bluuebunny/crossref_metadata_2025 huggingface dataset:
The stability of steady and time-dependent plane Poiseuille flow
Authors: Grosch Chester E. , Salwen Harold
Published: March 2006

The linear stability of plane Poiseuille flow has been studied both for the steady flow and also for the case of a pressure gradient that is periodic in time. The disturbance streamfunction is expanded in a complete set of functions that satisfy the boundary conditions. The expansion is truncated after N terms, yielding a set of N linear first-order differential equations for the time dependence of the expansion coefficients.

For the steady flow, calculations have been carried out for both symmetric and antisymmetric disturbances over a wide range of Reynolds numbers and disturbance wave-numbers. The neutral stability curve, curves of constant amplification and decay rate, and the eigenfunctions for a number of cases have been calculated. The eigenvalue spectrum has also been examined in some detail. The first N eigenvalues are obtained from the numerical calculations, and an asymptotic formula for the higher eigenvalues has been derived. For those values of the wave-number and Reynolds number for which calculations were carried out by L. H. Thomas, there is excellent agreement in both the eigenvalues and the eigenfunctions with the results of Thomas.

For the time-dependent flow, it was found, for small amplitudes of oscillation, that the modulation tended to stabilize the flow. If the flow was not completely stabilized then the growth rate of the disturbance was decreased. For a particular wave-number and Reynolds number there is an optimum amplitude and frequency of oscillation for which the degree of stabilization is a maximum. For a fixed amplitude and frequency of oscillation the wave-number of the disturbance and the Reynolds number has been varied and a neutral stability curve has been calculated. The neutral stability curve for the modulated flow shows a higher critical Reynolds number and a narrower band of unstable wave-numbers than that of the steady flow. The physical mechanism responsible for this stabiIization appears to be an interference between the shear wave generated by the modulation and the disturbance.

For large amplitudes, the modulation destabilizes the flow. Growth rates of the modulated flow as much as an order of magnitude greater than that of the steady unmodulated flow have been found.
Linearized analysis of constant-property duct flows
Authors: Williams F. A.
Published: March 2006

The conservation equations describing steady, incompressible flow in a variable-area duct with mass transfer occurring at its walls are simplified by linearizing the inertial and convective terms. Solutions to a large class of problems can be obtained by means of the general method which is presented. Particular examples considered are entrance flow with heat transfer to an isothermal wall in the presence of mass addition, constant rate of injection of a foreign gas through the wall, vaporization and sublimation of a volatile wall material, and gas-phase combustion of a fuel which enters the duct from its wall. Comparison of the present results with previous work and with new experimental results is discussed for the first of these applications. It is concluded that the present results for velocity and pressure fields are likely to be accurate for small values of the Reynolds number based on wall injection velocity, that the present results for temperature and composition fields are likely to be accurate in the absence of wall mass transfer, and that in the absence of wall mass transfer the linearization technique is likely to exhibit its highest accuracy for flows with uniform entrance conditions.
A problem of gravity wave drag at an interface
Authors: Warren F. W. G. , Mackinnon R. F.
Published: March 2006

A thin disk, immersed in a fluid consisting of two superposed homogeneous inviscid liquids, moves in its own vertical plane. A Green's function is found for the potential and a formula for the gravity wave energy is found and evaluated for a specific case.
The effects of interference and viscosity in the Kelvin ship-wave problem
Authors: Allen R. F.
Published: March 2006

A comparison is made between the effects of spreading the source (thus causing interference) and of viscosity in the Kelvin ship-wave problem. Two simple pressure distributions are considered and in both it is found that spreading, like viscosity, introduces a multiplicative damping factor into the inviscid pressure-point solution. In one case it is this factor, rather than that of viscosity, which dominates the decay of the wave profile, while, in the other, the effects alternate in importance as one travels along any particular wave crest.
The dynamic response of towed thermometers
Authors: Fabula A. G.
Published: March 2006

Small-scale temperature fields in water were used to test the dynamic response of towed thermometers of the platinum film resistance type. Laminar buoyant plumes rising from submerged heaters below the line of motion were the test temperature fields. The analysis of results was based on an approximate ‘diffusion-layer’ model of the dynamic heat-transfer process occurring near the platinum film on the probe tip. The model represents the linear heat transfer into a two-layer semi-infinite medium, with the platinum thin film located at the interface between a water layer of thickness Δ and a semi-infinite substrate of glass. The differences of the thermal properties of water and glass were found to be negligible. The characteristic time Δ2/D, where D is the thermal diffusivity of water, was determined by the ratio of actual to film-indicated plume-peak temperature, assuming a sinusoid approximation to the plume profile. The frequency response for the same operating conditions as the plume tests could then be obtained from the diffusion-layer model.
Diffraction of Kelvin waves at a sharp bend
Authors: Packham B. A. , Williams W. E.
Published: March 2006

