bluuebunny/crossref_metadata_2025 huggingface dataset:
A visual investigation of the wall region in turbulent flow
Authors: Corino E. R. , Brodkey Robert S.
Published: March 2006

The objective of this study is to investigate for turbulent flow the fluid motions very near a solid boundary, and to create a physical picture which relates these motions to turbulence generation and transport processes. An experimental technique was developed which permitted detailed observations of the regions very near a pipe wall, including the viscous sublayer, without requiring the introduction of any injection or measuring device into the flow. This technique involved suspending solid particles of colloidal size in a liquid, and photographing their motions with a high-speed motion picture camera moving with the flow. To provide greater detail, the field of view was magnified.

Fluid motions were observed to change in character with distance from the wall. The sublayer was continuously disturbed by small-scale velocity fluctuations of low magnitude and periodically disturbed by fluid elements which penetrated into the region from positions further removed from the wall. From a thin region adjacent to the sublayer, fluid elements were periodically ejected outward toward the centreline. Often there was associated with these events a zone of high shear at the interface between the mean flow and the decelerated region that gave rise to the ejected element. When the ejected element entered this shear zone, it interacted with the mean flow and created intense, chaotic velocity fluctuations. These ejections and resulting fluctuations were the most important feature of the wall region, and are believed to be a factor in the generation and maintenance of turbulence.
The effect of initial conditions and lateral boundaries on convection
Authors: Foster Theodore D.
Published: March 2006

A theoretical analysis of two-dimensional, finite-amplitude, thermal convection is made for a fluid which has an infinite Prandtl number. The vertical velocity disturbance is expanded in a double Fourier series which satisfies the horizontal and lateral boundary conditions. The resulting coupled sets of non-linear differential equations are solved numerically. It is found that for a particular Rayleigh number the number and size of the convection cells that form depend upon the ratio of the distance between the lateral boundaries to the depth of the fluid layer and on the initial conditions. The steady-state solutions are not unique and the solution for which the heat transport is a maximum is not necessarily the solution that results. Where there are no lateral boundaries, the lateral edges of the cells tend to tilt and the Nusselt number increases slightly.
Calculation of incompressible flow past a circular cylinder at moderate Reynolds numbers
Authors: Underwood Robert Leigh
Published: March 2006

The steady, two-dimensional, incompressible flow past a circular cylinder is calculated for Reynolds numbers up to ten. An accurate description of the flow field is found by employing the semi-analytical method of series truncation to reduce the governing partial differential equations of motion to a system of ordinary differential equations which can be integrated numerically. Results are given for Reynolds numbers between 0.4 and 10.0 (based on diameter). The Reynolds number at which separation first occurs behind the cylinder is found to be 5.75. Over the entire Reynolds number range investigated, characteristic flow parameters such as the drag coefficient, pressure coefficient, standing eddy length, and streamline pattern compare favourably with available experimental data and numerical solution results.
The dynamic balances of dissolved air and heat in natural cavity flows
Authors: Brennen C.
Published: March 2006

In steady, fully developed and unventilated cavity flows occurring in practice, air (originally dissolved in the water) and heat are diffused through the fluid towards the interface providing a continuous supply of air and vapour to the cavity. This must be balanced by the rate of entrainment of volume of air and vapour away from the cavity in the wake. These equilibria which determine respectively the partial pressure of air within the cavity and the temperature differences involved in the flow are studied in this paper. The particular case of the cavitating flow past a spherical headform has been investigated in detail. Measurements indicate a near-linear relation between the partial pressure of air in the cavity and the total air content of the water. From a second set of experiments, designed to estimate the volume rates of entrainment under various conditions by employing artificial ventilation, it appears that this is a function only of tunnel speed and cavity size within the range of the experiments. A simplified theoretical approach involving the turbulent boundary layer on the surface of the cavity is then used to estimate the rates of diffusion into the cavity. The resulting air balance yields a partial pressure of air/air content relation compatible with experiment. The water vapour or heat balance suggests that the temperature differences involved are likely to be virtually undetectable experimentally.
Turbulent boundary-layer flow on a rotating disk
Authors: Cham T-S. , Head M. R.
Published: March 2006

Calculations have been made of the development of the turbulent boundary layer on a disk rotating in free air, using circumferential and radial momentum-integral equations and an auxiliary equation of entrainment. In the calculations, circumferential velocity profiles are represented by Thompson's (1965) two-parameter family, while radial profiles are given by Mager's (1952) quadratic expression. The circumferential component of skin friction follows from the use of Thompson's profile family for the circumferential velocity. The entrainment, in dimensionless form, is assumed to be determined uniquely by the circumferential velocity profile in the same way as was proposed by Head (1958) for a two-dimensional turbulent boundary layer.

