bluuebunny/crossref_metadata_2025 huggingface dataset:
The decay of perturbations in an electrically conducting and thermally radiating gas
Authors: Sarma M. Srinivasa , Sarma L. V. K. V.
Published: March 2006

The decay of perturbations in an infinite, thermally radiating gas of perfect electrical conductivity in the presence of magnetic field is studied. Complete solutions for the decay of initial sinusoidal perturbations in the temperature, gas velocity and pressure are determined. The sinusoidal perturbations are superposed to yield solutions for the decay of initial ‘step’ temperature profiles consisting of a constant initial temperature perturbation inside a finite planar region, with zero temperature perturbation outside. For a broad range of small and intermediate Boltzmann numbers the cooling proceeds in time from being a constant-density cooling process to being a constant-pressure cooling process. The magnetic field causes slower temperature decay with time and makes the temperature perturbations tend to attain constant-pressure cooling values. It quickens the decay of velocity and pressure perturbations and thus the transition from a constant-density to a constant-pressure cooling process is hastened. This transition is produced by the magneto-acoustic waves generated near the profile edges by the radiative cooling.
Shock attenuation in a ‘gradual’ area expansion
Authors: Nettleton M. A.
Published: March 2006

The effect of the angle of divergence ϕ, the magnitude of the area ratio A21 and the specific-heat ratio of the gas γ on the attenuation of a shock in a two-dimensional area expansion has been experimentally determined. The results are compared with current theories relating shock strength and area ratio. At some distance from the expansion the change in shock strength, for shocks with strengths up to 10, is predicted reasonably accurately by an analysis of Chisnell (1957). Close to the area change surprisingly strong shocks were observed. This is shown to result from the diffracted shock undergoing a Mach reflexion at the end of the expansion.
Heat transfer from a sphere at low Reynolds numbers
Authors: Dennis S. C. R. , Walker J. D. A. , Hudson J. D.
Published: March 2006

The heat transfer due to forced convection from an isothermal sphere in a steady stream of viscous incompressible fluid is calculated for low values of the Reynolds number and Prandtl numbers ofO(1). The mean Nusselt number is compared with the results of experimental measurements. At very low Reynolds numbers, both the local and mean Nusselt numbers are compared with the results obtained from the theory of matched asymptotic expansions.
On an asymptotic solution of the Korteweg–de Vries equation with slowly varying coefficients
Authors: Johnson R. S.
Published: March 2006

The variable-coefficient Korteweg–de Vries equation
\[
H_X + {\textstyle\frac{3}{2}}d^{-\frac{7}{4}}HH_{\xi} + {\textstyle\frac{1}{6}}\kappa d^{\frac{1}{2}}H_{\xi\xi\xi} = 0
\]
with d = d(εX) is discussed for solitary-wave initial profiles. A straightforward asymptotic solution for ε → 0 is constructed and is shown to be non-uniform both ahead of and behind the solitary wave. The behaviour ahead is rectified by matching to the appropriate exponential form and, together with the use of conservation laws for the equation, the nature of the solution behind the solitary wave is discussed. This leads to the formulation of the solution in the oscillatory ‘tail’, which is again matched directly.

The results are applied to the development of the solitary wave into variable-depth water, and the predictions are compared with those obtained, for example, by Grimshaw (1970, 1971). Finally, the asymptotic behaviour of both the solitary wave and the oscillatory tail are assessed in the light of some numerical integrations of the equation.
The steady two-dimensional reflexion of an oblique partly dispersed shock wave from a plane wall
Authors: Buggisch H.
Published: March 2006

The steady two-dimensional problem of reflexion of an oblique partly dispersed plane shock wave from a plane wall is studied analytically. Viscosity, diffusion and heat conduction are neglected. The thermodynamic state of the gas is assumed to be determined by the instantaneous values of the specific entropy s, pressure p and a finite number of internal state variables. Results for the flow field behind the reflected shock are obtained by a perturbation method which is based on the assumption that the influence of relaxation is relatively weak.
The effects of wake splitter plates on the flow past a circular cylinder in the range 104<R<5×104
Authors: Apelt C. J. , West G. S. , Szewczyk Albin A.
Published: March 2006

Experiments were carried out using models having L/D [les ] 2 and the resulting pressure distributions and vortex shedding characteristics are presented. A simple visualization technique which provides explanations of some of the measured results is described. It is concluded that splitter planes reduce the drag markedly by stabilizing the separation points and produce a wake narrower than that for a plain cylinder, raise the base pressure by as much as 50% and affect the Strouhal number to a lesser degree. Careful measurement techniques have enabled these effects to be presented accurately.
Wind-induced turbulent entrainment across a stable density interface
Authors: Wu Jin
Published: March 2006

Turbulent entrainment across a density interface is studied in a laboratory wind–wave tank in which a stably stratified system consisting of two homogeneous fluid layers is introduced. The results indicate that the rate of change of the potential energy of the mixing layer is proportional to the rate of work done by the wind. However, only a very small fraction of the work done by the wind is used for interfacial mixing or developing a seasonal thermocline. A formula relating the entrainment rate to the density stratification and the wind-friction velocity is derived from the experimental results.
An internal wave in a viscous ocean stratified by both salt and heat
Authors: Thomas N. H. , Stevenson T. N.
Published: March 2006

