by σa2/μ, where a is the radius of the largest inscribed sphere). The two flows are equal at a critical flow rate Qc, whose value depends strongly on capillary geometry and bubble length. For the six polygonal capillaries studied, Qc [Lt ] 10−6. For Qc [Lt ] Q [Lt ] 1, the plug flow dominates, and the gradient in liquid pressure varies with Q2/3. For Q [Lt ] Qc, the corner flow dominates, and the pressure gradient varies linearly with Q. A transition at such low flow rates is unexpected and partly explains the complex rheology of foam flow in porous media.
Stationary perturbations of Couette–Poiseuille flow: the flow development in long cavities and channels
Authors: Stocker Jennifer R. , Duck Peter W.
Published: April 2006

We consider stationary perturbations to Couette–Poiseuille flows. These may be considered to be related to far downstream/upstream entry/end effects in flow inside long cavities and channels. Three distinct classes of basic flow are considered, all of which are exact solutions of the Navier–Stokes equations. We first study the problem in the case of Poiseuille flow, and are able to explain a previous discrepancy between fully numerical results, and asymptotic theory valid for large Reynolds numbers, R. The second case, which may be derived from a combination of an imposed streamwise pressure gradient and sliding of the upper channel wall, is for the particular situation where the flow on the lower surface is on the verge of reversing direction. The third case is relevant to the flow inside a long driven cavity (with closed ends, no imposed streamwise pressure gradient and no net mass flux). The flow is driven exclusively by a sliding top wall and mass conservation demands that the flow is no longer unidirectional.

For low Reynolds numbers, the stationary eigenvalues in all cases considered are complex (and hence are not monotonic in the streamwise direction). Indeed as R → 0 the eigenvalues become completely independent of the base profile. As the Reynolds number is increased, the eigenvalues generally undergo a number of branching processes switching between being complex and real (and vice versa) in nature, and at large Reynolds numbers fall broadly into three distinct categories, namely O(1), O(R−1/7) and O(1/R). In this limit the eigenvalues may be either complex or real (tending to monotonic eigensolutions in the streamwise direction).

Of particular interest are certain of the O(1) eigensolutions for the ‘driven-cavity’ problem, in the high-Reynolds-number limit; these turn out to be highly oscillatory (WKB-type) over much of the cavity section.

In all three cases, we use a combination of numerical and asymptotic techniques, and a thorough comparison between results thus obtained is made.
On penetrative convection at low Péclet number
Authors: Lister John R.
Published: April 2006

A new theoretical model is developed for the growth of a convecting fluid layer at the base of a stable, thermally stratified layer when heated from below. The imposed convective heat flux is taken to be comparable to the heat flux conducted down the background gradient so that diffusion ahead of the interface between the convecting and stable layers makes a significant contribution to the interfacial heat flux and to the rate of rise of the interface. Closure of the diffusion problem in the stable region requires the interfacial heat flux to be specified, and it is argued that this is determined by the ability of convective eddies to mix warmed fluid below the interface downwards. The interfacial velocity, which may be positive or negative, is then determined by the joint requirements of continuity of heat flux and temperature. A similarity solution is derived for the case of an initially linear temperature gradient and uniform heating. Solutions are also given for a heat flux that undergoes a step change and for a heat flux determined from a four-thirds power law with a fixed base temperature. Numerical calculations show that the predictions of the model are in good agreement with previously reported experimental measurements. Similar calculations are applicable to a wide range of geophysical problems in which the tendency for diffusive restratification is comparable to that for mixed-layer deepening by entrainment.
Stability of two-dimensional convection in a fluid-saturated porous medium
Authors: De La Torre Juárez M. , Busse F. H.
Published: April 2006

The range of existence and the stability of spatially periodic solutions has been studied for steady and oscillatory two-dimensional convection in a fluid-saturated porous medium. We have analysed the limit where viscous effects are dominant and Darcy's law can be applied. A Galerkin method has been used to obtain the steady and the centrosymmetric oscillatory solutions that appear in nonlinear convection at Rayleigh numbers up to 20 times the critical value. Their stability boundaries to arbitrary infinitesimal perturbations have been obtained. Above a given Rayleigh number stable oscillatory solutions are possible at wavenumbers close to the critical values. The stability of this oscillatory state with respect to infinitesimal perturbations of any wavenumber has also been studied. The resulting temporal dynamics in the different unstable regimes is briefly discussed. We show the existence of 3:1 spatial resonances of the steady roll solutions and the existence of stable centrosymmetric and non-centrosymmetric oscillatory solutions.
The rise velocity and shape of bubbles in pure water at high Reynolds number
Authors: Duineveld P. C.
Published: April 2006

The velocity and shape of rising bubbles, with an equivalent radius of 0.33–1.00 mm, in ‘hyper clean’ water, have been experimentally determined. For the small bubbles there is perfect agreement with theory, proving that this water can be considered as pure (no surfactants). For the larger bubbles there is a small discrepancy due to an overestimation in the theory.
Stratified circular Couette flow: instability and flow regimes
Authors: Boubnov B. M. , Gledzer E. B. , Hopfinger E. J.
Published: April 2006

The stability conditions of the flow between two concentric cylinders with the inner one rotating (circular Couette flow) have been investigated experimentally and theoretically for a fluid with axial, stable linear density stratification. The behaviour of the flow, therefore, depends on the Froude number Fr = Ω/N (where Ω is the angular velocity of the inner cylinder and N is the buoyancy frequency of the fluid) in addition to the Reynolds number and the non-dimensional gap width ε, here equal to 0.275.