Detailed measurements have been made of the development of the turbulent boundary layer on the rotating disk, and the calculations are found to be in excellent agreement with the results when a suitable adjustment is made to Head's two-dimensional entrainment curve.
An experimental investigation of the composition of jet noise
Authors: Krishnappa G. , Csanady G. T.
Published: March 2006

Noise intensity measurements per octave band have been carried out in the far field of a small circular nozzle, at angles 0-150° from the jet axis, with and without vortex generators. The results, interpreted in the light of recent theoretical conclusions, confirm Lighthill's original suggestion that the dominant noise radiator is the pressure-shearx-rquadrupole at all but the lowest frequencies. At very low frequencies Reynolds stress-shearx-xand x-r quadrupoles contribute comparable amounts of radiation. When the product of Strouhal number (based on nozzle diameter and jet velocity) and jet Mach number exceeds unity the geometrical acoustics approximation becomes valid, but even at such high frequencies the shear noise contribution dominates. Vortex generators, while reducing noise intensity, do not appear to modify drastically the above composition of noise radiators.
Slow oscillations in an ocean of varying depth Part 1. Abrupt topography
Authors: Rhines P. B.
Published: March 2006

This paper is part of a study of quasigeostrophic waves, which depend on the topography of the ocean floor and the curvature of the earth.

In a homogeneous, β-plane ocean, motion of the fluid across contours of constantf/hreleases relative vorticity (fis the Coriolis parameter andhthe depth). This well-known effect provides a restoring tendency for either Rossby waves (withhconstant) or topographic waves over a slope. The long waves in general obey an elliptic partial differential equation in two space variables. Because the equation has been integrated in the vertical direction, the exact inviscid bottom boundary condition appears in variable coefficients.

When the depth varies in only one direction the equation is separable at the lowest order in ω, the frequency uponf. With a simple slope, |[xdtri ]h/h| = constant, the transition from Rossby to topographic waves occurs at |[xdtri ]h| ∼h/Re, where Reis the radius of the earth. Isolated topographic features are considered in §2. It is found that a step of fractional height δ on an otherwise flat ocean floor reflects the majority of incident Rossby waves when δ > 2ω. In the ocean ω is usually small, due to continental barriers, so even slight depth variations are important. A narrowridgedoes not act as a great obstruction but calculations show, for example, that the Mid-Atlantic Ridge is broad enough to reflect all but the lowest mode Rossby waves in the North Atlantic.

Besides isolating oceanic plains from one another, steps and ridges support trapped topographic waves of greatest frequency ∼ δ/2, analogous to the potential well solutions in quantum mechanics. These waves cannot carry energy alongabrupttopography, but they disperse more rapidly over broader slopes; the phase and group speeds may be hundreds of cm/sec. The continentalshelf waves found by Robinson are an example of the latter case. There are many such wave guides, where thef/hcontours are crowded, in the deep ocean.

The theory suggests that measurement of Rossby waves will rarely be possible at the coast of a continent.
Slow oscillations in an ocean of varying depth Part 2. Islands and seamounts
Authors: Rhines P. B.
Published: March 2006

We consider slow oscillations trapped about axisymmetric islands and seamounts. ω is in the range [lsim ] δ/2 (ω is the frequency divided by f, the Coriolis parameter, and δ the fractional change in depth). The periods, for example, are [gsim ] 2·4 days for an island with a sloping ‘skirt’, h [vprop ] r½, where h is the depth and (r, θ) are polar co-ordinates in the plane tangent to the mean sea surface. Energy leaks slowly away from the topography in Rossby waves. In the limiting case of a cylindrical island with vertical walls there are no such trapped motions, but incident Rossby waves are scattered anisotropically. If γ, the ratio of the island radius, a, to the Rossby wavelength, is small, the scattering cross-section ∼ γ3a. The free oscillations at seamounts and islands with skirts allow much stronger scattering (with cross-section ∼ a/γ, ∼ a wavelength), when one of their frequencies is near that of the incident wave.

The theory suggests that measurements of Rossby waves will be possible at small islands, but that the many local oscillations in the same frequency range will add some confusion.
Hartmann flow in annular channels. Part 2. Numerical solutions for low to moderate Hartmann numbers
Authors: Apelt C. J.
Published: March 2006

Flow of a conducting fluid along the annular channel between two non-conducting circular cylinders is examined by a numerical method for concentric and eccentric cases. Solutions have been obtained for Hartmann numbers ranging from 0·1 to 40 and, for some of these, details of velocity distribution and of induced current are given. The results obtained enable the development of the patterns of velocity and of current flow to be traced as the Hartmann number increases. The details of the development of the current flow patterns for eccentric cylinders are particularly interesting and are discussed in detail. At the higher values of Hartmann number studied the solutions are in excellent agreement with the results of Todd's (1967) high Hartmann number analysis and it is possible to determine at what value of Hartmann number Todd's analysis becomes applicable within a specified accuracy. The effect of eccentricity of the cylinders on the flow rate at a fixed pressure gradient is shown to diminish rapidly with increasing Hartmann number. The net flow of current around the annulus, which occurs when the cylinders are eccentric, has a maximum value for each case studied at a Hartmann number of 3, approximately.
On the reflexion of wave characteristics from rough surfaces
Authors: Longuet-Higgins M. S.
Published: March 2006

The energy of internal waves tends to be propagated along certain characteristic paths inclined at fixed angles to the vertical direction; the angle of inclination depending only on the wave frequency and the density stratification (not on the wavelength). The reflexion of such waves by smooth plane surfaces has been discussed recently by Sandstrom (1966).

