bluuebunny/crossref_metadata_2025 huggingface dataset:
Development of a high-performance cryogenic shock tube
Authors: Cummings John C.
Published: March 2006

A cryogenic shock tube has been developed as a tool for research in fluid mechanics and low-temperature physics. The shock tube was designed to operate with the test section immersed in a cryogenic liquid. A unique diaphragm-changing mechanism makes this shock tube an economical and practical device. There are several advantages in operating a shock tube at cryogenic temperatures. Shock waves of very large Mach number can be produced. The flow field can be accurately calculated using ideal-shock-tube/perfect-gas theory. Boundary-layer effects are decreased, so that long test times are possible.

The cases that were studied are test-gas temperatures of 300, 77, 4·2 and 2·3 °K. Helium was used as both test and driver gas. The largest Mach numbers observed range from 2·4 at 300 °K to 32 at 2·3 °K (several runs at 1·46 °K produced Mach 40 shocks). As the temperature of the test gas was decreased, the observed Mach numbers approached those calculated using the ‘shock-tube equation’.

As a laboratory tool, the cryogenic shock tube may be applied in many areas and modified for use in even more. Shock waves with large Mach numbers and flows with large Reynolds numbers can be produced with this device. The rapid increase in temperature and pressure across the shock wave is useful for studies of sublimation, evaporation or chemical reactions. Quantum-mechanical effects in cryogenic materials, superconductors or superfluid helium can also be investigated.
Friction factors in the theory of bifurcating Poiseuille flow through annular ducts
Authors: Joseph D. D. , Chen T. S.
Published: March 2006

The objective of this paper is to show how to formulate a bifurcation theory for pipe flows in terms of the friction factor. We compute the slope of the friction factor vs. Reynolds number curve and the frequency change for the time-periodic solution which bifurcates from Poiseuille flow through annular ducts.
Nonlinear wave interactions in shear flows. Part 1. A variational formulation
Authors: Usher J. R. , Craik A. D. D.
Published: March 2006

A modified version of Bateman's variational formulation of the incompressible Navier-Stokes equations and boundary conditions (see Dryden, Murnaghan & Bateman 1956) is introduced. This is employed to examine a particular nonlinear problem of hydrodynamic stability which was treated previously, using a ‘direct’ approach, by Craik (1971). This problem concerns the resonant interaction at second order of a triad of wave modes in a parallel shear flow.

The present method is conceptually attractive; it also has the major advantage over the ‘direct’ method of a substantial reduction in algebraic complexity, which allows results to be derived far more readily. Also, some further improvements are made upon Craik's previous analysis. Such a variational approach may often be simpler than present conventional methods of tackling nonlinear viscous-flow problems. The present paper shows how other problems of nonlinear stability and wave interactions may be tackled in this way.
Experimental study of internal waves over a slope
Authors: Cacchione David , Wunsch Carl
Published: March 2006

Internal waves of the fundamental mode propagating into a shoaling region have been studied experimentally in a continuously stratified fluid. The waves divide into three classes depending upon the ratio of the bottom slope γ to the wave-characteristic slopec. For γ/c< 1, the amplitude and wavenumber changes of the waves over the slope are in reasonable accord with a simple inviscid linear theory, prior to wave breakdown near the intersection of the slope and surface. Considerable mixing occurs in this corner region. When γ/c= 1, a striking instability of the bottom boundary layer is observed and the waves are heavily damped. When γ/c> 1, the waves are inhomogeneous and have complex spatial dependence.
Vertical oscillations in a viscous and thermally conducting isothermal atmosphere
Authors: Lyons Patrick , Yanowitch Michael
Published: March 2006

