Papers
Viscous and inviscid centre modes in the linear stability of vortices: the vicinity of the neutral curves
- DAVID FABRE, STÉPHANE LE DIZÈS
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- 30 April 2008, pp. 1-38
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In a previous paper, We have recently that if the Reynolds number is sufficiently large, all trailing vortices with non-zero rotation rate and non-constant axial velocity become linearly unstable with respect to a class of viscous centre modes. We provided an asymptotic description of these modes which applies away from the neutral curves in the (q, k)-plane, where q is the swirl number which compares the azimuthal and axial velocities, and k is the axial wavenumber. In this paper, we complete the asymptotic description of these modes for general vortex flows by considering the vicinity of the neutral curves. Five different regions of the neutral curves are successively considered. In each region, the stability equations are reduced to a generic form which is solved numerically. The study permits us to predict the location of all branches of the neutral curve (except for a portion of the upper neutral curve where it is shown that near-neutral modes are not centre modes). We also show that four other families of centre modes exist in the vicinity of the neutral curves. Two of them are viscous damped modes and were also previously described. The third family corresponds to stable modes of an inviscid nature which exist outside of the unstable region. The modes of the fourth family are also of an inviscid nature, but their structure is singular owing to the presence of a critical point. These modes are unstable, but much less amplified than unstable viscous centre modes. It is observed that in all the regions of the neutral curve, the five families of centre modes exchange their identity in a very intricate way. For the q vortex model, the asymptotic results are compared to numerical results, and a good agreement is demonstrated for all the regions of the neutral curve. Finally, the case of ‘pure vortices’ without axial flow is also considered in a similar way. In this case, centre modes exist only in the long-wave limit, and are always stable. A comparison with numerical results is performed for the Lamb–Oseen vortex.
Local and global instabilities in the wake of a sphere
- BENOÎT PIER
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- 30 April 2008, pp. 39-61
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The global dynamics of open shear flows is closely related to the nature of their local instability characteristics, either convective or absolute. The present investigation revisits the wake of a sphere, obtains its global behaviour by direct numerical simulations and derives its local stability features, computed for the underlying basic flow under a quasi-parallel flow assumption. It is shown that both the axisymmetric and the planar symmetric basic flows exhibit domains of local absolute instability in the near-wake region. The largest absolute growth rates occur for instabilities developing on the non-axisymmetric basic wake and conserving its planar symmetry.
Evolution of clusters of sedimenting low-Reynolds-number particles with Oseen interactions
- G. SUBRAMANIAN, DONALD L. KOCH
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- Published online by Cambridge University Press:
- 30 April 2008, pp. 63-100
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A theoretical framework is developed to describe, in the limit of small but finite Re, the evolution of dilute clusters of sedimenting particles. Here, Re =aU/ν is the particle Reynolds number, where a is the radius of the spherical particle, U its settling velocity, and ν the kinematic viscosity of the suspending fluid. The theory assumes the disturbance velocity field at sufficiently large distances from a sedimenting particle, even at small Re, to possess the familiar source--sink character; that is, the momentum defect brought in via a narrow wake behind the particle is convected radially outwards in the remaining directions. It is then argued that for spherical clusters with sufficiently many particles, specifically with N much greater than O(R0U/ν), the initial evolution is strongly influenced by wake-mediated interactions; here, N is the total number of particles, and R0 is the initial cluster radius. As a result, the cluster first evolves into a nearly planar configuration with an asymptotically small aspect ratio of O(R0U/N ν), the plane of the cluster being perpendicular to the direction of gravity; subsequent expansion occurs with an unchanged aspect ratio. For relatively sparse clusters with N smaller than O(R0U/ν), the probability of wake interactions remains negligible, and the cluster expands while retaining its spherical shape. The long-time expansion in the former case, and that for all times in the latter case, is driven by disturbance velocity fields produced by the particles outside their wakes. The resulting interactions between particles are therefore mutually repulsive with forces that obey an inverse-square law. The analysis presented describes cluster evolution in this regime. A continuum representation is adopted with the clusters being characterized by a number density field (n(r, t)), and a corresponding induced velocity field (u (r, t)) arising on account of interactions. For both planar axisymmetric clusters and spherical clusters with radial symmetry, the evolution equation admits a similarity solution; either cluster expands self-similarly for long times. The number density profiles at different times are functions of a similarity variable η = (r/t1/3), r being the radial distance away from the cluster centre, and t the time. The radius of the expanding cluster is found to be of the form Rcl (t) = A (ν a)1/3N1/3t1/3, where the constant of proportionality, A, is determined from an analytical solution of the evolution equation; one finds A = 1.743 and 1.651 for planar and spherical clusters, respectively. The number density profile in a planar axisymmetric cluster is also obtained numerically as a solution of the initial value problem for a canonical (Gaussian) initial condition. The numerical results compare well with theoretical predictions, and demonstrate the asymptotic stability of the similarity solution in two dimensions for long times, at least for axisymmetric initial conditions.