In the present paper the role of surface roughness is examined. Surprisingly, it appears that quite small-scale irregularities can completely alter the reflecting properties of a surface; the tangential scale of the roughness elements may be much smaller than the wavelength of the incident or reflected waves. All scales of roughness are relevant, down to those comparable in magnitude to the thickness of the oscillatory boundary layer. For tidal waves in the ocean this thickness is of the order of 1 m.

The behaviour of the coefficient of transmission as a function of the angle of incidence appears at first sight to be extremely complicated. Some simple examples of periodic surface roughnesses are discussed and elucidated: a sawtooth, a square-topped wave and a simple sine-wave. The transmission coefficientTfor a sine-wave, for example, is shown in figure 9. An approximate expression forTis also derived in the case of a slowly modulated sine-wave (figure 10). These results are for a non-viscous fluid. The effects of viscosity are also considered qualitatively.
Spherical cap bubbles with laminar wakes
Authors: Parlange Jean-Yves
Published: March 2006

The steady rise of a given amount of gas in an infinite liquid under the action of buoyancy forces is examined. The shape of the bubble is that of a spherical cap. Behind the bubble there is a region of closed streamlines where the flow is assumed to be steady and rotational. The predicted velocity and geometry of the cap agree well with observation for a limited range of values of the Reynolds number.
Effects of sound on separated flows
Authors: Peterka Jon A. , Richardson Peter D.
Published: March 2006

Measurements of flow and fluctuating heat transfer were made for a circular cylinder in cross-flow with a transverse standing sound field imposed simultaneously. Reynolds numbers were of the order of 104, known to be in the disturbance-sensitive range, and sound intensities were as large as 140 db. The frequencies of the sound field were of the order of the disturbance frequency in the separated shear layers, reported first by Bloor.

With a sound field having its frequency matched sufficiently closely to that occurring naturally in the shear layer, the growth of the instability is enhanced with the processes of vortex fusion and possibly vortex breakdown being detectable. At the same time, the vortex street frequency is only very weakly affected, although the vortex formation region length is reduced when the instability in the shear layer is enhanced. It is suggested that the discretization of vorticity in the shear layers is one factor significant in reducing the formation length. Heat transfer at the rear of the cylinder fluctuates at frequencies centred on the shedding frequency. The fluctuation level increases as the formation region shortens.
On thermohaline convection with linear gradients
Authors: Baines P. G. , Gill A. E.
Published: March 2006

The thermohaline stability problem previously treated by Stern, Walin and Veronis is examined in greater detail. An error in an earlier paper is corrected and some new calculations made. It is shown, for instance, that direct convection can occur for thermal Rayleigh number R much less than 100 Rs when Rs [gsim ] 0·1, where Rs is the salinity Rayleigh number. A graphical presentation is devised to show the relative importance of the different terms in the equations of motion as a function of R and Rs. The most unstable mode over all wave-numbers for each R, Rs is found and it is shown that where both unstable direct and oscillating modes are present, the most unstable mode is direct in most cases.
Experiments on shock stand-off distance in non-equilibrium flow
Authors: Wegener Peter P. , Buzyna George
Published: March 2006

An experimental investigation of the non-equilibrium behaviour of the shock stand-off distance ahead of spheres at low supersonic Mach number is reported. An intermittent wind tunnel operating with a reacting gas mixture with one non-equilibrium mode is described. A non-equilibrium parameter after Damköhler is determined for the flow and the experiments cover a range of this variable from near-frozen to near-equilibrium states. The shock stand-off distance is measured and found to vary as expected between these two bounds with a low value at equilibrium. With all other variables governing shock standoff distance held constant, such measurements can also be used to determine the relaxation time of the non-equilibrium mode.
The flow induced by torsional oscillations of infinite planes
Authors: Jones A. F. , Rosenblat S.
Published: March 2006

A viscous fluid is confined between two parallel, infinite planes which perform torsional oscillations of small amplitude about a common axis. The resulting flow is studied for the case of high-frequency oscillations, when boundary layers form adjacent to moving surfaces. Particular analysis is made of the second-order, steady, radial-axial streaming. It is shown that in certain circumstances viscosity may be effective throughout the domain of flow, while in others there is a region in which viscosity is negligible.
On the generation of double Kelvin waves
Authors: Mysak L. A.
Published: March 2006

