bluuebunny/crossref_metadata_2025 huggingface dataset:
A finite element solution technique for the Boltzmann equation
Authors: Bramlette T. Taz , Mallett Robert H.
Published: March 2006

A new method is presented for solution of the Boltzmann equation governing the dynamic behaviour of gases. The essence of the method is idealization of the problem domain into subdomains called finite elements. Then, the Galerkin assumed-mode technique is employed as the basis for discretization of the individual finite elements and also for the assembly of the resulting algebraic models for these finite elements to form an algebraic model for the complete problem. The procedure is cast in a systematic matrix notation that makes evident the broad application potential of the analysis method. An illustrative application is presented for the problem of one-dimensional, linearized Couette flow. Numerical predictions of macroscopic flow velocity and viscous shear stress based upon the subject finite element method are compared with alternative analytical and numerical results. Special attributes of the finite element method are discussed in the context of this example problem. Applications to practical problems governed by generalized forms of the Boltzmann equation are projected on the basis of concepts established herein.
The lee-wave régime for a slender body in a rotating flow. Part 2
Authors: Miles John W.
Published: March 2006

The axisymmetric motion of an inviscid, rotating liquid over a prescribed stream surface, say S, is constructed from assumed values of the velocity and azimuthal vorticity on S. The hypothesis of unseparated flow, which implies continuity of the vorticity on S, is shown to imply that: (a) the azimuthal vorticity and azimuthal circulation (relative to the basic flow) must be simply proportional to the perturbation stream function in the exterior of S; (b) the exterior field exhibits a dipole behaviour far upstream of the body, thereby satisfying Long's hypothesis of no upstream disturbance.
On magnetohydrodynamic flow in rectangular ducts: an extension of the Hunt—Stewartson approach
Authors: Fahidy T. Z.
Published: March 2006

The Hunt–Stewartson technique of estimating fluid velocity and magnetic flux profiles in rectangular ducts is generalized for the entire secondary boundary layer.
High Reynolds number flow in a moving corner
Authors: Tabaczynski Rodney J. , Hoult David P. , Keck James C.
Published: March 2006

The problem of a piston moving in a cylinder is studied experimentally using flow visualization techniques. A vortex motion is observed at the piston face and cylinder wall interface as the cylinder wall moves toward the piston. Non-dimensional scaling parameters for the vortex size and stability are determined and semi-empirical theories for the size of the vortex are presented.
On the transition to turbulent convection. Part 2. The transition to time-dependent flow
Authors: Krishnamurti Ruby
Published: March 2006

The Rayleigh number at which steady convective flow changes to time-dependent flow is determined experimentally for several fluids with Prandtl numbers from 1 to 104. The time dependence is of two forms: (i) a slow tilting of the cell boundary, with time scale of the vertical thermal diffusion time, (ii) an oscillation with a faster time scale determined by the orbit time of the fluid around the cell. The nature of this oscillation is one of hot (or cold) spots advected with the original cellular motion. At a fixed point in the fluid this produces a time periodic oscillation of the temperature. A discrete change of slope of the heat flux curve accompanies this transition. As the Rayleigh number is increased, transition to disorder is seen to result from an increase in the frequency and number of these oscillations.
Kelvin–Helmholtz instability of finite amplitude
Authors: Drazin P. G.
Published: March 2006

Non-linear Kelvin–Helmholtz instability, of two parallel horizontal streams of inviscid incompressible fluids under the action of gravity, is studied theoretically. The lower stream is denser and there is surface tension between the streams. Some progressing waves of finite amplitude are found as the development of a slightly unstable wave of infinitesimal amplitude. In particular, the non-linear elevation of the interface between the fluids is calculated. The finite amplitude of the waves does not equilibrate to a constant after a long time, but varies periodically with time. In practice, slight dissipation should lead to equilibration at an amplitude close to a value given by the present theory.
Inviscid film flow over an inclined surface originated by strong fluid-injection
Authors: Kuiken H. K.
Published: March 2006

This paper considers inviscid film flows that originate by injection of fluid through the bounding surface. In the first part of the paper plates of an average inclination are considered. Power-law injection rates are studied in some detail. Later it is shown how the analysis has to be modified for almost horizontal plates.
The stability of stratified shear flows
Authors: Collyer M. R.
Published: March 2006

Small perturbations of a parallel shear flow U(z) in an inviscid, incompressible, stably stratified fluid of density ρ(z) are considered, for which the principal measure of stability is the Richardson number, R. For an arbitrary velocity and density profile we discuss the problem of determining whether a curve of neutral stability has adjacent unstable regions in an (α, R) plane, where α is the wave-number of the disturbance. Neutral curves bounding unstable regions are then obtained for a triangular jet flow in conjunction with various density distributions. A comparison is also made between the stability characteristics of jet and shear flows with corresponding density structures.
A free-streamline model of the two-dimensional sail
Authors: Dugan J. P.
Published: March 2006

