bluuebunny/crossref_metadata_2025 huggingface dataset:
A note on the expansion of turbulent wakes
Authors: Keffer J. F.
Published: March 2006

An analysis has been made of a two-dimensional turbulent wake which was subjected to a homogeneous strain. This was applied in such a way that the rate of growth of the wake tended to be accelerated but the main stream convective velocity remained constant. An attempt to describe the flow by classical self-preservation techniques was made. However, the results suggested that a detrainment of wake fluid would result. In lieu of this a more realistic approach was sought and it was found that in certain situations where the flow has had an opportunity to develop before the strain is applied, the growth of the wake in the strain field can be described successfully by a simple geometric distortion. It appears likely that in these cases the interaction between the external strain and the turbulent structure within the wake is negligible, the wake remaining essentially passive. Suggestions as to the mechanism of the turbulent mixing and transport under these conditions are made.
Steady flows in rectangular cavities
Authors: Pan Frank , Acrivos Andreas
Published: March 2006

This paper deals with the steady flow in a rectangular cavity where the motion is driven by the uniform translation of the top wall. Creeping flow solutions for cavities having aspect ratios from ¼ to 5 were obtained numerically by a relaxation technique and were shown to compare favourably with Dean & Montagnon's (1949) similarity solution, as extended by Moffatt (1964), in the region near the bottom corners of a square cavity as well as throughout the major portion of a cavity with aspect ratio equal to 5. In addition, for a Reynolds number range from 20 to 4000, flow patterns were determined experimentally by means of a photographic technique for finite cavities, as well as for cavities of effectively infinite depth. These experimental results suggest that, within finite cavities, the high Reynolds number steady flow should consist essentially of a single inviscid core of uniform vorticity with viscous effects being confined to thin shear layers near the boundaries, while, for cavities of infinite depth, the viscous and inertia forces should remain of comparable magnitude throughout the whole domain even in the limit of very large Reynolds number R.
Nonlinear effects in the rheology of dilute suspensions
Authors: Goddard J. D. , Miller Chester
Published: March 2006

An analysis is presented of the deformation of a solid-like, viscoelastic sphere suspended in the infinite Stokesian flow field of a Newtonian fluid undergoing an arbitrary time-dependent homogeneous deformation far from the particle. The results of the analysis are then used to deduce the macroscopic rheological behaviour of a dilute monodisperse suspension of slightly deformable spheres.

Even though inertial effects and second-order terms in the particle deformation are neglected, it is found that non-linear rheological effects can arise, because of the interaction between the deformed particle and the flow. As a consequence, the rheological relation obtained here differs from those presented earlier by Fröhlich & Sack (1946) and by Oldroyd (1955) through the appearance of certain terms which are non-linear in the deformation rate.

When the suspended particles are purely elastic in their behaviour the rheological equation presented here reduces for certain flows to a special case of Oldroyd's (1958) phenomenological model, with material constants which can be directly related to suspension properties.
Investigation of turbulent thermal convection between horizontal plates
Authors: Deardorff J. W. , Willis G. E.
Published: March 2006

Properties of turbulent thermal convection were measured in air between horizontal plates maintained at constant temperatures. Rayleigh numbers of 6·3 × 105, 2·5 × 106 and 1·0 × 107 were studied with a convection chamber designed to allow measurements to be taken along a horizontal path. Vertical profiles are presented of horizontally averaged temperature; r.m.s. fluctuations of temperature, horizontal velocity and vertical velocity; total heat flux; the correlation coefficient between vertical velocity and temperature; all terms in the thermal variance equation; and of terms in the turbulent kinetic energy balance.

One-dimensional spectra of temperature, horizontal velocity and vertical velocity at two different heights all disclose large intensity for waves of length 5 to 15 times the plate separation. Secondary peaks occur for scales of from 0·7 to 1·7 times the separation height.

Many of the observations are consistent with a thermal structure dominated by plumes extending most or all of the distance between plates.
The crossed-beam correlation technique
Authors: Fisher M. J. , Krause F. R.
Published: March 2006

This paper describes an experimental technique which allows estimates of local turbulent properties of a shear layer to be obtained without the necessity of inserting a probe into the flow field. The probe is replaced by two beams of radiation, which pass through the entire flow field in two mutually perpendicular directions. It is shown that, although each beam independently reflects only an integral of the flow properties along the entire path between the source and detector, the covariance between the two detected signals does yield local turbulent information.

To verify the validity of the technique, experimental results are presented for the shear layer of a subsonic jet and are shown to be in good agreement with published hot-wire data.
Wall-pressure fluctuations associated with subsonic turbulent boundary layer flow
Authors: Bull M. K.
Published: March 2006

Experimental results are given for various statistical properties of the fluctuating wall-pressure field associated with a subsonic turbulent equilibrium boundary layer, developed on a smooth wind tunnel wall after natural transition from laminar to turbulent flow.