A closed form solution is obtained for the linearized problem of the diffraction of Kelvin waves by a sharp bend on a rotating earth. It is shown that for the frequency ω of the incoming wave less than f, where ½f is the angular velocity of rotation, the wave is transmitted around the bend without change of amplitude. For ω > f the amplitude is in general reduced but is unaltered for the special angles π/(2n + 1). For these special angles the solution is obtained in elementary terms.
Theoretical study of the slow motion of a sphere and a fluid in a cylindrical tube
Authors: Greenstein T. , Happel J.
Published: March 2006

The results of a theoretical treatment are presented for the slow flow of a viscous fluid through a circular cylinder within which a small spherical particle is confined. The sphere is situated in an arbitrary position within the cylinder, rotates with an arbitrary constant angular velocity and moves at constant velocity parallel to the wall. Approximate expressions are presented which give the frictional force, torque, and permanent pressure drop caused by the presence of this obstacle in the original Poiseuillian field of flow.

An eccentricity function for the torque on a sphere in a circular cylinder was evaluated numerically. It can be used to predict the wall-effect for the torque as well as the angular velocity with which a ‘dense’ spherical particle will rotate. Expressions are presented which predict the angular velocity of ‘dense’ as well as neutrally buoyant hydrodynamically supported spherical particles.
Solutions for supersonic rotational flow around a corner using a new co-ordinate system
Authors: Adamson T. C.
Published: March 2006

A co-ordinate system consisting of the left-running characteristics (α = const.) and the streamlines (ϕ = const.) is used. The governing equations are derived in terms of α and ϕ for a two-dimensional steady supersonic rotational inviscid flow of a perfect gas. The equations are applied to the problem of an initially parallel supersonic rotational flow which expands around a convex corner. The velocity of the incoming flow at the wall is considered to be either supersonic (case 1) or sonic (case 2). For each case, solutions uniformly valid in the region near the leading characteristic and in the region near the corner, are found for the Mach angle and flow deflexion angle in terms of their values on the leading characteristic and at the corner. In case 2, a transonic similarity solution is found and composite solutions are constructed for each region. Comparisons are made with existing exact numerical results.
Long standing waves in a curved canal
Authors: Johns B. , Hamzah A. M. O.
Published: March 2006

The dynamics of long water waves are considered in a curved geometry representing a canal bend. The presence of the bend is found to produce a spectrum of transverse oscillations in the canal. The associated dominant amplitudes are evaluated for both tidal periods and higher frequencies representative of tsunamis. It is found that low-frequency waves do not lead to significant transverse amplitudes. For tsunamis, the presence of the bend may result in considerable changes in the local wave amplitude.
Laminar flow of a stably stratified fluid past a flat plate
Authors: Pao Yih-Ho
Published: March 2006

Laminar flow of a stably stratified fluid with uniform upstream velocity and density gradient past a flat plate is investigated experimentally and theoretically. In the experimental study, the flat plate, parallel to the direction of motion, is towed horizontally at uniform speeds in a tank of stratified salt water at uniform density gradients. The horizontal velocity in front of, above, and behind the flat plate is measured with a flow visualization technique. One of the striking phenomena is the strong upstream influence (the upstream wake) of an obstacle in a flow of stably stratified fluid when the gravity effect is comparable to, or more than the inertial effect. The velocity profiles of the upstream wake and boundary layer above the flat plate are wavy and found to be governed by the ratio of Reynolds number Be to Richardson number Ri. The problem is also analysed theoretically with Boussinesq's approximation and two-parameter perturbation expansions for the upstream wake and the boundary layer. The solutions for the upstream wake and for the boundary layer are compared with measurements.
On the hydrodynamics of pairs of spheres falling along their line of centres in a viscous medium
Authors: Steinberger E. H. , Pruppacher H. R. , Neiburger M.
Published: March 2006

The velocities, accelerations and drag forces experienced by two equal spheres falling along their line of centres in a viscous fluid were determined for three groups of Reynolds numbers R in the range where it is commonly assumed that Stokes's approximation applies. For all groups, with R ranging between 0·060 and 0·216, both spheres continually acclerated as they fell, and the upper sphere fell faster and accelerated more than the lower one. In contrast to Stimson & Jeffery's (1926) theory, which is based on the Stokes approximation, and to most earlier experimenters, the drag-force coefficients of the upper sphere computed from the experiments were significantly smaller than those for the lower sphere. Oseen's theory for this case agreed with the experiments in some respects, but contrary to it the drag-force coefficient varied with R for the upper sphere as well as the lower sphere.
Aerodynamic noise dependent on mean shear
Authors: Jones Ian S. F.
Published: March 2006

New factors illustrating the effect of convection on acoustic radiation are deduced from Lighthill's theory of aerodynamic noise. For sound dependent on mean velocity variations for its generation, that is shear noise, the new factors are (1 − M cos θ)−3 and (1 − M cos θ)−1 in contrast with (1 − M cos θ)−5 for self noise.

Comparison with the measured far-field directionality patterns reported by Howes (1960) shows that the factor (1 − M cos θ)−3 provides considerably better agreement with the experimental observations than the self noise factor.

