bluuebunny/crossref_metadata_2025 huggingface dataset:
Relation of turbulent mass transfer to a wall at high Schmidt numbers to the velocity field
Authors: Sirkar Kamalesh K. , Hanratty Thomas J.
Published: March 2006

Turbulent mass transfer to a wall at high Schmidt numbers is controlled by the velocity field within the viscous sublayer. Measurements have been obtained of the root-mean-square fluctuating mass transfer coefficient and the frequency spectrum of the fluctuating mass transfer coefficient for a Schmidt number of about 2300. From an order-of-magnitude analysis it is concluded that flow fluctuations in the direction of mean flow have little effect on the mass transfer fluctuations. A comparison of the mass transfer spectrum with the spectrum of the component of the velocity gradient in the transverse directionszreveals that the high-frequency portion of theszspectrum is not effective in transferring mass. Approximate relations between the mass transfer spectrum and theszspectrum are developed for high frequencies and for low frequencies.
Slow motion of two spheres in a shear field
Authors: Lin C. J. , Lee K. J. , Sather N. F.
Published: March 2006

The exact solution of the Stokes equations for the creeping motion of two spheres of arbitrary size and arbitrarily oriented with respect to a shear field is obtained by use of spherical bipolar co-ordinates. Numerical results are given for two special cases: (1) the free motion of two equal-sized spheres in simple shear flow and (2) the free motion of a sphere near a wall in the rotational shear field between two parallel disks rotating at different rates. The sphere trajectories calculated for the first of these problems are found to agree fairly well with those observed experimentally.
On stagnation-point conditions in non-equilibrium inviscid blunt-body flows
Authors: Vinokur Marcel
Published: March 2006

In non-equilibrium inviscid blunt-body flows, the state of the gas at the stagnation point is known to be in thermodynamic equilibrium for all finite relaxation times. The dependence of this state on the non-equilibrium processes and body geometry is investigated for the most general conditions. The stagnation-point state is always found to be in a narrow range bounded on one side by the state obtained in an equilibrium flow. The other bound, called the frozen limit, is far removed from the state obtained in an identically frozen flow (infinite relaxation times). For certain state variables, the frozen-limit value lies outside the range determined by frozen and equilibrium flow. Significant errors are found in several published predictions of the stagnation-point state, resulting from the non-analytic approach to equilibrium in nearly frozen flow.

The two bounds on the pressure are expressible in terms of the normal shock density ratios for equilibrium and frozen flow. The actual pressure for an arbitrary flow situation is found to depend only on the shock nose radius and the relation between density and time in the relaxation zone behind a normal shock wave. If the density law is given by a single relaxation model, a closed form expression for the pressure is obtained. The analysis is carried out for both plane and axisymmetric flows, and is also valid for non-equilibrium free-stream conditions.
The three-dimensional mixing layer between two grazing perpendicular streams
Authors: Hackett J. E. , Cox D. K.
Published: March 2006

The three-dimensional mixing layer between two grazing perpendicular streams is investigated both theoretically and experimentally. Similar shear velocity profiles are present about 40 initial momentum thicknesses downstream of the splitter edges. Thereafter the velocity component in the resultant direction of the two streams is shown to be constant through the layer; the profile perpendicular to this (the shear direction) has approximately an error function form. The ratio of the maximum shear stress to the square of the shear velocity difference is about 50% greater than in a two-dimensional layer.
The diffraction of internal waves by a semi-infinite barrier
Authors: Manton M. J. , Mysak L. A. , Mcgorman R. E.
Published: March 2006

We discuss the diffraction of internal waves by a semi-infinite vertical barrier in a uniformly rotating, stably stratified fluid of constant depth and Brunt–Väisälä frequency. N, For the frequency passband f < σ < N, where f and σ are respectively the inertial and wave frequencies, the presence of rotation gives rise to internal Kelvin waves which propagate without attenuation along the barrier. For the passband N < σ < f, however, the barrier generates waves which propagate without attenuation away from the barrier and which have amplitudes that fall off exponentially in the direction along the barrier.
Three-dimensional boundary layer near the plane of symmetry of a spheroid at incidence
Authors: Wang K. C.
Published: March 2006

This paper presents incompressible laminar boundary-layer results on both the leeside and windside of a prolate spheroid. The results are obtained by an implicit finite difference method of the Crank–Nicolson type. Particular attention has been given to the determination of separation and of embedded streamwise vortices. No restriction on the angle of attack or the thickness ratio is imposed, nor are there invoked any of the common assumptions such as similarity, conical flow and others. The results suggest an embedded vortex region existing between the regular boundary-layer region and the separated region. At higher angle of attack, the vortex region becomes so thick that it itself may be more appropriately called ‘separated’ also. The latter possibility leads to questions of applicability for existing theories on three-dimensional separation.
A variational principle for magnetohydrodynamic channel flow
Authors: Wenger Norman C.
Published: March 2006

