bluuebunny/crossref_metadata_2025 huggingface dataset:
Turbulent flow in smooth concentric annuli with small radius ratios
Authors: Rehme K.
Published: March 2006

Fully developed turbulent flow through three concentric annuli was investigated experimentally for a Reynolds-number rangeRe= 2 × 104−2 × 105. Measurements were made of the pressure drop, the positions of zero shear stress and maximum velocity, and the velocity distribution in annuli of radius ratios α = 0.02, 0.04 and 0.1, respectively. The results for the key problem in the flow through annuli, the position of zero shear stress, showed that this position is not coincident with the position of maximum velocity. Furthermore, the investigation showed the strong influence of spacers on the velocity and shear-stress distributions. The numerous theoretical and experimental results in the literature which are based on the coincidence of the positions of zero shear stress and maximum velocity are not in agreement with reality.
Evolution of unstable shear layers in a rotating fluid
Authors: Siegmann W. L.
Published: March 2006

The time development of non-axisymmetric disturbances on a shear layer in a uniformly rotating fluid is studied theoretically. It is assumed that the Rossby number, Ekman number, shear-layer length scale, and initial disturbance amplitude are small, and that disturbances grow if vorticity transfer from the shear flow exceeds Ekman-layer vorticity dissipation, a mechanism investigated by Busse (1968). Expressions for the ultimate instability amplitude are determined for specified relations among the small parameters, using the volume-integrated energy equation, the shape assumption, and approximations for the spatial dependence of disturbances. Disturbance growth is limited by modification of the initially-unstable shear-layer profile, and the eventual amplitude is shown to be fairly insensitive to the specific form of the profile. Using data from observations by Hide & Titman, the predicted maximum velocity amplitude of the non-axisymmetric motions for their experiments is approximately one-quarter of the velocity of the shear flow at the point of maximum gradient.
Experimental studies on the onset of thermohaline convection
Authors: Caldwell Douglas R.
Published: March 2006

Measurements of the heat flux and temperature gradient in a layer of saline solution stably stratified by Soret diffusion and heated from below exhibit the onset of convection by the growth of very small oscillations which eventually trigger finite amplitude convection. The critical Rayleigh numbers and periods of oscillation are consistent with published theoretical calculations. Several non-linear modes are observed, even for the same heat flux, near the point of onset.
Influence of viscous dissipation on Bénard convection
Authors: Turcotte D. L. , Hsui A. T. , Torrance K. E. , Schubert G.
Published: March 2006

The approximations implicit in Bénard convection have been modified to include viscous dissipation. It is shown that both the influence of an adiabatic temperature gradient and of viscous dissipation are governed by the same dimensionless parameter Di = αgh/cp. Numerical calculations of finite amplitude convection are given for finite values of Di. It is found that increasing Di decreases flow velocities and finally stabilizes the flow.
Hydromechanics of lunate-tail swimming propulsion
Authors: Chopra M. G.
Published: March 2006

This paper investigates the non-uniform motion of a thin plate of finite aspect ratio, with a rounded leading edge and sharp trailing edge, executing heaving and pitching oscillations at zero mean lift. Such vertical motions characterize the horizontal lunate tails with which cetacean mammals propel themselves, and the same motions, turned through 90° to become horizontal motions of sideslip and yaw, characterize the vertical lunate tails of certain fast-swimming fishes. An oscillating vortex sheet consisting of streamwise and spanwise components is shed to trail behind the body and it is this additional feature of the streamwise component resulting from the finiteness of the plate that makes this study a generalization of the two-dimensional treatment of lunate-tail propulsion by Lighthill (1970). The forward thrust, the power required, the energy imparted to the wake and the hydromechanical propulsive efficiency are determined for this general motion as functions of the physical parameters defining the problem: namely the aspect ratio, the reduced frequency, the feathering parameter and the position of the pitching axis. The dependence of the thrust coefficient and propulsive efficiency on these physical parameters, for the complete range of variation consistent with the assumptions of the problem, has been depicted graphically.
Investigation of turbulent convection under a rotational constraint
Authors: Chan Shu-Kwan
Published: March 2006

Turbulent convection for a rotating layer of fluid heated from below is studied in this paper. The boundaries of the fluid layer are taken to be free. The underlying principle, used is the Malkus hypothesis that the flow tends to transport the maximum amount of heat possible, subject to certain constraints. By taking the Prandtl number to be infinite, a linear differential constraint and an integral constraint are used. The variational problem that follows then depends on two dimensionless parameters, the Taylor number T and the Rayleigh number R.

