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Continuous spectra of the Batchelor vortex
- XUERUI MAO, SPENCER SHERWIN
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- Published online by Cambridge University Press:
- 31 May 2011, pp. 1-23
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The spectra of the Batchelor vortex are obtained by discretizing its linearized evolution operator using a modified Chebyshev polynomial approximation at a Reynolds number of 1000 and zero azimuthal wavenumber. Three types of eigenmodes are identified from the spectra: discrete modes, potential modes and free-stream modes. The discrete modes have been extensively documented but the last two modes have received little attention. A convergence study of the spectra and pseudospectra supports the classification that discrete modes correspond to discrete spectra while the other two modes correspond to continuous spectra. Free-stream modes have finite amplitude in the far field whilst potential modes decay to zero in the far field. The free-stream modes are therefore a limiting form of the potential modes when the radial decay rate of velocity components reduces to zero. The radial form of the free-stream modes with axial and radial wavenumbers is investigated and the penetration of the free-stream mode into the vortex core highlights the possibility of interaction between the potential region and the vortex core. A wavepacket pseudomode study confirms the existence of continuous spectra and predicts the locations and radial wavenumbers of the eigenmodes. The pseudomodes corresponding to the potential modes are observed to be in the form of one or two wavepackets while the free-stream modes are not observed to be in the form of wavepackets.
A two-dimensional model of low-Reynolds number swimming beneath a free surface
- DARREN CROWDY, SUNGYON LEE, OPHIR SAMSON, ERIC LAUGA, A. E. HOSOI
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- 29 June 2011, pp. 24-47
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Biological organisms swimming at low-Reynolds number are often influenced by the presence of rigid boundaries and soft interfaces. In this paper, we present an analysis of locomotion near a free surface with surface tension. Using a simplified two-dimensional singularity model and combining a complex variable approach with conformal mapping techniques, we demonstrate that the deformation of a free surface can be harnessed to produce steady locomotion parallel to the interface. The crucial physical ingredient lies in the nonlinear hydrodynamic coupling between the disturbance flow created by the swimmer and the free boundary problem at the fluid surface.
Direct and large-eddy simulations of internal tide generation at a near-critical slope
- BISHAKHDATTA GAYEN, SUTANU SARKAR
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- Published online by Cambridge University Press:
- 25 May 2011, pp. 48-79
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A numerical study is performed to investigate nonlinear processes during internal wave generation by the oscillation of a background barotropic tide over a sloping bottom. The focus is on the near-critical case where the slope angle is equal to the natural internal wave propagation angle and, consequently, there is a resonant wave response that leads to an intense boundary flow. The resonant wave undergoes both convective and shear instabilities that lead to turbulence with a broad range of scales over the entire slope. A thermal bore is found during upslope flow. Spectra of the baroclinic velocity, both inside the boundary layer and in the external region with free wave propagation, exhibit discrete peaks at the fundamental tidal frequency, higher harmonics of the fundamental, subharmonics and inter-harmonics in addition to a significant continuous part. The internal wave flux and its distribution between the fundamental and harmonics is obtained. Turbulence statistics in the boundary layer including turbulent kinetic energy and dissipation rate are quantified. The slope length is varied with the smaller lengths examined by direct numerical simulation (DNS) and the larger with large-eddy simulation (LES). The peak value of the near-bottom velocity increases with the length of the critical region of the topography. The scaling law that is observed to link the near-bottom peak velocity to slope length is explained by an analytical boundary-layer solution that incorporates an empirically obtained turbulent viscosity. The slope length is also found to have a strong impact on quantities such as the wave energy flux, wave energy spectra, turbulent kinetic energy, turbulent production and turbulent dissipation.
