bluuebunny/crossref_metadata_2025 huggingface dataset:
Random convection
Authors: Newell Alan C. , Lange C. G. , Aucoin P. J.
Published: March 2006

The main thrust of this work is to treat the initial convective phase of a fluid heated from below as a statistical initial value problem. The advantage of the approach is that it allows a continuous bandwidth of modes to be represented in the initial spectrum. We show that if the initial disturbance field is small and has a sufficiently smooth spectrum, then a natural statistical selection process chooses from the initial disorder a perfectly ordered field of single rolls. The scale of this roll is the scale corresponding to the most critical wave-number obtained from the linear stability problem. We relate this solution to the optimal solution which would be obtained by the upper bound procedures of Howard, Malkus and Busse. Moreover, we show in addition, that if the initial disturbance field is weighted in favour of a particular single roll whose scale is close to critical, the final solution reflects the initial condition providing a certain stability criterion is met. In the two-dimensional case we analyze, this turns out to be the Eckhaus stability condition previously obtained by a discrete multimodal analysis.
A wake source model for bluff body potential flow
Authors: Parkinson G. V. , Jandali T.
Published: March 2006

A theory is presented for two-dimensional incompressible potential flow external to a symmetrical bluff body and its wake. The desired flow-separation points are made the critical points of a conformal transformation to a complex plane in which surface sources in the wake create stagnation conditions at the critical points. The stagnation streamlines then transform to tangential separation streamlines in the physical plane, with separation at the desired pressure. The position and strength of the sources are determined by the requirements of separation position and pressure coefficient. The flow inside the separation streamlines is ignored and base pressure is assumed constant at the separation value. Features of the theoretical model include a finite wake width, a pressure distribution on the separation streamlines decreasing asymptotically towards the free stream value at infinity and a simple analytic expression for the pressure distribution on the body. Comparisons of the theory with experimental data and with other theories are presented for the normal plate, the circular cylinder, the 90° wedge, and the elliptical cylinder. Although simpler to apply than the other theories, the present theory produces at least as good agreement with the experimental data.
Inertial oscillations in a rotating fluid cylinder
Authors: McEwan A. D.
Published: March 2006

A study is described of the forced inertial oscillations appearing in an axially rotating completely filled circular cylinder with plane ends. Excitation is provided by causing the top end to rotate about an axis inclined slightly to the rotation axis. Experiments demonstrate the presence of numerous low mode resonances in a densely spaced range of ratios of net cylinder height to radius in close conformance with linear inviscid theory. Where geometry permits simple corner reflexion, characteristic surfaces are revealed which confirm in part the theoretical predictions concerning their scale and form.

Detailed measurements are given of the amplitude at one point within the cylinder for the condition in which the disturbance frequency equals the rotation frequency. Amplitude column height spectra are compared with theoretical estimates, and the evolution of amplitude for the simplest mode of resonant oscillation is studied. A non-linear theory based on the integral energy of large amplitude oscillation is derived whose broad features are in fair quantitative and qualitative agreement with these observations.

Some investigation is made of the phenomenon ofresonant collapse, in which larger amplitude resonant oscillations, after persisting in an apparently laminar form, degenerate abruptly into a state of agitation and disorder from which they do not recover. It is found that the time for emergence of this collapse after the introduction of the forcing disturbance has a close correspondence with the theoretical period of one ‘evolutionary’ cycle of momentum exchange between the main motion and the secondary oscillation.
Wall-pressure fluctuations beneath turbulent boundary layers on a flat plate and a cylinder
Authors: Willmarth W. W. , Yang C. S.
Published: March 2006

Measurements of the turbulent pressure field on the outer surface of a 3 in. diameter cylinder aligned with the flow were made at a point approximately 24 ft. downstream of the origin of the turbulent boundary layer in an air stream of 145 ft./sec. The boundary-layer thickness was 2·78 in. and the Reynolds number based on momentum thickness was 2·62 × 104. The wall-pressure measurements were made with pressure transducers constructed from 0·06 in. diameter lead–zirconate–titanate disks mounted flush with the wall. The measurements including root-mean-square, power spectrum, and correlations of the wall pressure are compared with the existing experimental results for the turbulent pressure field beneath a plane boundary layer. The streamwise convection speed deduced from longitudinal space-time correlation measurements was almost identical to that obtained in the plane boundary layer. The rate of decay of the maxima of the space-time correlation of the pressure produced by the convected eddies was double that in a plane boundary layer. The longitudinal and transverse scales of the pressure correlation were approximately equal (in a plane boundary layer the transverse scale is larger than longitudinal scale) and were one-half or less than the longitudinal scale in the plane boundary layer. It is concluded that the effect of the transverse curvature of the wall is an overall reduction in size of pressure-producing eddies. The reduction in transverse scale of the larger eddies is greater than that of the smaller eddies. In general, the smaller eddies decay more rapidly and produce greater spectral densities at high frequencies owing to the unchanged convection speed.
Bounds for turbulent shear flow
Authors: Busse F. H.
Published: March 2006