Asymptotic analysis for the turbulent regime shows that the flow arranges itself so as to tend to offset the stabilizing effect of the rotational constraint, at least in so far as the heat flux is concerned. The dimensionless heat flux, or the Nusselt number, has in general different dependence on T and R, depending on the particular region in the parameter space. For T [les ] O(R), the flow is essentially non-rotating. For O(R) [les ]T [les ] O(R4/3), the flow will always have finitely many horizontal wavenumbers, though the total number of modes increases as T increases in this region. For O(R4/3) [les ] T [les ] O (R3/2), the Nusselt number has a functional dependence proportional to R3/T2, having essentially infinitely many horizontal modes as both R and T increase indefinitely in this region. The last expression is particularly interesting, as it agrees qualitatively with results in finite-amplitude laminar convection. It is also linearly dependent on the layer thickness, as one might expect from dimensional argument. It is suggested that, in the context of the maximum principle, the result in this region of the parameter space may be applicable as well to the same fluid layer with rigid boundaries through the existence of an Ekman layer that is thinner than the thermal layer.
The evolution of under-ice melt ponds, or double diffusion at the freezing point
Authors: Martin Seelye , Kauffman Peter
Published: March 2006

In an experimental and theoretical study, we model a phenomenon observed in the summer Arctic, where a fresh-water layer at a temperature of 0°C floats both over a sea-water layer at its freezing point and under an ice layer. Our results show that the ice growth in this system takes place in three phases. First, because the fresh-water density decreases upon supercooling, the rapid diffusion of heat relative to salt from the fresh to the salt water causes a density inversion and thereby generates a high Rayleigh number convection in the fresh water. In this convection, supercooled water rises to the ice layer, where it nucleates into thin vertical interlocking ice crystals. When these sheets grow down to the interface, supercooling ceases. Second, the presence of the vertical ice sheets both constrains the temperatureTand salinitysto lie on the freezing curve and allows them to diffuse in the vertical. In the interfacial region, the combination of these processes generates a lateral crystal growth, which continues until a horizontal ice sheet forms. Third, because of theTandsgradients in the sea water below this ice sheet, the horizontal sheet both migrates upwards and increases in thickness. From one-dimensional theoretical models of the first two phases, we find that the heat-transfer rates are 5–10 times those calculated for classic thermal diffusion.
A theory of thermal oscillations in liquid metals
Authors: Gill A. E.
Published: March 2006

Thermal oscillations have been found to occur during crystal growing, thereby introducing undesirable striations in the solid crystal produced. A reason for the existence of such oscillations is given in this paper, and relevant experiments discussed.
Phase-angle measurements between hot-wire signals in the turbulent wake of a two-dimensional bluff body
Authors: Simmons J. E. L.
Published: March 2006

Measurements have been made of the phase difference between the signals of two hot wires separated in the stream direction and placed in and close to the turbulent wake of a two-dimensional bluff body. Results are presented in the form of the wavelength and the phase speed of perturbations at the vortex shedding frequency for various positions in the flow. It is shown that values of the phase speed may be obtained that are in agreement with published values of the vortex transport velocity. It is necessary to ensure that the hot wires are close to the paths of the centres of the vortices in the wake, otherwise spurious results caused by the influence of the bound vorticity at the model will be obtained.
The jet density exponent issue for the noise of heated subsonic jets
Authors: Mani R.
Published: March 2006

The subject of the present study is the question of how the sound power of a jet of constant exit velocity would vary if the jet exit density were varied. Changes in jet exit density would inevitably be accompanied in a real experiment by changes in the speed of sound (temperature) in the jet, so that both effects must be considered simultaneously. The point of view advanced at the end of the study is that experimentally observed results in this area seem to admit an explanation based on how the radiative efficiency of moving acoustic sources is affected by the shrouding effect of a jet flow whose velocity, temperature and density differ from those of the ambient fluid. This change in efficiency is calculated with the aid of a simple model as follows. We determine the acoustic power output of a convected monopole source, simple harmonic in its own frame of reference, moving along the axis of a plug-flow round jet whose velocity is the same as that of the source. The jet is doubly infinite and the source is assumed to have an infinite lifetime. The density and temperature of the jet are allowed to differ from those of the ambient fluid though the specific-heat ratio of the jet fluid is assumed to be the same as that of the ambient. The requirement of equality of the static pressure inside and outside the jet then calls for a certain restraint on how the jet density and temperature vary. For a specific value of the jet exit velocity, the variation of acoustic power with the ratio of jet to ambient density along with a simple assumption on how the source strength varies with jet density are employed to deduce theoretically the ‘jet density exponent for jets which are subsonic with respect to the ambient speed of sound. The jet density exponent is found to depend both on the jet Mach number and even more strongly on a source frequency parameter. The theoretical results are compared with some experimental studies of this problem. Encouraging agreement is obtained both for the detailed observed effects on the power spectrum and the exponent for the overall power.
Small-scale mixing in stably stratified fluids: a report on Euromech 51
Authors: Linden P. F. , Turner J. S.
Published: March 2006

The 51st Euromech Colloquium, on mixing in stratified fluids, was held at the University of Cambridge from 24-27 June 1974, with one of the present authors (J.S.T.) acting as Chairman of the Organizing Committee. There were fifty-two participants from twelve countries, and thirty-two contributions were presented and discussed during the four days of the meeting.
The stability of unsteady cylinder flows
Authors: Hall P.
Published: March 2006

First, the linear stability of the flow between two concentric cylinders when the outer one is a t rest and the inner has angular velocity Ω{1+ εcosωt} is considered. In the limit in which ε and ω tend to zero it is found that the critical Taylor number a t which instability first occurs is decreased by an amount of order ε2from its unmodulated value, the stabilizing effect a t order ε2ω2being slight. The limit in which ω tends to infinity with ε arbitrary is then studied. In this case it is found that the critical Taylor number is decreased by an amount of order ε2ω−3from its unmodulated value.

