bluuebunny/crossref_metadata_2025 huggingface dataset:
On the noise sources of the unsuppressed high-speed jet
Authors: Bishop K. A. , Williams J. E. Ffowcs , Smith W.
Published: March 2006

The paper describes an interpretation of jet-noise theory and scale-model experiments to highlight physical properties of jet-noise sources at very high speed. The study is prompted by current efforts to suppress the noise of supersonic transport aircraft.

The principal noise sources are shown to be very large-scale wave-like undulations of the jet flow that travel downstream at supersonic speed for a distance of several jet diameters. These motions are relatively well ordered and are probably more akin to recognizable instabilities of a laminar flow than the confused small-scale turbulence. Because of this we postulate a model of the noise generating motions as the instability products of a jet flow of low equivalent Reynolds number. This Reynolds number is based on an eddy viscosity and can be further reduced by artificially increasing the small-scale turbulence level. This step would tend to stabilize the flow and inhibit the formation of large-scale noise producing eddies.
Rotating flow over shallow topographies
Authors: Vaziri Arsalan , Boyer Don L.
Published: March 2006

The flow of a rotating homogeneous incompressible fluid over various shallow topographies is investigated. In the physical system considered, the rotation axis is vertical while the topography and its mirror image are located on the lower and upper of two horizontal plane surfaces. Upstream of the topographies and outside the Ekman layers on the bounding planes the fluid is in a uniform free-stream motion. An analysis is considered in which E [Lt ] 1, Ro ∼ E½, H/D ∼ E0, and h/D ∼ E½, where E is the Ekman number, Ro the Rossby number, H/D the fluid depth to topography width ratio and h/D the topography height-to-width ratio. The governing equation for the lowest-order interior motion is obtained by matching an interior geostrophic region with Ekman boundary layers along the confining surfaces. The equation includes contributions from the non-linear inertial, Ekman suction, and topographic effects. An analytical solution for a cosine-squared topography is given for the case in which the inertial terms are negligible; i.e. Ro [Lt ] E½. Numerical solutions for the non-linear equations are generated for both cosine-squared and conical topographies. Laboratory experiments are presented which are in good agreement with the theory advanced.
The flow due to an oscillating piston in a cylindrical tube: a comparison between experiment and a simple entrance flow theory
Authors: Gerrard J. H. , Hughes M. D.
Published: March 2006

The velocity on the axis of a circular tube was measured over a range of distances from a piston reciprocating in simple harmonic motion. These velocities become independent of axial distance sufficiently far from the piston. The method of calculating the developing flow is based on a comparison with steady laminar flow which, in the entry region of a circular tube, approaches the fully developed state exponentially with distance x from the entry. The steady flow is a function of xν/R2u0 where ν is the kinematic viscosity, R is the tube radius and u0 is the entry velocity. It is shown that within the limits of experimental error, an oscillating flow follows the steady flow development if u0 is the instantaneous entry velocity and if the characteristic length is changed from R to the oscillating boundary-layer thickness in the established flow.
Weak quadratic interactions of two-dimensional waves
Authors: Kim Young Yuel , Hanratty Thomas J.
Published: March 2006

This paper reports on weak quadratic interactions which can occur with two-dimensional waves on shallow water layers and in the capillary-gravity range on deep water layers. It supplies experimental support of theoretical predictions for resonant interactions, but, perhaps of more significance, it explores in detail interactions which occur under conditions near resonance.

Waves of approximately sinusoidal form are introduced on the surface of water in a long rectangular tank. For deep water a rapid distortion in the sinusoidal wave and sometimes additional crests are observed because of energy exchange among the first, second and third harmonics at frequencies where both surface tension and gravity are important (7·5–13 c/s). An even greater exchange of energy can be observed on shallow water layers at low frequencies. For example, a wave train with seven secondary crests can be observed when the wave maker is operated at 3·04 c/s in a water layer of 0·65 cm.

Measured amplitudes and phase angles of the Fourier components of the wave train are described by a system of equations using only quadratic interactions among participating harmonics. The exchange of energy among Fourier components under certain conditions is explained in terms of the rate of change of relative phase angles of the different harmonics.
The production of turbulence near a smooth wall in a turbulent boundary layer
Authors: Kim H. T. , Kline S. J. , Reynolds W. C.
Published: March 2006

The structure of the flat plate incompressible smooth-surface boundary layer in a low-speed water flow is examined using hydrogen-bubble measurements and also hot-wire measurements with dye visualization. Particular emphasis is placed on the details of the process of turbulence production near the wall. In the zone 0 < y+ < 100, the data show that essentially all turbulence production occurs during intermittent ‘bursting’ periods. ‘Bursts’ are described in some detail.