This paper considers the linear response of a homogeneous uniformly rotating ocean of infinite horizontal extent with a discontinuity in depth to a variable horizontal wind stress. It is shown that, for either a transient or time-periodic wind stress which is suddenly applied to an initially calm sea surface, the asymptotic response far from the forcing region is dominated by an outgoing dispersive wave which is trapped along the depth discontinuity, i.e. a double Kelvin wave. Plots of the forced wave patterns in the neighbourhood of the depth discontinuity itself are also presented.
Jets from two-dimensional symmetric nozzles of arbitrary shape
Authors: Larock Bruce E.
Published: March 2006

A unified approach to the problem of jet efflux from symmetrical channels of finite width and possessing a general curvilinear nozzle shape is presented. The nozzle may be composed of polygonal and/or curved-arc segments. Precise nozzle shapes cannot be initially prescribed, however. The solution is based on the combined use of conformal mapping and the Riemann-Hilbert solution to a mixed boundary-value problem. The selection of an appropriate curvature function is described; examples show possible applications.
Smoke observations of the formation of a Kármán vortex street
Authors: Zdravkovich M. M.
Published: March 2006

A smoke visualization technique was used to study the laminar wake behind a single cylinder. Observations of the wake revealed the nature of the process of formation of a Kármán vortex street: instability of the shear layers was coupled with the rolling up process induced by the concentration of vorticity.

An unexpected mass transfer was observed through the Kármán vortex street which cause mixing of the fluid surrounding the wake.
On vortex shedding from a circular cylinder in the critical Reynolds number régime
Authors: Bearman P. W.
Published: March 2006

The flow around a circular cylinder has been examined over the Reynolds number range 105 to 7·5 × 105, Reynolds number being based on cylinder diameter. Narrow-band vortex shedding has been observed up to a Reynolds number of 5·5 × 105, i.e. well into the critical régime. At this Reynolds number the Strouhal number reached the unusually high value of 0·46. Spectra of the velocity fluctuations measured in the wake are presented for several values of Reynolds number.
The calculation of three-dimensional turbulent boundary layers in incompressible flow
Authors: Nash J. F.
Published: March 2006

A method is described for calculating the development of a three-dimensional turbulent boundary layer, over a flat or developable surface, in incompressible flow. The method involves the numerical integration of the equations of motion by an explicit finite-difference method. The shear stress is determined by a parallel integration of the turbulent energy equation modified by the inclusion of empirical functions of a form which has proved successful in two dimensions, and the additional assumption is made that the turbulent shear stress acts in the direction of the rate of strain of the mean motion. The treatment of the turbulent energy equation follows closely the work of Bradshaw, Ferriss & Atwell (1967) in two dimensions.

Comparison with experiment is found to be substantially more difficult than in two dimensions. Particular difficulty is encountered in translating the recorded details of the experiment into boundary conditions for the calculation. The comparisons submitted here give some indication that the method as a whole performs satisfactorily, but they do not provide a definitive assessment of the validity of the basic assumptions. A plea is made for an experiment to supply data in a suitable form for making a more careful assessment of methods of this type.
On the penetration of a turbulent layer into stratified fluid
Authors: Kato H. , Phillips O. M.
Published: March 2006

An experiment is described in which a constant stress is applied to the surface of an initially quiescent tank of fluid with a uniform density gradient. The development of the turbulent layer by entrainment of the underlying fluid is described and it is found that the entrainment coefficient E, the ratio of the entrainment velocity ue to the friction velocity u* is given in terms of the depth D of the mixed layer and the density jump δρ across the entrainment interface by the relation
\[
E = \frac{u_e}{U_{*}} = 2.5\frac{\rho_0u^2_{*}}{g\delta\rho D}.
\]
The rate of increase of potential energy of the stratified fluid was found to be proportional to the rate of dissipation of kinetic energy per unit area in the turbulent layer. The form of these results is consistent with those found by Turner with an agitation tank, but the parameters used here allow direct application to entrainment in the ocean.
Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow
Authors: Williams Gareth P.
Published: March 2006

A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.

Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.

