Papers
Unsteady forces on an accelerating plate and application to hovering insect flight
- D. I. PULLIN, Z. JANE WANG
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- 07 June 2004, pp. 1-21
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The aerodynamic forces on a flat plate accelerating from rest at fixed incidence in two-dimensional power-law flow are studied analytically and numerically. An inviscid approximation is made in which separation at the two plate edges is modelled by growing spiral vortex sheets, whose evolution is determined by the Birkhoff–Rott equation. A solution based on a similarity expansion is developed, valid when the scale of the separated vortex is much smaller than the plate dimension. The leading order is given by the well-known similarity growth of a vortex sheet from a semi-infinite flat plate, while equations at the second order describe the asymmetric sweeping effect of that component of the free-stream parallel to the plate. Owing to subtle cancellation, the unsteady vortex force exerted on the plate during the starting motion is independent of the sweeping effect and is determined by the similarity solution, to the order calculated. This gives a mechanism for dynamic stall based on a combination of unsteady vortex lift and pure added mass; the incidence angle for maximum vortex lift is $\arccos \sqrt{3/8}\,{\approx}\,52.2^\circ$ independent of the acceleration profile. Circulation on the flat plate makes no direct contribution. Both lift and drag force predictions from the unsteady inviscid theory are compared with those obtained from numerical solutions of the two-dimensional unsteady Navier–Stokes equations for an ellipse of high aspect ratio, and with predictions of Wagner's classical theory. There is good agreement with numerical results at high incidence and moderate Reynolds number. The force per unit span predicted by the vortex theory is evaluated for parameters typical of insect wings and is found to be in reasonable agreement with numerical simulations. Estimates for the shed circulation and the size of the start-up vortices are also obtained. The significance of this flow as a mechanism for insect hovering flight is discussed.
The effect of two degrees of freedom on vortex-induced vibration at low mass and damping
- N. JAUVTIS, C. H. K. WILLIAMSON
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- 07 June 2004, pp. 23-62
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Although there are a great many papers dedicated to the problem of a cylinder vibrating transverse to a fluid flow ($Y$-motion), there are almost no papers on the more practical case of vortex-induced vibration in two degrees of freedom ($X,Y$ motion) where the mass and natural frequencies are precisely the same in both $X$- and $Y$-directions. We have designed the present pendulum apparatus to achieve both of these criteria. Even down to the low mass ratios, where $m^*\,{=}\,6$, it is remarkable that the freedom to oscillate in-line with the flow affects the transverse vibration surprisingly little. The same response branches, peak amplitudes, and vortex wake modes are found for both $Y$-only and $X,Y$ motion. There is, however, a dramatic change in the fluid–structure interactions when mass ratios are reduced below $m^*\,{=}\,6$. A new amplitude response branch with significant streamwise motion appears, in what we call the ‘super-upper’ branch, yielding massive amplitudes of 3 diameters peak-to-peak ($A^*_Y \,{\sim}\, 1.5$). We discover a corresponding periodic vortex wake mode, comprising a triplet of vortices being formed in each half-cycle, in what we define as a ‘2T’ mode. We qualitatively interpret the principal vortex dynamics and vortex forces which yield a positive rate of energy transfer ($\dot{e}_V$) causing the body vibration, using the following simple equation: \[\dot{e}_V = 2 {\Gamma}^* U^*_V \skew3\dot{Y}\] where $\Gamma^*$ is vortex strength, $U^*_V$ is the speed downstream of the dominant near-wake vorticity, and $\skew3\dot{Y}$ is the transverse velocity of the body. This simple approach suggests that the massive amplitude of vibration for the 2T mode is principally attributed to the energy transfer from the ‘third’ vortex of each triplet, which is not present in the lower-amplitude 2P mode. We also find two low-speed streamwise vibration modes, which is not unexpected, since they correspond to the first and second excitation modes of vibration for flexible cantilevers. By considering equations of motion for the two degrees of freedom, we find a critical mass, $m^*_{\hbox{\scriptsize\it crit}} \,{=}\, 0.52$, similar to recent $Y$-only studies, below which the large-amplitude vibrations persist to infinite flow velocity. We show that the critical mass $m^*_{\hbox{\scriptsize\it crit}}$ is the same for the $X$- and $Y$-directions, which ensures that the shapes of $X, Y$ trajectories can retain their form as the velocity becomes large. The extensive studies of vortex-induced vibration for $Y$-only body motions, built up over the last 35 years, remain of strong relevance to the case of two degrees of freedom, for $m^* \,{>}\, 6$. It is only for ‘small’ mass ratios, $m^* \,{<}\, 6$, that one observes a rather dramatic departure from previous results, which would suggest a possible modification to offshore design codes.
