Showing papers in "Journal of Fluid Mechanics in 1985"

Spectral and statistical properties of the equilibrium range in wind-generated gravity waves

Abstract: Recent measurements of wave spectra and observations by remote sensing of the sea surface indicate that the author's (1958) conception of an upper-limit asymptote to the spectrum, independent of wind stress, is no longer tenable. The nature of the equilibrium range is reexamined, using the dynamical insights into wave–wave interactions, energy input from the wind and wave-breaking that have been developed since 1960. With the assumption that all three of these processes are important in the equilibrium range, the wavenumber spectrum is found to be of the form , where p ∼ ½ and the frequency spectrum is proportional to u*gσ−4. These forms have been found by Kitaigorodskii (1983) on a quite different dynamical basis; the latter is consistent with the form found empirically by Toba (1973) and later workers. Various derived spectra, such as those of the sea-surface slope and of an instantaneous line traverse of the surface, are also given, as well as directional frequency spectra and frequency spectra of slope.The theory also provides expressions for the spectral rates of action, energy and momentum loss from the equilibrium range by wave-breaking and for the spectrally integrated rates across the whole range. These indicate that, as a wave field develops with increasing fetch or duration, the momentum flux to the underlying water by wave-breaking increases asymptotically to a large fraction of the total wind stress and that the energy flux to turbulence in the water, occurring over a wide range of scales, increases logarithmically as the extent of the equilibrium range increases. Interrelationships are pointed out among different sets of measurements such as the various spectral levels, the directional distributions, the total mean-square slope and the ratio of downwind to crosswind mean-square slopes.Finally, some statistical characteristics of the breaking events are deduced, including the expected length of breaking fronts (per unit surface area) with speeds of advance between c and c+dc and the number of such breaking events passing a given point per unit time. These then lead to simple expressions for the density of whitecapping, those breaking events that produce bubbles and trails of foam, the total number of whitecaps passing a given point per unit time and, more tenuously, the whitecap coverage.

Absolute and convective instabilities in free shear layers

Abstract: The absolute or convective character of inviscid instabilities in parallel shear flows can be determined by examining the branch-point singularities of the dispersion relation for complex frequencies and wavenumbers. According to a criterion developed in the study of plasma instabilities, a flow is convectively unstable when the branch-point singularities are in the lower half complex-frequency plane. These concepts are applied to a family of free shear layers with varying velocity ratio their average velocity. It is demonstrated that spatially growing waves can only be observed if the mixing layer is convectively unstable, i.e. when the velocity ratio is smaller than Rt = 1.315. When the velocity ratio is larger than Rt, the instability develops temporally. Finally, the implications of these concepts are discussed also for wakes and hot jets.

Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence

Abstract: In a three dimensional simulation higher order derivative correlations, including skewness and flatness factors, are calculated for velocity and passive scalar fields and are compared with structures in the flow. The equations are forced to maintain steady state turbulence and collect statistics. It is found that the scalar derivative flatness increases much faster with Reynolds number than the velocity derivative flatness, and the velocity and mixed derivative skewness do not increase with Reynolds number. Separate exponents are found for the various fourth order velocity derivative correlations, with the vorticity flatness exponent the largest. Three dimensional graphics show strong alignment between the vorticity, rate of strain, and scalar-gradient fields. The vorticity is concentrated in tubes with the scalar gradient and the largest principal rate of strain aligned perpendicular to the tubes. Velocity spectra, in Kolmogorov variables, collapse to a single curve and a short minus 5/3 spectral regime is observed.

The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation

Abstract: The newly developed simulation method known as Stokesian dynamics is used to investigate the rheological behaviour of concentrated suspensions. Both the detailed microstructure (e.g. pair-distribution function) and the macroscopic properties are determined for a suspension of identical rigid spherical particles in a simple shear flow. The suspended particles interact through both hydrodynamic and non-hydrodynamic forces. For suspensions with purely hydrodynamic forces, the increase in the suspension viscosity with volume fraction ϕ is shown to be caused by particle clustering. The cluster formation results from the lubrication forces, and the simulations of a monolayer of spheres show a scaling law for the cluster size: lc ∼ [1 − (ϕ/ϕm)½]−1, where ϕm is the maximum volume fraction that can shear homogeneously. The simulation results suggest that the suspension viscosity becomes infinite at the percolation-like threshold ϕm owing to the formation of an infinite cluster. The predicted simulation viscosities are in very good agreement with experiment. A suspension with short-range repulsive interparticle forces is also studied, and is seen to have a non-Newtonian rheology. Normal-stress differences arise owing to the anisotropic local structure created by the interparticle forces. The repulsive forces also reduce particle clustering, and as a result the suspension is shear-thickening.