As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.
The interaction of internal waves with an unsteady non-uniform current
Authors: Wang Yee-Chang
Published: March 2006

On the assumptions of incompressibility, and negligible thermal conduction, salinity diffusion and viscosity, simple expressions are derived for the conservation equations of mass, momentum and energy when internal waves encounter an unsteady non-uniform current. These expressions of conservation equations are valid for all kinds of internal waves without regard to their different characteristics. From the dynamical conservation equations, we find that a stress-like term, the ‘excess momentum flux tensor’, plays an important role in the interaction between internal waves and an unsteady non-uniform current. Furthermore, it is deduced from the energy balance equation that, in the encounter of interfacial waves with a steady non-uniform current in a two-liquid system, the waves are amplified in an adverse current but suppressed in an advancing current as a result of interaction of the waves with the current. This conclusion may explain the large amplitudes sometimes observed in internal waves near the confluence of currents and near fronts at the thermocline, the region in the ocean where the density gradient is a maximum.
Free convection at low Prandtl numbers
Authors: Kuiken H. K.
Published: March 2006

In this paper it is shown that the free convection boundary layer approaches a singular character if the Prandtl number tends to zero. The method of matched asymptotic expansions is used to integrate the equations for this extreme case. An expression is derived for the Nusselt—Grashof relation and the results are compared with those of previous investigations which attack the problem in a different way.
Peristaltic pumping with long wavelengths at low Reynolds number
Authors: Shapiro A. H. , Jaffrin M. Y. , Weinberg S. L.
Published: March 2006

Pumping by means of an infinite train of peristaltic waves is investigated under conditions for which the relevant Reynolds number is small enough for inertial effects to be negligible and the wavelength to diameter ratio is large enough for the pressure to be considered uniform over the cross-section. Theoretical results are presented for both plane and axisymmetric geometries, and for amplitude ratios ranging from zero to full occlusion. For a given amplitude ratio, the theoretical pressure rise per wavelength decreases linearly with increasing time-mean flow. An experiment with a quasi-two-dimensional apparatus confirmed the theoretical values.

Calculations of the detailed fluid motions reveal that under many conditions of operation the net time-mean flow is the algebraic difference between a forward time-mean flow in the core of the tube and a backward (‘reflux’) time-mean flow near the periphery. The percentage of reflux flow can be very high. This reflux phenomenon is probably of physiologic significance in the functioning of the ureter and the gastro-intestinal system. A second fluid-mechanical peculiarity with physiological implications is that of ‘trapping’: under certain conditions an internally circulating bolus of fluid, lying about the axis, is transported with the wave speed as though it were trapped by the wave.
Stability of natural convection in a vertical slot
Authors: Vest Charles M. , Arpaci Vedat S.
Published: March 2006

The stability of natural convection of a viscous fluid in a vertical slot having isothermal side walls of different temperatures is investigated analytically. Both the conduction and boundary-layer régimes are found to be unstable with respect to stationary disturbances in the form of multicellular secondary flows. Theoretical predictions of the critical Rayleigh number and of the form of the secondary flow are verified by experimental measurements.
Experiments on natural convection in enclosures with localized heating from below
Authors: Torrance K. E. , Orloff L. , Rockett J. A.
Published: March 2006

An experimental study was made of the steady-state natural convection induced in enclosures by a small hot spot centrally located on the floor. Enclosures of rectangular and circular floor plan were employed, with height equal to one-half the major dimension of the floor plan. The movement of air within the chambers was made visible by adding metaldehyde dust particles and illuminating them with an intense light beam. The Grashof number (Gr) based on hot-spot temperature and enclosure height ranged from 8 × 105 to 1 × 1010. Laminar flows were observed for Gr [lsim ] 1.2 × 109. The experimental flows in the circular chamber are compared in a companion paper with theoretically calculated flows (Torrance & Rockett 1969). In the region of laminar flows the agreement was excellent. The present paper notes certain similarities in the flows in rectangular and circular geometries. The disturbing effect of a slight heating of one wall of the rectangular enclosure was also investigated. Measurements were made of heat transfer from the hot spot to the air in the chamber.
Numerical study of natural convection in an enclosure with localized heating from below—creeping flow to the onset of laminar instability
Authors: Torrance K. E. , Rockett J. A.
Published: March 2006

An analytical study was made of the natural convection induced in an enclosure by a small hot spot centrally located on the floor. The enclosure was a circular cylinder, vertically oriented, with height equal to radius. A Prandtl number of 0.7 (air) was assumed; the Grashof number (Gr) was based on cylinder height and hot spot temperature. The equations of fluid flow in axisymmetric cylindrical co-ordinates were simplified with the Boussinesq approximation. The equations were solved numerically with a computationally stable, explicit method. The computation, starting from quiescent conditions, proceeded through the initial transient to the fully developed flow. Solutions were obtained for Gr from 4 × 104 to 4 × 1010. The theoretical flows are in excellent agreement with experimentally observed laminar flows (Gr [lsim ] 1.2 × 109) which are discussed in a companion paper, Torrance, Orloff & Rockett (1969). Turbulence was observed experimentally for Gr [gsim ] 1.2 × 109. When the theoretical calculations were extended to Gr = 4 × 1010, a periodic vortex shedding developed, suggestive of the onset of laminar instability. The theoretical results reveal a √Gr scaling for the initial flow transients and, at large Gr, the velocities and heat transfer rates.
Wave diffraction in a two-fluid system
Authors: Kelly R. E.
Published: March 2006