Oblique collisions and rebound of spheres from a wetted surface
- ADVAIT A. KANTAK, ROBERT H. DAVIS
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- 07 June 2004, pp. 63-81
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Plastic and metal spheres were dropped from various heights onto a quartz disk covered with a thin layer of viscous oil and inclined at various angles with the horizontal. Rebound was observed only above a critical approach velocity, similar to that observed for head-on collisions when the disk is horizontal. The tangential component of the sphere's velocity is reduced only a small amount by the collision, owing to sliding lubrication/friction forces that also impart a small rotational velocity to the sphere. In contrast, the normal component of velocity is reduced substantially by viscous losses, and so the rebound angle of the sphere relative to the surface of the disk is smaller than the impact angle. The normal component of restitution and the rebound angle increase with the normal Stokes number based on the normal component of the impact velocity.
Unidirectional stratified flow through a non-rectangular channel
- ANDERS ENGQVIST, ANDREW McC. HOGG
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- 07 June 2004, pp. 83-92
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A self-similar solution describing stratified flow through a non-rectangular channel is derived. The solution shown here is an extension of Wood's (1968) solution for stratified withdrawal through a rectangular channel. We consider a restricted set of geometries (where the bottom of the channel is constrained to be flat) and calculate the flow, assuming first multi-layer stratification, and second continuous stratification. In the case of two-layer flow we prove that the self-similar solution is the only possible solution. The analytical solutions are corroborated by three-dimensional numerical model simulations.
Trapped waves between submerged obstacles
- F. DIAS, J.-M. VANDEN-BROECK
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- 07 June 2004, pp. 93-102
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Free-surface flows past submerged obstacles in a channel are considered. The fluid is assumed to be inviscid and incompressible and the flow to be irrotational. In previous work involving a single obstacle (Dias & Vanden-Broeck 2002), new solutions called ‘generalized hydraulic falls’ were found. These solutions are characterized by a supercritical flow on one side of the obstacle and a train of waves on the other. However, in the case of a single submerged object, the generalized hydraulic falls are unphysical because the waves do not satisfy the radiation condition. In this paper new solutions for the flow past two obstacles of arbitrary shape are computed. These solutions are characterized by a train of waves ‘trapped’ between the obstacles. The generalized hydraulic falls are shown to describe locally the flow over one of the two obstacles when the distance between the two obstacles is large.
On a new surfactant-driven fingering phenomenon in a Hele-Shaw cell
- R. KRECHETNIKOV, G. M. HOMSY
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- 07 June 2004, pp. 103-124
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According to the Saffman–Taylor criterion there is no instability when a more viscous fluid is displacing a less viscous one in a Hele-Shaw cell. Yet an instability was observed experimentally (Chan et al. 1997) in the same classical set-up but with the inner walls of the cell coated with surfactant solution. Linear stability analysis is applied to reveal the basic mechanism of this new instability. Asymptotic theory for low capillary numbers allows us to predict the long-wave instability, along with the dependence of the critical parameters on the material properties of the surfactant.
Instability due to a discontinuity in magnetic diffusivity in the presence of magnetic shear
- S. N. BHATTACHARYYA, A. S. GUPTA
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- 07 June 2004, pp. 125-144
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The linear stability of two viscous electrically conducting quiescent fluids, separated by a plane interface, and permeated by a sheared magnetic field parallel to the interface is studied. An analytical study using a short-wavelength approximation shows that, in the absence of surface tension, if the magnetic field vanishes on the unperturbed interface, the configuration is always unstable provided the magnetic diffusivities of the two fluids are different. When the unperturbed magnetic field does not vanish on the interface it may stabilize or destabilize the configuration depending on the values of certain parameters. The growth rates for the instability obtained using a short-wavelength approximation are shown to be in good agreement with the results obtained by numerical solution. The numerical study further shows that the instability has maximum growth rate for wavenumbers of order unity and persists even for long-wavelength perturbations. A physical explanation for the instability is provided.