The uncertainties in the bubble data are large, but they have the distinct advantage of providing velocity profiles as a function of time and the time sequences of events. These data show that the velocity profiles during bursting periods assume a shape which is qualitatively distinct from the well-known mean profiles. The observations are also used as the basis for a discussion of possible appropriate mathematical models for turbulence production.
Non-linear effects in steady supersonic dissipative gasdynamics Part 1. Two-dimensional flow
Authors: Chong T. H. , Sirovich L.
Published: March 2006

Steady supersonic two-dimensional flows governed by the Navier–Stokes equations are considered. For flows past a thin body, the Oseen theory is shown to fail at large distances. An investigation of the equations bridging the linear and non-linear zones is made. From this, it follows that the resulting equations are a system of Burgers and diffusion equations. The Whitham theory is shown to result under the inviscid limit of our analysis. Various other limits are also obtained.

An explicit expression for flows past a thin airfoil is given, and the flow past a double wedge is exhibited in terms of known functions.
Velocity measurements in regions of upstream influence of a body in aligned-fields MHD flow
Authors: Lake Bruce M.
Published: March 2006

Experiments are described in which velocities were measured ahead of a semi-infinite Rankine body moving parallel to a uniform magnetic field in a conducting fluid. The flow disturbance in front of the body is found to increase in length asN½, whereNis the interaction parameter. In most of the experiments this parameter was varied from 4 to about 50. Measurements made along the axis of symmetry in the flow show that there is a relatively short region of stagnant fluid directly ahead of the body. The major part of the disturbance is found to consist of a much longer region in which the flow undergoes transition from conditions in the free stream to conditions near the body. Velocity profiles across the flow in this region show that for increasedN, at a fixed distance ahead of the body, the velocity defect increases and the disturbance becomes more confined radially. Although the radial gradients in the flow increase withN, they are found to be much smaller than would be expected in a flow containing thin current layers. A physical model of the flow which has currents and pressures consistent with these results is discussed.
The determination of the Kolmogoroff constants for velocity, temperature and humidity fluctuations from second- and third-order structure functions
Authors: Paquin J. E. , Pond S.
Published: March 2006

Second- and third-order structure functions were computed from velocity, temperature and humidity fluctuations in the wind over the ocean. Universal inertial-convective subrange constants (Kolmogoroff constants) were computed from these structure functions. The constant for velocity is consistent with other recent observations. The temperature and humidity constants are found to be equal within the measurement error and have values of about 0·8.
Experiments on internal intermittency and fine-structure distribution functions in fully turbulent fluid
Authors: Kuo Albert Yi-Shuong , Corrsin Stanley
Published: March 2006

Spatial ‘intermittency’ in the velocity field fine-structure of fully turbulent flow regions, first observed by Batchelor & Townsend (1949), is studied further here in grid-generated nearly isotropic turbulence and on the axis of a round jet. At large enough Reynolds numbers, appropriately filtered hot-wire anemometer signals appear intermittent as the turbulent patterns are convected past the hot wire by the mean flow. Measurements show that there is a decrease in the relative fluid volume (equal to the ‘intermittency factor’) occupied by fine-structure of given size as the turbulence Reynolds number is increased. They show also that, for a fixed Reynolds number, the relative volume is smaller for smaller fine-structure. The average linear dimension of the fine-structure regions turns out to be much larger than the sizes of fine-structure therein. At Rλ, = 110, for example, the ratio ranges from 15 to 30, decreasing with decreasing ‘eddy’ size. It appears to be approaching an asymptote with increasing Rλ.

The flatness factors and probability distributions of the first derivative, the second derivative, band-passed and high-passed velocity fluctuation signals were also measured. The turbulence Reynolds numbers Rλ ranged from 12 to 830. The flatness factors of the first and the second derivatives increase monotonically with Rλ. Those of the second derivative vary with Rλ0.25 for Rλ < 100, and with Rλ0.75 for Rλ > 300. No indication of asymptotic constant values was observed for Rλ up to the order of one thousand.

The probability distributions of velocity fluctuations and large-scale signals are nearly normal, while the small-scale signals are not. The flatness factor of the filtered band-pass velocity signal increases with increasing frequency.

At the larger Reynolds numbers, the square of the signal associated with large wave-numbers may be approximated by a log-normal probability distribution for amplitudes when probabilities fall between 0·3 and 0·95, in limited agreement with the theory of Kolmogorov (1962), Oboukhov (1962), Gurvich & Yaglom (1967).
Non-linear resonant instability in boundary layers
Authors: Craik Alex D. D.
Published: March 2006

An investigation is made of resonant triads of Tollmien-Schlichting waves in an unstable boundary layer. The triads considered are those comprising a two-dimensional wave and two oblique waves propagating at equal and opposite angles to the flow direction and such that all three waves have the same phase velocity in the downstream direction. For such a resonant triad remarkably powerful wave interations take place, which may cause a continuous and rapid transfer of energy from the primary shear flow to the disturbance. It appears that the oblique waves can grow particularly rapidly and it is suggested that such preferential growth may be responsible for the rapid development of three-dimensionality in unstable boundary layers. The non-linear energy transfer primarily takes place in the vicinity of the critical layer where the downstream propagation velocity of the waves equals the velocity of the primary flow.

