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Stability of helical tip vortices in a rotor far wake
- V. L. OKULOV, J. N. SØRENSEN
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- Published online by Cambridge University Press:
- 28 March 2007, pp. 1-25
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As a means of analysing the stability of the wake behind a multi-bladed rotor the stability of a multiplicity of helical vortices embedded in an assigned flow field is addressed. In the model the tip vortices in the far wake are approximated by infinitely long helical vortices with constant pitch and radius. The work is a further development of a model developed in Okulov (J. Fluid Mech., vol. 521, p. 319) in which the linear stability of N equally azimuthally spaced helical vortices was considered. In the present work the analysis is extended to include an assigned vorticity field due to root vortices and the hub of the rotor. Thus the tip vortices are assumed to be embedded in an axisymmetric helical vortex field formed from the circulation of the inner part of the rotor blades and the hub. As examples of inner vortex fields we consider three generic axial columnar helical vortices, corresponding to Rankine, Gaussian and Scully vortices, at radial extents ranging from the core radius of a tip vortex to several rotor radii.
The analysis shows that the stability of tip vortices largely depends on the radial extent of the hub vorticity as well as on the type of vorticity distribution. As part of the analysis it is shown that a model in which the vortex system is replaced by N tip vortices of strength Γ and a root vortex of strength − N/Γ is unconditionally unstable.
Langmuir turbulence in shallow water. Part 1. Observations
- ANN E. GARGETT, JUDITH R. WELLS
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- 28 March 2007, pp. 27-61
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During extended deployment at an ocean observatory off the coast of New Jersey, a bottom-mounted five-beam acoustic Doppler current profiler measured large-scale velocity structures that we interpret as Langmuir circulations filling the entire water column. These circulations are the large-eddy structures of wind-wave-driven turbulent flows that occur episodically when a shallow water column experiences prolonged strong wind forcing. Many observational characteristics agree with former descriptions of Langmuir circulations in deep water. The three-dimensional velocity field reveals quasi-organized structures consisting of pairs of surface-intensified counter-rotating vortices, aligned approximately downwind. Maximum downward velocities are stronger than upward velocities, and the downwelling region of each cell, defined as a pair of vortices, is narrower than the upwelling region. Maximum downward vertical velocity occurs at or above mid-depth, and scales approximately with wind speed. The estimated crosswind scale of cells is roughly 3–6 times their vertical scale, set under these conditions by water depth. The long axis of the cells appears to lie at an angle ∼10°–20° to the right of the wind. A major difference from deep-water observations is strong near-bottom intensification of the downwind ‘jets’ found typically centred over downwelling regions. Accessible observational features such as cell morphology and profiles of mean velocities, turbulent velocity variances, and shear stress components are compared with the results of associated large-eddy simulations (reported in Part 2) of shallow water flows driven by surface stress and the Craik–Leibovich vortex forcing generally used to represent generation of Langmuir cells. A particularly sensitive diagnostic for identification of Langmuir circulations as the energy-containing eddies of the turbulent flow is the depth trajectory of invariants of the turbulent stress tensor, plotted in the Lumley ‘triangle’ corresponding to realizable turbulent flows. When Langmuir structures are present in the observations, the Lumley map is distinctly different from that of surface-stress-driven Couette flow, again in agreement with the large-eddy simulations (LES). Unlike the LES, observed velocity fields contain two distinct and significant scales of variability, documented by wavelet analysis of observational records of vertical velocity. Variability with periods of many minutes is that expected from Langmuir cells drifting past the instrument at the slowly time-varying crosswind velocity. Shorter period variability, of the order of 1–2 min, has roughly the observed periodicity of surface wave groups, suggesting a connection with the wave groups themselves and/or the wave breaking associated with them in high wind conditions.