On the non-local geometry of turbulence
- IVÁN BERMEJO-MORENO, D. I. PULLIN
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- 30 April 2008, pp. 101-135
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A multi-scale methodology for the study of the non-local geometry of eddy structures in turbulence is developed. Starting from a given three-dimensional field, this consists of three main steps: extraction, characterization and classification of structures. The extraction step is done in two stages. First, a multi-scale decomposition based on the curvelet transform is applied to the full three-dimensional field, resulting in a finite set of component three-dimensional fields, one per scale. Second, by iso-contouring each component field at one or more iso-contour levels, a set of closed iso-surfaces is obtained that represents the structures at that scale. The characterization stage is based on the joint probability density function (p.d.f.), in terms of area coverage on each individual iso-surface, of two differential-geometry properties, the shape index and curvedness, plus the stretching parameter, a dimensionless global invariant of the surface. Taken together, this defines the geometrical signature of the iso-surface. The classification step is based on the construction of a finite set of parameters, obtained from algebraic functions of moments of the joint p.d.f. of each structure, that specify its location as a point in a multi-dimensional ‘feature space’. At each scale the set of points in feature space represents all structures at that scale, for the specified iso-contour value. This then allows the application, to the set, of clustering techniques that search for groups of structures with a common geometry. Results are presented of a first application of this technique to a passive scalar field obtained from 5123 direct numerical simulation of scalar mixing by forced, isotropic turbulence (Reλ = 265). These show transition, with decreasing scale, from blob-like structures in the larger scales to blob- and tube-like structures with small or moderate stretching in the inertial range of scales, and then toward tube and, predominantly, sheet-like structures with high level of stretching in the dissipation range of scales. Implications of these results for the dynamical behaviour of passive scalar stirring and mixing by turbulence are discussed.
Poiseuille flow in a fluid overlying a porous medium
- ANTONY A. HILL, BRIAN STRAUGHAN
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- 30 April 2008, pp. 137-149
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This paper numerically investigates the instability of Poiseuille flow in a fluid overlying a porous medium saturated with the same fluid. A three-layer configuration is adopted. Namely, a Newtonian fluid overlying a Brinkman porous transition layer, which in turn overlies a layer of Darcy-type porous material. It is shown that there are two modes of instability corresponding to the fluid and porous layers, respectively. The key parameters which affect the stability characteristics of the system are the depth ratio between the porous and fluid layers and the transition layer depth.
Self-similar solutions of unsteady ablation flows in inertial confinement fusion
- C. BOUDESOCQUE-DUBOIS, S. GAUTHIER, J.-M. CLARISSE
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- 30 April 2008, pp. 151-178
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We exhibit and detail the properties of self-similar solutions for inviscid compressible ablative flows in slab symmetry with nonlinear heat conduction which are relevant to inertial confinement fusion (ICF). These solutions have been found after several contributions over the last four decades. We first derive the set of ODEs – a nonlinear eigenvalue problem – which governs the self-similar solutions by using the invariance of the Euler equations with nonlinear heat conduction under the two-parameter Lie group symmetry. A sub-family which leaves the density invariant is detailed since these solutions may be used to model the ‘early-time’ period of an ICF implosion where a shock wave travels from the front to the rear surface of a target. A chart allowing us to determine the starting point of a numerical solution, knowing the physical boundary conditions, has been built. A physical analysis of these unsteady ablation flows is then provided, the associated dimensionless numbers (Mach, Froude and Péclet numbers) being calculated. Finally, we show that self-similar ablation fronts generated within the framework of the above hypotheses (electron heat conduction, growing heat flux at the boundary, etc.) and for large heat fluxes and not too large pressures at the boundary do not satisfy the low-Mach-number criteria. Indeed both the compressibility and the stratification of the hot-flow region are too large. This is, in particular, the case for self-similar solutions obtained for energies in the range of the future Laser MegaJoule laser facility. Two particular solutions of this latter sub-family have been recently used for studying stability properties of ablation fronts.