The theoretical analysis is initially carried out for a general primary velocity profile; then, in order to demonstrate the essential features of the results, precise interaction equations are derived for a particular profile consisting of a layer of constant shear bounded by a uniform flow. Some exact solutions of the general interaction equations are presented, one of which has the property that the wave amplitudes become indefinitely large at a finite time. The possible relevance of the present theoretical model to the experiments of Klebanoff, Tidstrom & Sargent (1962) is examined.
The stability of a water drop oscillating with finite amplitude in an electric field
Authors: Brazier-Smith P. R.
Published: March 2006

By assuming that an uncharged drop situated in a uniform electric fieldEretains a spheroidal shape while oscillating about its equilibrium configuration, two approximate equations of motion are derived for the deformation ratio γ expressed as the ratioa/bof the major and minor axis of the drop. Solutions of these equations of motion indicate that the stability of a drop of undistorted radiusRand surface tensionTdepends uponE(R/T)½and the initial displacement of γ from its equilibrium value. The predictions of the two equations are compared to assess the accuracy of the spheroidal assumption as applied to such a dynamical situation. The analysis is used to determine the stability criterion of a drop subject to a step function field. Finally, the limit of validity of the spheroidal assumption is discussed in terms of Rayleigh's criterion for the stability of charged spherical drops. By applying Rayleigh's criterion to the poles of a spheroidal drop, the stage at which the drop departs from spheroidal form to form conical jets was approximately determined.
A non-linear investigation of critical levels for internal atmospheric gravity waves
Authors: Breeding R. J.
Published: March 2006

The behaviour of internal gravity waves near a critical level is investigated by means of a transient two dimensional finite difference model. All the important non-linear, viscosity and thermal conduction terms are included, but the rotational terms are omitted and the perturbations are assumed to be incompressible. For Richardson numbers greater than 2·0 the interaction of the incident wave and the mean flow is largely as predicted by the linear theory–very little of the incident wave penetrates through the critical level and almost all of the wave's energy and momentum are absorbed by changes in the original wind. However, these changes in the wind are centred above the critical level, so that the change in the wind has only a small effect on the height of the critical level. For Richardson numbers less than 2·0 and greater than 0·25 a significant fraction of the incident wave is reflected, part of which could have been predicted by the linear theory. For these stable Richardson numbers a steady state is apparently reached where the maximum wind change continues to grow slowly, but the minimum Richardson number and wave magnitudes remain constant. This condition represents a balance between the diffusion outward of the added momentum and the rate at which it is absorbed. For Richardson numbers less than 0·25, over-reflexion, predicted from the linear theory, is observed, but because the system is dynamically unstable no over-reflecting steady state is ever reached.
Spin-up of a stratified fluid: theory and experiment
Authors: Buzyna George , Veronis George
Published: March 2006

Stratified spin-up, the process of adjustment of a uniformly rotating stratified fluid to an abrupt change in the rotation of the container, is important in many geophysical contexts. An experimental study of this process is presented here for the case where a linearly stratified salt solution is enclosed in a cylindrical container whose rotation rate is changed by a small amount. Results are presented for a limited range of values of B, the internal Froude number, which measures the ratio of the frequencies due to buoyancy and rotation. The experimental study is augmented by a theoretical treatment of idealized models which clarify the more fundamental physical processes that occur. The response of a stratified fluid is faster than that of a homogeneous fluid but the adjustment is limited to layers near the bottom and top boundaries the thickness of which is determined by the value of B. A comparison of the experimental results with the theories of Holton, Walin and Sakurai is also made and it is shown that for the present physical arrangement (insulated side walls) the theories of the latter two authors agree much more closely with experiment than does the theory of Holton. However, all three theories tend to over-estimate the azimuthal displacement in the regions near the upper and lower boundaries where the spin-up is most rapid. The Sweet-Eddington circulation, which accompanies the ideal state of rigid-body rotation, can be significant under normal laboratory conditions and it was necessary to correct some of the spin-up results for this effect. The circulation in the vertical plane is described qualitatively.
Hydromagnetic spin-up of a fluid confined by two flat electrically conducting boundaries
Authors: Loper David E.
Published: March 2006

The prototype linear spin-up problem consisting of a homogeneous viscous electrically conducting fluid confined between two infinite flat rotating electrically conducting plates in the presence of an applied axial magnetic field is studied in an effort to understand better the strength and nature of the coupling between a fluid and its rotating conducting container. It is assumed that the response time of the bounding plates to a magnetic perturbation is much less than the fluid spin-up time and that the plate conductivity is an arbitrary function of distance from the fluid-plate interface. The general Laplace transform solution is inverted and discussed for three special cases: magnetic diffusion regions thick compared with fluid depth during spin-up, arbitrary magnetic field strength and boundary conductance; magnetic diffusion regions thin, weak conductance, arbitrary field; magnetic diffusion regions thin, strong conductance, arbitrary field. In each case conductance of the boundary strengthens the coupling between fluid and boundary, thereby decreasing the spin-up time. The corresponding single plate analysis of Loper (1970a) is found to predict spin-up accurately only if the boundary conductance is much smaller than that of the fluid. The fluid possesses an oscillatory mode of spin-up if the magnetic diffusion regions are thin and boundary conductance is large. That is, the inviscid current-free core of fluid rotates significantly faster than the boundaries during a portion of the spin-up process.
An exact periodic solution of the energy equation
Authors: Ishigaki Hiroshi
Published: March 2006

