bluuebunny/crossref_metadata_2025 huggingface dataset:
Waves and turbulence on a beta-plane
Authors: Rhines Peter B.
Published: March 2006

Two-dimensional eddies in a homogeneous fluid at large Reynolds number, if closely packed, are known to evolve towards larger scales. In the presence of a restoring force, the geophysical beta-effect, this cascade produces a field of waves without loss of energy, and the turbulent migration of the dominant scale nearly ceases at a wavenumber kβ = (β/2U)½ independent of the initial conditions other than U, the r.m.s. particle speed, and β, the northward gradient of the Coriolis frequency.

The conversion of turbulence into waves yields, in addition, more narrowly peaked wavenumber spectra and less fine-structure in the spatial maps, while smoothly distributing the energy about physical space.

The theory is discussed, using known integral constraints and similarity solutions, model equations, weak-interaction wave theory (which provides the terminus for the cascade) and other linearized instability theory. Computer experiments with both finite-difference and spectral codes are reported. The central quantity is the cascade rate, defined as
\[
T = 2\int_0^{\infty} kF(k)dk/U^3\langle k\rangle ,
\]
where F is the nonlinear transfer spectrum and 〈k〉 the mean wavenumber of the energy spectrum. (In unforced inviscid flow T is simply U−1d〈k〉−1/dt, or the rate at which the dominant scale expands in time t.) T is shown to have a mean value of 3·0 × 10−2 for pure two-dimensional turbulence, but this decreases by a factor of five at the transition to wave motion. We infer from weak-interaction theory even smaller values for k [Lt ] kβ.

After passing through a state of propagating waves, the homogeneous cascade tends towards a flow of alternating zonal jets which, we suggest, are almost perfectly steady. When the energy is intermittent in space, however, model equations show that the cascade is halted simply by the spreading of energy about space, and then the end state of a zonal flow is probably not achieved.

The geophysical application is that the cascade of pure turbulence to large scales is defeated by wave propagation, helping to explain why the energy-containing eddies in the ocean and atmosphere, though significantly nonlinear, fail to reach the size of their respective domains, and are much smaller. For typical ocean flows, $k_{\beta}^{-1} = 70\,{\rm km} $, while for the atmosphere, $k_{\beta}^{-1} = 1000\,{\rm km}$. In addition the cascade generates, by itself, zonal flow (or more generally, flow along geostrophic contours).
On the role of density jumps in the reflexion and breaking of internal gravity waves
Authors: Delisi Donald P. , Orlanski Isidoro
Published: March 2006

A laboratory experiment is presented which examines the role of density jumps in the reflexion and breaking of internal gravity waves. It is found that the measured phase shift of the reflected wave and the measured amplitude of the density jump are in good agreement with linear theory. Local overturning occurs when wave amplitude becomes large, and there appears to be a critical amplitude above which overturning will occur and below which it will not. The overturning seems to be due to local gravitational instability, caused by the horizontal advection of density. Overturning changes the basic flow field in the region of interaction; and it results in smaller-scale motions.
On energy and enstrophy exchanges in two-dimensional non-divergent flow
Authors: Merilees Philip E. , Warn Helen
Published: March 2006

It is shown that, in two-dimensional non-divergent flow in a bounded region, roughly 70% of triad interactions exchange more energy with longer wavelengths than with shorter wavelengths whilst roughly 40% exchange more enstrophy with longer wavelengths than with shorter wavelengths.
A study of turbulent boundary-layer separation and reattachment
Authors: Perry A. E. , Fairlie B. D.
Published: March 2006

An experimental and theoretical study is made of a suddenly separating and reattaching two-dimensional turbulent boundary layer on a flat surface. A separation bubble is formed on the floor of a wide parallel-sided wind-tunnel duct with the pressure field causing the bubble formation produced by fixing the shape of the flexible roof of the duct. Boundary layers on the roof are controlled and remain attached. It is found that a very satisfactory model for the flow is an inviscid one.

The boundary layer on the floor of the duct is represented by a region of constant vorticity with slip at the boundary, and it is assumed that the separation process is dominated by the interaction between this ‘vortical’ region and the irrotational field between the vortical region and the roof (of prescribed shape). The interface between the rotational and irrotational regions is a free boundary and may be located when all necessary boundary conditions are given. These conditions include two characteristic parameters for the adverse-pressure-gradient turbulent boundary layer which is developing upstream of the region of interest.