Heat release rate correlation and combustion noise in premixed flames
- N. SWAMINATHAN, G. XU, A. P. DOWLING, R. BALACHANDRAN
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- 29 June 2011, pp. 80-115
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The sound emission from open turbulent flames is dictated by the two-point spatial correlation of the rate of change of the fluctuating heat release rate. This correlation in premixed flames can be represented well using Gaussian-type functions and unstrained laminar flame thermal thickness can be used to scale the correlation length scale, which is about a quarter of the planar laminar flame thermal thickness. This correlation and its length scale are observed to be less influenced by the fuel type or stoichiometry or turbulence Reynolds and Damkohler numbers. The time scale for fluctuating heat release rate is deduced to be about τc/34 on an average, where τc is the planar laminar flame time scale, using direct numerical simulation (DNS) data. These results and the spatial distribution of mean reaction rate obtained from Reynolds-averaged Navier–Stokes (RANS) calculations of open turbulent premixed flames employing the standard
model and an algebraic reaction rate closure, involving a recently developed scalar dissipation rate model, are used to obtain the far-field sound pressure level from open flames. The calculated values agree well with measured values for flames of different stoichiometry and fuel types, having a range of turbulence intensities and heat output. Detailed analyses of RANS results clearly suggest that the noise level from turbulent premixed flames having an extensive and uniform spatial distribution of heat release rate is low.
Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks
- NICHOLAS J. VAUGHAN, TAMER A. ZAKI
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- 31 May 2011, pp. 116-153
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The secondary instability of a zero-pressure-gradient boundary layer, distorted by unsteady Klebanoff streaks, is investigated. The base profiles for the analysis are computed using direct numerical simulation (DNS) of the boundary-layer response to forcing by individual free-stream modes, which are low frequency and dominated by streamwise vorticity. Therefore, the base profiles take into account the nonlinear development of the streaks and mean flow distortion, upstream of the location chosen for the stability analyses. The two most unstable modes were classified as an inner and an outer instability, with reference to the position of their respective critical layers inside the boundary layer. Their growth rates were reported for a range of frequencies and amplitudes of the base streaks. The inner mode has a connection to the Tollmien–Schlichting (T–S) wave in the limit of vanishing streak amplitude. It is stabilized by the mean flow distortion, but its growth rate is enhanced with increasing amplitude and frequency of the base streaks. The outer mode only exists in the presence of finite amplitude streaks. The analysis of the outer instability extends the results of Andersson et al. (J. Fluid Mech. vol. 428, 2001, p. 29) to unsteady base streaks. It is shown that base-flow unsteadiness promotes instability and, as a result, leads to a lower critical streak amplitude. The results of linear theory are complemented by DNS of the evolution of the inner and outer instabilities in a zero-pressure-gradient boundary layer. Both instabilities lead to breakdown to turbulence and, in the case of the inner mode, transition proceeds via the formation of wave packets with similar structure and wave speeds to those reported by Nagarajan, Lele & Ferziger (J. Fluid Mech., vol. 572, 2007, p. 471).
Direct numerical simulation of spatially developing turbulent boundary layers with uniform blowing or suction
- YUKINORI KAMETANI, KOJI FUKAGATA
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- 23 June 2011, pp. 154-172
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Direct numerical simulation (DNS) of spatially developing turbulent boundary layer with uniform blowing (UB) or uniform suction (US) is performed aiming at skin friction drag reduction. The Reynolds number based on the free stream velocity and the 99% boundary layer thickness at the inlet is set to be 3000. A constant wall-normal velocity is applied on the wall in the range, −0.01U∞ ≤ Vctr ≤ 0.01U∞. The DNS results show that UB reduces the skin friction drag, while US increases it. The turbulent fluctuations exhibit the opposite trend: UB enhances the turbulence, while US suppresses it. Dynamical decomposition of the local skin friction coefficient cf using the identity equation (FIK identity) (Fukagata, Iwamoto & Kasagi, Phys. Fluids, vol. 14, 2002, pp. L73–L76) reveals that the mean convection term in UB case works as a strong drag reduction factor, while that in US case works as a strong drag augmentation factor: in both cases, the contribution of mean convection on the friction drag overwhelms the turbulent contribution. It is also found that the control efficiency of UB is much higher than that of the advanced active control methods proposed for channel flows.