Langmuir turbulence in shallow water. Part 2. Large-eddy simulation
- A. E. TEJADA-MARTÍNEZ, C. E. GROSCH
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- 28 March 2007, pp. 63-108
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Results of large-eddy simulation (LES) of Langmuir circulations (LC) in a wind-driven shear current in shallow water are reported. The LC are generated via the well-known Craik–Leibovich vortex force modelling the interaction between the Stokes drift, induced by surface gravity waves, and the shear current. LC in shallow water is defined as a flow in sufficiently shallow water that the interaction between the LC and the bottom boundary layer cannot be ignored, thus requiring resolution of the bottom boundary layer. After the introduction and a description of the governing equations, major differences in the statistical equilibrium dynamics of wind-driven shear flow and the same flow with LC (both with a bottom boundary layer) are highlighted. Three flows with LC will be discussed. In the first flow, the LC were generated by intermediate-depth waves (relative to the wavelength of the waves and the water depth). The amplitude and wavelength of these waves are representative of the conditions reported in the observations of A. E. Gargett & J. R. Wells in Part 1 (J. Fluid Mech. vol .000, 2007, p. 00). In the second flow, the LC were generated by shorter waves. In the third flow, the LC were generated by intermediate waves of greater amplitude than those in the first flow. The comparison between the different flows relies on visualizations and diagnostics including (i) profiles of mean velocity, (ii) profiles of resolved Reynolds stress components, (iii) autocorrelations, (iv) invariants of the resolved Reynolds stress anisotropy tensor and (v) balances of the transport equations for mean resolved turbulent kinetic energy and resolved Reynolds stresses. Additionally, dependencies of LES results on Reynolds number, subgrid-scale closure, size of the domain and grid resolution are addressed.
In the shear flow without LC, downwind (streamwise) velocity fluctuations are characterized by streaks highly elongated in the downwind direction and alternating in sign in the crosswind (spanwise) direction. Forcing this flow with the Craik–Leibovich force generating LC leads to streaks with larger characteristic crosswind length scales consistent with those recorded by observations. In the flows with LC, in the mean, positive streaks exhibit strong intensification near the bottom and near the surface leading to intensified downwind velocity ‘jets’ in these regions. In the flow without LC, such intensification is noticeably absent. A revealing diagnostic of the structure of the turbulence is the depth trajectory of the invariants of the resolved Reynolds stress anisotropy tensor, which for a realizable flow must lie within the Lumley triangle. The trajectory for the flow without LC reveals the typical structure of shear-dominated turbulence in which the downwind component of the resolved normal Reynolds stresses is greater than the crosswind and vertical components. In the near bottom and surface regions, the trajectory for the flow with LC driven by wave and wind forcing conditions representative of the field observations reveals a two-component structure in which the downwind and crosswind components are of the same order and both are much greater than the vertical component. The two-component structure of the Langmuir turbulence predicted by LES is consistent with the observations in the bottom third of the water column above the bottom boundary layer.
Mean flow of turbulent–laminar patterns in plane Couette flow
- DWIGHT BARKLEY, LAURETTE S. TUCKERMAN
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- 28 March 2007, pp. 109-137
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A turbulent–laminar banded pattern in plane Couette flow is studied numerically. This pattern is statistically steady, is oriented obliquely to the streamwise direction, and has a very large wavelength relative to the gap. The mean flow, averaged in time and in the homogeneous direction, is analysed. The flow in the quasi-laminar region is not the linear Couette profile, but results from a non-trivial balance between advection and diffusion. This force balance yields a first approximation to the relationship between the Reynolds number, angle, and wavelength of the pattern. Remarkably, the variation of the mean flow along the pattern wavevector is found to be almost exactly harmonic: the flow can be represented via only three cross-channel profiles as U(x, y, z) ≈ U0(y) + Uc(y) cos(kz) + Us(y) sin(kz). A model is formulated which relates the cross-channel profiles of the mean flow and of the Reynolds stress. Regimes computed for a full range of angle and Reynolds number in a tilted rectangular periodic computational domain are presented. Observations of regular turbulent–laminar patterns in other shear flows – Taylor–Couette, rotor–stator, and plane Poiseuille – are compared.
