bluuebunny/crossref_metadata_2025 huggingface dataset:
Hydromechanics of low-Reynolds-number flow. Part 3. Motion of a spheroidal particle in quadratic flows
Authors: Chwang Allen T.
Published: March 2006

Exact solutions in closed form have been found using the singularity method for various quadratic flows of an unbounded incompressible viscous fluid at low Reynolds numbers past a prolate spheroid with an arbitrary orientation with respect to the fluid. The quadratic flows considered here include unidirectional paraboloidal flows, with either an elliptic or a hyperbolic velocity distribution, and stagnation-like quadratic flows as typical representations. The motion of a force-free spheroidal particle in a paraboloidal flow has been determined. It is shown that the spheroid rotates about three principal axes with angular velocities governed by a set of Jeffery orbital equations with the rate of shear evaluated at the centre of the spheroid. These angular velocities depend on the minor-to-major axis ratio of the spheroid and its instantaneous orientation, but are independent of its actual size. The spheroid also translates at a variable speed, depending on its orientation relative to the surrounding fluid, along a straight path parallel to the main flow direction without any side drift or migration. This ‘jerk’ motion obeys a trajectory equation which is size dependent.
Some experiments on the motion of an isolated laminar thermal
Authors: Shlien D. J. , Thompson D. W.
Published: March 2006

A novel technique for injecting buoyancy (heat) into a liquid is described and demonstrated. When buoyancy was injected for a short time a laminar vortex ring formed. Its vertical displacement was found to be only approximately proportional to the square root of time (measured from an apparent initial time). Approximate geometrical similarity was also observed although the Reynolds number decreased from 28 to about 14.
Boundary layer over a blunt body at low incidence with circumferential reversed flow
Authors: Wang K. C.
Published: March 2006

This paper investigates the three-dimensional laminar boundary layer over a blunt body (a prolate spheroid) at low incidence and with reversed flow. Results reflecting the general characteristics of such a problem are presented. More significant are the features relating to the circumferential flow reversal. Some of these features confirm our early hypotheses concerning the existence of a reversed region ahead of separation and the role of the zero-cfθ line in the general context of separation in three dimensions. Other features are unexpected, including the distribution of cfμ and the shape of the separation line. Here cfθ and cfμ denote, respectively, the circumferential and meridional components of the skin friction.
Electrohydrostatic instability in electrically stressed dielectric fluids. Part 2
Authors: Michael D. H. , Norbury J. , O'Neill M. E.
Published: March 2006

The paper is the second part of a study of the failure of the insulation of a layer of dielectric fluid of arbitrary volume, occupying a hole in a solid dielectric sheet, when stressed by an applied electric field. In part 1 symmetric and asymmetric equilibria were found for the two-dimensional problem, using an approximation given by Taylor (1968) for the electric field, which is valid for large holes. In this paper axisymmetric equilibria are given for a circular hole, under the same conditions. In addition the points of bifurcation of asymmetric solutions have been found, and provide sufficient information to give the stability characteristics. It is found that when the volume-excess fraction δ exceeds a value of approximately −0·3 instability occurs in an asymmetric form reported earlier for large holes by Michael, O'Neill & Zuercher (1971) in the case δ = 0. For δ < −0·3 the nature of the instability changes to an axisymmetric form of failure associated with a maximum of the loading parameter.

The analysis given shows that axisymmetric displacements of ‘sausage’ mode type, that is, symmetric about a centre-plane, are associated with small changes in the static pressure in the dielectric layer. Such modes have not previously been examined in this context, and in an appendix to this paper Michael & O'Neill give an analysis of them when δ = 0, valid for all hole sizes, by extending the small perturbation analysis of Michael, O'Neill & Zuercher. These modes however do not provide the most unstable displacements for any configuration, and do not therefore affect the stability from a physical point of view.
On the eigenvalues of the Orr-Sommerfeld equation
Authors: Gaster M. , Jordinson R.
Published: March 2006

The characteristic function defining the eigenvalues of the Orr-Sommerfeld equation is discussed and it is shown how the expected analytic properties of this function can be exploited to generate series expansions defining eigenvalues within the circle of convergence. This technique is applied to the modes arising in the Blasius flat-plate boundary layer (treated as a parallel flow), for which the complex wavenumber α can be expanded as a convergent power series in the complex frequency parameter β in various regions of the β plane. Such power series are effectively equivalent to Fourier expansions and the properties of the latter are used to find the coefficients.

A square-root singularity in the relationship between α and β is found and it is shown how α can, nevertheless, be expressed in terms of β as the sum of one regular series and the square root of a second regular series. The loci of the real and imaginary parts of α have been computed from these series and show the behaviour in the neighbourhood of the branch point.