The statistical quantities of the wall-pressure field investigated were root-mean-square pressure, frequency power spectrum and space-time correlations. Space-time correlation measurements were made in both broad and narrow frequency bands. The experiments were made at flow Mach numbers of 0·3 and 0·5 and covered a Reynolds number range of about 5 to 1.

The main conclusion to which the measurements lead is that the wall-pressure field has a structure produced by contributions from pressure sources in the boundary layer with a wide range of convection velocities, and comprises two families of convected wave-number components. One family is of high wave-number components and is associated with turbulent motion in the constant stress layer; the components are longitudinally coherent for times proportional to the times taken for them to be convected distances equal to their wavelengths and laterally coherent over distances proportional to their wavelengths. The other family comprises components of wavelength greater than about twice the boundary-layer thickness, which lose coherence as a group more or less independently of wavelength and are associated with large-scale eddy motion in the boundary layer, outside the constant stress layer. The evolution of the pressure field is discussed in terms of these two wave-number families.
A three-dimensional laminar boundary-layer calculation
Authors: Banks W. H. H.
Published: March 2006

Results are presented of a preliminary numerical investigation into a three-dimensional laminar boundary layer. It is assumed that the flow is over a developable surface and the boundary conditions at the outer edge of the layer are chosen to be u = U0 + xU1, v = yV1. This choice enables the governing equations to be written in terms of two, and not three, independent variables, viz. x and z. However, the three-dimensionality of the problem gives rise to a coupling of the equations which, not unnaturally, is still present after the elimination of y.

For appropriate values of U1 and V1 it is found possible to integrate the equations approximately from the ‘birth’ of the boundary layer (x = 0) right up to a saddle point of attachment. Calculations have already been made for flow at such attachment points and the comparison of the present results with them is extremely good.
Hydromagnetic planetary waves
Authors: Malkus Willem V. R.
Published: March 2006

A study is made of hydromagnetic oscillations in a rotating fluid sphere. The basic state is chosen as a uniform current parallel to the axis of rotation. It is found that the non-dissipative normal modes are described by a modified form of the Poincaré eigenvalue problem. For small rotation rates, the lowest non-axisymmetric modes are unstable. For rotation rates of geophysical interest all normal modes are stable. The introduction of ohmic dissipation leads to a hydro-magnetic boundary-layer problem. Solutions for the boundary layer are outlined indicating its role in altering the free periods, damping the oscillations and producing external magnetic fields. Dispersion relations are derived which establish that the zonal phase velocities of both ‘fast’ hydrodynamic and ‘slow’ hydro-magnetic waves can be of either sign. Observations of the secular variations of the earth's magnetic field indicate motion primarily towards the west. A mechanism for the selective excitation of the observed motion is discussed.
Inviscid radiating flow over a blunt body
Authors: Cheng Ping , Vincenti Walter G.
Published: March 2006

The effect of thermal radiation is investigated for the axisymmetric flow over the blunt body associated with a given paraboloidal shock wave. Radiative transfer is treated by means of the differential approximation, which applies to multidimensional flow and is valid throughout the entire range of temperature and optical thickness. The gas is assumed to be perfect and optically grey, and molecular-transport processes are neglected. A semi-analytical solution for the flow and radiation fields is obtained by the method of series truncation.

Results are presented, in the strong-shock approximation, for various values of the appropriate dimensionless variables. In general, radiation is found to have a significant influence on temperature and density, moderate effect on velocity, and little effect on pressure. The stand-off distance between the shock wave and the body is found to decrease significantly with increasing radiation; the body shape is less affected. The anomalous behaviour of the gas temperature on the body streamline as obtained by earlier investigators in the optically thin case does not appear in the present work. The results thus show correct physical behaviour throughout the flow field for all values of optical thickness. The detailed flow quantities exhibit a number of features of multidimensional radiating flow. They also provide a check on the special assumptions made in other, more approximate treatments. Similarities between radiating flow and non-equilibrium reactive flow over blunt bodies are apparent.
Viscous flow near a cusped corner
Authors: Schubert G.
Published: March 2006

The slow motion of fluid exterior to a cylinder lying on a wall is considered for a variety of boundary conditions. In particular, the solution is obtained for the case when the motion far from the cylinder is one of uniform shear. Calculations are made for the force and the moment exerted by the fluid on the cylinder. The asymptotic form of the flow both far from the cylinder and near the cusped corners is presented. The flow sufficiently near a cusp consists of a sequence of eddies of rapidly diminishing strength, and the solution of another boundary-value problem supports the view that the nature of this eddy system is independent of conditions far from the cusp. The nature of inviscid flow with uniform vorticity in cusped corners is also considered.
On the stability of an inviscid shear layer which is periodic in space and time
Authors: Kelly R. E.
Published: March 2006

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.