Second, the effect of taking nonlinear terms into account is investigated. It is found that equilibrium perturbations of small but finite amplitude can exist under slightly supercritical conditions in both the high and low frequency limits. Some comparisons with experimental results are made, but these indicate that further theoretical work is needed for a broad band of values of ω. In appendix B it is shown how this can be done by an alternative formulation of the problem.
Stability characteristics of a single-phase free convection loop
Authors: Creveling H. F. , De Paz J. F. , Baladi J. Y. , Schoenhals R. J.
Published: March 2006

Stability characteristics of a single-phase free convection loop are presented. In the experiments, water was placed inside a toroidal glass loop oriented in a vertical plane. The lower half of the loop was heated and the upper half was cooled. At low heat-transfer rates and also a t high heat-transfer rates the free convection flow was observed to be steady. For the intermediate range, however, the flow was found to be highly oscillatory. Stability predictions are also developed. The comparison between theory and experiment yields favourable agreement.

Observations of unstable behaviour have been reported previously for single-phase fluids in the vicinity of the thermodynamic critical point. In these situations it has been assumed that the unusual behaviour of the fluid properties in the near-critical region necessarily constitutes the underlying cause of such instabilities. In contrast t o this view, analyses by Keller (1966) and Welander (1967) indicate that instabilities can occur for ordinary fluids as well. Results of the present study confirm this contention, since instabilities were clearly observed for water at atmospheric pressure and moderate temperatures.
The nonlinear calculation of Taylor-vortex flow between eccentric rotating cylinders
Authors: Diprima R. C. , Stuart J. T.
Published: March 2006

This paper is concerned with the nonlinear stability of the flow between two long eccentric rotating cylinders. The problem, which is of interest in lubrication technology, is an extension both of the authors’ earlier work on the linear eccentric case and of still earlier work by Davey and others on the nonlinear concentric analysis of Taylor-vortex development. There are four parameters which are assumed small in the analysis; they are the mean clearance ratio, the eccentricity, the amount by which the Taylor number exceeds its critical value; and the Taylor-vortex amplitude. Following the earlier work mentioned above, relation-ships are specified between these parameters in order to develop a satisfactory perturbation scheme. Thus a non-local solution is obtained to the nonlinear stability problem, in which the whole flow field is taken into account.

Of some importance in the analysis is the fact that it is necessary to allow for the development of a pressure field substantially bigger than that associated with Taylor vortices in the concentric case, owing to the Reynolds lubrication effect in a viscous fluid moving through a converging passage. I n order to achieve this mathematically, it is necessary to solve the continuity equation to a higher order than is necessary for the momentum equations.

It is found that the angular position for maximum vortex activity, which is 90° downstream of the maximum gap in the linear case, can taken on any value between 0 and 90°, depending on the value of the supercritical Taylor number. For a particular experiment of Vohr (1968) acceptable agreement is obtained for this angle (50°), though the ‘small’ parameters are somewhat outside the expected range of perturbation theory. Formulae are obtained for the torque and forces acting on the inner cylinder.
Taylor instability of a non-uniform free-surface flow
Authors: Dagan G.
Published: March 2006

The evolution of a small disturbance in a three-dimensional steady free-surface flow is investigated. The radius of curvature of the free surface and the length scale characterizing the non-uniformity of the velocity are assumed to be of the same order of magnitude. It is shown that the local rate of growth of the amplitude of the disturbance depends on both the normal pressure gradient (as in the case of Taylor instability) and the rate of strain on the free surface. Application of the theory to rising gaseous bubbles and gravity water waves is discussed.
Dispersion by random velocity fields
Authors: Phythian R.
Published: March 2006

A simple approximation is proposed for the problem of the dispersion of marked particles in an incompressible fluid in random motion when the probability distribution of the velocity field is taken as Gaussian, homogeneous, isotropic, stationary and of zero mean. Approximations for the Lagrangian velocity correlation function and the dispersion are given and compared with exact computer calculations of Kraichnan. Agreement is found to be good except for time-independent velocity fields and singular wavenumber spectral functions.
Statistical dynamics of two-dimensional flow
Authors: Kraichnan Robert H.
Published: March 2006