The source terms discussed by Lighthill (1954) when considering shear amplification are examined and shown to contain the terms produced by the analysis of Csanady (1966) as well as a new term dependent on the second derivative of the mean velocity.
Wave propagation and boundary instability in erodible-bed channels
Authors: Gradowczyk Mario H.
Published: March 2006

Wave propagation in one-dimensional erodible-bed channels is discussed by using the shallow-water approximation for the fluid and a continuity equation for the bed. In addition to gravity waves, a third wave, which gives the velocity of propagation of a bed disturbance, is found. An appropriate dimensional analysis yields the quasi-steady approximation for the complete shallow-water equations.

The well-known linear stability analysis of free-surface flows is extended to include the erodibility of the bed. The critical Froude numberFcabove which the free-surface of the fluid may become unstable is obtained. It is shown that erodibility increases the stability of the free surface, in qualitative agreement with previous experiments ifqb>qs,qbandqsbeing respectively the contact-bed discharge and suspended-material discharge. The stability theory is also used to discuss coupled beds and surface waves. From it, five different configurations have been obtained: a sinusoidal wave pattern moving downstream, a transition zone and antidunes moving upstream, moving downstream and stationary. These bed forms are in agreement with experimental results; hence shallow-water theory seems to give a reasonable explanation of the boundary instability.

It is shown that the quasi-steady approximation and Kennedy's (1963) stability analysis will be in agreement if (kh)2[Lt ]1, wherekis the wave number, andhis the depth of the water. When the phase shift δ is introduced in the quasi-steady approximation, the five bed patterns derived from the full equations are found again.
Transformations of the boundary-layer equations for rotating flows
Authors: Rott N. , Ohrenberger J. T.
Published: March 2006

The boundary layer on an axisymmetric surface above which the flow is rotating about the axis of symmetry is considered. Transformations of the governing equations which permit the generalizations of a known solution for one meridian shape in incompressible flow to a family of meridian shapes are shown to exist. For compressible flow, a transformation of the Stewartson-Illingworth type was found which reduces a compressible flow problem to an incompressible case. Also, remarks are made concerning the invariance of the turbulent boundary-layer integral equations assuming particular semi-empirical shear laws.
Stability of three-dimensional laminar and turbulent shear layers
Authors: Lessen M. , Harpavat G. , Zien H. M.
Published: March 2006

The linear, normal mode instability of three-dimensional laminar and turbulent shear layers is studied. The flows consist of two streams semi-infinitely extended in the y-direction, flowing obliquely in the x-z plane. It is found that the stability of the flows depends on the main flow velocity components in the direction of the disturbance wave-number vector. Numerical calculations are performed to obtain the neutral stability curves. Under the usual parallel unperturbed flow assumption, the neutral stability curves pass through the origin of the α-R diagram, where α is the wave-number and R is the flow Reynolds number. It is also found that eddy viscosity has a destabilizing effect for small Reynolds numbers but a stabilizing effect at larger Reynolds numbers. Because any linear perturbation trajectory eventually leaves the unstable region of the α-R plane, it is probable that a Lin-Benney or Taylor-Goertler secondary instability ensues. Suitable components of the existing turbulence grow and develop into a large eddy system which causes rapid entrainment, giving rise to a turbulent burst. A first-order non-parallel correction is made to the neutral stability curves. The new curves have both upper and lower branches, and there exist minimum critical Reynolds numbers.
Viscous flow in a cylindrical tube containing a line of spherical particles
Authors: Wang Henry , Skalak Richard
Published: March 2006

The viscous, creeping flow through a cylindrical tube of a liquid, which contains rigid, spherical particles, is investigated analytically. The spheres are located on the axis of the cylinder and are equally spaced. Solutions are derived for particles in motion and fixed, with and without fluid discharge. Numerical results are presented for the drag on each sphere and the mean pressure drop for a wide range of sizes and spacings of the spheres. The study is motivated by possible application to blood flow in capillaries, where red blood cells represent particles of the same order of magnitude as the diameter of the capillary itself. The results may also be of interest in other applications, such as sedimentation and fluidized beds. It is shown that there is little interaction between particles if the spacing is more than one tube diameter, and that the additional pressure drop over that for Poiseuille flow is less than 50% if the sphere diameter is less than 0·8 of the tube diameter.
Viscous dissipation in external natural convection flows
Authors: Gebhart B. , Mollendorf J.
Published: March 2006

The effects of viscous dissipation are considered for external natural convection flow over a surface. A class of similar boundary-layer solutions is given and numerical results are presented for a wide range of the dissipation and Prandtl numbers. Several general aspects of similarity conditions for flow over surfaces and in convection plumes are discussed and their special characteristics considered. The general equations including the dissipation effect are given for the non-similar power law surface condition.
On the outer-edge problem of a hypersonic boundary layer
Authors: Lee Richard S. , Cheng H. K.
Published: March 2006

