# Showing papers in "Journal of Fluid Mechanics in 1971"

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TL;DR: In this paper, the authors show that a large-scale orderly pattern may exist in the noiseproducing region of a round subsonic jet by observing the evolution of orderly flow with advancing Reynolds number.

Abstract: Past evidence suggests that a large-scale orderly pattern may exist in the noiseproducing region of a jet. Using several methods to visualize the flow of round subsonic jets, we watched the evolution of orderly flow with advancing Reynolds number. As the Reynolds number increases from order 102 to 103, the instability of the jet evolves from a sinusoid to a helix, and finally to a train of axisymmetric waves. At a Reynolds number around 104, the boundary layer of the jet is thin, and two kinds of axisymmetric structure can be discerned: surface ripples on the jet column, thoroughly studied by previous workers, and a more tenuous train of large-scale vortex puffs. The surface ripples scale on the boundary-layer thickness and shorten as the Reynolds number increases toward 105. The structure of the puffs, by contrast, remains much the same: they form at an average Strouhal number of about 0·3 based on frequency, exit speed, and diameter.To isolate the large-scale pattern at Reynolds numbers around 105, we destroyed the surface ripples by tripping the boundary layer inside the nozzle. We imposed a periodic surging of controllable frequency and amplitude at the jet exit, and studied the response downstream by hot-wire anemometry and schlieren photography. The forcing generates a fundamental wave, whose phase velocity accords with the linear theory of temporally growing instabilities. The fundamental grows in amplitude downstream until non-linearity generates a harmonic. The harmonic retards the growth of the fundamental, and the two attain saturation intensities roughly independent of forcing amplitude. The saturation amplitude depends on the Strouhal number of the imposed surging and reaches a maximum at a Strouhal number of 0·30. A root-mean-square sinusoidal surging only 2% of the mean exit speed brings the preferred mode to saturation four diameters downstream from the nozzle, at which point the entrained volume flow has increased 32% over the unforced case. When forced at a Strouhal number of 0·60, the jet seems to act as a compound amplifier, forming a violent 0·30 subharmonic and suffering a large increase of spreading angle. We conclude with the conjecture that the preferred mode having a Strouhal number of 0·30 is in some sense the most dispersive wave on a jet column, the wave least capable of generating a harmonic, and therefore the wave most capable of reaching a large amplitude before saturating.

2,108 citations

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TL;DR: In this article, the Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm.

Abstract: The Orr-Sommerfeld equation is solved numerically using expansions in Chebyshev polynomials and the QR matrix eigenvalue algorithm. It is shown that results of great accuracy are obtained very economically. The method is applied to the stability of plane Poiseuille flow; it is found that the critical Reynolds number is 5772·22. It is explained why expansions in Chebyshev polynomials are better suited to the solution of hydrodynamic stability problems than expansions in other, seemingly more relevant, sets of orthogonal functions.

1,365 citations

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TL;DR: In this paper, the structure of the flat plate incompressible smooth-surface boundary layer in a low-speed water flow is examined using hydrogen-bubble measurements and also hot-wire measurements with dye visualization.

Abstract: The structure of the flat plate incompressible smooth-surface boundary layer in a low-speed water flow is examined using hydrogen-bubble measurements and also hot-wire measurements with dye visualization. Particular emphasis is placed on the details of the process of turbulence production near the wall. In the zone 0 < y+ < 100, the data show that essentially all turbulence production occurs during intermittent ‘bursting’ periods. ‘Bursts’ are described in some detail.The uncertainties in the bubble data are large, but they have the distinct advantage of providing velocity profiles as a function of time and the time sequences of events. These data show that the velocity profiles during bursting periods assume a shape which is qualitatively distinct from the well-known mean profiles. The observations are also used as the basis for a discussion of possible appropriate mathematical models for turbulence production.

1,004 citations

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TL;DR: In this paper, the Eulerian time correlation coefficient of turbulent velocities passed through matched narrow-band niters shows a strong dependence on nominal filter frequency (∼ wave-number at these small turbulence levels).