The two-dimensional sail is considered in a free-streamline model to complement the oft-considered airfoil model which is limited to small angles of attack. The shape of the sail, the lift and drag coefficients, and the moment are obtained for various angles of attack and states of tension.
Numerical solutions for steady flow past a circular cylinder at Reynolds numbers up to 100
Authors: Dennis S. C. R. , Chang Gau-Zu
Published: March 2006

Finite-difference solutions of the equations of motion for steady incompressible flow around a circular cylinder have been obtained for a range of Reynolds numbers from R = 5 to R = 100. The object is to extend the Reynolds number range for reliable data on the steady flow, particularly with regard to the growth of the wake. The wake length is found to increase approximately linearly with R over the whole range from the value, just below R = 7, at which it first appears. Calculated values of the drag coefficient, the angle of separation, and the pressure and vorticity distributions over the cylinder surface are presented. The development of these properties with Reynolds number is consistent, but it does not seem possible to predict with any certainty their tendency as R → ∞. The first attempt to obtain the present results was made by integrating the time-dependent equations, but the approach to steady flow was so slow at higher Reynolds numbers that the method was abandoned.
The development of horizontal boundary layers in stratified flow. Part 1. Non-diffusive flow
Authors: Kelly R. E. , Redekopp L. G.
Published: March 2006

The development of the boundary layer on the upper surface of a horizontal flat plate in a non-diffusive, stratified flow is described. It is shown that the flow can be characterized by two basic parameters, the Reynolds (RL) and Russell (RuL) numbers, and that, depending on the relative magnitude of these two parameters, three different régimes of flow can be defined. The delineation of these régimes and the description of the flow in each of them is obtained by deriving a uniformly valid first approximation to the Boussinesq equations of motion for a flow contained in the two-dimensional parameter spaceRuL> 0,RL> 1. The critical stratification for the self-blocking of a horizontal boundary layer is shown to be given by the conditionRuL=O(RL½).
Shallow three-dimensional flows with variable surface tension
Authors: Adler J. , Sowerby L.
Published: March 2006

The three-dimensional steady flow of a shallow viscous liquid with non-uniform surface tension has been considered when the variation in surface tension results from the presence of an insoluble chemical contaminant on the surface. Similarly solutions for the particular problem of a channel flowing into a semi-infinite lake have been obtained, the depth and surface concentration at infinity being specified.
Trailing-edge stall
Authors: Brown S. N. , Stewartson K.
Published: March 2006

A study is made of the laminar flow in the neighbourhood of the trailing edge of an aerofoil at incidence. The aerofoil is replaced by a flat plate on the assumption that leading-edge stall has not taken place. It is shown that the critical order of magnitude of the angle of incidence α* for the occurrence of separation on one side of the plate is$\alpha^{*} = O(R^{\frac{1}{16}})$, whereRis a representative Reynolds number, for incompressible flow, and α* =O(R−¼) for supersonic flow. The structure of the flow is determined by the incompressible boundary-layer equations but with unconventional boundary conditions. The complete solution of these fundamental equations requires a numerical investigation of considerable complexity which has not been undertaken. The only solutions available are asymptotic solutions valid at distances from the trailing edge that are large in terms of the scaled variable of orderR−⅜, and a linearized solution for the boundary layer over the plate which gives the antisymmetric properties of the aerofoil at incidence. The value of α* for which separation occurs is the trailing-edge stall angle and an estimate is obtained from the asymptotic solutions. The linearized solution yields an estimate for the viscous correction to the circulation determined by the Kutta condition.
Diffraction of shock waves by a moving thin wing
Authors: Ting L. , Gunzburger M.
Published: March 2006

An analytical solution is obtained for the flow field due to the impinging of a plane shock wave of arbitrary strength by a thin wing moving in the opposite direction. The planform and the thickness distribution of the wing can be arbitrary and the speed of the wing can be either supersonic or subsonic relative to the undisturbed stream ahead of the shock or to that behind the shock. The solution is a generalization of the previous solution of Ting & Ludloff for the diffraction of shock wave by a two-dimensional stationary airfoil to a three-dimensional wing moving with supersonic or subsonic speed relative to the stream ahead of or behind the shock. The solution is employed for the analysis of the changes in aerodynamic forces when an airplane encounters a blast wave or a shock wave of another airplane. It is also used to study the diffraction of a shock wave or anN-wave advancing over flat terrains.
On the development of the wake behind the trailing edge of a flat plate
Authors: Smith S. H.
Published: March 2006

A stream with constant velocity U is impulsively started at time t = 0 past the trailing edge of a semi-infinite flat plate. According to boundary-layer theory, it is found that the flow at a distance x downstream from the trailing edge is unaware of the presence of the plate when x > Ut; at time t = x/U there is then a discontinuity in the velocity normal to the plate. It is the neglect of diffusion parallel to the axis of the plate that introduces the discontinuity, and when the complete Navier–Stokes equations are considered for t ≃ x/U, a solution is found that can be matched with that gained from boundary-layer arguments.
On the generation of Kelvin-type waves by atmospheric disturbances
Authors: Thomson Richard E.
Published: March 2006