The series description provides a particularly simple and rapid method of evalauting eigenvalues and their derivatives within any given region.
Instability of flow in a curved alluvial channel
Authors: Engelund Frank
Published: March 2006

This paper deals with two main problems concerning flow in curved alluvial channels. First, the large-scale bottom geometry that develops through the interaction of flow and sediment motion is determined. Second, experiments in an annular flume indicate that the bed is unstable and that this particular instability leads to the formation of a certain number of scour holes. This is explained by a linear stability analysis.
Turbulence measurements in smooth concentric annuli with small radius ratios
Authors: Rehme K.
Published: March 2006

The structure of turbulence of fully-developed flow through three concentric annuli with small radius ratios was investigated experimentally for a Reynolds number range Re = 2 × 104−2 × 105. Turbulence intensities were measured in three directions, and turbulent shear stresses in the radial and azimuthal direction, in annuli of radius ratios α = 0·02, 0·04 and 0·1, respectively. The results showed that the structure of turbulence for these asymmetric flows is not the same as that for symmetrical flows (tubes and parallel plates). The main difference between symmetrical and asymmetric flows is that, for the latter, the diffusion of turbulent energy plays an important role. This is the reason not only for the non-coincidence of the positions of zero shear stress and maximum velocity, but also for the failure of most turbulence models in calculating asymmetric flows.
Hydromagnetic edge waves in a rotating stratified fluid
Authors: Andrews D. G. , Hide R.
Published: March 2006

The properties of edge waves confined by the interaction of buoyancy and Coriolis forces to the vicinity of a rigid plane boundary in a rotating, stratified, electrically conducting fluid pervaded by a magnetic field are established in some simple cases. The background shear is taken to be zero, the basic Alfvén velocity V and Brunt–Väisälä frequency N are assumed uniform, and all dissipative effects are taken to be vanishingly small. It is shown that waves trapped against the bounding wall can occur only if V is parallel to the wall. When the basic rotation vector Ω is also parallel to the wall, the hydromagnetic edge waves have a higher frequency and smaller spatial extent perpendicular to the wall than their non-hydromagnetic counterparts, but more complex behaviour is found when Ω possesses a component normal to the wall. There are conditions under which edge waves may exist even when the basic density stratification is top-heavy (i.e. when N2 < 0).
Closed-streamline flows past rotating single cylinders and spheres: inertia effects
Authors: Poe G. G. , Acrivos Andreas
Published: March 2006

The flow around a cylinder and a sphere rotating freely in a simple shear was studied experimentally for moderate values of the shear Reynolds numberRe.For a freely rotating cylinder, the data were found to be consistent with the results obtained numerically by Kossack & Acrivos (1974), at least for Reynolds numbers up to about 10. Rates of rotation of a freely suspended sphere were also obtained over the same range of Reynolds numbers and showed that, with increasingRe, the dimensionless angular velocity does not decrease as fast for a sphere as it does for a cylinder. In both cases, photographs of the streamline patterns around the objects were consistent with this behaviour. Furthermore, it was found in each case that the asymptotic solutions forRe[Lt ] 1 derived by Robertson & Acrivos (1970) for a cylinder and by Lin, Peery & Schowalter (1970) for a sphere are not valid for Reynolds numbers greater than about 0.1, and that the flow remains steady only up to values ofReof about 6.
Critical-level behaviour and wave amplification of a gravity wave incident upon a shear layer
Authors: Eltayeb I. A. , McKenzie J. F.
Published: March 2006

The properties of reflexion, refraction and absorption of a gravity wave incident upon a shear layer are investigated. It is shown that one must expect these properties to be very different depending upon the parameters (such as the Richardson number Ri, the wavelength normalized by the length scale of the shear and the ratio of the flow speed to the phase speed of the wave) characterizing the interaction of a gravity wave with a shear layer. In particular, it is shown that for all Richardson numbers there is a discontinuity in the net wave-action flux across the critical level, i.e. at a height where the flow speed matches the horizontal phase speed of the wave. When Ri > ¼, this is accompanied by absorption of part of the energy of the incident wave into the mean flow. In addition it is shown that the phenomenon of wave amplification (over-reflexion) can arise provided that the ultimate shear flow speed exceeds the horizontal phase speed of the wave and Ri is less than a certain critical value Ric ≃ 0·1129, in which case the reflected wave extracts energy from the streaming motion. It is also pointed out that wave amplification can lead to instability if the boundary conditions are altered in such a way that the system can behave like an ‘amplifier’.
On higher-order spectra of turbulence
Authors: Atta C. W. Van , Wyngaard J. C.
Published: March 2006

Measurements of higher-order spectra of turbulent velocity fluctuations in the atmospheric boundary layer over the open ocean and land produce the interesting result that, in the wavenumber range designated originally by Kolmogorov as an inertial subrange, the functional dependence of the spectra on wavenumber is practically independent of the order of the spectrum. These results confirm the observation of Dutton & Deaven that their extension by a dimensional similarity argument of the original Kolmogorov theory to higher-order spectra was not valid. In the present work, we derive an alternative generalization of the Kolmogorov ideas for spectra of arbitrary order. The results of this generalization describe the dependence upon wavenumber of the available data quite well. We also present theoretical calculations based on a Gaussian model for the fluctuating velocity field which furnish quantitative predictions for spectra of arbitrary order that are also in good agreement with the measurements, both in functional form and in absolute value.