The problem is solved by an electrical analog method. The theoretical size and shape of the bubble and positions of separation and reattachment are in agreement with observations. The advantage of the model over most previous attempts to predict separation is that the governing equations are elliptic rather than parabolic or hyperbolic and therefore the interaction between the boundary-layer flow and the irrotational free stream is included in the calculations.
Viscous incompressible flow between concentric rotating spheres. Part 3. Linear stability and experiments
Authors: Munson B. R. , Menguturk M.
Published: March 2006

The stability of flow of a viscous incompressible fluid contained between a stationary outer sphere and rotating inner sphere is studied theoretically and experimentally. Previous theoretical results concerning the basic laminar flow (part 1) are compared with experimental results. Small and large Reynolds number results are compared with Stokes-flow and boundary-layer solutions. The effect of the radius ratio of the two spheres is demonstrated. A linearized theory of stability for the laminar flow is formulated in terms of toroidal and poloidal potentials; the differential equations governing these potentials are integrated numerically. It is found that the flow is subcritically unstable and that the observed instability occurs at a Reynolds number close to the critical value of the energy stability theory. Observations of other flow transitions, at higher values of the Reynolds number, are also described. The character of the stability of the spherical annulus flow is found to be strongly dependent on the radius ratio.
A Galerkin method for a class of steady, two-dimensional, incompressible, laminar boundary-layer flows
Authors: Hsu Chen-Chi
Published: March 2006

A Galerkin method is proposed for a class of boundary-layer flow problems. In this method, the assumed solution is composed of an auxiliary function and a series solution. The representing functions used in the series solution are orthonormal eigenfunctions, closely related to that of the boundary-layer equation, and are independent of the initial condition, as well as the boundary conditions. The reduced system of stiff, first-order, nonlinear, ordinary differential equations then has diagonal dominance for the first part of the flow region. The proposed method has been tested on two representative flows. Numerical experiments show that highly accurate results can be obtained for the entire boundary-layer flow region, if the auxiliary function satisfying the initial and boundary conditions is chosen to satisfy the first compatibility condition of the upstream flow region. In fact the computation is rather simple, and the numerical integration of the reduced initial-value problem can be carried out up to separation with a fairly large step.
A multi-scaling analysis of the spin-up problem
Authors: St-Maurice Jean-Pierre , Veronis George
Published: March 2006

The response of a contained rotating fluid to a small, abrupt change in the rotation rate is analysed by multi-scaling methods. The procedure makes use of the fact that three different physical processes (inertial oscillations, spin-up response, diffusion) give rise to three different time scales. Since the flow is known to have a boundary-layer character, the variables are derived into interior and boundary-layer parts. The pertinent parameter separating the magnitudes of the amplitudes and the different time scales is the square root of the Ekman number E½, so an expansion in powers of E½ is used. The solution for a homogeneous fluid is derived first and is shown to be consistent with the solution of Green-span & Howard (1963). The results are given in two forms: one is a direct deduction of the expansion method and is valid to O(E); the other is obtained by regrouping the terms to derive a form apparently valid for indefinitely long times. When the fluid is stratified, the physical structure of the system is substantially more complicated, and so is the analysis. Exact results can be obtained for the case where the buoyancy N and the rotational Ω frequencies are the same. For the general case F = N/Ω ≠ 1, results valid for t [Gt ] 1 can be obtained (where t is measured in units of Ω−1). In both cases the exact lowest-order solution for the interior can be derived since it is independent of short time t. For the stratified fluid the elementary spin-up solution of Holton (1965) is part of the solution at O(E½). The remaining part includes the long-time behaviour towards which the system tends as diffusive processes become dominant. The formulation of the long-time problem is complete a t O(E), but parts of it emerge from the analysis at lower order, and it is necessary to treat the lower-order system to obtain a consistent formulation at O(E). In particular, it is possible to show that the thermal boundary condition, which does not affect the elementary spin-up solution, should be satisfied only by the long-time part of the problem. The complete, lowest-order response of the system includes a diffusive part which is quantitatively significant even for times of the order of one spin-up time. It is suggested here that the diffusive contribution may be responsible for parts of the discrepancy between elementary spin-up theory and recent experiments.
The instability of capillary jets
Authors: Sterling Arthur M. , Sleicher C. A.
Published: March 2006

At high jet velocity the aerodynamic interaction between a capillary jet and the surrounding medium leads to an enhanced growth rate of axisymmetric disturbances. The available theories which account for this effect fail to agree with experimental observations. The difference is attributed, in part, to the relaxation of the velocity profile in jets formed by fully developed laminar pipe flow. The profile relaxation has a destabilizing effect just as does the aerodynamic interaction. In the absence of velocity-profile relaxation it is shown that the available theories overestimate the aerodynamic effect. A consideration of the viscosity of the ambient fluid yields a semi-empirical modification to the theory which shows good agreement with experimental values.
Progress in the development of a Reynolds-stress turbulence closure
Authors: Launder B. E. , Reece G. J. , Rodi W.
Published: March 2006

The paper develops proposals for a model of turbulence in which the Reynolds stresses are determined from the solution of transport equations for these variables and for the turbulence energy dissipation rate ε. Particular attention is given to the approximation of the pressure-strain correlations; the forms adopted appear to give reasonably satisfactory partitioning of the stresses both near walls and in free shear flows.