This paper considers the surface response of a semi-infinite, uniformly rotating, constant depth, homogeneous ocean to a variable atmospheric force. For a general wind and pressure system it is shown that forced Kelvin-type waves can be generated and that only the longshore wind component and the pressure can generate them. In particular a semi-infinite wind and moving pressure pattern are shown to generate Kelvin waves that travel away from the force discontinuities at the speed of shallow-water waves. The waves in the latter case exhibit a frequency shift typical of non-dispersive waves from a moving source. Some numerical values for the amplitudes of the Kelvin waves are also given.
A lock exchange flow
Authors: Wood I. R.
Published: March 2006

In this paper the interchange flow between two reservoirs connected by a contraction and containing fluid of different densities is considered. The effect of the boundary layers on the floor and walls of the contraction on the depth of flow in the contraction is discussed for the case of single layer flowing from one reservoir to the other. Next the theory for a denser layer plunging under a stationary layer is developed. In this case there is a discontinuity at the point of intersection of the surfaces of the flowing and the stationary fluids and there are three possible flow régimes depending on whether this discontinuity occurs at, downstream of, or upstream of the contraction.

Finally, the case where there is an interchange flow with fluid flowing from each reservoir into the other is introduced. This latter theory parallels that developed by Wood (1968) for the case of two layers flowing from one reservoir through a contraction into another reservoir and as in this case there are two points of control, one at the position of minimum width and one (the virtual point of control) away from this position of minimum width.

Experiments are described for a single layer flowing through the contraction and the results of these are used to obtain an indication of the accuracy that could be expected from the experiments with the more complicated exchange flow. The experiments with the exchange flow verified the major elements of the theory.
A technique for phase speed measurements in turbulent flow
Authors: Stegen G. R. , Van Atta C. W.
Published: March 2006

A technique is described for measuring the local phase speed in a turbulent flow. The technique has been used to measure the phase speed of the Fourier components of the longitudinal velocity fluctuations in grid turbulence. These measurements are unique in that the probe spacing is only twice the Kolmogoroff length scale. The velocity fluctuations were measured with linearized constant-temperature hot-wire anemometers, the outputs of which were digitally sampled and recorded in real time. Digital Fourier analysis techniques were then used to calculate the cross-spectral density of the two velocity measurements. From this, the phase, phase speed, and coherence were calculated. The coherence has been used to estimate the variance of these measurements.
Steady currents induced by oscillations round islands
Authors: Longuet-Higgins M. S.
Published: March 2006

An oscillating current such as a tidal stream or an inertial oscillation may have a horizontal scale of the order of many times the local depth of water. Thus an island projecting from an otherwise uniform sea bed will give rise to a local, periodic disturbance near the island. It is shown that this disturbance may be resolved into two waves travelling in opposite senses round the island. If the particle orbits at large distances are circular, then only one of these waves has non-zero amplitude.

In addition to the oscillatory motion, however, there is a steady d.c. streaming, or mass-transport velocity, whose magnitude is of order u2/σa where u denotes the magnitude of the oscillatory velocity at large distances, σ denotes the radian frequency, and a is the radius of the island. In this paper the profile of the streaming velocity is calculated for circular islands, with or without shoaling regions offshore. It is shown that resonance with the free modes trapped by the shoaling regions may greatly increase the streaming velocity. Viscosity (or horizontal mixing) also tends to increase the streaming velocity close to the shoreline.

The conclusions are supported by some simple model experiments. It is suggested that such streaming may partly account for the observed pattern of currents near Bermuda.
On the turbulent flow over a wavy boundary
Authors: Davis Russ E.
Published: March 2006

Two hypotheses concerning the turbulent flow over an infinitesimal-amplitude travelling wave are investigated. One hypothesis, originally made by Miles, is that the wave does not affect the turbulence and therefore the turbulent Reynolds stresses are dependent only on height above the mean wave surface. Alternatively, the proposal that turbulent stresses are primarily dependent on height above the instantaneous wave surface is examined. Numerical solutions of the appropriate equations are compared with Stewart's recent experimental results and with the approximate solutions employed by Miles and others. No definite conclusion can be reached from comparison with experimental results since the predicted flows are quite sensitive to details of the mean velocity profile near the wave surface where no data was taken. It is found that the asymptotic results do not apply for the conditions investigated.
Laboratory studies of the velocity field over deep-water waves
Authors: Stewart Robert H.
Published: March 2006

The mean-velocity field over monochromatic, 1·96 Hz, deep-water waves was measured by means of hot-wire anemometers for a range of wind speeds (relative to wave speed) of 0·4 to 3·0. The mean-velocity profile, over waves 0·64 cm in amplitude, was the same as that over a rough plate; that is, the mean velocity varied as the logarithm of the height above the mean-water level, except very close to the water, where the effect of the viscous sublayer became important. The wave-induced perturbation-velocity field and its associated Reynolds stresses were also measured and compared with numerical solutions of various linear equations governing shearing flow over a wavy boundary. The comparison showed that the measured velocity field was not well predicted by these theories.
Observed flow patterns at the initiation of convection in a horizontal liquid layer heated from below
Authors: Somerscales E. F. C. , Dougherty T. S.
Published: March 2006