Existing analytical treatments of the hypersonic strong interaction problem adopt the two-region structure of the classical boundary-layer theory; however, the uneven heating and external vorticity created by the highly curved, leading-edge shock wave leads to the question of the uniform validity of the boundary-layer approximation near the boundary-layer edge. Recently Bush (1966), using a non-linear viscosity-temperature law (μ ∞ Tω, ω < 1) instead of the linear one (μ ∞ T) used by previous investigators, demonstrated the need to analyze separately a transitional layer intermediate between the inviscid region and the boundary layer. In this paper, an asymptotic analysis of the Navier-Stokes equations in von Mises's variables, allowing a three-region structure, is carried out for the case of μ ∞ T. Results, with the second-order effects associated with heating and vorticity accounted for, show that a separate analysis of the transitional region is not strictly necessary in this case, and hence the equivalence of the two-region approach is confirmed. On the other hand, it is shown that the second-order boundary-layer correction owing to the heating and vorticity effects, not considered by Bush, is necessary in order to determine a uniformly valid temperature distribution in the physical variables. Numerical results for an insulated and a cold flat plate, considerably different from those of others, are obtained.
Some observations on skin friction and velocity profiles in fully developed pipe and channel flows
Authors: Patel V. C. , Head M. R.
Published: March 2006

Measurements of skin friction and mean-velocity profiles have been made in fully developed flows in pipes and channels in the Reynolds number range 1000 <Re< 10000. These measurements, and observations of hot-wire signals, indicate rather remarkable differences between two-dimensional and axially symmetric flows and also make it difficult to give a precise definition of the term ‘fully developed turbulent flow’.
The effect of a parallel magnetic field on the stability of free boundary-layer type flows of low magnetic Reynolds number
Authors: Abas S.
Published: March 2006

Stability to infinitesimal disturbances, when a parallel magnetic field is imposed, is investigated for the free boundary-layer type flows, of low magnetic Reynolds number, between two unbounded parallel streams of a viscous, incompressible, electrically conducting fluid. Neutral stability curves are calculated for small wave-number making use of the limiting profile: previous results by another author are found to be incomplete. A qualitative neutral stability picture is conjectured for other values of the wave-number and, granted a certain part of this conjecture, the conclusion is that the critical Reynolds number remains zero until the parameter Q/R exceeds the value (Q/R)crit [eDot ] 0·0233. It is suggested that a sufficiently strong magnetic field can stabilize a flow of any finite Reynolds number.
The strength of vortex and swirling core flows
Authors: Morton B. R.
Published: March 2006

This note presents a discussion of the roles of axial momentum flux, flow force, angular momentum flux and circulation in determining the strength and hence characterizing the structure of such narrow rotating axisymmetric core flows as swirling jets, vortex jets, sink vortices and vortex wakes. The salient (though sometimes neglected) features of these core flows are that perturbation pressure plays an essential role both in the coupling of axial and azimuthal velocity fields and in the transmission of force along the core, and that flux of angular momentum is invariant only along cores with zero gross circulation. A number of existing solutions are brought into relationship by the discussion, including Long's similarity solution for draining vortices and Reynolds’ dimensional treatment of swirling wakes.
Internal wave propagation in an inhomogeneous fluid of non-uniform depth
Authors: Keller Joseph B. , Mow Van C.
Published: March 2006

An asymptotic solution is obtained to the problem of internal wave propagation in a horizontally stratified inhomogeneous fluid of non-uniform depth. It also applies to fluids which are not stratified, but in which the constant density surfaces have small slopes. The solution is valid when the wavelength is small compared to all horizontal scale lengths, such as the radius of curvature of a wavefront, the scale length of the bottom surface variations and the scale length of the horizontal density variations. The theory underlying the solution involves rays, a phase function satisfying the eiconal equation, and amplitude functions satisfying transport equations. All these equations are solved in terms of the rays and of the solution of the internal wave problem for a horizontally stratified fluid of constant depth. The theory is thus very similar to geometrical optics and its extensions. It can be used to treat problems of propagation, reflexion from vertical cliffs or from shorelines, refraction, determination of the frequencies and wave patterns of trapped waves, etc. For fluid of constant density, it reduces to the theory of Keller (1958). The theory is applied to waves in a fluid with an exponential density distribution on a uniformly sloping beach. The predicted wavelength is shown to agree well with the experimental result of Wunsch (1969). It is also applied to determine edge waves near a shoreline and trapped waves in a channel.
Scattering of surface waves by rectangular obstacles in waters of finite depth
Authors: Mei Chiang C. , Black Jared L.
Published: March 2006

The scattering of infinitesimal surface waves normally incident on a rectangular obstacle in a channel of finite depth is considered. A variational formulation is used as the basis of numerical computations. Scattering properties for bottom and surface obstacles of various proportions, including thin barriers and surface docks, are presented. Comparison with experimental and theoretical results by other investigators is also made.
Some measurements in the self-preserving jet
Authors: Wygnanski I. , Fiedler H.
Published: March 2006