Comparison of results based on the Gaussian model with laboratory measurements obtained in a free shear layer shows that the Gaussian theory predicts accurately all the available normalized higher-order spectra for all frequencies. When the corresponding measured higher-order moments are close to those expected for a Gaussian process, the Gaussian theory also correctly predicts the absolute magnitudes of the higher-order spectra.
The electron-beam fluorescence technique for measurements in hypersonic turbulent flows
Authors: Smith Jerome A. , Driscoll James F.
Published: March 2006

The electron-beam fluorescence technique has been employed in measuring simultaneous density and temperature fluctuations in a hypersonic (M ≃ 16), adiabatic wall boundary layer. The paper discusses this technique, as it is applied to the conditions of relatively high density associated with turbulent flows. It presents general considerations concerning the attainable frequency response and spatial resolution of the technique. It describes results from initial measurements in a boundary layer on a wind-tunnel wall. These results show that the r.m.s. of the density, temperature and pressure fluctuations are large, much larger than observed in supersonic boundary layers.
Double-diffusive convection in an inclined slot
Authors: Chen C. F.
Published: March 2006

We examine two-dimensional motion of a stably stratified fluid containing two solutes with different molecular diffusivities in an inclined slot. The two solutes have continuous opposing gradients with the slower-diffusing one more dense at the bottom. It is found that, in the steady state, there exists a slow upward flow along the slope driven by the slight buoyancy difference near the wall, not unlike the solution found by Phillips (1970) for a single solute. The magnitude of the flow is less than that in Phillips’ solution by a factor of approximately (1-λ)/(1-λτ), where λ is the ratio of the density gradient and τ−1 is the ratio of the diffusivity of the faster-diffusing solute to that of the slower-diffusing one. For the time-dependent flow resulting from switching on the diffusivities at t = 0, there may be a dramatic reversal of the flow near the walls depending on the relative magnitude of λ and τ. If λ is somewhat greater than τ, the initial flow is downward, along the slope, reaching a maximum magnitude about one order of magnitude greater than the steady-state value. Then the ‘reverse’ flow gradually diminishes and approaches the steady state rather slowly. For λ [gsim ] τ, the approach to the steady state is monotonic; there is no ‘reverse’ flow near the wall. The existence of the downward flow, which was observed by Turner & Chen (1974), may lead to double-diffusive instabilities which eventually result in horizontal convecting layers.
An experimental investigation of the stability of plane Poiseuille flow
Authors: Nishioka M. , A S. Iid , Ichikawa Y.
Published: March 2006

Stability experiments were made on plane Poiseuille flow generated in a long channel of a rectangular cross-section with a width-to-depth ratio of 27·4. By reducing the background turbulence down to a level of 0·05 %, we succeeded in maintaining the flow laminar at Reynolds numbers up to 8000, which is much larger than the critical Reynolds number of the linear theory, about 6000. The downstream development of the sinusoidal disturbance introduced by the vibrating ribbon technique was studied in detail at various frequencies in the range of Reynolds number from 3000 to 7500. This paper presents the experimental results and clarifies the linear stability, the nonlinear subcritical instability and the breakdown leading to the transition.
The asymptotic behaviour of a starting plume
Authors: Middleton Jason H.
Published: March 2006

A similarity solution is obtained for a model of the turbulent starting plume comprising a steady plume feeding mass, momentum and buoyancy into a vortex ring. Bulk equations representing the time rate of increase of ring momentum and ring buoyancy, together with equations (dependent on broad features of the ring structure) representing the velocity of propagation and time rate of circulation increase are used to determine the motion of the vortex ring. The similarity solution is found to exist only for diffuse distributions of vorticity and buoyancy within the ring. Further, the ratio of ring velocity to plume velocity, which is assumed to be constant, is found to take a value which agrees with that obtained from experimental observations.
Internal waves in a viscous atmosphere
Authors: Chang W. L. , Stevenson T. N.
Published: March 2006

The way in which internal waves change in amplitude as they propagate through an incompressible fluid or an isothermal atmosphere is considered. A similarity solution for the small amplitude isolated viscous internal wave which is generated by a localized two-dimensional disturbance or energy source was given by Thomas & Stevenson (1972). It will be shown how summations or superpositions of this solution may be used to examine the behaviour of groups of internal waves. In particular the paper considers the waves produced by an infinite number of sources distributed in a horizontal plane such that they produce a sinusoidal velocity distribution. The results of this analysis lead to a new small perturbation solution of the linearized equations.
Damped vibration of a string
Authors: Lin S. P.
Published: March 2006

The bounded solution of the unsteady Stokes equations is obtained for the flow of a viscous incompressible fluid about a circular cylinder which undergoes a linear translation starting from rest. A drag formula which consists of the known added-mass term and an additional term arising from the presence of viscosity is obtained. The drag obtained is applied locally in a study of damped vibration of a string. It is shown that the usual theory based on the quasi-steady drag formula overestimates considerably the period and the decay rate of damped vibration of a string in a viscous fluid.
Taylor-vortex instability and annulus-length effects
Authors: Cole J. A.
Published: March 2006