The equilibrium statistical mechanics of inviscid two-dimensional flow are re-examined both for a continuum truncated at a top wavenumber and for a system of discrete vortices. In both cases, there are negative-temperature equilibria for finite flows. But for spatially infinite flows, there are only positive-temperature equilibria, and both the continuum and discrete system exhibit proper, extensive, thermodynamic limits a t all realizable values of the energy and enstrophy density. The negative-temperature behaviours of the continuum and discrete system are semi-quantitatively the same, except for a supercondensation phenomenon in the discrete case a t the smallest realizable values of negative temperature. The supercondensed states have very large energy and in them all vortex cores of the same sign are clumped within an area small eompared with the mean area per vortex. The approach of the continuum system to absolute equilibrium by enstrophy cascade to high wavenumbers and energy cascade to low wavenumbers is examined. It is argued that the enstrophy cascade is closely analogous to distortion of a passive scalar field by straining of large spatial scale. This implies that high intermittency of spatial derivatives of the vorticity field can develop but that there is no associated change in the previously proposed log-corrected k−1 enstrophy spectrum law. On the other hand, intermittency build-up in the downward energy cascade can result in a change of the exponent in the energy spectrum law to a negative value of smaller magnitude than 5/3. Intermittency effects in the non-equilibrium energy cascade seem a more plausible explanation for vortex clumping observed in recent computer experiments than do the spatially smooth condensation phenomena associated with the negative-temperature absolute equilibria.
Entry flow in a curved pipe
Authors: Yao L.-S. , Berger S. A.
Published: March 2006

A secondary flow is set up when a fluid flows through a stationary curved pipe. The fluid in the middle of the pipe moves outwards and that near the wall inwards. Dean showed that the dynamical similarity of this fully developed flow depends on a non-dimensional parameter $D = 2(a/R)^{\frac{1}{2}}(a\overline{W}/\nu) $, where $\overline{W} $ is the mean velocity along the pipe, v is the coefficient of kinematic viscosity and a is the radius of the pipe, which is bent into a coil of radius R. Dean's analysis was limited to small values of D. Later, Barua developed an asymptotic boundary-layer theory for large values of D and showed for these values of D that the resistance coefficient γc is much larger than that for the corresponding straight pipe. The present work deals with the flow in a curved pipe as it develops from a uniformly distributed velocity at the entrance to a fully developed profile. Barua's results for the fully developed flow are adopted as downstream conditions in the present work. The ratio of the entry lengths of the curved ipe and the corresponding straight one is shown to be proportional to D−1/2 when D is large. Thus, the entry length for a curved pipe is much shorter than that for the corresponding straight pipe.
On the effect of a small counterpressure on constant-power spherical detonation waves as driven by focused radiation
Authors: George Y. H.
Published: March 2006

A small perturbation analysis is made of the flow inside a spherical explosion created in a gas absorbing energy from an array of lasers with a common focus. The influence of a small counterpressure outside the leading shock wave is investigated for power deposition constant in time. The departure from self-similarity is described by a power series in the variable y, the inverse square of a Mach number based on the wave-front velocity. To first order, the flow equations are solved numerically, the shock velocity is obtained and the radial profiles of velocity, density and pressure are presented graphically. The flow singularity at the radiation focus is discussed.
Starting vortex, separation bubbles and stall: a numerical study of laminar unsteady flow around an airfoil
Authors: Mehta Unmeel B. , Lavan Zalman
Published: March 2006

The stalling characteristics of an airfoil in laminar viscous incompressible fluid are investigated. The governing equations in terms of the vorticity and stream function are solved using an implicit finite-difference scheme and point successive relaxation procedure. The development of the impulsively started flow, the initial generation of circulation, and the behaviour of the forces at large times are studied.

Following the impulsive start, the lift is at first very large and then it rapidly drops. The subsequent growth of circulation and lift is associated with the starting vortex. After incipient separation, the lift increases owing to enlargement of the separation bubble and intensification of the flow rotation in it. The extension of this bubble beyond the trailing edge causes it to rupture and brings about the stalling characteristics of the airfoil. Subsequently, new bubbles are formed near the upper surface of the airfoil and are swept away. The behaviour of the lift acting on the airfoil is explained in terms of the strength and sense of these bubbles. The lift increases when attached clockwise bubbles grow and when counterclockwise bubbles are swept away and vice versa.
Scaling laws for pipe-flow turbulence
Authors: Perry A. E. , Abell C. J.
Published: March 2006

Using hot-wire-anemometer dynamic-calibration methods, fully developed pipe-flow turbulence measurements have been taken in the Reynolds-number range 80 × 103to 260 × 103. Comparisons are made with the results of previous workers, obtained using static-calibration methods. From the dynamic-calibration results, a consistent and systematic correlation for the distribution of turbulence quantities becomes evident, the resulting correlation scheme being similar to that which has previously been established for the mean flow. The correlations reported have been partly conjectured in the past by many workers but convincing experimental evidence has always been masked by the scatter in the results, no doubt caused by the difficulties associated with static-calibration methods, particularly the earlier ones. As for the mean flow, the turbulence intensity measurements appear to collapse to an inner and outer law with a region of overlap, from which deductions can be made using dimensional arguments. The long-suspected similarity of the turbulence structure and its consistency with the established mean-flow similarity appears to be confirmed by the measurements reported here.
On the correlation between temperature and velocity dissipation fields in a heated turbulent jet
Authors: Antonia R. A. , Van Atta C. W.
Published: March 2006