An experimental investigation has been made of the flow patterns at the initiation of convection in a layer of a high Prandtl number liquid confined between rigid, horizontal surfaces and heated from below. The experiment was designed to overcome the limitations of earlier experiments and to correspond closely to the conditions of the theory. In particular, the upper and lower rigid surfaces which enclosed the layer were made of copper which has a high thermal conductivity. To aid in the analysis of the experimental results some supplementary observations of the flow patterns were made using a glass upper plate. For small fluid depths and large temperature differences between the upper and lower surface the initial flow was in the form of hexagonal cells as predicted theoretically. With increasing Rayleigh number the cellular flow appeared to transform into rolls as predicted. For large fluid depths and small temperature differences only circular plan-form rolls were observed. This is in agreement with the results of other experimenters. It is tentatively proposed that this non-appearance of an initial cellular flow is due to the shape of the test chamber having a dominating influence on the flow pattern when the temperature gradient in the fluid is small. Measurements were also made of the development time for the flow patterns and the critical Rayleigh number.
Weakly non-linear waves in rotating fluids
Authors: Leibovich S.
Published: March 2006

The Korteweg–de Vries equation is shown to govern formation of solitary and cnoidal waves in rotating fluids confined in tubes. It is proved that the method must fail when the tube wall is moved to infinity, and the failure is corrected by singular perturbation procedures. The Korteweg–de Vries equation must then give way to an integro-differential equation. Also, critical stationary flows in tubes are considered with regard to Benjamin's vortex breakdown theories.
Measurements in a three-dimensional turbulent boundary layer induced by a swept, forward-facing step
Authors: Johnston James P.
Published: March 2006

An experiment is reported, in which turbulent shear-stresses as well as mean velocities have been measured in a three-dimensional turbulent boundary layer approaching separation. It is shown that even very close to the wall the stress vector does not align itself with the mean velocity gradient vector, as would be required by a scalar ‘eddy viscosity’ or ‘mixing length’ type assumption. The calculation method of Bradshaw (1969) is tested against the data, and found to give good results, except for the prediction of shear-stress vector direction.
Active water-wave absorbers
Authors: Milgram Jerome H.
Published: March 2006

The problem considered is that of absorbing two-dimensional water waves in a channel by means of a moving termination at the end of the channel. The problem is formulated for a semi-infinite channel and solutions are determined according to a linearized theory. The motion of the termination that is needed for absorption is determined in the form of a linear operation on the measured surface elevation at a fixed point in the channel so a self-actuating wave-absorbing system can be devised. A theoretical method of studying the stability of such a system is presented. A system of this type was built and experiments with it are described. Wave absorption is demonstrated both for monochromatic waves and for wave pulses. The absorption of a wave pulse is compared with the absorption of the same pulse by a fixed beach making a ten degree angle with the horizontal direction.
Simple shear flow round a rigid sphere: inertial effects and suspension rheology
Authors: Lin Chen-Jung , Peery James H. , Schowalter W. R.
Published: March 2006

An analysis is presented of the flow field near a neutrally-buoyant rigid spherical particle immersed in an in compressible Newtonian fluid which, at large distances from the particle, is undergoing simple shear flow. Subject to conditions of continuity of stress at the particle surface and to conditions of zero net torque and zero net force on the sphere, the effect of fluid inertia on the velocity and pressure fields in the vicinity of the particle has been computed to $O(R^{\frac{3}{2}})$, where R = a2G/ν is a shear Reynolds number, a being the sphere radius, G the velocity gradient in the free stream (taken to be a positive number), and ν the kinematic viscosity.

Some streamlines have been computed and plotted. These illustrate how the fore–aft symmetry of the creeping-motion solution is destroyed when one includes inertial effects.

Knowledge of the velocity and pressure fields enables one to compute the effect of inertial forces in suspension rheology. The results include a correction to the Einstein viscosity law to $O(R^{\frac{3}{2}})$ for a dilute (non-interacting) suspension of spheres. In addition it is found that inertial effects give rise to a non-isotropic normal stress.
Some cavitation experiments with dilute polymer solutions
Authors: Brennen Christopher
Published: March 2006

Experiments on fully developed cavity flows were carried out with the prime initial objective of investigating the effects of the addition of small quantities of ‘turbulent drag reducing additive’ upon the cavity-surface boundary-layer instability and transition reported in the previous paper (Brennen 1970). However, in most instances, the additives were found to cause an unforeseen instability in the wetted surface flow around the headform. Upon convection, the resulting disturbances dramatically disfigured the cavity surface, thus negating the original purpose. This new phenomenon warranted investigation and became the principal subject of this paper.
Characteristics of Ekman boundary layer instabilities
Authors: Caldwell D. R. , Atta C. W. Van
Published: March 2006