Critical speeds for the onset of Taylor vortices and for the later development of wavy vortices have been determined from torque measurements and visual observations on concentric cylinders of radius ratiosR1/R2= 0·894–0·954 for a range of values of the clearancecand lengthL:c/R1= 0·0478–0·119 andL/c= 1–107. Effectively zero variation of the Taylor critical speed with annulus length was observed. The speed at the onset of wavy vortices was found to increase considerably as the annulus length was reduced and theoretical predictions are realistic only forL/cvalues exceeding say 40. The results were similar for all four clearance ratios examined. Preliminary measurements on eccentrically positioned cylinders withc/R1= 0·119 showed corresponding effects.
The low frequency sound from multipole sources in axisymmetric shear flows. Part 2
Authors: Goldstein M. E.
Published: March 2006

A previous analysis of the acoustic radiation from multipole sources is extended to include additional components of the dipole and quadrupole sources. It is found that, unlike the components of the sources considered in the previous paper, the exponent of the Doppler factor now depends on the location of the sources within the jet.
Axisymmetric convection in a rotating sphere. Part 1. Stress-free surface
Authors: Weir A. D.
Published: March 2006

This paper examines large-scale nonlinear thermal convection in a rotating selfgravitating sphere of Boussinesq fluid containing a uniform distribution of heat sources. Conservative finite-difference forms of the equations of axisymmetric laminar motion are marched forward in time. The surface is assumed to be stress free and at constant temperature. Numerical solutions are obtained for Taylor numbers in the range 0 ≤ Λ ≤ 104 and Rayleigh numbers with
\[
R_c \leqslant R\lesssim 10R_c.
\]
For high Prandtl number (P > 5) the solutions are steady and most of them resemble the solutions of the linear stability equations, though other steady solutions are also found. For P [lsim ] 1, the steady solutions have horizontal wavenumber l = 1 and nearly uniform angular momentum per unit mass, rather than nearly uniform angular velocity. This rotation law seems to be independent of many details of the model and may hold in the convective core of a rotating star.
On the flow in an annulus surrounding a whirling cylinder
Authors: Brennen Christopher
Published: March 2006

When fluid in an annulus between two cylinders is set in motion by whirling movements of one or both of the cylinders, dynamic forces are imposed by the fluid on the cylinders. Knowledge of these forces is frequently important, indeed often critical, to the engineer designing rotor systems or journal bearings. Quite general solutions of the Navier-Stokes equations are presented for this problem and are limited only by restrictions on the amplitude of the whirl motion. From these solutions, the forces are derived under a wide variety of circumstances, including large and small annular widths, high and low Reynolds numbers and with and without a mean flow created by additional net rotation of one or both of the cylinders.
Experiments on transition to turbulence in an oscillatory pipe flow
Authors: Hino Mikio , Sawamoto Masaki , Takasu Shuji
Published: March 2006

Experiments on transition to turbulence in a purely oscillatory pipe flow were performed for values of the Reynolds number Rδ, defined using the Stokes-layer thickness δ = (2ν/ω)½ and the cross-sectional mean velocity amplitude Û, from 19 to 1530 (or for values of the Reynolds number Re, defined using the pipe diameter d and Û, from 105 to 5830) and for values of the Stokes parameter λ = ½d(ω/2ν)½ (ν = kinematic viscosity and ω = angular frequency) from 1·35 to 6·19. Three types of turbulent flow regime have been detected: weakly turbulent flow, conditionally turbulent flow and fully turbulent flow. Demarcation of the flow regimes is possible on Rλ, λ or Re, λ diagrams. The critical Reynolds number of the first transition decreases as the Stokes parameter increases. In the conditionally turbulent flow, turbulence is generated suddenly in the decelerating phase and the profile of the velocity distribution changes drastically. In the accelerating phase, the flow recovers to laminar. This type of partially turbulent flow persists even at Reynolds numbers as high as Re = 5830 if the value of the Stokes parameter is high.
Wakes in stratified flow past a hot or cold two-dimensional body
Authors: Robertson G. E. , Seinfeld J. H. , Leal L. G.
Published: March 2006

This paper considers the general problem of laminar, steady, horizontal, Oseen flow at large distances upstream and downstream of a two-dimensional body which is represented as a line source of horizontal or vertical momentum, or as a line heat source or heat dipole. The fluid is assumed to be incompressible, diffusive, viscous and stably stratified. The analysis is focused on the general properties of the horizontal velocity component, as well as on explicit calculation of the horizontal velocity profiles and disturbance stream-function fields for varying degrees of stratification. For stable stratifications, the flow fields for all four types of singularities exhibit the common feature of multiple recirculating rotors of finite thicknesses, which leads to an alternating jet structure both upstream and downstream for the horizontal velocity component and to leewaves downstream in the overall flow. The self-similar formulae for the velocity, temperature and pressure at very large distances upstream and downstream are also derived and compared with the Oseen solutions.
Vortex shedding from a cylinder vibrating in line with an incident uniform flow
Authors: Griffin Owen M. , Ramberg Steven E.
Published: March 2006

