bluuebunny/crossref_metadata_2025 huggingface dataset:
Non-linear Kelvin–Helmholtz instability
Authors: Nayfeh Ali Hasan , Saric William S.
Published: March 2006

A non-linear analysis is presented for the stability of a liquid film adjacent to a compressible gas and under the influence of a body force directed either outward from or toward the liquid. The effects of the liquid's surface tension are taken into account. The non-linear Rayleigh–Taylor instability is included as a special case. The analysis considers the case of an inviscid liquid adjacent to a subsonic flow and the case of a very viscous liquid adjacent to a subsonic or a supersonic flow. For a subsonic external flow, it is found that the cut-off wave-number is amplitude dependent in the inviscid case whereas it is amplitude independent in the viscous case. It is found that the non-linear motion of the gas may be stabilizing or destabilizing, whereas the non-linear motion of the liquid is found to be stabilizing in the viscous case. For a supersonic external flow and a viscous liquid, the cut-off wave-number is amplitude dependent. Moreover, unstable disturbances with wave-numbers near the cut-off wave-number do not grow indefinitely with time but achieve a steady-state amplitude.
Heat transfer at high Péclet number from a small sphere freely rotating in a simple shear field
Authors: Acrivos Andreas
Published: March 2006

The problem of heat transfer at high Péclet number Pe from a sphere freely rotating in a simple shear field is considered theoretically for the case of small shear Reynolds numbers. It is shown that the present problem is in many respects similar to that of heat transfer past a freely rotating cylinder which was recently solved by Frankel & Acrivos (1968). By taking advantage of the close analogy between these two problems, an approximate method of solution is developed according to which the asymptotic Nusselt number for Pe → ∞ is 9, i.e. 4½ times its value for pure conduction. As in the corresponding case of the cylinder, the fact that the asymptotic Nusselt number is independent of Pe results from the presence of a region of closed streamlines which completely surrounds the rotating sphere.
Resonant response of harbours: an equivalent-circuit analysis
Authors: Miles John W.
Published: March 2006

The surface-wave response of a harbour to a prescribed, incident wave is calculated on the hypotheses of shallow-water theory, an ideal fluid, and a narrow mouth,M.An equivalent electrical circuit is constructed, in which the incidentwave displacement inMappears as the input voltage and the flow throughMappears as the input current. This circuit contains aradiation impedance, ZM, which comprises resistive and inductive terms, and aharbour impedance, ZH, which comprises an infinite sequence of parallel combinations of inductance and capacitance that bear a one-to-one correspondence with the natural modes of the closed harbour, together with a single capacitor, which corresponds to the degenerate mode of uniform displacement and dominates the response of the harbour as a Helmholtz resonator. Variational approximations toZHandZMare developed. The equivalent circuit exhibits parallel resonance at the resonant frequencies of the closed harbour, ωn, and series resonance at a second set of frequencies,$\tilde{\omega}_n $, where$\tilde{\omega}_n\downarrow \omega_n > 0 $and$\tilde{\omega}_0\downarrow 0 $asM→ 0;$\tilde{\omega}_0 $corresponds to the Helmholtz mode. A narrow canal between the coastline and the harbour is represented by a four-terminal network betweenZMandZH. It is shown that narrowing the harbour mouth and/or increasing the length of the canal does not affect the mean response of the harbour to a broad-band, random input except in the Helmholtz mode, but that it does increase significantly the response in that mode, which may dominate tsunami response. The general results are applied to circular and rectangular harbours. The numerical calculation ofZHfor an arbitrarily shaped harbour is discussed.
A non-linear effect in the capillary instability of liquid jets
Authors: Rutland D. F. , Jameson G. J.
Published: March 2006

A recent theory for the instability of liquid jets predicted that because of nonlinear interactions, a swelling should occur between the crests of the primary disturbance waves. Experiments are described in which such swellings have been observed. The shape of the swellings is in qualitative agreement with the predictions.
The reflexion of internal/inertial waves from bumpy surfaces
Authors: Baines P. G.
Published: March 2006

When internal and/or inertial waves reflect from a smooth surface which is not plane, there is in general some energy flux which is ‘back-reflected’ in the opposite direction to that of the incident energy flux (in addition to that ‘transmitted’ in the direction of the reflected rays), provided only that the incident wavelength is sufficiently large in comparison with the length scales of the reflecting surface. The reflected wave motion due to an incident plane wave is governed by a Fredholm integral equation whose kernel depends on the form of the reflecting surface. Some specific examples are discussed, and the special case of the ‘linearized boundary’ is considered in detail. For an incoming plane wave incident on a sinusoidally varying surface of sufficiently small amplitude, in addition to the main reflected wave two new waves are generated whose wave-numbers are the sum and difference respectively of those of the surface perturbations and the incident wave. If the incident wave-number is the smaller, the difference wave is back-reflected.
Hydromechanics of swimming propulsion. Part 1. Swimming of a two-dimensional flexible plate at variable forward speeds in an inviscid fluid
Authors: Wu T. Yao-Tsu
Published: March 2006