In a laboratory study, the class of instabilities of the laminar Ekman layer called type II (or class A) are found to have the spectral characteristics of narrow-band noise (Q ∼ 5). The unperturbed laminar profile resembles very closely the ideal Ekman solution. The frequency of the spectral peak varies with the Reynolds number as predicted theoretically by Lilly (1966), but the measured frequencies are only 60% of the predicted value. The critical Reynolds number for this instability is found to be 56·7, in good agreement with Lilly's analysis. The measured boundary layer profile of the magnitude of the spectral peak has the behaviour predicted by Lilly. A sudden onset of turbulence is found at a Reynolds number of 148.
Pressure measurements in the water-entry cavity
Authors: Abelson H. I.
Published: March 2006

Significant experimental results from a study of pressure in the water-entry cavity are presented. Projectiles were fired into water at velocities up to 250 ft./sec and entry angles of 90°, 60°, and 45°. Pressure data obtained using underwater probes were correlated with high-speed motion pictures taken of the entries. Results indicate that the cavity pressure drop prior to surface closure is an order of magnitude greater than previously assumed. As the entry angle is decreased from 90°, the pressure drop decreases. The minimum cavity pressure decreases linearly with increasing entry velocity over the test range. As the entry angle is increased, the minimum entry velocity required to produce a measurable pressure drop becomes greater. An improved pressure-volume correlation is obtained if the volume enclosed by the cavity walls is corrected to account for re-entrant jet volume and air volume enclosed by the splash walls. Cavity pressure during the closed cavity phase behaves approximately according to the isentropic pressure–volume relation. Pressure drop and history are strongly dependent on projectile nose geometry. No appreciable cavity pressure gradient, axial or transverse, was found to exist. Deep closure or cavity collapse is accompanied by relatively high-pressure pulses.
Structure functions of turbulence in the atmospheric boundary layer over the ocean
Authors: Atta C. W. Van , Chen W. Y.
Published: March 2006

Structure functions of turbulent velocity fluctuations up to fourth order have been measured at several heights in the atmospheric boundary layer over the open ocean, and the results are compared with theoretical predictions for separations in the inertial subrange. The behaviour of second- and third-order quantities shows substantial agreement with the predictions of Kolmogorov's original theory over a wide range of separations, but the results of a recent modification of the theory, attempting to account for intermittency in the local dissipation rate, are also consistent with the data over somewhat shorter separation intervals. The behaviour of the measured fourth-order structure function disagrees with that predicted from Kolmogorov's original work, but good agreement is found with the results of the modified theory.
East coast ocean currents
Authors: Durance J. A. , Johnson J. A.
Published: March 2006

A three-dimensional barotropic model of the wind-driven ocean circulation is examined and the flow near the east coast of the ocean is considered in detail. The model is linear and has constant coefficients of eddy viscosity. It is shown that a strong current may exist in the eastern boundary layer when upwelling or downwelling is present. No net northwards transport is produced as an equal deep counter current also occurs. A consequence of the downwelling is that the principal interior gyre is prevented from reaching the east coast and a secondary gyre is formed in this region.
Grid-generated turbulence in dilute polymer solutions
Authors: Friehe Carl A. , Schwarz W. H.
Published: March 2006

Measurements were made of some of the properties of grid-generated, turbulence for several concentrations of a drag-reducing polymer additive in water. Reference measurements of the grid pressure drops, streamwise intensities and one-dimensional spectra in pure water agreed with previous measurements obtained in Newtonian fluids. Corresponding results for the polymer solutions showed that the grid pressure drops were generally lowered, the turbulence intensity levels were increased and the one-dimensional energy spectra were unchanged compared to the results in water. A dimensionless rate of decay correlation was introduced which removed the dependence of the rate of decay on the initial conditions of grid-generated turbulence for Newtonian fluids: the polymer solution decay results did not follow this correlation, indicating that the decay process is different in these fluids. The one-dimensional energy spectra of the turbulence in the polymer solutions were of the same shape as those in Newtonian fluids, and were normalized with modified Kolmogoroff variables using an effective viscosity (Lumley 1964) and the viscous dissipation in order to provide a quantitative comparison to Newtonian spectra. All of the normalized spectra in the polymer solutions collapsed to a single curve which coincided with the normalized Newtonian spectral curve. It was also found that the rate of decay was less than the viscous dissipation in the polymer solutions, in accordance with recent theoretical predictions.
Slender-body theory for particles of arbitrary cross-section in Stokes flow
Authors: Batchelor G. K.
Published: March 2006

