bluuebunny/crossref_metadata_2025 huggingface dataset:
The Rayleigh–Bénard problem in intermediate bounded domains
Authors: Stella F. , Guj G. , Leonardi E.
Published: April 2006

The stationary instabilities of flow patterns associated with Rayleigh–Bénard convection in a 3 × 1 × 9 rectangular container are extensively investigated by numerical simulation. Two types of spatial instabilities of the base convection rolls are predicted in the transition from steady two-dimensional flow to the unsteady oscillatory regime; these instabilities depend on the Prandtl number. For Pr = 0.71 the soft-roll instability is found at moderate Rayleigh number Ra. The results obtained confirm the importance of this flow pattern as a continuous mechanism for steady transition from one wavenumber to another. For Pr = 15, cross-roll instability is obtained, which at larger Ra leads to bimodal convection. For this value of Pr the soft-roll flow pattern is found at intermediate Ra. At higher Ra a new flow structure in which cross-rolls are superimposed on the soft roll is obtained. The effects of the various flow structures on the heat transfer are given. A quantitative comparison with previous experimental and theoretical findings is also presented and discussed.
Bjerknes forces between two bubbles. Part 1. Response to a step change in pressure
Authors: Pelekasis Nikolaos A. , Tsamopoulos John A.
Published: April 2006

It is well known from experiments in acoustic cavitation that two bubbles pulsating in a liquid may attract or repel each other depending on whether they oscillate in or out of phase, respectively. The forces responsible for this phenomenon are called ‘Bjerknes’ forces. When attractive forces are present the two bubbles are seen to accelerate towards each other and coalesce (Kornfeld & Suvorov 1944) and occasionally even breakup in the process. In the present study the response of two initially equal and spherical bubbles is examined under a step change in the hydrostatic pressure at infinity. A hybrid boundary–finite element method is used in order to follow the shape deformation and change in the potential of the two interfaces. Under the conditions mentioned above the two bubbles are found to attract each other always, with a force inversely proportional to the square of the distance between them when this distance is large, a result known to Bjerknes. As time increases the two bubbles continue accelerating towards each other and often resemble either the spherical-cap shapes observed by Davies & Taylor (1950), or the globally deformed shapes observed by Kornfeld & Suvorov (1944). Such shapes occur for sufficiently large or small values of the Bond number respectively (based on the average acceleration). It is also shown here that spherical-cap shapes arise through a Rayleigh–Taylor instability, whereas globally deformed shapes occur as a result of subharmonic resonance between the volume oscillations of the two bubbles and certain non-spherical harmonics (Hall & Seminara 1980). Eventually, in both cases the two bubbles break up due to severe surface deformation.
Bjerknes forces between two bubbles. Part 2. Response to an oscillatory pressure field
Authors: Pelekasis Nikolaos A. , Tsamopoulos John A.
Published: April 2006

The motion of two gas bubbles in response to an oscillatory disturbance in the ambient pressure is studied. It is shown that the relative motion of bubbles of unequal size depends on the frequency of the disturbance. If this frequency is between the two natural frequencies for volume oscillations of the individual bubbles, the two bubbles are seen to move away from each other; otherwise attractive forces prevail. Bubbles of equal size can only attract each other, irrespective of the oscillation frequency. When the Bond number, Bo (based on the average acceleration) lies above a critical region, spherical-cap shapes appear with deformation confined on the side of the bubbles facing away from the direction of acceleration. For Bo below the critical region shape oscillations spanning the entire bubble surface take place, as a result of subharmonic resonance. The presence of the oscillatory acoustic field adds one more frequency to the system and increases the possibilities for resonance. However, only subharmonic resonance is observed because it occurs on a faster timescale, O(1/ε), where ε is the disturbance amplitude. Furthermore, among the different possible periodic variations of the volume of each bubble, the one with the smaller period determines which Legendre mode will be excited through subharmonic resonance. Spherical-cap shapes also occur on a timescale O(1/ε). When the bubbles are driven below resonance and for quite large amplitudes of the acoustic pressure, ε ≈ 0.8, a subharmonic signal at half the natural frequency of volume oscillations is obtained. This signal is primarily associated with the zeroth mode and corresponds to volume expansion followed by rapid collapse of the bubbles, a behaviour well documented in acoustic cavitation experiments.
The hydraulics of steady two-layer flow over a fixed obstacle
Authors: Lawrence Gregory A.
Published: April 2006

This paper reports the results of a theoretical and experimental study of steady two-layer flow over a fixed two-dimensional obstacle. A classification scheme to predict the regime of flow given the maximum height of the obstacle, the total depth of flow, and the density and flow rate of each layer, is presented with experimental confirmation. There are differences between this classification scheme and that derived for flow over a towed obstacle by Baines (1984, 1987). These differences are due to the motion of upstream disturbances in towed obstacle flows. Approach-controlled flows, i.e. flows with an internal hydraulic control in the flow just upstream of the obstacle are studied in detail for the first time. This study reveals that non-hydrostatic forces, rather than a shock solution (called an internal hydraulic drop by previous investigators), need to be considered to explain the behaviour of Approach-controlled flows.
Shallow-water approach to the circular hydraulic jump
Authors: Bohr Tomas , Dimon Peter , Putkaradze Vakhtang
Published: April 2006