Dispersion in fixed beds

Abstract: A macroscopic equation of mass conservation is obtained by ensemble-averaging the basic conservation laws in a porous medium. In the long-time limit this ‘macro-transport’ equation takes the form of a macroscopic Fick's law with a constant effective diffusivity tensor. An asymptotic analysis in low volume fraction of the effective diffusivity in a bed of fixed spheres is carried out for all values of the Peclet number ℙ = Ua/Df, where U is the average velocity through the bed. a is the particle radius and Df is the molecular diffusivity of the solute in the fluid. Several physical mechanisms causing dispersion are revealed by this analysis. The stochastic velocity fluctuations induced in the fluid by the randomly positioned bed particles give rise to a convectively driven contribution to dispersion. At high Peclet numbers, this convective dispersion mechanism is purely mechanical, and the resulting effective diffusivities are independent of molecular diffusion and grow linearly with ℙ. The region of zero velocity in and near the bed particles gives rise to non-mechanical dispersion mechanisms that dominate the longitudinal diffusivity at very high Peclet numbers. One such mechanism involves the retention of the diffusing species in permeable particles, from which it can escape only by molecular diffusion, leading to a diffusion coefficient that grows as ℙ2. Even if the bed particles are impermeable, non-mechanical contributions that grow as ℙ ln ℙ and ℙ2 at high ℙ arise from a diffusive boundary layer near the solid surfaces and from regions of closed streamlines respectively. The results for the longitudinal and transverse effective diffusivities as functions of the Peclet number are summarized in tabular form in §6. Because the same physical mechanisms promote dispersion in dilute and dense fixed beds, the predicted Peclet-number dependences of the effective diffusivities are applicable to all porous media. The theoretical predictions are compared with experiments in densely packed beds of impermeable particles, and the agreement is shown to be remarkably good.

Creeping motion of a sphere through a Bingham plastic

Abstract: A solid sphere falling through a Bingham plastic moves in a small envelope of fluid with shape that depends on the yield stress. A finite-element/Newton method is presented for solving the free-boundary problem composed of the velocity and pressure fields and the yield surfaces for creeping flow. Besides the outer surface, solid occurs as caps at the front and back of the sphere because of the stagnation points in the flow. The accuracy of solutions is ascertained by mesh refinement and by calculation of the integrals corresponding to the maximum and minimum variational principles for the problem. Large differences from the Newtonian values in the flow pattern around the sphere and in the drag coefficient are predicted, depending on the dimensionless value of the critical yield stress Yg below which the material acts as a solid. The computed flow fields differ appreciably from Stokes’ solution. The sphere will fall only when Yg is below 0.143 For yield stresses near this value, a plastic boundary layer forms next to the sphere. Boundary-layer scalings give the correct forms of the dependence of the drag coefficient and mass-transfer coefficient on yield stress for values near the critical one. The Stokes limit of zero yield stress is singular in the sense that for any small value of Yg there is a region of the flow away from the sphere where the plastic portion of the viscosity is at least as important as the Newtonian part. Calculations For the approach of the flow field to the Stokes result are in good agreement with the scalings derived from the matched asymptotic expansion valid in this limit.

Observations of bispectra of shoaling surface gravity waves

Abstract: Aspects of the nonlinear dynamics of waves shoaling between 9 and 1 m water depths are elucidated via the bispectrum. The biphase values associated with significant bicoherence levels in 9 m depth are consistent with Stokes-like nonlinearities, but as the water depth decreases the waves evolve through a slightly skewed shape somewhat asymmetrical to a vertical axis toward a highly asymmetrical unskewed sawtooth shape. The real and imaginary parts of the bispectrum are the contributions to skewness and asymmetry from individual frequency pairs.