Numerical solutions of the model equations are presented for a selection of strained homogeneous shear flows and for two-dimensional inhomogeneous shear flows including the jet, the wake, the mixing layer and plane channel flow. In addition, it is shown that the closure does predict a very strong influence of secondary strain terms for flow over curved surfaces.
Experimental velocity profiles in laminar flow around spheres at intermediate Reynolds numbers
Authors: Seeley L. E. , Hummel R. L. , Smith J. W.
Published: March 2006

Normal and tangential velocities in the boundary layer and out into the free stream have been obtained using a non-disturbing flow visualization technique for uniform laminar flow around a sphere. The non-similar data are available in tables at 2.5° intervals from 20° from the front to about 15° past the separation point a t Reynolds numbers of 290, 750, 1300 and 3000. Stream functions calculated by LeClair using a numerical solution of the Navier-Stokes equation at Re 21 300 are not in good agreement with measured values from 30° to 60°, but are in much better agreement around the separation point. Too few grid points near the sphere where the tangential velocities rise to a maximum above free-stream values may account for the difference.
Energy transport in a nonlinear and inhomogeneous random gravity wave field
Authors: Willebrand Jürgen
Published: March 2006

Certain tertiary resonant interactions of gravity waves which have been found previously can be obtained more easily by using a simple extension of Whitham's formalism. The contribution of these interactions to the total energy transfer in an inhomogeneous random field of gravity waves is calculated. It is found to be small for open-ocean waves, but to be of some importance for shallow-water waves, where topography or mean shear currents may produce strong inhomogeneities. The nonlinear splitting of the group velocity is found to be unimportant in wave fields with sufficiently smooth spectra.
Measurements in an incompressible three-dimensional turbulent boundary layer, under infinite swept-wing conditions, and comparison with theory
Authors: Van Den Berg B. , Elsenaar A. , Lindhout J. P. F. , Wesseling P.
Published: March 2006

First a three-dimensional turbulent boundary-layer experiment is described. This has been carried out with the specific aim of providing a test-case for calculation methods. Much attention has been paid to the design of the test set-up. An infinite swept-wing flow has been simulated with good accuracy. The initially two-dimensional boundary layer on the test plate was subjected to an adverse pressure gradient, which led to three-dimensional separation near the trailing edge of the plate. Next, a calculation method for three-dimensional turbulent boundary layers is discussed. This solves the boundary-layer equations numerically by finite differences. The turbulent shear stress is obtained from a generalized version of Bradshaw's two-dimensional turbulent shear stress equation. The results of the calculations are compared with those of the experiment. Agreement is good over a considerable distance; but large discrepancies are apparent near the separation line.
Thermoacoustic effects in a resonance tube
Authors: Merkli P. , Thomann H.
Published: March 2006

New experiments with a gas-filled resonance tube have shown that not only heating, but also cooling of the tube wall is possible and that these phenomena are not restricted to oscillation amplitudes that generate shocks. The present paper concentrates on amplitudes outside the shock region. For this case, an extended acoustic theory is worked out. The results show cooling in the section of the tube with maximum velocity amplitude (and thus dissipation) and marked heating in the region of the velocity nodes. A strong dependence of these effects on the Prandtl number is noted. The results are in good agreement with experiments. Although the theory is not valid for proper resonance conditions, it nevertheless sheds some light on what happens when nonlinear effects dominate.

Closely related to the limit of validity of the thermoacoustic theory is the question of transition from laminar to turbulent flow in the viscous boundary layer (Stokes layer). This problem has also been investigated; the results are given in a separate paper (Merkli & Thomann 1975). In the present article laminar flow is assumed.
A proposed model of the bursting process in turbulent boundary layers
Authors: Offen G. R. , Kline S. J.
Published: March 2006

A model is proposed which attempts to explain the complete ‘burst cycle’. This model views the wall streak as a sub-boundary layer, within the conventionally defined boundary layer, and the lift-up stage of bursting either as an upwelling motion of this sub-boundary layer which is similar to a local, convected separation or, equivalently, as the consequence of a vortex roll-up. ‘Sweeps’ are thought to represent the passage of a previous burst from further upstream. They appear either to impress on the wall streak the temporary adverse pressure gradient required to bring about its lifting or, alternatively, to provide the outer vortex which rolls up with the vortex associated with the wall streak. The model is also used to explain how the interactions between a burst and a sweep bring about (i) breakup, as well as (ii) new wall streaks further downstream.