We show that the circular hydraulic jump can be qualitatively understood using simplified equations of the shallow-water type which include viscosity. We find that the outer solutions become singular at a finite radius and that this lack of asymptotic states is a general phenomenon associated with radial flow with a free surface. By connecting inner and outer solutions through a shock, we obtain a scaling relation for the radiusRjof the jump,Rj∼Q⅝v⅜g⅛, whereQis the volume flux,vis the kinematic viscosity andgis the gravitational acceleration. This scaling relation is valid asymptotically for largeQ. We discuss the corrections appearing at smallerQand compare with experiments.
Near-contact thermocapillary motion of two non-conducting drops
Authors: Loewenberg Michael , Davis Robert H.
Published: April 2006

The axisymmetric, thermocapillary-driven motion of a pair of non-conducting, spherical drops in near contact is analysed for conditions of small Reynolds and Marangoni numbers. The pairwise motion and an associated contact force are computed by considering touching drops in point contact. Relative motion for nearly touching drops results from the contact force balanced by a lubrication resistance. A new, analytical solution is obtained for the axisymmetric temperature field around an unequal pair of non-conducting, tangent spheres embedded in an ambient temperature gradient. Numerical results for the pairwise migration velocity, contact force, and the relative and individual drop velocities are presented for all size ratios and a wide range of viscosity ratios, and asymptotic formulae are derived for small size ratios. For nearly equisized drops, the ratio of the relative velocity for two drops in near contact to that for widely separated drops is similar for thermocapillary-driven and gravity-driven motion. For small and moderate size ratios, however, this ratio is much larger for thermocapillary-driven relative motion than for gravity-driven relative motion, indicating that the former represents a more efficient coalescence mechanism. An explanation for this finding is provided in terms of the thermocapillary motion of the interface of the larger drop aiding the withdrawal of continuous phase from between the two drops.
Direct simulations of low-Reynolds-number turbulent flow in a rotating channel
Authors: Kristoffersen Reidar , Andersson Helge I.
Published: April 2006

Direct numerical simulations of fully developed pressure-driven turbulent flow in a rotating channel have been performed. The unsteady Navier–Stokes equations were written for flow in a constantly rotating frame of reference and solved numerically by means of a finite-difference technique on a 128 × 128 × 128 computational mesh. The Reynolds number, based on the bulk mean velocity Um and the channel half-width h, was about 2900, while the rotation number Ro = 2|Ω|h/Um varied from 0 to 0.5. Without system rotation, results of the simulation were in good agreement with the accurate reference simulation of Kim, Moin & Moser (1987) and available experimental data. The simulated flow fields subject to rotation revealed fascinating effects exerted by the Coriolis force on channel flow turbulence. With weak rotation (Ro = 0.01) the turbulence statistics across the channel varied only slightly compared with the nonrotating case, and opposite effects were observed near the pressure and suction sides of the channel. With increasing rotation the augmentation and damping of the turbulence along the pressure and suction sides, respectively, became more significant, resulting in highly asymmetric profiles of mean velocity and turbulent Reynolds stresses. In accordance with the experimental observations of Johnston, Halleen & Lezius (1972), the mean velocity profile exhibited an appreciable region with slope 2Ω. At Ro = 0.50 the Reynolds stresses vanished in the vicinity of the stabilized side, and the nearly complete suppression of the turbulent agitation was confirmed by marker particle trackings and two-point velocity correlations. Rotational-induced Taylor-Görtler-like counter-rotating streamwise vortices have been identified, and the simulations suggest that the vortices are shifted slightly towards the pressure side with increasing rotation rates, and the number of vortex pairs therefore tend to increase with Ro.
Mixing of a viscoelastic fluid in a time-periodic flow
Authors: Niederkorn T. C. , Ottino J. M.
Published: April 2006

We present an experimental and computational investigation of mixing of a viscoelastic fluid in two-dimensional time-periodic flows generated in an eccentric cylindrical geometry. The objective of the study is to investigate the impact of fluid elasticity on the morphological structures produced by the advection of passive tracers in chaotic flows. The relevant dimensionless numbers that quantify the rheological differences with respect to the Newtonian fluid are the Deborah number (De), defined as the ratio of the fluid timescale to the flow timescale, and the Weissenberg number (We), defined as the product of the fluid timescale and the mean shear rate. The effects of elasticity are investigated in the limit of slow flows, De ≈ 0 and We < 0.1. The experimental window of We is limited to Newtonian behaviour on the low end and the transition to three-dimensional flow on the high end; experiments show that this window is small, 0.02 < We < 0.1. Typical values of the Reynolds number and the Strouhal number are O(0.001) and O(0.1), respectively.