The propagation of an internal wave in an ocean which is stratified in both temperature and salinity is considered. It is shown how the effects of viscosity, heat conduction and solute diffusion attenuate a cross-wave from an oscillatory disturbance. Under certain conditions there is no wave solution even though the oscillatory frequency is less than the Brunt–Väisälä frequency and the fluid is stably stratified.
The effect of a bevelled trailing edge on vortex shedding and vibration
Authors: Greenway M. E. , Wood C. J.
Published: March 2006

Experiments using a wind tunnel and a flow visualization technique in a towing tank were conducted to investigate the mechanism of vortex shedding from bevelled trailing edges. These reveal an important difference between the wake structures generated by heaving and steady motion. The suppression of vortex-excited vibration by means of bevelled trailing edges is attributed to the intermittency and rapid decay of the vortex trail resulting from an asymmetric circulation distribution in the vortex formation region.
An experimental and numerical study of the viscous stability of a round laminar vertical jet with and without thermal buoyancy for symmetric and asymmetric disturbances
Authors: Mollendorf J. C. , Gebhart B.
Published: March 2006

The fully viscous hydrodynamic stability equations for a round laminar vertical jet have been numerically solved using the proper boundary-layer base-flow velocity profile and for both symmetric and asymmetric disturbances. The symmetric mode is found to be unconditionally stable. The first asymmetric mode is found to be unstable and characteristics are compared with previous calculations. The computed critical Reynolds number for this mode is 9·4, which agrees with the calculations of Burridge (1968). Disturbance amplitude-ratio contours are also calculated and related to the convection of disturbances in the flow.

The effect of thermal buoyancy on jet stability is assessed by solving the fully viscous equations, coupled through buoyancy, and using a buoyancy-perturbed jet flow. A Prandtl number σ of 6·7 is used, and positive thermal buoyancy is found to have a destabilizing effect. Stability characteristics for the limiting case of buoyancy, a purely thermal point-source plume, are determined for a Prandtl number of 2.

Finally, an experiment was performed using water jets in water (σ ≈ 4·52–5·89) and a new method of jet production is described. The effect of varying amounts of thermal buoyancy on the laminar length of a jet undergoing naturally occurring transition was determined experimentally. These experiments confirm the calculated destabilizing effect of buoyancy. An empirical correlation is presented for the laminar length of a jet. Also, the effect of both symmetric and asymmetric artificially induced disturbances was determined experimentally. The disturbance amplitude ratio at which transition to turbulence takes place is found to be much less than for buoyant flows adjacent to a wall. The effects of frequency and amplitude of the artificial disturbances were experimentally determined and the trends are found to be consistent with the results of small disturbance theory.

The principal new result is that positive thermal buoyancy destabilizes jet flow, and consequently calls into question earlier experimental studies wherein jet flows were observed by density differences. Another new result is the calculation. of amplitude-ratio contours for a non-buoyant jet and the quantitative description of jet stability in terms of these contours along paths of constant physical frequency. A comparison of the non-buoyant theory with experimental jets containing varying amounts of thermal buoyancy indicates that transition did not occur at a well-defined value of the amplitude ratio. Perhaps experiments with truly non-buoyant jets and/or more detailed buoyant calculations could explain this remaining question.
Nonlinear penetrative convection
Authors: Moore D. R. , Weiss N. O.
Published: March 2006

Convection in water above ice penetrates into the stably stratified region above the density maximum at 4 °C. Two-dimensional penetrative convection in a Boussinesq fluid confined between free boundaries has been studied in a series of numerical experiments. These included cases with a constant temperature at both boundaries as well as cases with a fixed average flux at the lower boundary. Steady convection occurs at Rayleigh numbers below the critical value predicted by linear theory. At high Rayleigh numbers, resonant coupling between convection and gravitational modes in the stable layer excites finite amplitude oscillations. The problem can be described by a simplified model which allows for distortion of the mean temperature profile and balances the convected and conducted flux. This model explains the finite amplitude instability and predicts the Nusselt number as a function of Rayleigh number. These predictions are in excellent agreement with the computed results.
Topographic Rossby waves in a rough-bottomed ocean
Authors: Rhines Peter , Bretherton Francis
Published: March 2006

The object is to predict the nature of small-amplitude long-period oscillations of a homogeneous rotating fluid over a ‘sea bed’ that is nowhere level. Analytically, we are limited to special choices of bottom topography, such as sinusoidal corrugations or an undulating continental slope, so long as the topographic restoring effect equals or exceeds that due to planetary curvature (the beta-effect). (Very slight topographic features, on the other hand, provide weak, resonant interactions between Rossby waves.)