Abstract: Space-time correlation measurements in the roughly isotropic turbulence behind a regular grid spanning a uniform airstream give the simplest Eulerian time correlation if we choose for the upstream probe signal a time delay which just ‘cancels’ the mean flow displacement. The correlation coefficient of turbulent velocities passed through matched narrow-band niters shows a strong dependence on nominal filter frequency (∼ wave-number at these small turbulence levels). With plausible scaling of the time separations, a scaling dependent on both wave-number and time, it is possible to effect a good collapse of the correlation functions corresponding to wave-numbers from 0·5 cm−1, the location of the peak in the three-dimensional spectrum, to 10 cm−1, about half the Kolmogorov wave-number. The spectrally local time-scaling factor is a ‘parallel’ combination of the times characterizing (i) gross strain distortion by larger eddies, (ii) wrinkling distortion by smaller eddies, (iii) convection by larger eddies and (iv) gross rotation by larger eddies.

991 citations

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TL;DR: In this paper, a numerical method was proposed to solve the problem of balloon bubble collapse near a plane solid wall, using finite time steps and an iterative technique for applying the boundary conditions at infinity directly to the liquid at a finite distance from the free surface.

Abstract: Vapor bubble collapse problems lacking spherical symmetry are solved here using a numerical method designed especially for these problems. Viscosity and compressibility in the liquid are neglected. The method uses finite time steps and features an iterative technique for applying the boundary conditions at infinity directly to the liquid at a finite distance from the free surface. Two specific cases of initially spherical bubbles collapsing near a plane solid wall were simulated: a bubble initially in contact with the wall, and a bubble initially half its radius from the wall at the closest point. It is shown that the bubble develops a jet directed towards the wall rather early in the collapse history. Free surface shapes and velocities are presented at various stages in the collapse. Velocities are scaled like (Δp/ρ)^1/2 where ρ is the density of the liquid and Δp is the constant difference between the ambient liquid pressure and the pressure in the cavity. For Δp/ρ = 10^6 (cm/sec)^2 ~ 1 atm./density of water the jet had a speed of about 130 m/sec in the first case and 170 m/sec in the second when it struck the opposite side of the bubble. Such jet velocities are of a magnitude which can explain cavitation damage. The jet develops so early in the bubble collapse history that compressibility effects in the liquid and the vapor are not important.

890 citations

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TL;DR: In this article, the effects of different surface roughness conditions on the turbulence structure in the boundary region were investigated and it was shown that ejection phases corresponded with ejection of low momentum fluid outwards from the boundary whilst inrush phases were associated with the transport of high momentum fluid inwards towards the boundary.

Abstract: An experimental study of boundary-layer turbulence in a free surface channel flow is described. Attention is concentrated on the effects of different surface roughness conditions on the turbulence structure in the boundary region. Hydrogen bubble flow tracers and medium high-speed motion photography were used to obtain an instantaneous visual and quantitative description of the flow field. In particular it proved possible to record instantaneous longitudinal and vertical velocity profiles from which distributions of the instantaneous Reynolds stress contribution were computed.Two well-defined intermittent features of the flow structure were visually identified close to the boundary. These consisted of fluid ejection phases, previously reported by Kline et al. (1967) for smooth boundary flow, and fluid inrush phases. Conditional averaging of the instantaneous velocity data yielded quantitative confirmation that ejection phases corresponded with ejection of low momentum fluid outwards from the boundary whilst inrush phases were associated with the transport of high momentum fluid inwards towards the boundary. Inrush and ejection events were present irrespective of the surface roughness condition.Conditional averaging also indicated that both inrush and ejection sequences correlate with an extremely high contribution to Reynolds stress and hence turbulence production close to the boundary. Indeed the present results, taken with those from previous studies, suggest that turbulence production is dominated by the joint contribution from the inrush and ejection events. It is emphasized that these structural features are intermittent, forming important linked elements of a randomly repeating cycle of wall-region turbulence production which is apparently driven by some violent three-dimensional instability mechanism.Whilst the most coherent effects of the observed inrush phases appear to be mainly confined to a region close to the boundary, the influence of the ejection phases is far more extensive. The ejected low momentum fluid elements, drawn from the viscous sublayer and from between the interstices of the roughness elements, travel outwards from the boundary into the body of the flow and give rise to very large positive contributions to Reynolds stress at points remote from the boundary. This effect is sufficiently strong to prompt the suggestion that the ejection process could represent a universal and dominant mode of momentum transport outside the immediate wall region and possibly extending across the entire thickness of the boundary layer.A structural model based on the present observations is seen to exhibit consistency with many commonly visualized features and recorded average properties of turbulent boundary-layer flows in general.