Results from experiments with a constant-viscosity elastic fluid and computations using the upper-convected Maxwell constitutive equation are presented. Even though the streamlines for the elastic flow are nearly indistinguishable from the Newtonian flow, small deviations in the velocity field produce large effects on chaotically advected patterns. Elasticity affects both the asymptotic coverage of a dyed passive tracer and the rate at which the tracer is stretched. In all cases the tracer undergoes exponential stretching, but on a longer timescale as the elasticity increases. According to flow conditions, elasticity might increase or decrease the degree of regularity; however, island symmetry does not seem to be affected. Similar phenomena are observed in both the experiments and computations; therefore, an analysis of the chaotic dynamics of the periodic flow using numerical techniques is possible.
Compressibility effects on the growth and structure of homogeneous turbulent shear flow
Authors: Blaisdell G. A. , Mansour N. N. , Reynolds W. C.
Published: April 2006

Compressibility effects within decaying isotropic turbulence and homogeneous turbulent shear flow have been studied using direct numerical simulation. The objective of this work is to increase our understanding of compressible turbulence and to aid the development of turbulence models for compressible flows. The numerical simulations of compressible isotropic turbulence show that compressibility effects are highly dependent on the initial conditions. The shear flow simulations, on the other hand, show that measures of compressibility evolve to become independent of their initial values and are parameterized by the root mean square Mach number. The growth rate of the turbulence in compressible homogeneous shear flow is reduced compared to that in the incompressible case. The reduced growth rate is the result of an increase in the dissipation rate and energy transfer to internal energy by the pressure–dilatation correlation. Examination of the structure of compressible homogeneous shear flow reveals the presence of eddy shocklets, which are important for the increased dissipation rate of compressible turbulence.
The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number
Authors: Lovalenti Phillip M. , Brady John F.
Published: April 2006

The hydrodynamic force acting on a rigid spherical particle translating with arbitrary time-dependent motion in a time-dependent flowing fluid is calculated to O(Re) for small but finite values of the Reynolds number, Re, based on the particle's slip velocity relative to the uniform flow. The corresponding expression for an arbitrarily shaped rigid particle is evaluated for the case when the timescale of variation of the particle's slip velocity is much greater than the diffusive scale, a2/v, where a is the characteristic particle dimension and v is the kinematic viscosity of the fluid. It is found that the expression for the hydrodynamic force is not simply an additive combination of the results from unsteady Stokes flow and steady Oseen flow and that the temporal decay to steady state for small but finite Re is always faster than the t-½ behaviour of unsteady Stokes flow. For example, when the particle accelerates from rest the temporal approach to steady state scales as t-2.
Vortex rings impinging on walls: axisymmetric and three-dimensional simulations
Authors: Orlandi Paolo , Verzicco Roberto
Published: April 2006

Accurate numerical simulations of vortex rings impinging on flat boundaries revealed the same features observed in experiments. The results for the impact with a free-slip wall compared very well with previous numerical simulations that used spectral methods, and were also in qualitative agreement with experiments. The present simulation is mainly devoted to studying the more realistic case of rings interacting with a no-slip wall, experimentally studied by Walker et al. (1987). All the Reynolds numbers studied showed a very good agreement between experiments and simulations, and, at Rev > 1000 the ejection of a new ring from the wall was seen. Axisymmetric simulations demonstrated that vortex pairing is the physical mechanism producing the ejection of the new ring. Three-dimensional simulations were also performed to investigate the effects of azimuthal instabilities. These simulations have confirmed that high-wavenumber instabilities originate in the compression phase of the secondary ring within the primary one. The large instability of the secondary ring has been explained by analysis of the rate-of-strain tensor and vorticity alignment. The differences between passive scalars and the vorticity field have been also investigated.
Buoyancy-driven interactions between two deformable viscous drops
Authors: Manga Michael , Stone H. A.
Published: April 2006

Time-dependent interactions between two buoyancy-driven deformable drops are studied in the low Reynolds number flow limit for sufficiently large Bond numbers that the drops become significantly deformed. The first part of this paper considers the interaction and deformation of drops in axisymmetric configurations. Boundary integral calculations are presented for Bond numbers ℬ = Δρga2/σ in the range 0.25 ≤ ℬ < ∞ and viscosity ratios λ in the range 0.2 ≤ λ ≤ 20. Specifically, the case of a large drop following a smaller drop is considered, which typically leads to the smaller drop coating the larger drop for ℬ [Gt ] 1. Three distinct drainage modes of the thin film of fluid between the drops characterize axisymmetric two-drop interactions: (i) rapid drainage for which the thinnest region of the film is on the axis of symmetry, (ii) uniform drainage for which the film has a nearly constant thickness, and (iii) dimple formation. The initial mode of film drainage is always rapid drainage. As the separation distance decreases, film flow may change to uniform drainage and eventually to dimpled drainage. Moderate Bond numbers, typically ℬ = O(10) for λ = O(1), enhance dimple formation compared to either much larger or smaller Bond numbers. The numerical calculations also illustrate the extent to which lubrication theory and analytical solutions in bipolar coordinates (which assume spherical drop shapes) are applicable to deformable drops.