Bounds on the transport of momentum in turbulent shear flow are derived by variational methods. In particular, variational problems for the turbulent regimes of plane Couette flow, channel flow, and pipe flow are considered. The Euler equations resemble the basic Navier–Stokes equations of motion in many respects and may serve as model equations for turbulence. Moreover, the comparison of the upper bound with the experimental values of turbulent momentum transport shows a rather close similarity. The same fact holds with respect to other properties when the observed turbulent flow is compared with the structure of the extremalizing solution of the variational problem. It is suggested that the instability of the sublayer adjacent to the walls is responsible for the tendency of the physically realized turbulent flow to approach the properties of the extremalizing vector field.
The turbulent boundary layer over a wall with progressive surface waves
Authors: Kendall James M.
Published: March 2006

An experimental study of the interaction of a turbulent boundary layer with a wavy wall was conducted in a wind tunnel. A smooth neoprene rubber sheet comprising a portion of the floor of the tunnel was mechanically deformed into 12 sinusoidal waves which progressed upwind or down at controlled speed. The turbulent layer thickness was a little less than the wavelength. The mean velocity profile was linear on a semi-log plot over a substantial range of vertical height.

The wall pressure was observed to be asymmetrical about the wave profile, resulting in a pressure drag. Flow separation was not the cause of the drag. The drag was found to be larger than that predicted by the inviscid wave generation theory. The measurements indicate that the waves strongly modulate the turbulent structure. The phase of the turbulent stresses with respect to the waves varies with wave speed, indicating that the dynamical reaction time of the turbulence is not negligible in comparison with the wave period.
The two-dimensional mixing region
Authors: Wygnanski I. , Fiedler H. E.
Published: March 2006

The two-dimensional incompressible mixing layer was investigated by using constant-temperature, linearized hot wire anemometers. The measurements were divided into three categories: (1) the conventional average measurements; (2) time-average measurements in the turbulent and the non-turbulent zones; (3) ensemble average measurements conditioned to a specific location of the interface. The turbulent energy balance was constructed twice, once using the conventional results and again using the turbulent zone results. Some differences emerged between the two sets of results. It appears that the mixing region can be divided into two regions, one on the high velocity side which resembles the outer part of a wake and the other on the low velocity side which resembles a jet. The binding turbulent–non-turbulent interfaces seem to move independently of each other. There is a strong connexion between the instantaneous location of the interface and the axial velocity profile. Indeed the well known exponential mean velocity profile never actually exists at any given instant. In spite of the complexity of the flow the simple concepts of eddy viscosity and eddy diffusivity appear to be valid within the turbulent zone.
Analytical theories of turbulence
Authors: Orszag Steven A.
Published: March 2006

This paper surveys the current state of analytical attempts at a theory of turbulence. The formulation of the problem in terms of moments is discussed. The difficulty posed by the closure problem is examined in detail using the quasinormal approximation as an example. The notion of dynamical relaxation by non-linear scrambling leads to the introduction of eddy relaxation times and the direct-interaction approximation. The properties of the direct-interaction approximation are indicated. Finally, a comparison is made between numerical solution of the equations of turbulence their and direct numerical simulation of the Navier–Stokes equations.
Relationship between a Wiener–Hermite expansion and an energy cascade
Authors: Crow S. C. , Canavan G. H.
Published: March 2006

Meecham and his co-workers have developed a theory of turbulence involving a truncated Wiener–Hermite expansion of the velocity field. The randomness is taken up by a white-noise function associated, in the original version of the theory, with the initial state of the flow. The mechanical problem then reduces to a set of coupled integro-differential equations for deterministic kernels. We have solved numerically an analogous set for Burgers's model equation and have computed, for the sake of comparison, actual random solutions of the Burgers equation. We find that the theory based on the first two terms of the Wiener–Hermite expansion predicts an insufficient rate of energy decay for Reynolds numbers larger than two, because the equations for the kernels contain no convolution integrals in wave-number space and therefore permit no cascade of energy. An energy cascade in wave-number space corresponds to a cascade up through successive terms of the Wiener-Hermite expansion. Pictures of the Gaussian and non-Gaussian components of an actual solution of the Burgers equation show directly that only higher-order terms in the Wiener–Hermite expansion are capable of representing shocks, which dissipate the energy. Higher-order terms would be needed even for a nearly Gaussian field of evolving three-dimensional turbulence. ‘Gaussianity’, in the experimentalist's sense, has no bearing on the rate of convergence of a Wiener–Hermite expansion whose white-noise function is associated with the initial state. Such an expansion would converge only if the velocity field and its initial state were joint-normally distributed. The question whether a time-varying white-noise function can speed the convergence is treated in the paper following this one.
Some properties of a Lagrangian Wiener–Hermite expansion
Authors: Canavan Gregory H.
Published: March 2006

