# Showing papers in "Journal of Fluid Mechanics in 1983"

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TL;DR: In this paper, the velocity distribution and reattachment length of a single backward-facing step mounted in a two-dimensional channel were measured using laser-Doppler measurements.

Abstract: Laser-Doppler measurements of velocity distribution and reattachment length are reported downstream of a single backward-facing step mounted in a two-dimensional channel. Results are presented for laminar, transitional and turbulent flow of air in a Reynolds-number range of 70 < Re < 8000. The experimental results show that the various flow regimes are characterized by typical variations of the separation length with Reynolds number. The reported laser-Doppler measurements do not only yield the expected primary zone of recirculating flow attached to the backward-facing step but also show additional regions of flow separation downstream of the step and on both sides of the channel test section. These additional separation regions have not been previously reported in the literature.Although the high aspect ratio of the test section (1:36) ensured that the oncoming flow was fully developed and two-dimensional, the experiments showed that the flow downstream of the step only remained two-dimensional at low and high Reynolds numbers.The present study also included numerical predictions of backward-facing step flow. The two-dimensional steady differential equations for conservation of mass and momentum were solved. Results are reported and are compared with experiments for those Reynolds numbers for which the flow maintained its two-dimensionality in the experiments. Under these circumstances, good agreement between experimental and numerical results is obtained.

1,637 citations

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TL;DR: In this article, the authors focus on an idealized granular material comprised of identical, smooth, imperfectly elastic, spherical particles which is flowing at such a density and is being deformed at a rate that particles interact only through binary collisions with their neighbours.

Abstract: We focus attention on an idealized granular material comprised of identical, smooth, imperfectly elastic, spherical particles which is flowing at such a density and is being deformed at such a rate that particles interact only through binary collisions with their neighbours. Using general forms of the probability distribution functions for the velocity of a single particle and for the likelihood of binary collisions, we derive local expressions for the balance of mass, linear momentum and fluctuation kinetic energy, and integral expressions for the stress, energy flux and energy dissipation that appear in them. We next introduce simple, physically plausible, forms for the probability densities which contain as parameters the mean density, the mean velocity and the mean specific kinetic energy of the velocity fluctuations. This allows us to carry out the integrations for the stress, energy flux and energy dissipation and to express these in terms of the mean fields. Finally, we determine the behaviour of these fields as solutions to the balance laws. As an illustration of this we consider the shear flow maintained between two parallel horizontal plates in relative motion.

1,230 citations

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TL;DR: In this paper, an experimental investigation of transport processes in the near wake of a circular cylinder at a Reynolds number of 140000 was performed using X-array hot-wire probes mounted on a pair of whirling arms, which increases the relative velocity component along the probe axis and decreases the relative flow angle to usable values in regions where fluctuations in flow velocity and direction are large.

Abstract: This paper describes an experimental investigation of transport processes in the near wake of a circular cylinder at a Reynolds number of 140000. The flow in the first eight diameters of the wake was measured using X-array hot-wire probes mounted on a pair of whirling arms. This flying-hot-wire technique increases the relative velocity component along the probe axis and thus decreases the relative flow angle to usable values in regions where fluctuations in flow velocity and direction are large. One valuable fringe benefit of the technique is that rotation of the arms in a uniform flow applies a wide range of relative flow angles to the X-arrays, making them inherently self-calibrating in pitch. An analog circuit was used to generate an intermittency signal, and a fast surface-pressure sensor was used to generate a phase signal synchronized with the vortex-shedding process. The phase signal allowed sorting of the velocity data into 16 populations, each having essentially constant phase. An ensemble average for each population yielded a sequence of pictures of the instantaneous mean flow field, with the vortices frozen as they would be in a photograph. In addition to globally averaged data for velocity and stress, the measurements yield non-steady mean data (in the sense of an average a t constant phase) for velocity, intermittency, vorticity, stress and turbulent-energy production as a function of phase for the first eight diameters of the near wake. The stresses were resolved into a contribution from the periodic motion and a contribution from the random motion at constant phase. The two contributions are found to have comparable amplitudes but quite different geometries, and the time average of their sum (the conventional global Reynolds stress) therefore has a quite-complex structure. The non-steady mean-vorticity field is obtained with good resolution as the curl of the non-steady mean-velocity field. Less than half of the shed circulation appears in the vortices, and there is a slow decay of this circulation for each shed vortex as it moves downstream. In the discussion, considerable emphasis is put on the topology of the non-steady mean flow, which emerges as a pattern of centres and saddles in a frame of reference moving with the eddies. The kinematics of the vortex-formation process are described in terms of the formation and evolution of saddle points between vortices in the first few diameters of the near wake. One important conclusion is that a substantial part of the turbulence production is concentrated near the saddles and that the mechanism of turbulence production is probably vortex stretching at intermediate scales. Entrainment is also found to be closely associated with saddles and to be concentrated near the upstream-facing interface of each vortex.