The second part of this investigation considers the 'stability’ of axisymmetric drop configurations. Laboratory experiments and two-dimensional boundary integral simulations are used to study the interactions between two horizontally offset drops. For sufficiently deformable drops, alignment occurs so that the small drop may still coat the large drop, whereas for large enough drop viscosities or high enough interfacial tension, the small drop will be swept around the larger drop. If the large drop is sufficiently deformable, the small drop may then be ‘sucked’ into the larger drop as it is being swept around the larger drop. In order to explain the alignment process, the shape and translation velocities of widely separated, nearly spherical drops are calculated using the method of reflections and a perturbation analysis for the deformed shapes. The perturbation analysis demonstrates explicitly that drops will tend to be aligned for ℬ > O(d/a) where d is the separation distance between the drops.
Nonlinear temporal-spatial modulation of near-planar Rayleigh waves in shear flows: formation of streamwise vortices
Authors: Wu Xuesong
Published: April 2006

The nonlinear temporal-spatial modulation of a near-planar Rayleigh instability wave is studied. The amplitude of the wave is allowed to be a slowly varying function of spanwise position as well as of time (or streamwise variable in the spatial evolution case). It is shown that the development of the disturbance is controlled by critical-layer nonlinear effects when the linear growth rate decreases to O(ε⅖), where ε is the magnitude of the disturbance. Nonlinear interactions influence the evolution by producing spanwise dependent mean-flow distortions. The evolution is governed by an integro-partial-differential equation containing history-dependent nonlinear terms of Hickernell (1984) type. A notable feature of the amplitude equation is that the highest derivative with respect to spanwise position appears in the nonlinear terms. These terms are associated with three-dimensionality. The possible properties of the amplitude equation are discussed. Numerical solutions show that a disturbance initially centred at a spanwise position can propagate laterally to form concentrated, quasi-periodic streamwise vortices. This qualitatively captures the phenomena observed in experiments. The focusing of vorticity may be associated with a localized singularity which can occur at a finite distance downstream or within a finite time. It is noted that the amplitude equation is rather generic and applies to a broad class of shear flows which is inviscidly unstable.
Global linear stability analysis of weakly non-parallel shear flows
Authors: Monkewitz Peter A. , Huerre Patrick , Chomaz Jean-Marc
Published: April 2006

The global linear stability of incompressible, two-dimensional shear flows is investigated under the assumptions that far-field pressure feedback between distant points in the flow field is negligible and that the basic flow is only weakly non-parallel, i.e. that its streamwise development is slow on the scale of a typical instability wavelength. This implies the general study of the temporal evolution of global modes, which are time-harmonic solutions of the linear disturbance equations, subject to homogeneous boundary conditions in all space directions. Flow domains of both doubly infinite and semi-infinite streamwise extent are considered and complete solutions are obtained within the framework of asymptotically matched WKBJ approximations. In both cases the global eigenfrequency is given, to leading order in the WKBJ parameter, by the absolute frequency ω0(Xt) at the dominant turning pointXtof the WKBJ approximation, while its quantization is provided by the connection of solutions acrossXt. Within the context of the present analysis, global modes can therefore only become time-amplified or self-excited if the basic flow contains a region of absolute instability.
On the irreversibility of internal-wave dynamics due to wave trapping by mean flow inhomogeneities. Part 1. Local analysis
Authors: Badulin Sergei I. , Shrira Victor I.
Published: April 2006

The propagation of guided internal waves on non-uniform large-scale flows of arbitrary geometry is studied within the framework of linear inviscid theory in the WKB-approximation. Our study is based on a set of Hamiltonian ray equations, with the Hamiltonian being determined from the Taylor-Goldstein boundary-value problem for a stratified shear flow. Attention is focused on the fundamental fact that the generic smooth non-uniformities of the large-scale flow result in specific singularities of the Hamiltonian. Interpreting wave packets as particles with momenta equal to their wave vectors moving in a certain force field, one can consider these singularities as infinitely deep potential holes acting quite similarly to the ‘black holes’ of astrophysics. It is shown that the particles fall for infinitely long time, each into its own ‘black hole‘. In terms of a particular wave packet this falling implies infinite growth with time of the wavenumber and the amplitude, as well as wave motion focusing at a certain depth. For internal-wave-field dynamics this provides a robust mechanism of a very specific conservative and moreover Hamiltonian irreversibility.

This phenomenon was previously studied for the simplest model of the flow non-uniformity, parallel shear flow (Badulin, Shrira & Tsimring 1985), where the term ‘trapping’ for it was introduced and the basic features were established. In the present paper we study the case of arbitrary flow geometry. Our main conclusion is that although the wave dynamics in the general case is incomparably more complicated, the phenomenon persists and retains its most fundamental features. Qualitatively new features appear as well, namely, the possibility of three-dimensional wave focusing and of ‘non-dispersive’ focusing. In terms of the particle analogy, the latter means that a certain group of particles fall into the same hole.