Lubrication theory for electro-osmotic flow in a non-uniform electrolyte
- T. L. SOUNART, J. C. BAYGENTS
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- Published online by Cambridge University Press:
- 28 March 2007, pp. 139-172
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A lubrication theory has been developed for the electro-osmotic flow of non-uniform buffers in narrow rectilinear channels. The analysis applies to systems in which the transverse dimensions of the channel are large compared with the Debye screening length of the electrolyte. In contrast with related theories of electrokinetic lubrication, here the streamwise variations of the velocity field stem from, and are nonlinearly coupled to, spatiotemporal variations in the electrolyte composition. Spatially non-uniform buffers are commonly employed in electrophoretic separation and transport schemes, including iso-electric focusing (IEF), isotachophoresis (ITP), field-amplified sample stacking (FASS), and high-ionic-strength electro-osmotic pumping. The fluid dynamics of these systems is controlled by a complex nonlinear coupling to the ion transport, driven by an applied electric field. Electrical conductivity gradients, attendent to the buffer non-uniformities, result in a variable electro-osmotic slip velocity and, in electric fields approaching 1 kV cm−1, Maxwell stresses drive the electrohydrodynamic circulation. Explicit semi-analytic expressions are derived for the fluid velocity, stream function, and electric field. The resulting approximations are found to be in good agreement with full numerical solutions for a prototype buffer, over a range of conditions typical of microfluidic systems. The approximations greatly simplify the computational analysis, reduce computation times by a factor 4–5, and, for the first time, provide general insight on the dominant fluid physics of two-dimensional electrically driven transport.
A new proof on net upscale energy cascade in two-dimensional and quasi-geostrophic turbulence
- ELEFTHERIOS GKIOULEKAS, KA KIT TUNG
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- 28 March 2007, pp. 173-189
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A general proof that more energy flows upscale than downscale in two-dimensional turbulence and barotropic quasi-geostrophic (QG) turbulence is given. A proof is also given that in surface QG turbulence, the reverse is true. Though some of these results are known in restricted cases, the proofs given here are pedagogically simpler, require fewer assumptions and apply to both forced and unforced cases.
Cavitation microstreaming patterns in single and multiple bubble systems
- PAUL THO, RICHARD MANASSEH, ANDREW OOI
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- Published online by Cambridge University Press:
- 28 March 2007, pp. 191-233
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Cavitation microstreaming is a well-known phenomenon; however, few flow visualizations or measurements of the velocity fields have been conducted. In this paper micro-PIV (particle image velocimetry) measurements and streak photography were used to study the flow field around a single and two oscillating bubbles resting on a solid boundary. The mode of oscillation of the bubble was also measured in terms of the variation in the radius of the bubble and the movement of the bubble's centroid so that the streaming flow field could be accurately related to the bubble's oscillatory motion. The mode of oscillation was found to vary primarily with the applied acoustic frequency. Several modes of oscillation were investigated, including translating modes where the bubble's centroid moves along either a single axis, an elliptical orbit or a circular orbit. The flow field resulting from these oscillation modes contains closed streamlines representing vortical regions in the vicinity of the bubble. The translating modes were observed to occur in sequential order with the acoustic excitation frequency, changing from a translation along a single axis, to an elliptical orbit and finally to a circular orbit, or vice versa. Following this sequence, there is a corresponding transformation of the streaming pattern from a symmetrical flow structure containing four vortices to a circular vortex centred on the bubble. Despite some inconsistencies, there is general agreement between these streaming patterns and those found in existing theoretical models. Volume and shape mode oscillations of single bubbles as well as several different cases of multiple bubbles simultaneously oscillating with the same frequency and phase were also investigated and show a rich variety of streaming patterns.