879 citations

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TL;DR: In this paper, an attempt has been made to model the dynamics of ciliary propulsion through the concept of an "envelope" covering the ends of the numerous cilia of the microscopic organism.

Abstract: In this paper, an attempt has been made to model the dynamics of ciliary propulsion through the concept of an ‘envelope’ covering the ends of the numerous cilia of the microscopic organism. This approximation may be made in the case when the cilia are close together, as can occur in the case of the symplectic metachronal wave (i.e. the wave travels in the same direction as the effective beat). For simplicity, a spherical model has been chosen, and the analysis which follows is a correction to Lighthill's (1952) paper on squirming motions of a nearly spherical organism. The velocity and efficiency compared to the work done in pushing an inert organism are obtained, and compared to that of a ciliated organism.

742 citations

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TL;DR: In this paper, an almost-Markovian Galilean-in variant turbulence model was proposed for the k−3 enstrophy-transfer range in two-dimensional turbulence.

Abstract: A simple dynamical argument suggests that the k−3 enstrophy-transfer range in two-dimensional turbulence should be corrected to the form
\[
E(k) = C^{\prime} \beta^{\frac{21}{3}}k^{-3}[\ln (k/k_1)]^{-\frac{1}{3}}\quad (k \gg k_1),
\]
where E(k) is the usual energy-spectrum function, β is the rate of enstrophy transfer per unit mass, C′ is a dimensionless constant, and k1 marks the bottom of the range, where enstrophy is pumped in. Transfer in the energy and enstrophy inertial ranges is computed according to an almost-Markovian Galilean-in variant turbulence model. Transfer in the two-dimensional energy inertial range,
\[
E(k) = C\epsilon^{\frac{2}{3}}k^{-\frac{5}{3}},
\]
is found to be much less local than in three dimensions, with 60 % of the transfer coming from wave-number triads where the smallest wave-number is less than one-fifth the middle wave-number. The turbulence model yields the estimates C′ = 2·626, C = 6·69 (two dimensions), C = 1·40 (three dimensions).

716 citations

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TL;DR: In this article, the authors investigated the effect of interactions between rigid particles in a dilute suspension and showed that the contribution of particles to the bulk stress due to the interactions between them is relatively large, for volume fractions which are still small.

Abstract: In a pure straining motion, elongated rigid particles in suspension are aligned parallel to the direction of the greatest principal rate of extension, provided the effect of Brownian motion is weak. If the suspension is dilute, in the sense that the particles are hydrodynamically independent, each particle of length 2l makes a contribution to the bulk deviatoric stress which is of roughly the same order of magnitude as that due to a rigid sphere of radius l. The fractional increase in the bulk stress due to the presence of the particles is thus equal to the concentration by volume multiplied by a factor of order l2/b2, where 2b is a measure of the linear dimensions of the particle cross-section. This suggests that the stress due to the particles might be relatively large, for volume fractions which are still small, with interesting implications for the behaviour of polymer solutions. However, dilute-suspension theory is not applicable in these circumstances, and so an investigation is made of the effect of interactions between particles. It is assumed that, when the average lateral spacing of particles (h) satisfies the conditions b [Lt ] h [Lt ] l, the disturbance velocity vector is parallel to the particles and varies only in the cross-sectional plane. The velocity near a particle is found to have the same functional form as for an isolated particle, and the modification to the outer flow field for one particle is determined by replacing the randomly placed neighbouring particles by an equivalent cylindrical boundary. The resulting expression for the contribution to the bulk stress due to the particles differs from that for a dilute suspension only in a minor way, viz. by the replacement of log 2l/b by log h/b, and the above suggestion is confirmed. The relative error in the expression for the stress is expected to be of order (log h/b)−1. Some recent observations by Weinberger of the stress in a suspension of glass-fibre particles for which 2l/h = 7·4 and h/2b = 7·8 do show a particle stress which is much larger than the ambient-fluid stress, although the theoretical formula is not accurate under these conditions.

645 citations

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TL;DR: In this article, the authors describe some experiments in swirling flows in a diverging cylindrical tube in which various types of vortex breakdowns were observed, viz. mild (double helix) breakdown, spiral breakdown, and axisymmetric breakdown.

