Focus on Fluids
Cloud-top entrainment instability?
- B. STEVENS
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- Published online by Cambridge University Press:
- 16 September 2010, pp. 1-4
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Mixing processes at cloud boundaries are thought to play a critical role in determining cloud lifetime, spatial extent and cloud microphysical structure. High-fidelity direct numerical simulations by Mellado (J. Fluid Mech., 2010, this issue, vol. 660, pp. 5–36) show, for the first time, the character and potency of a curious instability that may arise as a result of molecular mixing processes at cloud boundaries, an instability which until now has been thought by many to control the distribution of climatologically important cloud regimes.
Papers
The evaporatively driven cloud-top mixing layer
- JUAN PEDRO MELLADO
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- 27 July 2010, pp. 5-36
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Direct numerical simulations of the turbulent temporally evolving cloud-top mixing layer are used to investigate the role of evaporative cooling by isobaric mixing locally at the stratocumulus top. It is shown that the system develops a horizontal layered structure whose evolution is determined by molecular transport. A relatively thin inversion with a constant thickness h = κ/we is formed on top and travels upwards at a mean velocity we ≃ 0.1(κ |bs|χc2)1/3, where κ is the mixture-fraction diffusivity, bs < 0 is the buoyancy anomaly at saturation conditions χs and χc is the cross-over mixture fraction defining the interval of buoyancy reversing mixtures. A turbulent convection layer develops below and continuously broadens into the cloud (the lower saturated fluid). This turbulent layer approaches a self-preserving state that is characterized by the convection scales constructed from a constant reference buoyancy flux Bs = |bs|we/χs. Right underneath the inversion base, a transition or buffer zone is defined based on a strong local conversion of vertical to horizontal motion that leads to a cellular pattern and sheet-like plumes, as observed in cloud measurements and reported in other free-convection problems. The fluctuating saturation surface (instantaneous cloud top) is contained inside this intermediate region. Results show that the inversion is not broken due to the turbulent convection generated by the evaporative cooling, and the upward mean entrainment velocity we is negligibly small compared to the convection velocity scale w* of the turbulent layer and the corresponding growth rate into the cloud.
Mean flow deformation in a laminar separation bubble: separation and stability characteristics
- OLAF MARXEN, ULRICH RIST
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- 17 August 2010, pp. 37-54
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The mutual interaction of laminar–turbulent transition and mean flow evolution is studied in a pressure-induced laminar separation bubble on a flat plate. The flat-plate boundary layer is subjected to a sufficiently strong adverse pressure gradient that a separation bubble develops. Upstream of the bubble a small-amplitude disturbance is introduced which causes transition. Downstream of transition, the mean flow strongly changes and, due to viscous–inviscid interaction, the overall pressure distribution is changed as well. As a consequence, the mean flow also changes upstream of the transition location. The difference in the mean flow between the forced and the unforced flows is denoted the mean flow deformation. Two different effects are caused by the mean flow deformation in the upstream, laminar part: a reduction of the size of the separation region and a stabilization of the flow with respect to small, linear perturbations. By carrying out numerical simulations based on the original base flow and the time-averaged deformed base flow, we are able to distinguish between direct and indirect nonlinear effects. Direct effects are caused by the quadratic nonlinearity of the Navier–Stokes equations, are associated with the generation of higher harmonics and are predominantly local. In contrast, the stabilization of the flow is an indirect effect, because it is independent of the Reynolds stress terms in the laminar region and is solely governed by the non-local alteration of the mean flow via the pressure.