920 citations

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TL;DR: In this paper, the behavior of granular material in motion is studied from a continuum point of view, where individual grains are treated as the "molecules" of a granular "fluid" and the resulting equations are nonlinear and coupled.

Abstract: The behaviour of granular material in motion is studied from a continuum point of view. Insofar as possible, individual grains are treated as the ‘molecules’ of a granular ‘fluid’. Besides the obvious contrast in shape, size and mass, a key difference between true molecules and grains is that collisions of the latter are inevitably inelastic. This, together with the fact that the fluctuation velocity may be comparable to the flow velocity, necessitates explicit incorporation of the energy equation, in addition to the continuity and momentum equations, into the theoretical description. Simple ‘microscopic’ kinetic models are invoked for deriving expressions for the ‘coefficients’ of viscosity, thermal diffusivity and energy absorption due to collisions. The ‘coefficients’ are not constants, but are functions of the local state of the medium, and therefore depend on the local ‘temperature’ and density. In general the resulting equations are nonlinear and coupled. However, in the limit s « d, where s is the mean separation between neighbouring grain surfaces and d is a grain diameter, the above equations become linear and can be solved analytically. An important dependent variable, in this formulation, in addition to the flow velocity u, is the mean random fluctuation (‘thermal’) velocity v of an individual grain. With a sufficient flux of energy supplied to the system through the boundaries of the container, v can remain non-zero even in the absence of flow. The existence of a non-uniform v is the means by which energy can be ‘conducted’ from one part of the system to another. Because grain collisions are inelastic, there is a natural (damping) lengthscale, governed by the value of d, which strongly influences the functional dependence of v on position. Several illustrative examples of static (u = 0) systems are solved. As an example of grain flow, various Couette-type problems are solved analytically. The pressure, shear stress, and ‘thermal’ velocity function v are all determined by the relative plate velocity U (and the boundary conditions). If v is set equal to zero at both plates, the pressure and stress are both proportional to U^2, i.e. the fluid is non-Newtonian. However, if sufficient energy is supplied externally through the walls (v ≠ 0 there), then the forces become proportional to the first power of U. Some examples of Couette flow are given which emphasize the large effect on the grain system properties of even a tiny amount of inelasticity in grain–grain collisions. From these calculations it is suggested that, for the case of Couette flow, the flow of sand is supersonic over most of the region between the confining plates.

845 citations

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TL;DR: In this paper, the mechanisms of displacement of one fluid by another were investigated in an etched network, where both fluids are simultaneously present in a duct, the wetting fluid remaining in the extreme corners of the cross-section.

Abstract: The mechanisms of displacement of one fluid by another are investigated in an etched network.Experiments show that both fluids are simultaneously present in a duct, the wetting fluid remaining in the extreme corners of the cross-section. Calculation of displacement pressures are in good agreement with experiments for drainage, imbibition and removal of blobs. The results may be related to some flow behaviour exhibited in porous media.

773 citations

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TL;DR: In this article, the dynamics of inviscid and viscous Taylor-Green (TG) vortex flows are investigated by both direct spectral numerical solution of the Navier-Stokes equations and by power-series analysis in time.

Abstract: The dynamics of both the inviscid and viscous Taylor–Green (TG) three-dimensional vortex flows are investigated. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier–Stokes equations (with up to 256 3 modes) and by power-series analysis in time. The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation in the form of vortex sheets located near the walls of this box. The flow is smooth but has complex-space singularities within a distance $\hat{\delta}(t)$ of real (physical) space which give rise to an exponential tail in the energy spectrum. It is found that $\hat{\delta}(t)$ decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place at later times, possibly leading to a real singularity ( $\hat{\delta}(t) = 0$ ) at a finite time. These direct integration results are consistent with new temporal power-series results that extend the Morf, Orszag & Frisch (1980) analysis from order t 44 to order t 80 . Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis. The viscous dynamics (decay) have been studied for Reynolds numbers R (based on an integral scale) up to 3000 and beyond the time t max at which the maximum energy dissipation is achieved. Early-time, high- R dynamics are essentially inviscid and laminar. The inviscidly formed vortex sheets are observed to roll up and are then subject to instabilities accompanied by reconnection processes which make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Near t max the small scales of the flow are nearly isotropic provided that R [gsim ] 1000. Various features characteristic of fully developed turbulence are observed near t max when R = 3000 and R λ = 110:
a k − n inertial range in the energy spectrum is obtained with n ≈ 1.6–2.2 (in contrast with a much steeper spectrum at earlier times); th energy dissipation has considerable spatial intermittency; its spectrum has a k −1+μ inertial range with the codimension μ ≈ 0.3−0.7. Skewness and flatness results are also presented.