At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.

Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.
The axisymmetric flow in a rotating annulus due to a horizontally applied temperature gradient
Authors: Hunter C.
Published: March 2006

The axisymmetric flow of fluid, confined within a rapidly rotating annulus and caused by a horizontally applied temperature gradient, is discussed under the assumptions of small thermal Rossby number and small Taylor number. It is initially assumed that the transfer of heat is purely conductive, and the flow is calculated for when the upper fluid surface is either rigid or free. The analysis of the problem in which the upper surface is rigid is extended so as to allow con-vective as well as conductive transfer of heat.
On the impulsively started rotating sphere
Authors: Barrett K. E.
Published: March 2006

The velocity field generated in a fluid of viscosity, v, by impulsively starting at time t = 0, a sphere of radius a spinning with angular velocity Ω about a diameter is described using a new expansion variable 2 √vt/r. It is first shown how the standard time-dependent boundary-layer equations can be modified to give series solutions satisfying all the boundary conditions. Next, that these new solutions are relevant when the Reynolds number R = a2Ω/v goes to infinity in such a way that $R^{\frac{1}{3}} \Omega t$ is large. Lastly, solutions are given, applicable at small times for non-zero Reynolds numbers. These last expansions show that the velocity components decay algebraically rather than exponentially at large distances.
An analytic approach to the method of series truncation for the supersonic blunt body problem
Authors: Kao Hsiao C.
Published: March 2006

A method for obtaining analytic solutions to the problem of blunt bodies in the supersonic stream of an ideal gas is presented. The solutions are written in terms of power series whose coefficients are elementary functions. These solutions are approximate, but the approximation is rational, i.e. any higher approximation can, in principle, be obtained. Some of these higher approximations have been calculated. Examples are presented for various free-stream conditions and prescribed body shapes. These are compared with results from standard numerical procedure and with available experimental measurements.
Long waves on a beach
Authors: Peregrine D. H.
Published: March 2006

Equations of motion are derived for long waves in water of varying depth. The equations are for small amplitude waves, but do include non-linear terms. They correspond to the Boussinesq equations for water of constant depth. Solutions have been calculated numerically for a solitary wave on a beach of uniform slope. These solutions include a reflected wave, which is also derived analytically by using the linearized long-wave equations.
Gravity currents and related phenomena
Authors: Benjamin T. Brooke
Published: March 2006

This paper presents a broad investigation into the properties of steady gravity currents, in so far as they can be represented by perfect-fluid theory and simple extensions of it (like the classical theory of hydraulic jumps) that give a rudimentary account of dissipation. As usually understood, a gravity current consists of a wedge of heavy fluid (e.g. salt water, cold air) intruding into an expanse of lighter fluid (fresh water, warm air); but it is pointed out in § 1 that, if the effects of viscosity and mixing of the fluids at the interface are ignored, the hydrodynamical problem is formally the same as that for an empty cavity advancing along the upper boundary of a liquid. Being simplest in detail, the latter problem is treated as a prototype for the class of physical problems under study: most of the analysis is related to it specifically, but the results thus obtained are immediately applicable to gravity currents by scaling the gravitational constant according to a simple rule.

In § 2 the possible states of steady flow in the present category between fixed horizontal boundaries are examined on the assumption that the interface becomes horizontal far downstream. A certain range of flows appears to be possible when energy is dissipated; but in the absence of dissipation only one flow is possible, in which the asymptotic level of the interface is midway between the plane boundaries. The corresponding flow in a tube of circular cross-section is found in § 3, and the theory is shown to be in excellent agreement with the results of recent experiments by Zukoski. A discussion of the effects of surface tension is included in § 3. The two-dimensional energy-conserving flow is investigated further in § 4, and finally a close approximation to the shape of the interface is obtained. In § 5 the discussion turns to the question whether flows characterized by periodic wavetrains are realizable, and it appears that none is possible without a large loss of energy occurring. In § 6 the case of infinite total depth is considered, relating to deeply submerged gravity currents. It is shown that the flow must always feature a breaking ‘head wave’, and various properties of the resulting wake are demonstrated. Reasonable agreement is established with experimental results obtained by Keulegan and others.
Flow properties for the continuum viewpoint of a non-equilibrium gas-particle mixture
Authors: Panton Ronald
Published: March 2006