These results indicate a robust tendency of the wave field towards an irreversible transformation into small spatial scales, due to the presence of large-scale flows and towards considerable wave energy concentration in narrow spatial zones.
The mean rise velocity of pairwise-interacting bubbles in liquid
Authors: Wijngaarden L. van
Published: April 2006

This paper is concerned with calculations regarding a collection of small gas bubbles rising under buoyancy in a clear liquid. For dilute mixtures interactions can be restricted to those between two bubbles. In the analysis of binary interactions it is assumed that the Reynolds number for relative motion between bubbles and liquid is large, that surfactants, if present, do not modify the condition of zero tangential stress at the bubble-liquid interface, and that bubbles bounce at an encounter.

A two-bubble probability density is derived from the analysis, valid on a short timescale associated with the interaction. It is shown that on a long timescale, based on viscous dissipation, clustering together of pairs takes place, most likely even when triple encounters are allowed for. An analysis is given of the vertical motion of pairs, followed by a calculation of the mean vertical bubble velocity with help of the (short timescale) probability density function. The result is compared with experimental data.
Turbulent shear flow over slowly moving waves
Authors: Belcher S. E. , Hunt J. C. R.
Published: April 2006

We investigate the changes to a fully developed turbulent boundary layer caused by the presence of a two-dimensional moving wave of wavelength L = 2π/k and amplitude a. Attention is focused on small slopes, ak, and small wave speeds, c, so that the linear perturbations are calculated as asymptotic sequences in the limit (u* + c)/UB(L) → 0 (u* is the unperturbed friction velocity and UB(L) is the approach-flow mean velocity at height L). The perturbations can then be described by an extension of the four-layer asymptotic structure developed by Hunt, Leibovich & Richards (1988) to calculate the changes to a boundary layer passing over a low hill.

When (u* + c)/UB(L) is small, the matched height, zm (the height where UB equals c), lies within an inner surface layer, where the perturbation Reynolds shear stress varies only slowly. Solutions across the matched height are then constructed by considering an equation for the shear stress. The importance of the shear-stress perturbation at the matched height implies that the inviscid theory of Miles (1957) is inappropriate in this parameter range. The perturbations above the inner surface layer are not directly influenced by the matched height and the region of reversed flow below zm: they are similar to the perturbations due to a static undulation, but the ‘effective roughness length’ that determines the shape of the unperturbed velocity profile is modified to zm = z0 exp (kc/u*).

The solutions for the perturbations to the boundary layer are used to calculate the growth rate of waves, which is determined at leading order by the asymmetric pressure perturbation induced by the thickening of the perturbed boundary layer on the leeside of the wave crest. At first order in (u* + c)/UB(L), however, there are three new effects which, numerically, contribute significantly to the growth rate, namely: the asymmetries in both the normal and shear Reynolds stresses associated with the leeside thickening of the boundary layer, and asymmetric perturbations induced by the varying surface velocity associated with the fluid motion in the wave; further asymmetries induced by the variation in the surface roughness along the wave may also be important.
The influence of swirl and confinement on the stability of counterflowing streams
Authors: Goddard J. D. , Didwania A. K. , Wu C.-Y.
Published: April 2006

A linear stability analysis of laterally confined swirling flow is given, of the type described by Long's equation in the inviscid limit or by the von Kármán similarity equations in the absence of lateral confinement. The flow of interest involves identical counterflowing fluid streams injected with equal velocityW0through opposing porous disks, rotating with angular velocities Ω and ±Ω, respectively, about a common normal axis. By means of mass transfer experiments on an aqueous system of this type we have detected an apparent hydrodynamic instability having the appearance of an inviscid supercritical bifurcation at a certain |Ω| > 0. As an attempt to elucidate this phenomenon, linear stability analyses are performed on several idealized flows, by means of a numerical Galerkin technique. An analysis of high-Reynolds-number similarity flow predicts oscillatory instability for all non-zero Ω. The spatial structure of the most unstable modes suggests that finite container geometry, as represented by the confining cylindrical sidewalls, may have a strong influence on flow stability. This is borne out by an inviscid stability analysis of a confined flow described by Long's equation. This analysis suggests a novel bifurcation of the inviscid variety, which serves qualitatively to explain the results of our mass transfer experiments.
A wake singularity potential flow model for airfoils experiencing trailing-edge stall
Authors: Yeung W. W. H. , Parkinson G. V.
Published: April 2006

