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The development of disturbances (two-dimensional non-linear and three-dimensional linear) in the entrance region of a circular pipe is studied in the limit of Reynolds number R → ∞ in the framework of triple-deck theory. It is found that lower-branch axisymmetric disturbances can interact in the resonant manner. Numerical calculations show that a two-dimensional nonlinear wave packet grows much more rapidly than that in the boundary layer on a flat plate, producing a spike-like solution which seems to become singular at a finite time. Large-sized, short-scaled disturbances are also studied. In this case the development of axisymmetric disturbances is governed by single one-dimensional equation in the form of the Korteweg-de Vries and Benjamin-Ono equations in the long- and short-wave limits respectively. The nonlinear interactions of these disturbances lead to the formation of solitons which can run both upstream and downstream. Linear three-dimensional wave packets are also calculated.
An analysis is constructed in order to estimate the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface. The main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves. It is demonstrated that, in general 1/cp = 1/CpO + k/ωmax + ∈(k) where cp is the wave phase speed for a particular mode, CpO is the phase speed at k = 0, ωmax is the maximum possible wave angular frequency and ωmax ≤ Nmax where Nmax is the maximum buoyancy frequency. Also, ∈(0) = 0, ∈(k) = o(k) for k large, and is bounded for finite k. In particular, when ∈(k) can be neglected, the dispersion relation for a lowest mode wave is approximately 1/cp ≈ (∫∞0N2(y)ydy)-½ + k/ωmax. The eigenvalue problem is analysed for a class of buoyancy frequency squared functions N2(x) which is taken to be a class of realvalued functions of a real variable x where O ≤ x ∞ such that N2(x) = O(e-βx) as x → ∞ and 1/β is an arbitrary length scale. It is demonstrated that N2(x) can be represented by a power series in e-βx. The eigenfunction equation is constructed for such a function and it is shown that there are two cases of the equation which have solutions in terms of known functions (Bessel functions and confluent hypergeometric functions). For these two cases it is shown that ∈(k) can be neglected and that, in addition, ωmax = Nmax. More generally, it is demonstrated that when k → ∞ it is possible to approximate the equation uniformly in such a way that it can be compared with the confluent hypergeometric equation. The eigenvalues are then, approximately, zeros of the Whittaker functions. The main result which follows from this approach is that if N2(x) is O(e-βx) as x → ∞ and has a maximum value N2max then a sufficient condition for 1/cp ∼ k/Nmax to hold for large k for the lowest mode is that N2(t)/t is convex for O ≤ t ≤ 1 where t = e-βx.
During the motion of a fluid interface undergoing Rayleigh-Taylor instability, vorticity is generated on the interface baronclinically. This vorticity is then subject to Kelvin-Helmholtz instability. For the related problem of evolution of a nearly flat vortex sheet without density stratification (and with viscosity and surface tension neglected), Kelvin-Helmholtz instability has been shown to lead to development of curvature singularities in the sheet. In this paper, a simple approximate theory is developed for Rayleigh-Taylor instability as a generalization of Moore's approximation for vortex sheets. For the approximate theory, a family of exact solutions is found for which singularities develop on the fluid interface. The resulting predictions for the time and type of the singularity are directly verified by numerical computation of the full equations. These computations are performed using a point vortex method, and singularities for the numerical solution are detected using a form fit for the Fourier components at high wavenumber. Excellent agreement between the theoretical predictions and the numerical results is demonstrated for small to medium values of the Atwood number A, i.e. for A between 0 and approximately 0.9. For A near 1, however, the singularities actually slow down when close to the real axis. In particular, for A = 1, the numerical evidence suggests that the singularities do not reach the real axis in finite time.
We consider the problem of nonlinear thermal-solutal convection in the mushy zone accompanying unstable directional solidification of binary systems. Attention is focused on possible nonlinear mechanisms of chimney formation leading to the occurrence of freckles in solid castings, and in particular the coupling between the convection and the resulting porosity of the mush. We make analytical progress by considering the case of small growth Péclet number, δ, small departures from the eutectic point, and infinite Lewis number. Our linear stability results indicate a small O(δ) shift in the critical Darcy-Rayleigh number, in accord with previous analyses. We find that nonlinear two-dimensional rolls may be either sub- or supercritical, depending upon a single parameter combining the magnitude of the dependence of mush permeability on solids fraction and the variations in solids fraction owing to melting or freezing. A critical value of this combined parameter is given for the transition from supercritical to subcritical rolls. Three-dimensional hexagons are found to be transcritical, with branches corresponding to upflow and lower porosity in either the centres or boundaries of the cells. These general results are discussed in relation to experimental observations and are found to be in general qualitative agreement with them.