A study has been made of the wake of a cylinder vibrating in line with an incident steady flow. The Reynolds number for the experiments was 190, and the vortex shedding was at all times synchronized with the vibrations of the cylinder, which were in a range of frequencies near twice the Strouhal shedding frequency for the stationary cylinder. Two distinct vortex wake patterns were encountered. The first is a complex regime in which two vortices are shed during each cycle of the vibration and form an alternating pattern of vortex pairs downstream. The second pattern is an alternating street which results from the shedding of a single vortex during each cycle of the cylinder's motion. The street geometry in the latter case shares many basic characteristics with the wake of a cylinder vibrating in cross-flow. These include the effects of vibration amplitude and frequency on the longitudinal and transverse spacing of the vortices. The results obtained from these experiments in air are in agreement with previous findings from free- and forced-vibration experiments in water at both higher and lower Reynolds numbers.
The instability of a stratified periodic boundary layer
Authors: Von Kerczek Christian , Davis Stephen H.
Published: March 2006

A vertical plate oscillating vertically in a statically stably-stratified fluid induces an internal wave damped by viscous forces. A two-dimensional linear stability analysis of this time-dependent state shows that the wave is highly unstable when the buoyancy and forcing frequencies are comparable. This gravitational (buoyancy) instability is due to the presence of the background stratification. The neutral curve is calculated and the system energetics are explored. Excellent agreement is obtained with the recent experimental observations of Robinson & McEwan.
The stability of the decaying flow in a suddenly blocked channel
Authors: Hall P. , Parker K. H.
Published: March 2006

The stability of the decaying laminar flow in a suddenly blocked channel is investigated. The partial differential system governing the stability of the flow is solved using a WKB type of approach. It is shown that the first term of the WKB expansion of the disturbance velocity field is just that obtained by a quasi-steady approach. The flow is found to be unstable at quite small Reynolds numbers. This instability is associated with the inflexional nature of the velocity profiles of the decaying flow.
Nonlinear ship-wave theories by continuous mapping
Authors: Noblesse F. , Dagan G.
Published: March 2006

The exact equations of steady inviscid flow past a ship hull are formulated in a reference domain, onto which the flow domain is mapped. A thin-ship perturbation analysis is performed in the reference domain, and the first- and second-order solutions are derived. The classical thin-ship theory is obtained as the consistent, mapping-independent, perturbation solution in the physical space. Guilloton's method is interpreted as an inconsistent, mapping-dependent, second-order approximation. A new inconsistent approximation is obtained by exploiting the freedom in the mapping of the flow domain onto the reference domain. Further improvements are suggested.
Experimental investigation of shock waves in liquid helium I and II
Authors: Cummings John C.
Published: March 2006

The flow field produced by a shock wave reflecting from a helium gas-liquid interface was investigated using a cryogenic shock tube. Incident and reflected shock waves were observed in the gas; transmitted first- and second-sound shocks were observed in the liquid. Wave diagrams are constructed to compare the data with theoretical wave trajectories. Qualitative agreement between data and theory is shown. Quantitative differences between data and theory indicate a need for further analysis of both the gas-liquid interface and the propagation of nonlinear waves in liquid helium.

This work was a first step in the experimental investigation of a complex non-equilibrium state. The results demonstrate clearly the usefulness of the cryogenic shock tube as a research tool. The well-controlled jump in temperature and pressure across the incident shock wave provides unique initial conditions for the study of dynamic phenomena in superfluid helium.
On the difference between a bounding surface and a material surface
Authors: Dussan V E. B.
Published: March 2006

The ideas of Lagrange, Poisson, Kelvin and Truesdell are reviewed. It is shown that in order for a bounding surface not to be a material surface either u. n = c must fail or more than one deformation can be associated with the velocity field. Examples are given.
Linear stability of modulated circular Couette flow
Authors: Riley P. J. , Laurence R. L.
Published: March 2006

The linear stability of modulated circular Couette flow to axisymmetric disturbances is examined in the narrow-gap limit. The outer cylinder is assumed stationary, while the inner is modulated both with and without a mean rotation. The equations governing the disturbance motion are solved by a Galerkin expansion with time-dependent coefficients, and the stability of the motion determined by Floquet theory. Modulation is found, in general, to destabilize the flow due to steady rotation, although weak stabilization is found for some modulation amplitudes at intermediate frequencies.
Hydromechanics of low-Reynolds-number flow. Part 4. Translation of spheroids
Authors: Chwang Allen T. , Wu Theodore Y.
Published: March 2006

The problem of a uniform transverse flow past a prolate spheroid of arbitrary aspect ratio at low Reynolds numbers has been analysed by the method of matched asymptotic expansions. The solution is found to depend on two Reynolds numbers, one based on the semi-minor axis b, Rb = Ub/v, and the other on the semi-major axis a, Ra = Ua/v (U being the free-stream velocity at infinity, which is perpendicular to the major axis of the spheroid, and v the kinematic viscosity of the fluid). A drag formula is obtained for small values of Rb and arbitrary values of Ra. When Ra is also small, the present drag formula reduces to the Oberbeck (1876) result for Stokes flow past a spheroid, and it gives the Oseen (1910) drag for an infinitely long cylinder when Ra tends to infinity. This result thus provides a clear physical picture and explanation of the ‘Stokes paradox’ known in viscous flow theory.
The equilibrium statistical mechanics of simple quasi-geostrophic models
Authors: Salmon Rick , Holloway Greg , Hendershott Myrl C.
Published: March 2006