Spatial optimal disturbances in swept attachment-line boundary layers
- ALAN GUÉGAN, PETER J. SCHMID, PATRICK HUERRE
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- 30 April 2008, pp. 179-188
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A formulation based on direct and adjoint parabolized equations is developed to account for the spatial evolution of perturbations in swept attachment-line boundary layers. For sweep Reynolds numbers larger than Re = 100 the dynamics is dominated by a lift-up mechanism which is responsible for large energy amplification by transforming spanwise vortices into spanwise streaks. This mechanism favours steady perturbations with a chordwise scale that quantitatively matches its counterpart for classical Blasius boundary layers.
Instabilities of plane Poiseuille flow with a streamwise system rotation
- S. MASUDA, S. FUKUDA, M. NAGATA
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- 30 April 2008, pp. 189-206
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We analyse the stability of plane Poiseuille flow with a streamwise system rotation. It is found that the instability due to two-dimensional perturbations, which sets in at the well-known critical Reynolds number, Rc = 5772.2, for the non-rotating case, is delayed as the rotation is increased from zero, showing a stabilizing effect of rotation. As the rotation is increased further, however, the laminar flow becomes most unstable to perturbations which are three-dimensional. The critical Reynolds number due to three-dimensional perturbations at this higher rotation case is many orders of magnitude less than the corresponding value due to two-dimensional perturbations. We also perform a nonlinear analysis on a bifurcating three-dimensional secondary flow. The secondary flow exhibits a spiral vortex structure propagating in the streamwise direction. It is confirmed that an antisymmetric mean flow in the spanwise direction is generated in the secondary flow.
The influence of secondary flows induced by normal stress differences on the shear-induced migration of particles in concentrated suspensions
- ARUN RAMACHANDRAN, DAVID T. LEIGHTON, JR
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- 30 April 2008, pp. 207-243
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It was first demonstrated experimentally by H. Giesekus in 1965 that the second normal stress difference in polymers can induce a secondary flow within the cross-section of a non-axisymmetric conduit. In this paper, we show through simulations that the same may be true for suspensions of rigid non-colloidal particles that are known to exhibit a strong negative second normal stress difference. Typically, the magnitudes of the transverse velocity components are small compared to the average axial velocity of the suspension; but the ratio of this transverse convective velocity to the shear-induced migration velocity is characterized by the shear-induced migration Péclet number χ which scales as B2/a2, B being the characteristic length scale of the cross-section and a being the particle radius. Since this Péclet number is kept high in suspension experiments (typically 100 to 2500), the influence of the weak circulation currents on the concentration profile can be very strong, a result that has not been appreciated in previous work. The principal effect of secondary flows on the concentration distribution as determined from simulations using the suspension balance model of Nott & Brady (J. Fluid Mech. vol. 275, 1994, p. 157) and the constitutive equations of Zarraga et al. (J. Rheol. vol. 44, 2000, p. 185) is three-fold. First, the steady-state particle concentration distribution is no longer independent of particle size; rather, it depends on the aspect ratio B/a. Secondly, the direction of the secondary flow is such that particles are swept out of regions of high streamsurface curvature, e.g. particle concentrations in corners reach a minimum rather than the local maximum predicted in the absence of such flows. Finally, the second normal stress differences lead to instabilities even in such simple geometries as plane-Poiseuille flow.