The axisymmetric turbulent incompressible and isothermal jet was investigated by use of linearized constant-temperature hot-wire anemometers. It was established that the jet was truly self-preserving some 70 diameters downstream of the nozzle and most of the measurements were made in excess of this distance. The quantities measured include mean velocity, turbulence stresses, intermittency, skewness and flatness factors, correlations, scales, low-frequency spectra and convection velocity. The r.m.s. values of the various velocity fluctuations differ from those measured previously as a result of lack of self-preservation and insufficient frequency range in the instrumentation of the previous investigations. It appears that Taylor's hypothesis is not applicable to this flow, but the use of convection velocity of the appropriate scale for the transformation from temporal to spatial quantities appears appropriate. The energy balance was calculated from the various measured quantities and the result is quite different from the recent measurements of Sami (1967), which were obtained twenty diameters downstream from the nozzle. In light of these measurements some previous hypotheses about the turbulent structure and the transport phenomena are discussed. Some of the quantities were obtained by two or more different methods, and their relative merits and accuracy are assessed.
Turbulent buoyant convection from a source in a confined region
Authors: Baines W. D. , Turner J. S.
Published: March 2006

This paper considers the effect of continuous convection from small sources of buoyancy on the properties of the environment when the region of interest is bounded. The main assumptions are that the entrainment into the turbulent buoyant region is at a rate proportional to the local mean upward velocity, and that the buoyant elements spread out at the top of the region and become part of the non-turbulent environment at that level. Asymptotic solutions, valid at large times, are obtained for the cases of plumes from point and line sources and also periodically released thermals. These all have the properties that the environment is stably stratified, with the density profile fixed in shape, changing at a uniform rate in time at all levels, and everywhere descending (with ascending buoyant elements).

The analysis is carried out in detail for the point source in an environment of constant cross-section. Laboratory experiments have been conducted for this case, and these verify the major predictions of the theory. It is then shown how the method can be extended to include more realistic starting conditions for the convection, and a general shape of bounded environment. Finally, the model is applied quantitatively to a variety of problems in engineering, the atmosphere and the ocean, and the limitations on its use are discussed.
Longitudinal vortices in natural convection flow on inclined plates
Authors: Sparrow E. M. , Husar R. B.
Published: March 2006

Experiments are performed to demonstrate the occurrence and explore the characteristics of a secondary flow superposed upon the natural convection main flow on an inclined plate. A flow visualization technique is employed whereby the flow pattern is made visible by local changes of colour of the fluid itself, the colour change being brought about by a change in pH. The secondary flow consists of longitudinal vortices or rolls distributed more or less periodically across the width of the plate. The number of such vortices increases with the temperature difference between the surface and the ambient fluid, but appears to be relatively insensitive to the inclination angle of the plate. The secondary flow results from the destabilizing effect of the buoyancy force component, which acts normal to the plate surface. The longitudinal vortices are the first stage of the laminarturbulent transition process. This is in contrast to the case of natural convection on a vertical plate, where the first stage of transition is Tollmien-Schlichting waves.
Axisymmetric inertial oscillations of a fluid in a rotating spherical container
Authors: Aldridge Keith D. , Toomre Alar
Published: March 2006

This paper describes an experiment performed with a fluid-filled sphere whose rotation speed about a fixed axis was forcibly varied in a slight but sinusoidal manner about a non-zero mean value. The object of this experiment was both to excite axisymmetric inertial eigen-oscillations within the relatively low viscosity fluid through the mild pumping action of the oscillatory Ekman boundary layer near the wall, and to measure and compare with theory some of the properties of such modes.

Seven distinct fluid resonances were detected via pressure measurements made along the axis for various ratios of the excitation to the mean rotation frequency. For the three most pronounced of those modes, the observed frequency ratios agree within ½ of 1% with the corresponding ratios predicted from linear, small viscosity theory. The response amplitudes at the various resonances and the rates of decay upon switching off the excitation also compare favourably with theory, although the observed amplitudes are systematically lower and the decays more rapid by a few per cent to several tens of per cent.

The theory referred to above is largely that of Greenspan (1964, 1968). It is in part rederived here from energy considerations.
On the structure of a class of aerothermodynamic shocks
Authors: Blythe P. A. , Petty D. G. , Schofield D. A. , Wilson J. L.
Published: March 2006

Some recent work on the existence of vibrational de-excitation shocks (δ-shocks) in expanding non-equilibrium nozzle flows is extended to include situations in which an adiabatic shock (δ-shocks) may be embedded within the de-excitation shock. A discussion of some further properties of the shock solution is given and some examples are worked out. Numerical solutions of the full equations are also presented. These solutions confirm the existence of the δ-shocks but bring to light certain anomalies in the simple approximate solution. The modifications necessary to remove these discrepancies are outlined, and the implications of the numerical results are briefly discussed. Finally, some comments on the nature of the asymptotic solution for an arbitrary rate process are made.
Variations in filtration velocity due to random large-scale fluctuations of porosity
Authors: Buyevich Yu. A. , Leonov A. I. , Safrai V. M.
Published: March 2006

The local porosity of a real porous material (obtained by averaging over volumes containing a sufficiently large number of pores) is different from the mean porosity of the material as a whole. This difference is caused by large-scale defects of the porous structure and can be treated as a random function of position in the porous material. Such a random deviation of the local porosity from the average value causes random local flows superimposed upon the mean filtration flow. The characteristic scale of such motion is much larger than that of the flow of a fluid through individual pores. The phenomenon appears to play an important role in transport processes in filtration.