An incompressible inviscid flow theory for single and two-element airfoils experiencing trailing-edge stall is presented. For the single airfoil the model requires a simple sequence of conformal transformations to map a Joukowsky airfoil, partially truncated on the upper surface, onto a circle over which the flow problem is solved. Source and doublet singularities are used to create free streamlines simulating shear layers bounding the near wake. The model's simplicity permits extension of the method to airfoil-flap configurations in which trailing-edge stall is assumed on the flap. Williams’ analytical method to calculate the potential flow about two lifting bodies is incorporated in the Joukowsky-arc wake-singularity model to allow for flow separation. The theoretical pressure distributions from these models show good agreement with wind-tunnel measurements.
Isotropy of the small scales of turbulence at low Reynolds number
Authors: Kim J. , Antonia R. A.
Published: April 2006

Spectral local isotropy tests are applied to direct numerical simulation data, mainly at the centreline of a fully developed turbulent channel flow. Despite the small Reynolds number of the simulation, the high-wavenumber behaviour of velocity and vorticity spectra is consistent with local isotropy. This consistency is verified by the relationship between streamwise wavenumber spectra and spanwise wavenumber spectra. The high-wavenumber behaviour of the pressure spectrum is also consistent with local isotropy and compares favourably with the calculation of Batchelor (1951), which assumes isotropy and joint normality of the velocity field at two points in space. The latter assumption is validated by the shape but not the magnitude of the quadruple correlation of the streamwise velocity fluctuation at small separations. There is only partial support for local spectral isotropy away from the centreline as the magnitude of the mean strain rate increases.
Shear-induced particle diffusion and longitudinal velocity fluctuations in a granular-flow mixing layer
Authors: Hsiau S. S. , Hunt M. L.
Published: April 2006

In flows of granular material, collisions between individual particles result in the movement of particles in directions transverse to the bulk motion. If the particles were distinguishable, a macroscopic overview of the transverse motions of the particles would resemble a self-diffusion of molecules as occurs in a gas. The present granular- flow study includes measurements of the self-diffusion process, and of the corresponding profiles of the average velocity and of the streamwise component of the fluctuating velocity. The experimental facility consists of a vertical channel fed by an entrance hopper that is divided by a splitter plate. Using differently-coloured but otherwise identical glass spheres to visualize the diffusion process, the flow resembles a classic mixing-layer experiment. Unlike molecular motions, the local particle movements result from shearing of the flow; hence, the diffusion experiments were performed for different shear rates by changing the sidewall conditions of the test section, and by varying the flow rate and the channel width. In addition, experiments were also conducted using different sizes of glass beads to examine the scaling of the diffusion process. A simple analysis based on the diffusion equation shows that the thickness of the mixing layer increases with the square-root of downstream distance and depends on the magnitude of the velocity fluctuations relative to the mean velocity. The results are also consistent with other studies that suggest that the diffusion coefficient is proportional to the particle diameter and the square-root of the granular temperature.
Suspensions of prolate spheroids in Stokes flow. Part 1. Dynamics of a finite number of particles in an unbounded fluid
Authors: Claeys Ivan L. , Brady John F.
Published: April 2006

A new simulation method is presented for low-Reynolds-number flow problems involving elongated particles in an unbounded fluid. The technique extends the principles of Stokesian dynamics, a multipole moment expansion method, to ellipsoidal particle shapes. The methodology is applied to prolate spheroids in particular, and shown to be efficient and accurate by comparison with other numerical methods for Stokes flow. The importance of hydrodynamic interactions is illustrated by examples on sedimenting spheroids and particles in a simple shear flow.
Suspensions of prolate spheroids in Stokes flow. Part 2. Statistically homogeneous dispersions
Authors: Claeys Ivan L. , Brady John F.
Published: April 2006

The simulation method for prolate spheroids in Stokes flow introduced in a companion paper (Claeys & Brady 1993a) is extended to handle statistically homogeneous unbounded dispersions. The convergence difficulties associated with the slow decay of velocity disturbances at zero Reynolds number are overcome by applying O';Brien's renormalization procedure. The Ewald summation technique is employed to accelerate the evaluation of all mobility interactions. As a first application of this new method, the hydrodynamic transport properties of equilibrium hard-ellipsoid structures are calculated for aspect ratios ranging from 3 to 50. Calculated viscosities in the isotropic phase agree reasonably well with published experimental measurements.
Suspensions of prolate spheroids in Stokes flow. Part 3. Hydrodynamic transport properties of crystalline dispersions
Authors: Claeys Ivan L. , Brady John F.
Published: April 2006

The short-time limit of the hydrodynamic transport properties is calculated for crystalline dispersions of parallel prolate spheroids using a moment expansion technique similar in concept to the simulation method known as Stokesian dynamics. The concentration dependence of the sedimentation rate, the hindered diffusivity and the Theological behaviour of face-centred lattices are examined for concentrations up to regular close packing (74% by volume). The influence of the detailed microstructure of the dispersion is also investigated by considering different arrangements of parallel ellipsoids. Useful reference configurations are proposed as standard geometries for regular arrays of prolate spheroids.
Baroclinic instability of large-amplitude geostrophic flows
Authors: Benilov E. S.
Published: April 2006