The presence of dissipation in an isothermal atmosphere may cause upwardpropagating small amplitude waves to be reflected downward. For an atmosphere with small dynamic viscosity μ this was demonstrated in Yanowitch (1967b); this will be referred to as case II. Here two problems will be investigated: (i) a thermally conducting atmosphere with small conductivity k (case III) and (ii) a viscous and thermally conducting atmosphere with small k and μ, and a small ratio μ/k, i.e. small Prandtl number (case IV). It will be shown that the validity of the model in case III is questionable. The solution for case IV is determined from the conditions that the average rate of energy dissipation and of entropy increase in a column of fluid be finite, but a radiation condition is required in case III. The solution for case III does not approximate the one for case IV uniformly, and the reflexion coefficient for case IV does not tend to the one for case III as the Prandtl number Pr → 0, but varies periodically with log Pr. Numerical results show that when the Prandtl number is not small the reflexion coefficient can be approximated by the asymptotic value obtained from case II.
On transition mechanisms in vertical natural convection flow
Authors: Jaluria Yogesh , Gebhart Benjamin
Published: March 2006

An experimental investigation has been made of the processes occurring during the natural transition from laminar to turbulent flow of natural convection flow of water adjacent to a flat vertical surface where the surface heat flux is uniform. Measurements of both the velocity and temperature fields were made over wide ranges of the heat flux and at various downstream locations. Of principal interest were the definitions of the boundaries of the transition regime and their determination at several values of the surface heat flux. The interaction of the velocity and temperature fields during transition was measured. Our results show that transition events are not correlated in terms of the Grashof numberG*. The formG*/xn, wherenis of order ½ was found to give satisfactory correlations. Measurements of the frequency and growth rate of disturbances indicate the primacy of the velocity field during transition and show that the growth of turbulence in the temperature field lags behind that in the velocity field. The study of the turbulence growth, in terms of intermittency factors in both the velocity and temperature fields, resulted in unambiguous criteria for the boundaries of the transition regime. Our results suggest a kinetic energy flux parameterEand a single value closely correlates both our measurements of the onset of transition as well as those from all past studies known to us, for both different fluids and heating conditions.
Slow viscous flow of an incompressible stratified fluid past a sphere
Authors: Chadwick R. S. , Zvirin Y.
Published: March 2006

The steady, uniform, horizontal flow of a vertically stratified, non-diffusive fluid over a sphere is considered. A correction to the Stokes drag formula is obtained which is valid for small values of a stratification parameter α, Re [Lt ] |α| 1/3 and Fr2 [Lt ] |α|−1/3. To the order of the calculations, the sphere has no tendency to rotate, nor does it experience a lift force.
Rotatory Brownian motion of a rigid dumbbell
Authors: Szu Shousun C. , Hermans J. J.
Published: March 2006

The frictional torque on a dumbbell rotating with time-dependent angular velocity is calculated from the hydrodynamic interaction between the two ends of the dumbbell. This leads to the correlation function for the random torque in rotatory Brownian motion. Although the motion of each dumbbell end has the characteristics of a translational motion, the correlation function at large times decays like t−5/2, as in the case of a solid sphere. The correlation function may be calculated for the limiting case of very small angular displacements. The results for displacements of arbitrary magnitude are the same provided that terms quadratic in the angular velocity are negligible.
On the effects of boundary-layer growth on flow stability
Authors: Gaster M.
Published: March 2006

The stability of small travelling-wave disturbances in the flow over a flat plate is discussed. An iterative method is used to generate an asymptotic series solution in inverse powers of the Reynolds number Rx = Ux/v to the power one half. The neutral-stability boundaries given by the first two terms of this series are obtained and compared with experimental data. It is shown that the parallel flow approximation leads to a valid solution at very large Reynolds numbers.
A boundary layer developing in an increasingly adverse pressure gradient
Authors: Samuel A. E. , Joubert P. N.
Published: March 2006

This paper deals with a survey of mean flow and fluctuating quantities in a turbulent boundary layer developing on a smooth wall in a pressure domain P(x), where both dP/dx and d2P/dx2 are positive (increasingly adverse). The two-dimensional nature of the flow field was checked by momentum balance, as well as velocity traverses either side of the working section centre-line. Using the integrated form of the momentum integral equation, it was found that the skinfriction term and the summed momentum and pressure terms differed by at most 19%; but for the majority of measuring points they differed by less than 14%. The off-centre-line velocity profiles were indistinguishable from those taken on the centre-line. The flow field was also surveyed for fluctuating components $(\overline{u^2_1})^{\frac{1}{2}}, (\overline{u^2_2})^{\frac{1}{2}}, (\overline{u^2_3})^{\frac{1}{2}}$, and $\overline{u_1u_2}$, as well as for u1 spectra. Wherever possible, the results were compared with existing models of boundary-layer development. These comparisons indicated that the only all-embracing model for boundary-layer development is the law of the wall.
The effect of overtaking disturbances on the motion of converging shock waves
Authors: Yousaf M.
Published: March 2006