Large-amplitude topographic waves in 2D stratified flow
- DAVID J. MURAKI
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- 16 June 2011, pp. 173-192
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Our fundamental understanding of steady, stratified flow over two-dimensional (2D) topography rests on the pioneering works of G. Lyra and R. Long. Within linear theory, Lyra established the far-field radiation conditions that determine the downstream pattern of buoyancy waves. Soon after, Long discovered that the steady, nonlinear streamfunction for special cases of stratified, 2D flow could satisfy the same equations as linear theory, subject to an exact topographic boundary condition. Fourier methods are currently used to compute solutions to Long's theory for arbitrary topography in the near-hydrostatic or small-amplitude topographic parameter regimes. It is not generally appreciated however, that these methods encounter difficulties for flows that are both strongly non-hydrostatic and beyond linear amplitudes. By recasting Long's theory into a linear integral equation, this difficulty is shown to be a computational barrier associated with an ill-conditioning of the Fourier method. The problem is overcome through the development of a boundary integral computation which relies on some lesser known solutions from Lyra's original analysis. This method is well-conditioned for strongly non-hydrostatic flows, and is used to extend the exploration of critical overturning flows over Gaussian and bell-shaped ridges. These results indicate that the critical value of the non-dimensional height (
) asymptotes to a finite value with increasing non-hydrostatic parameter ( ).
Experimental investigation of laminar turbulent intermittency in pipe flow
- DEVRANJAN SAMANTA, ALBERTO DE LOZAR, BJÖRN HOF
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- 02 June 2011, pp. 193-204
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In shear flows, turbulence first occurs in the form of localized structures (puffs/spots) surrounded by laminar fluid. We here investigate such spatially intermittent flows in a pipe experiment showing that turbulent puffs have a well-defined interaction distance, which sets their minimum spacing as well as the maximum observable turbulent fraction. Two methodologies are employed. Starting from a laminar flow, puffs are first created by locally injecting a jet of fluid through the pipe wall. When the perturbation is applied periodically at low frequencies, as expected, a regular sequence of puffs is observed where the puff spacing is given by the ratio of the mean flow speed to the perturbation frequency. At large frequencies however puffs are found to interact and annihilate each other. Varying the perturbation frequency, an interaction distance is determined which sets the highest possible turbulence fraction. This enables us to establish an upper bound for the friction factor in the transitional regime, which provides a well-defined link between the Blasius and the Hagen-Poiseuille friction laws. In the second set of experiments, the Reynolds number is reduced suddenly from fully turbulent to the intermittent regime. The resulting flow reorganizes itself to a sequence of constant size puffs which, unlike in Couette and Taylor–Couette flow are randomly spaced. The minimum distance between the turbulent patches is identical to the puff interaction length. The puff interaction length is found to be in agreement with the wavelength of regular stripe and spiral patterns in plane Couette and Taylor–Couette flow.
Instability of streaks in wall turbulence with adverse pressure gradient
- MATTHIEU MARQUILLIE, UWE EHRENSTEIN, JEAN-PHILIPPE LAVAL
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- 02 June 2011, pp. 205-240
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A direct numerical simulation of a turbulent channel flow with a lower curved wall is performed at Reynolds number Reτ ≈ 600. Low-speed streak structures are extracted from the turbulent flow field using methods known as skeletonization in image processing. Individual streaks in the wall-normal plane averaged in time and superimposed to the mean streamwise velocity profile are used as basic states for a linear stability analysis. Instability modes are computed at positions along the lower and upper wall and the instability onset is shown to coincide with the strong production peaks of turbulent kinetic energy near the maximum of pressure gradient on both the curved and the flat walls. The instability modes are spanwise-symmetric (varicose) for the adverse pressure gradient streak base flows with wall-normal inflection points, when the total average of the detected streaks is considered. The size and shape of the counter-rotating streamwise vortices associated with the instability modes are shown to be reminiscent of the coherent vortices emerging from the streak skeletons in the direct numerical simulation. Conditional averages of streaks have also been computed and the distance of the streak's centre from the wall is shown to be an essential parameter. For the upper-wall weak pressure gradient flow, spanwise-antisymmetric (sinuous) instability modes become unstable when sets of highest streaks are considered, whereas varicose modes dominate for the streaks closest to the wall.