This paper examines the large-scale dynamics of a layer of stratified fluid on the β-plane. A three-dimensional asymptotic system is derived which governs geostrophic flows with large displacement of isopycnal surfaces. This is then reduced to a two-dimensional set of equations which describe the interaction of a baroclinic ‘quasi-mode’ with arbitrary vertical profile and barotrophic motion. The baroclinic instability of large-amplitude zonal flows with vertical shear is studied within the framework of these equations. In the case where the displacement of isopycnal surfaces is small, the results obtained should overlap with the ‘traditional’ baroclinic instability of quasi-geostrophic (small-amplitude) flows. In order to compare the two types of instability, the quasi-geostrophic boundary-value problem is solved asymptotically for the case of long-wave disturbances and weak β-effect (the latter limit of quasi-geostrophic theory has not been considered previously). The instability that is found is linked to the Hamiltonian structure of the governing equations. The equations derived are generalized for the case of more than one baroclinic quasi-mode.
First-order resonance in the reflection of baroclinic Rossby waves
Authors: Graef Federico
Published: April 2006

The nonlinear interaction between an incident and a reflected Rossby wave produces a steady flow parallel to the (non-zonal) reflecting wall and a transient flow oscillating at twice the frequency of the incident-reflected pair. If the transient forcing is resonant, i.e. a free Rossby wave, the resonant response must have zero amplitude at the wall in order to fulfil the boundary condition there; a straightforward expansion predicts a linear growth of its amplitude in the offshore direction y. Resonance is possible only if 0 < |sin α| ≤ $\frac13$, where α is the angle between the wall and the easterly direction. This requirement is met by several boundaries in the ocean. A simple graphical method to find a resonant triad is described.

Using the method of multiple scales, it is shown that the wave amplitudes of the triad are slowly varying periodic functions of y, such that the energy flux of the triad through any plane parallel to the wall vanishes, as required by energy conservation. The waves participating in the resonant triad become wave packets. The three waves do not exchange energy in time due to the additional constraint on the motion imposed by the boundary condition at the wall. It is shown that the wave amplitudes cannot be slowly varying functions of v and time.

As a possible oceanic application of the theoretical findings, the distance from the wall where one would expect to find large semi-annual amplitudes if annual Rossby waves are impinging on the boundary is of the order of 100 km. Motivated by similar studies (Plumb 1977; Mysak 1978), there are speculations on what would happen if three incident-reflected Rossby wave pairs (or modes) are taken, allowing each mode amplitude to be slowly varying in time.
The vaporization of a liquid front moving through a hot porous rock
Authors: Woods Andrew W. , Fitzgerald Shaun D.
Published: April 2006

We develop an analytical model to describe the generation of vapour as water moves through a hot porous rock, as occurs in hot, geothermal reservoirs. Typically the isotherms in the liquid lag behind the water-vapour interface and so water is supplied to the interface at the interface temperature. This temperature is lower than that in the rock far ahead of the interface. Therefore, as the hot porous rock is invaded with water, it cools and the heat released is used to vaporize some of the water. At low injection rates, vapour formed from the injected liquid may readily move ahead of the advancing liquid-vapour interface and so the interfacial pressure remains close to that in the far field ahead of the interface. The mass fraction that vaporizes is then limited by the superheat of the rock. For larger injection rates, the interfacial vapour pressure becomes considerably greater than that in the far field in order to drive the vapour ahead of the moving interface. As a result, the interfacial temperature increases. The associated reduction in the thermal energy available for vaporization results in a decrease in the mass fraction of vapour produced.

Since the vapour is compressible, the motion of the vapour ahead of the interface is governed by a nonlinear diffusion equation. Therefore, the geometry of injection has an important effect upon the mass fraction of water that vaporizes. We show that with a constant supply of water from (i) a point source, the mass fraction of water which vaporizes increases towards the maximum permitted by the superheat of the rock; (ii) a line source, a similarity solution exists in which the mass fraction vaporizing is constant; and (iii) a planar source, the liquid-vapour interface steadily translates through the rock with a very small fraction of the injected water vaporizing.
Simple shear flow of a suspension of fibres in a dilute polymer solution at high Deborah number
Authors: Harlen O. G. , Koch Donald L.
Published: April 2006

The behaviour of fibre suspensions in dilute polymer solutions at high Deborah numbers is analysed. In particular, we calculate the change to the extension of the polymers and the orientation of the fibres caused by hydrodynamic interactions between the polymers and the fibres. At a sufficiently high Deborah number the combined effect of the fibre velocity disturbances and the mean shear flow produce a dramatic increase in the extension of the polymers, similar to the coil-stretch transition observed in extensional flow.