A variational formulation is presented for a class of magnetohydrodynamic (MHD) channel flow problems. This formulation yields solutions for the fluid velocity and the induced electric potential in a channel with a uniform transverse static magnetic field. The channel cross-section is constant but arbitrary, and the channel walls can be either insulators or conductors with finite electrical conductivity. Electric currents are permitted to enter and leave the channel walls so that the solutions are suitable for MHD generator and pump applications. An example of a square channel with conducting walls is solved as an illustration.
Motion produced by combination of gravity and pressure differential when a gate in a tank of liquid is broken
Authors: Chiu W. S. , Jaffrin M. Y. , Shapiro A. H.
Published: March 2006

A two-dimensional, semi-infinite tank, closed by a vertical wall at one end, is initially filled to a certain level with an in viscid liquid at rest. A gate in the lower part of the wall is suddenly opened to a region where the gas pressure is greater or less than the gas pressure over the free liquid surface in the tank. The liquid escapes (or moves inwardly) through the gate under the combined action of gravity and the driving pressure differential. The fluid motion, the shapes of the two free surfaces, and the discharge through the gate are calculated for small times after the opening.
The approach to normality of the concentration distribution of a solute in a solvent flowing along a straight pipe
Authors: Chatwin P. C.
Published: March 2006

Taylor (1953, 1954a) showed that, when a cloud of solute is injected into a pipe through which a solvent is flowing, it spreads out, so that the distribution of concentration C is eventually a Gaussian function of distance along the pipe axis. This paper is concerned with the approach to this final form. An asymptotic series is derived for the distribution of concentration based on the assumption that the diffusion of solute obeys Fick's law. The first term is the Gaussian function, and succeeding terms describe the asymmetries and other deviations from normality observed in practice. The theory is applied to Poiseuille flow in a pipe of radius a and it is concluded that three terms of the series describe C satisfactorily if Dt/a2 > 0·2 (where D is the coefficient of molecular diffusion), and that the initial distribution of C has little effect on the approach to normality in most cases of practical importance. The predictions of the theory are compared with numerical work by Sayre (1968) for a simple model of turbulent open channel flow and show excellent agreement. The final section of the paper presents a second series derived from the first which involves only quantities which can be determined directly by integration from the observed values of C without knowledge of the velocity distribution or diffusivity. The latter series can be derived independently of the rest of the paper provided the cumulants of C tend to zero fast enough as t → ∞, and it is suggested, therefore, that the latter series may be valid in flows for which Fick's law does not hold.
Low-frequency modulation of thermal instability
Authors: Rosenblat S. , Herbert D. M.
Published: March 2006

A Boussinesq fluid is heated from below. The applied temperature gradient is the sum of a steady component and a low-frequency sinusoidal component. An asymptotic solution is obtained which describes the behaviour of infinitesimal disturbances to this configuration. The solution is discussed from the viewpoint of the stability or otherwise of the basic state, and possible stability criteria are analyzed. Some comparison is made with known experimental results.
Thermal boundary-layer theory near the stagnation point in three-dimensional fluctuating flow
Authors: Ghoshal S. , Ghoshal A.
Published: March 2006

The equations of motion and energy governing a three-dimensional fluctuating flow of an incompressible fluid in the vicinity of a stagnation point on a regular surface have been integrated analytically. The velocity of the oncoming flow relative to the body oscillates in magnitude but not in direction.

It has also been shown that the analysis of Lighthill for the two-dimensional fluctuating flow may be extended to the three-dimensional flow (both chordwise and spanwise), namely for each point on the body there is a critical frequency ω0 such that for frequencies ω > ω0 the oscillations are to a close approximation ordinary ‘shear waves’, unaffected by the mean flow; the phase advance in the skin friction is then 45°. For frequencies ω < ω0 the oscillations may be closely approximated by the sum of two parts: one quasi-steady part and the other proportional to the acceleration of the oncoming stream. The phase advance in the skin friction is then tan−1 (ω/ω0).
The theory of induced voltage electromagnetic flowmeters
Authors: Bevir M. K.
Published: March 2006

The performance of an electromagnetic flowmeter head is assessed in terms of a weight vector W such that the output voltage ∝ ∫ v. Wdτ, where v is the velocity and τ the flowmeter volume. The condition curl W = 0 with W → 0 at ∞ is shown to be necessary and sufficient for the velocity to depend only on the flow rate and not on the flow pattern. A class of such ‘ideal’ meters is described. It is shown that meters with point electrodes can never be ideal but may, with considerable complication of the magnetic field, be made immune to asymmetric velocity-profile variations if the flow is rectilinear.
An optical study of turbulence
Authors: Roe G. E.
Published: March 2006

This paper describes theoretical and experimental work carried out at the Cavendish Laboratory of the University of Cambridge. The main object of the work was to develop a new technique for measuring the structure of fluid turbulence.