Integral properties of the equations, and computer experiments reported elsewhere, verify the following results found in the analytical models: typical frequencies of oscillation are [lsim ]fδ, where f is the Coriolis frequency and δ measures the fractional height of the bottom bumps; an initially imposed flow pattern of large scale will rapidly shrink in scale over severe roughness (even the simplest analytical model shows this rapid change in spatial structure with time); and energy propagation can be severely reduced by roughness of the medium, the energy velocity being of order fδa, where a is the horizontal topographic scale (although in an exceptional case, the sinusoidal bottom, the group velocity remains finite for vanishingly small values of a).
Stability of a circular cylinder oscillating in uniform flow or in a wake
Authors: Tanida Y. , Okajima A. , Watanabe Y.
Published: March 2006

The lift and drag forces were measured on both a single circular cylinder and tandem circular cylinders in uniform flow at Reynolds numbers from 40 to 104, to investigate the stability of an oscillating cylinder. A cylinder (the downstream one in the tandem case) was made to oscillate in either the transverse or longitudinal direction (perpendicular or parallel to the stream). In the case of a single cylinder, its oscillation causes the so-called synchronization in a frequency range around the Strouhal frequency (transverse mode) or double the Strouhal frequency (longitudinal mode). The aerodynamic damping for transverse oscillation becomes negative in the synchronization range. In the case of tandem cylinders, at low Reynolds numbers in the pure Kármán range synchronization was observed to occur only when the downstream cylinder oscillated inside the vortex-formation region of the upstream one, and at high (low subcritical) Reynolds numbers synchronization occurred irrespective of the cylinder spacing in either oscillating mode. In the tandem case, too, the transverse oscillation of the downstream cylinder becomes unstable in the range of synchronization.
The stability of a layer of binary gas mixture heated below
Authors: Lawson M. L. , Yang Wen-Jei
Published: March 2006

This paper investigates the Bénard problem in a binary mixture of dilute gases in which an imposed vertical temperature gradient induces a concentration gradient owing to the thermal diffusion effect. The transfer equations are derived by first-order perturbation theory which leads to instability criteria. Numerical results indicate that instability will set in only as stationary convection. This is distinctly different from the cases of liquids and concentrated gases, in which the thermal diffusion (or Soret) effect gives rise to oscillatory instability. It is disclosed in the study that the destabilization of the dilute gas-mixture layer is enhanced by an increase in the thermal diffusion ratio and/or the molecular weight ratio of the species.
Scattering of Poincaré waves by an irregular coastline
Authors: Howe M. S. , Mysak L. A.
Published: March 2006

This paper discusses the theory of the reflexion and scattering by an irregular coastline of a Poincaré-type wave on a rotating ocean. It is assumed that the coast is straight except for small deviations from the rectilinear form, and that these deviations may be regarded as a random function of position along the coast. The rigorous theory of energy-transfer processes in random media is applied to determine the power flux from the incident Poincard wave into the scattered Kelvin wave, which propagates in a unique direction along the coast, and into Poincard ocean wave noise. The relative efficiencies of generation of these waves is examined in some detail, and studied in particular for varying ranges of values of certain non-dimensional parameters characterizing the coastal configuration. Detailed estimates are given for a shoreline whose irregularities are specified by a Gaussian spectrum of Fourier components, and the results extra-polated in the concluding section of the paper to give a general qualitative discussion of the effects of an arbitrary coastline on an incident wave.
A model of flow separation at a free surface
Authors: Longuet-Higgins M. S.
Published: March 2006

Flow separation can be observed (1) at the leading edge of a spilling breaker or ‘white-cap’, (2) at the lower edge of a tidal bore or hydraulic jump and (3) upstream of an obstacle abutting a steady free-surface flow. At the point of flow separation there is a discontinuity in the slope of the free surface. The flow upstream of this point is relatively smooth; the flow downstream of the discontinuity is turbulent.

In this note, a local solution for the flow in the neighbourhood of the discontinuity is derived. The turbulence is represented by a constant eddy viscosity N, and the tangential stress across the interface between the laminar and turbulent zones is expressed in terms of a drag coefficient C. It is shown that the inclinations of the free surface of the two sides of the discontinuity depend on C only, and are independent of N and g. As C increases from zero to large values, so the inclination of the free surface in the turbulent zone increases from 10° 54′ to 30°. In the laminar zone the inclination of the free surface simultaneously decreases from 10° 54′ to 0°, the densities in the two zones being assumed equal.

Owing to the possible entrainment of air at the separation point, the effective density ρ′ in the turbulent zone may be less than the density ρ in the laminar zone. When these densities are allowed to be different it is found that the possible flows are of two distinct types. Flows of the first type, called ‘quasi-static’, are contiguous to a state of rest. Flows of the second type, called ‘dynamic’, are contiguous with the frictional flows described above, for which ρ′ = ρ At a given positive value of C there exists generally only one quasi-static solution. There is also just one dynamic solution provided ρ′/ρ > 0·50012. On the other hand, if ρ′/ρ < 0·5 there may be either two or no dynamic flows, depending on the value of C; and when 0·5 < ρ′/ρ > 0·50012 there may be three such flows.