Analytic study of developing flows in a tube laden with non-evaporating and evaporating drops via a modified linearization of the two-phase momentum equations
- S. KHOSID, Y. TAMBOUR
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- Published online by Cambridge University Press:
- 30 April 2008, pp. 245-270
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A novel modification of the classical Langhaar linearization of the mutually coupled momentum equations for developing two-phase flows in circular ducts is presented. This modification enables us to treat: (i) flows developing from spatially periodic initial velocity distributions without the presence of droplets, and (ii) two-phase flows in which monosize, non-evaporating and evaporating droplets suspended in a developing gas flow of an initially uniform velocity distribution exchange momentum with the host-gas flow. New solutions are presented for the downstream evolution in the velocity profiles which develop from spatially periodic initial velocity distributions that eventually reach the fully developed Poiseuille velocity profile. These solutions are validated by employing known numerical procedures, providing strong support for the physical underpinnings of the present modified linearization. New solutions are also presented for the evolution in drop velocities and vapour spatial distributions for evaporating droplets suspended in an initially uniform velocity profile of the host gas. Asymptotic solutions are presented for the flow region which lies very close to the inlet of the tube, where the relative velocity between the droplets and the host gas is high, and thus the velocity fields of the two phases are mutually coupled. These solutions provide new explicit formulae for the droplet velocity field as a function of the initial conditions and droplet diameter (relative to the tube diameter) for non-evaporating drops, and also as a function of evaporation rate for evaporating drops.
Convective instability and transient growth in flow over a backward-facing step
- H. M. BLACKBURN, D. BARKLEY, S. J. SHERWIN
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- 30 April 2008, pp. 271-304
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Transient energy growths of two- and three-dimensional optimal linear perturbations to two-dimensional flow in a rectangular backward-facing-step geometry with expansion ratio two are presented. Reynolds numbers based on the step height and peak inflow speed are considered in the range 0–500, which is below the value for the onset of three-dimensional asymptotic instability. As is well known, the flow has a strong local convective instability, and the maximum linear transient energy growth values computed here are of order 80×103 at Re = 500. The critical Reynolds number below which there is no growth over any time interval is determined to be Re = 57.7 in the two-dimensional case. The centroidal location of the energy distribution for maximum transient growth is typically downstream of all the stagnation/reattachment points of the steady base flow. Sub-optimal transient modes are also computed and discussed. A direct study of weakly nonlinear effects demonstrates that nonlinearity is stablizing at Re = 500. The optimal three-dimensional disturbances have spanwise wavelength of order ten step heights. Though they have slightly larger growths than two-dimensional cases, they are broadly similar in character. When the inflow of the full nonlinear system is perturbed with white noise, narrowband random velocity perturbations are observed in the downstream channel at locations corresponding to maximum linear transient growth. The centre frequency of this response matches that computed from the streamwise wavelength and mean advection speed of the predicted optimal disturbance. Linkage between the response of the driven flow and the optimal disturbance is further demonstrated by a partition of response energy into velocity components.
Shallow-water modons on the f-plane
- Z. KIZNER, G. REZNIK, B. FRIDMAN, R. KHVOLES, J. McWILLIAMS
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- 30 April 2008, pp. 305-329
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Solutions for steadily translating localized vortical structures, or modons, are sought in the framework of a 1½-layer rotating shallow-water (RSW) model on the f-plane. In this model, the fluid is assumed to rotate at a constant rate and to be composed of an active finite-depth layer and a passive infinitely deep layer. The focus is on the smooth intense modons, in which the potential vorticity field is continuous, and the pressure (hence, the active-layer thickness) and velocity are smooth, while inertial effects and deviations of the active-layer thickness from the static level are considerable. The problem is solved numerically employing a Newton--Kantorovich iterative procedure, Fourier–Chebyshev spectral expansion and collocations. The numerics are preceded by a theoretical modon design discussion that includes: derivation of fundamental modon invariants; distinction between the flow in the trapped-fluid region and the flow outside it; and the boundary conditions at the separatrix, the streamline demarcating the two regions. Also, some basic distinctions from the quasi-geostrophic modons are discussed, and an asymptotic analysis of the RSW modon far-field characteristics is carried out. This analysis reveals that an RSW modon must propagate more slowly than inertia–gravity waves. In smooth modons, the requirement that the active-layer thickness should be positive imposes an even stronger restriction on the allowed translational speed. To enable the use of Fourier–Chebyshev series, only the modons with circular separatrices are considered. The numerical iterative procedure is initialized by an analytical quasi-geostrophic dipolar modon solution; accordingly, the obtained RSW modons appear as cyclone–anticyclone pairs. Computations show that the allowed maximal translational speed monotonically decreases as a function of the modon size and, for reasonable sizes, is appreciably smaller than the gravity-wave limit. As distinct from quasi-geostrophic modons, the RSW modons with circular separatrices display nonlinearity of the potential vorticity (PV) vs. streamfunction relation, and the cyclone–anticyclone asymmetry: while the integral mass anomaly in the modon is zero, the cyclone is more intense and compact than the anticyclone.