H2 optimal actuator and sensor placement in the linearised complex Ginzburg–Landau system
- KEVIN K. CHEN, CLARENCE W. ROWLEY
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- 20 June 2011, pp. 241-260
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The linearised complex Ginzburg–Landau equation is a model for the evolution of small fluid perturbations, such as in a bluff body wake. By implementing actuators and sensors and designing an H2 optimal controller, we control a supercritical, infinite-domain formulation of this system. We seek the optimal actuator and sensor placement that minimises the H2 norm of the controlled system, from flow disturbances and sensor noise to a cost on the perturbation and input magnitudes. We formulate the gradient of the H2 squared norm with respect to the actuator and sensor placements and iterate towards the optimal placement. When stochastic flow disturbances are present everywhere in the spatial domain, it is optimal to place the actuator just upstream of the origin and the sensor just downstream. With pairs of actuators and sensors, it is optimal to place each actuator slightly upstream of each corresponding sensor, and scatter the pairs throughout the spatial domain. When disturbances are only introduced upstream, the optimal placement shifts upstream as well. Global mode and Gramian analyses fail to predict the optimal placement; they produce H2 norms about five times higher than at the true optimum. The wavemaker region is a better guess for the optimal placement.
A study of Mach wave radiation using active control
- M. KEARNEY-FISCHER, J.-H. KIM, M. SAMIMY
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- 13 June 2011, pp. 261-292
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Mach wave radiation is one of the better understood sources of jet noise. However, the exact conditions of its onset are difficult to determine and the literature to date typically explores Mach wave radiation well above its onset conditions. In order to determine the conditions for the onset of Mach wave radiation and to explore its behaviour during onset and beyond, three ideally expanded jets with Mach numbers Mj = 0.9, 1.3 and 1.65 and stagnation temperature ratios ranging over To/T∞ = 1.0–2.5 (acoustic Mach number 0.83–2.10) were used. Data are collected using a far-field microphone array, schlieren imaging and streamwise two-component particle image velocimetry. Using arc filament plasma actuators to force the jet provides an unprecedented tool for detailed examination of Mach wave radiation. The response of the jet to various forcing parameters (combinations of one azimuthal mode m = 0, 1 and 3 and one Strouhal number StDF = 0.09–3.0) is explored. Phase-averaged schlieren images clearly show the onset and evolution of Mach wave radiation in response to both changes in the jet operating conditions and forcing parameters. It is observed that Mach wave radiation is initiated as a coalescing of the near-field hydrodynamic pressure fluctuations in the immediate vicinity of the large-scale structures. As the jet exit velocity increases, the hydrodynamic pressure fluctuations coalesce, first into a curved wavefront, then flatten into the conical wavefronts commonly associated with Mach wave radiation. The results show that the largest and most coherent structures (e.g. forcing with m = 0 and StDF ~ 0.3) produce the strongest Mach wave radiation. Conversely, Mach wave radiation is weakest when the structures are the least coherent (e.g. forcing with m = 3 and StDF > 1.5).