Flow properties for the non-equilibrium two-phase flow of a gas-particle mixture are formulated from the theoretical standpoint. A quasi-one-dimensional flow containing an arbitrary volume of particles is considered, and mass transfer between the phases is allowed. It is shown that meaningful definitions of the flow properties of each phase can be constructed as area-averages of (time-averaged local flow-field properties). Special definitions of averages overcome the difficulties introduced by the fact that one phase does not occupy the entire region at all times. Conservation equations for the newly defined properties are given and criteria for their validity determined. The results give fresh interpretation to several aspects of two-phase flow: the particle-phase pressure is associated with the internal particle pressure, whereas Reynolds-stress terms are introduced by fluctuations in particle velocity. Reynolds stresses for both phases are important in laminar as well as turbulent flow and provide a significant particlephase viscous effect. The interphase momentum transfer because of condensation or vaporization is shown to be characterized by the particle-phase velocity irrespective of the direction of the mass transfer.
A note on a swimming problem
Authors: Tuck E. O.
Published: March 2006

The rate of self-propulsion of a doubly-infinite flexible sheet due to transverse waving oscillations in a viscous fluid is shown to decrease with increasing frequency, at a fixed (small) wave amplitude. This result differs from that of Reynolds (1965) who included local inertia, and thereby predicted that the swimming speed increases above the limiting value given by Taylor (1951) at zero frequency. The error in Reynolds’ work is due to his neglect of the simultaneous effect of convection, which induces a non-uniform mean second-order flow, with direction such as to oppose propulsion. Some other results concerning swimming sheets are presented.
Magnetohydrodynamic flow along cylindrical pipes under non-uniform transverse magnetic fields
Authors: Todd L.
Published: March 2006

The unidirectional flow of an incompressible, electrically conducting, viscous fluid along cylindrical pipes is considered. An external magnetic field, B0, which lies in the plane transverse to the flow is applied. It is shown that the governing equations, written in the co-ordinate system traced out by B0, are mathematically very similar to those for a uniform field.

The paper deals mainly with ducts whose walls are insulators. Though exact solutions (valid for all values of the Hartmann number) are derived, the limit of high Hartmann number is taken for detailed discussion. Transition layers (or, loosely, ‘wakes’) can arise which are centred on curved field lines. In some cases, reversed flow occurs in part of the core (‘radial-type’ fields). Situations also arise where the magnitude (and sign) of the velocity remains the same as for B0 = 0, whatever the strength of the applied, transverse (azimuthal) magnetic field.
Penetrative convection
Authors: Musman Steven
Published: March 2006

A penetrative convection problem which depends on the density maximum of water near 4 °C is solved numerically. When a layer of water has its lower boundary maintained at 0 °C and its upper boundary at some temperature above 4 °C, the layer will be divided into a lower convectively unstable region and an upper convectively stable region. Steady-state finite-amplitude solutions to this problem are obtained using the mean field approximation and free boundaries.

Convective mixing alters the temperature structure of the layer so that the temperature of a large fraction of the layer is slightly below the temperature of maximum density, in agreement with laboratory measurements. The largest motions are found in a principal convective cell which extends from the lower boundary to a temperature of 7° or 8 °C. Above the principal cell one or more counter cells may form, depending on the temperature of the upper boundary. The velocities in the counter cells are substantially less than those in the principal cell and fall off rapidly going upward. When the temperature of the upper boundary is 7 °C or higher, convection first takes place at a finite amplitude and at a Rayleigh number less than that predicted by the linear theory. When the temperature of the upper boundary is 10 °C or higher, the upper boundary is no longer important dynamically to the system.

As the Rayleigh number is increased above critical stability the velocities in the principal cell, heat transport and the distortion of the temperature field all increase. In addition, the principal cell becomes more slender and fills a greater fraction of the layer. Also, as the Rayleigh number increases the counter cells become more flattened, fill a smaller fraction of the layer, and the velocities in the counter cells decrease relative to those in the principal cell.

The most important penetration of convective motions takes place in the form of nearly horizontal motions in the lowest part of the stable region, corresponding to the upper part of the principal cell. The velocities of these motions are a large fraction of the largest vertical velocities in the unstable region.
On the trapping of waves along a discontinuity of depth in a rotating ocean
Authors: Longuet-Higgins M. S.
Published: March 2006

It is shown that, according to the linearized theory of long waves in a rotating, unbounded sea, if there is a discontinuity in depth along a straight line separating two regions each of uniform depth, then wave motions may exist which are propagated along the discontinuity and whose amplitude falls off exponentially to either side. Thus the discontinuity acts as a kind of wave-guide.

The period of the waves is always greater than the inertial period. The wave period also exceeds the period of Kelvin waves in the deeper medium. As the ratio of the depth tends to infinity, the wave period tends to the inertial period or to the Kelvin wave period, whichever is the greater. On the other hand as the wavelength decreases (within the limits of shallow-water theory) so the waves tend to the non-divergent planetary waves found recently by Rhines.