Wiener proposed that the turbulent velocity be expanded in Hermite functionals of a Gaussian white noise random function advected by the fluid. This paper describes the mechanics of converting his suggestion into a computable model, and assesses its range of validity as an approximation for incompressible, homogeneous, and isotropic turbulence. The terms retained are a linear term, v1, representing an arbitrary Gaussian velocity, and a quadratic term, v2, representing a non-Gaussian contribution to the velocity needed for energy transfer. The requirement that advection by the dependent velocity v = v1 + v2 does not alter the statistics of the base necessitates a further truncation of the base to antisymmetric quadratic basis elements. Realizability of any statistics of v is common to all Wiener–Hermite expansions. The projected equations for the Lagrangian expansion conserve energy by non-linear interaction, preserve the inviscid Gaussian equipartition ensemble, and are invariant to random Galilean transformations. Numerical calculations with an approximate form of these equations reveal that irreversible relaxation to the inviscid equipartition solution is not a property of the Lagrangian model, and that the rapid convergence advanced as the original motivation for studying Wiener–Hermite expansions does not survive closure by truncation. The dynamics of the model is not inconsistent with the existence of an inertial range. A simple numerical search routine failed to produce a solution corresponding to such an equilibrium ensemble.
Toward a turbulent constitutive relation
Authors: Lumley J. L.
Published: March 2006

In an attempt to explain the failure of the various pure homogeneous strain experiments to reach equilibrium (and consequently to support the contention of Townsend of an equilibrium structure of the Reynolds stress dependent only on geometry), the nature of the general Reynolds stress-mean velocity relation is examined. It is shown that if homogeneous flows become asymptotically independent of initial conditions, and if the Reynolds stress bearing structure can be characterized by a single time scale (i.e.–at sufficiently high Reynolds number) then these flows behave like classical non-linear viscoelastic media, with the Reynolds stress structure dependent on the (strain-rate) (time scale) product. Thus, the existence of an equilibrium structure implies the existence of an equilibrium time scale and a universal value of the product. The ideas permitting Reynolds stress and mean velocity to be related are applied to the dissipative structure in homogeneous flows, and it is found that in such flow the time scale never ceases to grow, so that these flows can never reach an equilibrium structure. With the aid of an ad-hoc assumption these flows are examined in some detail, and the results of experiments are predicted with considerable accuracy. It is suggested that (inhomogeneous) flows having an equilibrium time scale may, in the homogeneous limit, be expected to display a universal structure. The small departure from universality induced by the large eddies associated with inhomogeneity may be adequately predicted by this same ad-hoc model.
A numerical study of three-dimensional turbulent channel flow at large Reynolds numbers
Authors: Deardorff James W.
Published: March 2006

The three-dimensional, primitive equations of motion have been integrated numerically in time for the case of turbulent, plane Poiseuille flow at very large Reynolds numbers. A total of 6720 uniform grid intervals were used, with sub-grid scale effects simulated with eddy coefficients proportional to the local velocity deformation. The agreement of calculated statistics against those measured by Laufer ranges from good to marginal. The eddy shapes are examined, and only theu-component, longitudinal eddies are found to be elongated in the downstream direction. However, the lateralveddies have distinct downstream tilts. The turbulence energy balance is examined, including the separate effects of vertical diffusion of pressure and local kinetic energy.

It is concluded that the numerical approach to the problem of turbulence at large Reynolds numbers is already profitable, with increased accuracy to be expected with modest increase of numerical resolution.
An investigation of the stability of a magnetohydrodynamic shear layer
Authors: Malcolm D. G.
Published: March 2006

This paper presents an experimental investigation of the conditions of instability and post-critical flow in an electrically driven shear layer which has been previously studied both theoretically and experimentally. The shear layer forms as a primarily azimuthal flow in an electrically conducting fluid between the edges of two parallel and coaxial circular electrodes, mounted in insulating planes, when a current is passed between them in the presence of a strong, axially applied magnetic field. The aximuthal velocity at any radial position in the layer is maximum near the electrode planes and decreases to zero at the centreplane between the electrodes. Radial and axial secondary velocity components apparently exist, though smaller in magnitude than the azimuthal velocity throughout most of the layer.

The experiments employ a miniature hot-film sensor as an instantaneous velocity indicator in mercury. The following experimental results are reported.
\begin{eqnarray*}
& R_c = 2420,\quad 150 \leqslant M \leqslant 350,\\
& R_c = 345\,M^{\frac{1}{3}},\quad 350\leqslant M \leqslant 650,
\end{eqnarray*}
where Rc is the critical Reynolds number of the shear layer, based upon the maximum velocity in the layer and the layer thickness, and M is the Hartmann number based upon the distance from the electrodes to the centreplane. For values of M below about 300, oscillograms of the hot-film signal show that the layer goes unstable in a chaotic fashion. For M ≥ 300, however, the instabilities grow to produce a wave-like post-critical flow of steady amplitude. The waves are apparently parallel to the applied magnetic field and drift in the azimuthal direction at a constant proportion (10–15%) of the critical velocity. The maximum intensity of the peak to peak fluctuations from the measured mean azimuthal velocity appears to be about 0·05. These fluctuations decrease to zero at the centreplane. By an approximate measurement of wavelength it is determined that the integral number of waves in the post-critical flow increases with M.