The inclination of the free surface is studied as a function of C and ρ′/ρ.
Secondary flow in a curved tube
Authors: Greenspan D.
Published: March 2006

The work of Dean and that of McConalogue & Srivastava on the steady motion of an incompressible fluid through a curved tube of circular cross-section is extended through the entire range of Reynolds numbers for which the flow is laminar. The coupled nonlinear system of partial differential equations which defines the motion is solved numerically by finite differences. Computer calculations are described and physical implications are discussed.
A family of steady vortex rings
Authors: Norbury J.
Published: March 2006

Axisymmetric vortex rings which propagate steadily through an unbounded ideal fluid at rest at infinity are considered. The vorticity in the ring is proportional to the distance from the axis of symmetry. Recent theoretical work suggests the existence of a one-parameter family, [npar ]2 ≥ α ≥ 0 (the parameter α is taken as the non-dimensional mean core radius), of these vortex rings extending from Hill's spherical vortex, which has the parameter value α = [npar ]2, to vortex rings of small cross-section, where α → 0. This paper gives a numerical description of vortex rings in this family. As well as the core boundary, propagation velocity and flux, various other properties of the vortex ring are given, including the circulation, fluid impulse and kinetic energy. This numerical description is then compared with asymptotic descriptions which can be found near both ends of the family, that is, when α → [npar ]2 and α → 0.
Bounds for heat transport in a porous layer
Authors: Gupta V. P. , Joseph D. D.
Published: March 2006

Strongly nonlinear heat transport across a porous layer is studied using Howard's (1963) variational method. The analysis explores a bifurcation property of Busse's (1969) multi-a solution of this variational problem and complements the 1972 study of Busse & Joseph by further restricting the fields which are allowed to compete for the maximum heat transported a t a given temperature difference. The restriction arises, as in the case of infinite Prandtl number convection studied by Chan (1971), from letting a parameter tend to infinity from the outset; here, however, the parameter which is assumed infinitely large (the Prandtl-Darcy number) is actually seldom smaller than O(107).

The theoretical bounding heat-transport curve is computed numerically. The maximizing Nusselt number (Nu) curve is given a t first by a functional of the single-a solution; then this solution bifurcates and the Nusselt number functional is maximized for an interval of Rayleigh numbers (R) by the two-a solution. The agreement between the numerical analysis and recent experiments is striking. The theoretical heat-transport curve is found to be continuously differentiable but has piecewise discontinuous second derivatives.

The results of an asymptotic (R → ∞) analysis following Chan (1971) are in qualitative agreement with the results of numerical analysis and give the asymptotic law Nu = 0.016R. This law is consistent with the result of the porous version of the well-known dimensional argument leading to the one-third power law for regular convection. The asymptotic results, however, do not appear to be in good quantitabive agreement with the numerical results.
Conditional sampling and other measurements in a plane turbulent wake
Authors: Thomas R. M.
Published: March 2006

A series of hot-wire measurements has been carried out in a plane wake to investigate the structure of the turbulence boundary and the relation of its instantaneous position to the behaviour in the core of the flow. The principal measured quantities are as follows: mean velocity profile; intermittency factor; burst rate; mean of the longitudinal component of velocity conditioned upon various specified interface positions; autocorrelation of the intermittency signal; probability densities a t the half-intermittency point for the time between bursts and the duration of a burst; probability density for the longitudinal velocity component and its time derivative at various points across the wake; probability density a t the half-intermittency point for the same quantities in the turbulent and irrotational zones separately. In addition, the profile of the second moment of the probability density for the time between bursts has been obtained indirectly and part of the theory of Phillips (1955) has been shown to be applicable in the intermittent region.

The present measurements appear to indicate that the turbulence boundary in the wake resembles that in other plane flows more closely than has been supposed hitherto. The theory of normally distributed random noise was found to explain many of the observed statistical properties of the turbulence boundary.
Some aspects of fully developed turbulent flow in non-circular ducts
Authors: Kacker S. C.
Published: March 2006

An experimental study of fully developed uniform-density turbulent flow in a circular pipe containing one or two rods located off-centre is described. The friction factor in both cases was found to be approximately 5 % higher than the simple pipe friction factor. The shear stress distribution on the rod surface was determined using calibrated boundary-layer fences. The normalized shear stress distributions were independent of Reynolds number in the range 3·7 × 104to 2·15 × 105. Mean-velocity measurements were obtained to check the validity of the universal law of the wall near the rod surface. Secondary-flow velocities were measured by a hot-wire anemometer and integrated to yield the secondary-flow stream function. Secondary-flow velocities of the order of 1 % of the mean velocity were observed. In the gap between the two pins, however, the secondary-flow velocities were only ½% of the mean velocity. It is demonstrated that the secondary flow cannot be neglected if a force balance is used to determine the shear stress distribution on the rod surface.
Turbulence in a conical diffuser with fully developed flow at entry
Authors: Okwuobi P. A. C. , Azad R. S.
Published: March 2006

An experimental study of the structure of turbulence in a conical diffuser having a total divergence angle of 8° and an area ratio of 4: 1 with fully developed flow at entry is described. Theresearch has been done for pipe entry Reynolds numbers of 152 000 and 293000 of profiles of the mean pressure, mean velocity, turbulence intensities, correlation coefficients and the one-dimensional energy spectra, but owing to similar behaviour for these two Reynolds numbers, data are presented for a Reynolds number of 293 000.