Approximations to wave scattering by an ice sheet of variable thickness over undulating bed topography
- D. PORTER, R. PORTER
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- 07 June 2004, pp. 145-179
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An investigation is carried out into the effect on wave propagation of an ice sheet of varying thickness floating on water of varying depth, in three dimensions. By deriving a variational principle equivalent to the governing equations of linear theory and invoking the mild-slope approximation in respect of the ice thickness and water depth variations, a simplified form of the problem is obtained from which the vertical coordinate is absent. Two situations are considered: the scattering of flexural–gravity waves by variations in the thickness of an infinite ice sheet and by depth variations; and the scattering of free-surface gravity waves by an ice sheet of finite extent and varying thickness, again incorporating arbitrary topography. Numerical methods are devised for the two-dimensional versions of these problems and a selection of results is presented. The variational approach that is developed can be used to implement more sophisticated approximations and is capable of producing the solution of full linear problems by taking a large enough basis in the Rayleigh–Ritz method. It is also applicable to other situations that involve wave scattering by a floating elastic sheet.
Sediment-entraining suspension clouds: a model of powder-snow avalanches
- M. RASTELLO, E. J. HOPFINGER
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- 07 June 2004, pp. 181-206
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A dense cloud model of avalanches is presented which includes large density difference effects as well as sediment entrainment along the path of the cloud. This model demonstrates the importance of sediment entrainment in the evolution of the front velocity. Without sediment entrainment the cloud first accelerates and then decelerates, a behaviour known from previous studies of cloud or thermal motions. With sediment entrainment the cloud is mostly in an accelerating state. The closure coefficients in the model concerning the cloud shape and air entrainment are obtained from laboratory experiments. These coefficients can be considered generic in the Boussinesq limit. A correction for inertial effects which need to be taken into account when applied to large density difference clouds such as avalanches, is proposed. An expression for the sediment entrainment coefficient is derived, taking into account the flow parameters and the sediment layer properties. The model predictions are in good agreement with recent measurement of the front velocity of a powder-snow avalanche. A presentation, in terms of dimensionless variables, of avalanche and laboratory Boussinesq cloud velocities shows clearly the similarities and differences between the two.
Compressibility effects and turbulence scalings in supersonic channel flow
- H. FOYSI, S. SARKAR, R. FRIEDRICH
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- 07 June 2004, pp. 207-216
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Turbulence in supersonic channel flow is studied using direct numerical simulation. The ability of outer and inner scalings to collapse profiles of turbulent stresses onto their incompressible counterparts is investigated. Such collapse is adequate with outer scaling when sufficiently far from the wall, but not with inner scaling. Compressibility effects on the turbulent stresses, their anisotropy, and their balance equations are identified. A reduction in the near-wall pressure–strain, found responsible for the changed Reynolds-stress profiles, is explained using a Green's-function-based analysis of the pressure field.
Induced-charge electro-osmosis
- TODD M. SQUIRES, MARTIN Z. BAZANT
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- 07 June 2004, pp. 217-252
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We describe the general phenomenon of ‘induced-charge electro-osmosis’ (ICEO) – the nonlinear electro-osmotic slip that occurs when an applied field acts on the ionic charge it induces around a polarizable surface. Motivated by a simple physical picture, we calculate ICEO flows around conducting cylinders in steady (DC), oscillatory (AC), and suddenly applied electric fields. This picture, and these systems, represent perhaps the clearest example of nonlinear electrokinetic phenomena. We complement and verify this physically motivated approach using a matched asymptotic expansion to the electrokinetic equations in the thin-double-layer and low-potential limits. ICEO slip velocities vary as $u_s \,{\propto}\,E_0^2 L$, where $E_0$ is the field strength and $L$ is a geometric length scale, and are set up on a time scale $\tau_c \,{=}\,\lambda_D L/D$, where $\lambda_D$ is the screening length and $D$ is the ionic diffusion constant. We propose and analyse ICEO microfluidic pumps and mixers that operate without moving parts under low applied potentials. Similar flows around metallic colloids with fixed total charge have been described in the Russian literature (largely unnoticed in the West). ICEO flows around conductors with fixed potential, on the other hand, have no colloidal analogue and offer further possibilities for microfluidic applications.
Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography
- P. H. GASKELL, P. K. JIMACK, M. SELLIER, H. M. THOMPSON, M. C. T. WILSON
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- 07 June 2004, pp. 253-280
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A range of two- and three-dimensional problems is explored featuring the gravity-driven flow of a continuous thin liquid film over a non-porous inclined flat surface containing well-defined topography. These are analysed principally within the framework of the lubrication approximation, where accurate numerical solution of the governing nonlinear equations is achieved using an efficient multigrid solver.
Results for flow over one-dimensional steep-sided topographies are shown to be in very good agreement with previously reported data. The accuracy of the lubrication approximation in the context of such topographies is assessed and quantified by comparison with finite element solutions of the full Navier–Stokes equations, and results support the consensus that lubrication theory provides an accurate description of these flows even when its inherent assumptions are not strictly satisfied. The Navier–Stokes solutions also illustrate the effect of inertia on the capillary ridge/trough and the two-dimensional flow structures caused by steep topography.
Solutions obtained for flow over localized topography are shown to be in excellent agreement with the recent experimental results of Decré & Baret (2003) for the motion of thin water films over finite trenches. The spread of the ‘bow wave’, as measured by the positions of spanwise local extrema in free-surface height, is shown to be well-represented both upstream and downstream of the topography by an inverse hyperbolic cosine function.
An explanation, in terms of local flow rate, is given for the presence of the ‘downstream surge’ following square trenches, and its evolution as trench aspect ratio is increased is discussed. Unlike the upstream capillary ridge, this feature cannot be completely suppressed by increasing the normal component of gravity. The linearity of free-surface response to topographies is explored by superposition of the free surfaces corresponding to two ‘equal-but-opposite’ topographies. Results confirm the findings of Decré & Baret (2003) that, under the conditions considered, the responses behave in a near-linear fashion.
Vortex motion in a weak background shear flow
- KONRAD BAJER, ANDREW P. BASSOM, ANDREW D. GILBERT
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- 07 June 2004, pp. 281-304
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A point vortex is introduced into a weak background vorticity gradient at finite Reynolds number. As the vortex spreads viscously so the background vorticity becomes wrapped around it, leading to enhanced diffusion of vorticity, but also giving a feedback on the vortex and causing it to move. This is investigated in the linear approximation, using a similarity solution for the advection of weak vorticity around the vortex, at finite and infinite Reynolds number. A logarithmic divergence in the far field requires the introduction of an outer length scale $L$ and asymptotic matching. In this way results are obtained for the motion of a vortex in a weak vorticity field modulated on the large scale $L$ and these are confirmed by means of numerical simulations.
Penetrative convection in a superposed porous-medium–fluid layer via internal heating
- MAGDA CARR
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- 07 June 2004, pp. 305-329
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Penetrative convection in a two-layer system in which a layer of fluid overlies and saturates a porous medium is simulated via internal heating. The motion in the porous medium is described via Darcy's law and in the fluid layer by the Navier–Stokes equations with a Boussinesq approximation. The lower porous surface is held fixed at a temperature $T_L$, while the upper fluid surface is stress free and held at $T_U\,{>}\,T_L$. Internal heating takes place in both layers and allows the model to describe penetrative convection. The strength of heating has a dramatic effect on both the onset of convection and the nature of the ensuing convection cells. It is found that a heat source/sink $Q$ in the fluid layer has a destabilizing effect on the porous layer whereas one in the porous medium $Q_m$ has a stabilizing influence on the fluid. The effect of $Q$ and $Q_m$ on their respective layers, however, depends strongly upon the temperature difference $T_U\,{-}\,T_L$, and the strength and type of heating in the opposite layer. When $Q$ and $Q_m$ are varied, a range of streamlines are presented that exhibit novel behaviour. The model is compared with an alternative in which the density is assumed to have quadratic temperature dependence and there is no internal heating. When the two models are mathematically adjoint they are shown to yield the same critical instability threshold but different eigenfunctions. It is also shown that the initiating cell is not necessarily the strongest one. This curious behaviour is explained and illustrated with a range of streamlines for variable permeability.