A parallel beam of light is passed through the turbulent region, containing refractive index fluctuations, and analyzed on exit by gratings of periodic transmissivity. Two forms of analysis yield (a) the spatial power spectrum of the refractive index fluctuations in the turbulence, and (b) the velocity distribution within the beam aperture. The method does not disturb the fluid physically, does not depend on the existence of a mean flow velocity, and works well in liquids.

One of the limitations of this single-beam method is that it produces information averaged along the path length of the beam in the turbulence, and to overcome this a cross-beam technique, using two beams intersecting at right-angles, has been developed in theory. This method gives the spatial power spectrum of the refractive index fluctuations, as does the single beam method, but the results are characteristic only of the volume of intersection of the beams.

The paper first discusses the theory of the single-beam and crossed-beam techniques, and then experimental results obtained with the single-beam method.

The turbulent region investigated was a rectangular tank of water, heated from below and cooled from above, producing convective turbulence of high Rayleigh number (4·1 × 108), a system difficult to analyze by conventional methods of measurement, such as the hot-wire anemometer.

Spectral density functions (power spectra) of refractive index, and hence in this case temperature fluctuations, have been measured, as have velocity distributions. Statistical analysis of the results also gives useful information about the Eulerian time scale of the turbulent field.
The fluid mechanics of the ureter from a lubrication theory point of view
Authors: Lykoudis Paul S. , Roos Rudolf
Published: March 2006

The function of a healthy ureter is analyzed in terms of a fluid-mechanical model. To the extent that the Reynolds number is of the order of one, the fundamental equations are shown to reduce to those of the theory of lubrication. It is found that from the point of view of the pressure variation with time (the urometrogram) the important part of the peristaltic wave is the constricting part. For this reason this part of the wave is represented with an algebraic expression of the form h ∼ xn making it possible to find closed form solutions. Using Fourier analysis in defining the complete wave shape of the ureter it was also possible to obtain numerical solutions. For both cases it is shown that there is good agreement between the theoretical and experimental pressure distributions, this not being the case for sinusoidal wave shapes. An approximate equation for the flux is developed and a universal relation is presented connecting the maximum pressure, flux and kinematic behaviour of the ureter.
Scattering of aerodynamic noise by a semi-infinite compliant plate
Authors: Crighton D. G. , Leppington F. G.
Published: March 2006

The acoustic scattering properties of a semi-infinite compliant plate immersed in turbulent flow are considered in the context of Lighthill's theory of aerodynamic noise. The turbulent eddies are replaced by a volume distribution of quadrupoles, and the reciprocal theorem used to transform the quadrupole scattering problem into one of the diffraction of a plane acoustic wave. This problem is solved by the Wiener–Hopf technique for the case when elastic forces in the plate are negligible, so that a local impedance condition relates the plate velocity to the pressure difference across the plate. Strong scattering of the near-field into propagating sound occurs when certain types of quadrupole lie sufficiently close to the plate edge, and we derive explicit expressions for the scattered fields in various cases. When fluid loading effects are small, and the plate relatively rigid, the results of Ffowcs Williams & Hall (1970) are recovered, in particular the U5 law for radiated intensity. A quite different behaviour is found in the case of high fluid loading, when the plate appears to be relatively limp. The radiated intensity then increases with flow velocity U according to a U6 law. In aeronautical situations, surface compliance is negligible in its effect on the scattering process, and the U5 law must then apply provided the surface is sufficiently large. On the other hand, the effect of appreciable surface compliance is to greatly inhibit the near-field scattering from the surface edge. This weaker scattering is likely to be observed in underwater applications, where fluid loading effects are generally so high as to render unattainable the condition for a plate to be effectively rigid.
Similarity rules for thin aerofoils in non-stationary subsonic flows
Authors: Graham J. M. R.
Published: March 2006

Similarity rules are constructed for the load distributions induced on a thin two-dimensional wing at subsonic speeds by sinusoidal gusts whose wave fronts are at an angle to the leading edge of the wing. It is shown that these rules divide into two groups according to the value of a parameter dependent on the Mach number and the angle between the gust front and the wing. The similarity rules for each group relate all the members of the group to a simpler problem whose solution can be found by existing methods. The similarity between the two groups is also discussed in terms of the two methods of solution available and it is shown that each method of solution is applicable in all cases.
Wave drag caused by a swirling flow through a convergent-divergent nozzle
Authors: Chow Chuen-Yen , Lai Ying-Chung
Published: March 2006