The most effective movements of swimming aquatic animals of almost all sizes appear to have the form of a transverse wave progressing along the body from head to tail. The main features of this undulatory mode of propulsion are discussed for the case of large Reynolds number, based on the principle of energy conservation. The general problem of a two-dimensional flexible plate, swimming at arbitrary, unsteady forward speeds, is solved by applying the linearized in viscid flow theory. The large-time asymptotic behaviour of an initial-value harmonic motion shows the decay of the transient terms. For a flexible plate starting with a constant acceleration from at rest, the small-time solution is evaluated and the initial optimum shape is determined for the maximum thrust under conditions of fixed power and negligible body recoil.
Waves in an electrically conducting rotating liquid
Authors: Rao Adabala Ramachandra
Published: March 2006

The waves generated by the steady motion of an obstacle along the axis of a uniformly rotating, electrically conducting homogeneous fluid have been studied by Lighthill's technique. The wave-number surface consists of a sphere and four coincident planes. The waves corresponding to points on the sphere travel ahead or trail behind the obstacle according as a1 the Alfvén velocity, is greater or less than U, the velocity of the obstacle. By drawing the appropriate normals to the four planes, it is seen that the formation of a Taylor column ahead of the obstacle is possible even at large Rossby numbers when U < a1 in contrast with the non-magnetic case and the case with U > a1.
Flow and pressure drop in systems of repeatedly branching tubes
Authors: Pedley T. J. , Schroter R. C. , Sudlow M. F.
Published: March 2006

The airways of the lung form a rapidly diverging system of branched tubes, and any discussion of their mechanics requires an understanding of the effects of the bifurcations on the flow downstream of them. Experiments have been carried out in models containing up to two generations of symmetrical junctions with fixed branching angle and diameter ratio, typical of the human lung. Flow visualization studies and velocity measurements in the daughter tubes of the first junction verified that secondary motions are set up, with peak axial velocities just outside the boundary layer on the inner wall of the junction, and that they decay slowly downstream. Axial velocity profiles were measured downstream of all junctions at a range of Reynolds numbers for which the flow was laminar.

In each case these velocity profiles were used to estimate the viscous dissipation in the daughter tubes, so that the mean pressure drop associated with each junction and its daughter tubes could be inferred. The dependence of the dissipation on the dimensional variables is expected to be the same as in the early part of a simple entrance region, because most of the dissipation will occur in the boundary layers. This is supported by the experimental results, and the ratio Z of the dissipation in a tube downstream of a bifurcation to the dissipation which would exist in the same tube if Poiseuille flow were present is given by
\[
Z = (C/4\surd{2})(Re\,d/L)^{\frac{1}{2}},
\]
where L and d are the length and diameter of the tube, Re is the Reynolds number in it, and the constant C (equal to one for simple entry flow) is equal to 1·85 (the average value from our experiments). In general, C is expected to depend on the branching angles and diameter ratios of the junctions used. No experiments were performed in which the flow was turbulent, but it is argued that turbulence will not greatly affect the above results at Reynolds numbers less than and of the order of 10000. Many more experiments are required to consolidate this approach, but predictions based upon it agree well with the limited number of physiological experiments available.
Zero and negative entrainment in turbulent shear flow
Authors: Head M. R. , Bradshaw P.
Published: March 2006

In certain accelerated flows the entrainment in the boundary layer, as normally defined, may be either zero or negative; on the other hand, there is no reason to suppose, on physical grounds, that the spread of mean or fluctuating vorticity should cease or become negative in such flows. This paradox is resolved in the present paper. It is also shown that in the equilibrium turbulent sink-flow boundary layer, where the entrainment as normally defined is zero, the reduced advection along streamlines in the outer part of the layer comes about mainly through increased dissipation: there is no reason to assume any radical change in the turbulence structure.
On the effect of particle couples on the motion of a dilute suspension of spheroids
Authors: Leal L. G.
Published: March 2006

This paper is concerned with the effect on the bulk motion of applying torque by external means to the particles of a suspension. Our investigation is based upon theoretical consideration of the motion of a dilute suspension of permanently magnetized spheroids, in the presence of a uniform magnetic field, for cases in which the motion of a Newtonian fluid would be uni-directional. For convenience, we have divided the work into two parts: in the first, the particles are assumed to be perfectly spherical and the magnetic field strength to be arbitrary; in the second, the particles are taken as arbitrary spheroids (with their magnetic dipole axis coincident with the axis of revolution), but the magnetic field is assumed to be strong enough to ensure that the magnetic dipole axis of each particle is effectively aligned with the field vector H. We concentrate attention on the development of general criteria which allow an a priori determination as to whether the bulk motion remains uni-directional in the presence of particle couples, and (when the motion is uni-directional) whether the resultant velocity profile has the same form as for a Newtonian fluid. In addition, we evaluate the effective viscosity of the suspension for several representative cases in which the velocity field is Newtonian in form. Finally, as an example of the general situation in which the bulk velocity field does not remain uni-directional, we obtain the solution for the motion through a circular tube when the magnetic particles are spherical and the magnetic field is applied at right angles to the tube axis.
On waves generated in rotating stratified liquids by travelling forcing effects
Authors: Rao V. Subba , Rao G. V. Prabhakara
Published: March 2006