The connection between centrifugal instability and Tollmien–Schlichting-like instability for spiral Poiseuille flow
- DAVID L. COTRELL, ARNE J. PEARLSTEIN
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- 07 June 2004, pp. 331-351
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For spiral Poiseuille flow with radius ratio $\eta\,{\equiv}\,R_i\slash R_o\,{=}\,0.5$, we have computed complete linear stability boundaries for several values of the rotation rate ratio $\mu\equiv{\it\Omega}_o\slash{\it\Omega}_i$}, where $R_i$ and $R_o$ are the inner and outer cylinder radii, respectively, and ${\it \Omega}_i$ and ${\it \Omega}_o$ are the corresponding (signed) angular speeds. The analysis extends the previous range of Reynolds number $Re$ studied computationally by more than eightyfold, and accounts for arbitrary disturbances of infinitesimal amplitude over the entire range of $Re$ for which spiral Poiseuille flow is stable for some range of the Taylor number $Ta$. We show how the centrifugally driven instability (beginning with steady or azimuthally travelling-wave bifurcation of circular Couette flow at $Re\,{=}\,0$ when $\mu\,{<}\,\eta^2$) connects, as conjectured by Reid (1961) in the narrow-gap limit, to a non-axisymmetric Tollmien–Schlichting-like instability of non-rotating annular Poiseuille flow at $Ta\,{=}\,0$. For $\mu\,{>}\,\eta^2$, we show that there is no instability for $0\,{\leq}\,Re\,{\leq}\,Re_{min}$. For $\mu\,{=}\,0.5$, $Re_{min}$ corresponds to a turning point, beyond which exists a range of $Re$ for which there are two critical values of $Ta$, with spiral Poiseuille flow being stable below the lower one and above the upper one, and unstable in between. For the special case $\mu\,{=}\,1$, with the two cylinders having the same angular velocity, $Re_{min}$ corresponds to a vertical asymptote smaller than found by Meseguer & Marques (2002), whose results for $\mu\,{>}\,\eta^2$ fail to account for disturbances with a sufficiently wide range of azimuthal wavenumbers.
Computational assessment of subcritical and delayed onset in spiral Poiseuille flow experiments
- DAVID L. COTRELL, SARMA L. RANI, ARNE J. PEARLSTEIN
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- 07 June 2004, pp. 353-378
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For spiral Poiseuille flow with radius ratios $\eta \equiv R_i/R_o = 0.77$ and 0.95, we have computed complete linear stability boundaries, where $R_i$ and $R_o$ are the inner and outer cylinder radii, respectively. The analysis accounts for arbitrary disturbances of infinitesimal amplitude over the entire range of Reynolds numbers $Re$ for which the flow is stable for some range of Taylor number $Ta$, and extends previous work to several non-zero rotation rate ratios $\mu \equiv \Omega_o/\Omega_i$, where $\Omega_i$ and $\Omega_o$ are the (signed) angular speeds. For each combination of $\mu$ and $\eta$, there is a wide range of $Re$ for which the critical $Ta$ is nearly independent of $Re$, followed by a precipitous drop to $Ta = 0$ at the $Re$ at which non-rotating annular Poiseuille flow becomes unstable with respect to a Tollmien–Schlichting-like disturbance. Comparison is also made to a wealth of experimental data for the onset of instability. For $Re > 0$, we compute critical values of $Ta$ for most of the $\mu = 0$ data, and for all of the non-zero-$\mu$ data. For $\mu = 0$ and $\eta = 0.955$, agreement with data from an annulus with aspect ratio (length divided by gap) greater than 570 is within 3.2% for $Re \leq 325$ (based on the gap and mean axial speed), strongly suggesting that no finite-amplitude instability occurs over this range of $Re$. At higher $Re$, onset is delayed, with experimental values of $Ta_{\hbox{\scriptsize{\it crit}}}$ exceeding computed values. For $\mu = 0$ and smaller $\eta$, comparison to experiment (with smaller aspect ratios) at low $Re$ is slightly less good. For $\eta = 0.77$ and a range of $\mu$, agreement with experiment is very good for $Re < 135$ except at the most positive or negative $\mu$ (where $Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it expt}}} > Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it comp}}}$), whereas for $Re \geq 166$, $Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it expt}}} > Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it comp}}}$ for all but the most positive $\mu$. For $\eta = 0.9497$ and 0.959 and all but the most extreme values of $\mu$, agreement is excellent (generally within 2%) up to the largest $Re$ considered experimentally (200), again suggesting that finite-amplitude instability is unimportant.
Addendum
ADDENDUM
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- 07 June 2004, p. 379
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A ‘win–win’ mechanism for low-drag transients in controlled two-dimensional channel flow and its implications for sustained drag reduction BY THOMAS R. BEWLEY & OLE MORTEN AAMOJournal of Fluid Mechanics, vol. 499 (2004), pp. 183–196