A general analysis for the electrohydrodynamic instability of stratified immiscible fluids
- J. ZHANG, J. D. ZAHN, H. LIN
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- 29 June 2011, pp. 293-310
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A general solution approach for the electrohydrodynamic instability of stratified immiscible fluids is presented. The problems of two and three fluid layers subject to normal electric fields are analysed. Analytical solutions are obtained by employing the transfer relations (Melcher 1981 Continuum Electromechanics. MIT Press) relating the disturbance stresses to the flow variables at the interface(s). This approach provides a convenient alternative to the direct solution of the linearized problem. The results assume a general format. Both new dispersion relations and those from various previous works are shown to be special cases when proper simplifications are considered. As a specific example, the instability behaviour of a three-layer channel flow is investigated in detail using this framework. This work provides a unifying method to treat a generic class of instability problems.
Bend theory of river meanders with spatial width variations
- ROSSELLA LUCHI, GUIDO ZOLEZZI, MARCO TUBINO
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- 20 June 2011, pp. 311-339
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The present work revisits the classical, uniform-width bend theory with the aim to understand whether and how spatial width oscillations can affect the process of linear bend stability that initiates meander planform evolution. Although longitudinal oscillations of channel width are common along many meandering streams, little investigation of their properties and dynamic effects has been pursued so far. The theory therefore accounts for width variations as a geometrical forcing in a depth-averaged model of meander morphodynamics by assuming the potential interaction with the classical curvature forcing effect. A first quantification of width variations is made by referring to a freely evolving meandering river, which shows that the dimensionless amplitude of width variations is a ‘small’ parameter with comparable magnitude to that of curvature variations, thus suggesting the use of a two-parameter perturbation expansion. Moreover, it is reasonable to assume that channel width oscillates in space with a double frequency relative to curvature, which implies that one nonlinear interaction between the two forcing effects is enough to reproduce the effect of spatial width variations on the process of bend stability. Overall, width variations consistently promote the instability of shorter bends with respect to meanders with uniform width: on average, this predicted tendency is supported by analysis of field data referring to hundreds of natural meander bends. The effect on meander wavelength selection depends on the location of the widest section relative to the bend apex. Under typical formative conditions of gravel-bed rivers, with large-enough channel aspect ratios, two distinct most unstable longitudinal modes develop. Such behaviour is absent when the width is uniform, and suggests a mechanistic interpretation for the reach-scale occurrence of chute cutoffs that can be observed more frequently in wider-at-bends than in equiwidth meandering channels.
Experiments and modelling of premixed laminar stagnation flame hydrodynamics
- JEFFREY M. BERGTHORSON, SEAN D. SALUSBURY, PAUL E. DIMOTAKIS
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- 23 June 2011, pp. 340-369
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The hydrodynamics of a reacting impinging laminar jet, or stagnation flame, is studied experimentally and modelled using large activation energy asymptotic models and numerical simulations. The jet-wall geometry yields a stable, steady flame and allows for precise measurement and specification of all boundary conditions on the flow. Laser diagnostic techniques are used to measure velocity and CH radical profiles. The axial velocity profile through a premixed stagnation flame is found to be independent of the nozzle-to-wall separation distance at a fixed nozzle pressure drop, in accord with results for non-reacting impinging laminar jet flows, and thus the strain rate in these flames is only a function of the pressure drop across the nozzle. The relative agreement between the numerical simulations and experiment using a particular combustion chemistry model is found to be insensitive to both the strain rate imposed on the flame and the relative amounts of oxygen and nitrogen in the premixed gas, when the velocity boundary conditions on the simulations are applied in a manner consistent with the formulation of the streamfunction hydrodynamic model. The analytical model predicts unburned, or reference, flame speeds that are slightly lower than the detailed numerical simulations in all cases and the observed dependence of this reference flame speed on strain rate is stronger than that predicted by the model. Experiment and simulation are in excellent agreement for near-stoichiometric methane–air flames, but deviations are observed for ethylene flames with several of the combustion models used. The discrepancies between simulation and experimental profiles are quantified in terms of differences between measured and predicted reference flame speeds, or position of the CH-profile maxima, which are shown to be directly correlated. The direct comparison of the measured and simulated reference flame speeds, ΔSu, can be used to infer the difference between the predicted flame speed of the combustion model employed and the true laminar flame speed of the mixture, ΔSOf, i.e. ΔSu=ΔSOf, consistent with recently proposed nonlinear extrapolation techniques.