A model of wind-blown sand transport is described with particular emphasis on the feedback between the grain cloud and the near-surface wind. The results from this model are used to develop Owen's (1964) hypothesis that ‘the grain layer behaves, so far as the flow outside it is concerned, as increased aerodynamic roughness whose height is proportional to the thickness of the layer’. The hypothesis is developed to show the influence this dynamic roughness has on the turbulent boundary layer above the saltation layer. Two processes are identified which influence the path of the system towards equilibrium. The first is the feedback between the near-surface wind and the grain cloud in which the quantity of sand transported is limited by the carrying capacity of the wind. The second is due to the temporal development of an internal boundary layer in response to the additional roughness imposed on the flow above the grain layer by the grain cloud. A similarity is noted between the temporal response of a turbulent boundary layer to sand transport and the spatial response of a turbulent boundary layer downstream of a step increase in surface roughness. Finally it is noted that the work may have important implications for transport rate prediction in unsteady winds.
This study deals with turbulent oscillatory boundary-layer flows over a plane bed with a sudden spatial change in roughness. Two kinds of ‘change in the roughness’ were investigated: in one, the roughness changed from a smooth-wall roughness to a roughness equal to 4.8 mm, and in the other, it changed from a roughness equal to 0.35 mm to the same roughness as in the previous experiment (4.8 mm). The free-stream flow was a purely oscillating flow with sinusoidal velocity variation. Mean flow and turbulence properties were measured. The Reynolds number was 6 × 106 for the major part of the experiments, with a maximum velocity of approximately 2 m/s and the stroke of the motion about 6 m. The response of the boundary layer to the sudden change in roughness was found to occur over a transitional length of the flow. The bed shear stress over this transitional length attains a peak value over the bed section with the larger roughness. It was found that the amplification in the bed shear stress due to this peak could be up to 2.5 times its asymptotic value. Also, it was found that the turbulence is quantitatively different in the two half periods; a much stronger turbulence is experienced in the half period where the flow is towards the less-rough section. The present experiments further showed that a constant streaming occurs near the bed in the neighbourhood of the junction between the two bed sections. This streaming is directed towards the section with the larger roughness.
The fluid flow induced by a cascade of circular cylinders which oscillates harmonically in an unbounded, incompressible, viscous fluid which is otherwise at rest is investigated both numerically and experimentally. Attention in this paper is mainly concentrated on the induced steady streaming flow which occurs when the ratio of the amplitude of the oscillation of the cascade to the size of the cylinder, ε, is very small. The leading-order flow is then governed by the steady Navier-Stokes equations. In order to solve these equations numerically we first generate numerically a grid system using the boundary element method and then use a finite-difference scheme on the newly generated rectangular grid system. Numerical results show that for small values of the streaming Reynolds number Rs there are four recirculating flows of equal strength around each circular cylinder of the cascade. At large values of Rs symmetry breaks down and numerical solutions are found for asymmetrical flows. Numerically, a critical value of Rs, Rso say, is identified such that the flow is symmetrical when Rs < Rso and asymmetrical when Rs > Rso and these results are in reasonable agreement with experimental results, which are also presented in this paper.
In random wind-generated wave motion on the sea surface, extreme wave events have been shown theoretically to occur within groups with a well-defined configuration and time history that can be specified in terms of the space-time autocovariances of the surface displacement. The predictions of the theory have been tested in a field experiment in the Straits of Messina in which an array of nine wave gauges and nine pressure transducers supported by vertical piles provided space-time information on waves generated over a fetch of approximately 10 km. It was confirmed that the general configuration of the extreme wave groups measured was consistent with the theoretical predictions in terms of the measured space-time autocovariance. During the development stage of a group, as the height of the central (outstanding) wave grows to a maximum, the width of the wave front reduces to a minimum. As an individual wave passes through the group, its wavelength decreases as the wave height increases towards the apex, after which the wavelength increases again as the wave moves towards the front of the group and abates.
The behaviour of fibre suspensions in dilute polymer solutions at high Deborah numbers is analysed. In particular, we calculate the change to the extension of the polymers and the orientation of the fibres caused by hydrodynamic interactions between the polymers and the fibres. At a sufficiently high Deborah number the combined effect of the fibre velocity disturbances and the mean shear flow produce a dramatic increase in the extension of the polymers, similar to the coil-stretch transition observed in extensional flow.
The non-Newtonian stresses caused by the polymers produce a perturbation to the angular velocity of the fibres, giving rise to a net drift across Jeffery orbits towards the vorticity axis. Unlike the second-order-fluid analysis of Leal (1975), this effect does not depend on the second-normal-stress difference.