Arguments are presented to demonstrate that the three kinds of oscillatory growth reported by Kim, Kline & Reynolds (1971) may be associated with just one type of flow structure: the stretched and lifted vortex described by Klineet al.(1967).
Wave-induced vorticity in free-surface boundary layers: application to mass transport in edge waves
Authors: Dore B. D.
Published: March 2006

The time-averaged vorticity field within the free-surface boundary layer associated with a general class of propagating gravity waves is considered. The principal results are applied in a calculation of the mass transport velocity field for edge waves.
Internal gravity waves: critical layer absorption in a rotating fluid
Authors: Grimshaw R.
Published: March 2006

The propagation of internal gravity waves in a shear flow in a rotating fluid is examined for the case when the rotation vector is inclined to the vertical. It is shown that internal gravity waves approaching a critical level, where ω*, the Doppler-shifted frequency, equals 2ΩV, the vertical component of the Coriolis parameter, will be either transmitted or absorbed according as
\[
W_g\omega^{*}2\Omega_V\{m(U_z + 2\Omega_H)-lV_z\}\lessgtr 0;
\]
here Wg is the vertical group velocity, 2ΩH is the horizontal component of the Coriolis parameter, l and m are the easterly and northerly wavenumber components, and Uz and Vz are the shear rates of the easterly and northerly components of the mean flow. Between critical levels, wave action flux is conserved. However, for a wave absorbed at a critical level, the wave action flux is attenuated by a factor exp { − 2π|m(Uz + 2ΩH) − lVz]/(lUz + mVz)|}. The phenomenon is also analysed using a WKBJ approximation.
Mean and fluctuating flow measurements of a fully-developed, non-adiabatic, hypersonic boundary layer
Authors: Owen F. K. , Horstman C. C. , Kussoy M. I.
Published: March 2006

Extensive boundary-layer measurements have been made on a cone-ogive-cylinder model at a free-stream Mach number of 7·0 and momentum-thickness Reynolds number of 8500. Mean flow transformations and calculated turbulence correlations are presented which are in good agreement with previous incompressible results. New quantitative turbulence measurements including measurements of the first higher moment and probability density of fluctuations in mass flow and total temperature in hypersonic flow are also presented. The higher moment and probability density data show that the characters of the fluctuation modes of the mass flow and total temperature are significantly different in the wall region and in the outer part of the boundary layer. These differences together with data on the turbulence scale and lifetime obtained from autocorrelation and space-time correlation measurements are discussed.
Nonlinear wave interactions in shear flows. Part 2. Third-order theory
Authors: Usher J. R. , Craik A. D. D. , Hendriks F.
Published: March 2006

The temporal evolution of a resonant triad of wave components in a parallel shear flow has been investigated at second order in the wave amplitudes by Craik (1971) and Usher & Craik (1974). The present work extends these analyses to include terms of third order and thereby develops the resonance theory to the same order of approximation as the non-resonant third-order theory of Stuart (1960, 1962).

Asymptotic analysis for large Reynolds numbers reveals that the magnitudes of the third-order interaction coefficients, like certain of those at second order, are remarkably large. The implications of this are discussed with particular reference to the roles of resonance and of three-dimensionality in nonlinear instability and to the range of validity of the perturbation analysis.
Wind-tunnel simulation of the atmospheric boundary layer: a report on Euromech 50
Authors: Hunt J. C. R. , Fernholz H.
Published: March 2006

The 50th Euromech Colloquium, on wind-tunnel simulation of the atmospheric boundary layer, was held in Berlin from 23–25 September 1974. Thirty-eight participants from eleven countries were present. Papers were presented describing and analysing different methods of simulation of neutral, stable and unstable atmospheric conditions in various types of wind tunnel. Numerous applications of wind-tunnel simulations were described or mentioned in the papers and the discussion sessions. Some conclusions about the validity, the techniques, the limitations and future developments of wind-tunnel simulations were reached in discussion. Tables are presented in appendix A listing the institutes in Europe and the U.S.A. of actual or invited participants where wind tunnels are used for simulation work; also listed are the characteristics of the wind tunnels and relevant measurements of the simulated atmospheric boundary layers, to enable comparisons to be made between different techniques.
Unsteady laminar boundary layers in a compressible stagnation flow
Authors: Vimala C. S. , Nath G.
Published: March 2006

The unsteady laminar compressible boundary-layer flow in the immediate vicinity of a two-dimensional stagnation point due to an incident stream whose velocity varies arbitrarily with time is considered. The governing partial differential equations, involving both time and the independent similarity variable, are transformed into new co-ordinates with finite ranges by means of a transformation which maps an infinite interval into a finite one. The resulting equations are solved by converting them into a matrix equation through the application of implicit finite-difference formulae. Computations have been carried out for two particular unsteady free-stream velocity distributions: (i) a constantly accelerating stream and (ii) a fluctuating stream. The results show that in the former case both the skin-friction and the heat-transfer parameter increase steadily with time after a certain instant, while in the latter they oscillate, thus responding to the fluctuations in the free-stream velocity.
Measurements in adverse-pressure-gradient turbulent boundary layers with a step change in surface roughness
Authors: Schofield W. H.
Published: March 2006