Unsteady boundary-layer transition in low-pressure turbines
- JOHN D. COULL, HOWARD P. HODSON
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- Published online by Cambridge University Press:
- 01 July 2011, pp. 370-410
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This paper examines the transition process in a boundary layer similar to that present over the suction surfaces of aero-engine low-pressure (LP) turbine blades. This transition process is of significant practical interest since the behaviour of this boundary layer largely determines the overall efficiency of the LP turbine. Modern ‘high-lift’ blade designs typically feature a closed laminar separation bubble on the aft portion of the suction surface. The size of this bubble and hence the inefficiency it generates is controlled by the transition between laminar and turbulent flow in the boundary layer and separated shear layer. The transition process is complicated by the inherent unsteadiness of the multi-stage machine: the wakes shed by one blade row convect through the downstream blade passages, periodically disturbing the boundary layers. As a consequence, the transition to turbulence is multi-modal by nature, being promoted by periodic and turbulent fluctuations in the free stream and the inherent instabilities of the boundary layer. Despite many studies examining the flow behaviour, the detailed physics of the unsteady transition phenomena are not yet fully understood. The boundary-layer transition process has been studied experimentally on a flat plate. The opposing test-section wall was curved to impose a streamwise pressure distribution typical of modern high-lift LP turbines over the flat plate. The presence of an upstream blade row has been simulated by a set of moving bars, which shed wakes across the test section inlet. Further upstream, a grid has been installed to elevate the free-stream turbulence to a level believed to be representative of multi-stage LP turbines. Extensive particle imaging velocimetry (PIV) measurements have been performed on the flat-plate boundary layer to examine the flow behaviour. In the absence of the incoming bar wakes, the grid-generated free-stream turbulence induces relatively weak Klebanoff streaks in the boundary layer which are evident as streamwise streaks of low-velocity fluid. Transition is promoted by the streaks and by the inherent inflectional (Kelvin–Helmholtz (KH)) instability of the separation bubble. In unsteady flow, the incoming bar wakes generate stronger Klebanoff streaks as they pass over the leading edge, which convect downstream at a fraction of the free-stream velocity and spread in the streamwise direction. The region of amplified streaks convects in a similar manner to a classical turbulent spot: the leading and trailing edges travel at around 88% and 50% of the free-stream velocity, respectively. The strongest disturbances travel at around 70% of the free-stream velocity. The wakes induce a second type of disturbance as they pass over the separation bubble, in the form of short-span KH structures. Both the streaks and the KH structures contribute to the early wake-induced transition. The KH structures are similar to those observed in the simulation of separated flow transition with high free-stream turbulence by McAuliffe & Yaras (ASME J. Turbomach., vol. 132, no. 1, 2010, 011004), who observed that these structures originated from localised instabilities of the shear layer induced by Klebanoff streaks. In the current measurements, KH structures are frequently observed directly under the path of the wake. The wake-amplified Klebanoff streaks cannot affect the generation of these structures since they do not arrive at the bubble until later in the wake cycle. Rather, the KH structures arise from an interaction between the flow disturbances in the wake and localised instabilities in the shear layer, which are caused by the weak Klebanoff streaks induced by the grid turbulence. The breakdown of the KH structures to small-scale turbulence occurs a short time after the wake has passed over the bubble, and is largely driven by the arrival of the wake-amplified Klebanoff streaks from the leading edge. During this process, the re-attachment location moves rapidly upstream. The minimum length of the bubble occurs when the strongest wake-amplified Klebanoff streaks arrive from the leading edge; these structures travel at around 70% of the free-stream velocity. The bubble remains shorter than its steady-flow length until the trailing edge of the wake-amplified Klebanoff streaks, travelling at 50% of the free-stream velocity, convect past. After this time, the reattachment location moves aft on the surface as a consequence of a calmed flow region which follows behind the wake-induced turbulence.