Evolution of a single wake behind a pair of bluff bodies

Abstract: In this paper the flow behind a pair of bluff bodies placed side by side in a stream is studied using a variety of flow-visualization methods. Above a critical gap size between the bodies, vortex-shedding synchronization occurs, either in phase or in antiphase. It has previously been assumed that such synchronization forms a wake comprising two parallel vortex streets in phase and in antiphase respectively. In the present paper we find that two antiphase streets are indeed formed, although in-phase shedding leads to the development of a single large-scale wake. The vortices which are formed simultaneously at the cylinders rotate around one another downstream, each pair forming a ‘binary vortex’. The combined wake comprises a street of such vortices, which we term a binary vortex street. Below a critical gap size between the bluff bodies the flow becomes asymmetric. We observe in this regime certain harmonic modes of vortex shedding whereby the shedding frequency on one side of the wake is a multiple of that on the other. Again, a large-scale wake is formed downstream. The present observations lead to a new interpretation of hot-wire-frequency data from other studies in terms of the harmonic modes.

A unified bar–bend theory of river meanders

Abstract: A two-dimensional model of flow and bed topography in sinuous channels with erodible boundaries is developed and applied in order to investigate the mechanism of meander initiation. By reexamining the problem recently tackled by Ikeda, Parker & Sawai (1981), a previously undiscovered ‘resonance’ phenomenon is detected which occurs when the values of the relevant parameters fall within a neighbourhood of certain critical values. It is suggested that the above resonance controls the bend growth, and it is shown that it is connected in some sense with bar instability. In fact, by performing a linear stability analysis of flow in straight erodible channels, resonant flow in sinuous channels is shown to occur when curvature ‘forces’ a ‘natural’ solution represented by approximately steady perturbations of the alternate bar type. A comparison with experimental observations appears to support the idea that resonance is associated with meander formation.

Computer simulation of granular shear flows

Abstract: Detailed understanding of flowing granular materials is severly hampered by the deficiencies of present experimental methods. To help increase the information base, a computer simulation has been developed to describe two-dimensional unidirectional flows of inelastic fully rough particles. This paper presents the results of a Couette shear-flow simulation. The results include distributions of velocity, density and granular temperature (a measure of the kinetic energy contained in the random particle motions). The effects of density and shear rate on the granular temperature are explored. Shear and normal forces on the solid walls are compared with experimental and theoretical results. The behaviour of the particles in the simulated flow is examined and assessments are made of the collision angle and velocity distributions. The development of a distinct, 'layered' microstructure is observed in high-density granular flows.

Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 1. Fundamentals

Abstract: The well-known analogy between the Euler equations for steady flow of an inviscid incompressible fluid and the equations of magnetostatic equilibrium in a perfectly conducting fluid is exploited in a discussion of the existence and structure of solutions to both problems that have arbitrarily prescribed topology. A method of magnetic relaxation which conserves the magnetic-field topology is used to demonstrate the existence of magnetostatic equilibria in a domain [Dscr ] that are topologically accessible from a given field B0(x) and hence the existence of analogous steady Euler flows. The magnetostatic equilibria generally contain tangential discontinuities (i.e. current sheets) distributed in some way in the domain, even although the initial field B0(x) may be infinitely differentiable, and particular attention is paid to the manner in which these current sheets can arise. The corresponding Euler flow contains vortex sheets which must be located on streamsurfaces in regions where such surfaces exist. The magnetostatic equilibria are in general stable, and the analogous Euler flows are (probably) in general unstable.The structure of these unstable Euler flows (regarded as fixed points in the function space in which solutions of the unsteady Euler equations evolve) may have some bearing on the problem of the spatial structure of turbulent flow. It is shown that the Euler flow contains blobs of maximal helicity (positive or negative) which may be interpreted as ‘coherent structures’, separated by regular surfaces on which vortex sheets, the site of strong viscous dissipation, may be located.