A rigid body whose length (2l) is large compared with its breadth (represented by R0) is straight but is otherwise of arbitrary shape. It is immersed in fluid whose undisturbed velocity, at the position of the body and relative to it, may be either uniform, corresponding to translational motion of the body, parallel or perpendicular to the body length, or a linear function of distance along the body length, corresponding to an ambient pure straining motion or to rotational motion of the body. Inertia forces are negligible. It is possible to represent the body approximately by a distribution of Stokeslets over a line enclosed by the body; and then the resultant force required to sustain translational motion, the net stresslet strength in a straining motion, and the resultant couple required to sustain rotational motion, can all be calculated. In the first approximation the Stokeslet strength density F(x) is independent of the body shape and is of order μUε, where U is a measure of the undisturbed velocity and ε = (log 2l/R0)−1. In higher approximations, F(x) depends on both the body cross-section and the way in which it varies along the length. From an investigation of the ‘inner’ flow field near one section of the body, and the condition that it should join smoothly with the ‘outer’ flow which is determined by the body as a whole, it is found that a given shape and size of the local cross-section is equivalent, in all cases of longitudinal relative motion, to a circle of certain radius, and, in all cases of transverse relative motion, to an ellipse of certain dimensions and orientation. The equivalent circle and the equivalent ellipse may be found from certain boundary-value problems for the harmonic and biharmonic equations respectively. The perimeter usually provides a better measure of the magnitude of the effect of a non-circular shape of a cross-section than its area. Explicit expressions for the various integral force parameters correct to the order of ε2 are presented, together with iterative relations which allow their determination to the order of any power of ε. For a body which is ‘longitudinally elliptic’ and has uniform cross-sectional shape, the force parameters are given explicitly to the order of any power of ε, and, for a cylindrical body, to the order of ε3.
Thermal instabilities in rapidly rotating systems
Authors: Busse F. H.
Published: March 2006

Thermal instabilities of a contained fluid are investigated for a fairly general class of problems in which the dynamics are dominated by the effects of rotation. In systems of constant depth in the direction of the axis of rotation the instability develops when the buoyancy forces suffice to overcome the stabilizing effects of thermal conduction and of viscous dissipation in the Ekman boundary layers. Owing to the Taylor–Proudman theorem, a slight gradient in depth exerts a strongly stabilizing influence. The theory is applied to describe the instability of the ‘lower symmetric régime’ in the rotating annulus experiments at high rotation rates. An example of geophysical relevance is the instability of a self-gravitating, internally heated, rotating fluid sphere. The results of the perturbation theory for this problem agree reasonably well with the results of an extension of the analysis by Roberts (1968).
On the stability of two-dimensional stagnation flow
Authors: Kestin J. , Wood R. T.
Published: March 2006

The paper examines the stability of the uniform flow which approaches a two-dimensional stagnation region formed when a cylinder or a two-dimensional blunt body of finite curvature is immersed in a crossflow. It is shown that such a flow is unstable with respect to three-dimensional disturbances. This conclusion is reached on the basis of a mathematical analysis of a simplified form of the disturbance equation for the stream-wise component of the vorticity vector. The ultimate, or stable, flow pattern is governed by a singular Sturm–Liouville problem whose solution possesses a single eigenvalue. The resulting flow is one in which a regularly distributed system of counter-rotating vortices is super-imposed on the basic, Hiemenz-like pattern of streamlines. The spacing of the vortices is a unique function of the characteristics of the flow, and a theoretical estimate for it agrees well with experimental results. The analysis is extended heuristically to include the effect of free-stream turbulence on the spacing.

The problem is similar to the classical Görtler–Hämmerlin study of the stability of stagnation flow against an infinite flat plate, which revealed the existence of a spectrum of eigenvalues for the disturbance equation. The present analysis yields the same result when an infinite radius of curvature is assumed for the blunt body.
Numerical study of steady flow past spheroids
Authors: Masliyah Jacob H. , Epstein Norman
Published: March 2006

Numerical methods have been used to investigate the steady incompressible flow past oblate and prolate spheroids for Reynolds numbers up to 100. The ratio of minor to major axis of the spheroids investigated were 0·9, 0·5 and 0·2, together with 1·0, which represents the limiting case of a sphere. The pressure distribution and the skin and form drag coefficients were numerically evaluated for the various Reynolds numbers. Streamlines, equi-vorticity lines and equivelocity lines are presented and show in detail the flow characteristics.
Static-pressure probes that are theoretically insensitive to pitch, yaw and Mach number
Authors: Smith A. M. O. , Bauer A. B.
Published: March 2006

The idea of distributing static probe cross-sectional areas so as to render the probe insensitive to Mach number is combined here with that of using non-circular cross-sections to render probes insensitive to yaw and angle of attack. Appropriate non-circular cross-sections are described in detail, and a general means of designing blunt or slender probes to have zero sensitivity to yaw and angle of attack in potential flow is described. Four experimental probes have been tested, and test results are presented. These results show that the probes are quite insensitive to yaw and angle of attack within certain limiting angles, which are assumed to correspond to the onset of flow separation.
The effect of buoyancy on the boundary layer about a heated horizontal circular cylinder in axial streaming
Authors: Switzer Karen Plain
Published: March 2006