The wave drag coefficient is computed approximately for a nozzle containing a swirling flow as a function of R0−1, the inverse of the Rossby number. When R0−1 < λ1 and R0−1 = λn, where λn denotes the nth zero of the Bessel function J1, there is no wave in the flow and the wave drag is zero. The drag coefficient is found to be sub-divided into different regions between R0−1 = λn and λn+1, with n = 1,2,3,.… When each λn is exceeded, the drag coefficient jumps from zero to a value which is one order higher than its values in the previous region (except in the case n = 1), and then decreases to zero as R0−1 increases toward λn+1. Very high wave drag can be expected in flows of large swirl ratios.
Measurements and computations of flow in pipe bends
Authors: Rowe M.
Published: March 2006

Measurements are given of the total pressure and yaw relative to the pipe axis of flow in various pipe bends attached to the end of a long straight pipe, for a pipe Reynolds number of 2·36 × 105. The behaviour of the induced secondary flows is presented in detail. In a 180° bend it is found that, from the start of the bend, the secondary flows increase to a maximum and then decrease to a steady value. This effect is explained by relating the local total pressure gradient to the production of streamwise vorticity. The same flow mechanism is used to predict qualitatively the flow pattern in an S-bend in which it is found that the secondary flows cause a complete interchange of the slow moving wall fluid and the faster central core. Results of a computer program for calculating the secondary flow from a knowledge of the upstream velocity profile are included. Although real fluid effects, other than those causing the upstream velocity profile, are omitted from the computations, the measured flow pattern is reasonably well predicted in a long bend up to a bend angle of about 75°.
On the spin-up of an electrically conducting fluid Part 2. Hydromagnetic spin-up between infinite flat insulating plates
Authors: Loper David E. , Benton Edward R.
Published: March 2006

The linear spin-up of a homogeneous electrically conducting fluid confined between infinite flat insulating plates is analyzed for the case in which a uniform magnetic field is applied normal to the boundaries. As in part 1 (Benton & Loper 1969), complete hydromagnetic interaction is embraced even within linearized equations. Approximate inversion of the exact Laplace transform solution reveals the presence of several flow structures: two thin Ekman–Hartmann boundary layers (one on each plate), which are quasi-steady on the time scale of spin-up, two thicker continuously growing magnetic diffusion regions, and an essentially inviscid, current-free core, which may or may not be present on the spin-up time scale, depending upon the growth rate of the magnetic diffusion regions. When a current-free core exists, it is found to spin-up at the same rate as the fluid within magnetic diffusion regions, although different physical mechanisms are at play. As a result, a single hydromagnetic spin-up time is derived, independently of the thickness of magnetic diffusion regions; this time is shorter than in the non-magnetic problem.
The flat plate boundary layer. Part 2. The effect of increasing thickness on stability
Authors: Barry M. D. J. , Ross M. A. S.
Published: March 2006

Numerical analysis has been used to find the neutral stability curve for the flat plate boundary layer in zero pressure gradient when the main terms representing the growth of boundary-layer thickness are either included or excluded. The boundary layer is found to be slightly less stable when the extra terms are included. The calculations give a critical Reynolds number of 500.
The extension of the Miles-Howard theorem to compressible fluids
Authors: Chimonas G.
Published: March 2006

A statically stable, gravitationally stratified compressible fluid containing a parallel shear flow is examined for stability against infinitesimal adiabatic perturbations. It is found that the Miles–Howard theorem of incompressible fluids may be generalized to this system, so that n2 ≥ ¼U′2 throughout the flow is a sufficient condition for stability. Here n2 is the Brunt–Väissälä frequency and U’ is the vertical gradient of the flow speed. Howard's upper bound on the growth rate of an unstable mode also generalizes to this compressible system.
The source-sink flow in a rotating system and its oceanic analogy
Authors: Kuo Han-Hsiung , Veronis George
Published: March 2006

Laboratory analogues of theoretical models of wind-driven ocean circulation are based on ideas presented by Stommel (1957). A particularly simple demonstration of the applicability of these ideas is contained in a paper by Stommel, Arons & Faller (1958). The present work develops the source-sink laboratory analogue of ocean circulation models to a point where chosen parametric values allow one to simulate the theoretical models of Stommel (1948) and Munk (1950) exactly. The investigation of the flow in a rotating cylinder generated by a source of fluid near the outer wall leads to a detailed description of the roles of the various boundary layers which occur. This knowledge is used to analyse the more complex source-sink flow in a pie-shaped basin. The laboratory analogue to the Stommel circulation model is analyzed in detail. It is shown that the change in the flow pattern brought about by a radial variation of the position of the eastern boundary in the pie-shaped basin is confined to the interior flow and the boundary layer is largely unaffected. When the bottom of the pie-shaped container slopes, the circulation pattern is changed significantly. For the particular case treated, the depth of the basin along the western boundary is unchanged and the maximum depth occurs at the southeast corner. The circulation generated by a source introduced at the apex of the pie has a gyre whose centre is shifted more toward the southwest corner than the corresponding centre of the gyre for a flat-bottomed basin. Two experiments are reported showing that the western boundary may separate because of the effect of bottom topography or because of the pressure of a cyclonic and an anti-cyclonic gyre generated by suitably placed sources and sinks.
A study of free jet impingement. Part 2. Free jet turbulent structure and impingement heat transfer
Authors: Donaldson Coleman Dup. , Snedeker Richard S. , Margolis David P.
Published: March 2006