In an infinite ocean of uniform depth free waves with period greater than a pendulum-day cannot normally be propagated without attenuation (if the Coriolis parameter is constant). But non-uniformities of depth provide a means whereby such energy may be channelled over great distances with little attenuation.

It is suggested that a gradually diminishing discontinuity will act as a chromatograph, each position along the discontinuity being marked by waves of a particular period.
A note on shock dynamics relative to a moving frame
Authors: Whitham G. B.
Published: March 2006

In this paper it is shown how a previous theory for treating shocks moving into fluid at rest (Whitham 1957, 1959) can be changed to a form suitable for treating propagation into a steady uniform flow. It is necessary only to transform the original theory to a moving frame but the details are not trivial.
A note on shock diffraction by a supersonic wedge
Authors: Huppert H. E. , Miles J. W.
Published: March 2006

An earlier treatment of the diffraction of a shock wave advancing into a region of uniform flow, based on Chisnell's (1965) extension of Whitham's (1957) rule for shock diffraction, is corrected for an algebraic error and then compared with an analogous treatment based on the more recent extension derived by Whitham (1968). The basis for comparison is the pressure just behind a shock wave that is diffracted by a thin wedge travelling at supersonic speed. The approximation provided by Whitham's extension is both simpler than, and typically superior to, that provided by Chisnell's extension (although the numerical differences are small in the Mach-number régime considered).
The laminar wall-jet over a curved surface
Authors: Wygnanski I. J. , Champagne F. H.
Published: March 2006

The laminar flow of a wall jet over a curved surface is considered. A unique similarity solution is obtained for both concave and convex surfaces when the local radius of curvature is proportional to x3/4. This solution satisfies a similar invariant condition to the one derived by Glauert for the wall jet over a plane surface. The variation of the shape of the velocity profile, the skin friction, and the surface pressure as a function of curvature is given.
The structure of a weak shock wave undergoing reflexion from a wall
Authors: Lesser M. B. , Seebass R.
Published: March 2006

The Navier–Stokes equations are used to study the unsteady structure of a weak shock wave reflecting from a plane wall. Both an adiabatic and an isothermal wall are considered. Incident and reflected shock structures are found by expanding the dependent variables in asymptotic series in the shock strength; the first-order terms are shown to satisfy an equation analogous to Burgers equation. The structure of the wave during reflexion is obtained from an expansion in which the first-order terms satisfy the acoustic equations. The isothermal wall boundary condition requires the introduction of a thermal layer adjacent to the wall. In this case viscosity and convection play a role secondary to the wall temperature boundary condition in determining the structure of the reflected wave. The presentation is simplified by introducing a generalized Burgers equation that gives the same first-order results as the Navier–Stokes equations. Correct second-order results are obtained from this equation simply by applying a correction to the result for the temperature.
The form drag of two-dimensional bluff-plates immersed in turbulent boundary layers
Authors: Good M. C. , Joubert P. N.
Published: March 2006

Measurements of the distributions of pressure on a bluff flat plate (fence) have been correlated with the characteristics of the smooth-wall boundary layer in which it is immersed. For zero pressure-gradient flows, correlations are obtained for the variation of form drag with plate heighthwhich are analogous in form to the ‘law of the wall’ and the ‘velocity-defect law’ for the boundary-layer velocity profile. The data for adverse pressure-gradient flows is suggestive of a ‘law of the wake’ type correlation. Pressures on the upstream face of the bluff-plate are determined by a wall-similarity law, even forh/δ > 1, and are independent of the pressure-gradient history of the flow; the separation induced upstream is apparently of the Stratford-Townsend type. The effects of the history of the boundary layer are manifested only in the flow in the rear separation bubble, and then only forh/δ > ½. The base pressure is also sensitive to free-stream pressure gradients downstream of the bluff-plate. The relative extent of upstream influence of the bluff-plate on the boundary layer is found to increase rapidly ash/δ decreases. One set of measurements of the mean flow field is also presented.
On the instability of rapidly rotating shear flows to non-axisymmetric disturbances
Authors: Pedley T. J.
Published: March 2006

The stability is considered of the flow with velocity components
\[
\{0,\Omega r[1+O(\epsilon^2)],\;2\epsilon\Omega r_0f(r/r_0)\}
\]
(where f(x) is a function of order one) in cylindrical polar co-ordinates (r, ϕ, z), bounded by the rigid cylinders r/r0 = x1 and r/r0 = 1 (0 [les ] x1 < 1). When ε [Lt ] 1, the flow is shown to be unstable to non-axisymmetric inviscid disturbances of sufficiently large axial wavelength. The case of Poiseuille flow in a rotating pipe is considered in more detail, and the growth rate of the most rapidly growing disturbance is found to be 2εΩ.
Long wavelength disturbances to non-planar parallel flow
Authors: Hocking L. M.
Published: March 2006