In this paper the statistical characteristics of the random fields under consideration are determined on the basis of the assumption that the local porosity is a random function of position with independent increments. Expressions for correlations of various quantities are obtained in terms of the characteristics of porosity fluctuations and the effective coefficients of diffusion caused by the random motions under study are estimated.
Rough wall turbulent boundary layers
Authors: Perry A. E. , Schofield W. H. , Joubert P. N.
Published: March 2006

This paper describes a detailed experimental study of turbulent boundary-layer development over rough walls in both zero and adverse pressure gradients. In contrast to previous work on this problem the skin friction was determined by pressure tapping the roughness elements and measuring their form drag.

Two wall roughness geometries were chosen each giving a different law of behaviour; they were selected on the basis of their reported behaviour in pipe flow experiments. One type gives a Clauser type roughness function which depends on a Reynolds number based on the shear velocity and on a length associated with the size of the roughness. The other type of roughness (typified by a smooth wall containing a pattern of narrow cavities) has been tested in pipes and it is shown here that these pipe results indicate that the corresponding roughness function does not depend on roughness scale but depends instead on the pipe diameter. In boundary-layer flow the first type of roughness gives a roughness function identical to pipe flow as given by Clauser and verified by Hama and Perry & Joubert. The emphasis of this work is on the second type of roughness in boundary-layer flow. No external length scale associated with the boundary layer that is analogous to pipe diameter has been found, except perhaps for the zero pressure gradient case. However, it has been found that results for both types of roughness correlate with a Reynolds number based on the wall shear velocity and on the distance below the crests of the elements from where the logarithmic distribution of velocity is measured. One important implication of this is that a zero pressure gradient boundary layer with a cavity type rough wall conforms to Rotta's condition of precise self preserving flow. Some other implications of this are also discussed.
The motion of a fluid in a cylindrical container with a free surface following vertical impact
Authors: Milgram Jerome H.
Published: March 2006

The effect of a sudden change in vertical velocity of a vessel partially filled with fluid is considered. It is shown that very small free surface disturbances can be amplified so strongly by a velocity change that instability can occur. This instability frequently causes a jet to emanate from the free surface. Conditions causing free surface disturbances in a vessel in free fall are considered and it is shown that a contact angle between the fluid and the wall of the vessel different from ½π radians can distort the surface in such a way that the amplification and instability upon vertical impact results in a central jet. The results of experiments of this effect are shown. The generation of waves due to transient side wall motions, such as those that might result from fluid pressure at impact, are considered in the appendix. It is shown that such waves would have a different form than those observed in the experiments.
A law of the wall for compressible turbulent boundary layers with air injection
Authors: Squire L. C.
Published: March 2006

A considerable body of experimental data now exists concerning turbulent boundary layers with air injection at the wall, both at subsonic and at supersonic speeds. In the present report these data for Mach numbers up to 6·5 have been analyzed to find the parameters which occur in the law of the wall as deduced from mixing-length theory. Although the absolute values of the parameters are subject to error because of the lack of accurate skin-friction measurements, the trends of these parameters with Mach number and injection mass flow are clearly defined.
On Howard's upper bound for heat transport by turbulent convection
Authors: Busse F. H.
Published: March 2006

The variational problem introduced by Howard (1963) for the derivation of an upper bound on heat transport by convection in a layer heated from below is analyzed for the case in which the equation of continuity is added as constraint for the velocity field. Howard's conjecture that the maximizing solution of the Euler equations is characterized by a single horizontal wave-number is shown to be true only for a limited range of the Rayleigh number, Ra. A new class of solutions with a multiple boundary-layer structure is derived. The upper bound for the Nusselt number, Nu, given by these solutions is Nu ≤ (Ra/1035)½ in the limit when the Rayleigh number tends to infinity. The comparison of the maximizing solution with experimental observations by Malkus (1954a) and Deardorff & Willis (1967) emphasizes the similarity pointed out by Howard.
Initial propagation of impulsively generated converging cylindrical and spherical shock waves
Authors: Bach Glen G. , Lee John H.
Published: March 2006