Three-dimensional instabilities in the boundary-layer flow over a long rectangular plate
- HEMANT K. CHAURASIA, MARK C. THOMPSON
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- 16 June 2011, pp. 411-433
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A detailed numerical study of the separating and reattaching flow over a square leading-edge plate is presented, examining the instability modes governing transition from two- to three-dimensional flow. Under the influence of background noise, experiments show that the transition scenario typically is incompletely described by either global stability analysis or the transient growth of dominant optimal perturbation modes. Instead two-dimensional transition effectively can be triggered by the convective Kelvin–Helmholtz (KH) shear-layer instability; although it may be possible that this could be described alternatively in terms of higher-order optimal perturbation modes. At least in some experiments, observed transition occurs by either: (i) KH vortices shedding downstream directly and then almost immediately undergoing three-dimensional transition or (ii) at higher Reynolds numbers, larger vortical structures are shed that are also three-dimensionally unstable. These two paths lead to distinctly different three-dimensional arrangements of vortical flow structures. This paper focuses on the mechanisms underlying these three-dimensional transitions. Floquet analysis of weakly periodically forced flow, mimicking the observed two-dimensional quasi-periodic base flow, indicates that the two-dimensional vortex rollers shed from the recirculation region become globally three-dimensionally unstable at a Reynolds number of approximately 380. This transition Reynolds number and the predicted wavelength and flow symmetries match well with those of the experiments. The instability appears to be elliptical in nature with the perturbation field mainly restricted to the cores of the shed rollers and showing the spatial vorticity distribution expected for that instability type. Indeed an estimate of the theoretical predicted wavelength is also a good match to the prediction from Floquet analysis and theoretical estimates indicate the growth rate is positive. Fully three-dimensional simulations are also undertaken to explore the nonlinear development of the three-dimensional instability. These show the development of the characteristic upright hairpins observed in the experimental dye visualisations. The three-dimensional instability that manifests at lower Reynolds numbers is shown to be consistent with an elliptic instability of the KH shear-layer vortices in both symmetry and spanwise wavelength.
Quasi-static magnetohydrodynamic turbulence at high Reynolds number
- B. FAVIER, F. S. GODEFERD, C. CAMBON, A. DELACHE, W. J. T. BOS
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- 21 June 2011, pp. 434-461
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We analyse the anisotropy of homogeneous turbulence in an electrically conducting fluid submitted to a uniform magnetic field, for low magnetic Reynolds number, in the quasi-static approximation. We interpret contradictory earlier predictions between linearized theory and simulations: in the linear limit, the kinetic energy of transverse velocity components, normal to the magnetic field, decays faster than the kinetic energy of the axial component, along the magnetic field (Moffatt, J. Fluid Mech., vol. 28, 1967, p. 571); whereas many numerical studies predict a final state characterized by dominant energy of transverse velocity components. We investigate the corresponding nonlinear phenomenon using direct numerical simulation (DNS) of freely decaying turbulence, and a two-point statistical spectral closure based on the eddy-damped quasi-normal Markovian (EDQNM) model. The transition from the three-dimensional turbulent flow to a ‘two-and-a-half-dimensional’ flow (Montgomery & Turner, Phys. Fluids, vol. 25, 1982, p. 345) is a result of the combined effects of short-time linear Joule dissipation and longer time nonlinear creation of polarization anisotropy. It is this combination of linear and nonlinear effects which explains the disagreement between predictions from linearized theory and results from numerical simulations. The transition is characterized by the elongation of turbulent structures along the applied magnetic field, and by the strong anisotropy of directional two-point correlation spectra, in agreement with experimental evidence. Inertial equatorial transfers in both DNS and the model are presented to describe in detail the most important equilibrium dynamics. Spectral scalings are maintained in high-Reynolds-number turbulence attainable only with the EDQNM model, which also provides simplified modelling of the asymptotic state of quasi-static magnetohydrodynamic (MHD) turbulence.