The instability of unbounded parallel inviscid flows which are neither plane nor axisymmetric is studied for disturbances with long wavelengths in the flow direction. The details of the variation of the flow velocity on any scale smaller than the wavelength are shown to have no effect on these disturbances and it is only the non-uniformity of the flow at infinity which is relevant. There is a class of disturbance which can only exist because of the non-uniformity, and it is governed by an equation similar to the Rayleigh equation for inviscid plane parallel flow. A number of the properties of the solution can be found and a large class of flows can be shown to be unstable. When the flow at infinity is linear in sectors it is easy to find simple solutions. Particular flows, which are either uniform in sectors with vortex sheets along the dividing radii or are continuous, but with a linear variation in each sector, are studied in detail and are shown to be unstable, even when the non-uniformity is confined to a narrow sector.
Some electrically driven flows in magnetohydrodynamics. Part 1. Theory
Authors: Hunt J. C. R. , Williams W. E.
Published: March 2006

The magnetohydrodynamic flow between two parallel conducting planes is investigated for the case when the flow is driven by the current produced by electrodes placed one in each plane, the applied magnetic field being perpendicular to the planes. An exact solution is presented for the case of line electrodes placed opposite each other and an approximate method of determining the flow in this case for large Hartman number (M) is also given. It is shown that for large M the asymptotic form of the exact solution agrees with that obtained by the approximate method. The results are generalized to cover the case of line electrodes displaced relative to each other. A similar investigation is carried out for the case of two point electrodes opposite each other.
Instability of unsteady flows or configurations Part 1. Instability of a horizontal liquid layer on an oscillating plane
Authors: Yih Chia-Shun
Published: March 2006

A layer of viscous liquid with a free surface is set in motion by the lower boundary moving simple-harmonically in its own plane. The stability of this motion is investigated. Since the primary flow is time-dependent, the time variable cannot be separated from at least one space variable, and a new approach must be used to investigate the problem. In this paper the stability of long waves is studied by a perturbation method which has not been applied before to problems of stability of unsteady flows, and it is found that the flow under consideration can be unstable for long waves.
Shallow water wave generation by unsteady flow
Authors: Williams J. E. Ffowcs , Hawkings D. L.
Published: March 2006

Small amplitude waves on a shallow layer of water are studied from the point of view used in aerodynamic sound theory. It is shown that many aspects of the generation and propagation of water waves are similar to those of sound waves in air. Certain differences are also discussed. It is concluded that shallow water simulation can be employed in the study of some aspects of aerodynamically generated sound.
Experimental studies in magneto-fluid dynamics: pressure distribution measurements around a sphere
Authors: Maxworthy T.
Published: March 2006

Measurements of the pressure distribution around a sphere placed in aligned magnetic and velocity fields show that an increase in drag is mainly due to a decrease in the pressure on the base of the body. When magnetic forces are large compared to inertia forces, this decrease is due to a loss in total pressure along streamlines just outside the surface boundary layer and an acceleration of the flow to a velocity much larger than the reference velocity. Separation of a viscous boundary layer takes place behind the equator and still, to a large extent, controls the magnitude of the base pressure and the drag experienced by the sphere. A model consistent with these findings is presented.
Note on the numerical solution for unsteady viscous flow past a circular cylinder
Authors: Ingham D. B.
Published: March 2006

The starting flow of a viscous fluid past a circular cylinder at Reynolds numbers 40 and 100 has been obtained by a numerical method. The method used is that developed by Payne (1957) but it has been extended here to cover a larger time interval.

At Reynolds number 40 Payne's result for the drag coefficient at time t = 6 is in reasonable agreement with Kawaguti's (1953) result for the steady case but if Payne's computation, is extended to time t ≈ 24, the result is in better agreement with Apelt's (1961) result for the steady case. Also, a further investigation into the case R = 100 shows that Payne's mesh size is too crude. Similar observations can be made concerning the size of the standing vortices downstream of the circular cylinder and how they grow in time.
The transient for Stokes's oscillating plate: a solution in terms of tabulated functions
Authors: Panton Ronald
Published: March 2006

The motion of a semi-infinite incompressible fluid caused by the sinusoidal oscillation of a plane flat plate is termed Stokes's problem. When the plate starts from rest in a still fluid a transient solution must be added to Stokes's well-known steady-state result. This paper presents a closed-form expression for the transient solution. Previous answers have contained a non-standard integral which could not be evaluated. The answer presented herein contains exponentials and error functions of a complex argument. These functions are readily available in newer mathematical tables. Graphs of the transient solution are presented for both sin (T) and – cos (T) boundary conditions. Velocity distributions in the fluid are also plotted and it is found that the transient period is essentially complete in one-half cycle for the cosine oscillation and in a full cycle for the sine wave case.
Heat transfer from a sphere in a stream of small Reynolds number
Authors: Rimmer P. L.
Published: March 2006