A few statistical properties of fine-scale velocity and temperature fluctuations have been measured on the axis of symmetry of a heated turbulent round jet. The probability density of ∂θ/∂x, the streamwise derivative of the temperature fluctuation, is strongly negatively skewed, indicating a lack of isotropy for the fine-scale temperature structure. An estimate of the correlation between the velocity and temperature dissipation fields has been obtained by assuming that the dissipation of velocity and dissipation of temperature can be approximated by (∂θ/∂x)2, whereuis the streamwise velocity fluctuation, and (∂θ/∂x)2rrespectively. The correlation between the quantities (∂θ/∂x)2rand (∂θ/∂x)2raverages over a volume of linear dimensionr, is fairly high and depends on the choice ofr.An analysis shows that this correlation plays a vital role in the prediction of high-order structure functions ofuand θ. The assumed lognormality of the probability density of (∂θ/∂x)2rand (∂θ/∂x)2rand of their joint density is found to be reasonable over a range ofrcorresponding to the inertial subrange.
Boundary-layer growth at a three-dimensional rear stagnation point
Authors: Howarth J. A.
Published: March 2006

The theory of boundary-layer growth at a rear stagnation point, first presented by Proudman & Johnson, is here extended to cover fully three-dimensional rear stagnation points. Supporting numerical solutions of the full initial-value problem establish the relevance of the in viscid similarity solutions obtained.
Electrohydrodynamics of a pair of liquid drops
Authors: Sozou C.
Published: March 2006

In this paper we consider the flow field generated by a uniform electrostatic field in and about a pair of identical liquid drops immersed in a conducting fluid. It is assumed that the undisturbed electric field is parallel to the line joining the centres of the two drops. The flow field is due to the tangential electric stress over the surfaces of the drops and here this stress and the flow field are expressed in terms of bispherical harmonics. When the distance between the centres of the drops is many drop diameters the tangential electric stress and the flow field in and about one drop are unaffected by the presence of the other drop, as expected. When the distance between the centres of the drops is of the order of two drop diameters there is a substantial modification in the tangential electric stress at the surfaces of the drops and in the induced flow field, especially in the region between the planes through the drop centres perpendicular to the undisturbed electric field.
Finite amplitude analysis of a flow-structure interaction problem
Authors: Hilaire A. O. St , Vaidya P. G.
Published: March 2006

The interaction between a vibrating structure and an unsteady potential flow-field disturbance induced by the motion of the structure itself is investigated, and is shown to be a significant source of both nonlinear excitation and nonlinear dissipation. An approximate analysis, based on small nonlinear disturbance theory, is presented of the forces that influence the characteristic behaviour of self-excited harmonium reeds vibrating at finite amplitudes. It is demonstrated that the ideas brought forth by this example can be generalized to apply to other flow-induced vibrating systems, regardless of the excitation mechanism, provided that certain basic assumptions about the flow can be made. For the case of the harmonium reed, it is shown that, taken by itself, an account of the feedback forces arising from induced higher-order unsteady disturbances in the surrounding potential flow field is sufficient for predicting the net nonlinear dissipative force that eventually causes the reed to reach and maintain a finite limiting amplitude. In particular, it is demonstrated that the nonlinear energy drain from the motion of the reed is a consequence of the net effect of the higher-harmonic disturbances that are generated near the structure.

A result of the analysis is the development of a functional dependence of the interactive forces on the system geometry and the flow velocity. One of the advantages of obtaining a functional expression is the ability to carry out parametric studies in the context of vibration and noise control.
Some remarks on the initiation of inertial Taylor columns
Authors: Huppert Herbert E.
Published: March 2006

The quasi-geostrophic flow over an obstacle placed on the lower of two horizontal planes in rapid rotation about the vertical z axis is considered. The flow field is calculated in the limit of vanishing viscosity after assuming the flow far upstream of the obstacle to be uniform, of magnitude V. The effects that occur in homogeneous flow are compared with those that occur in stratified flow.

If the flow is homogeneous, there is a region of closed streamlines if
\[
h_0R^{-1} > \min_r\left[r\left/\int_0^r xh(x)dx\right. \right],
\]
where the obstacle is assumed to be cylindrically symmetric and given by z = Hh0h(r/L), H is the distance between the planes and R is the Rossby number V/(fL). For any obstacle the right-hand side of (1) is greater than zero and hence h0 must be positive for a closed-streamline region to occur. It is argued, and illustrated by a particular example, that because (1) involves an integral of h(x) a representative flow pattern can be obtained for obstacles of less than critical height by considering the special case of a flat-topped obstacle, as is done by Ingersoll (1969).