Abstract: This paper describes some experiments in swirling flows in a diverging cylindrical tube in which various types of vortex breakdowns were observed.In one set of experiments, the position of the breakdown, axial component of the velocity of the vortex core, swirl angle distribution ahead of the breakdown, and the pressure distribution along the tube were determined for various flow rates and for various values of circulation imparted to the fluid (water). Basically, three types of vortex breakdown were observed, viz. mild (double helix) breakdown, spiral breakdown (followed by turbulent mixing), and axisymmetric breakdown (followed by a thicker vortex core, then a spiral breakdown, and finally by turbulent mixing). The type and the location of the stationary breakdowns were found to be dependent, for the particular vortex tube used, upon the Reynolds and circulation numbers of the flow. In a spiral breakdown, the vortex core filament maintained the same sense of rotation as the upstream fluid elements. In an axisymmetric breakdown, the bubble included an inclined vortex-ring whose axis gyrated about the axis of the tube.In a second set of experiments, the response of the abrupt structural change along the axis of flow to gradual and abrupt changes in the upstream and downstream flow conditions was examined. The axisymmetric breakdown responded in a manner analogous to the hydraulic jump in open-channel flow before if reached a new stationary position along the axis of the tube.The observations reported and the evidence presented herein revealed that the axisymmetric breakdown is a finite transition between two sequent states of flow as proposed by Benjamin (1962, 1965, 1967) on theoretical grounds.

554 citations

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TL;DR: In this article, the mean properties of the flow field are established for three types of jet flow issuing from a circular convergent nozzle, and measured distributions of surface pressure are given which result when the jets impinge both normally and obliquely at various distances on several surface shapes.

Abstract: In this, the first part of a two-part experimental study of the behaviour of impinging jets, the mean properties of the flow field are established. Velocity profiles are given for three types of jet flow issuing from a circular convergent nozzle. Measured distributions of surface pressure are given which result when the jets impinge both normally and obliquely at various distances on several surface shapes. The pressure distributions are used to compute the radial velocity gradient at the impingement stagnation point. It is found that for normal impingement this gradient correlates with the free jet centreline velocity and half-radius at the same axial location. A fall-off in the correlated value is noted as the impingement is made oblique. Measurements of the azimuthal distribution of momentum flux in the resulting wall jet are also given. The general behaviour of all three types of jet is found to be similar at locations downstream of any local effects due to the shock waves present in the under-expanded types. A special study of the close-range impingement of an under-expanded jet containing a normal shock disk reveals a region of separated flow surrounding the stagnation point. This condition results in a negative value of the radial velocity gradient at the centreline.

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TL;DR: In this article, particle velocity correlations and the Eulerian spatial correlation were coincident within experimental error when the separation was non-dimensionalized by the respective integral scale, where L is the Lagrangian time integral scale and u is the turbulence intensity.

Abstract: Particle velocity autocorrelations of single spherical beads (46·5 μhollow glass, 87 μ glass, 87 μ corn pollen, and 46·5 μ copper) were measured in a grid-generated turbulence. The hollow glass beads were small and light enough to behave like fluid points; the other types had significant inertia and ‘crossing trajectories’ effects. The autocorrelations decreased much faster for heavier particles, in contradiction to previous experimental results. The integral scale for the copper beads was 1/3 of that for the hollow glass beads. The particle velocity correlations and the Eulerian spatial correlation were coincident within experimental error when the separation was non-dimensionalized by the respective integral scale. The data generated by the hollow glass beads can be used to estimate Lagrangian fluid properities. The Lagrangian time integral scale is approximated by L/u′, where L is the Eulerian integral scale and u′ is the turbulence intensity.

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TL;DR: In this paper, the initial value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given for values of the Reynolds number slightly greater than the critical value, above which perturbation may grow.

Abstract: The initial-value problem for linearized perturbations is discussed, and the asymptotic solution for large time is given. For values of the Reynolds number slightly greater than the critical value, above which perturbations may grow, the asymptotic solution is used as a guide in the choice of appropriate length and time scales for slow variations in the amplitude A of a non-linear two-dimensional perturbation wave. It is found that suitable time and space variables are et and e½(x+a1rt), where t is the time, x the distance in the direction of flow, e the growth rate of linearized theory and (−a1r) the group velocity. By the method of multiple scales, A is found to satisfy a non-linear parabolic differential equation, a generalization of the time-dependent equation of earlier work. Initial conditions are given by the asymptotic solution of linearized theory.