The physical mechanism governing the onset of instability has not been established. The manner in which hydrodynamic and electromagnetic forces interact with the destabilizing disturbances cannot be fully understood without further theoretical analysis.
The stress system in a suspension of force-free particles
Authors: Batchelor G. K.
Published: March 2006

The purpose of the paper is to consider in general terms the properties of the bulk stress in a suspension of non-spherical particles, on which a couple (but no force) may be imposed by external means, immersed in a Newtonian fluid. The stress is sought in terms of the instantaneous particle orientations, and the problem of determining these orientations from the history of the motion is not considered. The bulk stress and bulk velocity gradient in the suspension are defined as averages over an ensemble of realizations, these averages being equal to integrals over a suitably chosen volume of ambient fluid and particles together when the suspension is statistically homogeneous. Without restriction on the type of particle or the concentration or the Reynolds number of the motion, the contribution to the bulk stress due to the presence of the particles is expressed in terms of integrals involving the stress and velocity over the surfaces of particles together with volume integrals not involving the stress. The antisymmetric part of this bulk stress is equal to half the total couple imposed on the particles per unit volume of the suspension. When the Reynolds number of the relative motion near one particle is small, a suspension of couple-free particles of constant shape is quasi-Newtonian; i.e. the dependence of the bulk stress on bulk velocity gradient is linear. Two significant features of a suspension of non-spherical particles are (1) that this linear relation is not of the Newtonian form and (2) that the effect of exerting a couple on the particles is not confined to the generation of an antisymmetrical part of the bulk stress tensor. The role of surface tension at the particle boundaries is described.

In the case of a dilute suspension the contributions to the bulk stress from the various particles are independent, and the contributions arising from the bulk rate of strain and from the imposed couple are independent for each particle. Each particle acts effectively as a force doublet (i.e. equal and opposite adjoining ‘Stokeslets’) whose tensor strength determines the disturbance flow far from the particle and whose symmetrical and antisymmetrical parts are designated as a stresslet and a couplet. The couplet strength is determined wholly by the externally imposed couple on the particle; but the stresslet strength depends both on the bulk rate of strain and, for a non-spherical particle, on the rate of rotation of the particle relative to the fluid resulting from the imposed couple. The general properties of the stress system in a dilute suspension are illustrated by the specific and complete results which may be obtained for rigid ellipsoidal particles by use of the work by Jeffery (1922).
Electrohydrodynamic instability in plane layers of fluid
Authors: Michael D. H. , O'Neill M. E.
Published: March 2006

The instability of a plane layer of non-conducting fluid which is in hydrostatic equilibrium between two semi-infinite conducting fluids with surface charges is discussed for both inviscid and viscous fluid models. It is shown that for both the inviscid and viscous fluid cases, the criteria for instability are the same. Consideration is given to the relevance of the results in explaining the mechanism by which the presence of an electric field promotes more readily the coalescence of water droplets on a water surface by viewing the onset of disruption of the air film as the instability of the air film under the action of the electrostatic field produced by the surface charges on the water surfaces.
The Cameron—Martin—Wiener method in turbulence and in Burgers’ model: general formulae, and application to late decay
Authors: Kahng W-H. , Siegel A.
Published: March 2006

We apply the Cameron—Martin—Wiener (formerly ‘Wiener—Hermite’) expansion of a random velocity field to the analytical study of turbulence. The kernels of this expansion contain all statistical information about the ensemble. Complete expressions are derived for constructing statistical quantities in terms of the kernels, and for the equations of motion of the kernels. We rigorously prove the Gaussian trend of the velocity field of the Navier—Stokes equation in the very late stage when the non-linear term is neglected. Then-dependence (nis the order of derivative) of the flatness factor, minus three for derivatives of the velocity field, shows a rapid increase withnin this stage.