Synoptic velocity and pressure fields at the water–sediment interface of streambeds
- M. DETERT, V. NIKORA, G. H. JIRKA
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- 16 August 2010, pp. 55-86
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This paper presents a comprehensive study of the near-bed hydrodynamics at non-moving streambeds based on laboratory experiments in open-channel flows. Pressure and velocity measurements were made with an array of up to 15 miniaturized piezo-resistive pressure sensors within the bed and slightly above it, and a two-dimensional particle-image-velocimetry (PIV) system measuring in streamwise vertical or horizontal planes. Three different types of bed materials were studied covering typical natural streambed conditions. The range of the global Reynolds number covered in the experiments was from 20000 to 200000. This study provides new insights into the flow structure over gravel beds based on the PIV measurements in both streamwise vertical and horizontal planes. In a streamwise vertical plane, large-scale wedge-like flow structures were observed where a zone of faster fluid over-rolled a zone with slower fluid. The resulting shear layer was inclined along the flow at an angle of 10°–25° to the bed, and was populated with clockwise rotating eddies. This mechanism occurred with sufficient frequency and shape to leave an ‘imprint’ in the velocity statistics. Typically, the described flow pattern is formed near the bed and is approximately scaled with the height of the logarithmic layer, although the biggest structures extended over the whole flow depth. In a horizontal near-bed plane, turbulent structures formed a patched ‘chessboard’ pattern with regions of lower and higher velocities that were elongated in the streamwise direction. Their lateral extension was typically two to four times the equivalent sand roughness with lengths up to several water depths. The dimensions of the regions were increasing linearly with the distance from the bed. These findings are consistent with conceptual models originally developed for smooth-wall flows. They also support observations made in rough-bed flume experiments, numerical simulations and natural rivers. Spatial fields of bed-pressure fluctuations were reconstructed by applying Taylor's frozen turbulence hypothesis on time data obtained with an array of pressure sensors. Based on the conditional sampling of velocity patterns associated with pressure-drop events a distinct bed-destabilizing flow-pressure pattern was identified. If a high-speed fluid in the wake of a large-scale wedge-like flow structure reaches the vicinity of the bed, a phenomenon akin to a Bernoulli effect leads to a distinctive low-pressure pattern. The resulting force may exceed the particles' submerged weight and is assumed to be able to give an initial lift to the particle. As a result, the exposed area of a particle is amplified and its angle of repose is reduced, increasing the probability for entrainment.
Experimental study of a tip leakage flow: wavelet analysis of pressure fluctuations
- R. CAMUSSI, J. GRILLIAT, G. CAPUTI-GENNARO, M. C. JACOB
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- 05 August 2010, pp. 87-113
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A wavelet-based conditional analysis of unsteady flow and sound signals highlights the role of intermittent perturbations both in the sound generation and the unsteady field of an aerofoil tip leakage flow experiment. It is shown how the most probable flow perturbations generated at the pressure side tip edge are convected through the gap and swept downstream along the suction side past the trailing edge tip corner, where they radiate sound. The nascent sound sources are identified and localized in the clearance between 40% and 60% of the chord. It is also found that the time dependence of the averaged intermittent structures scales with the inverse of the square root of the mean velocity and a physical interpretation based on a simple potential vortex model is proposed. The data are retrieved from an experiment that has been carried out at low Mach number (Ma < 0.3) in an anechoic test facility. A single motionless instrumented NACA 5510 aerofoil was mounted into the potential core of an open rectangular jet between two plates with an adjustable clearance. The tip leakage flow was ensured by the 5% camber and a 15° angle of attack. A large database obtained by a variety of measurement techniques is thus available for the present analysis. More specifically, the conditional approach is applied to joint far field, wall pressure and particle image velocimetry (PIV) measurements. The wall pressure probes are located along the suction side tip edge and on the tip inside the gap, whereas the PIV plane is parallel to the mid-gap plane. Additional joint wall pressure and single hot-wire anemometry (HWA) measurements are also analysed with a hot-wire probe located near the trailing edge tip corner. The conditional averaging is triggered by high-energy wavelet events selected in a reference signal by setting a threshold to the so-called local intermittency measure.
A study on boundary-layer transition induced by free-stream turbulence
- A. C. MANDAL, L. VENKATAKRISHNAN, J. DEY
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- 15 July 2010, pp. 114-146
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Boundary-layer transition at different free-stream turbulence levels has been investigated using the particle-image velocimetry technique. The measurements show organized positive and negative fluctuations of the streamwise fluctuating velocity component, which resemble the forward and backward jet-like structures reported in the direct numerical simulation of bypass transition. These fluctuations are associated with unsteady streaky structures. Large inclined high shear-layer regions are also observed and the organized negative fluctuations are found to appear consistently with these inclined shear layers, along with highly inflectional instantaneous streamwise velocity profiles. These inflectional velocity profiles are similar to those in the ribbon-induced boundary-layer transition. An oscillating-inclined shear layer appears to be the turbulent spot-precursor. The measurements also enabled to compare the actual turbulent spot in bypass transition with the simulated one. A proper orthogonal decomposition analysis of the fluctuating velocity field is carried out. The dominant flow structures of the organized positive and negative fluctuations are captured by the first few eigenfunction modes carrying most of the fluctuating energy. The similarity in the dominant eigenfunctions at different Reynolds numbers suggests that the flow prevails its structural identity even in intermittent flows. This analysis also indicates the possibility of the existence of a spatio-temporal symmetry associated with a travelling wave in the flow.