Scattering of acoustic waves into Tollmien-Schlichting waves by small streamwise variations in surface geometry

Abstract: By using the triple-deck scaling of Stewartson (1969) and Messiter (1970) it is shown that small but relatively sudden surface geometry variations that produce only very weak static pressure variations can nevertheless produce strong, i.e. O(1), coupling between an externally imposed acoustic disturbance and a spatially growing Tollmien-Schlichting wave. The analysis provides a qualitative explanation of the Leehey and Shapiro (1979) boundary-layer receptivity measurements and is in good quantitative agreement with the Aizin and Poliakov (1979) experiment. It may also explain why small 'trip wires' can promote early transition.

The hydrodynamic stability of flow over Kramer-type compliant surfaces. Part 1. Tollmien-Schlichting instabilities

Abstract: The hydrodynamic stability of flows over Kramer-type compliant surfaces is studied. Two main types of instability are considered. First, there are those which could not exist without viscosity, termed Tollmien–Schlichting Type Instabilities (TSI). Secondly, there are Flow-Induced Surface Instabilities (FISI), that depend fundamentally on surface flexibility and could exist with an inviscid fluid flow. Part 1, the present paper, deals mainly with the first type. The original Kramer experiments and the various subsequent attempts to confirm his results are reviewed, together with experimental studies of transition in flows over compliant surfaces and theoretical work concerned with the hydrodynamic stability of such flows.The Kramer-type compliant surface is assumed to be an elastic plate supported by springs which are modelled by an elastic foundation. It is also assumed that the plate is backed by a viscous fluid substrate having, in general, a density and viscosity different from the mainstream fluid. The motion of the substrate fluid is assumed to be unaffected by the presence of the springs and is determined by solving the linearized Navier–Stokes equations. The visco-elastic properties of the plate and springs are taken into account approximately by using a complex elastic modulus which leads to complex flexural rigidity and spring stiffness. Values for the various parameters characterizing the surface properties are estimated for the actual Kramer coatings.The boundary-layer stability problem for a flexible surface is formulated in a similar way to that of Landahl (1962) whereby the boundary condition at the surface is expressed in terms of an equality between the surface and boundary-layer admittances. This form of the boundary condition is exploited to develop an approximate theory which determines whether a particular change to the mechanical properties of the surface will be stabilizing or destabilizing with respect to the TSI. It is shown that a reduction in flexural rigidity and spring stiffness, an increase in plate mass, and the presence of an inviscid fluid substrate are all stabilizing, whereas viscous and visco-elastic damping are destabilizing.Numerical solutions to the Orr–Sommerfeld equation are also obtained. Apart from Kramer-type compliant surfaces, solutions are also presented for the rigid wall, for the spring-backed tensioned membrane with damping, previously considered by Landahl & Kaplan (1965), and for some of the compliant surfaces investigated experimentally by Babenko and his colleagues. The results for the Kramer-type compliant surfaces on the whole confirm the predictions of the simple theory. For a free-stream speed of 18 m/s the introduction of a viscous substrate leads to a complex modal interaction between the TSI and FISI. A single combined unstable mode is formed in the case of highly viscous substrate fluids and in this case increased damping has a stabilizing effect. When the free-stream speed is reduced to 15 m/s the modal interaction no longer occurs. In this case the effects of combined viscous and visco-elastic damping are investigated. It is found that damping tends to have a strong stabilizing effect on the FISI, in the form of travelling-wave flutter, but a weaker destabilizing effect on the TSI. The opposing effects of damping on the two modes of instability forms the basis of a possible explanation for Kramer's empirical observation of an optimum substrate viscosity. Results obtained using the e9 method also indicate that a substantial transition delay is theoretically possible for flows over Kramer's compliant coatings.

Large-scale structures in a forced turbulent mixing layer

Abstract: The large-scale structures that occur in a forced turbulent mixing layer at moderately high Reynolds numbers have been modelled by linear inviscid stability theory incorporating first-order corrections for slow spatial variations of the mean flow. The perturbation stream function for a spatially growing time-periodic travelling wave has been numerically evaluated for the measured linearly diverging mean flow. In an accompanying experiment periodic oscillations were imposed on the turbulent mixing layer by the motion of a small flap at the trailing edge of the splitter plate that separated the two uniform streams of different velocity. The results of the numerical computations are compared with experimental measurements.When the comparison between experimental data and the computational model was made on a purely local basis, agreement in both the amplitude and phase distribution across the mixing layer was excellent. Comparisons on a global scale revealed, not unexpectedly, less good accuracy in predicting the overall amplification.