The late decay problem of the Burgers model of turbulence is studied analytically with a view to obtaining suggestive guidelines for fitting the non-linear aspects of the model turbulence. We can divide the energy spectrum density into two parts, the larger of which is a kind of steady solution, which we call the ‘equilibrium state’, which remains self-similar in time in terms of an appropriate variable. The deviation from this ‘equilibrium solution’ satisfies the Kármán—Howarth equation. As initial velocity field, we take two particular cases: (a) a pure Gaussian, and (b) a non-Gaussian velocity field. With these two cases a detailed spectral analysis has been obtained. The energy spectrum deviation from ‘equilibrium’ declines exponentially to zero for all wave-numbers. The Gaussian case shows that the flatness factor minus three increases rapidly withn, while the non-Gaussian case does not show any marked dependence onn.
Visualization of heaving aerofoil wakes including the effect of a jet flap
Authors: Wood C. J. , Kirmani S. F. A.
Published: March 2006

Theoretical solutions exist for the flow about aerofoils in various modes of unsteady motion. This is also true for aerofoils whose lift is augmented by a jet flap. A common feature of these theories is the assumption of small amplitude displacements from the mean path and the generation of a continuous vortex sheet wake. However, in practice when the motion is oscillatory and of finite amplitude, an alternating trail of individual vortices is often found. The present visual investigation includes the measurement of some vortex street parameters associated with heaving aerofoils. The results are compared with those of conventional thin aerofoil theory in order to assess the significance of this departure from the conventional small displacement flow model.
Unsteady flow of a viscous fluid from a source in a wall
Authors: Tuck E. O.
Published: March 2006

A problem with possible physiological applications concerns the escape of a viscous fluid through a small hole in a wall. The solution presented here is for a line source of sinusoidally pulsating strength located at the originx=y= 0, where the planey= 0 is a rigid wall and the fluid is at rest aty= + ∞. The linearized Navier—Stokes equations are solved, and results in the form of streamline plots are shown and discussed.
Gravity wave refraction by islands
Authors: Lautenbacher Conrad C.
Published: March 2006

The refractive influence on tsunami run-up of the offshore bottom topography of islands is analyzed. Shallow water theory is used to treat problems in which the geometry resembles that of individual Hawaiian islands and in which the incident wave is plane and monochromatic.

Mathematically, the differential equation for long-wave propagation is converted into an integral equation to which numerical methods are applied. Results of practical importance include the run-up on island coastal areas. The results are used in conjunction with earlier one-dimensional analyses to estimate the total tsunami run-up.
An unsteady two-cell vortex solution of the Navier—Stokes equations
Authors: Bellamy-Knights P. G.
Published: March 2006

The steady two-cell viscous vortex solution of Sullivan (1959) is extended to yield unsteady two-cell viscous vortex solutions which behave asymptotically as certain analogous unsteady one-cell solutions of Rott (1958). The radial flux is a parameter of the solution, and the effect of the radial flow on the circumferential velocity, is analyzed. The work suggests an explanation for the eventual dissipation of meteorological flow systems such as tornadoes.
Note on slow rotation or rotary oscillation of axisymmetric bodies in hydrodynamics and magnetohydrodynamics
Authors: Kanwal R. P.
Published: March 2006

Results for three interrelated problems are obtained by making use of solutions of boundary-value problems obtained in a different context. The first one concerns a thin rigid circular disk rotating in a slow stream of viscous fluid, both when the fluid is conducting and when it is non-conducting. For the case of a conducting fluid formulae are given for both small and large Hartmann numbers. The second problem concerns a disk performing simple harmonic rotary oscillations about its axis of symmetry in a non-conducting viscous fluid which is at rest at infinity. The last problem is that of an arbitrary axisymmetric solid oscillating about its axis of symmetry in a bounded viscous fluid, and the solution is illustrated by the case of an oscillating disk.
On bubbles rising in line at large Reynolds numbers
Authors: Harper J. F.
Published: March 2006

Conditions for two gas bubbles in a liquid to rise steadily in a vertical line are derived theoretically with these assumptions: large Reynolds number, no surface contamination, spherical shape, negligible gas density and viscosity. Drag coefficients are found, and are lower than for single bubbles. The bubbles have equilibrium distances apart, which are calculated to a first approximation. The equilibrium is shown to be stable to small vertical disturbances but unstable to horizontal ones. Similar results exist for lines of more than two bubbles, but are not calculated in detail.
Two-phase convex-type corner flows
Authors: Healy J. Val.
Published: March 2006

The flow of an inviscid incompressible fluid, with imbedded identical spherical particles, around an arbitrary corner is treated by the method of small perturbations. Application is made to convex flows only and the approximate effects of separation are also considered. The assumption of arbitrary initial particle density k0 leads to a complex system of equations, which appears to have no simple solution in general, and is not considered. For small k0 the particle density distribution is governed by a first-order partial differential equation which, when solved by the method of characteristics, yields ordinary differential equations, whose solutions are simple and analytic for unseparated flow and numerical only when separation is considered. In the former flow, spiral type curves partially encircling the plate tip, and then trailing downstream, delineate the particle-free zones, and it is found that the particle density increases monotonically in the downstream direction on all particle streamlines. In the separated flow, the most noteworthy effect is the disappearance of the infinite velocity at the origin and the consequent considerable reduction in the magnitude of the perturbation.
A wave-interaction model for the generation of windrows
Authors: Craik Alex D. D.
Published: March 2006