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TL;DR: In this article, the authors used a hot wire in a turbulent boundary layer in air to measure the frequent periods of activity (to be called "bursts") noticed in turbulent signal that has been passed through a narrow band-pass filter.

Abstract: Using a hot wire in a turbulent boundary layer in air, an experimental study has been made of the frequent periods of activity (to be called ‘bursts’) noticed in a turbulent signal that has been passed through a narrow band-pass filter. Although definitive identification of bursts presents difficulties, it is found that a reasonable characteristic value for the mean interval between such bursts is consistent, at the same Reynolds number, with the mean burst periods measured by Kline et al. (1967), using hydrogen-bubble techniques in water. However, data over the wider Reynolds number range covered here show that, even in the wall or inner layer, the mean burst period scales with outer rather than inner variables; and that the intervals are distributed according to the log normal law. It is suggested that these ‘bursts’ are to be identified with the ‘spottiness’ of Landau & Kolmogorov, and the high-frequency intermittency observed by Batchelor & Townsend. It is also concluded that the dynamics of the energy balance in a turbulent boundary layer can be understood only on the basis of a coupling between the inner and outer layers.

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TL;DR: In this article, an analysis of resonant triads of Tollmien-Schlichting waves in an unstable boundary layer is made, and exact solutions of the general interaction equations are presented for a particular profile consisting of a layer of constant shear bounded by a uniform flow.

Abstract: An investigation is made of resonant triads of Tollmien-Schlichting waves in an unstable boundary layer. The triads considered are those comprising a two-dimensional wave and two oblique waves propagating at equal and opposite angles to the flow direction and such that all three waves have the same phase velocity in the downstream direction. For such a resonant triad remarkably powerful wave interations take place, which may cause a continuous and rapid transfer of energy from the primary shear flow to the disturbance. It appears that the oblique waves can grow particularly rapidly and it is suggested that such preferential growth may be responsible for the rapid development of three-dimensionality in unstable boundary layers. The non-linear energy transfer primarily takes place in the vicinity of the critical layer where the downstream propagation velocity of the waves equals the velocity of the primary flow.The theoretical analysis is initially carried out for a general primary velocity profile; then, in order to demonstrate the essential features of the results, precise interaction equations are derived for a particular profile consisting of a layer of constant shear bounded by a uniform flow. Some exact solutions of the general interaction equations are presented, one of which has the property that the wave amplitudes become indefinitely large at a finite time. The possible relevance of the present theoretical model to the experiments of Klebanoff, Tidstrom & Sargent (1962) is examined.

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TL;DR: In this article, the influence of surface roughness on the cross-flow around a circular cylinder is investigated in a high-pressure wind tunnel, thus high Reynolds numbers up to Re = 3 × 106 could be obtained.

Abstract: The influence of surface roughness on the cross-flow around a circular cylinder is the subject of the present experimental work. The investigations were carried out in a high-pressure wind tunnel, thus high Reynolds numbers up to Re = 3 × 106 could be obtained. Local pressure and skin friction distributions were measured. These quantities were evaluated to determine the total drag coefficient and the percentage of friction as functions of Reynolds number and roughness parameter. In addition the local skin friction distribution yields the angular position of boundary-layer transition from laminar to turbulent flow and the location of boundary-layer separation.

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TL;DR: In this paper, an experimental study of the internal layer which grows down-stream from a rough-to-smooth surface change is presented, which is essentially different from that studied by Antonia & Luxton (1971 b) for the case of a smooth-torough perturbation.

Abstract: An experimental study of the structure of the internal layer which grows down-stream from a rough-to-smooth surface change shows it to be essentially different from that studied by Antonia & Luxton (1971 b) for the case of a smooth-to-rough perturbation. The rate of growth of the internal layer is less than that for the smooth-to-rough step and it appears that the more intense initial rough-wall flow dictates the rate of diffusion of the disturbance for a considerable distance. Inside the internal layer the mixing length I is increased relative to the equilibrium distribution I = KY. A turbulent energy budget shows that the advection is comparable with the production or dissipation, whilst there seems to be some diffusion of energy into the internal-layer region close to the wall. The boundary layer, as a whole, recovers much more slowly following a rough-to-smooth change than following a smooth-to-rough change, and at the last measuring station (16 boundary-layer thicknesses from the start of the smooth surface) the distributions of mean velocity and Reynolds shear stress are far from self-preserving.