An experimental study of jet impingement is completed with the presentation of the measured turbulent characteristics of the circular subsonic jet and the heat transfer rates measured when this jet impinges normal to a flat plate. The data suggest that for impingement very close to the stagnation point, the heat transfer can be computed by applying a turbulent correction factor to the laminar value calculated for a flow having the same pressure distribution as that present in the impingement region. The correction factor is found to be a function of the axial distance and not of Reynolds number. Farther away, the measurements agree well with the heat transfer estimated using the method of Rosenbaum & Donaldson (1967). At large distances from the stagnation point, the heat transfer falls off in inverse proportion with the distance.

The documentation of the turbulent jet flow field includes measurements of the radial and axial velocity fluctuations and their spectra, as well as the radial distribution of turbulent shear$\overline{w^{\prime}u^{\prime}}$. In addition, measurements of the turbulence near the stagnation point and the total pressure fluctuation at the stagnation point are presented.
Baroclinic stability under non-hydrostatic conditions
Authors: Stone Peter H.
Published: March 2006

Eady's model for the stability of a thermal wind in an inviscid, stratified, rotating system is modified to allow for deviations from hydrostatic equilibrium. The stability properties of the flow are uniquely determined by two parameters: the Richardson number Ri, and the ratio of the aspect ratio to the Rossby number δ. The latter parameter may be taken as a measure of the deviations from hydrostatic equilibrium (δ = 0 in Eady's model). It is found that such deviations decrease the growth rates of all three kinds of instability which can occur in this problem: ‘geostrophic’ baroclinic instability, symmetric instability, and Kelvin–Helmholtz instability. The unstable wavelengths for ‘geostrophic’ and Kelvin–Helmholtz instability are increased for finite values of δ, while the unstable wavelengths for symmetric instability are unaffected. The ‘non-hydrostatic’ effects (δ ≠ 0) are significant for symmetric and Kelvin–Helmholtz instability when δ [gsim ] 1, but not for ‘geostrophic’ instability unless δ [Gt ] 1. Consequently, the first two types of instability tend to be suppressed relative to ‘geostrophic’ instability by ‘non-hydrostatic’ conditions. Figure 3 summarizes the different instability régimes that can occur. In laboratory experiments symmetric instability can be studied best when δ [lsim ] 1, while Kelvin–Helmholtz instability can be studied best when δ [Lt ] 1.
Finite-amplitude stability of pipe flow
Authors: Davey A. , Nguyen H. P. F.
Published: March 2006

In this paper we present some results concerning the stability of flow in a circular pipe to small but finite axisymmetric disturbances. The flow is unstable if the amplitude of a disturbance exceeds a critical value, the equilibrium amplitude, which we have calculated for a wide range of wave-numbers and Reynolds numbers. For large values of the Reynolds number, R, and for a real value of the wave-number, α, we indicate that the energy density of a critical disturbance is of order c2i, where −ααci is the damping rate of the associated infinitesimal disturbance. The energy, per unit length of the pipe, of a critical disturbance which is concentrated near the axis of the pipe is of order R−2, and the wave-number α is of order R1/3 For a critical disturbance which is concentrated near the wall of the pipe the energy is of order $R^{-\frac{3}{2}}$ and α is of order R½. This suggests that non-linear instability is most likely to be caused by a ‘centre’ mode rather than by a ‘wall’ mode. The wall mode solution is also essentially the solution for the problem of plane Couette flow when αR is large. We compare it with the true solution.

In an appendix Dr A. E. Gill indicates how some of the results of this paper may be inferred from a simple scale analysis.
Shock-wave reflexion in a relaxing gas
Authors: Hanson Ronald K.
Published: March 2006

The transient situation which follows when a plane normal shock wave is reflected (from a coplanar wall) into its own relaxation zone is examined theoretically. Approximate inner and outer solutions for the wall-pressure history are employed to establish the timewise variations in the thermodynamic state of the gas adjacent to the wall. Results for chemically relaxing O2 and vibrationally relaxing CO2 are compared with previous numerical solutions based on the method of characteristics, and the agreement is found to be excellent. The approximate technique is simple and requires only a minimum of computing time.
Convection in a rotating annulus uniformly heated from below
Authors: Davies-Jones Robert P. , Gilman Peter A.
Published: March 2006

We present a linear stability analysis, to second order in initial amplitude, of Bénard convection of a Boussinesq fluid in a thin rotating annulus for modest Taylor numbers T ([les ] 104). The work is motivated in part by the desire to study further a mechanism for maintaining, through horizontal Reynolds stresses induced in the convection, the sun's ‘equatorial acceleration’, which has been demonstrated for a rotating convecting spherical shell by Busse & Durney. The annulus is assumed to have stress free, perfectly conducting top and bottom (which allows separation of the equations) and non-conducting non-slip sides. A laboratory experiment which fulfills these conditions (except perhaps the free bottom) is being developed with H. Snyder.