An exact periodic solution of the unsteady energy equation for an incompressible fluid with constant properties is derived to illustrate the effect of an oscillation through the viscous dissipation on a temperature field. The flow field used here is a generalization of the well-known Couette flow solution of steady flow, in which one wall is at rest and the other wall oscillates in its own plane about a constant mean velocity. The solution is subject to two boundary conditions that correspond to the heat-transfer and thermometer problems. In order to have some suggestions about the approximate solutions, the solution is compared with its own approximate form. The temperature field consists of a time-mean, first and second harmonic fluctuation. The time-mean temperature profiles show the large influence of oscillation. The time-mean heat flux into or the time-mean temperature of the oscillating wall increases with frequency, and is ultimately proportional to the square root of the frequency. In § 4 the present exact solution of the Couette flow is compared with the formerly obtained approximate solution of the flat plate boundary-layer flow in terms of the wall characteristic values at high frequencies.
Blast wave propagation in an inhomogeneous atmosphere
Authors: Sachdev P. L.
Published: March 2006

The Brinkley–Kirkwood theory (1947) is modified to determine the law of propagation of a blast wave in an arbitrary inhomogeneous medium for spherically and cylindrically symmetric cases. The shock path is obtained in terms of a simple quadrature. The numerical results for the shock path and the entire flow region behind the shock, propagating in an exponential atmosphere, show excellent agreement with the exact numerical solution.
A numerical investigation into the effects of compressibility and total enthalpy difference on the development of a laminar free shear layer
Authors: Carpenter Peter W.
Published: March 2006

A method is presented for integrating numerically the equations of motion for a compressible free shear layer developing from a boundary-layer profile of arbitrary shape. Sutherland's law is used to determine the coefficient of viscosity and the Prandtl number is taken as 0·72. Calculated results are reported for free-stream Mach numbers ranging from 0 to 10 and for stagnation-enthalpy ratios ranging from 0 to 5·0. The effects of varying the initial boundary-layer profile and of a discontinuity in temperature at the origin are also studied. The results include graphs showing the development of dividing-streamline velocity, of local Nusselt number, and of dividing-streamline location.
Numerical solutions for steady symmetric viscous flow past a parabolic cylinder in a uniform stream
Authors: Dennis S. C. R. , Walsh J. D.
Published: March 2006

Numerical solutions are presented for steady two-dimensional symmetric flow past a parabolic cylinder in a uniform stream parallel to its axis. The solutions cover the range R = 0·25 to ∞, where R is the Reynolds number based on the nose radius of the cylinder. For large R, the calculated skin friction near the nose of the cylinder is compared with known theoretical results obtained from second-order boundary-layer theory. Some discrepancy is found to exist between the present calculations and the second-order theory. For small R, it is possible to obtain a reasonably consistent check with a recent theoretical prediction for the limit of the skin friction near the nose of the cylinder as R → 0.
Vibration of hot-wire anemometer filaments
Authors: Perry A. E. , Morrison G. L.
Published: March 2006

Hot-wire filament vibration generated by fluctuating aerodynamic loads is found to occur in turbulent flow fields. The motion of the filament relative to the stream can cause errors in the indicated velocity fluctuations. Analysis and experiment show that these errors are probably serious only when the velocity fluctuations are confined within a narrow bandwidth spectrum such as in a Kármán vortex street. Under such conditions the filament goes into a skipping or ‘whirling’ mode of motion and the errors in the velocity perturbation reading are substantial. Errors in normally encountered boundary-layer turbulence are suspected not to be serious but this is not yet certain. A convenient experimental method for detecting these vibrations is outlined.

A study is made of the effect of filament vibration on the direct frequency-response test for hot-wire systems. These tests include shaking the probe at high frequencies in a steady stream and the Kármán vortex street method.
Finite amplitude instability of time-dependent flows
Authors: Davis Stephen H.
Published: March 2006

An extension of the Stuart–Watson technique for examination of the non-linear hydrodynamic instability of time-dependent flows is proposed. An example in Bénard convection is used to illustrate the method. Extensions to more general problems are indicated.
On hydromagnetic precession in a cylinder
Authors: Gans Roger F.
Published: March 2006

The hydrodynamic theory of the resonant cylinder (Gans 1970) is extended to include the effects of a magnetic field parallel to the rotation axis. The linear response is modified by a change in boundary-layer suction and a change in the resonant length. These effects are of equal importance. The theory is valid for small container conductivity and for amplitudes such that the cube of the amplitude is less than the dimensionless precession rate. The importance of container conductivity is assessed. The free modes of the system are given in an appendix. These modes move both east and west.