The boundary-layer flow over a semi-infinite horizontal circular cylinder heated to a constant temperature and immersed in a uniform axial free stream is discussed in five situations corresponding to successively greater displacements from the leading edge. In the first three cases the drift velocity due to buoyancy is assumed small compared to the axial velocity component. Close to the leading edge of the cylinder the techniques of Seban & Bond are extended to include the drift velocity; far from the leading edge the asymptotic series methods of Stewartson, of Glauert & Lighthill, and of Eshghy & Hornbeck are employed to obtain a solution for the drift velocity. In the intermediate zone where the series solutions do not apply the appropriate partial differential equations are solved numerically. Still further downstream than the region where the ‘asymptotic’ solutions hold it is assumed that the boundary-layer flow is primarily convective and that the boundary layer is thin compared with the radius of the cylinder. A series solution is obtained which is valid near the lowest generator of the cylinder. Numerical methods are used to advance this solution upwards around the cylinder by solving the full boundary-layer equations step-by-step.
An experimental investigation of natural convection in a horizontal cylinder
Authors: Brooks Irving H. , Ostrach Simon
Published: March 2006

This work deals with an experimental investigation of natural convection inside a horizontal cylinder. The fluid, geometry, and thermal boundary condition were chosen so as to have a high Prandtl number and unit-order Grashof number.

The thermal boundary condition was established by imposing temperatures at two points, 180° apart, on the circumference of the cylinder. The resulting boundary condition for the full 360° was found experimentally and is presented. The apparatus was constructed so that the entire cylinder could be rotated in order to introduce an arbitrary heating angle into the boundary condition.

Temperature profiles and streamline patterns were observed at steady state for various values of this heating angle and for various initial conditions. For some cases, velocity profiles were also plotted. It was found that the interior ‘core’ of fluid was thermally stratified when the diameter containing the two imposed temperatures was horizontal. The flow occurred primarily in the region close to the cylinder wall. The cylinder was rotated so that the diameter containing the two imposed temperatures made an angle with the horizontal. The hotter of the imposed temperatures was always below the horizontal diameter of the cylinder. As this angle of rotation increased, it was found that the velocities encountered in the fluid increased and the degree of thermal stratification in the core region decreased. It was also found that the steady-state results were identical for the different initial conditions imposed. The results of this study are compared with previous work, both analytic and experimental.
On the motion of ν-fluids
Authors: Proudman Ian
Published: March 2006

The paper is concerned with a class of non-Newtonian fluids, ν-fluids, all of whose properties are determined by a single dimensional constant of the same dimensions as a viscosity. A regular nth-order ν-fluid is then defined to be one whose nth order time derivative of stress is a regular function of the local stress and flow fields and any of their space and time derivatives. The regularity condition determines the constitutive relation of such a fluid completely in terms of a finite set of non-dimensional constants which define the fluid.

An obvious property of these fluids is that their motions obey the same principles of Reynolds number similarity as those of a Newtonian fluid, and the primary aim of the paper is to examine the extent to which their flow properties are the same as those of the mean turbulent flow of a Newtonian fluid.

It is shown that a third-order fluid is the simplest ν-fluid which shares enough properties with turbulent motion to be worth further consideration in this context. At infinite Reynolds number, the constitutive relation for such a fluid reduces to the form
\[
AS\ddot{S} + B\dot{S}^2 + CS^2 S^{\prime\prime} + DSS^{\prime 2} + Eu^{\prime 2}S^2 = 0,
\]
where A,B, …,E, are isotropic tensor constants of the fluid, S is the stress tensor, u’ is the total rate of strain tensor, dots denote total time derivatives, and primes denote space derivatives. A number of illustrative examples of the properties of such a constitutive relation are then considered, representing the decay of a homogenous stress field, the effect of rigid-body rotation on such a decay, the structure of the equilibrium stress field in the presence of homogeneous rate of strain, both with and without vorticity, and the nature of flow near a plane boundary. In all cases, the results appear to be consistent with known properties of turbulent motion, to the extent that the analysis is taken.

Finally, the effect of finite Reynolds number on the decay of an isotropic and homogeneous stress field is shown to be consistent with observations on the decay of isotropic turbulence.
The limiting behaviour of the turbulent transverse velocity component close to a wall
Authors: Sirkar Kamalesh K. , Hanratty Thomas J.
Published: March 2006

Electrochemical techniques are used to measure the circumferential component of the velocity gradient sz at the wall of a pipe through which a turbulent fluid is flowing. The ratio of the root-mean-squared value of sz to the time-averaged velocity gradient is found to be 0·09 or 0·1, depending on whether corrections are made for frequency response. The frequency spectrum is similar to that for the component of the wall velocity gradient in the direction of mean flow. The amplitude distribution function for sz is very roughly approximated by a Gaussian distribution.
An experimental study of end effects for rectangular resonators on narrow channels
Authors: James W.
Published: March 2006