Resonant reflection of surface water waves by periodic sandbars

Abstract: One of the possible mechanisms of forming offshore sandbars parallel to a coast is the wave-induced mass transport in the boundary layer near the sea bottom. For this mechanism to be effective, sufficient reflection must be present so that the waves are partially standing. The main part of this paper is to explain a theory that strong reflection can be induced by the sandbars themselves, once the so-called Bragg resonance condition is met. For constant mean depth and simple harmonic waves this resonance has been studied by Davies (1982), whose theory, is however, limited to weak reflection and fails at resonance. Comparison of the strong reflection theory with Heathershaw's (1982) experiments is made. Furthermore, if the incident waves are slightly detuned or slowly modulated in time, the scattering process is found to depend critically on whether the modulational frequency lies above or below a threshold frequency. The effects of mean beach slope are also studied. In addition, it is found for periodically modulated wave groups that nonlinear effects can radiate long waves over the bars far beyond the reach of the short waves themselves. Finally it is argued that the breakpoint bar of ordinary size formed by plunging breakers can provide enough reflection to initiate the first few bars, thereby setting the stage for resonant reflection for more bars.

Observations of rapidly flowing granular-fluid materials

Abstract: The rapid shearing of a mixture of cohesionless glass spheres and air or water was studied in an annular, parallel-plate shear cell designed after Savage (1978). Two types of flow were observed. In the first type of flow the entire mass of the granular material was mobilized. At high shear rates the shear and normal stresses were found to be quadratically dependent upon the mean shear rate (at constant volume concentration), in general agreement with the observations of Bagnold (1954) and Savage & Sayed (1984), and the ‘kinetic’ theory of Jenkins & Savage (1983). The stresses were found to be weakly dependent on the volume concentration up to approximately 0.5, and strongly dependent above this concentration. For flows in which water was the interstitial fluid, the ratio of the shear stress to the normal stress was slightly higher (than in air), and the stresses at lower shear rates were found to be more nearly linearly related to the shear rate. It is suggested that these effects are contributed to by the viscous dampening of grain motions by the water. The second type of flow was distinguished by the existence of an internal boundary above which the granular material deformed rapidly, but below which the granular material remained rigidly locked in place. The thickness of the shearing layer was measured to be between 5 and 15 grain diameters. The stress ratio at the bottom of the shearing layer was found to be nearly constant, suggesting the internal boundary is a consequence of the immersed weight of the shearing grains, and may be described by a Coulomb yield criterion. A scaled concentration is proposed to compare similar data obtained using different-sized materials or different apparatus. An intercomparison of the two types of flow studied, along with a comparison between the present experiments and those of Bagnold (1954) and Savage & Sayed (1984), suggests that the nature of the boundaries can have a significant effect upon the dynamics of the entire flow.

Effective equations for wave propagation in bubbly liquids

Abstract: We derive a system of effective equations for wave propagation in a bubbly liquid. Starting from a microscopic description, we obtain the effective equations by using Foldy's approximation in a nonlinear setting. We discuss in detail the range of validity of the effective equations as well as some of their properties.

Sinusoidal flow relative to circular cylinders

Abstract: The motions of vortices around single cylinders and around pairs of cylinders in relative sinusoidal flow are investigated in this paper. Using simultaneous flow visualization and force measurements, the vortex motions are related to the fluid-induced lift and in-line forces. For the single cylinder, several repeatable patterns of vortex shedding are identified within particular ranges of flow amplitude. The process of pairing of vortices from a previous half cycle with those in a present half cycle is fundamental to all the patterns. Visualization is shown to be more effective in a reference frame which is fixed with respect to the undisturbed fluid rather than with respect to the cylinders. For this reason, the examples of vortex motions are taken from a rig in which vertical cylinders are oscillated in a tank of fluid. By oscillating a pair of cylinders over a range of gaps, orientations and amplitudes, it is found that the vortex-shedding patterns identified for a single cylinder can synchronize either in phase or in antiphase between the two cylinders. Such observations help to explain how lift and in-line forces are influenced by cylinder proximity and in some cases these forces are significantly magnified. Force coefficients are evaluated for both the single cylinder and the pair of cylinders.