Experimental apparatus capable of producing magnetic Reynolds numbers of the order of 20 for indefinite lengths of time is described. The apparatus was used to assess the linear theory, though not designed for this purpose. Experiments beyond the range of linear theory are described. The results show finite amplitude effects similar to those previously observed in precessing spheroids in the absence of magnetic effects (Malkus 1968). Additional structure attributable to magnetic effects is observed.
The three-dimensional distributions of tangential velocity and total-temperature in vortex tubes
Authors: Linderstrøm-Lang C. U.
Published: March 2006

The axial and radial gradients of the tangential velocity distribution are calculated from prescribed secondary flow functions on the basis of a zero-order approximation to the momentum equations developed by Lewellen. It is shown that secondary flow functions may be devised which meet pertinent physical requirements and which at the same time lead to realistic tangential velocity gradients.

The total-temperature distribution in both the axial and radial directions is calculated from such secondary flow functions and corresponding tangential velocity results on the basis of an approximate turbulent energy equation. The method employed for the solution of this equation stresses the equivalence of the vortex tube to counter-current systems with transverse diffusion such as distillation columns and heat exchangers.

An availability function is derived that permits the evaluation of vortex tube performance on the basis of velocity data.

Turbulent diffusivities resulting from the quantitative use of the tangential velocity approximation are shown to agree with those derived from the total-temperature calculations.
Viscous flow in a deformable rotating container
Authors: Suess Steven T.
Published: March 2006

The steady-state flow in a rotating container is examined when the container is deformed into an ellipsoidal figure which is fixed in inertial space. The analysis is carried out for a fluid of small viscosity, and to the second order in Rossby number. Results show a vortex existing along the rotation axis, this being an interior manifestation of non-linear boundary-layer effects. These effects are a direct consequence of the forced oscillation being at twice the rotational frequency. The configuration is chosen to suggest the effects of gravitational body tides on the dynamics of the core of the earth. This problem is related to the similar problem of a precessing spheroid, and an experiment is described which tests the present theory and confirms previous experimental data from precessing spheroids.
A note on vortex streets behind circular cylinders at low Reynolds numbers
Authors: Tritton D. J.
Published: March 2006

A discussion is given of the current state of knowledge of vortex streets behind circular cylinders in the Reynolds number range 50 to 160. This was prompted by Gaster's (1969) report that he could not find the transition at a Reynolds number of about 90 observed by Tritton (1959) and Berger (1964a). A further brief experiment confirming the existence of the transition is described Reasons for rejecting Gaster's interpretation are advanced. Possible (mutually alternative) explanations of the discrepant observations are suggested.
The flow and stability of thin liquid films on a rotating disk
Authors: Charwat A. F. , Kelly R. E. , Gazley C.
Published: March 2006

Measurements of the thickness and the stability of thin films of liquid (1–150 μmthick) formed on a rotating horizontal disk are presented and correlated in terms of an asymptotic-expansion solution of the thin-film equations. Water, various alcohols and water with wetting activities were used to cover a range of viscosity (1-2.5cP) and surface tension (20-72 dynes/cm). Smooth flow was found to occur in a region defined by the flow rate, rotational speed and physical properties of the liquid. Outside this region various wave patterns were observed (concentric, spiral and irregular waves). A linear theory of the stability of the film based on an extension of classical stability theories for plane films on inclined planes is given and contrasted with the experimental results. Surface phenomena associated with the use of wetting agents were found to have a strong effect on the stability of the film.
Jets induced by oscillatory motion
Authors: Davidson B. J. , Riley N.
Published: March 2006

The flow induced by a cylinder performing transverse periodic oscillations in a fluid which is otherwise at rest is investigated. The motion of the cylinder is not necessarily harmonic. Attention is focused upon the induced steady streaming. In particular, the jet-like flow which arises when the streaming Reynolds number is large is studied. The theoretical results obtained are applied to a class of elliptic cylinders, and a simple experimental investigation is carried out which substantiates the main theoretical conclusions.
An approach to a dynamic theory of dynamo action in a rotating conducting fluid
Authors: Moffatt H. K.
Published: March 2006