Interactions of suitable pairs of gravity waves in a shear flow are found to give rise to aperiodic or weakly periodic secondary motions. These secondary flows resemble the ‘Langmuir vortices’ which are associated with the formation of windrows. It seems likely that such wave interactions will play a substantial part in determining the quasi-steady structure of the flow when wind blows over a water surface.
On cross-waves
Authors: Garrett C. J. R.
Published: March 2006

Cross-waves are standing waves with crests at right angles to a wave-maker. They generally have half the frequency of the wave-maker and reach a steady state at some finite amplitude. A second-order theory of the modes of oscillation of water in a tank with a free surface and wave-makers at each end leads to a form of Mathieu's equation for the amplitude of the cross-waves, which are thus an example of parametric resonance and may be excited at half the wave-maker frequency if this is within a narrow band. The excitation depends on the amplitude of the wave-maker at the surface and the integral over depth of its amplitude. Cross-waves may be excited even if the mean free surface is stationary. The effects of finite amplitude are that the cross-waves approach a steady state such that a given amplitude is achieved at a frequency greater than that for free waves by an amount proportional to the amplitude of the wave-maker. The theory agrees reasonably well with the experimental results of Lin & Howard (1960). The amplification of the cross-waves may be understood in terms of the rate of working of the wave-maker against transverse stresses associated with the cross-waves, one located at the surface and the other equal to Miche's (1944) depth-independent second-order pressure. The theory applies to the situation where the primary motion consists of standing waves and the cross-waves are constant in amplitude away from the wave-maker, but certain generalizations may be made to the situation where the primary waves are progressive and the cross-waves decay away from the wave-maker.
The turbulent boundary layer on a rotating cylinder in an axial stream
Authors: Cham T-S. , Head M. R.
Published: March 2006

Calculations are presented of the development of the turbulent boundary layer on a rotating cylinder with its axis along the stream, and the results are compared with measurements made by Parr. With the choice of a suitable co-ordinate system, the boundary layer downstream of the nose of the cylinder approximates closely to a condition of two-dimensionality, and a simple integral method of solution can be applied. The only evidence of three-dimensionality lies in the destabilizing effect of rotation on the turbulence structure of the layer and an analysis of this instability has been made which relates changes in mixing length and entrainment to an instability parameter in the form of a Richardson number.

In spite of the changes in shear stress distribution and entrainment brought about by rotation, mean velocity profiles and skin friction values are found to be related to H and Rθ in the same way as for the stationary flat plate.
Absolute measurements of static-hole error using flush transducers
Authors: Franklin R.E. , Wallace James M.
Published: March 2006

Experiments are reported in which the error of a sharp-edged, circular static pressure hole normal to the boundary of a moving fluid is deduced from measurements made with pressure transducers, the diaphragms of which were set flush with the boundary surface. An error curve is presented which covers the range of hole Reynolds number up to a value of 2000.
The oseenlet as a model for separated flow in a rotating viscous liquid
Authors: Miles John W.
Published: March 2006

The disturbance induced in the uniform flow of a viscous, rotating liquid by an axial point force −D is studied under the restrictions that the Ekman number, E = 2Ων/U2, be small and that D = O(1/logE) as E → 0. The method of matched asymptotic expansions is invoked to obtain inner and outer (with reference to the dimensionless axial co-ordinate x refered to the length U/(2Ω)) approximations to the solution of the Oseen equations as E → 0. The outer approximation, E → 0 with Ex fixed, is also an outer approximation to the solution of the Navier–Stokes equations. The mass flow across any transverse plane, which is equal to D/U for an oseenlet in a non-rotating flow, vanishes in this approximation. The corresponding inner limit yields a non-uniform, cylindrical flow far upstream of the force in the inviscid limit, E → 0, if and only if D ∝ 1/(log E + const.). This cylindrical flow is a one-term, inner approximation to the solution of the Navier–Stokes equations and suffices to show that separation implies the failure of Long's hypothesis of no upstream influence for inviscid, rotating flow past a finite body. A two-term inner representation of the solution is related to Stewartson's solution of the Oseen equations for a moving source in an inviscid, rotating fluid.
Instability of erodible beds
Authors: Engelund Frank
Published: March 2006

The stability of a sand bed in an alluvial channel is investigated by a two-dimensional mathematical model, based on the vorticity transport equation. The model takes account of the internal friction and describes the non-uniform distribution of the suspended sediment. It turns out that the inclusion of the friction and of a definite model of the sediment transport mechanism leads to results rather different from those obtained previously by potential-flow analysis.
The effect of trapped field on stationary waves formed at the interface of a conduction fluid and a magnetic field
Authors: Savage M. D.
Published: March 2006

A two-dimensional, moving, incompressible and electrically conducting fluid having trapped magnetic field (i.e. field threading the fluid) is confined by a uniform vacuum magnetic field. The fluid-vacuum field interface is horizontal and assumed to be free from instabilities. Dissipation by viscosity and resistivity is neglected, hence, for steady motion, the trapped field may be regarded as frozen into the fluid, in the sense that both move together. A pressure disturbance is introduced into the system, and it is found that a harmonic stationary wave is formed upstream provided certain criteria hold.
On the transition to turbulent convection. Part 1. The transition from two- to three-dimensional flow
Authors: Krishnamurti Ruby
Published: March 2006