If the flow is stratified with constant Brunt-Väisälä frequency N, the condition for the existence of closed streamlines is shown to be
\[
h_0R^{-1} > \min_r \left[B\int_0^{\infty}\int_0^{\infty}xt\cot h(Bt)h(x)J_0(tx)J_1(tr)\,dt\,dx \right]^{-1},
\]
where B = NH/fL. In contrast to the homogeneous situation, the right-hand side of (2) can be zero and is so if the obstacle is somewhere vertical. Such obstacles will produce a closed-streamline region no matter now small their height and will hence not lead to patterns representative of smooth obstacles. This is because a stratified column of fluid cannot be stretched or compressed over an infinitesimal distance. Instead, the column bends markedly and the fluid flows around the obstacle. The critical conditions (1) and (2) for a number of specific obstacles are calculated and discussed.
The macrodynamics of α-effect dynamos in rotating fluids
Authors: Maekus W. V. R. , Proctor M. R. E.
Published: March 2006

Past study of the large-scale consequences of forced small-scale motions in electrically conducting fluids has led to the ‘α-effect’ dynamos. Various linear kinematic aspects of these dynamos have been explored, suggesting their value in the interpretation of observed planetary and stellar magnetic fields. However, large-scale magnetic fields with global boundary conditions can not be force free and in general will cause large-scale motions as they grow. I n this paper the finite amplitude behaviour of global magnetic fields and the large-scale flows induced by them in rotating systems is investigated. In general, viscous and ohmic dissipative mechanisms both play a role in determining the amplitude and structure of the flows and magnetic fields which evolve. In circumstances where ohmic loss is the principal dissipation, it is found that determination of a geo- strophic flow is an essential part of the solution of the basic stability problem. Nonlinear aspects of the theory include flow amplitudes which are independent of the rotation and a total magnetic energy which is directly proportional to the rotation. Constant a is the simplest example exhibiting the various dynamic balances of this stabilizing mechanism for planetary dynamos. A detailed analysis is made for this case to determine the initial equilibrium of fields and flows in a rotating sphere.
A note on the total reflexion or transmission of surface waves in the presence of parallel obstacles
Authors: Evans D. V.
Published: March 2006

It is shown how a two-dimensional surface wave can be either totally reflected or totally transmitted in the presence of two parallel vertical barriers each containing a small gap. Total transmission of a surface wave past obstacles has been known to occur in many situations in water-wave theory, but total reflexion is a comparatively new phenomenon which could be of practical use in the design of breakwaters.
Anisotropic surface waves under a vertical electromagnetic force. Part 2. Experimental demonstration
Authors: Robinson I. S.
Published: March 2006

It was shown theoretically in part 1 (Shercliff 1969) that, when a horizontal magnetic field and electric current field are imposed upon a conducting liquid with a free surface, waves excited at the surface have an anisotropic dispersion relation. Here the possibility of demonstrating this phenomenon in the laboratory is considered. The analysis is completed for waves a t the interface between a conducting and non-conducting fluid of similar density, taking full account of surface tension, and then viscosity. Surface tension is found to have a considerable influence in reducing the degree of anisotropy to be expected. The problem of choosing suitable experimental parameters is discussed, and an apparatus is described in which it was possible to demonstrate the existence of anisotropic surface waves, and to compare the phase velocity with the theoretical results. Also, an experiment is described which verifies the relationship between the orientation of the anisotropy and the relative orientation of the imposed magnetic and current fields.
On the effects of a gravitational field on the turbulent transport of heat and momentum
Authors: Launder B. E.
Published: March 2006

This paper suggests a simple way of including gravitational effects in the pres-sure-containing correlations that appear in the equations for the transport of Reynolds stress and heat flux. The predicted changes in structure due to the gravitational field are shown to agree closely with Webster's (1964) measurements in a stably stratified shear flow.
Prediction of turbulent flow in curved pipes
Authors: Patankar S. V. , Pratap V. S. , Spalding D. B.
Published: March 2006

A finite-difference procedure is employed to predict the development of turbulent flow in curved pipes. The turbulence model used involves the solution of two differential equations, one for the kinetic energy of the turbulence and the other for its dissipation rate. The predicted total-velocity contours for the developing flow in a 180° bend are compared with the experimental data. Predictions of fully developed velocity profiles for long helically wound pipes are also presented and compared with experimental measurements.
On the gravitationally forced motions of a compressible fluid within a horizontally rotating cylinder
Authors: Gans Roger F.
Published: March 2006

The effect of gravity acting perpendicularly to the stratification induced by rotation in a compressible fluid is investigated. The γ = 1 approximation used by Gans (1974) in an investigation of the free modes of a rotating gas is used to find an inviscid solution. The boundary layers are calculated, and their suctions are shown to lead to a resonance. The amplitude of the resonance mode is calculated and seen to be of the same order as the simple inviscid solution, so that a corrected solution can be obtained.

The stability of the system is discussed in a simple fashion. It seems likely that a turbulent core will form in a cylinder. The interaction of such a core with the remainder of the fluid is beyond the scope of this paper. Finally a procedure by which a zonal flow field can be found is given.
Numerical models of hydromagnetic dynamos
Authors: Jepps S. A.
Published: March 2006

The magnetic induction equation is solved numerically in a sphere for a variety of prescribed fluid flows. The models considered are the so-called ‘αω dynamos’, in which both small-scale turbulence and large-scale shearing play a significant role. Solutions are obtained by marching the finite–difference equations forward in time from some initial field. For a critical value of the magnetic Reynolds numberRmsolutions which neither grow nor decay are found.