In this paper an exact formulation of the strength of the disturbance overtaking a shock is presented. The similarity solution is used to find the five interaction terms at all points of the flow. This work confirms that when the strength of the overtaking disturbance is known the CCW approximation may be modified to become an exact theory.
Stability and transition of buoyancy-induced flows in a stratified medium
Authors: Jaluria Yogesh , Gebhart Benjamin
Published: March 2006

An experimental and theoretical investigation has been carried out to determine the effect of a stable ambient thermal stratification on the developing buoyancyinduced flow adjacent to a flat vertical surface dissipating a uniform heat flux. The nature of the resulting base flow and its instability characteristics, linearized for two-dimensional disturbances, were analysed for Prandtl numbersProf 6·7 and 0·733, for several levels of ambient stratification. Stratification was found to cause initial stabilization of the flow but later destabilization downstream. Disturbance growth rates, frequency filtering and amplitude distributions across the boundary region were calculated. These aspects of the disturbance field were measured in a flow generated by an electrically heated metal foil, with artificially introduced two-dimensional disturbances, in water (Pr= 6–7). The experimental results are in very good agreement with the calculations. Measurements of natural transition indicate that a stable ambient stratification delays the onset of transition. A tentative transition-correlating parameter is generalized to include the effect of stratification.
Wave breaking in the presence of wind drift and swell
Authors: Phillips O. M. , Banner M. L.
Published: March 2006

Wind, blowing over a water surface, induces a thin layer of high vorticity in which the wind stress is supported by molecular viscosity; the magnitude of the surface drift, the velocity difference across the layer, being of the order of 3% of the wind speed. When long waves move across the surface, there is a nonlinear augmentation of the surface drift near the long-wave crests, so that short waves, superimposed on the longer ones, experience an augmented drift in these regions. This is shown to reduce the maximum amplitude that the short waves can attain when they are at the point of incipient breaking.

Theoretical estimates of the reduction are compared with measurements in wind-wave tanks by the authors and by Mitsuyasu (1966) in which long mechanically generated waves are superimposed on short wind-generated waves. The reductions measured in the energy density of the short waves by increasing the slope of the longer ones at constant wind speed are generally consistent with the predictions of the theory in a variety of cases.
The pressure field due to a large circular capped air bubble rising in water
Authors: Lazarek G. M. , Littman H.
Published: March 2006

The pressure field due to a large circular capped air bubble rising in water has been determined experimentally. The results verify the Davies & Taylor cap boundary condition and the frontal pressure field is well approximated by that due to irrotational flow around an oval body.

The pressure field extends axially as far as ten bubble half-widths below the bubble floor. Immediately below the floor the pressure is constant for about two-thirds of a bubble height. The wake is closed and contains symmetric pressure minima. For the bubbles studied, turbulence, as well as the diffusion of vorticity, probably controls the momentum distribution and energy dissipation in the wake.
Interferometric study of two-dimensional Bénard convection cells
Authors: Farhadieh R. , Tankin R. S.
Published: March 2006

A Mach-Zehnder interferometer was used to stud two-dimensional Bénard convection cells. The experiments were performed with distilled water and sea water in the region where density is a linear function of temperature. Two-dimensional convection rolls were formed with Rayleigh numbers as great as 23400. Reversal in the temperature profile was obtained for R/Rc ≥ 3·8, and an overshoot of about 6% was observed at R/Rc = 9·2 and 13·8. This agrees with the values predicted theoretically by Veronis (1966) for stress-free boundaries and Royal (1969) for rigid boundaries. This disagrees with the experimental results of Gille (1967), who reports an overshoot of only 1 ½% at R/Rc = 16. Many of the other results agree with those of other experimenters, such as the relation between the cell height-to-width ratio and Rayleigh number, the relation between the Nusselt number and Rayleigh number, and the value of the critical Rayleigh number.
The dispersion of contaminant released from instantaneous sources in laminar flow near stagnation points
Authors: Chatwin P. C.
Published: March 2006