Galilean invariance of subgrid-scale stress models in the large-eddy simulation of turbulence

Abstract: The modelling of the subgrid-scale stresses in the large-eddy simulation of turbulence is examined from a theoretical standpoint. While there are a variety of approaches that have been proposed, it is demonstrated that one of the more recent models gives rise to equations of motion for the large eddies of turbulence which are not Galilean-invariant. Consequently, this model cannot be of any general applicability, since it is inconsistent with the basic physics of the problem, which requires that the description of the turbulence be the same in all inertial frames of reference. Alternative models that have been proposed which are properly invariant are discussed and compared.

Wave generation by an oscillating surface-pressure and its application in wave-energy extraction

Abstract: A two-dimensional analysis, based on linear surface-wave theory, is developed for an oscillating-water-column wave-energy device in water of arbitrary constant depth. The immersed part of the structure is assumed of shallow draught except for a submerged vertical reflecting wall. Both the cases of linear and nonlinear power take-off are considered. The results show that air compressibility can be important in practice, and its effects may in general be satisfactorily represented by linearization. The analysis indicates that using a turbine whose characteristic exhibits a phase difference between pressure and flow rate may be a method of strongly reducing the chamber length and turbine size with little change in the capability of energy extraction from regular waves. It was found in two examples of devices with strongly nonlinear power take-off that the maximum efficiency is only marginally inferior to what can be achieved in the linear case.

Some observations of the transition process on the windward face of a long yawed cylinder

Abstract: An experiment has been performed to determine the effect of yaw upon transition in the boundary layer formed on the windward face of a long cylinder. The china-clay-evaporation and surface-oil-flow techniques have been used to study the development of the fixed-wavelength stationary disturbances which are characteristic of cross-flow instability. It has been found that the boundary layer is also susceptible to time-dependent disturbances which grow to very large amplitudes prior to the onset of transition. These disturbances have been studied with a hot-wire anemometer. The conditions necessary for the onset and completion of transition have been determined by the use of surface Pitot tubes. Data from the experiment have been compared with the simple criteria for instability and transition which were proposed by Owen & Randall over thirty years ago. In general it has been found that these criteria are inadequate, and, where possible, improvements have been proposed. The raw data are presented in sufficient detail for them to be used to test, or calibrate, future theoretical models of the transition process in three-dimensional boundary-layer flows.

Boundary effects on electrophoretic motion of colloidal spheres

Abstract: An analysis is presented for electrophoretic motion of a charged non-conducting sphere in the proximity of rigid boundaries. An important assumption is that κa → ∞, where a is the particle radius and κ is the Debye screening parameter. Three boundary configurations are considered: single flat wall, two parallel walls (slit), and a long circular tube. The boundary is assumed a perfect electrical insulator except when the applied field is directed perpendicular to a single wall, in which case the wall is assumed to have a uniform potential (perfect conductor). There are three basic effects causing the particle velocity to deviate from the value given by Smoluchowski's classic equation: first, a charge on the boundary causes electro-osmotic flow of the suspending fluid; secondly, the boundary alters the interaction between the particle and applied electric field; and, thirdly, the boundary enhances viscous retardation of the particle as it tries to move in response to the applied field. Using a method of reflections, we determine the particle velocity for a constant applied field in increasing powers of λ up to O(λ6), where λ is the ratio of particle radius to distance from the boundary. Ignoring the O(λ0) electro-osmotic effect, the first effect attributable to proximity of the boundary is O(λ3) for all boundary configurations, and in cases when the applied field is parallel to the boundaries the electrophoretic velocity is proportional to ζp − ζw, the difference in zeta potential between the particle and boundary.