The strato-rotational instability of Taylor–Couette and Keplerian flows
- S. LE DIZÈS, X. RIEDINGER
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- 23 August 2010, pp. 147-161
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The linear inviscid stability of two families of centrifugally stable rotating flows in a stably stratified fluid of constant Brunt–Väisälä frequency N is analysed by using numerical and asymptotic methods. Both Taylor–Couette and Keplerian angular velocity profiles ΩTC = (1 − μ)/r2 + μ and ΩK = (1 − λ)/r2 + λ/r3/2 are considered between r = 1 (inner boundary) and r = d > 1 (outer boundary, or without boundary if d = ∞). The stability properties are obtained for flow parameters λ and μ ranging from 0 to +∞, and different values of d and N. The effect of the gap size is analysed first. By considering the potential flow (λ = μ = 0), we show how the instability associated with a mechanism of resonance for finite-gap changes into a radiative instability when d → ∞. Numerical results are compared with large axial wavenumber results and a very good agreement is obtained. For infinite gap (d = ∞), we show that the most unstable modes are obtained for large values of the azimuthal wavenumber for all λ and μ. We demonstrate that their properties can be captured by performing a local analysis near the inner cylinder in the limit of both large azimuthal and axial wavenumbers. The effect of the stratification is also analysed. We show that decreasing N is stabilizing. An asymptotic analysis for small N is also performed and shown to capture the properties of the most unstable mode of the potential flow in this limit.
Short- to long-wave resonance and soliton formation in boundary-layer interaction with a liquid film
- M. VLACHOMITROU, N. PELEKASIS
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- 12 July 2010, pp. 162-196
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Dynamic interaction between a boundary layer of air and a liquid film is investigated in this paper. The low air-to-film-viscosity ratio is considered in which case the boundary layer is quasi-steady on the time scale within which interfacial waves develop. The base flow consists of a boundary layer that drags a film of constant shear. Linear analysis, in the context of triple-deck theory, predicts the formation of a wavepacket of capillary waves that advances and spreads with time. The Froude number of de-/anti-icing fluids or water interacting with air falls well within the supercritical regime, i.e. Fr > FrCr. Numerical simulations of such flow systems were performed in the context of triple-deck theory, and they do not exhibit wave saturation or formation of uniform wavetrains. The long-term interaction is mainly dependent on film inertia as this is characterized by parameter
= (μ/μf)2(ρf/ρ), which involves film and air viscosity and density ratios, and the dimensionless film thickness, H0, and shear, λ, provided by the base flow. Weakly nonlinear analysis taking into consideration mean drift, i.e. generation of long waves, due to self-interaction of the linear wave to O(ϵ2) in amplitude of the initial disturbance, reveals resonance between the wavepacket predicted by linear theory and long waves when the group velocity of the former happens to coincide with the phase velocity, H0λ, of long interfacial waves. Numerical simulations with anti-icing fluids and water verify this pattern. In both cases, long waves eventually dominate the dynamics and, as they are modulated with time, they lead to soliton-type structures. Anti-icing fluids eventually exhibit oscillatory spikes whose mean value never exceeds 2H0, roughly. Water films exhibit a single spike that keeps growing, thus generating a large separation bubble.