The non-Newtonian stresses caused by the polymers produce a perturbation to the angular velocity of the fibres, giving rise to a net drift across Jeffery orbits towards the vorticity axis. Unlike the second-order-fluid analysis of Leal (1975), this effect does not depend on the second-normal-stress difference.
Energy growth in viscous channel flows
Authors: Reddy Satish C. , Henningson Dan S.
Published: April 2006

In recent work it has been shown that there can be substantial transient growth in the energy of small perturbations to plane Poiseuille and Couette flows if the Reynolds number is below the critical value predicted by linear stability analysis. This growth, which may be as large as O(1000), occurs in the absence of nonlinear effects and can be explained by the non-normality of the governing linear operator - that is, the non-orthogonality of the associated eigenfunctions. In this paper we study various aspects of this energy growth for two- and three-dimensional Poiseuille and Couette flows using energy methods, linear stability analysis, and a direct numerical procedure for computing the transient growth. We examine conditions for no energy growth, the dependence of the growth on the streamwise and spanwise wavenumbers, the time dependence of the growth, and the effects of degenerate eigenvalues. We show that the maximum transient growth behaves like O(R2), where R is the Reynolds number. We derive conditions for no energy growth by applying the Hille–Yosida theorem to the governing linear operator and show that these conditions yield the same results as those derived by energy methods, which can be applied to perturbations of arbitrary amplitude. These results emphasize the fact that subcritical transition can occur for Poiseuille and Couette flows because the governing linear operator is non-normal.
Linear acoustic properties of bubbly liquids near the natural frequency of the bubbles using numerical simulations
Authors: Sangani A. S. , Sureshkumar R.
Published: April 2006

We consider the problem of determining linear acoustic properties of bubbly liquids near the natural frequency of the bubbles. Since the effective wavelength and attenuation length are of the same order of magnitude as the size of the bubbles, we devise a numerical scheme to determine these quantities by solving exactly the multiple scattering problem among many interacting bubbles. It is shown that the phase speed and attenuation are finite at natural frequency even in the absence of damping due to viscous, thermal, nonlinear, and liquid compressibility effects, thus validating a recent theory (Sangani 1991). The results from the numerical scheme are in good agreement with the theory but considerably higher than the experimental values for frequencies greater than the natural frequency. The discrepancy with experiments remains even after accounting for the effect of polydispersity, finite liquid compressibility, and non-adiabatic thermal changes.
Recriprocal theorem for concentric compound drops in arbitrary Stokes flows
Authors: Haj-Hariri H. , Nadim A. , Borhan A.
Published: April 2006

The Lorentz reciprocal theorem is generalized and applied to the study of the quasisteady motion of a concentric spherical (CS) compound drop at zero Reynolds number. Using this result, the migration velocities of a force-free CS compound drop placed in a general ambient Stokes flow, as well as the forces on each drop when subjected to specified migration velocities, are calculated. The latter constitutes a generalization of Faxén's law to the case of a CS compound drop. Also some earlier results on the thermocapillary migration of such drops (Borhan et al. 1992) are rederived more simply and in greater generality.
On the nonlinear dynamics of free bars in straight channels
Authors: Schielen R. , Doelman A. , de Swart H. E.
Published: April 2006

A simple morphological model is considered which describes the interaction between a unidirectional flow and an erodible bed in a straight channel. For sufficiently large values of the width-depth ratio of the channel the basic state, i.e. a uniform current over a flat bottom, is unstable. At near-critical conditions growing perturbations are confined to a narrow spectrum and the bed profile has an alternate bar structure propagating in the downstream direction. The timescale associated with the amplitude growth is large compared to the characteristic period of the bars. Based on these observations a weakly nonlinear analysis is presented which results in a Ginzburg-Landau equation. It describes the nonlinear evolution of the envelope amplitude of the group of marginally unstable alternate bars. Asymptotic results of its coefficients are presented as perturbation series in the small drag coefficient of the channel. In contrast to the Landau equation, described by Colombini et al. (1987), this amplitude equation also allows for spatial modulations due to the dispersive properties of the wave packet. It is demonstrated rigorously that the periodic bar pattern can become unstable through this effect, provided the bed is dune covered, and for realistic values of the other physical parameters. Otherwise, it is found that the periodic bar pattern found by Colombini et al. (1987) is stable. Assuming periodic behaviour of the envelope wave in a frame moving with the group velocity, simulations of the dynamics of the Ginzburg-Landau equation using spectral models are carried out, and it is shown that quasi-periodic behaviour of the bar pattern appears.
Tollmien-Schlichting-wave resonant mechanism for subharmonic-type transition
Authors: Zelman M. B. , Maslennikova I. I.
Published: April 2006

Disturbance interactions in wave triads and multiwave systems of various configurations are investigated to reveal the mechanism of laminar-turbulent transition in Blasius and pressure-gradient boundary layers. The averaging method of weakly nonlinear instability theory in quasi-parallel flows is applied. Tollmien-Schlichting-wave resonant interaction is shown to be the only leading mechanism of subharmonic (S)-type transition. The mechanism universally dominates in boundary layers excited by sufficiently small initial disturbances. The role of any other mode is inefficient. Weakly nonlinear models are concluded not to explain the K-type transition scenario. The results of the study are employed to interpret physical and numerical experimental data.