This paper considers the dispersion of a cloud of passive contaminant released from an instantaneous source in the steady two-dimensional laminar flow near the forward stagnation point on a bluff body. The body is replaced by its tangent plane y = 0 with x measuring distance along the plane. Far away from y = 0 the flow is irrotational with velocity potential ½l(x2 – y2), where l is a positive constant. When the boundary layer is ignored the equation governing the distribution of concentration can be solved exactly. Consequences of this solution are that for large times the centre of mass moves parallel to the body at a speed proportional to exp (lt) while the cloud spreads out along the body symmetrically about the centre of mass with the magnitude of the spread also proportional to exp (lt). However, this solution is unrealistic because most of the contaminant is confined to a layer adjoining the body of thickness of order (k/l)½, where k is the molecular diffusivity, and this layer normally lies within the boundary layer, which is of thickness of order (v/l)½, where v is the kinematic viscosity. An approximate analysis, based on ideas similar to those supporting the Pohlhausen method in boundary-layer theory, suggests that when the boundary layer is taken into account the conclusions above remain true provided that exp (lt) is replaced by exp (βlt), where β is a constant depending on v/k. Calculations give values of β ranging from 0·73 when v/k = 0·5 to 0·10 when v/k = 103.
On the torque of wavy vortices
Authors: Eagles P. M.
Published: March 2006

Numerical calculations are made of the torque required to sustain a wavy-vortex flow between rotating cylinders. The results are found to agree well with experimental work of Donnelly (1958), and give further confirmation of the validity of Davey, DiPrima & Stuart's (1968) analysis.
Turbulent vortex streets and the entrainment mechanism of the turbulent wake
Authors: Papailiou Demosthenes D. , Lykoudis Paul S.
Published: March 2006

The results of an experimental investigation of a turbulent vortex street in the range 103 [lsim ] Re [lsim ] 2 × 104 are presented. The vortex street was created by the motion of a circular cylinder in a motionless fluid (mercury). Photographs obtained showed that the turbulent street, created by the vortex shedding behind the cylinder, persisted at longer downstream distances and higher Reynolds numbers than previously reported in the literature. A theory was developed to account for the experimental measurements pertaining to the change of the geometrical characteristics, h (distance between the two rows of vortices) and α (longitudinal distance between two consecutive vortices on the same row), of the street in the downstream direction. The implications of the structure of the vortex street on the entrainment mechanism of the turbulent wake are discussed. Some observations of the flow in the formation region of the vortices are discussed in relation to existing work.
Physical characteristics of subsonic jets in a cross-stream
Authors: Chassaing P. , George J. , Claria A. , Sananes F.
Published: March 2006

This paper deals with local flow characteristics of subsonic turbulent jets in the presence of a cross-flow. For the various types of jet considered (a cylindrical jet and coaxial jets) the experimental results concern the axes and the velocity profiles in the plane of symmetry of the flow. In the case of the cylindrical jet, the shape of the universal axial velocity profile is defined, as are the law of velocity decay along the axis and the laws of variation of the thicknesses of the jet. Finally, the existence of a link between the axis equation and the law of axial velocity decay in the zone of similarity of the velocity profiles is established.
Turbulent shear flow in a curved duct
Authors: Ellis L. B. , Joubert P. N.
Published: March 2006

Experiments were conducted in two curved rectangular ducts of different radii and the results compared with flow in a straight duct. The turbulent boundary layers developing on the curved walls were examined for mean flow properties. The law of the wall takes a modified form and the usual logarithmic portion applies over a very small range of the profiles on the inner walls and a slightly larger range on the outer walls. Similarity based on a defect law did not exist for any part of the flow examined.
A resolution of the blow-off singularity for similarity flow on a flat plate
Authors: Kassoy D. R.
Published: March 2006