The results show that the rate of turbulent energy production approximately reaches a maximum value at the edge of the wall layer extending to the point of maximum u1-fluctuation. It is found that, within the layer,$\overline{u^2_1}$varies linearly with the distance from the wall and the linear range grows with distance in the downstream direction.

The turbulent kinetic energy balance indicates that the magnitude of the energy convective diffusion due to kinetic and pressure effects is comparable with that of the energy production.
On the radiation and scattering of short surface waves. Part 2
Authors: Leppington F. G.
Published: March 2006

A short-wave asymptotic analysis is undertaken for problems concerned with the radiation and scattering of surface waves by a cylinder whose cross-section S intersects the free surface normally. It is assumed that S is locally smooth and convex at the two intersection points with the fluid, which may be of infinite or finite depth. For both the scattering and radiation problem, a matched expansion technique is used to provide asymptotic estimates, in terms of relatively simple wave-free limit potentials, for the amplitudes of the surface wave trains that propagate from S. Explicit details are given for some particular geometries, confirming and extending earlier results. The method can, in principle, be extended to deal with other geometries.
On the creeping motion of two arbitrary-sized touching spheres in a linear shear field
Authors: Nir Avinoam , Acrivos Andreas
Published: March 2006

The Stokes equations describing the creeping motion of two arbitrary-sized touching spheres are solved exactly through the use of tangent-sphere coordinates. For the case of a linear shear field at infinity, explicit results covering the entire range of size ratios are presented for: (a) the forces and torques on the aggregate; (b) the hydrodynamic forces on the individual spheres comprising a freely suspended aggregate, which are in general non-zero; (c) the contribution of the pair to the bulk stress of a dilute suspension; and (d) under free suspension conditions, the velocity of any material point relative to that of the undisturbed flow.
The damping of surface gravity waves in a bounded liquid
Authors: Mei C. C. , Liu L. F.
Published: March 2006

In deducing the viscous damping rate in surface waves confined by side walls, Ursell found in an example that two different calculations, one by energy dissipation within and the other by pressure working on the edge of the side-wall boundary layers, gave different answers. This discrepancy occurs in other examples also and is resolved here by examining the energy transfer in the neighbourhood of the free-surface meniscus. With due care near the meniscus a boundary-layer–Poincaré method is employed to give an alternative derivation for the rate of attenuation and to obtain in addition the frequency (or wave-number) shift due to viscosity. Surface tension is not considered.
On Burgers' model equations for turbulence
Authors: Murray J. D.
Published: March 2006

Burgers’ (1939) model equations for turbulence are considered analytically using a singular perturbation and nonlinear wave approach. The results indicate that there is an ultimate steady turbulent state. This is in agreement with the numerical results of Lee (1971) but not with Case & Chiu (1969): the last two papers start with a Fourier series approach.

A consequence of this model is that small disturbances ultimately grow into a single large domain of relatively smooth flow, accompanied by a vortex sheet in which strong vorticity is concentrated. This makes the results from the model different from those usually expected for turbulent flow fields. The model, as a result of its simplicity, has retained a degree of regularity which is not found in most forms of turbulence.
The effect of shear and stratification on the stability of a rotating fluid layer
Authors: Brunsvold A. R. , Vest C. M.
Published: March 2006

The stability of a layer of Newtonian fluid confined between two horizontal disks which rotate with different angular velocities is studied. Both isothermal and adversely stratified fluids are considered for small shear rates at low to moderate Taylor numbers. The linearized formulation of the stability problem is given a finite-difference representation, and the resulting algebraic eigenvalue problem is solved using efficient numerical techniques. The critical parameters and disturbance orientations are determined as a function of the Taylor number for the isothermal flow, and for the stratified flow for Prandtl numbers of 0·025, 1·0 and 6·0.

At high Taylor numbers, the unstratified fluid flows in Ekman-like layers near the disks, and two modes of instability are noted: the viscous-type ‘class A’ travelling wave, whose existence depends on Coriolis forces, and the inflexional ‘class B’ mode, which is nearly stationary with respect to the nearer bounding disk. As the Taylor number is decreased, the Ekman layers coalesce to form a fully developed flow. In this regime there is a Taylor number below which the class A waves are always damped. The critical Reynolds number for the class B waves increases rapidly as the Taylor number approaches zero.