The present paper deals with the temperature field past an isothermal sphere. Acrivos & Taylor (1962) have obtained results for the case R [les ] 1 with no restriction on the Prandtl number. The results of the present paper are for the case R < 1 in which the Prandtl number is of order unity. An expansion for the Nusselt number is given up to and including the term of order R2.
Calculations for anemometry with fine hot wires
Authors: Wood W. W.
Published: March 2006

The heat transfer appropriate to low Reynolds number hot-wire anemometry is calculated from the full non-linear equations of motion and of heat transfer by an iterative method starting with the Oseen solution and its heat flux analogue. The second and third iterates yield close agreement with measured data.
On high-frequency oscillatory viscous flows
Authors: Wang Chang-Yi
Published: March 2006

The equations governing high-frequency oscillatory viscous flows are investigated through the separation of the steady and the unsteady parts. All Reynolds number ranges are studied and the orders of magnitude of the steady streaming produced by the Reynolds stresses are established.

The oscillating circular cylinder at low Reynolds numbers is studied through the method of inner and outer expansions. Steady recirculating cells exist near the cylinder. The results compare very well with experiments. Analytic expressions for the streamfunction and the drag coefficient are obtained.

The oscillating flow towards an infinite plate is investigated in detail. The steady streaming is caused by the steady component of the Reynolds stress. The pressure gradient always causes reverse flow near the solid boundary.
The toroidal bubble
Authors: Pedley T. J.
Published: March 2006

It has been observed by Walters & Davidson (1963) that release of a mass of gas in water sometimes produces a rising toroidal bubble. This paper is concerned with the history of such a bubble, given that at the initial instant the motion is irrotational everywhere in the water. The variation of its overall radius a with time may be predicted from the vertical impulse equation, and it should be possible to make the same prediction by equating the rate of loss of combined kinetic and potential energy to the rate of viscous dissipation. This is indeed seen to be the case, but not before it is recognized that in a viscous fluid vorticity will continually diffuse out from the bubble surface, destroying the irrotationality of the motion, and necessitating an examination of the distribution of vorticity. The impulse equation takes the same form as in an inviscid fluid, but the energy equation is severely modified. Other results include an evaluation of the effect of a hydrostatic variation in bubble volume, and a prediction of the time which will have elapsed before the bubble becomes unstable under the action of surface tension.
Turbulent Rayleigh shear flow
Authors: Crow Steven
Published: March 2006

Townsend has derived a relation between mean vorticity and Reynolds stress valid in the wall layer of a turbulent flow. The vorticity Ω appears as a function of the local stress τ and its gradient. Such a relation is better suited for use in the vorticity equation than in the momentum equation. The Rayleigh problem, whose vorticity equation is simply ∂Ω/∂t = ∂2τ/∂y2, is introduced as a setting for Townsend's theory. Certain wall speed programmes are shown to generate Rayleigh layers that are exactly self-similar in the fully turbulent part of the flow. Those layers correspond to Clauser's equilibrium boundary layers. A formal analogy between the two families is found; the analogy becomes quantitatively exact in the limit of infinite Reynolds number. The Rayleigh problem is posed in similarity form. A composite non-linear, ordinary differential equation for the stress profile is deduced from a two-layer model incorporating Townsend's relation for the wall layer and Clauser's constant eddy-viscosity assumption for the outer layer. The profile depends on the wall-speed programme selected and on two empirical constants: the combination λ = √(k)/k of Clauser's k and Kármán's k, and Townsend's constant B. Closed-form solutions for arbitrary λ and B are obtained in two important cases: constant wall stress, analogous to constant pressure above a boundary layer, and zero wall stress, corresponding to continuous separation. The velocity profile in the wall region of a continuously separating Rayleigh layer is found to depend sensitively on B.
On the irregular refraction of a plane shock wave at a Mach number interface
Authors: Henderson L. F. , Macpherson A. K.
Published: March 2006

The paper considers the refraction of a plane shock wave at an interface between two streams of different Mach number. Particular attention is paid to the irregular wave systems. It is found that when the interface is slow-fast, that is when the speed of sound a0 in the first or incident wave medium is less than the speed of sound aB in the second or transmitted wave medium, then there are two irregular systems, one being a double Mach reflexion type and the other being a four-wave confluence type. There are also two irregular systems when the refraction is fast-slow; these are a single Mach reflexion type and an expansion wave type. This last system has a central expansion wave when the flow is steady and a continuous band expansion wave when the flow is self-similar. Only two of the irregular wave systems have been observed experimentally in the fully developed state. Possible degeneracies are discussed.
The spatial distribution of spheres falling in a viscous liquid
Authors: Smith T. N.
Published: March 2006