We study primarily annuli with gap-width to depth ratios a of order unity. The close, non-slip side-walls produce a number of effects not present in the infinite plane case, including overstability at high Prandtl numbers P, and multiple minima in Rayleigh number R on the stability boundary. The latter may give rise to vacillation. The eigenfunctions for stationary convection for a = 2, T [lsim ] 2000 clearly show momentum of the same sense as the rotation is transported from the inner to the outer half of the annulus, corresponding to transport toward equatorial latitudes on the sphere. The complete second-order solutions for the induced circulations indeed give faster rotation in the outer half, except for large P (> 102), in which case thermal stresses dominate. At all P, this differential rotation is qualitatively a thermal wind. Overstable convective cells, and stationary cells at higher T, induce more complicated differential rotations.
Calculation of boundary-layer development using the turbulent energy equation: compressible flow on adiabatic walls
Authors: Bradshaw P. , Ferriss D. H.
Published: March 2006

The basic method described by Bradshaw, Ferriss & Atwell (1967) is extended to compressible flow in two-dimensional boundary layers in arbitrary pressure gradient (excluding shock waves and expansion fans) by invoking Morkovin's hypothesis (Favre 1964) that the turbulence structure is unaffected by compressibility. Using the same empirical functions as in incompressible flow, skin friction in zero pressure gradient is predicted to within 3% of Spalding & Chi's (1964) correlation for free-stream Mach numbers less than 5. Comparisons with experiments in pressure gradient are restricted by the lack of data, but, since Morkovin's hypothesis does not depend on pressure gradient, methods which use it (of which the present method seems to be the first) can be checked fairly adequately by comparisons with data in zero pressure gradient.

No ‘compressibility transformations’ are needed, although the Crocco relation is used, provisionally, for the temperature: since the calculations take only about 20% longer than in incompressible flow, Morkovin's hypothesis does as much as any transformation could do. It is pointed out that, in supersonic flow, surface curvature which is large enough to induce a significant longitudinal pressure gradient is also large enough to have a very significant effect on the turbulence structure.
Wave forces on a circular dock
Authors: Garrett C. J. R.
Published: March 2006

The scattering of surface gravity waves by a circular dock is considered in order to determine the horizontal and vertical forces and torque on the dock. An incident plane wave is expanded in Bessel functions, and for each mode the problem is formulated in terms of the potential on the cylindrical surface containing the dock and extending to the bottom. The solution is shown to have phase independent of depth and so may be obtained from an infinite set of real equations, which are solved numerically by Galerkin's method. The convergence of the solution is discussed, and some numerical results are presented.

This problem has been investigated previously by Miles & Gilbert (1968) by a different method, but their work contained errors.
A note on variational principles for surface-wave scattering
Authors: Miles John W.
Published: March 2006

Complementary variational formulations are developed for the scattering of a gravity wave by a circular dock. These formulations, which are based on assumed distributions of the radial velocity and the potential, respectively, on the projection of the cylindrical boundary, yield lower and upper bounds to an impedance parameter that determines the difference between the scattered wave for the dock and the corresponding wave for a circular cylinder. Numerical results, using trial functions based on the incident wave, are compared with the results implied by a Galerkin solution (Garrett 1971). The maximum errors in the variational approximations to the total scattering cross-section are found to be of the order of 2% for a typical depth/radius ratio, draft/depth ratios of 0, ½ and 1, and all wavelengths. The axisymmetric component of the scattering cross-section is found to be very close to the value for scattering by a circular cylinder (dock extending to bottom). The intensity of the scattered wave on the forward axis for long wavelengths and a certain range of the geometric parameters is significantly less than that for a circular cylinder, and may vanish for critical combinations of these parameters.
Skin friction and surface temperature of an insulated flat plate fixed in a fluctuating stream
Authors: Ishigaki Hiroshi
Published: March 2006

The time-mean skin friction of the laminar boundary layer on a flat plate which is fixed at zero incidence in a fluctuating stream is investigated analytically. Flow oscillation amplitude outside the boundary layer is assumed constant along the surface. First, the small velocity-amplitude case is treated, and approximate formulae are obtained in the extreme cases when the frequency is low and high. Next, the finite velocity-amplitude case is treated under the condition of high frequency, and it is found that the formula obtained for the small-amplitude and high-frequency case is also valid. These results show that the increase of the mean skin friction reduces with frequency and is ultimately inversely proportional to the square of frequency.