Scattering of surface gravity waves by bottom topography with a current
- FABRICE ARDHUIN, RUDY MAGNE
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- Published online by Cambridge University Press:
- 28 March 2007, pp. 235-264
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A theory is presented that describes the scattering of random surface gravity waves by small-amplitude topography, with horizontal scales of the order of the wavelength, in the presence of an irrotational and almost uniform current. A perturbation expansion of the wave action to order η2 yields an evolution equation for the wave action spectrum, where η = max(h)/H is the small-scale bottom amplitude normalized by the mean water depth. Spectral wave evolution is proportional to the bottom elevation variance at the resonant wavenumbers, representing a Bragg scattering approximation. With a current, scattering results from a direct effect of the bottom topography, and an indirect effect of the bottom through the modulations of the surface current and mean surface elevation. For Froude numbers of the order of 0.6 or less, the bottom topography effects dominate. For all Froude numbers, the reflection coefficients for the wave amplitudes that are inferred from the wave action source term are asymptotically identical, as η goes to zero, to previous theoretical results for monochromatic waves propagating in one dimension over sinusoidal bars. In particular, the frequency of the most reflected wave components is shifted by the current, and wave action conservation results in amplified reflected wave energies for following currents. Application of the theory to waves over current-generated sandwaves suggests that forward scattering can be significant, resulting in a broadening of the directional wave spectrum, while back-scattering should be generally weaker.
Integral force acting on a body due to local flow structures
- J.-Z. WU, X.-Y. LU, L.-X. ZHUANG
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- 28 March 2007, pp. 265-286
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The forces exerted on a body moving through a fluid depend strongly on the local dynamic processes and structures generated by the body motion, such as flow separation, vortices, etc. A detailed and quantitative understanding of the effects of these processes and structures on the instantaneous overall force characteristics is of fundamental significance, and may improve our capabilities for flow analysis and control. In the present study, some unconventional force expressions based on ‘derivative-moment transformations’, which can have a rich variety of forms for the same flow field, are used to provide better insight into local dynamics. In particular, we apply jointly three alternative unconventional force expressions to analyse two numerical solutions of unsteady and viscous circular-cylinder flows. The results confirm the exactness of the expressions and, more importantly, provide a unified understanding of the specific influence on the force of each individual flow structure at its different evolution stages.
Inertial effects in droplet spreading: a comparison between diffuse-interface and level-set simulations
- HANG DING, PETER D. M. SPELT
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- Published online by Cambridge University Press:
- 28 March 2007, pp. 287-296
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Axisymmetric droplet spreading is investigated numerically at relatively large rates of spreading, such that inertial effects become important. Results from two numerical methods that use different means to alleviate the stress singularity at moving contact lines (a diffuse interface, and a slip-length-based level-set method) are shown to agree well. An initial inertial regime is observed to yield to a regime associated with Tanner's law at later times. The spreading rate oscillates during the changeover between these regimes. This becomes more significant for a fixed (effective) slip length when decreasing the value of an Ohnesorge number. The initial, inertia-dominated regime is characterized by a rapidly extending region affected by the spreading, giving the appearance of a capillary wave travelling from the contact line. The oscillatory behaviour is associated with the rapid collapse that follows the point at which this region extends to the entire droplet. Results are presented for the apparent contact angle as a function of dimensionless spreading rate for various values of Ohnesorge number, slip length and initial conditions. The results indicate that there is no such universal relation when inertial effects are important.
Overturning in a filling box
- N. B. KAYE, G. R. HUNT
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- 28 March 2007, pp. 297-323
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Overturning in a cylindrical filling box driven by a turbulent plume is examined theoretically and experimentally. We establish the initial penetration depth (h) of the buoyant flow that intrudes vertically up the sidewall as a function of the box radius (R) and height (H). Dimensional arguments reduce the problem to finding η = h/H as a function of the aspect ratio Φ = R/H. The flow is modelled in two parts, the radial outflow from the plume along the base of the box and the flow up the sidewall. The outflow is modelled as a forced radial gravity current with constant buoyancy flux while the sidewall flow is modelled as a line fountain. Two regimes were found: first, when the plume outflow is adjusting toward a pure gravity current on impact with the vertical wall and the rise height is given by η ∼ Φ−1/3; secondly, when the outflow is fully developed on, or before, impact and the rise height is given by η ∼ Const. Experimental results show good agreement with these scalings and allow the constants of proportionality to be established.