The pattern and propagation of waves excited by forcing effects that may oscillate with a frequency σo and are travelling with a uniform velocity U along the axis of rotation in a rotating stratified liquid are studied by employing the technique of Lighthill. The fluid is assumed to be an unbounded, in viscid, incompressible, non-diffusive, rotating stratified liquid, rotating with a constant angular velocity Ω about a vertical axis and with a density decreasing vertically upwards. It is found that the effect of rotation on stratification or vice versa is to reduce the region of disturbance and to split the wave crests for Ω0 ≠ 0. The periodic nature of the forcing effect excites various systems of waves which are otherwise coincident.
Laminar convection of a radiating gas in a vertical channel
Authors: Greif R. , Habib I. S. , Lin J. C.
Published: March 2006

An exact solution is obtained for the problem of fully-developed, radiating, laminar convective flow in a vertical heated channel. The effect of radiation is to decrease the temperature difference between the gas and the wall, thereby reducing the influence of natural convection. Thus, the reduction in velocity occurring in a heated upflow is less for a radiating gas. Graphs are presented for the dimensionless velocity and temperature profiles and for the volume and heat fluxes.
Viscous tails in Hele-Shaw flow
Authors: Wood C. J.
Published: March 2006

An experiment has been performed, using pulsed dye injection on an aerofoil in a Hele-Shaw cell. The purpose was to observe the form of the trailing-edge flow when the Reynolds number was high enough to permit separation and the initiation of a Kutta condition. The experiment provides a successful confirmation of the existence of a ‘viscous tail’ as predicted by Buckmaster (1970) although there is an unexplained quantitative discrepancy.
On the scattering of aerodynamic noise
Authors: Crighton D. G. , Leppington F. G.
Published: March 2006

According to the Lighthill acoustic analogy, the sound induced by a region of turbulence is the same as that due to an equivalent distribution of quadrupole sources within the fluid. It is known that the presence of scattering bodies situated near such multipoles can convert some of their intense near field energy into the form of sound waves whose amplitude is far greater than that of the incident field. Calculations are here presented to determine the extent of this conversion, for hard and soft bodies of various shapes, making use of the reciprocal theorem to recast the problem into one of finding the field, near the obstacle, induced by an incident plane wave. If the obstacle is small compared with a wavelength, then its presence is equivalent to an additional dipole (or source) whose greater efficiency as a sound radiator implies that the familiar intensity law I ∝ U8, for far field intensity I against typical turbulence velocity U for an unbounded flow, is replaced by I ∝ U6 (or I ∝ U4) for a hard (or soft) body. For the situation where the scatterer is large compared with wavelength, the prototype problem of a wedge of exterior angle (p/q)π is shown to yield an intensity law I ∝ U4+2q/p for both hard and soft surfaces. This result is shown to hold for the more general ‘wedge-like’ surfaces, whose dimensions are large scale and whose edges may be smoothed out on a small scale, compared with wavelength. The method used involves the matching of an incompressible flow, on the fine scales typical of the edge geometry, to an outer flow determined by the large scale features of the surface. Favourable comparisons are made with previous results pertaining to the two-dimensional semi-infinite duct and to the half-plate of finite thickness.
The solitary wave in water of variable depth. Part 2
Authors: Grimshaw R.
Published: March 2006

This paper examines the deformation of a solitary wave due to a slow variation of the bottom topography. Differential equations which determine the slow variation of the parameters of a solitary wave are derived by a certain averaging process applied to the exact in viscid equations. The equations for the parameters are solved when the bottom topography varies only in one direction, and when the wave evolves from a region of uniform depth. The variation of amplitude with depth is determined and compared with some recent experimental results.
Measurements in a subsonic turbulent jet using a quantitative schlieren technique
Authors: Davis M. R.
Published: March 2006

The development of optical methods for the quantitative study of the fluctuating properties of turbulent flows can provide a supplement to conventional hot-body anemometry techniques. In particular, the study of high-speed flows by hot-wire or hot-film anemometry is often difficult owing to the presence of temperature and velocity fluctuations in the flow, thereby complicating the correct interpretation of measured signals. In addition, restrictions are placed on the application of such anemometers by their physical strength, frequency-response characteristics and the introduction of disturbances by the measuring probes into the flow.