684 citations

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TL;DR: In this article, a high-speed video system and hydrogen bubble-wire flow visualization was used to investigate the characteristics of low-speed streaks which occur in the near-wall region of turbulent boundary layers.

Abstract: Employing a high-speed video system and hydrogen bubble-wire flow visualization, the characteristics of the low-speed streaks which occur in the near-wall region of turbulent boundary layers have been examined for a Reynolds-number range of 740 [les ] Reθ < 5830. The results indicate that the statistics of non-dimensional spanwise streak spacing are essentially invariant with Reynolds number, exhibiting consistent values of and remarkably similar probability distributions conforming to lognormal behaviour. Further studies show that streak spacing increases with distance from the wall owing to a merging and intermittency process which occurs for y+ [simg ] 5. An additional observation is that, although low-speed streaks are not fixed in time and space, they demonstrate a tremendous persistence, often maintaining their integrity and reinforcing themselves for time periods up to an order of magnitude longer than the observed bursting times associated with wall region turbulence production. A mechanism for the formation of low-speed streaks is suggested which may explain both the observed merging behaviour and the streak persistence.

621 citations

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TL;DR: In this paper, a steady shear flow is set up in a planar liquid layer bounded below by a rigid plate, and above by an interface with a passive gas, by imposing a temperature gradient along the layer and driving the motion by thermocapillarity.

Abstract: A steady shear flow is set up in a planar liquid layer bounded below by a rigid plate, and above by an interface with a passive gas, by imposing a temperature gradient along the layer and driving the motion by thermocapillarity. The thermal-convective instabilities to which this dynamic state is susceptible include stationary longitudinal rolls involving the classical Marangoni instability, and unsteady hydrothermal waves, whose new instability mechanism derives its energy from the horizontal temperature gradients. The thermal stability characteristics for liquid layers with and without return flow profiles are presently given as functions of the Prandtl number for the liquid and the Biot number for the interface, and comparisons are made with available experimental observations.

600 citations

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TL;DR: In this article, the steady and unsteady forces were measured by means of a piezobalance, which features a high natural frequency, low interferences and a large dynamic range.

Abstract: Force measurements were conducted in a pressurized wind tunnel from subcritical up to transcritical Reynolds numbers 2.3 × 104 [les ] Re [les ] 7.1 × 106 without changing the experimental arrangement. The steady and unsteady forces were measured by means of a piezobalance, which features a high natural frequency, low interferences and a large dynamic range. In the critical Reynolds-number range, two discontinuous transitions were observed, which can be interpreted as bifurcations at two critical Reynolds numbers. In both cases, these transitions are accompanied by critical fluctuations, symmetry breaking (the occurrence of a steady lift) and hysteresis. In addition, both transitions were coupled with a drop of the CD value and a jump of the Strouhal number. Similar phenomena were observed in the upper transitional region between the super- and the transcritical Reynolds-number ranges. The transcritical range begins at about Re ≈ 5 × 106, where a narrow-band spectrum is formed with Sr(Re = 7.1 × 106) = 0.29.

555 citations

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TL;DR: In this paper, it was shown that two-dimensional, finite amplitude waves are exponentially unstable to infinitimal three-dimensional disturbances, and that the threedimensional instability requires that a threshold 2-dimensional amplitude be achieved.

Abstract: The present analysis of a secondary instability in a wide class of wall-bounded parallel shear flows indicates that two-dimensional, finite amplitude waves are exponentially unstable to infinitessimal three-dimensional disturbances. The instability appears to be the prototype of transitional instability in such flows as Poiseuille flow, Couette flow, and flat plate boundary layers, in that it has the convective time scales observed in the typical transitions. The energetics and vorticity dynamics of the instability are discussed, and it is shown that the two-dimensional perturbation without directly providing energy to the disturbance. The three-dimensional instability requires that a threshold two-dimensional amplitude be achieved. It is found possible to identify experimental features of transitional spot structure with aspects of the nonlinear two-dimensional/linear three-dimensional instability.

539 citations

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TL;DR: In this paper, the cross-correlations between surface-pressure and velocity fluctuations are found to be useful for the study of large-scale vortex structure in the separation bubble formed along the sides of a blunt flat plate with right-angled corners.