For Prandtl numbers of 1·0 and 6·0, the adversely stratified flow exhibits two distinct types of instability: convective and dynamical. At low Reynolds numbers, a stationary mode associated with Bénard convection in a rotating fluid is critical. It is stabilized and given orientation by the shear. At higher Reynolds numbers, the critical mode is a travelling wave of the nature of either the class A or class B waves, depending upon the Taylor number. For a Prandtl number of 0·025, the critical mode resembles oscillatory convection at small Reynolds numbers and a class A wave at larger shear rates.
A note on the solution of the Navier-Stokes equations for a spherically symmetric expansion into a very low pressure
Authors: Freeman N. C. , Kumar S.
Published: March 2006

It is shown that, for a spherically symmetric expansion of a gas into a low pressure, the shock wave with area change region discussed earlier (Freeman & Kumar 1972) can be further divided into two parts. For the Navier–Stokes equation, these are a region in which the asymptotic zero-pressure behaviour predicted by Ladyzhenskii is achieved followed further downstream by a transition to subsonic-type flow. The distance of this final region downstream is of order (pressure)−2/3 × (Reynolds number)−1/3.
Flows of thin streams with free boundaries
Authors: Keller Joseph B. , Geer James
Published: March 2006

A method is developed for determining any thin steady two-dimensional potential flow with free and/or rigid boundaries in the presence of gravity. The flow is divided into a number of parts and in each part the flow and its free boundaries are represented as asymptotic series in powers of the slenderness ratio of the stream. There are three basic flows, having two, one and no free boundaries and called jet flow, wall flow and channel flow, respectively. First the three expansions for these flows are found, extending results of Keller & Weitz (1952). They are called outer expansions to distinguish them from the inner expansions which apply near the ends of the stream or at the junction of two different types of flow. The inner and outer expansions must be matched at a junction to find how the emerging flow is related to the entering flow. This process can be continued to build up any complex flow involving thin streams. The method is illustrated in the case of a wall flow that leaves the wall to become a jet, which includes the case of a waterfall treated by Clarke (1965) in a similar way. In part 2 (to be published) other inner expansions are found and matched to outer expansions, providing the ingredients for the construction of the solutions of many flow problems.
Asymptotic behaviour of a scalar in an axisymmetric final period turbulent wake
Authors: O'Brien Edward E.
Published: March 2006

The behaviour of a conserved scalar field in the final period of an axisymmetric turbulent wake is investigated theoretically. Explicit formulae are derived for the behaviour, during and just prior to the scalar field final period, of quantities such as mean concentration, concentration correlations and velocity–scalar or vorticity–scalar cross-correlations. In particular, if τ is the time measured from a virtual origin it is shown that scalar intensity, which decays as τ−5/2, is more persistent asymptotically than turbulence intensity, which displays a τ−7/2decay. By deriving the scalar field structure as perturbed by the Reynolds stresses acting on mean scalar gradients in the final period it is shown, e.g., that the mean field perturbation dies out at least as rapidly as τ−4, whereas the mean field itself decays as τ−1. Numerical results are presented to display the spatial structure of typical scalar correlations and velocity–scalar cross-correlations, which are compared where possible with non-asymptotic measurements in the wake of a heated sphere.
Viscous flow over a cone at moderate incidence. Part 2. Supersonic boundary layer
Authors: Lin T. C. , Rubin S. G.
Published: March 2006

A finite-difference method recently developed to study three-dimensional viscous flow is applied here to the supersonic boundary layer on a sharp cone at moderate angles of incidence (α/θ [les ] 2, angle of attack α, cone half-angle θ). The present analysis differs from previous investigations of this region in that (i) boundary-layer similarity is not assumed, (ii) the system of governing equations incorporates lateral diffusion and centrifugal force effects, and (iii) an improved numerical scheme for three-dimensional viscous flows of the type considered here is used. Solutions are shown to be non-similar at the separation streamline with local shear-layer formation. Detailed flow structure, including surface heat transfer, boundary-layer profiles and thickness, and the formation of swirling pairwise symmetric vortices, associated with cross-flow separation, are obtained. Good agreement is obtained between the present theoretical results and the existing experimental data.
Instability of an elliptic jet
Authors: Crighton D. G.
Published: March 2006

The dispersion equation for waves on an infinite uniform jet column of elliptic cross-section is derived, and approximated for large eccentricity with the aid of new asymptotics for the modified Mathieu functions. It is shown that the effect of appreciable eccentricity on lateral disturbances is greatly to reduce their growth rates below those for a circular jet, regardless of whether the disturbance grows spatially or temporally. For ‘vertical’ disturbances it is shown that the behaviour of waves of general length is qualitatively similar to that of long waves on a two-dimensional jet. Thus the mode symmetric about the major axis has small growth rate whether the mode grows temporally or spatially, while the mode antisymmetric about the major axis has small growth rate if temporally growing, but large growth rate if spatially growing. Comments are made as to the relevance of these results to the mode of action of jet silencers which squash a round jet into a flat ‘fish-tail’ shape.
Pulsating flow in a curved tube
Authors: Zalosh R. G. , Nelson W. G.
Published: March 2006