The operation of an optical detection system depends primarily on the mechanism by which the detected radiation intensity is modulated by the flow. Methods which have been used successfully include scattering or absorption of incident light by tracer constituents or particles, the absorption or emission of infra-red radiation by the flow and quantitative adaptations of the schlieren and interferometer systems which are sensitive to the flow density structure. All these systems detect a summation of signals from different parts of the flow and in consequence it is necessary to consider in detail the relation of the integrated signal to the local properties of the flow. This paper deals in particular with the application of the schlieren principle to an axisymmetric turbulent jet.
Boundary-layer separation at a free streamline Part 2. Numerical results
Authors: Ackerberg R. C.
Published: March 2006

An asymptotic solution of the boundary-layer equations, valid just upstream of a free streamline attached to the sharp trailing edge of a body, is compared with a numerical solution for the boundary-layer flow on a finite flat plate set perpendicular to a uniform stream. An arbitrary multiplicative constant in the asymptotic expansion, arising from an eigenfunction, is evaluated by requiring the skin friction to agree with a numerical value close to the free streamline. Using this value, the velocity profiles, computed from the asymptotic expansion, are in excellent agreement with the numerical solution.
Jet flow instability of an inviscid compressible fluid
Authors: Blumen William
Published: March 2006

A linear analysis of the perturbations associated with the jet flow, $\overline{u}(y) = {\rm sech}\,y$, of an inviscid compressible fluid is considered. Some subsonic stable solutions, not associated with stability boundaries, are first determined. Then a subsonic neutral solution is found and used as an aid in determining stability boundaries of the symmetric and antisymmetric disturbance modes. Numerical methods are also used to determine instability characteristics, including the Reynolds stress distributions. Comparisons are made with previous results obtained for the hyperbolic-tangent velocity profile and with unstable characteristics of the Bickley jet.
Vortex shedding from circular cylinders at low Reynolds numbers
Authors: Gaster M.
Published: March 2006

Experiments on slightly tapered models of circular cross-section have shown that the vortex wake structure exists in a number of discrete cells having different shedding frequencies. Within each cell shedding is regular and periodic, the frequency being somewhat lower than that from a parallel cylinder of the same diameter. A similar type of wake behaviour has also been observed on a parallel model in a non-uniform mean flow. These results suggest that the discontinuities in the shedding law observed by Tritton could arise through non-uniformities in the flow.
Directional acoustic radiation from a supersonic jet generated by shear layer instability
Authors: Tam Christopher K. W.
Published: March 2006

A theory on the generation mechanism of directional acoustic radiation from a supersonic jet is proposed. The theory is based on the concept of instability of the shear layer at the boundary of the jet close to the nozzle. Theoretical prediction of the directional wave pattern is found to agree with shadowgraphic observation.
The boundary layer on a flat plate moving transversely in a rotating, stratified fluid
Authors: Redekopp Larry G.
Published: March 2006

The motion of a horizontal plate moving in its own plane in a rotating, stratified fluid is studied to establish the parameter conditions specifying the onset of boundary-layer blocking for the entire range of Rossby and Russell numbers. Régimes in Rossby–Russell number space defining the range of applicability of the inertia–viscous, buoyancy–viscous, and Coriolis–viscous boundary-layer balances are presented, and similarity solutions valid over a limited region of that space are derived. The plate drag and heat transfer are computed from the similarity solutions.
Convection in a fluid with two phases
Authors: Busse F. H. , Schubert G.
Published: March 2006

The gravitational instability of a horizontal fluid layer with a univariant phase transition is considered. It is found that the layer can be unstable even when the less dense phase lies above the dense phase and can be stable in the opposite case. Applications of the theory to convection with phase transitions in astrophysical and geophysical problems are briefly discussed.
The stress generated in a non-dilute suspension of elongated particles by pure straining motion
Authors: Batchelor G. K.
Published: March 2006

In a pure straining motion, elongated rigid particles in suspension are aligned parallel to the direction of the greatest principal rate of extension, provided the effect of Brownian motion is weak. If the suspension is dilute, in the sense that the particles are hydrodynamically independent, each particle of length 2l makes a contribution to the bulk deviatoric stress which is of roughly the same order of magnitude as that due to a rigid sphere of radius l. The fractional increase in the bulk stress due to the presence of the particles is thus equal to the concentration by volume multiplied by a factor of order l2/b2, where 2b is a measure of the linear dimensions of the particle cross-section. This suggests that the stress due to the particles might be relatively large, for volume fractions which are still small, with interesting implications for the behaviour of polymer solutions. However, dilute-suspension theory is not applicable in these circumstances, and so an investigation is made of the effect of interactions between particles. It is assumed that, when the average lateral spacing of particles (h) satisfies the conditions b [Lt ] h [Lt ] l, the disturbance velocity vector is parallel to the particles and varies only in the cross-sectional plane. The velocity near a particle is found to have the same functional form as for an isolated particle, and the modification to the outer flow field for one particle is determined by replacing the randomly placed neighbouring particles by an equivalent cylindrical boundary. The resulting expression for the contribution to the bulk stress due to the particles differs from that for a dilute suspension only in a minor way, viz. by the replacement of log 2l/b by log h/b, and the above suggestion is confirmed. The relative error in the expression for the stress is expected to be of order (log h/b)−1. Some recent observations by Weinberger of the stress in a suspension of glass-fibre particles for which 2l/h = 7·4 and h/2b = 7·8 do show a particle stress which is much larger than the ambient-fluid stress, although the theoretical formula is not accurate under these conditions.
Separation in oscillating laminar boundary-layer flows
Authors: Despard R. A. , Miller J. A.
Published: March 2006