Wave diffraction due to a step change in bottom topography is considered for the case of two superimposed fluids of different, but constant, densities. The interface lies below the upper surface of the step. Shallow water theory is shown to be applicable only if the ratio of a non-dimensional frequency parameter to the departure of the density ratio from unity is sufficiently small. An approximate solution of the full equations, obtained by a method applied by Miles (1967) to surface wave diffraction, yields results limited only by the condition that the frequency parameter be small.
A class of solutions for steady stratified flows
Authors: Yih Chia-Shun
Published: March 2006

Under the assumption that the horizontal scales of the flow of a stratified fluid are much greater than the vertical scale, it can be shown that the pressure distribution in the fluid is nearly hydrostatic and that the solution for steady flows can be reduced to the solution of a non-linear partial differential equation with only horizontal co-ordinates as the space variables. The theory built on the basic assumption is the shallow-water theory for stratified fluids. Transformations are explicitly given with which a class of solutions for steady three-dimensional flows of a fluid of arbitrary stratification, continuous or discontinuous, issuing from a large reservoir can be found from a corresponding solution for a homogeneous fluid, provided a free surface is present and the shallow-water theory is applicable. A few examples of exact solutions according to the shallow-water theory are given and the parallel flow in a horizontal canal issuing from a large reservoir with the same horizontal bottom, which has some bearing on previous works on stratified flows, is discussed. But it is emphasized that the class is a very special one and that there are other solutions not belonging to this class. The conditions under which a solution belonging to this class is valid are discussed.
Finite-amplitude stability of a parallel flow with a free surface
Authors: Lin S. P.
Published: March 2006

The linearized problem of the instability of a layer of liquid flowing down an inclined plane was formulated by Yih (1954) and was solved by Benjamin (1957). It was found that the instability of such a film flow is initially due to long surface waves of infinitesimally small amplitudes. In the present study, a closed-form expression for the non-linear development of these long surface waves is obtained. It is shown that in the neighbourhood of the neutral curve an exponentially growing infinitesimal disturbance may develop into supercritically stable wave motion of small but finite amplitude if the surface tension of the liquid is sufficiently large. Theoretically obtained amplitudes of such waves are consistent with Kapitza's (1949) observation. The approach used in this analysis is a modification of the method used by Reynolds & Potter (1967), who extended the method of Stuart (1960) and Watson (1960) in their study of the non-linear instability of plane Poiseuille and Poiseuille-Couette flow.
The two-dimensional flow of a stratified fluid over an obstacle
Authors: Davis Russ E.
Published: March 2006

The two-dimensional stratified flow over an obstacle placed in a channel of finite height is examined to determine the extent to which Long's model provides an adequate description of real flows. A simple numerical method of solving Long's model for obstacles of arbitrary shape is used to calculate predicted streamline patterns which are compared with experimental observations of the flow over two bluff obstacles. If only a few lee-wave modes are excited there is qualitative agreement between theory and experiment, but, if the flow is subcritical with respect to several lee-wave modes, the effects of turbulence become dominant and the inviscid model is no longer useful. The theory predicts that the drag on an obstacle can increase with decreasing speed owing to the momentum transfer to lee-wave motion. Direct measurement of drag indicates that there are conditions under which the drag does increase with decreasing speed, but under these conditions the wake is dominated by turbulence and no lee waves can be detected.
Boundary-value problems in the kinetic theory of gases Part I. Slip flow
Authors: Williams M. M. R.
Published: March 2006

A new method for treating boundary-value problems in gas-kinetic theory has been developed. The new method has the advantage of reproducing the bulk or asymptotic flow properties accurately whilst giving a realistic description of the behaviour of the molecular distribution function in the neighbourhood of a wall. As an example, the Kramers, or slip-flow, problem is solved for a general specular-diffuse boundary condition and some new expressions for the slip coefficient, flow speed and molecular distribution function at the surface are derived.

A brief discussion of the eigenvalue spectrum of the associated Boltzmann equation is given and its physical significance pointed out.

Certain analogies between this problem and the Milne problem in neutron transport theory are demonstrated.
A flow reversal due to edge effects
Authors: Griffiths D. F. , Jones D. T. , Walters K.
Published: March 2006

An experiment is described which indicates a flow reversal due to fluid elasticity in the case of a rotating-plate experiment. An attempt to predict this reversal analytically on the ‘infinite’ plate assumption fails and recourse is made to a numerical procedure to solve the ‘finite’ plate problem. It is shown that the reversal is due to edge effects.
Hydromagnetic stability of dissipative flow between rotating permeable cylinders. Part 2. Oscillatory critical modes and asymptotic results
Authors: Chang Tien Sun , Sartory Walter K.
Published: March 2006

In considering the onset of instability of an electrically conducting fluid between rotating permeable perfectly conducting cylinders in an applied axial magnetic field, it is found that oscillatory axially symmetric modes occur at large values of Hartmann number, in addition to the usual stationary modes. Results are presented showing the effect of the oscillatory modes on the criterion of onset of instability.