Structure of large-scale vortices and unsteady reverse flow in the reattaching zone of a turbulent separation bubble

Abstract: This paper describes experiments concerning the structure of large-scale vortices and the unsteady reverse-flow properties in the reattaching zone of a nominally two-dimensional separation bubble formed at the leading edge of a blunt flat plate with right-angled corners. The experiment was performed in a wind tunnel with a constant Reynolds number 2.6 × 104 (based on the main-flow velocity and the thickness of the plate). Split-film probes, being sensitive to instantaneous reversals of flow direction, were extensively employed. An important feature of this study is a judicious use of surface-pressure fluctuations as a conditioning signal to educe the structure of the large-scale vortices.Distributions of fluctuating-velocity vectors and contour lines of high-frequency turbulent energy in a few space–time domains are presented and discussed. The most economical interpretation of these space-time distributions is that the large-scale vortices in the reattaching zone are hairpin vortices whose configuration is sketched in the text. The unsteady flow in the reattaching zone is mainly governed by two agents; the motion of the large-scale vortices and the low-frequency unsteadiness. The unsteady flow is clarified in terms of the motion (in a space–time domain) of zeros of the longitudinal velocity close to the surface of the plate; the effects of the two agents on this motion are presented separately. On the basis of these results, a mathematical model of the unsteady flow in the reattaching zone is suggested and found to yield good comparison with measured reverse-flow intermittency and frequency of local-flow reversals. It appears that the separation bubble experiences shrinkage and enlargement in connection with the low-frequency unsteadiness and that the speed of shrinkage is much greater than that of enlargement. The strength of the large-scale vortices in the reattaching zone seems to be dependent on the phase of the low-frequency unsteadiness.

A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation

Abstract: In existing experiments it is known that the slow evolution of nonlinear deep-water waves exhibits certain asymmetric features. For example, an initially symmetric wave packet of sufficiently large wave slope will first lean forward and then split into new groups in an asymmetrical manner, and, in a long wavetrain, unstable sideband disturbances can grow unequally to cause an apparent downshift of carrier-wave frequency. These features lie beyond the realm of applicability of the celebrated cubic Schrodinger equation (CSE), but can be, and to some extent have been, predicted by weakly nonlinear theories that are not limited to slowly modulated waves (i.e. waves with a narrow spectral band). Alternatively, one may employ the fourth-order equations of Dysthe (1979), which are limited to narrow-banded waves but can nevertheless be solved more easily by a pseudospectral numerical method. Here we report the numerical simulation of three cases with a view to comparing with certain recent experiments and to complement the numerical results obtained by others from the more general equations.

Effect of rotation on isotropic turbulence - Computation and modelling

Abstract: This paper uses numerical simulation to analyse the effects of uniform rotation on homogeneous turbulence. Both large-eddy and full simulations were made. The results indicate that the predominant effect of rotation is to decrease the rate of dissipation of the turbulence and increase the lengthscales, especially those along the axis of rotation. These effects are a consequence of the reduction, due to the generation of inertial waves, of the net energy transfer from large eddies to small ones. Experiments are also influenced by a more complicated interaction between the rotation and the wakes of the turbulence-generating grid which modifies the nominal initial conditions in the experiment. The latter effect is accounted for in simulations by modifying the initial conditions. Finally, a two-equation model is proposed that accounts for the effects of rotation and is able to reproduce the experimental decay of the turbulent kinetic energy.

The structure of the vorticity field in turbulent channel flow. I - Analysis of instantaneous fields and statistical correlations

Abstract: An investigation into the existence of hairpin vortices in turbulent channel flow is conducted using a database generated by the large-eddy simulation technique. It is shown that away from the wall the distribution of the inclination angle of vorticity vector gains its maximum at about 45° to the wall. Two-point correlations of velocity and vorticity fluctuations strongly support a flow model consisting of vortical structures inclined at 45° to the wall. The instantaneous vorticity vectors plotted in planes inclined at 45° show that the flow contains an appreciable number of hairpins. Vortex lines are used to display the three-dimensional structure of hairpins, which are shown to be generated from deformation (or roll-up) of sheets of transverse vorticity.

Chaotic mode competition in parametrically forced surface waves

Abstract: Vertical forcing of a fluid layer leads to standing waves by means of a subharmonic instability. When the driving amplitude and frequency are chosen to be near the intersection of the stability boundaries of two nearly degenerate modes, we find that they can compete with each other to produce either periodic or chaotic motion on a slow timescale. We utilize digital image-processing methods to determine the time-dependent amplitudes of the competing modes, and local-sampling techniques to study the onset of chaos in some detail. Reconstruction of the attractors in phase space shows that in the chaotic regime the dimension of the attractor is fractional and at least one Lyapunov exponent is positive. The evidence suggests that a theory incorporating four coupled slow variables will be sufficient to account for the mode competition.