Dynamo action associated with the motion generated by a random body force f(x, t) in a conducting fluid rotating with uniform angular velocity ω is considered. It is supposed that, in the Fourier decomposition off, only waves having a phase velocity V satisfying V. ω > 0 are present and that the Fourier amplitudes of f are isotropically distributed. The resulting velocity field then lacks reflexional symmetry, and energy is transferred to a magnetic field h0(x, t) provided the scale L of h0 is sufficiently large. Attention is focused on a particular distribution of h0 (x, t) (a circularly polarized wave) for which this dynamo action is most efficient. Under these conditions, the mean stresses acting on the fluid are irrotational and no mean flow develops. It is supposed that
\[
\lambda \ll \Omega l^2,\quad \lambda\ll h_0l\quad{\rm and}\quad\nu/\lambda = O(1)\quad\hbox{or smaller},
\]
where l ([Lt ] L) is the scale of the f-field, and ν and λ are the kinematic viscosity and magnetic diffusivity of the fluid. The response to f is then dominated by resonant contributions near the natural frequencies of the free undamped system. As h0 grows in strength, these frequencies change, and the dynamo process is rendered less efficient. Ultimately the magnetic energy M (and also the kinetic energy E) asymptote to steady values. Expressions for these values are obtained for the particular situation when ν [Lt ]λ and when the frequency ω0 characteristic of the f-field is small compared with other relevant frequencies, notably ω and h0/l; under these conditions, it is shown that
\[
\frac{M}{E}\sim C\left(\frac{\Omega}{\omega_0}\right)^{\frac{1}{2}}\left(\frac{\nu}{\lambda}\right)^{\frac{1}{2}}\frac{L}{l},
\]
where C is a number of order unity determined by the spectral properties of the f-field. The implications for the terrestrial dynamo are discussed.
Some numerical solutions of a variable-coefficient Korteweg-de Vries equation (with applications to solitary wave development on a shelf)
Authors: Johnson R. S.
Published: March 2006

Some numerical solutions of a variable-coefficient Korteweg-de Vries equation are presented. This particular equation was derived by the author recently (Johnson 1972) in an attempt to describe the development of a single solitary wave moving onto a shelf. Soliton production on the shelf was predicted and this is confirmed here. Results for two and three solitons are reproduced and two intermediate shelf depths are also considered. In these latter two cases both solitons and an oscillatory wave occur. One of the profiles corresponds to the integrations performed by Madsen & Mei (1969) and a comparison is made.
On steady convection in a porous medium
Authors: Palm Enok , Weber Jan Erik , Kvernvold Oddmund
Published: March 2006

For convection in a porous medium the dependence of the Nusselt number on the Rayleigh number is examined to sixth order using an expansion for the Rayleigh number proposed by Kuo (1961). The results show very good agreement with experiment. Additionally, the abrupt change which is observed in the heat transport at a supercritical Rayleigh number may be explained by a breakdown of Darcy's law.
Nonlinear internal gravity waves in a slowly varying medium
Authors: Grimshaw R.
Published: March 2006

Nonlinear internal gravity waves in an inviscid incompressible fluid are discussed for the case when the properties of the medium vary slowly on a scale determined by the local wave structure. A two-time-scale technique is used to obtain transport equations which describe the slowly varying modulations of the waves. Various solutions of these transport equations are discussed.
Propagation of Alfvén-gravitational waves in a stratified perfectly conducting flow with transverse magnetic field
Authors: Rudraiah N. , Venkatachalappa M.
Published: March 2006

Alfvén-gravitational waves are found to propagate in a Boussinesq, inviscid, adiabatic, perfectly conducting fluid in the presence of a uniform transverse magnetic field in which the mean horizontal velocity U is independent of vertical height z. The governing wave equation is a fourth-order ordinary differential equation with constant coefficients and is not singular when the Doppler-shifted frequency Ωd = 0, but is singular when the Alfvén frequency ΩA = 0.If Ω2d < Ω2A the waves are attenuated by a factor exp − [2ΩA(N2−Ω2d)½−Ω2d + Ω2A]z, which tends to zero as z → ∞. This attenuation is similar to the viscous attenuation of waves discussed by Hughes & Young (1966). The interpretation of upward and downward propagation of waves is given.
The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments
Authors: Reynolds W. C. , Hussain A. K. M. F.
Published: March 2006

The dynamical equations governing small amplitude wave disturbances in turbulent shear flows are derived. These equations require additional equations or assumptions about the wave-induced fluctuations in the turbulence Reynolds stresses before a closed system can be obtained. Some simple closure models are proposed, and the results of calculations using these models are presented. When the predictions are compared with our data for channel flow, we find it essential that these oscillations in the Reynolds stresses be included in the model. A simple eddy-viscosity representation serves surprisingly well in this respect.
Roll-diameter dependence in Rayleigh convection and its effect upon the heat flux
Authors: Willis G. E. , Deardorff J. W. , Somerville R. C. J.
Published: March 2006

The average roll diameter in Rayleigh convection for 2000 < R < 31000, where R is the Rayleigh number, has been measured from photographs of three convecting fluids: air, water and a silicone oil with a Prandtl number σ of 450. For air the average dimensionless roll diameter was found to depend uniquely upon R and to increase especially rapidly in the range 2000 < R < 8000. The fluids of larger σ exhibited strong hysteresis but also had average roll diameters tending to increase with R. The increase in average roll diameter with R tended to decrease with σ. Through use of two-dimensional numerical integrations for the case of air it was found that the increase in average roll diameter with R provides an explanation for the usual discrepancy in heat flux observed between experiment and two-dimensional numerical calculations which prescribe a fixed wavelength.
Sting-free measurements of sphere drag in laminar flow
Authors: Vlajinac M. , Covert E. E.
Published: March 2006