An analysis is presented of laminar fully developed flow in a curved tube of circular cross-section under the influence of a pressure gradient oscillating sinusoidally in time. The governing equations are linearized by an expansion valid for small values of the parameter (a/R) [Ka/ων]2, where a is the radius of the tube cross-section, R is the radius of curvature, ν is the kinematic viscosity of the fluid and K and ω are the amplitude and frequency, respectively, of the pressure gradient. A solution involving numerical evaluation of finite Hankel transforms is obtained for arbitrary values of the parameter α = a(ω/ν)½. In addition, closed-form analytic solutions are derived for both small and large values of α. The secondary flow is found to consist of a steady component and a component oscillatory at a frequency 2ω, while the axial velocity perturbation oscillates at ω and 3ω. The small-α flow field is similar to the low Dean number steady flow configuration, whereas the large-α flow field is altered and includes secondary flow directed towards the centre of curvature.
On the no-slip boundary condition
Authors: Richardson S.
Published: March 2006

It has been argued that the no-slip boundary condition, applicable when a viscous fluid flows over a solid surface, may be an inevitable consequence of the fact that all such surfaces are, in practice, rough on a microscopic scale: the energy lost through viscous dissipation as a fluid passes over and around these irregularities is sufficient to ensure that it is effectively brought to rest. The present paper analyses the flow over a particularly simple model of such a rough wall to support these physical ideas.
The stability of pendent liquid drops. Part 1. Drops formed in a narrow gap
Authors: Pitts E.
Published: March 2006

We consider a drop of liquid hanging from a horizontal support and sandwiched between two vertical plates separated by a very narrow gap. Equilibrium profiles of such ‘two-dimensional’ drops were calculated by Neumann (1894) for the case when the angle of contact between the liquid and the horizontal support is zero. This paper gives the equilibrium profiles for other contact angles and the criterion for their stability. Neumann showed that, as the drop height increases, its cross-sectional area increases until a maximum is reached. Thereafter, as the height increases, the equilibrium area decreases. This behaviour is shown to be typical of all contact angles. When the maximum area is reached, the total energy is a minimum. It is shown that the drops are stable as long as the height and the area increase together.
A note on the growth of Kelvin-Helmholtz waves on thin liquid sheets
Authors: Crapper G. D. , Dombrowski N. , Jepson W. P. , Pyott G. A. D.
Published: March 2006

A photographic study has been made of Kelvin-Helmholtz waves on thin liquid sheets and it has been found that, contrary to two-dimensional theory, wave growth is critically dependent upon sheet velocity and distance from the origin. This is attributed to boundary-layer separation and to subsequent vortex growth.
Entry flow in a channel. Part 3. Inlet in a uniform stream
Authors: Kapila A. K. , Ludford G. S. S. , Olunloyo V. O. S.
Published: March 2006

This paper complements earlier papers by Van Dyke (1970) and by Wilson (1971) which have appeared under the same title. Second-order boundary-layer theory is used to examine the region near the entrance to a single channel placed in a uniform stream. It is found that there are additional effects to those present in the three models treated by Van Dyke and Wilson. In particular, the cascade model misses the leading term in the separation force while the irrotational-entry model misses that in the skin friction.

There are also two new effects far downstream: logarithmic terms appear (apparently for the first time in second-order theory); and a resonance with the first eigensolution occurs.
Laminar flow over a small hump on a flat plate
Authors: Smith F. T.
Published: March 2006

A boundary layer flows over a flat plate which has on it a small hump situated downstream of the leading edge. The description of the boundary-layer flow, based upon a triple-deck structure, shows how the presence of the hump generates an interaction between the inviscid region just outside the layer and the viscous region near the hump. The pressure force dominant in the boundary layer and the connexion of the local flow with the main stream develop together and are self-perpetuating, and both remain of primary significance for a wide range of hump sizes, even for a hump buried well inside the boundary layer. By consideration of the limiting cases of very small and very large humps, a consistent account of the nature of the disturbances due to the various sizes of hump is produced. The forces and couples on the hump are also evaluated.
Amplification and decay of long nonlinear waves
Authors: Leibovich S. , Randall J. D.
Published: March 2006

The interaction of weakly nonlinear waves with slowly varying boundaries is considered. Special emphasis is given to rotating fluids, but the analysis applies with minor modifications to waves in stratified fluids and shallow-water aves. An asymptotic solution of a variant of the Korteweg–de Vries equation with variable coefficients is developed that produces a ‘Green's law’ for the amplification of waves of finite amplitude. For shallow-water waves in water of variable depth, the result predicts wave growth proportional to the $-\frac{1}{3}$ power of the depth.
Axisymmetric waves in compressible Newtonian liquids contained in rigid tubes: steady-periodic mode shapes and dispersion by the method of eigenvalleys
Authors: Scarton H. A. , Rouleau W. T.
Published: March 2006

In this paper the first thirty-two axisymmetric modes for steady-periodic waves in viscous compressible liquids contained in rigid, impermeable, circular tubes are calculated. These results end long speculation over the effects of viscosity on guided acoustic waves. Sixteen of the modes belong to a family of rotation-dominated modes whose existence was previously unknown. The thirty-two modes were computed for a wide range of frequencies, viscosities and wave-lengths.