In recent work it has been shown that there can be substantial transient growth in the energy of small perturbations to plane Poiseuille and Couette flows if the Reynolds number is below the critical value predicted by linear stability analysis. This growth, which may be as large as O(1000), occurs in the absence of nonlinear effects and can be explained by the non-normality of the governing linear operator - that is, the non-orthogonality of the associated eigenfunctions. In this paper we study various aspects of this energy growth for two- and three-dimensional Poiseuille and Couette flows using energy methods, linear stability analysis, and a direct numerical procedure for computing the transient growth. We examine conditions for no energy growth, the dependence of the growth on the streamwise and spanwise wavenumbers, the time dependence of the growth, and the effects of degenerate eigenvalues. We show that the maximum transient growth behaves like O(R2), where R is the Reynolds number. We derive conditions for no energy growth by applying the Hille–Yosida theorem to the governing linear operator and show that these conditions yield the same results as those derived by energy methods, which can be applied to perturbations of arbitrary amplitude. These results emphasize the fact that subcritical transition can occur for Poiseuille and Couette flows because the governing linear operator is non-normal.
We consider the problem of determining linear acoustic properties of bubbly liquids near the natural frequency of the bubbles. Since the effective wavelength and attenuation length are of the same order of magnitude as the size of the bubbles, we devise a numerical scheme to determine these quantities by solving exactly the multiple scattering problem among many interacting bubbles. It is shown that the phase speed and attenuation are finite at natural frequency even in the absence of damping due to viscous, thermal, nonlinear, and liquid compressibility effects, thus validating a recent theory (Sangani 1991). The results from the numerical scheme are in good agreement with the theory but considerably higher than the experimental values for frequencies greater than the natural frequency. The discrepancy with experiments remains even after accounting for the effect of polydispersity, finite liquid compressibility, and non-adiabatic thermal changes.
The Lorentz reciprocal theorem is generalized and applied to the study of the quasisteady motion of a concentric spherical (CS) compound drop at zero Reynolds number. Using this result, the migration velocities of a force-free CS compound drop placed in a general ambient Stokes flow, as well as the forces on each drop when subjected to specified migration velocities, are calculated. The latter constitutes a generalization of Faxén's law to the case of a CS compound drop. Also some earlier results on the thermocapillary migration of such drops (Borhan et al. 1992) are rederived more simply and in greater generality.
In this paper the flow resulting from the release of buoyant material within a long tunnel is investigated. The source fluid is discharged through a nozzle of small radius with sufficiently high flow rate to ensure that the axial lengthscale of the buoyant jet (subsequently called the ‘jet-length’) is several times the depth of the tunnel, d. The ends of the tunnel may be either open or closed and a number of ventilation points may exist along it. Consideration of a source with high momentum is an important development in confined jet flow models, as most previous models have assumed that the source has little or no initial momentum. It is found that circulation cells are driven near to the source and that the concentration within them increases to a steady-state maximum. At a distance of about 2.5d from the source the buoyancy forces are then sufficiently strong to drive a two-layered stratified counterflow. The steady-state conservation equations are analysed in order to calculate the mean flow variables. The flow past a ventilation point and the characteristics of the secondary outflow are derived, enabling the calculation of the total number of vents needed to vent the buoyant layer. The time dependence of the mean concentration in the circulation cell near to the source is also deduced. This could be used to calculate time-dependent solutions for the other mean flow variables. All of the theoretical results are compared with experimental measurements.
The nonlinear instability of the boundary layer on a heated flat plate placed in an oncoming flow is investigated. Such flows are unstable to stationary vortex instabilities and inviscid travelling wave disturbances governed by the Taylor-Goldstein equation. For small temperature differences the Taylor-Goldstein equation reduces to Rayleigh's equation. When the temperature difference between the wall and free stream is small the preferred mode of instability is a streamwise vortex. It is shown in this case that the vortex, assumed to be of small wavelength, restructures the underlying mean flow to produce a profile which can be massively unstable to inviscid travelling waves. The mean state is shown to be destabilized if the Prandtl number is less than unity.
A simple morphological model is considered which describes the interaction between a unidirectional flow and an erodible bed in a straight channel. For sufficiently large values of the width-depth ratio of the channel the basic state, i.e. a uniform current over a flat bottom, is unstable. At near-critical conditions growing perturbations are confined to a narrow spectrum and the bed profile has an alternate bar structure propagating in the downstream direction. The timescale associated with the amplitude growth is large compared to the characteristic period of the bars. Based on these observations a weakly nonlinear analysis is presented which results in a Ginzburg-Landau equation. It describes the nonlinear evolution of the envelope amplitude of the group of marginally unstable alternate bars. Asymptotic results of its coefficients are presented as perturbation series in the small drag coefficient of the channel. In contrast to the Landau equation, described by Colombini et al. (1987), this amplitude equation also allows for spatial modulations due to the dispersive properties of the wave packet. It is demonstrated rigorously that the periodic bar pattern can become unstable through this effect, provided the bed is dune covered, and for realistic values of the other physical parameters. Otherwise, it is found that the periodic bar pattern found by Colombini et al. (1987) is stable. Assuming periodic behaviour of the envelope wave in a frame moving with the group velocity, simulations of the dynamics of the Ginzburg-Landau equation using spectral models are carried out, and it is shown that quasi-periodic behaviour of the bar pattern appears.