Aeromechanics of passive rotation in flapping flight
- J. P. WHITNEY, R. J. WOOD
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- 27 July 2010, pp. 197-220
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Flying insects and robots that mimic them flap and rotate (or ‘pitch’) their wings with large angular amplitudes. The reciprocating nature of flapping requires rotation of the wing at the end of each stroke. Insects or flapping-wing robots could achieve this by directly exerting moments about the axis of rotation using auxiliary muscles or actuators. However, completely passive rotational dynamics might be preferred for efficiency purposes, or, in the case of a robot, decreased mechanical complexity and reduced system mass. Herein, the detailed equations of motion are derived for wing rotational dynamics, and a blade-element model is used to supply aerodynamic force and moment estimates. Passive-rotation flapping experiments with insect-scale mechanically driven artificial wings are conducted to simultaneously measure aerodynamic forces and three-degree-of-freedom kinematics (flapping, rotation and out-of-plane deviation), allowing a detailed evaluation of the blade-element model and the derived equations of motion. Variations in flapping kinematics, wing-beat frequency, stroke amplitude and torsional compliance are made to test the generality of the model. All experiments showed strong agreement with predicted forces and kinematics, without variation or fitting of model parameters.
On Lagrangian drift in shallow-water waves on moderate shear
- W. R. C. PHILLIPS, A. DAI, K. K. TJAN
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- 16 July 2010, pp. 221-239
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The Lagrangian drift in an O(ϵ) monochromatic wave field on a shear flow, whose characteristic velocity is O(ϵ) smaller than the phase velocity of the waves, is considered. It is found that although shear has only a minor influence on drift in deep-water waves, its influence becomes increasingly important as the depth decreases, to the point that it plays a significant role in shallow-water waves. Details of the shear flow likewise affect the drift. Because of this, two temporal cases common in coastal waters are studied, viz. stress-induced shear, as would arise were the boundary layer wind-driven, and a current-driven shear, as would arise from coastal currents. In the former, the magnitude of the drift (maximum minus minimum) in shallow-water waves is increased significantly above its counterpart, viz. the Stokes drift, in like waves in otherwise quiescent surroundings. In the latter, on the other hand, the magnitude decreases. However, while the drift at the free surface is always oriented in the direction of wave propagation in stress-driven shear, this is not always the case in current-driven shear, especially in long waves as the boundary layer grows to fill the layer. This latter finding is of particular interest vis-à-vis Langmuir circulations, which arise through an instability that requires differential drift and shear of the same sign. This means that while Langmuir circulations form near the surface and grow downwards (top down), perhaps to fill the layer, in stress-driven shear, their counterparts in current-driven flows grow from the sea floor upwards (bottom up) but can never fill the layer.
Elliptic instability of a stratified fluid in a rotating cylinder
- D. GUIMBARD, S. LE DIZÈS, M. LE BARS, P. LE GAL, S. LEBLANC
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- 16 July 2010, pp. 240-257
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In this paper, we analyse the characteristics of the elliptic instability in a finite cylinder in the presence of both background rotation and axial stratification. A general formula for the linear growth rate of the stationary sinuous modes is derived including viscous and detuning effects in the limit of small eccentricity. This formula is discussed and compared to experimental results which are obtained in a cylinder filled with salted water for two different eccentricities by varying the stratification, the background rotation and the cylinder rotation. A good agreement with the theory concerning the domain of instability of the sinuous modes is demonstrated. Other elliptic instability modes, oscillating at the cylinder angular frequency are also evidenced together with a new type of instability mode, which could be connected to a centrifugal instability occurring during the experimental phase of spin-up. The nonlinear regime of the elliptic instability is also documented. In contrast with the homogeneous case, no cycle involving growth, breakdown and re-laminarization is observed in the presence of strong stratification. The elliptic instability in a stratified fluid seems to yield either a persistent turbulent state or a weakly nonlinear regime.