The response of turbulent boundary layers to sudden changes in surface roughness under adverse-pressure-gradient conditions has been studied experimentally. The roughness used was in the ‘d’ type array of Perry, Schofield & Joubert (1969). Two cases of a rough-to-smooth change in surface roughness were considered in the same arbitrary adverse pressure gradient. The two cases differed in the distance of the surface discontinuity from the leading edge and gave two sets of flow conditions for the establishment and growth of the internal layer which develops downstream from a change in surface roughness. These conditions were in turn different from those in the zero-pressure-gradient experiments of Antonia & Luxton. The results suggest that the growth of the new internal layer depends solely on the new conditions at the wall and scales with the local roughness length of that wall. Mean velocity profiles in the region after the step change in roughness were accurately described by Coles’ law of the wall-law of the wake combination, which contrasts with the zero-pressure-gradient results of Antonia & Luxton. The skin-friction coefficient after the step change in roughness did not overshoot the equilibrium distribution but made a slow adjustment downstream of the step. Comparisons of mean profiles indicate that similar defect profile shapes are produced in layers with arbitrary adverse pressure gradients at positions where the values of Clauser's equilibrium parameter β (= δ*τ−10dp/dx) are similar, provided that the pressure-gradient history and local values of the pressure gradient are also similar.
An unfinished dialogue with G. I. Taylor
Authors: Batchelor G. K.
Published: March 2006

Sir Geoffrey Ingram Taylor, O.M., F.R.S., who died on 27 June 1975 at the age of 89, was one of the great men of our subject. He was a likeable happy man with an uncomplicated character and a razor-sharp mind for which scientific investigation was a natural activity. He was engaged in research throughout the whole of his life – not only the ‘working'part of it – and the fruits of his enquiries are described in over 200 papers published between 1909 and 1974. Nearly all these papers have been republished by Cambridge University Press in the four volumes of G. I. Taylor: Scientific Papers, three of which are on the mechanics of fluids and one on the mechanics of solids. These four volumes are his legacy to us, and will be a store-house of information and a source of illumination for many years to come.
Numerical simulation of two-dimensional compressible convection
Authors: Graham Eric
Published: March 2006

A procedure for obtaining numerical solutions to the equations describing thermal convection in a compressible fluid is outlined. The method is applied to the case of a perfect gas with constant viscosity and thermal conductivity. The fluid is considered to be confined in a rectangular region by fixed slippery boundaries and motions are restricted to two dimensions. The upper and lower boundaries are maintained at fixed temperatures and the side boundaries are thermally insulating. The resulting convection problem can be characterized by six dimension-less parameters. The onset of convection has been studied both by obtaining solutions to the nonlinear equations in the neighbourhood of the critical Rayleigh number Rc and by solving the linear stability problem. Solutions have been obtained for values of the Rayleigh number up to 100Rc and for pressure variations of a factor of 300 within the fluid. In some cases the fluid velocity is comparable to the local sound speed. The Nusselt number increases with decreasing Prandtl number for moderate values of the depth parameter. Steady finite amplitude solutions have been found in all the cases considered. As the horizontal dimension A of the rectangle is increased, the length of time needed to reach a steady state also increases. For large values of A the solution consists of a number of rolls. Even for small values of A, no solutions have been found where one roll is vertically above another.
The laminar flow of dilute polymer solutions through porous media
Authors: James David F. , Mclaren D. R.
Published: March 2006

Measurements of the pressure drop and flow rate were obtained for dilute solutions of polyethylene oxide flowing through beds of packed beads. When the velocity was sufficiently high, the pressure drop was above that for a Newtonian fluid of equal viscosity, often considerably above, and this viscoelastic effect was explored by varying the concentration and molecular weight of the polymer, by testing solutions over a wide range of flow rates, and by using several bead sizes. The non-Newtonian behaviour was most pronounced at moderate flow rates; at the highest velocities, the data were pseudo-Newtonian in character, i.e. the pressure drop still exceeded that for a Newtonian fluid, but was linearly related to the velocity. For some solutions, the large deviation from Newtonian values occurred over such a short range of flow rates that there was an interval in which the pressure drop decreased with velocity. It was not possible, therefore, to obtain steady-state measurements in this regime and a gap appears in the data curve of pressure vs. velocity.

The pressure drop was monitored in steps along the test section, so that it was possible to detect molecular degradation of the solutions as they flowed through the porous media. In general, degradation was not extensive and the solutions became stably degraded by the midpoint of the test section. Degradation increased with velocity and, quite surprisingly, became more severe as the bead size increased.