In a horizontal convecting layer several distinct transitions occur before the flow becomes turbulent. These are studied experimentally for several Prandtl numbers from 1 to 104. Cell size plan form, transitions in plan form, transition to time-dependence, as well as the heat flux, are measured for Rayleigh numbers from 103to 105. The second transition, occurring at around 12 times the critical Rayleigh number, is one from steady two-dimensional rolls to a steady regular cellular pattern. There is associated with this a discrete change of slope of the heat flux curve, coinciding with the second transition observed by Malkus. Transitions to time-dependence will be discussed in part 2.
A theory of turbulence in stratified fluids
Authors: Long Robert R.
Published: March 2006

An effort is made to understand turbulence in fluid systems like the oceans and atmosphere in which the Richardson number is generally large. Toward this end, a theory is developed for turbulent flow over a flat plate which is moved and cooled in such a way as to produce constant vertical fluxes of momentum and heat. The theory indicates that in a co-ordinate system fixed in the plate the mean velocity increases linearly with heightzabove a turbulent boundary layer and the mean density decreases asz3, so that the Richardson number is large far from the plate. Near the plate, the results reduce to those of Monin & Obukhov.

Thecurvatureof the density profile is essential in the formulation of the theory. When the curvature is negative, a volume of fluid, thoroughly mixed by turbulence, will tend to flatten out at a new level well above the original centre of mass, thereby transporting heat downward. When the curvature is positive a mixed volume of fluid will tend to fall a similar distance, again transporting heat downward. A well-mixed volume of fluid will also tend to rise when the density profile is linear, but this rise is negligible on the basis of the Boussinesq approximation. The interchange of fluid of different, mean horizontal speeds in the formation of the turbulent patch transfers momentum. As the mixing in the patch destroys the mean velocity shear locally, kinetic energy is transferred from mean motion to disturbed motion. The turbulence can arise in spite of the high Richardson number because the precise variations of mean density and mean velocity mentioned above permit wave energy to propagate from the turbulent boundary layer to the whole region above the plate. At the levels of reflexion, where the amplitudes become large, wave-breaking and turbulence will tend to develop.

The relationship between the curvature of the density profile and the transfer of heat suggests that the density gradient near the level of a point of inflexion of the density curve (in general cases of stratified, shearing flow) will increase locally as time goes on. There will also be a tendency to increase the shear through the action of local wave stresses. If this results in a progressive reduction in Richardson number, an ultimate outbreak of Kelvin–Helmholtz instability will occur. The resulting sporadic turbulence will transfer heat (and momentum) through the level of the inflexion point. This mechanism for the appearance of regions of low Richardson number is offered as a possible explanation for the formation of the surfaces of strong density and velocity differences observed in the oceans and atmosphere, and for the turbulence that appears on these surfaces.
Shallow water waves on shear flows
Authors: Freeman N. C. , Johnson R. S.
Published: March 2006

An equation for waves on the surface of a flow with shear is deduced and shown to reduce by suitable scaling to the classical equation of Korteweg & de Vries, which describes such motions on a stationary flow. For steady flows the corresponding theory of cnoidal waves is obtained and the results of Benjamin (1962) for a solitary wave recovered.
Asymptotic theory of turbulent shear flows
Authors: Yajnik Kirit S.
Published: March 2006

A theory is proposed in this paper to describe the behaviour of a class of turbulent shear flows as the Reynolds number approaches infinity. A detailed analysis is given for simple representative members of this class, such as fully developed channel and pipe flows and two-dimensional turbulent boundary layers. The theory considers an underdetermined system of equations and depends critically on the idea that these flows consist of two rather different types of regions. The method of matched asymptotic expansions is employed together with asymptotic hypotheses describing the order of various terms in the equations of mean motion and turbulent kinetic energy. As these hypotheses are not closure hypotheses, they do not impose any functional relationship between quantities determined by the mean velocity field and those determined by the Reynolds stress field. The theory leads to asymptotic laws corresponding to the law of the wall, the logarithmic law, the velocity defect law, and the law of the wake.
On the oscillations of harbours of arbitrary shape
Authors: Hwang Li-San , Tuck Ernest O.
Published: March 2006

A theory is developed for calculating oscillations of harbours of constant depth and arbitrary shape. This theory is based on the solution of a singular integral equation. Numerical results have been calculated for rectangular harbours so as to check the accuracy of the method. Examples for wave amplification factor and velocity field for both rectangular and actual complex-shaped harbours are given.
On the instability of natural convection flow on inclined plates
Authors: Lloyd J. R. , Sparrow E. M.
Published: March 2006