Particle focusing in a suspension flow through a corrugated tube
- G. F. HEWITT, J. S. MARSHALL
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- 21 July 2010, pp. 258-281
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A computational study is performed of the transport of a particulate suspension through a corrugated tube using a discrete-element method (DEM). The tube is axisymmetric with a radius that varies sinusoidally along the tube length, which, in the presence of a mean suspension flow, leads to periodic inward and outward acceleration of the advected particles. The oscillations in radial acceleration and straining rate lead to a net radial drift, with mean acceleration measuring about an order of magnitude smaller than the instantaneous radial acceleration, which over time focuses small particles within the tube. The foundations of particle focusing in this flow are examined analytically using lubrication theory, together with a low-Stokes-number approximation for the particle drift. This lubrication-theory solution provides the basic scaling for how the particle drift will vary with wave amplitude and wavelength. Computations are then performed using a finite-volume method for a fluid flow in the tube at higher Reynolds numbers over a range of amplitudes, wavelengths and Reynolds numbers, examining the effect of each of these variables on the averaged radial fluid acceleration. A DEM is used to simulate particle behaviour at finite Stokes numbers, and the results are compared to an asymptotic approximation valid for low Stokes numbers. At low tube Reynolds number (e.g. Re = 10), the drift velocity induced by the tube corrugations focuses the particles onto the tube centreline, in accordance with the low-Stokes-number approximation based on the axial-averaged fluid radial acceleration. At higher tube Reynolds numbers (e.g. Re = 100), the correlation between the particle radial oscillation and the fluid acceleration field leads the outermost particles to drift into a ring at a finite radius from the tube centre, with little net motion of the particles in the innermost part of the tube. At larger Stokes numbers, particles can be dispersed to the outer regions of the tube due to particle outward dispersion from the large instantaneous radial acceleration. The effects of eddy formation within the corrugation crests on particle focusing are also examined.
Investigation of the subgrid-scale fluxes and their production rates in a convective atmospheric surface layer using measurement data
- QINGLIN CHEN, SHUAISHUAI LIU, CHENNING TONG
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- 19 July 2010, pp. 282-315
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The subgrid-scale (SGS) potential temperature flux and stress in the atmospheric surface layer are studied using field measurement data. We analyse the mean values of the SGS temperature flux, the SGS temperature flux production rate, the SGS temperature variance production rate, the SGS stress and the SGS stress production rate conditional on both the resolvable-scale velocity and temperature, which must be reproduced by SGS models for large-eddy simulation to reproduce the one-point resolvable-scale velocity–temperature joint probability density function (JPDF). The results show that the conditional statistics generally depend on the resolvable-scale velocity and temperature fluctuations, indicating that these conditional variables have strong influences on the resolvable-scale statistics. The dependencies of the conditional SGS stress and the SGS stress production rate, which are partly due to the effects of flow history and buoyancy, suggest that model predictions of the SGS stress also affect the resolvable-scale temperature statistics. The results for the conditional flux and the conditional flux production rate vectors have similar trends. These conditional vectors are also well aligned. The positive temperature fluctuations associated with updrafts are found to have a qualitatively different influence on the conditional statistics than the negative temperature fluctuations associated with downdrafts. The conditional temperature flux and the temperature flux production rate predicted using several SGS models are compared with measurements in statistical a priori tests. The predictions using the nonlinear model are found to be closely related to the predictions using the Smagorinsky model. Several potential effects of the SGS model deficiencies on the resolvable-scale statistics, such as the overprediction of the vertical mean temperature gradient and the underprediction of the vertical temperature flux, are identified. The results suggest that efforts to improve the LES prediction of a resolvable-scale statistic must consider all the relevant SGS components identified using the JPDF equation and the surface layer dynamics. This study also provides impetus for further investigations of the JPDF equation, especially analytical studies on the relationship between the JPDF and the SGS terms that govern its evolution.
Weakly nonlinear dynamics of thermoconvective instability involving viscoplastic fluids
- C. MÉTIVIER, C. NOUAR, J.-P. BRANCHER
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- 04 August 2010, pp. 316-353
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In this paper, a weakly nonlinear stability of viscoplastic fluid flow is performed. The system consists of a plane Rayleigh–Bénard–Poiseuille (RBP) flow of a Bingham fluid. The basic flow is characterized by a central plug zone, of 2yb width, in which the stresses are smaller than the Bingham number B, the dimensionless yield stress. The Bingham model assumes that inside this zone the material moves as a rigid solid, and that outside this zone it behaves as a viscous fluid. The aim of this study is to investigate the influence of the yield stress on the instability conditions. The linear stability analysis is performed using a modal method and provides critical values of Rayleigh and wavenumbers, from which the system becomes unstable. The critical mode, i.e. the least stable mode, is also determined. This mode, also called the fundamental mode, creates perturbation harmonics which cannot be neglected above criticality. The weakly nonlinear analysis is performed for small-amplitude perturbations. In this study, the quadratic modes of the perturbation are determined. Results indicate that the nonlinear modes perturbation can attain high maximal values, which is the consequence of the high variations of viscosity in the flow. The characterization of the complex Landau equation sheds light on a transition in terms of the bifurcation nature above a critical Péclet number Pec = O(1). Below Pec, it is found that a supercritical equilibrium state could exist, such as in the Newtonian case, while above Pec, the bifurcation becomes subcritical. One observes a sharp transition from supercritical to subcritical bifurcation as the Péclet value is increased. A dependence of Pec on the yield stress is highlighted since the subcritical bifurcation is first observed for weak values of yb (yb < O(10−1)). For this range of values, the transition is mainly due to the presence of the unyielded region via non-homogeneous boundary conditions at the yield surfaces. Then for yb > O(10−1), the change of the bifurcation nature is due to the variations of the effective viscosity in the unyielded regions.