Abstract: Flow in the separation bubble formed along the sides of a blunt flat plate with right-angled corners has been studied in terms of extensive single- and two-point measurements of velocity and surface-pressure fluctuations. The cross-correlations between the surface-pressure and velocity fluctuations are found to be useful for the study of large-scale vortex structure in the bubble. Large-scale vortices are shed downstream from the separation bubble with a frequency of about 0.6U∞/xR, where U∞ is the approaching velocity and xR is the time-mean length of the bubble. On top of this regular vortex shedding, there exists a large-scale unsteadiness in the bubble. Vortices which are much larger than the regular vortices are shed with frequencies less than about 0.2U∞/xR. The large-scale unsteadiness is accompanied by enlargement and shrinkage of the bubble and also by a flapping motion of the shear layer near the separation line. The intermittent nature of the flow in the bubble is clarified in some detail. The distributions of the cross-correlations between the pressure and velocity fluctuations demonstrate the vortex structure in the reattaching zone. The longitudinal distance between the vortices is estimated to be (0.7–0.8) xR and their convection velocity is about 0.5U∞ near the reattachment line. The cross-correlations also suggest the existence of a longitudinal counter-rotating system in the bubble. The distance between the axes of the rotation is of the order of 0.6xR. Variations of timescales, lengthscales and phase velocities of the vortices are presented and discussed.

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TL;DR: In this paper, a finite volume of homogeneous fluid was released instantaneously into another fluid of slightly lower density and the experiments were performed in a channel of rectangular cross-section, and the two fluids used were salt water and fresh water.

Abstract: Results of laboratory experiments are presented in which a finite volume of homogeneous fluid was released instantaneously into another fluid of slightly lower density The experiments were performed in a channel of rectangular cross-section, and the two fluids used were salt water and fresh water As previously reported, the resulting gravity current, if viscous effects are negligible, passes through two distinct phases: an initial adjustment phase, during which the initial conditions are important, and an eventual self-similar phase, in which the front speed decreases as t−1/3 (where t is the time measured from release) The experiments reported herein were designed to emphasize the inviscid motion From our observations we argue that the current front moves steadily in the first phase, and that the transition to the inviscid self-similar phase occurs when a disturbance generated at the endwall (or plane of symmetry) overtakes the front If the initial depth of the heavy fluid is equal to or slightly less than the total depth of the fluid in the channel, the disturbance has the appearance of an internal hydraulic drop Otherwise, the disturbance is a long wave of depression Measurements of the duration of the initial phase and of the speed and depth of the front during this phase are presented as functions of the ratio of the initial heavy fluid depth to the total fluid depth These measurements are compared with numerical solutions of the shallow-water equations for a two-layer fluid

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TL;DR: In this paper, the effect of the phase velocity of the running disturbance on the aspect ratio of the column was investigated and a unique non-dimensional wavenumber ≈ 2·2 (near Mac) of the distortion was found.

Abstract: In liquid columns (Prandtl number 8·9) with free cylindrical surface heated from above, strong thermocapillary convection (TC) has been observed. Stationary thermocapillary convection exists in the form of a single axially symmetric roll bound to the free surface. For aspect ratios l/a < 1 the radial extension of the roll equals the zone length. The stream velocities and the temperature distribution were measured.The influence of buoyant forces due to horizontal temperature gradients in the experiments was also studied. Buoyant forces become obvious for a contaminated free surface and in bulk regions far from the cylinder surface.The thermocapillary convection shows a transition to time-dependent oscillatory motion when a critical Marangoni number Mac is exceeded. A unique Mac = 7 × 103 has been found for zones with lengths l < 3·5 mm. The oscillatory state of thermocapillary convection has experimentally been proved to be a distortion of the laminar state in form of a wave travelling in the azimuthal direction. A unique non-dimensional wavenumber ≈ 2·2 (near Mac) of the distortion has been found. The non-dimensional frequency of the temperature oscillations is independent of zone size if the aspect ratio is held constant. However, the non-dimensional frequency of temperature oscillations increases linearly with the aspect ratio of the zone. This result is interpreted as a dependence of the phase velocity of the running disturbance on the aspect ratio.

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TL;DR: In this article, the shape of the largest drop or bubble that can remain attached to an inclined solid surface and the shape and speed at which it moves along the surface when these conditions are exceeded are investigated.

Abstract: It is common knowledge that relatively small drops or bubbles have a tendency to stick to the surfaces of solids. Two specific problems are investigated: the shape of the largest drop or bubble that can remain attached to an inclined solid surface; and the shape and speed at which it moves along the surface when these conditions are exceeded. The slope of the fluid-fluid interface relative to the surface of the solid is assumed to be small, making it possible to obtain results using analytic techniques. It is shown that from both a physical and mathematical point of view contact-angle hysteresis, i.e. the ability of the position of the contact line to remain fixed as long as the value of the contact angle θ lies within the interval θR [les ] θ [les ] θA, where θA [nequiv ] θR, emerges as the single most important characteristic of the system.