The asymptotic behaviour of the stability criterion is considered in the limit of very large radial Reynolds number, and also in the limit where both the radial Reynolds number and the Hartmann number are large.
The stability of Poiseuille flow in a pipe
Authors: Davey A. , Drazin P. G.
Published: March 2006

Numerical calculations show that the flow of viscous incompressible fluid in a circular pipe is stable to small axisymmetric disturbances at all Reynolds numbers. These calculations are linked with known asymptotic results.
The viscous vortex induced by a sink on the axis of a circulating fluid in the presence of a plane free surface
Authors: Pedley T. J.
Published: March 2006

The flow under discussion represents an idealization of the bath-tub vortex; distortions of the free surface, finite sink size, and all rigid boundaries have been eliminated from the problem in order to isolate the effect of the non-uniform stretching of vortex lines produced by the sink flow. A boundary-layer type of approximation is made about the axis, which requires that the meridional Reynolds number (N) be large, and since the problem is still intractable, an expansion is made in powers ofK=R2/N(whereRis the swirl Reynolds number), which measures the strength of the interaction between the swirl and meridional velocity fields. In the limit of zeroKthe flow is a modified Burgers vortex whose radius decreases to zero at the sink. For non-zeroK, the interaction is not restricted to the vortex core, because the presence of the vortex modifies the outer irrotational flow, inducing a radial mass flux into the core, whose dependence on the axial co-ordinate is calculated to the first order inK.The structure of the core is obtained, again to the first order inK, from two co-ordinate expansions, one near the stagnation point on the axis, and the other near the sink, although only the first few terms of the latter can be determined explicitly. It is shown how the methods can be extended not only to higher orders inK, but also to any other narrow viscous vortex in which the vortex lines are stretched non-uniformly away from an internal stagnation point.
Buoyancy-driven convection in cylindrical geometries
Authors: Liang S. F. , Vidal A. , Acrivos Andreas
Published: March 2006

Numerical solutions to the Boussinesq equations containing a temperature-dependent viscosity are presented for the case of axisymmetric buoyancy-driven convective flow in a cylindrical cell. Two solutions, one with upflow and the other with downflow at the centre of the cell, were found for each set of boundary conditions that were considered. The existence of these two steady-state régimes was verified experimentally for the case of a cylindrical cell having rigid insulating lateral boundaries and isothermal top and bottom planes.

Using a perturbation expansion it is also shown that only one of these solutions remains stable in the subcritical régime. This, however, seems to be confined to a very narrow range of Rayleigh numbers, beyond which, according to all the evidence presently at hand, both solutions are equally stable for those values of the Rayleigh and Prandtl numbers for which axisymmetric motions occur.

Finally, certain fundamental differences between the problem considered here and that of thermal convection in a layer of infinite horizontal extent are briefly discussed.
On the non-linear interaction of inertial modes
Authors: Greenspan H. P.
Published: March 2006

It is shown that the principal non-linear interaction of inviscid inertial modes does not produce a resonant response in the steady geostrophic circulation. The rectified geostrophic flow manifested in a closed rotating container by a periodic excitation seems to result from a combination of viscous and non-linear effects within the boundary layers.
The lee-wave régime for a slender body in a rotating flow
Authors: Miles John W.
Published: March 2006

The uniform motion of a closed, axisymmetric body along the axis of an unbounded, rotating, inviscid, incompressible fluid is considered on Long's hypotheses that: the flow is steady; the flow is uniform far upstream of the body; the inertial waves excited by the body cannot propagate upstream. The appropriate similarity parameters arek, an inverse Rossby number based on the body length, and δ, the slenderness ratio of the body. It is conjectured that an upper bound to the parametric régime in which the solution implied by Long's hypotheses remains valid, saykδ≡k<kc, is determined by the first occurrence, with increasingk, of a local reversal of the flow.

A general solution for the stream function is established in terms of an assumed distribution of dipoles along the axis of the body. The disturbance upstream of the body is found to be proportional to the product ofk2and the dipole moment (total dipole strength) and to fall off only as the inverse distance, as compared with the inverse cube of the distance for a potential flow. The corresponding wave drag is found to depend on the power spectrum of the dipole distribution in the axial wave-number interval (0,k) and to be a monotonically decreasing function of the axial velocity for fixed angular velocity. Asymptotic solutions for prescribed bodies are established in the following limits: (i)K→ 0 with δ fixed; (ii) δ → 0 withkfixed; (iii)k→ ∞ withkδfixed. Both the upstream disturbance and the wave drag in the limit (i) depend essentially on the dipole moment of the body with respect to a uniform, potential flow. The limit (ii) is analogous to conventional slender-body theory and yields a dipole density that is proportional to the cross-sectional area of the body. The limit (iii) leads to a singular integral equation that is solved to determinekcand the dipole moment for a slender body.