The results of an experimental investigation of separation in oscillating laminar boundary layers is reported. Instantaneous velocity profiles obtained with multiple hot-wire anemometer arrays reveal that the onset of wake formation is preceded by the initial vanishing of shear at the wall, or reverse flow, throughout the entire cycle of oscillation. Correlation of the experimental data indicates that the frequency, Reynolds number and dynamic history of the boundary layer are the dominant parameters and oscillation amplitude has a negligible effect on separation-point displacement.
Instability due to heat diffusion in a stably stratified fluid
Authors: Welander Pierre , Holmåker Kjell
Published: March 2006

A thermally conducting body floating in a fluid that is stably stratified by heat may develop unstable oscillations provided the body temperature shows a large enough time-lag relative to the fluid temperature. A gravitationally stable, non-Boussinesq fluid may itself become unstable in an oscillatory way through a similar diffusive time-lag. One case, that is investigated in the present work, occurs when the basic vertical temperature gradient β = dT/dz and the thermal expansion coefficient α vary in an opposite sense: β is large when α is small and vice versa. The variation in β is here assumed to be caused by internal heat sources and sinks.

If vertical oscillations are started in such a fluid temperature variations will be produced in the regions of large β but the buoyancy forces do not develop until these perturbations have diffused to regions of large α. With an appropriate lag, the buoyancy forces may give a positive work and the oscillations can grow.

Two models are investigated. The first one is a non-viscous two-layer model with β = 0, α = α0 in one layer and β = β0, α = 0 in the other layer. For this model analytical results are derived. The second model is more realistic, having continuous profiles β(z) and α(z), viscosity and horizontal boundaries. The case is studied by a numerical technique, solving the equations directly in time.

A discussion of the numerical method is given in an appendix by K. Holmaker.
Stability of circular Couette flow of binary mixtures
Authors: Saric William S. , Lavan Zalman
Published: March 2006

The hydrodynamic stability of an ideal mixture of two viscous, dissimilar liquids contained between two concentric rotating cylinders is analyzed. The basic flow of the mixture is determined by coupling the mass and momentum equations with an equation for the equilibrium concentration distribution. Infinitesimal, axisymmetric disturbances are assumed, and the disturbance equations are written for the limiting case of large Schmidt numbers (no diffusion).

The presence of density and viscosity variations leads to a twelfth-order eigenvalue problem with two-point boundary conditions that has the appearance of a combined Taylor and density-stratified shear flow problem. A numerical technique is devised to determine the stability boundary and to calculate Taylor numbers and oscillation frequencies for different growth rates.

It is found that very small mean density gradients alter the critical Taylor number and that oscillations occurring in both the growing and neutral solutions are the dominant mode.
Comparison of theory and experiment in peristaltic transport
Authors: Yin F. C. P. , Fung Y. C.
Published: March 2006

Various theoretical aspects of peristaltic motion have attracted considerable attention in the recent literature. There also exists a difference of opinion concerning the definition of peristaltic reflux. Thus far, there has been no completely satisfactory experimental verification of any of the theoretical analyses. The present article describes some details of an experimental work which was carried out to verify the theoretical analysis of peristaltic transport in a two-dimensional channel with one fixed wall and one flexible wall on which a travelling sine wave is imposed. An approximate correction for finite width was introduced. Comparison of the theoretical and experimental results shows a good agreement and gives verification to the theoretical analysis. In the light of the experimental results, a discussion of the different definitions of peristaltic reflux is presented.
Electrohydrodynamics of a current-carrying semi-insulating jet
Authors: Melcher J. R. , Warren E. P.
Published: March 2006

A quasi-one-dimensional non-linear model is developed for the axisymmetric dynamics. Streaming is coaxial with a cylindrical ‘wall’ supporting a potential having a linear axial dependence. In addition to a tangential field due to an axial current, the stream surface supports charges in proportion to the stream-wall potential difference; hence it is driven by normal and shear electric stresses. Free charge and polarization waves compete with the destabilizing effect of capillarity. With supercritical steady flow (the local jet velocity exceeds the wave velocity), it is found that the stream accelerates or decelerates in accordance with whether an equivalent longitudinal force density is respectively positive or negative. With subcritical flow, the effect of the force is reversed. Experiments demonstrate accelerating and decelerating flow régimes. Model and experiment are in agreement with regard to choking at a critical radius, and the dependence of radius and potential on position. Hysteretic switching between flow régimes is obtained by adjustment of stream and wall potentials, and is explained in terms of the model.
A model of the mature hurricane
Authors: Carrier G. F. , Hammond A. L. , George O. D.
Published: March 2006