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TL;DR: The growth of Gortler vortices in boundary layers on concave walls is investigated in this article, and it is shown that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the gortler problem except for asymptotically small wavelengths.

Abstract: The Growth of Gortler vortices in boundary layers on concave walls is investigated. It is shown that for vortices of wavelength comparable to the boundary-layer thickness the appropriate linear stability equations cannot be reduced to ordinary differential equations. The partial differential equations governing the linear stability of the flow are solved numerically, and neutral stability is defined by the condition that a dimensionless energy function associated with the flow should have a maximum or minimum when plotted as a function of the downstream variable X. The position of neutral stability is found to depend on how and where the boundary layer is perturbed, so that the concept of a unique neutral curve so familiar in hydrodynamic-stability theory is not tenable in the Gortler problem, except for asymptotically small wavelengths. The results obtained are compared with previous parallel-flow theories and the small-wavelength asymptotic results of Hall (1982a, b), which are found to be reasonably accurate even for moderate values of the wavelength. The parallel-flow theories of the growth of Gortler vortices are found to be irrelevant except for the small-wavelength limit. The main deficiency of the parallel-flow theories is shown to arise from the inability of any ordinary differential approximation to the full partial differential stability equations to describe adequately the decay of the vortex at the edge of the boundary layer. This deficiency becomes intensified as the wavelength of the vortices increases and is the cause of the wide spread of the neutral curves predicted by parallel-flow theories. It is found that for a wall of constant radius of curvature a given vortex imposed on the flow can grow for at most a finite range of values of X. This result is entirely consistent with, and is explicable by the asymptotic results of, Hall (1982a).

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TL;DR: In this paper, the authors considered the linear stability of the cocurrent flow of two fluids of different viscosity in an infinite region and solved the problem using both numerical and asymptotic techniques.

Abstract: We consider the linear stability of the cocurrent flow of two fluids of different viscosity in an infinite region (the viscous analogue of the classical Kelvin-Helmholtz problem). Attention is confined to the simplest case, Couette flow, and we solve the problem using both numerical and asymptotic techniques. We find that the flow is always unstable (in the absence of surface tension). The instability arises at the interface between the two fluids and occurs for short wavelengths, when viscosity rather than inertia is the dominant physical effect.

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TL;DR: In this article, the homogenization of a passive tracer in a flow with closed mean streamlines is studied, where the initial values of the tracer are replaced by their (generalized) average about a streamline.

Abstract: The homogenization of a passive ‘tracer’ in a flow with closed mean streamlines occurs in two stages: first, a rapid phase dominated by shear-augmented diffusion over a time ≈P1/3(L/U), where the Peclet number P=LU/κ (L,U and κ are lengthscale, velocity scale and diffusivity), in which initial values of the tracer are replaced by their (generalized) average about a streamline; second, a slow phase requiring the full diffusion time ≈ L2/κ. The diffusion problem for the second phase, where tracer isopleths are held to streamlines by shear diffusion, involves a generalized diffusivity which is proportional to κ, but exceeds it if the streamlines are not circular. Expressions are also given for flow fields that are oscillatory rather than steady.

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TL;DR: In this article, the evolution of unsheared grid-generated turbulence in a stably stratified fluid was investigated in a closed-loop salt-stratified water channel, where simultaneous single point measurements of the horizontal and vertical velocity and density fluctuations were obtained, including turbulent mass fluxes central in understanding the energetics of the fluctuating motion.