Quarter-wavelength resonators for harbour entrances and similar applications are usually designed and built without regard to the end effect. This paper describes tests on rectangular resonators using water waves of unique frequency in a long narrow wave channel fitted with a wave generator at one end, a wave absorber at the other end and suitable wave filters. In these tests the wavelength was kept constant while the geometry of the rectangular resonant branch canal was systematically varied. Wave transmission across the resonator, as well as wave reflexion upstream of the resonator was measured. The results clearly indicate a considerable effect of main channel width on optimum resonator width and resonator length, invalidating the usual resonance theory (which predicts complete reflexion of the incident wave train when the length of the branch canal is one-quarter of a wavelength). For example, it is found that, under certain conditions, quarter-wavelength resonators may not have any effect on the incident wave. The work described is limited to the first resonant mode and to semi-infinite domains on each side of the resonator.
Internal waves in a wedge-shaped region
Authors: Hurley D. G.
Published: March 2006

The Green's functions are found for a line source of internal waves in a wedge of stratified fluid of constant Brunt–Väisälä frequency, and are used to discuss the diffraction of internal waves by a wedge in all cases when the vertex angle of the wedge of fluid exceeds the acute angle between a characteristic and the horizontal. Robinson's (1970) results are confirmed and extended.

It is found that the diffracted waves are as important as the incident and reflected ones at all points that lie within a quarter-wavelength or so of either characteristic that passes through the apex. Also, in cases when all the reflected waves are inclined forwards, the diffracted waves lead to a positive backscatter of energy. When the vertex angle of the fluid wedge is less than the characteristic angle, the diffraction problem appears to be ill-posed, and, instead, the motion due to a vibrating body in the wedge of fluid is considered.

A general conclusion is that the so-called ray theory for internal waves, in which the incident and reflected waves alone are considered, has similar limitations to the geometrical theory of optics. Both theories involve the assumption that the typical dimensions in the problem are large compared to the wavelength.
Slow flow of a fluid carrying a uniform current past a conducting ellipsoid of revolution
Authors: Sozou C.
Published: March 2006

The method of matched asymptotic expansions is employed for investigating the effect of a uniform current on the velocity field of a viscous, incompressible, conducting fluid streaming past a stationary conducting ellipsoid of revolution, assuming that the Reynolds number is small. It is also assumed that at infinity the velocity and uniform current are parallel to the axis of the ellipsoid. It is found that the presence of the current increases or decreases the drag coefficient, depending on whether the fluid conductivity is larger than that of the ellipsoid or vice versa. It is suggested that this effect of the current on the drag coefficient holds for all axisymmetric bodies that are also symmetric about a plane perpendicular to their axis. The case of a circular disk broadside on the undisturbed current, obtained as a special case of a planetary ellipsoid, is slightly different; when the conductivity of the disk is non-zero the electromagnetically induced flow field vanishes.
The effect of weak stratification and geometry on the steady motion of a contained rotating fluid
Authors: Allen J. S.
Published: March 2006

The effect of a density stratification on the steady, mechanically driven motion of a viscous fluid in a rotating cylinder with axis aligned with the rotation and gravity vectors and with parallel top and bottom surfaces that slope with respect to the plane perpendicular to the rotation vector is studied by a linear theory. Primary attention is given to a study of the alteration of the characteristics of the flow of a homogeneous fluid by the addition of a weak stratification. It is found, for example, that in the range $E^{\frac{3}{2}} < \sigma S < E$, where E = ν/ΩL2 and σS = ναgΔT0/κΩ2L, and with a homogeneous boundary condition on the perturbation temperature, the interior velocity is parallel to the direction perpendicular to the plane determined by the vector normal to the top surface and the rotation vector. The circulation closes in an inviscid, but heat-conducting, boundary layer of thickness E¾(σS)−½ on the side wall. Thus, with stratification, the steady flow in this configuration differs markedly from the corresponding flow in a cylinder where the top and bottom surfaces lie in planes perpendicular to the rotation vector. The difference is caused by the fact that in the container with sloping surfaces the basic stratification interacts with the geostrophic flow whereas, in the other case, the interaction is with the much smaller Ekman layer suction velocities.
Experimental investigations of the stability of channel flows. Part 1. Flow of a single liquid in a rectangular channel
Authors: Kao Timothy W. , Park C.
Published: March 2006

The stability of the laminar flow in a rectangular channel with aspect ratio 1:8 was investigated experimentally, with and without artificial excitation. The critical Reynolds number based on the hydraulic diameter and the average velocity was found to be 2600. Behaviour of damped and growing waves, using artificial excitation, was examined in detail. In particular the progress of growing disturbances was followed. Breaking was found to be the ultimate fate of a growing wave. Spectra of growing and damped waves were also obtained. Measurements were made for wavelengths, wave speeds and amplification or damping rates. The neutral stability boundary in the αr, R plane was determined. In the damped region, comparison of several aspects of the behaviour of the measured disturbances with the plane Poiseuille theory for spatial decay yielded good agreement.

Three-dimensionality and non-linear subcritical instability were briefly examined. Neutral subcritical waves at low Reynolds numbers appeared possible when the exciter amplitude was quadrupled.

The possible bearings of the present study on the stability of plane Poiseuille flow are suggested.