Zigzag instability of vortex pairs in stratified and rotating fluids. Part 1. General stability equations.
- PAUL BILLANT
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- 21 July 2010, pp. 354-395
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In stratified and rotating fluids, pairs of columnar vertical vortices are subjected to three-dimensional bending instabilities known as the zigzag instability or as the tall-column instability in the quasi-geostrophic limit. This paper presents a general asymptotic theory for these instabilities. The equations governing the interactions between the strain and the slow bending waves of each vortex column in stratified and rotating fluids are derived for long vertical wavelength and when the two vortices are well separated, i.e. when the radii R of the vortex cores are small compared to the vortex separation distance b. These equations have the same form as those obtained for vortex filaments in homogeneous fluids except that the expressions of the mutual-induction and self-induction functions are different. A key difference is that the sign of the self-induction function is reversed compared to homogeneous fluids when the fluid is strongly stratified: |
max| < N (where N is the Brunt–Väisälä frequency and max the maximum angular velocity of the vortex) for any vortex profile and magnitude of the planetary rotation. Physically, this means that slow bending waves of a vortex rotate in the same direction as the flow inside the vortex when the fluid is stratified-rotating in contrast to homogeneous fluids. When the stratification is weaker, i.e. | max| > N, the self-induction function is complex because the bending waves are damped by a viscous critical layer at the radial location where the angular velocity of the vortex is equal to the Brunt–Väisälä frequency. In contrast to previous theories, which apply only to strongly stratified non-rotating fluids, the present theory is valid for any planetary rotation rate and when the strain is smaller than the Brunt–Väisälä frequency: Γ/(2πb2) ≪ N, where Γ is the vortex circulation. Since the strain is small, this condition is met across a wide range of stratification: from weakly to strongly stratified fluids. The theory is further generalized formally to any basic flow made of an arbitrary number of vortices in stratified and rotating fluids. Viscous and diffusive effects are also taken into account at leading order in Reynolds number when there is no critical layer. In Part 2 (Billant et al., J. Fluid Mech., 2010, doi:10.1017/S002211201000282X), the stability of vortex pairs will be investigated using the present theory and the predictions will be shown to be in very good agreement with the results of direct numerical stability analyses. The existence of the zigzag instability and the distinctive stability properties of vortex pairs in stratified and rotating fluids compared to homogeneous fluids will be demonstrated to originate from the sign reversal of the self-induction function.
Zigzag instability of vortex pairs in stratified and rotating fluids. Part 2. Analytical and numerical analyses.
- P. BILLANT, A. DELONCLE, J.-M. CHOMAZ, P. OTHEGUY
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- 21 July 2010, pp. 396-429
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The three-dimensional stability of vertical vortex pairs in stratified and rotating fluids is investigated using the analytical approach established in Part 1 and the predictions are compared to the results of previous direct numerical stability analyses for pairs of co-rotating equal-strength Lamb–Oseen vortices and to new numerical analyses for equal-strength counter-rotating vortex pairs. A very good agreement between theoretical and numerical results is generally found, thereby providing a comprehensive description of the zigzag instability. Co-rotating and counter-rotating vortex pairs are most unstable to the zigzag instability when the Froude number Fh = Γ/(2πR2N) (where Γ is the vortex circulation, R the vortex radius and N the Brunt–Väisälä frequency) is lower than unity independently of the Rossby number Ro = Γ/(4πR2Ωb) (Ωb is the planetary rotation rate). In this range, the maximum growth rate is proportional to the strain Γ/(2πb2) (b is the separation distance between the vortices) and is almost independent of Fh and Ro. The most amplified wavelength scales like Fhb when the Rossby number is large and like Fhb/|Ro| when |Ro| ≪ 1, in agreement with previous results. While the zigzag instability always bends equal-strength co-rotating vortex pairs in a symmetric way, the instability is only quasi-antisymmetric for finite Ro for equal-strength counter-rotating vortex pairs because the cyclonic vortex is less bent than the anticyclonic vortex. The theory is less accurate for co-rotating vortex pairs around Ro ≈ −2 because the bending waves rotate very slowly for long wavelength. The discrepancy can be fully resolved by taking into account higher-order three-dimensional effects.