Experiments are carried out to establish the relationship between the nature of the flow instability and the inclination angle of the plate. The angular dependence of the Rayleigh number characterizing the onset of instability is also determined. An electrochemical flow visualization technique is utilized to expose the patterns of fluid motion. It is found that for inclination angles of less than 14° (relative to the vertical), waves are the mode of instability. On the other hand, for inclination angles in excess of 17°, the instability is characterized by longitudinal vortices. The range between 14° and 17° is a zone of continuous transition, with the two modes of instability co-existing.
Extensions of extremum principles for slow viscous flows
Authors: Skalak Richard
Published: March 2006

Several generalizations of theorems of the types originally stated by Helmholtz concerning the dissipation of energy in slow viscous flow have been given recently by Keller, Rubenfeld & Molyneux (1967). These generalizations included cases in which the fluid contains one or more solid bodies and drops of another liquid assuming the drops do not change shape. Some further extensions are given herein which allow for drops which may be deformed by the flow and include the effect of surface tension. The admissible boundary conditions have also been extended and particular theorems applicable to infinite domains, spatially periodic flows and to flows in infinite cylindrical pipes are derived. Uniqueness theorems are also proved.
The flow of a tubular film Part 2. Interpretation of the model and discussion of solutions
Authors: Pearson J. R. A. , Petrie C. J. S.
Published: March 2006

The equations governing the free-surface flow of a tubular film of liquid are derived from physical arguments, which throw some light on the formal process described in part 1. The solutions of the equations are discussed, in particular with reference to the film-blowing process for the manufacture of thin sheets of thermoplastic material. The qualitative adequacy of a model based on the dominance of viscous forces is demonstrated, and the effect of surface tension, air drag and non-isothermal flow is discussed briefly.
The solitary wave in water of variable depth
Authors: Grimshaw R.
Published: March 2006

Equations are derived for two-dimensional long waves of small, but finite, amplitude in water of variable depth, analogous to those derived by Boussinesq for water of constant depth. When the depth is slowly varying compared to the length of the wave, an asymptotic solution of these equations is obtained which describes a slowly varying solitary wave; also differential equations for the slow variations of the parameters describing the solitary wave are derived, and solved in the case when the solitary wave evolves from a region of uniform depth. For small amplitudes it is found that the wave amplitude varies inversely as the depth.
A note on Hamilton's principle for perfect fluids
Authors: Bretherton Francis P.
Published: March 2006

A derivation is given of the Eulerian equations of motion directly from the Lagrangian formulation of Hamilton's principle. The circulation round a circuit of material particles of uniform entropy appears as a constant of the motion associated with the indistinguishability of fluid elements with equal density, entropy and velocity. A discussion is given of the Lin constraint, and it is pointed out that, for a barotropic fluid, the variational principle recently suggested by Seliger & Whitham does not permit velocity fields in which the vortex lines are knotted.
Cavity surface wave patterns and general appearance
Authors: Brennen Christopher
Published: March 2006

Observations were made of the appearance of hydrodynamic cavities behind a series of axisymmetric headforms. Among the phenomena investigated was the transition of the interfacial or separated boundary layer on the cavity surface. The first stage of this process, namely the spatial growth of instability waves could be distinguished by means of high-speed photography. Comparison is made with a theoretical instability analysis.
The calculation of inviscid hypersonic flow past the lower surface of a delta wing
Authors: Akinrelere E. A.
Published: March 2006

Kennett (1963) calculated the hypersonic flow fields past the lower (compression) surface of a delta wing, using the one-strip approximation of the method of integral relations. He obtained solutions only for wings with detached shocks. In this paper, his solutions are extended to wings with attached shocks. Here, the sonic point is inboard of the leading edge which makes the problem mixed. The solutions compare very well with the numerical solutions of the full equations by Babaev (1963a) both in the shock shapes and pressure distributions for various Mach numbers.
The evolution of time-periodic long waves of finite amplitude
Authors: Madsen O. S. , Mei C. C. , Savage R. P.
Published: March 2006

The breakdown of shallow water waves into forms exhibiting several secondary crests is analyzed by numerical computations based on approximate equations accounting for the effects of non-linearity and dispersion. From detailed results of two cases it is shown that when long waves are such that the parameter σ = ν*L*2/h*3 is of moderate magnitude, either due to initially steep waves generated at a wave-maker or due to forced amplification by decreasing depth, waves periodic in time do not remain simply periodic in space. Numerical results are compared with experiments for waves propagating past a slope and onto a shelf.
Spherical vortex motions of a conducting fluid
Authors: Ranger K. B.
Published: March 2006

Some exact solutions of the steady M.H.D. equations for an inviscid infinitely conducting fluid are discussed, and a generalization of the Prendergast model for an idealized magnetic star is obtained, in which there is a finite azimuthal swirl velocity of the fluid. The flow in all examples is bounded by a sphere and expressions are found for the distribution of energy between the fluid motion and magnetic field.