A schematic picture of the structure of a mature hurricane is hypothesized. The heat and moisture transfer, the boundary-layer dynamics and the overall dynamic and thermodynamic balances are analysed crudely, but sufficiently accurately to provide support for the hypothesis.
A tracer dispersion study of the drag-reduction effect in a turbulent pipe flow
Authors: Bryson A. W. , Arunachalam Vr. , Fulford G. D.
Published: March 2006

Remarkable differences in dispersion of a tracer material injected into turbulent pipe flows of water and water containing as little as 2·5 parts per million by weight of a soluble high-molecular-weight drag-reducing polyoxyethylene additive have been measured. Analysis of the tracer response curves in terms of a simple one-parameter model shows that the observed results are compatible with a drag-reduction mechanism based on thickening of the viscous sublayer adjoining the wall. Other experiments, reported briefly, suggest that polymer adsorption on to the wall is responsible for this thickening.
Stability of convection in containers of arbitrary shape
Authors: Joseph Daniel D.
Published: March 2006

When a container of fluid of arbitrary shape is heated from below and the temperature gradient exceeds a critical value ([Rscr ]c2) the conduction solution with no motion becomes unstable and is replaced by convection. The convection may have two forms: one with ‘upflow’ at the centre of the container and one with ‘downflow’ there. Here we study the stability of the two forms of convection. Both forms are here shown to be stable to infinitesimal disturbances. When the viscosity varies with the temperature or the conduction profile is not linear, etc., the steady convection can be driven with finite amplitudes |ε| at subcritical values of the temperature contrast ([Rscr ]2< [Rscr ]c2). This subcritical convection is stable when the convection is strong (|ε| > |ε*| > 0) but is unstable when the convection is feeble (|ε| > |ε*|). Hence, when |ε| > |ε*| and [Rscr ]2< [Rscr ]c2either ‘upflow’ or ‘downflow’, but not both, is stable. When [Rscr ]2> [Rscr ]c2, however, both the ‘upflow’ and the ‘downflow’ can be stable. This contrasts with the corresponding situation which is known to hold when the container is an unbounded layer. In the layer there is only one stable form of convection. The difference between the bounded domain with two forms of convection and the layer with just one stable form is traced to the mathematical property of simplicity of [Rscr ]c2when viewed as an eigenvalue of the linear stability problem for the conduction solution. It is argued that [Rscr ]c2is a simple eigenvalue in most domains, but in the layer [Rscr ]c2can have infinite multiplicity. The explanation of the transition from the bounded domain to the unbounded layer is sought (1) in the chaotic conditions which frequently prevail at the edges of a ‘bounded’ layer and (2) in the fact that in the layer of large horizontal extent, the higher eigenvalues crowd [Rscr ]c2. In the course of the explanation, a new exact solution of the linear Bénard problem in a cylinder with a rigid side wall and a stress-free top and bottom is derived.
Collapse of an initially spherical vapour cavity in the neighbourhood of a solid boundary
Authors: Plesset Milton S. , Chapman Richard B.
Published: March 2006

Vapour bubble collapse problems lacking spherical symmetry are solved here using a numerical method designed especially for these problems. Viscosity and compressibility in the liquid are neglected. Two specific cases of initially spherical bubbles collapsing near a plane solid wall were simulated: a bubble initially in contact with the wall, and a bubble initially half its radius from the wall at the closest point. It is shown that the bubble develops a jet directed towards the wall rather early in the collapse history. Free surface shapes and velocities are presented at various stages in the collapse. Velocities are scaled like (Δp/ρ)½ where ρ is the density of the liquid and Δp is the constant difference between the ambient liquid pressure and the pressure in the cavity. For
\[
\Delta p/\rho = 10^6 {\rm cm}^2/\sec^2 \approx 1\, \hbox{atm/density of water}
\]
the jet had a speed of about 130m/sec in the first case and 170m/sec in the second when it struck the opposite side of the bubble. Such jet velocities are of a magnitude which can explain cavitation damage. The jet develops so early in the bubble collapse history that compressibility effects in the liquid and the vapour are not important.
Strong streaming induced by a moving thermal wave
Authors: Hinch E. J. , Schubert Gerald
Published: March 2006

The motion induced in a layer of Boussinesq fluid by a moving thermal wave is studied in the case where the mean velocity could exceed the wave speed. A nonlinear boundary-layer theory shows that strong streaming is possible for small viscosity. Velocity fluctuations are limited in magnitude by their short time scale, while viscosity alone, assumed to be relatively weak, limits the mean flow.
Burgers turbulence model for a channel flow
Authors: Lee Jon
Published: March 2006

The truncated Burgers models have a unique equilibrium state which is defined continuously for all the Reynolds numbers and attainable from a realizable class of initial disturbances. Hence, they represent a sequence of convergent approximations to the original (untruncated) Burgers problem. We have pointed out that consideration of certain degenerate equilibrium states can lead to the successive turbulence-turbulence transitions and finite-jump transitions that were suggested by Case & Chiu. As a prototype of the Navier–Stokes equations, Burgers model can simulate the initial-value type of numerical integration of the Fourier amplitude equations for a turbulent channel flow. Thus, the Burgers model dynamics display certain idiosyncrasies of the actual channel flow problem described by a truncated set of Fourier amplitude equations, which includes only a modest number of modes due to the limited capability of the computer at hand.
The non-linear evolution of Stokes waves in deep water
Authors: Chu Vincent H. , Mei Chiang C.
Published: March 2006