When Fh is increased above unity, the growth rate of the zigzag instability is strongly reduced because the bending waves of each vortex are damped by a critical layer at the radius where the angular velocity of the vortex is equal to the Brunt–Väisälä frequency. The zigzag instability, however, continues to exist and is dominant up to a critical Froude number, which mostly depends on the Rossby number. Above this threshold, equal-strength co-rotating vortex pairs are stable with respect to long-wavelength bending disturbances whereas equal-strength counter-rotating vortex pairs become unstable to a quasi-symmetric instability resembling the Crow instability in homogeneous fluids. However, its growth rate is lower than in homogeneous fluids because of the damping by the critical layer. The structure of the critical layer obtained in the computations is in excellent agreement with the theoretical solution.
Physically, the different stability properties of vortex pairs in stratified and rotating fluids compared to homogeneous fluids are shown to come from the reversal of the direction of the self-induced motion of bent vortices.
Numerical study of a vortex ring impacting a flat wall
- MING CHENG, JING LOU, LI-SHI LUO
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- 16 August 2010, pp. 430-455
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We numerically study a vortex ring impacting a flat wall with an angle of incidence θ ≥ 0°) in three dimensions by using the lattice Boltzmann equation. The hydrodynamic behaviour of the ring–wall interacting flow is investigated by systematically varying the angle of incidence θ in the range of 0° ≤ θ ≤ 40° and the Reynolds number in the range of 100 ≤ Re ≤ 1000, where the Reynolds number Re is based on the translational speed and initial diameter of the vortex ring. We quantify the effects of θ and Re on the evolution of the vortex structure in three dimensions and other flow fields in two dimensions. We observe three distinctive flow regions in the θ–Re parameter space. First, in the low-Reynolds-number region, the ring–wall interaction dissipates the ring without generating any secondary rings. Second, with a moderate Reynolds number Re and a small angle of incidence θ, the ring–wall interaction generates a complete secondary vortex ring, and even a tertiary ring at higher Reynolds numbers. The secondary vortex ring is convected to the centre region of the primary ring and develops azimuthal instabilities, which eventually lead to the development of hairpin-like small vortices through ring–ring interaction. And finally, with a moderate Reynolds number and a sufficiently large angle of incidence θ, only a secondary vortex ring is generated. The secondary vortex wraps around the primary ring and propagates from the near end of the primary ring, which touches the wall first, to the far end, which touches the wall last. The rings develop a helical structure. Our results from the present study confirm some existing experimental observations made in the previous studies.
The unidirectional emptying box
- C. J. COFFEY, G. R. HUNT
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- 01 September 2010, pp. 456-474
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A theoretical description of the turbulent mixing within and the draining of a dense fluid layer from a box connected to a uniform density, quiescent environment through openings in the top and the base of the box is presented in this paper. This is an extension of the draining model developed by Linden et al. (Annu. Rev. Fluid Mech. vol. 31, 1990, pp. 201–238) and includes terms that describe localized mixing within the emptying box at the density interface. Mixing is induced by a turbulent flow of replacement fluid into the box and as a consequence we predict, and observe in complementary experiments, the development of a three-layer stratification. Based on the data collated from previous researchers, three distinct formulations for entrainment fluxes across density interfaces are used to account for this localized mixing. The model was then solved numerically for the three mixing formulations. Analytical solutions were developed for one formulation directly and for a second on assuming that localized mixing is relatively weak though still significant in redistributing buoyancy on the timescale of the draining process. Comparisons between our theoretical predictions and the experimental data, which we have collected on the developing layer depths and their densities show good agreement. The differences in predictions between the three mixing formulations suggest that the normalized flux turbulently entrained across a density interface tends to a constant value for large values of a Froude number FrT, based on conditions of the inflow through the top of the box, and scales as the cube of FrT for small values of FrT. The upper limit on the rate of entrainment into the mixed layer results in a minimum time (tD) to remove the original dense layer. Using our analytical solutions, we bound this time and show that 0.2tE ≲ tD ≲ tE, i.e. the original dense layer may be depleted up to five times more rapidly than when there is no internal mixing and the box empties in a time tE.