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TL;DR: In this paper, measurements of the noise field from a 25 mm diameter subsonic air jet were analyzed in some detail by determining both the jet velocity dependence and the directivity of the intensity of the radiation in 1/3-octave bands at particular values of the frequency parameter, showing that the predicted variations overestimate the measurements and it appears that the convective amplification predicted by the theory is much reduced.

Abstract: Measurements of the noise field from a 25 mm diameter subsonic air jet are presented. These results are analysed in some detail by determining both the jet velocity dependence and the directivity of the intensity of the radiation in 1/3-octave bands at particular values of the frequency parameter,
\[
(fD/V_J)(1-M_c\cos\theta).
\]
This procedure should ensure that a particular source in a geometrically similar position in the jet is always observed, whatever the jet velocity, diameter and emission angle.These results are compared with the predictions of Lighthil's (1952) theory of convected quadrupoles. It is shown that the theory predicts the variation of the intensity with jet velocity and emission angle provided that the observed frequency is below a certain critical value, which depends on jet diameter and emission angle and is independent of jet velocity. Above this critical frequency, the predicted variations overestimate the measurements and it appears that the convective amplification predicted by the theory is much reduced. The variation of this critical frequency is explained by assuming that substantial interaction occurs between the radiated sound and the jet flow when the wavelength of the sound becomes shorter than the sound path length in the jet flow.

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TL;DR: In this paper, the main features of this undulatory mode of propulsion are discussed for the case of large Reynolds number, based on the principle of energy conservation, and the general problem of a two-dimensional flexible plate, swimming at arbitrary, unsteady forward speeds, is solved by applying the linearized inviscid flow theory.

Abstract: The most effective movements of swimming aquatic animals of almost all sizes appear to have the form of a transverse wave progressing along the body from head to tail. The main features of this undulatory mode of propulsion are discussed for the case of large Reynolds number, based on the principle of energy conservation. The general problem of a two-dimensional flexible plate, swimming at arbitrary, unsteady forward speeds, is solved by applying the linearized inviscid flow theory. The large-time asymptotic behaviour of an initial-value harmonic motion shows the decay of the transient terms. For a flexible plate starting with a constant acceleration from at rest, the small-time solution is evaluated and the initial optimum shape is determined for the maximum thrust under conditions of fixed
power and negligible body recoil.

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TL;DR: In this article, an artificial mathematical model of a porous medium is proposed for which the flow can be calculated both inside and outside the surface, and the experimental results agree with the calculations.

Abstract: In problems where a viscous fluid flows past a porous solid it has frequently been assumed that the tangential component of surface velocity is zero. When the porous solid has an open structure with large pores the external surface stress may produce a tangential flow below the surface. Recently, Beavers & Joseph (1967) have assumed that the surface velocity UB depends on the mean tangential stress in the fluid outside the porous solid through the relation
\[
\left[\mu\frac{d\overline{u}}{dy}\right]_{y=0} = \frac{\mu\alpha}{k^{\frac{1}{2}}}(U_B-Q),
\]
where Q is the volume flow rate per unit cross-section within the porous material due to the pressure gradient, k is the Darcy constant and α is a constant which depends only on the nature of the porosity. An artificial mathematical model of a porous medium is proposed for which the flow can be calculated both inside and outside the surface. This conceptual model was materialized and the experimental results agree with the calculations. The calculated values of α so found are not quite independent of the external means of producing the external tangential stress.

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TL;DR: In this paper, the role of small Brownian couples in establishing a steady-state probability distribution for a particle being on any particular orbit was considered, and an expression for the equilibrium distribution was derived, which was then used to calculate some bulk properties for a suspension of such particles.

Abstract: Axisymmetric particles in zero Reynolds number shear flow execute closed orbits. In this paper we consider the role of small Brownian couples in establishing a steady-state probability distribution for a particle being on any particular orbit. After presenting the basic equations, we derive an expression for the equilibrium distribution. This result is then used to calculate some bulk properties for a suspension of such particles, and these predicted properties are compared with available experimental observation.

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TL;DR: In this paper, the stability of convection rolls with varying wave-number was investigated and the results confirmed the theoretical predictions by Busse (1967a) and showed two distinct types of instability in the form of nonoscillatory disturbances.