A visual examination of the flow field revealed that the streamline pattern for the polymer solutions was the same as that for water. The large non-Newtonian effects were therefore due to changes in the stress field, and in an effort to understand these effects, an analysis was carried out which examined how the stresses generated by each component of the deformation, i.e. by shear and pure strain, influence the pressure drop. This analysis, combined with a study of onset data, indicates that onset and the sudden large departures from Newtonian values are probably due to an interaction between extensional and shearing deformation, and that the reduced viscoelastic effect of higher flow rates may be due to the dominance of extensional stresses.
The motion generated by a body moving through a stratified fluid at large Richardson numbers
Authors: Barnard B. J. S. , Pritchard W. G.
Published: March 2006

Experiments are described in which a body was towed at large Richardson numbers through a uniformly stratified fluid, with particular attention being given to the parameter range in which diffusive effects are likely to play an important role. When the Richardson number is only moderately large (i.e. less than about 106) it appears that a linear viscous diffusive theory fails to model the experimental observations, but at larger Richardson numbers it seems that such a theory is a strong candidate for modelling the physical situation. At Richardson numbers of about 103 a non-diffusive viscous model (see the theories of Graebel 1969; Janowitz 1971) appears to give a fairly good description of the experimental results, except for some discrepancies between the theories and the observed flows in the wake downstream from the body. However, the conditions under which the experiments were carried out were not completely favourable for the application of these theories.

A brief survey and new interpretations of some of the work on the rotatingfluid counterparts of the present experiments are also given. In view of the theoretical similarities between the two situations we have tried to assess and compare (where possible) the results of experimental investigations in each field and we feel that the observations in the rotating-fluid experiments are in good qualitative agreement with the present results. No measurements of the drag acting on the body were made in the present experiments but this has been done carefully in the rotating-fluid experiments, and the results of those measurements are in poor agreement with the theoretical predictions. Because of the close similarities between the axisymmetric model for viscous rotating fluids (cf. Moore & Saffman 1969) and the two-dimensional model for diffusive stratified fluids (Freund & Meyer 1972), an explanation for the discrepancy should be sought before the linear viscous diffusive model can be applied with any confidence to the flow of stratified fluids at very large Richardson numbers.
Convection in boxes: an experimental investigation in vertical cylinders and annuli
Authors: Stork K. , Müller U.
Published: March 2006

Convective flow of the cellular type has been investigated in vertical cylinders of axisymmetric and annular shape. The critical Rayleigh numbers were determined as functions of the box diameter and the gap width respectively. A distinct influence of side walls with different thermal conductivities was observed.
Nonlinear processes in rotating fluids: a report on Euromech 56
Authors: Acheson D. J. , Andrews D. G. , Walton I. C. , Hocking L. M.
Published: March 2006

A European Mechanics Colloquium (Euromech 56) on ‘Nonlinear processes in rotating fluids’ was held at University College, London from 14–18 April 1975. It was attended by 77 mathematicians, engineers and physicists from 14 countries, and 33 papers were presented and discussed. The topics included motions in homogeneous and stratified rotating fluids, with applications to the atmosphere, the oceans, the earth's core and centrifuges. Brief summaries of the papers presented are contained in this report.
Shear layer instability of an inviscid compressible fluid. Part 2
Authors: Blumen W. , Drazin P. G. , Billings D. F.
Published: March 2006

The linear stability of a shear layer of an inviscid compressible fluid is considered. It is shown that there is instability of two-dimensional disturbances at all values of the Mach number, contrary to previous results for a vortex sheet. The difference arises from the discovery of a second unstable mode. This mode is supersonic, decays weakly with distance from the shear layer, and is not governed by the principle of exchange of stabilities. Detailed numerical and asymptotic results are given for the hyperbolic-tangent shear layer.
Three-dimensional theory on supercavitating hydrofoils near a free surface
Authors: Furuya Okitsugu
Published: March 2006

Supercavitating hydrofoils of large aspect ratio operating near a free surface are investigated, assuming an inviscid and irrotational flow with the effects of gravity and surface tension neglected. The flow near the foil, treated as two-dimensional, is solved by a nonlinear free-streamline theory, then a three-dimensional ‘downwash’ correction is made using Prandtl's lifting-line theory. The strength of the lifting-line vortex is determined by information from the two-dimensional solution through a matching procedure, in which the inverse of aspect ratio is used as a small parameter for asymptotic expansions. The analysis incorporates a free-surface reference level to determine the submergence depth of the foil. The present method can be applied to any type of foil having an arbitrary planform or profile shape, including a rounded leading edge, a twist and even a small dihedral angle, within the assumption of large aspect ratio. Numerical computations made on rectangular flat-plate hydrofoils show excellent agreement of results with existing experimental data, even for large angles of attack and relatively low aspect ratios. The pressure distributions, shapes of the cavity and free surface are also calculated as a function of spanwise position.
The deepening of a mixed layer in a stratified fluid
Authors: Linden P. F.
Published: March 2006