An analytical description for the initial phases of collapse of a spherical or cylindrical shock wave in a perfect gas is given in the present paper. The shock wave is initiated by the instantaneous and uniform deposition of a finite quantity of energy per unit surface area at a finite radiusR0. For the initial shock motion wherexs= (Rs-R0)/R0is small, analytical solutions are obtained by a power-series expansion of the dependent variables inxs. The classical self-similar solution for a strong planar blast wave is recovered as the present zero-order solution. Non-similar effects arising from both finite shock strengths and the presence of a characteristic lengthR0are accounted for simultaneously in the present perturbation scheme. The analysis is carried out up to third order inxs. For very large values of the initiation energy where the shock wave remains strong throughout its collapse, it is found that the present perturbation solution can adequately describe a significant portion of the collapse processes. The solution indicates that the shock decays rather rapidly initially and later begins to accelerate as a result of the additional adiabatic compression of the shocked states due to flow-area convergence. However, for weak initiation where the energy released is small, the present perturbation solution is an asymptotic series and diverges very rapidly as the shock propagates away from the wall. The range of validity then is limited to very small values ofxs.
Distortion of sonic bangs by atmospheric turbulence
Authors: Crow S. C.
Published: March 2006

Recorded pressure signatures of supersonic aircraft often show intense, spiky perturbations superimposed on a basicN-shaped pattern. A first-order scattering theory, incorporating both inertial and thermal interactions, is developed to explain the spikes. Scattering from a weak shock is studied first. The solution of the scattering equation is derived as a sum of three terms: a phase shift corresponding to the singularity found by Lighthill; a small local compression or rarefaction; a surface integral over a paraboloid of dependence, whose focus is the observation point and whose directrix is the shock. The solution is found to degenerate at the shock into the result given by ray acoustics, and the surface integral is identified with the scattered waves that make up the spikes. The solution is generalized for arbitrary wave-forms by means of a superposition integral. Eddies in the Kolmogorov inertial subrange are found to be the main source of spikes, and Kolmogorov's similarity theory is used to show that, for almost all timestafter a sonic-bang shock passes an observation point, the mean-square pressure perturbation equals$(\Delta p)^2 (t_c/t)^{\frac{7}{6}}$, where Δpis the pressure jump across the shock andtcis a critical time predicted in terms of meteorological conditions. For an idealized model of the atmospheric boundary layer,tcis calculated to be about 1 ms, a figure consistent with the qualitative data currently available. The mean-square pressure perturbation just behind the shock itself is found to be finite but enormous, according to first-order scattering theory. It is conjectured that a second-order theory might explain the shock thickening that actually occurs.
Particle collision efficiencies for a sphere
Authors: Michael D. H. , Norey P. W.
Published: March 2006

Trajectories are calculated for small particles introduced upstream into a fluid flowing past a fixed sphere. Unseparated potential flow is taken as the velocity profile for the fluid, and the effect of gravity is included in the formulation when it acts along the axis of symmetry. Using a numerical procedure, particle trajectories which graze the sphere, and the corresponding collision efficiencies, are calculated for values of the Stokes number σ. When gravity is neglected, an analytic solution is obtained for large values of σ which is in good agreement with the numerical results for σ as low as 5. These results are compared with those of Sell (1931) and Langmuir & Blodgett (1946). When gravity is included, a critical value of the Stokes number σc is calculated for which no collisions occur until σ > σc.
The deformation of a drop in a general time-dependent fluid flow
Authors: Cox R. G.
Published: March 2006

A theoretical method is given for the determination of the shape of a fluid drop in steady and unsteady flows by making an expansion in terms of the drop deformation. Effects of fluid viscosity and interfacial tension are taken into account. Examples given include the determination of the shape of a drop in shear and in hyperbolic flow when each is started impulsively from rest.
A numerical solution of axisymmetric cavity flows
Authors: Brennen Christopher
Published: March 2006

In the first part of the paper a method is developed for the relaxation or numerical solution of axisymmetric fully cavitating flows. Employing the technique suggested in a paper by Woods (1951 a) of working in a transformed (ϕ ϕ)-plane, solutions are obtained for cavities behind a disk and a sphere in different sizes of solid wall tunnel. Under certain conditions flow ‘choking’ occurs.

The results of a series of experiments carried out with such headforms are then reported. The apparent viscous effect on the position of separation from the sphere and thus on the drag proves to be of particular interest.
Spin down problem of rotating stratified fluid in thermally insulated circular cylinders
Authors: Sakurai Takeo
Published: March 2006

A response of viscous heat-conducting compressible fluid to an abrupt change of angular velocity of a containing thermally insulated circular cylinder under the existence of stable distribution of the temperature is investigated within the framework of the Boussinesq approximation for a time duration of the order of the homogeneous-fluid spin down time in order to resolve the Holton-Pedlosky controversy. The explicit expression of the solution is obtained by the standard method and Holton's conclusion is confirmed. The secondary meridional current induced by the Ekman layers spins the fluid down to a quasi-steady state within the present time scale. However, unlike the homogeneous case, the quasi-steady state is not one of solid body rotation. The final approach to the state of rigid rotation is achieved via the viscous diffusion in the time scale of the usual diffusion time.
Some aspects of turbulence measurement in liquid mercury using cylindrical quartz-insulated hot-film sensors
Authors: Malcolm D. G.
Published: March 2006