Centre modes in inviscid swirling flows and their application to the stability of the Batchelor vortex
- C. J. HEATON
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- 28 March 2007, pp. 325-348
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We identify a family of centre-mode disturbances to inviscid swirling flows such as jets, wakes and other vortices. The centre modes form an infinite family of modes, increasingly concentrated near to the symmetry axis of the mean flow, and whose frequencies accumulate to a single point in the complex plane. This asymptotic accumulation allows analytical progress to be made, including a theoretical stability boundary, in O(1) parameter regimes. The modes are located close to the continuous spectrum of the linearized Euler equations, and the theory is closely related to that of the continuous spectrum. We illustrate our analysis with the inviscid Batchelor vortex, defined by swirl parameter q. We show that the inviscid instabilities found in previous numerical studies are in fact the first members of an infinite set of centre modes of the type we describe. We investigate the inviscid neutral curve, and find good agreement of the neutral curve predicted by the analysis with the results of numerical computations. We find that the unstable region is larger than previously reported. In particular, the value of q above which the inviscid vortex stabilizes is significantly larger than previously reported and in agreement with a long-standing theoretical prediction.
The role of forcing in the local stability of stationary long waves. Part 1. Linear dynamics
- DANIEL HODYSS, TERRENCE R. NATHAN
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- 28 March 2007, pp. 349-376
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The local linear stability of forced, stationary long waves produced by topography or potential vorticity (PV) sources is examined using a quasi-geostrophic barotropic model. A multiple scale analysis yields coupled equations for the background stationary wave and low-frequency (LF) disturbance field. Forcing structures for which the LF dynamics are Hamiltonian are shown to yield conservation laws that provide necessary conditions for instability and a constraint on the LF structures that can develop. Explicit knowledge of the forcings that produce the stationary waves is shown to be crucial to predicting a unique LF field. Various topographies or external PV sources can be chosen to produce stationary waves that differ by asymptotically small amounts, yet the LF instabilities that develop can have fundamentally different structures and growth rates. If the stationary wave field is forced solely by topography, LF oscillatory modes always emerge. In contrast, if the stationary wave field is forced solely by PV, two LF structures are possible: oscillatory modes or non-oscillatory envelope modes. The development of the envelope modes within the context of a linear LF theory is novel.
An analysis of the complex WKB branch points, which yields an analytical expres-sion for the leading-order eigenfrequency, shows that the streamwise distribution of absolute instability and convective growth is central to understanding and predicting the types of LF structures that develop on the forced stationary wave. The location of the absolute instability region with respect to the stationary wave determines whether oscillatory modes or envelope modes develop. In the absence of absolute instability, eastward propagating wavetrains generated in the far field can amplify via local convective growth in the stationary wave region. If the stationary wave region is streamwise symmetric (asymmetric), the local convective growth results in a local change in wave energy that is transient (permanent).