We have applied the methods of classical statistical mechanics to derive the inviscid equilibrium states for one- and two-layer nonlinear quasi-geostrophic flows, with and without bottom topography and variable rotation rate. In the one-layer case without topography we recover the equilibrium energy spectrum given by Kraichnan (1967). In the two-layer case, we find that the internal radius of deformation constitutes an important dividing scale: at scales of motion larger than the radius of deformation the equilibrium flow is nearly barotropic, while at smaller scales the stream functions in the two layers are statistically uncorrelated. The equilibrium lower-layer flow is positively correlated with bottom topography (anticyclonic flow over seamounts) and the correlation extends to the upper layer at scales larger than the radius of deformation. We suggest that some of the statistical trends observed in non-equilibrium flows may be looked on as manifestations of the tendency for turbulent interactions to maximize the entropy of the system.
Vortex shedding from a blunt trailing edge with equal and unequal external mean velocities
Authors: Boldman D. R. , Brinich P. F. , Goldstein M. E.
Published: March 2006

A flow-visualization study has shown that strong Kármán vortices develop behind the blunt trailing edge of a plate when the free-stream velocities over both surfaces are equal and that the vortices tend to disappear when the surface velocities are unequal. This observation provides an explanation for the occurrence and disappearance of certain discrete tones often found to be present in the noise spectra of coaxial jets. Both the vortex formation and the tones occur at a Strouhal number based on the lip thickness and the average of the external steady-state velocities of about 0.2.

Results from theoretical calculations of the vortex formation, based on an inviscid incompressible analysis of the motion of point vortices, were in good agreement with the experimental observations.
Multi-valued solutions of steady-state supersonic flow. Part 1. Linear analysis
Authors: Henderson L. F. , Atkinson J. D.
Published: March 2006

The shock wave equations for a perfect gas often provide more than one solution to a problem. In an attempt to find out which solution appears in a given physical situation, we present a linearized analysis of the equations of motion of a flow field with a shock boundary. It is found that a solution will be stable when there is supersonic flow downstream of the shock, and asymptotically unstable when there is subsonic flow downstream of it. It is interesting that both flows are found to be stable against disturbances of the d'Alembert type which grow from point sources; it is only when larger-scale line sources are considered that one can discriminate between the stabilities of the two types of flow. The results are applicable to supersonic flow over flat plates at incidence, to wedges, and to some cases of regular reflexion, diffraction and refraction of shocks.
On the Oseen linearization of the swirling-flow boundary layer
Authors: Tam K. K.
Published: March 2006

The modified Oseen linearization of the swirling-flow boundary layer of a mature hurricane is discussed. Upper and lower solutions for the radial and azimuthal velocity components are constructed in a subdomain by using a comparison theorem. It is shown that the error incurred in these velocity components by using the linear solution is no more than 30% in the subdomain. It is then inferred that the vertical velocity is also approximated to that order.
On the short surface waves due to an oscillating, partially immersed body
Authors: Davis A. M. J.
Published: March 2006

A body whose boundary C is vertical at the free surface F oscillates at high frequency in some prescribed manner. Asymptotic forms in terms of limit potentials are obtained for the upward force and the moment about a line in F of the pressure forces on the body. The cylindrical waves radiated to infinity are found to be asymptotically equivalent to those obtained in a two-dimensional solution of the Helmholtz equation. Some illustrative examples are given, principally the horizontal ellipsoid, in which case comparison with strip theory is possible.
Equilibrium and near-equilibrium turbulent boundary layers
Authors: Head M. R.
Published: March 2006

A well-tested integral method has been used to calculate turbulent boundary-layer development for the distribution of external velocity given byU∝x−0.255. The results suggest that different values of the initial momentum thickness, so long as this is below some critical value, produce a range of equilibrium layers having widely different values of the form parameterG. For values of the initial momentum thickness greater than the critical value, layers are produced which proceed more or less rapidly to separation. These results provide a plausible explanation for conflicting experimental observations made in the past.

Additional calculations for the flowsU∝x−0.15andU∝x−0.35suggest that, in the first case, a unique equilibrium condition is approached whatever the initial momentum thickness unless this exceeds some critical value; in the second case no equilibrium condition appears possible.
The calculation of near-wake flows
Authors: Pope S. B. , Whitelaw J. H.
Published: March 2006