Abstract: The evolution of unsheared grid-generated turbulence in a stably stratified fluid was investigated in a closed-loop salt-stratified water channel. Simultaneous single-point measurements of the horizontal and vertical velocity and density fluctuations were obtained, including turbulent mass fluxes central in understanding the energetics of the fluctuating motion. When the buoyancy lengthscale was initially substantially larger than the largest turbulent scales, the initial behaviour of the velocity and density fields was similar to that in the non-stratified case. With further downstream development, the buoyancy lengthscale decreased while the turbulence scale grew. Deviations from neutral behaviour occurred when these two lengthscales became of the same order, after the initially larger inertial forces associated with the initial kinetic energy had become weaker and buoyancy forces became important.Buoyancy forces produced anisotropy in the largest scales first, preventing them from overturning, while smaller-scale isotropic turbulent motions remained embedded within the larger-scale wave motions. These small-scale motions exhibited classical turbulent behaviour and scaled universally with Kolmogorov length and velocity scales. Eventually even the smallest scales of the decaying turbulence were affected by buoyancy, all isotropic motions disappeared, and Kolmogorov scaling failed. The turbulent vertical mass flux decreased to zero for this condition, indicating that the original turbulent field had been completely converted to random internal wave motions.The transition from a fully turbulent state to one of internal waves occurred rapidly in a time less than the characteristic time of the turbulence based on the largest-scale eddies found in the flow at transition. The dissipation rate for complete transition to a wave field was found to be of the order of et = 24.5νN2, where ν is the kinematic viscosity and N the Brunt-Vaisala frequency. This is in fairly good agreement with the value 30νN2 predicted by Gibson (1980, 1981).The present experiments have determined quantitative limits on the range of active turbulent scales in homogeneous stratified turbulence, in terms of an upper limit near the buoyancy lengthscale and a lower limit determined by viscosity in the usual way. This description has been used here to help explain and assimilate the results from the earlier stratified grid-turbulence experiments of Lin & Veenhuizen (1975) and Dickey & Mellor (1980). While some of the features of the present observations may be qualitatively seen in the numerical simulations of the problem of Riley, Metcalfe & Weissman (1981), there are fundamental differences, probably due in part to large differences in initial lengthscale ratios and in the limited range of scales attainable in numerical simulations. The present experiments may serve as a useful test case for future modelling and interpretation of the behaviour of turbulence in stratified flows observed in the oceans and atmosphere.

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TL;DR: In this article, the authors used matched asymptotic expansions to study the generation of Tollmien-Schlichting waves by free-stream disturbances incident on a flat-plate boundary layer.

Abstract: The method of matched asymptotic expansions is used to study the generation of Tollmien-Schlichting waves by free-stream disturbances incident on a flat-plate boundary layer. Near the leading edge, the motion is governed by the unsteady boundary-layer equation, while farther downstream it is governed (to lowest order) by the Orr-Sommerfeld equation with slowly varying coefficients. It is shown that there is an overlap domain where the Tollmien-Schlichting wave solutions to the Orr-Sommerfeld equation and appropriate asymptotic solutions of the unsteady boundary-layer equation match, in the matched-asymptotic-expansion sense. The analysis explains how long-wavelength free-stream disturbances can generate Tollmien-Schlichting waves of much shorter wavelength. It also leads to a set of scaling laws for the asymptotic structure of the unsteady boundary layer.

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TL;DR: In this paper, the inviscid instability of columnar vortex flows in unbounded domains to three-dimensional perturbations is considered and a sufficient condition for columnar vortices is formulated.

Abstract: The inviscid instability of columnar vortex flows in unbounded domains to three-dimensional perturbations is considered. The undisturbed flows may have axial and swirl velocity components with a general dependence on distance from the swirl axis. The equation governing the disturbance is found to simplify when the azimuthal wavenumber n is large. This permits us to develop the solution in an asymptotic expansion and reveals a class of unstable modes. The asymptotic results are confirmed by comparisons with numerical solutions of the full problem for a specific flow modelling the trailing vortex. It is found that the asymptotic theory predicts the most-unstable wave with reasonable accuracy for values of n as low as 3, and improves rapidly in accuracy as n increases. This study enables us to formulate a sufficient condition for the instability of columnar vortices as follows. Let the vortex have axial velocity W(r), azimuthal velocity V(r), where r is distance from the axis, let Ω be the angular velocity V/r, and let Γ be the circulation rV. Then the flow is unstable if
$
V\frac{d\Omega}{dr}\left[ \frac{d\Omega}{dr}\frac{d\Gamma}{dr} + \left(\frac{dW}{dr}\right)^2\right] < 0.$

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TL;DR: In this article, the effects of crossing trajectories and inertia on the dispersion of particles suspended in a field of grid-generated turbulence were investigated experimentally, and the effect of particle trajectories crossing the trajectories of fluid elements, under the influence of a potential field (usually gravity), is to force the particles from one region of highly correlated flow to another.

Abstract: The effects of ‘crossing trajectories’ and inertia on the dispersion of particles suspended in a field of grid-generated turbulence were investigated experimentally. The effect of particle trajectories crossing the trajectories of fluid elements, under the influence of a potential field (usually gravity), is to force the particles from one region of highly correlated flow to another. In this manner, particles lose velocity correlation more rapidly than the corresponding fluid points and as a result disperse less.A homogeneous decaying turbulent field was created behind a square biplanar grid in a wind tunnel. Particles were charged by a corona discharge then passed into the test section through a small plastic tube. A uniform electric field within the test section was used to simulate the effect of gravity, forcing the charged particles out of regions of correlated fluid at a higher than normal rate, therefore inducing the effects of crossing trajectories. Two sizes of glass beads (5 μm and 57 μm diameter) were employed in order to observe inertial effects. Laser-Doppler anemometry was used to measure particle mean-square displacement, autocorrelation coefficient, and mean-square velocity, from which dispersion coefficients were calculated.For the two particle sizes used in the tests, it was found that the particle diffusion coefficient, after a suitably long time from their release, was influenced primarily by the effect of crossing trajectories. Only in the particle mean-square velocity was the particle inertia seen to have any effect. The ratio of the particle relaxation time to the Kolmogoroff timescale was found to be a good indicator for the effects of particle inertia.