In this paper two aspects of the deepening of a mixed layer in a stratified fluid are examined in the laboratory. The first is the deepening of a layer into a region of constant density gradient. Turbulence is produced by an oscillating grid which generates a horizontally homogeneous field of motion with no significant mean flow. It is found that the rate at which the potential energy of the basic stratification is increased by the mixing does not bear a simple relationship to the rate of energy input by the grid. On the other hand, when allowance is made for the decay of turbulent energy away from the grid and only that portion to reach the bottom of the mixed layer is considered, the rate of potential energy increase is found to be proportional to this available energy. The second aspect to be discussed is the effect of energy radiation by internal waves in the region below the mixed layer. Estimates are made of the possible loss of energy to these waves, which reduces the amount available to deepen the layer. An experimental demonstration of up to 50 % reduction in the mixing rate due to the presence of internal waves is given. Finally, the implications of these results are discussed in the light of current theoretical models of the deepening process.
The development of a partially mixed region in a stratified shear flow
Authors: Hartman R. J.
Published: March 2006

The influence of a mean uniform shear on the development of a partially mixed region in an unbounded stratified fluid is considered. Results are given for the very near field and the far field in the large-J (Richardson number) limit. A qualitative discussion for J [gsim ] 1 is also included. The near-field motion is modified on a time scale given by Rτ ∼ J−¼, where R = dU/dz is the mean shear strength. This modification is shown to be a consequence of the kinematic distortion of the wake profile by the mean shear. The radiation field is largely anticipated by the wave-packet discussion of an earlier paper.
Constitutive equations in suspension mechanics. Part 1. General formulation
Authors: Hinch E. J. , Leal L. G.
Published: March 2006

Neither the phenomenological nor the structural approach to the determination of constitutive equations has yet shown itself to be capable of producing useful and predictive descriptions of the majority of technologically important complex fluids. In the present paper we explore the suggestion that significant progresscanbe made when these two complementary approaches to rheology are combined. For this initial study we restrict our attention to materials which can be modelled as a suspension of particles in a Newtonian fluid, thereby including most polymer solutions while excluding polymer melts. By applying phenomenological techniques to the basic formulation of suspension mechanics we are able to deduce a common simplified constitutive model for all suspension-like materials and to reveal its physical origin. The present analysis demonstrates that the constitutive model of Hand (1962), involving a single second-order tensor, is not sufficiently general for a rigorous description of the majority of suspension-like materials. Consideration of the constitutive forms for the limiting cases of near-equilibrium and strongly non-equilibrium microstructure suggests, however, that Hand's model may provide a reasonable approximation to the exact constitutive behaviour which is useful over the whole range of flow strengths.
Nonlinear internal gravity waves in a rotating fluid
Authors: Grimshaw R.
Published: March 2006

The interaction between internal gravity waves in a rotating frame and the mean flow is discussed for the case when the properties of the mean flow vary slowly on a scale determined by the local wave structure. The principle of conservation of wave action is established. It is shown that the main effect of the waves on the Lagrangian mean velocity is due to an appropriate ‘radiation stress’ tensor. A circulation theorem and a potential-vorticity equation are derived for the mean velocity.
On the longitudinal dispersion of passive contaminant in oscillatory flows in tubes
Authors: Chatwin P. C.
Published: March 2006

The paper examines how a passive contaminant disperses along the axis of a tube in which the flow is driven by a longitudinal pressure gradient varying harmonically with time. This problem is of intrinsic interest and is relevant to some important practical problems. Two examples are dispersion in estuaries and in the blood stream. By means both of statistical arguments and an analysis like that used by Taylor (1953) in the case of a steady pressure gradient it is shown that eventually the mean distribution of concentration satisfies a diffusion equation (and is therefore a Gaussian function of distance along the axis) with an effective longitudinal diffusion coefficient K(t) which is a harmonic function of time with a period equal to one half of that of the imposed pressure gradient. Contrary to the supposition made in most previous work on this problem it is shown by examining some special cases that the harmonic terms in K(t) may have a noticeable effect on the dispersion of the contaminant and in particular on the rate at which it is spreading axially. The size of the effect depends on both the frequency and the Schmidt number and is particularly large at low frequencies. The paper concludes with an analysis of a model of dispersion in estuaries which has been used frequently and it is concluded that here too oscillatory effects may often be noticeable.
Circulations and density distributions in a deep, strongly stratified, two-layer estuary
Authors: Long Robert R.
Published: March 2006

The paper discusses a theoretical model of statistically steady flow in a strongly stratified estuary. A halocline is assumed to be present and the lower layer is taken to be deep and non-turbulent. The outflowing upper fluid mixes with the salty lower fluid and the flux of the brackish water increases with distance from the head of the estuary. The mixing is assumed to be similar to that in laboratory models of mixing across density interfaces.

Two equations of mass conservation are used, one for the steady-state mass flux across a vertical section from top to bottom of the channel and one for the mass flux into a section of the upper fluid. A buoyancy conservation equation is used for the buoyancy flux across a vertical section. A final equation is obtained by integrating the horizontal equation of motion across a section of the upper fluid. The flow in this layer is assumed to be opposed by a frictional force proportional to the square of the velocity averaged over the layer. The pressure-gradient force arising from the slope of the free surface is solved for in terms of the thickness of the upper layer, the buoyancy difference across the interface, the slope of the interface and the horizontal density gradient in the upper layer. The derivation shows that the horizontal pressure-gradient force vanishes in the lower layer.