Further calculations are performed with a ‘cut-off effect’ in which an attempt is made to simulate the effect of the Lorentz forces on the turbulence. For supercritical values of R, the magnetic field now stabilizes a t a finite value instead of increasing indefinitely. The form of the final field is compared with that produced at criticalRmin the absence of the cut-off effect.
The expansion of a hypersonic turbulent boundary layer at a sharp corner
Authors: Bloy A. W.
Published: March 2006

A sharp wedge expansion flap was tested in the von Kámán Institute Long-shot tunnel at Mach 16 and data on the wall pressure and heat transfer were obtained. Pitot pressure measurements in the boundary layer just ahead of the expansion flap were also made. The surface data are compared with predictions from a characteristics solution for the boundary-layer expansion and from a simple heat-transfer theory.
On rotationally symmetric flow above an infinite rotating disk
Authors: Bodonyi R. J.
Published: March 2006

The similarity equations for rotationally symmetric flow above an infinite counter–rotating disk are investigated both numerically and analytically. Numerical solutions are found when α, the ratio of the disk's angular speed to that of the rigidly rotating fluid far from it, is greater than −0.68795. It is deduced that there exists a critical value αcr, of α above which finite solutions are possible. The value of α and the limiting structure as α → αcrare found using the method of matched asymptotic expansions. The flow structure is found to consist of a thin viscous wall region above which lies a thick inviscid layer and yet another viscous transition layer. Furthermore, this structure is not unique: there can be any number of thick inviscid layers, each separated from the next by a viscous transition layer, before the outer boundary conditions on the solution are satisfied. However, comparison with the numerical solutions indicates that a single inviscid layer is preferred.
Parametric instability of internal gravity waves
Authors: McEwan A. D. , Robinson R. M.
Published: March 2006

A continuously stratified fluid, when subjected to a weak periodic horizontal acceleration, is shown to be susceptible to a form of parametric instability whose time dependence is described, in its simplest form, by the Mathieu equation. Such an acceleration could be imposed by a large-scale internal wave field. The growth rates of small-scale unstable modes may readily be determined as functions of the forcing-acceleration amplitude and frequency. If any such mode has a natural frequency near to half the forcing frequency, the forcing amplitude required for instability may be limited in smallness only by internal viscous dissipation. Greater amplitudes are required when boundaries constrain the form of the modes, but for a given bounding geometry the most unstable mode and its critical forcing amplitude can be defined.

An experiment designed to isolate the instability precisely confirms theoretical predictions, and evidence is given from previous experiments which suggest that its appearance can be the penultimate stage before the traumatic distortion of continuous stratifications under internal wave action.

A preliminary calculation, using the Garrett & Munk (197%) oceanic internal wave spectrum, indicates that parametric instability could occur in the ocean at scales down to that of the finest observed microstructure, and may therefore have a significant role to play in its formation.
The refraction of head seas by a long ship
Authors: Ursell F.
Published: March 2006

It is known that head seas cannot travel without deformation along a horizontal cylinder of full constant cross-section. Calculations are given which indicate that the waves are refracted away from the axis of the cylinder. Similar refraction effects are found for waves generated by a pulsating source on the cylinder, and also for the Kelvin wave pattern generated by a long ship of nearly constant cross-sectiop moving with constant speed in the axial direction.
Lee waves in stratified flows with simple harmonic time dependence
Authors: Bell T. H.
Published: March 2006

The process of internal gravity wave generation by the simple harmonic flow (U=U0, cos ω0t) of a stably stratified fluid (Brunt–Väisälä frequencyN) over an obstacle is investigated in some detail. Attention is primarily directed to the behaviour of the solution in various limiting cases, and to estimating the flux of energy into the internal wave field. In general, waves are generated not only at the fundamental frequency ω0, but also at all of its harmonics. But, for values of ω0/Ngreater than about one half, the waves of fundamental frequency are dominant. For values of ω0/N, less than about one half, the quasi-static approximation, in which the problem is considered as a slowly-varying version of the classical lee wave problem, is found to provide a viable estimate for the wave field. The general solution is found to compare favourably with the limited available experimental data.
Draining from rapidly spinning tubes
Authors: Collins R. , Hoath M. T.
Published: March 2006

When a liquid-filled tube of radius b which is spinning about its axis with angular velocity ω is allowed to drain by opening one end, a cavity propagates in the tube with velocity U. This velocity has been measured with the tube in both vertical and horizontal positions for large values of the rat,io ω2b/g of the centrifugal to the gravitational force, g being the gravitational acceleration. It is found that the Rossby number Uωb becomes constant and independent of ω2/g in both cases, taking the value U/ωb = 0.52 in the vertical case and U/Ωb ≃ 0.4 in the horizontal case. Experiments conducted in a tube which is slightly inclined to the horizontal confirm that, although the Rossby number is constant when ω2b/g is large, the value adopted depends upon α, the inclination of the tube to the gravitational field.
Hydromechanics of low-Reynolds-number flow. Part 2. Singularity method for Stokes flows
Authors: Chwang Allen T. , Wu T. Yao-Tsu
Published: March 2006