On impulsively generated inviscid axisymmetric surface jets, waves and drops
- K. K. TJAN, W. R. C. PHILLIPS
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- 28 March 2007, pp. 377-403
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The evolution of an unbounded inviscid free surface subjected to a velocity potential of Gaussian form and also to the influence of inertial, interfacial and gravitational forces is considered. This construct was motivated by the occurrence of lung haemorrhage resulting from ultrasonic imaging and pursues the notion that bursts of ultrasound act to expel droplets that puncture the soft air-filled sacs in the lung plural surface, allowing them to fill with blood. The tissue adjacent to the sacs is modelled as a liquid and the air–tissue interface in the sacs as a free surface. The evolution of the free surface is described by a boundary-integral formulation and, since the free surface evolves slowly relative to the bursts of ultrasound, they are realized as an impulse at the free surface, represented by the velocity potential. As the free surface evolves, it is seen to form axisymmetric surface jets, waves or droplets, depending upon the levels of gravity and surface tension. Moreover the droplets may be spherical and ejected away from the surface or an inverted tear shape and fall back to the surface. These conclusions are expressed in a phase diagram of inverse Froude number Fr−1 versus inverse Weber number We−1. Specifically, while axisymmetric surface jets form in the absence of surface tension and gravity, gravity acts to bound their height, rendering them waves, although instability overrides the calculation prior to its reaching that bound. Surface tension acts to suppress the instability (provided that We−1 > 0.045) and to form drops; if sufficiently strong it can also damp the evolving wave, causing it to collapse. The pinchoff which effects spherical drops is of power-law type with exponent 2/3, and the universal constant that relates the necking radius to the time from pinchoff, thereby realizing a finite-time singularity, has the value . Finally, drops can occur once the mechanical index, a dimensional measure used in ultrasonography, exceeds 0.5.
On the formation of geophysical and planetary zonal flows by near-resonant wave interactions
- YOUNGSUK LEE, LESLIE M. SMITH
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- Published online by Cambridge University Press:
- 28 March 2007, pp. 405-424
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Numerical simulations on a β-plane are used to further understand the formation of zonal flows from small-scale fluctuations. The dynamics of ‘reduced models’ are computed by restricting the nonlinear term to include a subset of triad interactions in Fourier space. Reduced models of near-resonant triads are considered, as well as the complement set of non-resonant triads. At moderately small values of the Rhines number, near-resonant triad interactions are shown to be responsible for the generation of large-scale zonal flows from small-scale random forcing. Without large-scale drag, both the full system and the reduced model of near resonances produce asymmetry between eastward and westward jets, in favour of stronger westward jets. When large-scale drag is included, the long-time asymmetry is reversed in the full system, with eastward jets that are thinner and stronger than westward jets. Then the reduced model of near resonances exhibits a weaker asymmetry, but there are nevertheless more eastward jets stronger than a threshold value.
Effect of streamwise-periodic wall transpiration on turbulent friction drag
- M. QUADRIO, J. M. FLORYAN, P. LUCHINI
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- Published online by Cambridge University Press:
- 28 March 2007, pp. 425-444
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In this paper a turbulent plane channel flow modified by a distributed transpiration at the wall, with zero net mass flux, is studied through direct numerical simulation (DNS) using the incompressible Navier–Stokes equations. The transpiration is steady, uniform in the spanwise direction, and varies sinusoidally along the streamwise coordinate. The transpiration wavelength is found to dramatically affect the turbulent flow, and in particular the frictional drag. Long wavelengths produce large drag increases even with relatively small transpiration intensities, thus providing an efficient means for improved turbulent mixing. Shorter wavelengths, on the other hand, yield an unexpected decrease of turbulent friction. These opposite effects are separated by a threshold of transpiration wavelength, shown to scale in viscous units, related to a longitudinal length scale typical of the near-wall turbulence cycle. Transpiration is shown to affect the flow via two distinct mechanisms: steady streaming and direct interaction with turbulence. They modify the turbulent friction in two opposite ways, with streaming being equivalent to an additional pressure gradient needed to drive the same flow rate (drag increase) and direct interaction causing reduced turbulent activity owing to the injection of fluctuationless fluid. The latter effect overwhelms the former at small wavelengths, and results in a (small) net drag reduction. The possibility of observing large-scale streamwise-oriented vortical structures as a consequence of a centrifugal instability mechanism is also discussed. Our results do not demonstrate the presence of such vortices, and the same conclusion can be arrived at through a stability analysis of the mean velocity profile, even though it is possible that a higher value of the Reynolds number is needed to observe the vortices.