The mathematical problem reduces to two ordinary differential equations for the flux in the upper layer and its thickness. Attention is confined to the solution for subcritical flow, in which the interface falls with distance from the head, reaching a maximum depth at a certain section of the estuary. Beyond this the interface rises. At the mouth, where, by definition, the width of the estuary increases rapidly, it is shown that there must be a transition from subcritical to supercritical flow. This condition, applied to the solution for uniform width, determines a remaining unknown related to the depth of the halocline at the head of the estuary and the complete solution is obtained as a function of the freshwater influx per unit width, the r.m.s. turbulent velocity, the estuary length and the buoyancy of sea water.

The solution is complicated but has reasonable behaviour for variations of the given parameters of the problem. A basic feature for values of the constants appropriate to fjord-type estuaries is the dominance of friction, omitted in an earlier, incomplete investigation by Stommel. This is also revealed by the large drop in the free surface over the length of the estuary.

A comparison with two estuaries, Oslofjord and Knight Inlet, British Columbia, indicates that the former is very different from the model of this paper but that the latter may have a similar nature.
Mean and turbulence characteristics of three-dimensional wall jets
Authors: Swamy N. V. Chandrasekhara , Bandyopadhyay P.
Published: March 2006

This paper reports experimental investigations on the characteristic decay and the radial-type decay regions of a three-dimensional isothermal turbulent wall jet in quiescent surroundings. The velocity and the length scale behaviour for both the longitudinal and the transverse directions are studied, and compared with the results of other workers. The estimated skin friction is discussed in relation to the available data from earlier investigations. Wall jet expansion rates and the behaviour of skin friction are also discussed. The rate of approach of turbulence components to a self-similar form is found to be influenced by the fact that the expansion rate of the wall jet in the longitudinal direction is different from that in the transverse.
Sound radiation into uniformly flowing fluid by compact surface vibration
Authors: Williams J. E. Ffowcs , Lovely D. J.
Published: March 2006

This paper describes a model problem where compact surface vibration radiates sound into a subsonically flowing fluid. There are two distinct acoustic effects. First, the radiation is increased by flow by an amount proportional to 5M2 and that increase is shown by a general argument to arise from an enhanced surface damping and work done by the flow to overcome drag in the ratio 2:1. Second, the acoustic source strength is affected and resonance frequencies are significantly modified by flow. The main effect is that flow induces on the surface a force proportional to the displacement which opposes the action of natural surface elasticity. A critical velocity exists beyond which the surface is unstable; the stability limit is determined. The surface motion might be regarded as an acoustic monopole, but since aerodynamic fields are determined by the rate of change of the rate of mass outflow, the frequency dependence is more that of a quadrupole. Convective amplification of the sound is also shown to be that characteristic of quadrupole sources. This result indicates that real simple fields may be more sensitive to convection than might be expected from past studies of simple inhomogeneities satisfying a convected wave equation.
Three-dimensional turbulent boundary layers in external flows: a report on Euromech 60
Authors: Fanneløp T. K. , Krogstad P. Å.
Published: March 2006

Euromech 60 is the third in a series of European Mechanics Colloquia dealing primarily with three-dimensional turbulent boundary layers. The Colloquium was held in Trondheim at the Technical University (NTH) from 14–16 April 1975 with forty-two participants from ten different countries, and was organized by L. N. Persen and T. K. Fanneløp. A total of 23 papers were presented, dealing with both experiments on and predictions of turbulent boundary-layer flows and related topics. Those concerned with the development of prediction methods were challenged by a set of boundary-layer flow problems defined well in advance in order to be calculated prior to the Colloquium. The results show close agreement for some calculated variables and surprisingly large discrepancies in others. A brief account of this exercise is included and it will also be the subject of a special report.
Patterns of convection in spherical shells
Authors: Busse F. H.
Published: March 2006

The problem of the pattern of motion realized in a convectively unstable system with spherical symmetry can be considered without reference to the physical details of the system. Since the solution of the linear problem is degenerate because of the spherical homogeneity, the nonlinear terms must be taken into account in order to remove the degeneracy. The solvability condition leads to the selection of patterns distinguished by their symmetries among spherical harmonics of even order. It is shown that the corresponding convective motions set in as subcritical finite amplitude instabilities.
A note on natural convection in a vertical slot
Authors: de Vahl Davis G. , Mallinson G. D.
Published: March 2006

The secondary and tertiary motions which were observed by Elder (1965) during his experimental study of natural convection in a vertical cavity have been obtained in a numerical solution of the Boussinesq equations. Aspect ratios up to 20, Rayleigh numbers up to 3000000, a Prandtl number of 1000, and up to almost 11000 effective mesh points were used.