Experimental study of the initial stages of wind waves' spatial evolution
- DAN LIBERZON, LEV SHEMER
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- 24 June 2011, pp. 462-498
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Despite a significant progress and numerous publications over the last few decades a comprehensive understanding of the process of waves' excitation by wind still has not been achieved. The main goal of the present work was to provide as comprehensive as possible set of experimental data that can be quantitatively compared with theoretical models. Measurements at various air flow rates and at numerous fetches were carried out in a small scale, closed-loop, 5 m long wind wave flume. Mean airflow velocity and fluctuations of the static pressure were measured at 38 vertical locations above the mean water surface simultaneously with determination of instantaneous water surface elevations by wave gauges. Instantaneous fluctuations of two velocity components were recorded for all vertical locations at a single fetch. The water surface drift velocity was determined by the particle tracking velocimetry (PTV) method. Evaluation of spatial growth rates of waves at various frequencies was performed using wave gauge records at various fetches. Phase relations between various signals were established by cross-spectral analysis. Waves' celerities and pressure fluctuation phase lags relative to the surface elevation were determined. Pressure values at the water surface were determined by extrapolating the measured vertical profile of pressure fluctuations to the mean water level and used to calculate the form drag and consequently the energy transfer rates from wind to waves. Directly obtained spatial growth rates were compared with those obtained from energy transfer calculations, as well as with previously available data.
On the onset of dissipation thermal instability for the Poiseuille flow of a highly viscous fluid in a horizontal channel
- A. BARLETTA, M. CELLI, D. A. NIELD
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- 20 June 2011, pp. 499-514
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The thermal instability of the plane Poiseuille flow as a consequence of the effect of viscous dissipation is investigated. No external temperature difference is assumed in the environment; the lower boundary is considered adiabatic, while the upper boundary is isothermal. Thus, the sole cause of the unstable thermal stratification is the flow rate, through the volumetric heating induced by the viscous dissipation. A linear stability analysis is carried out numerically by the analysis of normal modes perturbing the basic flow with different inclinations. The study of cases with different Prandtl numbers and Gebhart numbers suggests that the most unstable perturbations are the longitudinal rolls, namely the normal modes with a wave vector perpendicular to the basic flow direction. A possible comparison with the hydrodynamic instability of the plane Poiseuille flow, described by the Orr–Sommerfeld eigenvalue problem is proposed. This comparison, when referred to high values of the Prandtl number, reveals that the dissipation instability may be effective at a Reynolds number smaller than that needed for the onset of the hydrodynamic instability.
Transitions to three-dimensional flows in a cylinder driven by oscillations of the sidewall
- C. PANADES, F. MARQUES, J. M. LOPEZ
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- 24 June 2011, pp. 515-536
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The transition from two-dimensional to three-dimensional flows in a finite circular cylinder driven by an axially oscillating sidewall is explored in detail. The complete symmetry group of this flow, including a spatio-temporal symmetry related to the oscillating sidewall, is Z2 × O(2). Previous studies in flows with the same symmetries, such as symmetric bluff-body wakes and periodically forced rectangular cavities, were unable to obtain the theoretically predicted bifurcation to modulated travelling waves. In the simpler cylindrical geometry, where the azimuthal direction is physically periodic, we have found these predicted modulated travelling waves as stable fully saturated nonlinear solutions for the first time. A careful analysis of the base states and their linear stability identifies different parameter regimes where three-dimensional states are either synchronous with the forcing or quasi-periodic, corresponding to different symmetry-breaking processes. These results are in good agreement with theoretical predictions and previous results in similar flows. These different regimes are separated by three codimension-two bifurcation points that are yet to be fully analysed theoretically. Finally, the saturated nonlinear states and their properties in different parameter regimes are analysed.