The results are applied to a sphere and a slender ellipsoid. The upstream axial velocity and the drag coefficient based on Stewartson's results for a sphere are found to differ significantly from Maxworthy's (1969) measurements, presumably in consequence of viscous separation effects. Maxworthy's measured values of upstream axial velocity are found to agree with the theoretical values for an equivalent ellipsoid, based on the sphere plus its upstream wake, fork[lsim ]kc.
A study of Bénard convection with and without rotation
Authors: Rossby H. T.
Published: March 2006

An experimental study of the response of a thin uniformly heated rotating layer of fluid is presented. It is shown that the stability of the fluid depends strongly upon the three parameters that described its state, namely the Rayleigh number, the Taylor number and the Prandtl number. For the two Prandtl numbers considered, 6·8 and 0·025 corresponding to water and mercury, linear theory is insufficient to fully describe their stability properties. For water, subcritical instability will occur for all Taylor numbers greater than 5 × 104, whereas mercury exhibits a subcritical instability only for finite Taylor numbers less than 105. At all other Taylor numbers there is good agreement between linear theory and experiment.

The heat flux in these two fluids has been measured over a wide range of Rayleigh and Taylor numbers. Generally, much higher Nusselt numbers are found with water than with mercury. In water, at any Rayleigh number greater than 104, it is found that the Nusselt number will increase by about 10% as the Taylor number is increased from zero to a certain value, which depends on the Rayleigh number. It is suggested that this increase in the heat flux results from a perturbation of the velocity boundary layer with an ‘Ekman-layer-like’ profile in such a way that the scale of boundary layer is reduced. In mercury, on the other hand, the heat flux decreases monotonically with increasing Taylor number. Over a range of Rayleigh numbers (at large Taylor numbers) oscillatory convection is preferred although it is inefficient at transporting heat. Above a certain Rayleigh number, less than the critical value for steady convection according to linear theory, the heat flux increases more rapidly and the convection becomes increasingly irregular as is shown by the temperature fluctuations at a point in the fluid.

Photographs of the convective flow in a silicone oil (Prandtl number = 100) at various rotation rates are shown. From these a rough estimate is obtained of the dominant horizontal convective scale as a function of the Rayleigh and Taylor numbers.
Numerical studies of the laminar boundary layer for Mach numbers up to 15
Authors: Fitzhugh Henry A.
Published: March 2006

A comprehensive set of exact solutions to the first-order boundary-layer equations has been computed using the finite difference computer programme of Sells, with and without wall cooling. The effects of Prandtl number, wall cooling and Mach number on separation point location were studied. Values of displacement thickness, skin friction coefficient and Stanton number are displayed graphically for the supersonic flow over a circular concave arc, for a subsonic cooled cylinder and for the case of a linearly retarded velocity distribution. The influence of pressure gradient on recovery factor was studied. Velocity and temperature profiles are shown for four cold wall cases. The exact computer results show the errors in many of the more approximate methods available for the case whereUe=U∞(1 -X/L). The importance of second-order effects and the applicability of a first-order solution are discussed briefly.
Axially symmetric motion of a stratified, rotating fluid in a spherical annulus of narrow gap
Authors: Pedlosky J.
Published: March 2006

A linear theory is presented for the steady, axially symmetric motion of a stratified fluid in a narrow, rotating spherical annulus with a spherically symmetric gravitational field.

The fluid is driven by a combination of differential rotation of the two shells and differential heating applied at the surfaces of the spheres.

It is shown that the effect of stratification becomes increasingly important at lower latitudes with the Ekman layers on the spheres’ surfaces fading in strength as the geostrophic interior velocities themselves tend toward the shell speeds at lower latitudes.

The Singularities In The Geostrophic Solutions At The Equator Are Removed By A Boundary Layer Whose Detailed Structure Depends On The Ratio Of Horizontal To Vertical Mixing Coefficients Of Momentum And Heat.
Laminar convection cells at high Rayleigh number
Authors: Wesseling P.
Published: March 2006

The asymptotic behaviour for large Rayleigh number and Prandtl number of O(1) of two-dimensional convection cells in a fluid between horizontal plates heated from below has been discussed by Pillow (1952) and more recently by Robinson (1967). The flow models derived by Pillow and Robinson differ from each other. The purpose of the present paper is to point out the likelihood of an inconsistency in Robinson's flow model, and to discuss qualitatively a few hitherto unnoticed features of the flow.