Dynamics of sheared inelastic dumbbells
- K. ANKI REDDY, J. TALBOT, V. KUMARAN
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- 16 August 2010, pp. 475-498
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We study the dynamical properties of the homogeneous shear flow of inelastic dumbbells in two dimensions as a first step towards examining the effect of shape on the properties of flowing granular materials. The dumbbells are modelled as smooth fused disks characterized by the ratio of the distance between centres (L) and the disk diameter (D), with an aspect ratio (L/D) varying between 0 and 1 in our simulations. Area fractions studied are in the range 0.1–0.7, while coefficients of normal restitution (en) from 0.99 to 0.7 are considered. The simulations use a modified form of the event-driven methodology for circular disks. The average orientation is characterized by an order parameter S, which varies between 0 (for a perfectly disordered fluid) and 1 (for a fluid with the axes of all dumbbells in the same direction). We investigate power-law fits of S as a function of (L/D) and (1−en2). There is a gradual increase in ordering as the area fraction is increased, as the aspect ratio is increased or as the coefficient of restitution is decreased. The order parameter has a maximum value of about 0.5 for the highest area fraction and lowest coefficient of restitution considered here. The mean energy of the velocity fluctuations in the flow direction is higher than that in the gradient direction and the rotational energy, though the difference decreases as the area fraction increases, due to the efficient collisional transfer of energy between the three directions. The distributions of the translational and rotational velocities are Gaussian to a very good approximation. The pressure is found to be remarkably independent of the coefficient of restitution. The pressure and dissipation rate show relatively little variation when scaled by the collision frequency for all the area fractions studied here, indicating that the collision frequency determines the momentum transport and energy dissipation, even at the lowest area fractions studied here. The mean angular velocity of the particles is equal to half the vorticity at low area fractions, but the magnitude systematically decreases to less than half the vorticity as the area fraction is increased, even though the stress tensor is symmetric.
Effect of compressibility on the global stability of axisymmetric wake flows
- P. MELIGA, D. SIPP, J.-M. CHOMAZ
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- Published online by Cambridge University Press:
- 19 August 2010, pp. 499-526
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We study the linear dynamics of global eigenmodes in compressible axisymmetric wake flows, up to the high subsonic regime. We consider both an afterbody flow at zero angle of attack and a sphere, and find that the sequence of bifurcations destabilizing the axisymmetric steady flow is independent of the Mach number and reminiscent of that documented in the incompressible wake past a sphere and a disk (Natarajan & Acrivos, J. Fluid Mech., vol. 254, 1993, p. 323), hence suggesting that the onset of unsteadiness in this class of flows results from a global instability. We determine the boundary separating the stable and unstable domains in the (M, Re) plane, and show that an increase in the Mach number yields a stabilization of the afterbody flow, but a destabilization of the sphere flow. These compressible effects are further investigated by means of adjoint-based sensitivity analyses relying on the computation of gradients or sensitivity functions. Using this theoretical formalism, we show that they do not act through specific compressibility effects at the disturbance level but mainly through implicit base flow modifications, an effect that had not been taken into consideration by previous studies based on prescribed parallel base flow profiles. We propose a physical interpretation for the observed compressible effects, based on the competition between advection and production of disturbances, and provide evidence linking the stabilizing/destabilizing effect observed when varying the Mach number to a strengthening/weakening of the disturbance advection mechanism. We show, in particular, that the destabilizing effect of compressibility observed in the case of the sphere results from a significant increase of the backflow velocity in the whole recirculating bubble, which opposes the downstream advection of disturbances.