This investigation demonstrates the measurement of low turbulence intensities in a flow of mercury using the constant-temperature hot-film technique. A simple equation for the determination of turbulence intensity is derived on the basis of observed calibration data for short, cylindrical, quartz-insulated, platinum hot-film sensors. This equation has the advantage of being independent of the changeable effects at the sensor-fluid interface and of the usually troublesome drift in anemometer output signal. The flow speeds considered are low, so that the Péclet number based on the outside diameter of the insulated sensor is less than unity. Various problems associated with the application of hot-film anemometry to mercury flows are discussed.
A note on the damping and oscillations of a fluid drop moving in another fluid
Authors: Subramanyam S. V.
Published: March 2006

The oscillations of a drop moving in another fluid medium have been studied at low values of Reynolds number and Weber number by taking into consideration the shape of the drop and the viscosities of the two phases in addition to the interfacial tension. The deformation of the drop modifies the Lamb's expression for frequency by including a correction term while the viscous effects split the frequency into a pair of frequencies—one lower and the other higher than Lamb's. The lower frequency mode has ample experimental support while the higher frequency mode has also been observed. The two modes almost merge with Lamb's frequency for the asymptotic cases of a drop in free space or a bubble in a dense viscous fluid but the splitting becomes large when the two fluids have similar properties. Instead of oscillations, aperiodic damping modes are found to occur in drops with sizes smaller than a critical size ($\sim\hat{\rho}\hat{\nu}^2/T $). With the help of these calculations, many of the available experimental results are analyzed and discussed.
On the flow past a sphere at low Reynolds number
Authors: Chester W. , Breach D. R. , Proudman Ian
Published: March 2006

The flow of an incompressible, viscous fluid past a sphere is considered for small values of the Reynolds number. In particular the drag is found to be given by
\[
D = D_s\{1+{\textstyle\frac{3}{8}}R+{\textstyle\frac{9}{40}}R^2(\log R+\gamma + {\textstyle\frac{5}{3}}\log 2 - {\textstyle\frac{323}{360}})+{\textstyle\frac{27}{80}}R^3\log R+O(R^3)\},
\]
where Ds is the Stokes drag, R is the Reynolds number and γ is Euler's constant.
On the instability of viscous flow in a rapidly rotating pipe
Authors: Pedley T. J.
Published: March 2006

The stability of almost fully developed viscous flow in a rotating pipe is considered. In cylindrical polar co-ordinates (r, ø, z) this flow has the velocity components\[ \{W_0o(1),\quad\Omega r[1+o(\epsilon)],\quad W_0[1-r^2/r^2_0+o(1)]\},_{+}^{+} \]where ε =Wo/2Ωr0and is bounded externally by the rigid cylinderr = r0, which rotates about its axis with angular velocity Ω. In the limit of small ε, the disturbance equations can be solved in terms of Bessel functions and it is shown that, in that limit, the flow is unstable for Reynolds numbersR=Wor0/vgreater thanRc[asymp ] 82[sdot ]9. The unstable disturbances take the form of growing spiral waves, which are stationary relative to the rotating cylinder and the critical disturbance atR = Rchas azimuthal wave-number 1 and axial wavelength 2πr0/ε. Furthermore, it is shown that the most rapidly growing disturbance forR > Rchas an azimuthal wave-number which increases withR. Some of the problems involved in testing the results by experiment are discussed and a possible application to the theory of vortex breakdown is mentioned. In an appendix this instability is shown to be an example of inertial instability.
Progressive internal waves on slopes
Authors: Wunsch Carl
Published: March 2006

The refraction of progressive internal waves on sloping bottoms is treated for the case of constant Brunt—Väisälä frequency. In two dimensions simple, explicit expressions for the changing wavelengths and amplitudes are found. For small slopes, the solutions reduce to simple propagating waves at infinity.

The singularity along a characteristic is shown to be removable, though the solutions are now inhomogeneous waves. The viscous boundary layers of the wedge geometry are briefly considered with the inviscid solutions remaining as interior solutions.

A theory valid for small slopes is obtained for three-dimensional waves. The waves are refracted in the usual manner, turning parallel to the beach in shallow water.
Measurement of the relaxation frequency of the asymmetric stretching mode of carbon dioxide
Authors: Hodgson J. P. , Hine R. J.
Published: March 2006

The vibrational relaxation frequency of carbon dioxide has been determined by measuring the rate of change of thermal emission in shock waves near 4±3μ. This method of measuring the relaxation frequency depends mainly on the degree of excitation of the asymmetric stretching mode of the molecule, and the results are compared with those of earlier density measurements made in the same shock tube. The gas samples used are not optically thin, and it is shown that self-absorption can be taken into account. The results imply that the relaxation frequency of the asymmetric stretching mode is about 70% of that of the bending mode.