We look at two classes of contained flow: swirling flow and buoyancy-driven flow. We note that the strong links between these arise from the way in which vorticity is generated and propagated within each. We take advantage of this shared behaviour to investigate the structure of steady-state solutions of the governing equations. First, we look at flows with a small but finite viscosity. Here we find that, Batchelor regions apart, the steady state for each type of flow must consist of a quiescent stratified core, bounded by high-speed wall jets. (In the case of swirling flow, this is a radial stratification of angular momentum.) We then give a general, if approximate, method for finding these steady-state flow fields. This employs a momentum-integral technique for handling the boundary layers. The resulting predictions compare favourably with numerical experiments. Finally, we address the problem of inviscid steady states, where there is a well-known class of steady solutions, but where the question of the stability of these solutions remains unresolved. Starting with swirling flow, we use an energy minimization technique to show that stable solutions of arbitrary net azimuthal vorticity do indeed exist. However, the analogy with buoyancy-driven flow suggests that these solutions are all of a degenerate, stratified form. If this is so, then the energy minimization technique, which conserves vortical invariants, may mimic the stratification of temperature or angular momentum in a turbulent flow.
A linear stability analysis of the inviscid, parallel flow of air over water leads to an eigenvalue problem for the wave speed, which is solved numerically for air profiles typical of both laminar and turbulent flows. Comparison is made with Miles’ (1957) theory; growth rates differ from those predicted from the Miles (1957) formula but are in agreement with Conte & Miles’ (1959) computations for turbulent flow profiles. In the limit of a highly sheared wind profile the numerical computations retrieve the Kelvin-Helmholtz instability.
The problem of flow along a horizontal semi-infinite flat plate moving in its own plane through a viscous liquid just below the free surface is considered. The method of matched asymptotic expansions is used to analyse the interaction between the free surface and the boundary layer formed on the plate. It is found that, due to viscosity, small-amplitude gravity waves on the free surface can be formed. The formulae for the resistance of the plate containing the free-surface effect and for the lift, appearing as a new phenomenon, are derived.
In a previous paper we analysed the stability to small disturbances of stationary stratified fluid which is unbounded. Various forms of the undisturbed density distribution were considered, including a sinusoidal profile and a function of the vertical coordinate z which is constant outside a central horizontal layer. Both these types of stratification are so unstable that the critical Rayleigh number is zero. In this sequel we make the study more complete and more useful by taking account of the effect of a vertical circular cylindrical boundary of radius a which is rigid and impermeable. As in the previous paper we assume that the undisturbed density distribution is steady.
The case of fluid in a vertical tube with a uniform density gradient is useful for comparison, and so we review and extend the available results, in particular obtaining growth rates for a disturbance which is neither z-independent nor axisymmetric. A numerical finite-difference method is then developed for the case in which dρ/dz = ρ0 kA cos kz. When ka [Lt ] 1 the relation between growth rate and Rayleigh number approximates to that for a uniform density gradient of magnitude ρ0 kA; and when ka [Gt ] 1 the tilting-sliding mechanism identified in the previous paper is relevant and the results approximate to those for an unbounded fluid, except that the smallest Rayleigh number for a neutral disturbance is not zero but is of order (ka-1. In the case of an undisturbed density which varies only in a central layer of thickness l, the same mechanism is at work when the horizontal lengthscale of the disturbance is large compared with l, resulting in high growth rates and a critical Rayleigh number which vanishes with l/a. Estimates of the growth rate are given for some particular density profiles.
Disturbance interactions in wave triads and multiwave systems of various configurations are investigated to reveal the mechanism of laminar-turbulent transition in Blasius and pressure-gradient boundary layers. The averaging method of weakly nonlinear instability theory in quasi-parallel flows is applied. Tollmien-Schlichting-wave resonant interaction is shown to be the only leading mechanism of subharmonic (S)-type transition. The mechanism universally dominates in boundary layers excited by sufficiently small initial disturbances. The role of any other mode is inefficient. Weakly nonlinear models are concluded not to explain the K-type transition scenario. The results of the study are employed to interpret physical and numerical experimental data.