Based on a set of modulation equations derived in a previous paper, the nonlinear evolution of wave envelope in deep water is studied numerically. It is found that the wave envelope tends to disintegrate to multiple groups of waves each of which approaches a stable permanent envelope representing dynamical equilibrium between the amplitude dispersion and the frequency dispersion. Qualitative agreement with the experimental measurements of Feir (1967) is also observed.
Dynamical evolution of two-dimensional unstable shear flows
Authors: Zabusky N. J. , Deem G. S.
Published: March 2006

A direct numerical integration of the time-dependent, incompressible Navier-Stokes equations is used to treat the nonlinear evolution of perturbed, linearly unstable, nearly parallel shear flow profiles in two dimensions. Calculations have been made for infinite (inviscid) and finite Reynolds numbers. The latter results are compared with laboratory measurements of Sato & Kuriki for wakes behind thin flat plates, and many of the detailed features are in excellent agreement, including mean flow profiles with ‘overshoot’ development, first harmonic energy profiles with off-axis nulls, and first harmonic phase profiles. The linear instability saturates by forming a vortex street consisting of elliptical vortex pairs. The solutions are followed for times up to eleven linear exponentiation times of the unstable disturbance. A new low-frequency non-linear oscillation is found, which explains the features of the above experiment, including the nearly periodic phase inversions in the first harmonic component of the longitudinal velocity. It results from a nutation of the elliptical vortices with respect to the mean flow direction. Inertial range spectral energy properties are also examined. Inviscid solutions have large wave-number spectral energies obeying the approximate power law, Ek ∼ k−μ, where μ lies between 3 and 4.
Energy stability of the Ekman boundary layer
Authors: Dudis Joseph J. , Davis Stephen H.
Published: March 2006

The critical value RE of the Reynolds number R is predicted by the application of the energy theory. When R < RE, the Ekman layer is the unique steady solution of the Navier-Stokes equations and the same boundary conditions, and is, further, stable in a slightly weaker sense than asymptotically stable in the mean. The critical value RE is determined by numerically integrating the relevant Euler-Lagrange equations. An analytic lower bound to RE is obtained. Comparisons are made between RE and RL, the critical value of R according to linear theory, in order to demark the region of parameter space, RE < R < RL, in which subcritical instabilities are allowable.
Structure of jets in rotating systems
Authors: Gadgil Sulochana
Published: March 2006

An investigation of the structure of jets in rotating systems is presented. The fluid is assumed to be homogeneous and the flow laminar and quasi-geostrophic. When the rate of rotation is small, the dynamics is shown to be identical to that of jets in non-rotating systems, being a balance between inertial terms and lateral dissipation. As the rate of rotation increases the Ekman layers become important and in the strongly rotating case the friction in the Ekman layers dominates lateral dissipation. The jet in a non-rotating system entrains fluid at its edges and the downstream momentum flux is independent of the distance downstream. In the strongly rotating case, however, the jet ejects fluid at its edges and the downstream momentum flux decreases with downstream distance due to dissipation in the Ekman layers. A similarity solution for the general case with both types of friction is obtained and the transition from a jet in which lateral dissipation dominates to one in which Ekman friction is more important is discussed. General features of jets in strongly rotating systems are studied and implications for the Gulf Stream are mentioned.
Response behaviour of hot wires in shear flow
Authors: Gessner F. B. , Moller G. L.
Published: March 2006

The response characteristics of a hot wire operated at constant temperature and exposed to a mean-velocity gradient along its length are examined both analytically and experimentally. The shear sensitivity of local wire temperature distributions, as measured with an infrared microscope, are compared with predicted temperature distributions in order to select a convective heat transfer correlation which can be applied locally along a wire in shear flow. On the basis of this correlation, the steady-state and dynamic response behaviour of platinum and tungsten wires in shear flow are examined by means of computer-generated data. Response curves of general applicability are presented which can be used to correct local mean-velocity and turbulence intensity measurements whenever a mean-velocity gradient exists along a wire.
Forced axial flow between rotating concentric cylinders
Authors: Barcilon Victor , Berg Howard C.
Published: March 2006

Forced axial flow in an annular gap of a cylindrical rotor is investigated analytically and experimentally. At small rotation rates and narrow gap widths, the axial flow is a simple Poiseuille flow over most of the rotor. The distance required for this Poiseuille flow to get established is estimated. An instability is observed at large rotation rates with certain input geometries.
Turbulent flow in wavy pipes
Authors: Hsu Sheng-Tien , Kennedy John F.
Published: March 2006