Calculated flow properties are compared with measurements obtained in two-dimensional isothermal wakes with and without recirculation. The equations of continuity and momentum were solved numerically together with equations which formed a turbulence model. Calculations were made using three turbulence models: the first comprised transport equations for turbulence kinetic energy and the rate of turbulence dissipation; the second and third comprised equations for the rate of turbulence dissipation and two forms of Reynolds-stress equations characterized by different redistribution terms. The results show that, for wakes without recirculation, the particular turbulence model is less important than the boundary condition assumed in the plane of the trailing edge of the body; though the Reynolds-stress models do, of course, provide a better representation of the individual normal stresses. In the case of wakes with recirculation, both the length of the recirculation region and the rate of spread of the downstream wake are underestimated. The second discrepancy is particularly evident and appears to stem from the form of the dissipation equation. A suggestion for improving the modelling of this equation is provided together with necessary justification.
Solid-particle motion in two-dimensional peristaltic flows
Authors: Hung Tin-Kan , Brown Thomas D.
Published: March 2006

Some insight into the mechanism of solid-particle transport by peristalsis is sought experimentally through a two-dimensional model study (§ 2). The peristaltic wave is characterized by a single bolus sweeping by the particle, resulting in oscillatory motion of the particle. Because of fluid-particle interaction and the significant curvature in the wall wave, the peristaltic flow is highly nonlinear and time dependent.

For a neutrally buoyant particle propelled along the axis of the channel by a single bolus, the net particle displacement can be either positive or negative. The instantaneous force acting upon the particle and the resultant particle trajectory are sensitive to the Reynolds number of the flow (§ 3 and 4). The net forward movement of the particle increases slightly with the particle size but decreases rapidly as the gap width of the bolus increases. The combined dynamic effects of the gap width and Reynolds number on the particle displacement are studied (§ 5). Changes in both the amplitude and the form of the wave have significant effects on particle motion. A decrease in wave amplitude along with an increase in wave speed may lead to a net retrograde particle motion (§ 6). For a non-neutrally buoyant particle, the gravitational effects on particle transport are modelled according to the ratio of the Froude number to the Reynolds number. The interaction of the particle with the wall for this case is also explored (§ 7).

When the centre of the particle is off the longitudinal axis, the particle will undergo rotation as well as translation. Lateral migration of the particle is found to occur in the curvilinear flow region of the bolus, leading to a reduction in the net longitudinal transport (§ 8). The interaction of the curvilinear flow field with the particle is further discussed through comparison of flow patterns around a particle with the corresponding cases without a particle (§ 9).
Structural development of gas-liquid mixture flows
Authors: Herringe R. A. , Davis M. R.
Published: March 2006

The paper describes an experimental study of the structure of air-water mixtures flowing vertically. Resistivity probe techniques were applied to measurements of local void properties, including void fraction, gas-phase convection velocity, bubble size distributions and space-time correlation functions. The axial development of flows for six different air-water mixing conditions were examined. Measurements up to 108 diameters from inlet indicated that, while flow patterns for the different mixers may be significantly different initially, the flows tend to develop towards a common structure determined only by the flux rates of the two phases. This was evidenced by the convergence of the void and velocity profiles, and particularly the bubble size distributions, as the flow developed. Estimates of bubble sizes for these more developed conditions, based on a balance of energy between the interfacial structure and the turbulent structure, gave values of diameters which were on average 13% above experimental values. The void distributions obtained for bubbly flow conditions, after an adequate settling length, appear to be characterized by a local minimum at the centre of the pipe.
Propagation of internal Alfvén—acoustic—gravity waves in a perfectly conducting isothermal compressible fluid
Authors: Rudraiah N. , Venkatachalappa M. , Kandaswamy P.
Published: March 2006

Internal Alfvén-acoustic-gravity waves propagating in an isothermal, perfectly electrically conducting, plane stratified, inviscid, compressible atmosphere permeated by a horizontal stratified magnetic field in which the mean horizontal velocity U(z) depends on the height z only exhibit singular properties at the Doppler-shifted frequencies
\[
\Omega_{d} = 0,\quad\pm\Omega_A,\quad\pm\Omega_A/(1+M^2)^{\frac{1}{2}},\quad\pm (\Omega_c/2^{\frac{1}{2}})[1+M^2\pm \{(1+M^2)^2 - 4\Omega^2_A/\Omega^2_c\}^{\frac{1}{2}}]^{\frac{1}{2}}
\]
where ΩA is the Alfvén frequency, Ωc the sonic frequency and M the magnetic Mach number. The phenomenon of critical-layer absorption is studied using the momentum-transport approach of Booker & Bretherton (1967), the wave-packet approach (which is a consequence of the WKBJ approximation) of Bretherton (1966) and the technique involving wave normal curves of McKenzie (1973). The absorption effects are also illustrated, following Acheson (1972), by drawing ray trajectories. We find that the waves are absorbed at the critical levels Ωd = ± ΩA and ± ΩA/(1 + M2)½, and in particular we observe that these levels do not act like valves as observed by Acheson (1972). We also conclude that the combined effect of velocity shear and density and magnetic-field stratification is to increase the number of absorption levels.
Stability of Hagen-Poiseuille flow with superimposed rigid rotation
Authors: Mackrodt P.-A.
Published: March 2006