Experiments show that stratification has a stabilizing effect on the flow with the critical Reynolds number depending on N, in agreement with linear stability theory. The selected, most amplified, vertical wavelength at onset of instability is reduced by the stratification effect and is for the geometry considered only about half the gap width. Furthermore, the observed instability is non-axisymmetric. The resulting vortex motion causes some mixing and this leads to layer formation, clearly visible on shadowgraph images, with the height of the layer being determined by the vertical vortex size. This regime of vertically reduced vortex size is referred to as the S-regime.

For larger Reynolds and Froude numbers the role of stratification decreases and the most amplified vertical wavelength is determined by the gap width, giving rise to the usual Taylor vortices (we call this the T-regime). The layers which emerge are determined by these vortices. For relatively small Reynolds number when Fr ≈ 1 the Taylor vortices are stable and the layers have a height h equal to the gap width; for larger Reynolds number or Fr ≈ 2 the Taylor vortices interact in pairs (compacted Taylor vortices, regime CT) and layers of twice the gap width are predominant. Stratification inhibits the azimuthal wavy vortex flow observed in homogeneous fluid. By further increasing the Reynolds number, turbulent motions appear with Taylor vortices superimposed like in non-stratified fluid.

The theoretical analysis is based on a linear stability consideration of the axisymmetric problem. This gives bounds of instability in the parameter space (Ω, N) for given vertical and radial wavenumbers. These bounds are in qualitative agreement with experiments. The possibility of oscillatory-type instability (overstability) observed experimentally is also discussed.
The temporal behaviour of the hydrodynamic force on a body in response to an abrupt change in velocity at small but finite Reynolds number
Authors: Lovalenti Phillip M. , Brady John F.
Published: April 2006

A modification to the O(Re)-accurate expression for the hydrodynamic force acting on a body in arbitrary time-dependent motion, determined by Lovalenti & Brady (1993), is presented. This simple modification captures the O(Re2) transient behaviour of the force, which has been recently shown to dominate at large time (Lawrence & Mei 1995), while maintaining the overall O(Re) accuracy.
A general theory for two-dimensional vortex interactions
Authors: Dritschel David G.
Published: April 2006

A general theory for two-dimensional vortex interactions is developed from the observation that, under slowly changing external influences, an individual vortex evolves through a series of equilibrium states until such a state proves unstable. Once an unstable equilibrium state is reached, a relatively fast unsteady evolution ensues, typically involving another nearby vortex. During this fast unsteady evolution, a fraction of the original coherent circulation is lost to filamentary debris, and, remarkably, the flow reorganizes into a set of quasi-steady stable vortices.

The simplifying feature of the proposed theory is its use of adiabatic steadiness and marginal stability to determine the shapes and separation distance of vortices on the brink of an inelastic interaction. As a result, the parameter space for the inelastic interaction of nearby vortices is greatly reduced. In the case of two vortex patches, which is the focus of the present work, inelastic interactions depend only on a single parameter: the area ratio of the two vortices (taking the vorticity magnitude inside each to be equal). Without invoking adiabatic steadiness and marginal stability, one would have to contend with the additional parameters of vortex separation and shape, and the latter is actually an infinitude of parameters.
Measurements of spanwise scale change in a forced mixing layer
Authors: Leboeuf Richard L. , Mehta Rabindra D.
Published: April 2006

Spanwise scale changes of the streamwise vortical structure in a plane forced mixing layer have been investigated through direct measurements. Detailed three-dimensional phase-averaged measurements were obtained of the spanwise and streamwise vorticity in a forced mixing layer undergoing three spanwise roller pairings. A two-stream mixing layer with a velocity ratio (U2/U1) of 0.6 and laminar initial boundary layers was generated in a mixing-layer wind tunnel. Acoustic forcing, consisting of a fundamental roll-up frequency and its first, second and third subharmonics, was used to phase-lock the initial development and the first three pairings of the spanwise rollers. Although the overall spanwise scale remained unchanged through the first two roller pairings, some (cyclic) ‘readjustment’ of the weaker streamwise structures was observed. The overall spanwise scale doubled during the third roller pairing. For the first time, one of the proposed mechanisms for the scale change has been identified and its details measured directly. The weakest (positive) streamwise vortex is split into two and displaced by stronger neighbouring (negative) vortices. These two vortices (of the same sign) then merge together, thus doubling the spanwise scale and circulation of the resulting streamwise vortical structure.
Statistical interpretation of the turbulent dissipation rate in wall-bounded flows
Authors: Jovanović J. , Ye Q.-Y. , Durst F.
Published: April 2006

Statistical analysis was performed for interpreting the dissipation correlations in turbulent wall-bounded flows. The fundamental issues related to the formulation of the closure assumptions are discussed. Using the two-point correlation technique, a distinction is made between the homogeneous and inhomogeneous parts of the dissipation tensor. It is shown that the inhomogeneous part contributes half of the dissipation rate at the wall and vanishes remote from the wall region. The structure of an analytically derived equation was analysed utilizing the results of direct numerical simulations of turbulent channel flow at low Reynolds number.