The present study further explores the fundamental singular solutions for Stokes flow that can be useful for constructing solutions over a wide range of free-stream profiles and body shapes. The primary singularity is the Stokeslet, which is associated with a singular point force embedded in a Stokes flow. From its derivatives other fundamental singularities can be obtained, including rotlets, stresslets, potential doublets and higher-order poles derived from them. For treating interior Stokes-flow problems new fundamental solutions are introduced; they include the Stokeson and its derivatives, called the roton and stresson.

These fundamental singularities are employed here to construct exact solutions to a number of exterior and interior Stokes-flow problems for several specific body shapes translating and rotating in a viscous fluid which may itself be providing a primary flow. The different primary flows considered here include the uniform stream, shear flows, parabolic profiles and extensional flows (hyper-bolic profiles), while the body shapes cover prolate spheroids, spheres and circular cylinders. The salient features of these exact solutions (all obtained in closed form) regarding the types of singularities required for the construction of a solution in each specific case, their distribution densities and the range of validity of the solution, which may depend on the characteristic Reynolds numbers and governing geometrical parameters, are discussed.
On the continuum theory for the large Reynolds number spherical expansion into a near vacuum
Authors: Freeman N. C. , Johnson R. S. , Kumar S. , Bush W. B.
Published: March 2006

The steady, spherically symmetric flow of a compressible gas is considered. The gas is both viscous and heat-conducting. In the limit of very high Reynolds number (= α−1, α → 0) and correspondingly low pressure at infinity, the structure of the whole flow field is discussed. The five regions that arise by virtue of the limit α → 0 are briefly considered. Special care is given to the matching across the overlap domains and the first region (close to, but outside, the sonic point) and the fifth (where the pressure adjusts to its ambient value) are carefully examined. It is argued that the application of appropriate matching principles, together with judicious use of numerical solutions, allows an arbitrary pressure and temperature to be assigned to the background gas.
Modal equations for cellular convection
Authors: Gough D. O. , Spiegel E. A. , Toomre Juri
Published: March 2006

We expand the fluctuating flow variables of Boussinesq convection in the planform functions of linear theory. Our proposal is to consider a drastic truncation of this expansion as a possibly useful approximation scheme for studying cellular convection. With just one term included, we obtain a fairly simple set of equations which reproduces some of the qualitative properties of cellular convection and whose steady-state form has already been derived by Roberts (1966). This set of ‘modal equations’ is analysed at slightly supercritical and at very high Rayleigh numbers. In the latter regime the Nusselt number varies with Rayleigh number just as in the mean-field approximation with one horizontal scale when the boundaries are rigid. However, the Nusselt number now depends also on the Prandtl number in a way that seems compatible with experiment. The chief difficulty with the approach is the absence of a deductive scheme for deciding which planforms should be retained in the truncated expansion.
Variational inequalities for a body in a viscous shearing flow
Authors: Nir A. , Weinberger H. F. , Acrivos A.
Published: March 2006

The slow motion of a body in a viscous shearing field is examined. Variational principles are used to derive inequalities which approximate the elements of the shearing matrix M of a body of arbitrary shape, where M is the matrix relating the force, torque and stresslet exerted by the body on the fluid to the relative translational and rotational velocities of the body and the rate of deformation of the undisturbed linear field. An upper bound for the elements of M is obtained by showing that the quadratic form of M increases monotonically with B, the region occupied by the body, while a lower bound for this form is given in terms of the electrostatic properties of a conductor and a dielectric of the same shape as B. Particular attention is paid to bodies of revolution, for which certain more definitive results are obtained: for example, their resistance to a rotation with axial symmetry is always less than twice their resistance to a rotation perpendicular to their axis.
The viscous nonlinear symmetric baroclinic instability of a zonal shear flow
Authors: Walton I. C.
Published: March 2006

The stability of a baroclinic zonal current to symmetric perturbations on a meridionally unboundedf-plane is considered. The lower boundary is at rest but the upper one moves with a constant velocity in keeping with the velocity of the zonal current. Following Stone (1966) a horizontal length scaleO(Ro) is taken, whereRois the Rossby number, with the Richardson numberRi=O(1). Instability sets in when the wavelength isO(E1/3), whereEis the Ekman number based on the distance between the rigid horizontal boundaries, which corresponds to Stone's inviscid value zero, and to McIntyre's (1970) value infinity on a length scaleO(E½).

A nonlinear analysis about the point of onset of instability yields the result that for the monotonic mode zonal momentum is convected polewards. The possible implications of this result for the dynamics of Jupiter's atmosphere are discussed.