The modes were found through the use of the method of eigenvalleys, which also led to the discovery of backward-propagating waves, an exact analytical expression for the zeroth rotational mode eigenvalue, definitive boundaries between low and intermediate frequencies and between intermediate and high frequencies, and a new type of boundary layer, called a dilatational boundary layer.
Low-Reynolds-number fall of slender cylinders near boundaries
Authors: de Mestre N. J.
Published: March 2006

The motion of bodies through fluid at low Reynolds number is appreciably affected by the container walls. Consequently the Stokes-flow theory due to Batchelor (1970) and others for a slender body falling in an unbounded fluid is difficult to test experimentally unless it is extended to take account of nearby boundaries. Theoretical expressions are given here for certain drag coefficients of a circular cylindrical slender rod of finite length falling close to a single plane wall or falling midway between two parallel plane walls. Experiments with a very viscous liquid are described in which cylinders of small thickness-to-length ratios (ranging from 1:10 to 1:100 approximately) are made to fall in suitable orientations. From their times of fall over a measured distance experimental drag coefficients are determined and compared with the corresponding theoretical value from the extension of Batchelor's theory. For rods falling in a horizontal orientation the theoretical and experimental results are consistent within the order of accuracy of the experiments. However, when results are compared for rods falling in a vertical orientation there is a significant difference for which possible explanations are presented.
Wave velocity in parallel flows of a viscous fluid
Authors: Yih Chia-Shun
Published: March 2006

It is shown in this note that the velocity of unstable or neutral waves in plane Poiseuille flow or plane Couette-Poiseuille flow, or of axisymmetric waves in Poiseuille flow, stable or unstable, must lie within the range of the velocity of the flow.
Perturbation methods in boundary-layer theory
Authors: Banks W. H. H. , Drazin P. G.
Published: March 2006

To solve a mathematical problem involving a small parameter, it is customary to expand the solution in powers of that parameter. In singular cases the resultant linearized problem may be insoluble, and in some such cases it is appropriate to expand the solution in powers of thesquare-rootof the small parameter. These cases are associated with bifurcation of the solution. The method is illustrated by applying it to the Falkner-Skan equation and to a problem in hydrodynamic instability. In particular, Hartree's conjecture, that near separation the skin friction vanes like the square-root of the appropriate parameter of the Falkner-Skan equation, is substantiated.
Hydromagnetic stability of parallel flow of an ideal heterogeneous fluid
Authors: Gedzelman Stanley David
Published: March 2006

The magnetohydrodynamic and Boussinesq approximations are used on a fully ionized ideal fluid with a longitudinal magnetic field. The flow is in a direction normal to the gravity vector and all variations in velocity, density and magnetic field. The stability characteristics, mainly for normal-mode perturbations, are investigated. Two simple problems with discontinuous profiles are solved analytically. For a double shear layer, an appropriate range of magnetic field values destabilizes the flow. A long wave theory is presented and applied to several problems, some of which are destabilized by an appropriate magnetic field. Finally, the solution for continuous profiles is presented and shown to decay algebraically in time for any stable stratification.
The measurement of turbulence intensities using real-time laser-Doppler velocimetry
Authors: George William K.
Published: March 2006

A technique by which the effect of the random phase fluctuations or Doppler ambiguity can be removed from measurements of turbulence intensities by the laser-Doppler velocimeter is proposed. Because the ambiguity is often much broader band than the turbulence, it is proposed that a sequence of low-pass filters at frequencies above those of the turbulence be used in the measurement of turbulence intensities. The true turbulence intensity is given by the extrapolation of these measurements to zero frequency.
Interaction of water waves with two closely spaced vertical obstacles
Authors: Newman J. N.
Published: March 2006

Two-dimensional waves are incident upon a pair of vertical flat plates intersecting the free surface in a fluid of infinite depth. An asymptotic theory is developed for the resulting wave reflexion and transmission, assuming that the separation between the plates is small. The fluid motion between the plates is a uniform vertical oscillation, matched to the outer wave field by a local flow at the opening beneath the plates. It is shown that the reflexion and transmission coefficients undergo rapid changes, ranging from complete reflexion to complete transmission, in the vicinity of a critical wavenumber where the fluid column between the obstacles is resonant.
Boundary-layer-induced potential flow on an elliptic cylinder
Authors: Rubin Stanley G. , Mummolo Frank J.
Published: March 2006

The application of slender-body theory to the evaluation of the three-dimensional surface velocities induced by a boundary layer on an elliptic cylinder is considered. The method is applicable when the Reynolds number is sufficiently large so that the thin-boundary-layer approximation is valid. The resulting potential problem is reduced to a two-dimensional consideration of the flow over an expanding cylinder with porous boundary conditions. The limiting solutions for a flat plate of finite span and a nearly circular cross-section are obtained in a simple analytic form. In the former case, within the limitations of slender-body theory, the results are in exact agreement with the complete three-dimensional solution for this geometry.