Abstract: An experiment on the stability of convection rolls with varying wave-number is described in extension of the earlier work by Chen & Whitehead (1968). The results agree with the theoretical predictions by Busse (1967a) and show two distinct types of instability in the form of non-oscillatory disturbances. The ‘zigzag instability’ corresponds to a bending of the original rolls; in the ‘cross-roll instability’ rolls emerge at right angles to the original rolls. At Rayleigh numbers above 23,000 rolls are unstable for all wave-numbers and are replaced by a three-dimensional form of stationary convection for which the name ‘bimodal convection’ is proposed.

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TL;DR: Galerkin (spectral) methods for numerical simulation of incompressible flows within simple boundaries are shown to possess many advantages over existing finite-difference methods as mentioned in this paper, and the accuracy of Galerkin approximations obtained from truncated Fourier expansions is explored.

Abstract: Galerkin (spectral) methods for numerical simulation of incompressible flows within simple boundaries are shown to possess many advantages over existing finite-difference methods. In this paper, the accuracy of Galerkin approximations obtained from truncated Fourier expansions is explored. Accuracy of simulation is tested empirically using a simple scalar-convection test problem and the Taylor–Green vortex-decay problem. It is demonstrated empirically that the Galerkin (Fourier) equations involving Np degrees of freedom, where p is the number of space dimensions, give simulations at least as accurate as finite-difference simulations involving (2N)p degrees of freedom. The theoretical basis for the improved accuracy of the Galerkin (Fourier) method is explained. In particular, the nature of aliasing errors is examined in detail. It is shown that ‘aliasing’ errors need not be errors at all, but that aliasing should be avoided in flow simulations. An eigenvalue analysis of schemes for simulation of passive scalar convection supplies the mathematical basis for the improved accuracy of the Galerkin (Fourier) method. A comparison is made of the computational efficiency of Galerkin and finite-difference simulations, and a survey is given of those problems where Galerkin methods are likely to be applied most usefully. We conclude that numerical simulation of many of the flows of current interest is done most efficiently and accurately using the spectral methods advocated here.

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TL;DR: In this paper, a model equation of Langevin type for the turbulent velocity field is constructed, in which the nonlinear terms of the Navier-Stokes equation are replaced by a dynamical damping term and a random forcing term, with strength parameters determined by the past history of the energy spectrum.

Abstract: A model equation of Langevin type for the turbulent velocity field is constructed, in which the non-linear terms of the Navier–Stokes equation are replaced by a dynamical damping term and a random forcing term, with strength parameters determined by the past history of the energy spectrum The model leads to a closed set of first-order differential equations in time for the evolution of two functions: the energy spectrum and the effective memory times for the interaction of mode triads Invariance of the energy transfer to random Galilean transformation is achieved by using the interaction between solenoidal and compressive parts of a convected test field to determine the memory–time functions The model equation is developed from the direct-interaction approximation as starting-point At an intermediate stage, before the Galilean invariance is introduced, a model representation of Edwards's (1964) theory is obtained which extends the latter to statistically non-stationary states

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TL;DR: In this paper, the authors investigated the Soret effect on overstable solutions of the thermosolutal Rayleigh-Jeffreys problem theoretically and experimentally and observed critical Rayleigh numbers and temporal frequencies are consistent with theory.

Abstract: The suggestion made by the authors in a previous paper (Hurle & Jakeman 1969) that the Soret effect could give rise to overstable solutions of the thermosolutal Rayleigh–Jeffreys problem is investigated theoretically and experimentally.Oscillatory instability is shown to occur in initially homogeneous layers of water-methanol mixtures when they are heated from below. This instability triggers a finite-amplitude steady mode. The magnitude and sign of the Soret coefficient was changed by varying the composition of the mixture; as predicted, overstable modes were observed when the sign of the coefficient was such as to produce a stabilizing contribution to the density gradient. The observed critical Rayleigh numbers and temporal frequencies are consistent with theory.

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TL;DR: In this article, the scattering of surface gravity waves by a circular dock is considered in order to determine the horizontal and vertical forces and torque on the dock, and the solution is shown to have phase independent of depth and so may be obtained from an infinite set of real equations.