The corresponding energy equation is also studied simultaneously under the condition of zero heat transfer between the fluid and the surface. It is confirmed that the time-mean surface temperature increases with frequency and tends to be proportional to the square root of frequency. Moreover, it is shown that the timemean recovery factor can be several times as large as that without flow oscillation.
A spherical envelope approach to ciliary propulsion
Authors: Blake J. R.
Published: March 2006

In this paper, an attempt has been made to model the dynamics of ciliary propulsion through the concept of an ‘envelope’ covering the ends of the numerous cilia of the microscopic organism. This approximation may be made in the case when the cilia are close together, as can occur in the case of the symplectic metachronal wave (i.e. the wave travels in the same direction as the effective beat). For simplicity, a spherical model has been chosen, and the analysis which follows is a correction to Lighthill's (1952) paper on squirming motions of a nearly spherical organism. The velocity and efficiency compared to the work done in pushing an inert organism are obtained, and compared to that of a ciliated organism.
Experiments on the instability of stratified shear flows: miscible fluids
Authors: Thorpe S. A.
Published: March 2006

Earlier papers have described a technique whereby a stratified shear flow may be produced under controlled conditions in the laboratory. A comparison between the experiments and theory is made here for small-amplitude unstable disturbances in an accelerating stratified free shear layer at the diffuse interface between layers of brine and water. In the early stages of the observed growth of the instability, which takes the form of growing waves, three measurable quantities can be compared with predicted values: the wavelength of the small amplitude waves, the time, which determines the flow conditions, and the growth rates of the waves. Some observations of the development of the disturbances to finite amplitude, the transition to turbulence and the resulting turbulence are reported.
Influence of surface roughness on the cross-flow around a circular cylinder
Authors: Achenbach Elmar
Published: March 2006

The influence of surface roughness on the cross-flow around a circular cylinder is the subject of the present experimental work. The investigations were carried out in a high-pressure wind tunnel, thus high Reynolds numbers up toRe= 3 × 106could be obtained. Local pressure and skin friction distributions were measured. These quantities were evaluated to determine the total drag coefficient and the percentage of friction as functions of Reynolds number and roughness parameter. In addition the local skin friction distribution yields the angular position of boundary-layer transition from laminar to turbulent flow and the location of boundary-layer separation.
Calculation of three-dimensional turbulent boundary layers
Authors: Bradshaw P.
Published: March 2006

The two-dimensional prediction method of Bradshaw, Ferriss & Atwell (1967), which was based on the empirical conversion of the turbulent energy equation into a ‘transport’ equation for shear stress, is extended to three-dimensional flows satisfying the boundary-layer approximation (which excludes flows near bluff obstacles or streamwise corners). Predictions, using exactly the same empirical data as in two-dimensional flow, agree to within the likely experimental error with a variety of experiments on ‘infinite’ swept wings.
Measurements of subsonic jet noise and comparison with theory
Authors: Lush P. A.
Published: March 2006

Measurements of the noise field from a 25 mm diameter subsonic air jet are presented. These results are analysed in some detail by determining both the jet velocity dependence and the directivity of the intensity of the radiation in 1/3-octave bands at particular values of the frequency parameter,
\[
(fD/V_J)(1-M_c\cos\theta).
\]
This procedure should ensure that a particular source in a geometrically similar position in the jet is always observed, whatever the jet velocity, diameter and emission angle.

These results are compared with the predictions of Lighthil's (1952) theory of convected quadrupoles. It is shown that the theory predicts the variation of the intensity with jet velocity and emission angle provided that the observed frequency is below a certain critical value, which depends on jet diameter and emission angle and is independent of jet velocity. Above this critical frequency, the predicted variations overestimate the measurements and it appears that the convective amplification predicted by the theory is much reduced. The variation of this critical frequency is explained by assuming that substantial interaction occurs between the radiated sound and the jet flow when the wavelength of the sound becomes shorter than the sound path length in the jet flow.
Hydromechanics of swimming propulsion. Part 2. Some optimum shape problems
Authors: Wu T. Yao-Tsu
Published: March 2006

The optimum shape problems considered in this part are for those profiles of a two-dimensional flexible plate in time-harmonic motion that will minimize the energy loss under the condition of fixed thrust and possibly also under other isoperimetric constraints. First, the optimum movement of a rigid plate is completely determined; it is necessary first to reduce the original singular quadratic form representing the energy loss to a regular one of a lower order, which is then tractable by usual variational methods. A favourable range of the reduced frequency is found in which the thrust contribution coming from the leading-edge suction is as small as possible under the prescribed conditions, outside of which this contribution becomes so large as to be hard to realize in practice without stalling. This optimum solution is compared with the recent theory of Lighthill (1970); these independently arrived-at conclusions are found to be virtually in agreement.