A primarily experimental investigation was undertaken to determine the internal structure of steady, quasi-uniform, non-separated, axisymmetric flows in circular pipes with sinusoidal wall profiles. The quantities measured include radial and longitudinal distributions of mean velocity, pressure, and total head; the Reynolds shear stress and all three components of turbulence velocity; and boundary shear stress and pressure. Two different wall-wave steepnesses were investigated, and a constant Reynolds number of 1·13 × 105 (based on the average pipe diameter) was maintained in most experiments. The boundary shear stress was found to be shifted upstream relative to the boundary wave, whereas the wall pressure is shifted slightly downstream. The turbulence measurements revealed that there is a central core extending over some 60% of the pipe radius in which the turbulence quantities are constant along the pipe. Near the boundary, however, the turbulence velocities and stress vary periodically along the boundary waves. The longitudinal component of mean velocity was found to be distributed radially according to the power law, but with an exponent that varies along each wave; a simple analytical model is constructed to predict the variation of the exponent. It was not found possible to relate the local boundary shear stress to just the local flow characteristics, since the convective or ‘history’ effects play a significant role in its determination. An empirical formula is derived relating the local boundary shear stress to the local velocity distribution and the first two derivatives of the boundary profile.
An almost-Markovian Galilean-invariant turbulence model
Authors: Kraichnan Robert H.
Published: March 2006

A model equation of Langevin type for the turbulent velocity field is constructed, in which the non-linear terms of the Navier–Stokes equation are replaced by a dynamical damping term and a random forcing term, with strength parameters determined by the past history of the energy spectrum. The model leads to a closed set of first-order differential equations in time for the evolution of two functions: the energy spectrum and the effective memory times for the interaction of mode triads. Invariance of the energy transfer to random Galilean transformation is achieved by using the interaction between solenoidal and compressive parts of a convected test field to determine the memory–time functions. The model equation is developed from the direct-interaction approximation as starting-point. At an intermediate stage, before the Galilean invariance is introduced, a model representation of Edwards's (1964) theory is obtained which extends the latter to statistically non-stationary states.
The effect of oscillation on flat plate heat transfer
Authors: Ishigaki Hiroshi
Published: March 2006

The time-mean heat transfer of the incompressible laminar boundary layer on a flat plate under the influence of oscillation is studied analytically. Flow oscillation amplitude outside the boundary layer is assumed constant along the surface and the viscous dissipation effect is considered. First, the small velocity–amplitude case is treated and the approximate formulae are obtained in the extreme cases when the frequency is low and high. Next, the finite velocity–amplitude case is treated under the condition of high frequency and it is found that the formulae obtained for the small amplitude and high frequency case are also valid. These results show that, when the oscillation is of high frequency, the time-mean heat flux to the wall can be several times as large as that without oscillation. This is due wholly to the viscous dissipation effect combined with oscillation.
Stability of the flow in a differentially heated inclined box
Authors: Hart John E.
Published: March 2006

The effect of sloping boundaries on thermal convection is studied theoretically and in the laboratory in the context of a model in which fluid is contained in a differentially heated rectangular box of small aspect ratio (depth/length), inclined at an angle δ to the vertical. Like its two limiting cases, Bénard convection and convection in the vertical slot, a basic state which exists for low Rayleigh numbers becomes unstable as this parameter is increased. The types of instability and indeed the manner in which the motions become turbulent depend crucially on δ. In our work with water the following general picture of the primary instabilities applies:

For 90° > δ > 10° with the bottom plate hotter, the instabilities are stationary longitudinal convectively driven rolls with axes oriented up the slope. Near δ = 10° there is an upper and lower Rayleigh number cut off. If the Rayleigh number is too small diffusion damps the instabilities, but if it is too large they are damped by the development of a stable upslope temperature gradient in the mean flow.

For 10° > δ > −10° (negative angles imply a hotter upper plate), transverse travelling waves oriented across the slope are the first instabilities of the mean flow. They obtain their kinetic energy via the working of the upslope buoyancy force.

For - 10° > δ > −85° longitudinal modes are again observed. These are rather curious in that they may exist when the stratification $-\hat{g}\cdot\nabla T $ is everywhere positive. The necessary energy for these modes comes out of the mean velocity field and out of the mean available potential energy.

Agreement between the stability theory and the experiments is generally quite good over the whole range of δ, considering the approximations involved in finding a suitable basic flow solution.

For Rayleigh numbers less than ∼ 106 turbulence is only possible for positive angles. For 85° > δ > 20° the development of unsteadiness involves the occurrence and the breaking of wavy longitudinal vortices in a manner reminiscent of the development of turbulence in cylindrical Couette flow.
A study of the constant-temperature hot-wire anemometer
Authors: Perry A. E. , Morrison G. L.
Published: March 2006

The conventional ‘bridge-feedback amplifier’ constant-temperature hot-wire anemometer is analysed to determine its static and dynamic response. The effects of moderate feedback amplifier gain, bridge imbalance, stray bridge reactance, amplifier offset voltage, lack of common mode rejection, amplifier frequency response and departure from constant transconducture are included. The root loci of the system are mapped out and the consequences of the analysis are discussed from the viewpoint of both the operator and the designer.