Abstract: The scattering of surface gravity waves by a circular dock is considered in order to determine the horizontal and vertical forces and torque on the dock. An incident plane wave is expanded in Bessel functions, and for each mode the problem is formulated in terms of the potential on the cylindrical surface containing the dock and extending to the bottom. The solution is shown to have phase independent of depth and so may be obtained from an infinite set of real equations, which are solved numerically by Galerkin's method. The convergence of the solution is discussed, and some numerical results are presented.This problem has been investigated previously by Miles & Gilbert (1968) by a different method, but their work contained errors.

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TL;DR: In this paper, the effect of sloping boundaries on thermal convection is studied theoretically and in the laboratory in the context of a model in which fluid is contained in a differentially heated rectangular box of small aspect ratio (depth/length), inclined at an angle δ to the vertical.

Abstract: The effect of sloping boundaries on thermal convection is studied theoretically and in the laboratory in the context of a model in which fluid is contained in a differentially heated rectangular box of small aspect ratio (depth/length), inclined at an angle δ to the vertical. Like its two limiting cases, Benard convection and convection in the vertical slot, a basic state which exists for low Rayleigh numbers becomes unstable as this parameter is increased. The types of instability and indeed the manner in which the motions become turbulent depend crucially on δ. In our work with water the following general picture of the primary instabilities applies:
For 90° > δ > 10° with the bottom plate hotter, the instabilities are stationary longitudinal convectively driven rolls with axes oriented up the slope. Near δ = 10° there is an upper and lower Rayleigh number cut off. If the Rayleigh number is too small diffusion damps the instabilities, but if it is too large they are damped by the development of a stable upslope temperature gradient in the mean flow.For 10° > δ > −10° (negative angles imply a hotter upper plate), transverse travelling waves oriented across the slope are the first instabilities of the mean flow. They obtain their kinetic energy via the working of the upslope buoyancy force.For - 10° > δ > −85° longitudinal modes are again observed. These are rather curious in that they may exist when the stratification is everywhere positive. The necessary energy for these modes comes out of the mean velocity field and out of the mean available potential energy.Agreement between the stability theory and the experiments is generally quite good over the whole range of δ, considering the approximations involved in finding a suitable basic flow solution.For Rayleigh numbers less than ∼ 106 turbulence is only possible for positive angles. For 85° > δ > 20° the development of unsteadiness involves the occurrence and the breaking of wavy longitudinal vortices in a manner reminiscent of the development of turbulence in cylindrical Couette flow.

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TL;DR: A theory of combustion noise is developed in this paper, which follows rigorously from the principles of fluid mechanics, and the sound radiated from open, turbulent flames is found to depend strongly upon the structure of such flames; at present their structure is not well known.

Abstract: Upon review of past experimental results and theoretical efforts it is apparent that the mechanism by which combustion noise is generated is not well understood. A theory of combustion noise is developed in this paper which follows rigorously from the principles of fluid mechanics. Lighthill's approach, used in his studies of aerodynamic noise, is closely followed in the present work. The sound radiated from open, turbulent flames is found to depend strongly upon the structure of such flames; at present their structure is not well known. However, meaningful bounds and scaling rules for the sound power output and spectral content are derived based upon the present limited knowledge. A framework is developed which explains past experimental work and the origin of combustion noise.

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TL;DR: In this paper, the deformation of a solitary wave due to a slow variation of the bottom topography was examined and the variation of amplitude with depth was determined and compared with some recent experimental results.

Abstract: This paper examines the deformation of a solitary wave due to a slow variation of the bottom topography. Differential equations which determine the slow variation of the parameters of a solitary wave are derived by a certain averaging process applied to the exact in viscid equations. The equations for the parameters are solved when the bottom topography varies only in one direction, and when the wave evolves from a region of uniform depth. The variation of amplitude with depth is determined and compared with some recent experimental results.

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TL;DR: In this article, the flow in the wakes behind two-dimensional perforated plates has been investigated in the Reynolds number range 2·5 × 104 to 9·0 × 104 and the results indicate the existence of two distinct types of flows: one appropriate to high and the other to low values of plate porosity.

Abstract: The flow in the wakes behind two-dimensional perforated plates has been investigated in the Reynolds number range 2·5 × 104 to 9·0 × 104.Measurements of drag and shedding frequency were made and a pulsed hotwire anemometer was used to measure the mean velocity and turbulent intensity variations in the highly turbulent regions immediately behind the plates.The results indicate the existence of two distinct types of flows: one appropriate to high and the other to low values of plate porosity.