Unsteady separation in a boundary layer produced by an impinging jet

Abstract: A vortex-induced unsteady separation was investigated experimentally in the laminar boundary layer produced by an axisymmetric jet impinging normally onto a flat plate. By forcing the air jet, primary ring vortices were periodically generated in the jet shear layer. Phase-locked flow visualization showed that the wall-jet boundary layer separated periodically and evolved into a secondary vortex counter rotating with respect to the primary vortex. The unsteady separation is induced by the primary vortex and moves downstream in the radial mean-flow direction. Phase-averaged hot-wire measurements using a parallel-wire sensor in the vicinity of the unsteady separation provided data for locating the onset of separation in space and time. The data revealed that the unsteady separation originated from a local shear layer which was initiated by the unsteady adverse pressure gradient produced by the primary vortex.

Do Rossby-wave critical layers absorb, reflect, or over-reflect?

Abstract: The Stewartson-Warn-Warn (SWW) solution for the time evolution of an inviscid, nonlinear Rossby-wave critical layer, which predicts that the critical layer will alternate between absorbing and over-reflecting states as time goes on, is shown to be hydrodynamically unstable. The instability is a two-dimensional shear instability, owing its existence to a local reversal of the cross-stream absolute vorticity gradient within the long, thin Kelvin cat’s eyes of the SWW streamline pattern. The unstable condition first develops while the critical layer is still an absorber, well before the first over-reflecting stage is reached. The exponentially growing modes have a two-scale cross-stream structure like that of the basic SWW solution. They are found analytically using the method of matched asymptotic expansions, enabling the problem to be reduced to a transcendental equation for the complex eigenvalue. Growth rates are of the order of the inner vorticity scale Sq, i.e. the initial absolute vorticity gradient dq,/dy times the critical-layer width scale. This is much faster than the time evolution of the SWW solution itself, albeit much slower than the shear rate du,/dy of the basic flow. Nonlinear saturation of the growing instability is expected to take place in a central region of width comparable to the width of the SWW cat’s-eye pattern, probably leading to chaotic motion there, with very large ‘eddy-viscosity ’ values. Those values correspond to critical-layer Reynolds numbers A-’ Q 1, suggesting that for most initial conditions the time evolution of the critical layer will depart drastically from that predicted by the SWW solution. A companion paper (Haynes 1985) establishes that the instability can, indeed, grow to large enough amplitudes for this to happen. The simplest way in which the instability could affect the time evolution of the critical layer would be to prevent or reduce the oscillations between over-reflecting and absorbing states which, according to the SWW solution, follow the first onset of perfect reflection. The possibility that absorption (or over-reflection) might be prolonged indefinitely is ruled out, in many cases of interest (even if the ‘eddy viscosity’ is large), by the existence of a rigorous, general upper bound on the magnitude of the time-integrated absorptivity a(t). The bound is uniformly valid for all time t. The absorptivity a(t) is defined aa the integral over all past t of the jump in the wave-induced Reynolds stress across the critical layer. In typical cases the bound implies that, no matter how large t may become, I a(t) I cannot greatly exceed the rate of absorption predicted by linear theory multiplied by the timescale on which linear theory breaks down, say the time for the cat’s-eye flow to twist up the absolute vorticity contours by about half a turn. An alternative statement is that I a(t) I cannot greatly exceed the initial absolute vorticity gradient dq,/dy times the cube of the

Determining modes and fractal dimension of turbulent flows

Abstract: Research on the abstract properties of the Navier–Stokes equations in three dimensions has cast a new light on the time-asymptotic approximate solutions of those equations. Here heuristic arguments, based on the rigorous results of that research, are used to show the intimate relationship between the sufficient number of degrees of freedom describing fluid flow and the bound on the fractal dimension of the Navier–Stokes attractor. In particular it is demonstrated how the conventional estimate of the number of degrees of freedom, based on purely physical and dimensional arguments, can be obtained from the properties of the Navier–Stokes equation. Also the Reynolds-number dependence of the sufficient number of degrees of freedom and of the dimension of the attractor in function space is elucidated.