An aerodynamic investigation was conducted to determine the laminar-flow drag coefficient of spheres of various sizes in a subsonic wind tunnel. The tests were conducted using the M.I.T.-N.A.S.A. prototype magnetic-balance system. By measuring the drag of different sized spheres without model support interference the tunnel wall effect can be deduced. The present results indicate that the classical wind tunnel correction does not completely account for the effects of model size and wall interference. That is, the corrected drag coefficient data for the different sphere sizes differ among themselves in the region of Reynolds number overlap.

A comparison of the present sphere drag results with those of numerous other investigations including free-flight and ballistic-range data is given. The drag coefficients presented here are slightly lower than those of other workers for Reynolds numbers ranging from 20 000 to 150 000, but fall between the limits of experimental scatter for Reynolds numbers from 150 000 to 260 000.

An analysis of the estimated error in the present data indicates the primary source to be measurement of the wind tunnel parameters rather than errors resulting from the balance system.
Non-local effects in the stability of flow between eccentric rotating cylinders
Authors: Diprima R. C. , Stuart J. T.
Published: March 2006

In this paper the linear stability of the flow between two long eccentric rotating circular cylinders is considered. The problem, which is of interest in lubrication technology, is an extension of the classical Taylor problem for concentric cylinders. The basic flow has components in the radial and azimuthal directions and depends on both of these co-ordinates. As a consequence the linearized stability equations arepartial differential equationsrather than ordinary differential equations. Thus standard methods of stability theory are not immediately useful. However, there are two small parameters in the problem, namely δ, the clearance ratio, and ε, the eccentricity. By letting these parameters tend to zero in such a way that δ½ is proportional to ε, a global solution to the stability problem is obtained without recourse to the concept of ‘local instability’, or ‘parallel-flow’ approximation, so commonly used in boundary-layer stability theory. It is found that the predictions of the present theory are at variance with what is given by a ‘local’ theory. First, the Taylor-vortex amplitude is found to be largest at about 90° downstream of the region of ‘maximum local instability’. This result is given support by some experimental observations of Vohr (1968) with δ = 0·1 and ε = 0·475, which yield a corresponding angle of about 50°. Second, the critical Taylor number rises with ε, rather than initially decreasing with ε as predicted by local stability theory using the criteria of maximum local instability. The present prediction of the critical Taylor number agrees well with Vohr's experiments for ε up to about 0·5 when δ = 0·01 and for ε up to about 0·2 when δ = 0.1.
Propagation of weak shock waves through turbulence
Authors: Plotkin Kenneth J. , George A. R.
Published: March 2006

The effect of turbulence on the structure of weak shock waves is investigated. The equilibrium structure is shown to be governed by a balance between nonlinear steepening and the turbulent scattering of acoustic energy out of the main wave direction. The scattered energy appears as perturbations behind the shock front. For conditions typical of sonic booms in atmospheric turbulence the wave structure is governed by a Burgers equation similar to that describing viscous shocks, except that parameters related to the turbulence appear instead of the viscosity coefficient. The magnitude of the perturbations following a shock is estimated from first-order scattering applied to a thickened shock. Predictions of shock thicknesses and perturbations compare favourably with available experimental data. The approach used in the analysis of shock structure is to account for energy scattered from a single wave propagating a long distance through turbulence. This avoids difficulties of physical interpretation which arise if an ensembleaveraged structure is calculated, which is the usual approach in turbulent scattering analysis.
Nonlinear energy transfer in gravity–capillary wave spectra, with applications
Authors: Valenzuela G. R. , Laing M. B.
Published: March 2006

The energy flux in gravity-capillary wave spectra has been obtained using Hasselmann's (1962) perturbation analysis for a homogeneous Gaussian sea. As expected, resonant interactions now appear at second order, and a third-order perturbation analysis shows that energy is redistributed from waves with intermediate wavelengths (in the neighbourhood of 1·7 cm) toward gravity and capillary waves. Numerical computations are also obtained for the energy flux and the interaction time of a sharply peaked spectrum consisting of wavenumbers concentrated around a single wavenumber, superposed on a smooth background spectrum. The range of validity of the inviscid results is discussed.
On the breaking of standing internal gravity waves
Authors: Orlanski Isidoro
Published: March 2006