The linear stability of Hagen-Poiseuille flow (Poiseuille pipe flow) with superimposed rigid rotation against small three-dimensional disturbances is examined at finite and infinite axial Reynolds numbers. The neutral curve, which is obtained by numerical solution of the system of perturbation equations (derived from the Navier-Stokes equations), has been confirmed for finite axial Reynolds numbers by a few simple experiments. The results suggest that, at high axial Reynolds numbers, the amount of rotation required for destabilization could be small enough to have escaped notice in experiments on the transition to turbulence in (nominally) non-rotating pipe flow.
Experimental investigation of turbulent shear flow with quadratic mean-velocity profiles
Authors: Richards H. K. , Morton J. B.
Published: March 2006

Three turbulent shear flows with quadratic mean-velocity profiles are generated by using an appropriately designed honeycomb and parallel-rod grids with adjustable rod spacing. The details of two of the flow fields, with quadratic mean-velocity profiles with constant positive mean-shear gradients ($\partial^2\overline{U}_1/\partial X^2_2 >0$), are obtained, and include, in the mean flow direction, the development and distribution of mean velocities, fluctuating velocities, Reynolds stresses, microscales, integral scales, energy spectra, shear correlation coefficients and two-point spatial velocity correlation coefficients. A third flow field is generated with a quadratic mean velocity profile with constant negative mean-shear gradient ($\partial^2\overline{U}_1/\partial X^2_2 < 0$), to investigate in the mean flow direction the effect of the change in sign on the resulting field. An open-return wind tunnel with a 2 × 2 × 20 ft test-section is used.
Long-wave decay due to convective turbulence
Authors: Green Theodore , Kang See Whan
Published: March 2006

Long waves are generated in a laboratory-size rectangular basin, which is heated uniformly from below. Their subsequent decay is measured, and the decay component due to the action of convective turbulence isolated, using a combination of existing theories and interpretation techniques. An expression is proposed for the turbulent decay decrement as a function of the bulk Rayleigh number. The results agree as well as can be expected with a simple model based on a Reynolds-stress decay estimate obtained by superposing convective thermals on the oscillating flow associated with the long wave.
Relation between Nusselt number and Rayleigh number in turbulent thermal convection
Authors: Long Robert R.
Published: March 2006

A theory is developed for the dependence of the Nusselt number on the Rayleigh number in turbulent thermal convection in horizontal fluid layers. The theory is based on a number of assumptions regarding the behaviour in the molecular boundary layers and on the assumption of a buoyancy-defect law in the interior analogous to the velocity-defect law in flow in pipes and channels. The theory involves an unknown constant exponentsand two unknown functions of the Prandtl number. For eithers= ½ ors= 1/3, corresponding to two different theories of thermal convection, and for a given Prandtl number, constants can be chosen to give excellent agreement with existing data over nearly the whole explored range of Rayleigh numbers in the turbulent case. Unfortunately, comparisons with experiment do not permit a definite choice ofs, but consistency with the chosen form of the buoyancy-defect law seems to suggests= 1/3, corresponding to similarity theory.
An experimental investigation of the detention of airborne smoke in the wake bubble behind a disk
Authors: Humphries W. , Vincent J. H.
Published: March 2006

Experiments have been performed in a low-speed wind tunnel to determine the detention time of airborne smoke particles that become trapped in the wake vortex (or bubble) region behind flat disks placed perpendicular to the flow. Using a laser transmissometer to detect the smoke, its detention time was obtained from the time-dependent decay of the smoke in the disk bubble during the time immediately following the removal of the source of smoke. The dimensionless groupH, the product of the detention time and the mainstream air velocity divided by the disk diameter, is seen to be a constant equal to 7.44 ± 0.52 for Reynolds numbers in the range 2000–40000. This result is compatible with a simple fluid-mechanical model which describes the transport of fluid-borne scalar entities across the bubble boundary by turbulent diffusion. The investigation suggests thatHshould be unique for the flow about a disk over a wide range of conditions, and further suggests the possibility that similar unique values forHcan exist for flow about other obstacles. The numberHhas potential applications in a number of physical and engineering research areas.
A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer
Authors: Mack Leslie M.
Published: March 2006

A numerical study is made of the temporal eigenvalue spectrum of the Orr-Sommerfeld equation for the Blasius boundary layer. Unlike channel flows, there is no mathematical proof that this flow has an infinite spectrum of discrete eigenvalues. The Orr-Sommerfeld equation is integrated numerically, and the eigenvalues located by tracing out the contour lines in the complex wave velocity (c = cr + ici) plane on which the real and imaginary parts of the secular determinant are zero. This method gives only a finite and small number of discrete eigenvalues for a wide range of Reynolds numbers and wavenumbers. The spectrum of plane Poiseuille flow is used as a guide to study the spectrum of an artificial two wall flow which consists of two Blasius boundary layers. As the upper boundary of this flow moves to infinity, it is found that the portion of the spectrum with an infinite number of eigenvalues moves towards cr = 1 and the spacing between eigenvalues goes to zero. It is concluded, on the basis of this result and the contour method, that the original few eigenvalues found are the only discrete eigenvalues that exist for Blasius flow over a wide portion of the c plane for cr < 1 and cr > 1. It is suggested that the discrete spectrum is supplemented by a continuous spectrum which lies along the cr = 1 axis for ci < −α/R.