The present theory is further applied to predict the movement of a porpoise tail of large aspect-ratio and is found in satisfactory agreement with the experimental measurements. A qualitative discussion of the wing movement in flapping flight of birds is also given on the basis of optimum efficiency.

The optimum shape of a flexible plate is analysed for the most general case of infinite degrees of freedom. It is shown that the solution can be determined to a certain extent, but the exact shape is not always uniquely determinate.
Hydromechanics of swimming propulsion. Part 3. Swimming and optimum movements of slender fish with side fins
Authors: Wu T. Yao-Tsu
Published: March 2006

This paper seeks to evaluate the swimming flow around a typical slender fish whose transverse cross-section to the rear of its maximum span section is of a lenticular shape with pointed edges, such as those of spiny fins, so that these side edges are sharp trailing edges, from which an oscillating vortex sheet is shed to trail the body in swimming. The additional feature of shedding of vortex sheet makes this problem a moderate generalization of the paper on the swimming of slender fish treated by Lighthill (1960a). It is found here that the propulsive thrust depends not only on the virtual mass of the tail-end section, but also on an integral effect of variations of the virtual mass along the entire body segment containing the trailing side edges, and that this latter effect can greatly enhance the thrust-making.

The optimum shape problem considered here is to determine the transverse oscillatory movements a slender fish can make which will produce a prescribed thrust, so as to overcome the frictional drag, at the expense of the minimum work done in maintaining the motion. The solution is for the fish to send a wave down its body at a phase velocity c somewhat greater than the desired swimming speed U, with an amplitude nearly uniform from the maximum span section to the tail. Both the ratio U/c and the optimum efficiency are found to depend upon two parameters: the reduced wave frequency and a ‘proportionalloading parameter’, the latter being proportional to the thrust coefficient and to the inverse square of the wave amplitude. The basic mechanism of swimming is examined in the light of the principle of action and reaction by studying the vortex wake generated by the optimum movement.
Wake characteristics of two-dimensional perforated plates normal to an air-stream
Authors: Castro I. P.
Published: March 2006

The flow in the wakes behind two-dimensional perforated plates has been investigated in the Reynolds number range 2·5 × 104 to 9·0 × 104.

Measurements of drag and shedding frequency were made and a pulsed hotwire anemometer was used to measure the mean velocity and turbulent intensity variations in the highly turbulent regions immediately behind the plates.

The results indicate the existence of two distinct types of flows: one appropriate to high and the other to low values of plate porosity.
The fluid mechanics of the ureter
Authors: Roos Rudolf , Lykoudis Paul S.
Published: March 2006

The effect of the presence of a catheter upon the pressure distribution inside the ureter is considered. Under the assumption of Stokes flow and long wavelengths it is shown that during the contraction a thin lubrication-type layer is formed between the catheter and the ureteral wall, capable of sustaining high pressures. Furthermore, it is found that the insertion of a catheter does not change the pressure distribution inside the ureter appreciably, leading to the conclusion that a urometrogram obtained with a catheter gives a good representation of the pressure inside an undisturbed ureter.
Three-dimensional MHD duct flows with strong transverse magnetic fields. Part 2. Variable-area rectangular ducts with conducting sides
Authors: Walker J. S. , Ludford G. S. S. , Hunt J. C. R.
Published: March 2006

In this paper the general analysis, developed in part 1, of three-dimensional duct flows subject to a strong transverse magnetic field is used to examine the flow in diverging ducts of rectangular cross-section. It is found that, with the magnetic field parallel to one pair of the sides, the essential problem is the analysis of the boundary layers on these (side) walls. Assuming that they are highly conducting and that those perpendicular to the magnetic field are non-conducting, the flow is found to have some interesting properties: if the top and bottom walls diverge, the side walls remaining parallel, then an O(1) velocity overshoot occurs in the side-wall boundary layers; but if the top and bottom walls remain parallel, the side walls diverging, these boundary layers have conventional velocity profiles. The most interesting flows occur when both pairs of walls diverge, when it is found that large, 0(M½), velocities occur in the side-wall boundary layers, either in the direction of the mean flow or in the reverse direction, depending on the geometry of the duct and the external electric circuit!

The mathematical analysis involves the solution of a formidable integral equation which, however, does have analytic solutions for some special types of duct.
The effect of weak Brownian rotations on particles in shear flow
Authors: Leal L. G. , Hinch E. J.
Published: March 2006

Axisymmetric particles in zero Reynolds number shear flow execute closed orbits. In this paper we consider the role of small Brownian couples in establishing a steady-state probability distribution for a particle being on any particular orbit. After presenting the basic equations, we derive an expression for the equilibrium distribution. This result is then used to calculate some bulk properties for a suspension of such particles, and these predicted properties are compared with available experimental observation.