Double-diffusive instabilities of autocatalytic chemical fronts
- J. D'HERNONCOURT, A. DE WIT, A. ZEBIB
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- 28 March 2007, pp. 445-456
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Convective instabilities of an autocatalytic propagating chemical front in a porous medium are studied. The front creates temperature and concentration gradients which then generate a density gradient. If the front propagates in the direction of the gravity field, adverse density stratification can lead to Rayleigh–Taylor or Rayleigh–Bénard instabilities. Differential diffusivity of mass and heat can also destabilize the front because of the double-diffusive phenomena. We compare the stability boundaries for the classical hydrodynamic case of a bounded layer without reaction and for the chemical front in the parameter space spanned by the thermal and solutal Rayleigh numbers. We show that chemical reactions profoundly affect the stability boundaries compared to the non-reactive situation because of a delicate coupling between the double-diffusive and Rayleigh–Taylor mechanisms with localized density perturbations driven by the reaction. In the reactive case, a linear stability analysis identifies three distinct stationary branches of the instability. They bound a region of stability that shrinks with increasing Lewis number, in marked contrast to the classical double-diffusive layer. In particular a region of global and local stable stratification is susceptible to a counter-intuitive mechanism of convective instability driven by chemistry and double-diffusion. The other two regions display an additional contribution of localized Rayleigh–Taylor instabilities. Displaced-particle arguments are employed in support of and to elucidate the entire stability boundary.
Scaling of the wall-normal turbulence component in high-Reynolds-number pipe flow
- RONGRONG ZHAO, ALEXANDER J. SMITS
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- 28 March 2007, pp. 457-473
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Streamwise and wall-normal turbulence components are obtained in fully developed turbulent pipe over a Reynolds number range from 1.1 × 105 to 9.8 × 106. The streamwise intensity data are consistent with previous measurements in the same facility. For the wall-normal turbulence intensity, a constant region in v'r.m.s. is found for the region 200 ≤ y+ ≤ 0.1R+ for Reynolds numbers up to 1.0 × 106. An increase in v'r.m.s. is observed below about y+ ∼ 100, although additional measurements will be required to establish its generality. The wall-normal spectra collapse in the energy-containing region with inner scaling, but for the low-wavenumber region a y/R dependence is observed, which also indicates a continuing influence from the outer flow on the near-wall motions.
The effect of disturbances on the flows under a sluice gate and past an inclined plate
- B. J. BINDER, J.-M. VANDEN-BROECK
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- Published online by Cambridge University Press:
- 28 March 2007, pp. 475-490
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Free surface potential flows past disturbances in a channel are considered. Three different types of disturbance are studied: (i) a submerged obstacle on the bottom of a channel; (ii) a pressure distribution on the free surface; and (iii) an obstruction in the free surface (e.g. a sluice gate or a flat plate). Surface tension is neglected, but gravity is included in the dynamic boundary condition. Fully nonlinear solutions are computed by boundary integral equation methods. In addition, weakly nonlinear solutions are derived. New solutions are found when several disturbances are present simultaneously. They are discovered through the weakly nonlinear analysis and confirmed by numerical computations for the fully nonlinear problem.
The linear stability of high-frequency flow in a torsionally oscillating cylinder
- P. J. BLENNERHASSETT, ANDREW P. BASSOM
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- Published online by Cambridge University Press:
- 28 March 2007, pp. 491-505
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The linear stability of the Stokes layer induced in a fluid contained within a long cylinder oscillating at high frequency about its longitudinal axis is investigated. The disturbance equations are derived using Floquet theory and the resulting system solved using pseudo-spectral methods. Both shear modes and axially periodic centripetal disturbance modes are examined and neutral stability curves and corresponding critical conditions for instability identified. For sufficiently small cylinder radius it is verified that the centripetal perturbations limit the stability of the motion but that in larger-radius configurations the shear modes associated with the Stokes layer take over this role. These results suggest a possible design, free of entry-length effects, for experiments intended to examine the breakdown of oscillatory boundary layers.