A solution has been found for the transient behaviour of resonant growing standing waves by using a perturbation expansion. Comparison with laboratory experiments as well as a numerical nonlinear solution of the same problem leads to the conclusion that: (i) the transient behaviour and the nonlinear tendency of the standing waves are described well by the analytic expression; (ii) the numerical results describe the solution very well until the wave starts to break; (iii) from the laboratory experiments and the numerical results, the standing internal gravity waves break owing to local gravitational instability at a critical amplitude which is similar to the one predicted by the expansion theory; (iv) the critical amplitude seems to be the maximum amplitude that a wave can reach; (v) when the generation of turbulence is violent, the small eddies begin forcing a secondary flow characterized by layers of strong jets separated by patches of turbulence.
On the mechanism of blocking in a stratified fluid
Authors: Freund D. D. , Meyer R. E.
Published: March 2006

The steady, two-dimensional motion which can occur when a body moves horizontally at large Richardson number is examined. Theoretical evidence is presented for two propositions: (i) The nature of the motion depends on whether the vertical thickness of the body is large compared with an intrinsic length scale of the motion. (ii) If the body is sufficiently thick, then diffusion or heat conduction are important, even if the Schmidt or Prandtl number is large. The notion of ‘near-similar’ solutions (§4) is used to obtain a description of the motion past a thick body which is likely to approximate the real motion everywhere except fairly close to the body surface (§5). It predicts a very long wake, at the core of which is a blocking column, both fore and aft of the body (§5). The same prediction is implied for the two-dimensional Taylor column in a rotating system (§6).
The cumulants of the distribution of concentration of a solute dispersing in solvent flowing through a tube
Authors: Chatwin P. C.
Published: March 2006

The mean distribution of concentration of a solute over the cross-section of a tube through which a solvent is flowing is given approximately by the solution of an equation of the heat-conduction type (Taylor 1953, 1954). One solution of this equation is a Gaussian curve but observed distributions are normally not Gaussian. On the basis of an equation derived by Chatwin (1970) it is shown here that the deviation of an observed distribution from the Gaussian curve with the same mean and variance is not determined by the heat-conduction equation, although the practical importance of this remark may be small if the initial distribution of solute is greatly elongated along the tube axis.
The entrainment interface
Authors: Phillips O. M.
Published: March 2006

A theory is developed to describe the evolution of the entrainment interface in turbulent flow, in which the surface is convoluted by the large-scale eddies of the motion and at the same time advances relative to the fluid as a result of the micro-scale entrainment process. A pseudo-Lagrangian description of the process indicates that the interface is characterized by the appearance of ‘billows’ of negative curvature, over which surface area is, on average, being generated, separated by re-entrant wedges (lines of very large positive curvature) where surface area is consumed. An alternative Eulerian description allows calculation of the development of the interfacial configuration when the velocity field is prescribed. Several examples are considered in which the prescribed velocity field in the z direction is of the general form w = Wf(x – Ut), where the maximum value of the function f is unity. These indicate the importance of leading points on the surface which are such that small disturbances in the vicinity will move away from the point in all directions. The necessary and sufficient condition for the existence of one or more leading points on the surface is that U [les ] V, the speed of advance of an element of the surface relative to the fluid element at the same point. The existence of leading points is accompanied by the appearance of line discontinuities in the surface slope re-entrant wedges, In these circumstances, the overall speed of advance of the convoluted surface is found to be W + (V2 – U2)½, where W is the maximum outwards velocity in the region; this result is independent of the distribution f.

When the speed U with which an ‘eddy’ moves relative to the outside fluid is greater than the speed of advance V of an element of the front, the interface develops neither leading points nor discontinuities in slope; the amplitude of the surface convolutions and the overall entrainment speed are both reduced greatly. In a turbulent flow, therefore, the large-scale motions influencing entrainment are primarily those that move slowly relative to the outside fluid (with relative speed less than V). The experimental results of Kovasznay, Kibens & Blackwelder (1970) are reviewed in the light of these conclusions. It appears that in their experiments the entrainment speed V is of the order fifteen times the Kolmogorov velocity, the large constant of proportionality being apparently the result of augmentation by micro-convolutions of the interface associated with small and meso-scale eddies of the turbulence.
Examples of steady vortex rings of small cross-section in an ideal fluid
Authors: Fraenkel L. E.
Published: March 2006

The existence theory for steady vortex rings of small cross-section is used to derive asymptotic formulae that describe the shape and overall properties of such rings. A certain two-parameter family of rings is studied in detail, to a first approximation; for members of this family, the ratio ω/r(of vorticity to cylindrical radius) falls from a positive maximum at a central point of the core cross-section to a value at the core boundary that can be substantially smaller or even negative. The case of uniform ω/ris considered to a higher order of approximation, and the formulae given for this case appear to be useful for quite substantial cross-sections.
Gas and solids motion around deformed and interacting bubbles in fluidized beds
Authors: Clift R. , Grace J. R. , Cheung L. , Do T. H.
Published: March 2006

Previous analyses of gas and particle motion around bubbles in fluidized beds have concentrated on idealized isolated bubbles. In this paper three non-idealities are considered using the theoretical models of Davidson and Murray. Gas flow patterns are derived for indented and elongated bubbles and for pairs of interacting bubbles. Cloud boundaries are predicted for these situations and some effects on gas-solid contacting are discussed.