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TL;DR: In this paper, the rate of mass transfer of a diffusing substance along a pipe is augmented by an oscillatory motion of the ambient fluid in the pipe, and the increase of the flux is evaluated for the cases of a circular pipe and of a two-dimensional channel.

Abstract: The rate of mass transfer of a diffusing substance along a pipe is augmented by an oscillatory motion of the ambient fluid in the pipe. The increase of the flux is evaluated for the cases of a circular pipe and of a two-dimensional channel. Results are given for a general cross-section in the limiting cases of slow and fast oscillations of the flow.

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TL;DR: In this article, the bursting frequency in turbulent boundary layers has been measured over the Reynolds-number range 103 < U∞/ν < 104, and the non-dimensional frequency was constant independent of Reynolds number.

Abstract: The bursting frequency in turbulent boundary layers has been measured over the Reynolds-number range 103 < U∞/ν < 104. When scaled with the variables appropriate for the wall region, the non-dimensional frequency was constant independent of Reynolds number. A strong effect of the sensor size was noted on the measured bursting frequency. Only sensors having a spatial scale less than twenty viscous lengthscales were free from spatial-averaging effects and yielded consistent results. The spatial-resolution problem was apparently the reason for erroneous results reported in the past.

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TL;DR: In this paper, the existence and role of solitary waves in the instability of a fluid layer flowing down an inclined plane was studied and the long-term evolution was shown to be a slow relaxation to a steady state in a moving frame.

Abstract: We study the existence and the role of solitary waves in the instability of a fluid layer flowing down an inclined plane. The approach presented is fully nonlinear. Solitary waves steady in a moving frame are described by homoclinic trajectories of an associated ordinary differential equation. They are searched numerically for a given value of viscosity and surface tension. Several kinds of solitary waves can exist, characterized by their number n of humps. We investigate the stability of these waves by integrating the initial-value problem directly. Solitary waves with more than 1 hump did not appear in the simulation, and moreover a catastrophic behaviour took place for too large a Reynolds number (R [gsim ] R*1) or too large an amplitude, suggesting a finite-time singularity. The long-term evolution is shown to be a very slow relaxation to a steady state in a moving frame. The relation to the experimental observation of localized wavetrains is also discussed.

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TL;DR: In this article, the wall-pressure fluctuations in a turbulent boundary layer were investigated in a wind tunnel designed for flow-acoustic measurements, and it was shown that the dimensionless diameter d+ = 19 of the transducers is sufficient to resolve the essential structures of the turbulent pressure fluctuations.

Abstract: In a wind tunnel designed for flow-acoustic measurements, the wall-pressure fluctuations beneath a turbulent boundary layer have been investigated. The measurements were carried out with variously sized pressure transducers (19 [les ] d+ [les ] 333) and with an array of four small transducers (separation distance Δx+ = 75). It is shown that the dimensionless diameter d+ = 19 of the transducers is sufficient to resolve the essential structures of the turbulent pressure fluctuations. The power spectrum Φ(ω+) measured with the smallest transducer d+ = 19 partly exhibits power-law decay , which has been theoretically predicted for locally isotropic turbulence. By visual analysis and signal averaging in the time domain, pressure structures with high amplitudes could be detected which have the shape of short wavetrains or pulses. Their characteristic frequency and longitudinal wavelength have the mean values ω+ = 0.52 and λ+ = 145 respectively, and their mean convection velocity amounts to uc/u∞ = 0.53. It was calculated from the measured probability density that these characteristic structures play an important role, although the probability of their occurrence is low. The sources of these wall-pressure structures can be located in the buffer layer of the boundary layer.

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TL;DR: In this paper, the authors studied the far-field acoustic spectrum of an airfoil profile placed in a uniform laminar flow and showed that it is composed of a broadband contribution around frequency fs and a discrete contribution at equidistant frequencies fn, which follow power laws of the forms fs ∼ U1.5 and fn ∼ U0.85.