The spatial distribution of uniformly sized spheres falling in a viscous liquid is investigated experimentally for a solid volume fraction of 0·025 and a single sphere Reynolds number of 0·6. The observed spatial distribution agrees closely with a random distribution based on allocation of spheres to space cells according to a binomial probability mechanism.
Use of the Wiener—Hermite expansion for nearly normal turbulence
Authors: Meecham William C. , Jeng Dah-Teng
Published: March 2006

The turbulence problem is formulated using the Wiener stochastic expansion. The expansion is useful for processes which are in some sense nearly normal, and can be used for non-linear non-Gaussian processes such as many turbulent fluid flows. Here we present the general formulation for statistically inhomogeneous and anistropic processes.

The transfer term in the energy equation, or equivalently the third-order velocity correlation, forms a sensitive measure of the amount of non-Gaussianity present in real fluid flows. Experimental evidence shows that in many flows this component is small compared with the Gaussian part. It is shown that a homogeneous and isotropic flow which has but a small non-Gaussian part possesses a distribution at one point which is Gaussian to terms of second order. The experiments suggest that immediately behind a grid in a wind tunnel the flow is very nearly normal. The non-Gaussian part grows at a moderate rate, at least within the range of distance downstream (or decay time) available in the usual experiments. This growth is probably due to the relative increase in the amount of energy in the smallest eddies, which are non-normal.

A necessary criterion for the validity of the zero-fourth-cumulant approximation is suggested: the transfer term in dimensionless form should be small. It is shown that calculations using the zero-fourth-cumulant approximation have given negative energy spectra when this condition is violated, probably for the reason that the process is no longer nearly Gaussian. However, even when this condition is fulfilled, it is shown that that approximation must be corrected.

It is suggested that the present theory is valid for quite large times of decay if initial energy spectra are chosen which are not too far from the actual physical values for fluid in turbulent flow. Equations are given for the next-higher-order term in a nearly normal approximation. The expansion is also used in § 6 to describe turbulent mixing problems and is compared with the zero-fourth-cumulant approximation for these problems. Computational results are presented in § 7 and compared with experiments by Stewart and Townsend.
On slow transverse flow past obstacles in a rapidly rotating fluid
Authors: Hide R. , Ibbetson A. , Lighthill M. J.
Published: March 2006

This paper describes an experimental and theoretical study of the complicated disturbance (Taylor column) due to the slow relative motion between a spherical, or short cylindrical, rigid object and an incompressible fluid of low viscosity in which the object is immersed, when the motion of the object is that of steady revolution with angular speed Ω rad/see about an axis (the Z-axis) whose perpendicular distance from the centre of the object,$\overline{R}$, is much greater than a typical linear dimension of the object,L, and the undisturbed fluid motion is one of steady rotation about the same axis with angular speed (Ω+Uϑ/R) and zero relative vorticity (i.e.d(UϑR)/dR= 0). It extends earlier experimental work on Taylor columns to systems of sufficiently large axial dimensions for Z variations in the disturbance pattern to be perceptible. Over the ranges of Rossby and Ekman numbers (based onL) covered by the experiments, namely ε = 1·89 × 10−3to 2·36 ×10−1and γ = 1·30 × 10−3to 2·03 × 10−2respectively, the axis of the Taylor column is found to trail in the downstream direction at a small angle ϕ = tan−1(Kε) to the line parallel to the Z-axis through the centre of the object, whereK= (1·54 ± 0·04) for a sphere. The variation with Z of the amplitude of the disturbance is roughly linear and the scale-length of this variation, Zc, is close toL/γ¼over the limited range of γ covered by the experiments.

The experimental value ofKis remarkably close to the theoretical value derived by Prof. Lighthill in the appendix, where he applies his general linear theory of waves generated in a dispersive system by travelling forcing effects to the problem of describing a Taylor column at large distances from the moving object when the fluid is inviscid and unbounded.
On the laminar flow in a free jet of liquid at high Reynolds numbers
Authors: Tillett J. P. K.
Published: March 2006

This paper is concerned with the jet of liquid, open to the atmosphere, that emerges from a two-dimensional channel in which there is Poiseuille flow far upstream, the flow being driven by an applied pressure gradient. The problem is discussed with the aid of the method of matched asymptotic expansions; the small parameter involved is the inverse Reynolds number. A boundary layer forms adjacent to the free surface, and a classical boundary-layer analysis is applied to find the flow there (for moderate distances downstream); the influence of this boundary layer on the flow in the core of the jet is then investigated. Higher-order boundary-layer effects, such as indeterminacy and eigensolutions, are also discussed. The first few terms are found of an asymptotic expansion for the equation of the free surface, and considerations of momentum balance are applied to find the asymptotic contraction ratio of the jet.