The wake structure and thrust performance of a rigid low-aspect-ratio pitching panel
- JAMES H. J. BUCHHOLZ, ALEXANDER J. SMITS
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- 30 April 2008, pp. 331-365
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Thrust performance and wake structure were investigated for a rigid rectangular panel pitching about its leading edge in a free stream. For ReC = O(104), thrust coefficient was found to depend primarily on Strouhal number St and the aspect ratio of the panel AR. Propulsive efficiency was sensitive to aspect ratio only for AR less than 0.83; however, the magnitude of the peak efficiency of a given panel with variation in Strouhal number varied inversely with the amplitude to span ratio A/S, while the Strouhal number of optimum efficiency increased with increasing A/S. Peak efficiencies between 9% and 21% were measured. Wake structures corresponding to a subset of the thrust measurements were investigated using dye visualization and digital particle image velocimetry. In general, the wakes divided into two oblique jets; however, when operating at or near peak efficiency, the near wake in many cases represented a Kármán vortex street with the signs of the vortices reversed. The three-dimensional structure of the wakes was investigated in detail for AR = 0.54, A/S = 0.31 and ReC = 640. Three distinct wake structures were observed with variation in Strouhal number. For approximately 0.20 < St < 0.25, the main constituent of the wake was a horseshoe vortex shed by the tips and trailing edge of the panel. Streamwise variation in the circulation of the streamwise horseshoe legs was consistent with a spanwise shear layer bridging them. For St > 0.25, a reorganization of some of the spanwise vorticity yielded a bifurcating wake formed by trains of vortex rings connected to the tips of the horseshoes. For St > 0.5, an additional structure formed from a perturbation of the streamwise leg which caused a spanwise expansion. The wake model paradigm established here is robust with variation in Reynolds number and is consistent with structures observed for a wide variety of unsteady flows. Movies are available with the online version of the paper.
Coherent structures in bypass transition induced by a cylinder wake
- CHONG PAN, JIN JUN WANG, PAN FENG ZHANG, LI HAO FENG
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- 30 April 2008, pp. 367-389
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Flat-plate boundary layer transition induced by the wake vortex of a two-dimensional circular cylinder is experimentally investigated. Combined visualization and velocity measurements show a different transition route from the Klebanoff mode in free-stream turbulence-induced transition. This transition scenario is mainly characterized as: (i) generation of secondary transverse vortical structures near the flat plate surface in response to the von Kármán vortex street of the cylinder; (ii) formation of hairpin vortices due to the secondary instability of secondary vortical structures; (iii) growth of hairpins which is accelerated by wake-vortex induction; (iv) formation of hairpin packets and the associated streaky structures. Detailed investigation shows that during transition the evolution dynamics and self-sustaining mechanisms of hairpins, hairpin packets and streaks are consistent with those in a turbulent boundary layer. The wake vortex mainly plays the role of generating and destabilizing secondary transverse vortices. After that, the internal mechanisms become dominant and lead to the setting up of a self-sustained turbulent boundary layer.
On the rotational compressible Taylor flow in injection-driven porous chambers
- BRIAN A. MAICKE, JOSEPH MAJDALANI
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- 30 April 2008, pp. 391-411
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This work considers the compressible flow field established in a rectangular porous channel. Our treatment is based on a Rayleigh–Janzen perturbation applied to the inviscid steady two-dimensional isentropic flow equations. Closed-form expressions are then derived for the main properties of interest. Our analytical results are verified via numerical simulation, with laminar and turbulent models, and with available experimental data. They are also compared to existing one-dimensional theory and to a previous numerical pseudo-one-dimensional approach. Our analysis captures the steepening of the velocity profiles that has been reported in several studies using either computational or experimental approaches. Finally, explicit criteria are presented to quantify the effects of compressibility in two-dimensional injection-driven chambers such as those used to model slab rocket motors.