Abstract: The present paper is devoted to the experimental study of the noise generated by an airfoil profile placed in a uniform laminar flow. The far-field acoustic spectrum is shown to be composed of a broadband contribution around frequency fs and a discrete contribution at equidistant frequencies fn, which follow power laws of the forms fs ∼ U1.5 and fn ∼ U0.85. Both contributions can be accounted for by a simple model derived from the original suggestions of Tam (1974) and Fink (1975). It is essentially assumed that the diffraction of the Tollmien-Schlichting instabilities by the trailing edge generates acoustic waves which propagate in the far field and also trigger an aeroacoustic feedback loop, whose length is equal to the distance between the trailing edge and the maximum velocity point of the airfoil.

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TL;DR: In this article, a porous wavemaker theory is developed to analyse small-amplitude surface waves on water of finite depth, produced by horizontal oscillations of a porous vertical plate, and analytical solutions in closed forms are obtained for the surface-wave profile, the hydrodynamic-pressure distribution and the total force on the wavemaker.

Abstract: A porous-wavemaker theory is developed to analyse small-amplitude surface waves on water of finite depth, produced by horizontal oscillations of a porous vertical plate. Analytical solutions in closed forms are obtained for the surface-wave profile, the hydrodynamic-pressure distribution and the total force on the wavemaker. The influence of the wave-effect parameter C and the porous-effect parameter G, both being dimensionless, on the surface waves and on the hydrodynamic pressures is discussed in detail.

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TL;DR: In this paper, a set of equations that are suitable for numerical solution for the mean flow properties of any round bubble plume is presented and a comparison of one of these with existing experimental data is given.

Abstract: Previous experimental studies are reviewed and those whose data are deemed reliable are identified. New experiments at larger scale are described and the results are reported. These are combined with the reliable previous studies to form a data set covering heights from 3.66 to 50 m and gasflow rates from 0.0002 to 0.59 normal m3/s. These wide-ranging data are combined with an integral theory for bubble plumes to determine functional relationships between local plume properties and the entrainment coefficient and the fraction of the momentum flux that is carried in the turbulent velocity fluctuations. These relationships together with the integral theory provide a set of equations that are suitable for numerical solution for the mean flow properties of any round bubble plume. Examples of the numerical solutions are presented and a comparison of one of these with existing experimental data is given. The relationships between the local plume properties and the entrainment coefficient and the momentum flux carried by the turbulence are interpreted to provide a qualitative understanding of the parameters involved and their influences on the plume.

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TL;DR: In this paper, the surface-height profile and the vertical distributions of velocity and total head were measured behind a two-dimensional fully submerged hydrofoil moving horizontally at constant speed and angle of attack.

Abstract: Measurements of the surface-height profile and the vertical distributions of velocity and total head were made behind a two-dimensional fully submerged hydrofoil moving horizontally at constant speed and angle of attack. These measurements were used to resolve the drag on the foil into two parts: one associated with the turbulent breaking region that is sometimes present on the forward face of the first wave, and the other associated with the remaining non-breaking wavetrain. It was found that at ‘incipient breaking’ the first wave existed in either a breaking or a non-breaking state depending on the starting procedures. It was possible to induce steady breaking when the wave slope was 17° or higher. The wake survey measurements showed that the drag associated with breaking reached more than three times the maximum drag that could theoretically be obtained with non-breaking waves. The drag associated with breaking was found to be proportional to the downslope component of the weight of the breaking region.

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TL;DR: In this paper, a reciprocating oscillatory turbulent flow in a rectangular duct is investigated experimentally by making use of a laser-Doppler velocimeter, hot-wire anemometers as well as electronic digital sampling and processing equipments.

Abstract: A reciprocating oscillatory turbulent flow in a rectangular duct is investigated experimentally by making use of a laser-Doppler velocimeter, hot-wire anemometers as well as electronic digital sampling and processing equipments.The profiles of the mean velocity, the turbulence intensities, the Reynolds stress and the turbulent-energy production rate are compared for the accelerating and decelerating phases.The characteristics of such a flow are quite different from wall turbulence which is steady in the mean. In the accelerating phase, turbulence is triggered by the shear instability at a slight distance from the wall but is suppressed and cannot develop. However, with the beginning of flow deceleration, turbulence grows explosively and violently and is maintained by the bursting type of motion.The turbulent-energy production becomes exceedingly high in the decelerating phase, but the turbulence is reduced to a very low level at the end of the decelerating phase and in the accelerating stage of reversal flow. Spectra and spatial correlations for the various phases are compared. The spectral decay in the high-frequency range for the decelerating phase with high turbulence is far steeper than that of Kolmogorov's −5/3 power law, indicating remarkably high energy dissipation by high-frequency turbulence.Notwithstanding the great difference between the ensemble-averaged characteristics of the oscillatory flow and those of steady wall turbulence, its basic processes such as ejection, sweep and interactions directed towards and away from the wall are the same as those of ‘steady’ wall turbulence.