On the far wake and induced drag of aircraft
- PHILIPPE R. SPALART
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- Published online by Cambridge University Press:
- 30 April 2008, pp. 413-430
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A set of matched asymptotic expansions is proposed for the flow far behind an aircraft, with the primary purpose of identifying lift, thrust and drag, particularly induced drag, in a unified manner in integral statements of the momentum equation. The fluid in the far wake is inviscid and incompressible, and variations of total pressure are allowed, as are vortex sheets. A notable feature is that the Trefftz-plane approximation is not invoked; instead the wake is taken as fully rolled-up, and the analysis proceeds without the assumption of light loading. Attention is paid to the absolute convergence of integrals over infinite domains and handling of discontinuities. The expansion includes a sink term, which appears new, so that the mass flux through a transverse plane is non-zero, as is the flux of mechanical energy. The lift can be formally attributed to the velocity induced by the bound vortex of the wing, which is at odds with some treatments, although consistent with Prandtl's analysis over a ground plane. The drag contains the integral of ρ(v2 + w2 − u2)/2, as in many treatments of the subject, u being the perturbation velocity along the wake. The negative sign for u2 appears paradoxical on two counts, one of which is resolved here. First, its very presence instead of the + sign, which would lead to the perturbation kinetic energy and therefore a compelling explanation of induced drag, is explained by the longitudinal energy flux. This energy, the integral of ρu2, is continuously provided by the unsteady starting-vortex system and was deposited earlier by the aircraft. Second, it appears that negative drag could be predicted by this equation. This is shown to be impossible, because of inequalities between the integrals of (v2 + w2) and of u2, but the proof is valid only if the vorticity is of only one sign on each side. A general proof of positivity has not been derived, because of nonlinearities, but neither has a counter-example.
Modelling film flows down a fibre
- C. RUYER-QUIL, P. TREVELEYAN, F. GIORGIUTTI-DAUPHINÉ, C. DUPRAT, S. KALLIADASIS
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- 30 April 2008, pp. 431-462
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Consider the gravity-driven flow of a thin liquid film down a vertical fibre. A model of two coupled evolution equations for the local film thickness h and the local flow rate q is formulated within the framework of the long-wave and boundary-layer approximations. The model accounts for inertia and streamwise viscous diffusion. Evolution equations obtained by previous authors are recovered in the appropriate limit. Comparisons to experimental results show good agreement in both linear and nonlinear regimes. Viscous diffusion effects are found to have a stabilizing dispersive effect on the linear waves. Time-dependent computations of the spatial evolution of the film reveal a strong influence of streamwise viscous diffusion on the dynamics of the flow and the wave selection process.
Unsteady shock wave dynamics
- P. J. K. BRUCE, H. BABINSKY
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- 30 April 2008, pp. 463-473
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An experimental study of an oscillating normal shock wave subject to unsteady periodic forcing in a parallel-walled duct has been conducted. Measurements of the pressure rise across the shock have been taken and the dynamics of unsteady shock motion have been analysed from high-speed schlieren video (available with the online version of the paper). A simple analytical and computational study has also been completed. It was found that the shock motion caused by variations in back pressure can be predicted with a simple theoretical model. A non-dimensional relationship between the amplitude and frequency of shock motion in a diverging duct is outlined, based on the concept of a critical frequency relating the relative importance of geometry and disturbance frequency for shock dynamics. The effects of viscosity on the dynamics of unsteady shock motion were found to be small in the present study, but it is anticipated that the model will be less applicable in geometries where boundary layer separation is more severe. A movie is available with the online version of the paper.
Review
Computational Methods for Multiphase Flow. By A. Prosperetti & G. Tryggvason. Cambridge University Press, 2007. 470 pp. ISBN 13 978 0521 84764 3. £ 55 (hardback).
- Jie Li
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- Published online by Cambridge University Press:
- 30 April 2008, pp. 474-475
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