A study is made of uniform flow past a semi-infinite flat plate with a similarity injection distribution of boundary-layer magnitude. Attention is focused on a solution at exactly the critical injection rate for which classical boundary-layer theory predicts the blow-off singularity. Following a description of the more recent interaction analyses which also fail at the critical rate, a new theory is developed which leads to physically meaningful results. In particular, it is shown that the non-monotonic variation in wall shear with increasing injection rate near the critical value, noted by Klemp & Acrivos (1972), is real. A delicate interplay of weak pressure interactions and viscous effects is shown to be responsible for this surprising phenomenon.
Combined dye-streak and hydrogen-bubble visual observations of a turbulent boundary layer
Authors: Offen G. R. , Kline S. J.
Published: March 2006

The turbulent boundary layer over a flat plate was observed visually using two dye injectors and a normal bubble wire. One dye injector was a standard wall slot; the other one was a small Pitot probe which could be placed anywhere in the flow. The purpose of these experiments was to improve understanding of the relationships between the bursting of low-speed streaks near the wall and the flow field farther away from the wall.

On the basis of these observations, it seems that (i) each lift-up is associated with a disturbance which originates in the logarithmic region and is characterized by a mean motion towards the wall, and that (ii) such disturbances are generated by the interaction of an earlier burst from further upstream with the fluid motion in the logarithmic region.
Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier
Authors: Mandelbrot Benoit B.
Published: March 2006

Kolmogorov's ‘third hypothesis’ asserts that in intermittent turbulence the average ε of the dissipation ε, taken over any domainD, is ruled by the lognormal probability distribution. This hypothesis will be shown to be inconsistent, save under assumptions that are extreme and unlikely. Further, a widely used justification of lognormality, due to Yaglom and based on probabilistic argument involving a self-similar cascade, will be discussed. In this model, lognormality indeed applies strictly when D is ‘an eddy’, typically a three-dimensional box embedded in a self-similar hierarchy, and may perhaps remain a reasonable approximation when D consists of a few such eddies. On the other hand, the experimental situation is described better by considering averages taken over essentially one-dimensional domainsD. The first purpose of this paper is to carry out Yaglom's cascade argument, labelled as ‘microcanonical’, for such averaging domains. The second is to replace Yaglom's model by a different, less constrained one, based upon the concept of ‘canonical cascade’. It will be shown, both for one-dimensional domains in a microcanonical cascade, and for all domains in canonical cascades, that in every non-degenerate caJe the distribution of ε differs from the lognormal distribution. Depending upon various parameters, the discrepancy may be either moderate, or considerable, or even extreme. In the latter two cases, high-order moments of E turn out to be infinite. This avoids various paradoxes (to be explored) that are present in Kolmogorov's and Yaglom's approaches. The third purpose is to note that high-order moments become infinite only when the number of levels of the cascade tends to infinity, meaning that the internal scale η tends to zero. Granted the usual value of η, this number of levels is actually small, so the representativity of the limit is questionable. This issue was investigated through computer simulation. The results bear upon the question of the extent to which Kolmogorov's second hypothesis applies in the face of intermittency. The fourth purpose is as follows. Yaglom noted that the cascade model predicts that dissipation only occurs in a portion of space of very small total volume. In order to describe the structure of this portion of space, it will be shown useful to introduce the concept of the ‘intrinsic fractional dimension’ A of the carrier of intermittent turbulence. The fifth purpose is to study the relations between the parameters ruling the distribution of η, and those ruling its spectral and dimensional properties. Both conceptually and numerically, these various parameters turn out to be distinct, which opens up several problems for empirical study.
Stress work effects in laminar flat-plate natural convection
Authors: Ackroyd J. A. D.
Published: March 2006

The effects of both viscous and pressure stress work are considered in laminar natural convection on flat surfaces. Whereas in previous investigations only viscous stress work effects were studied, it is established here that pressure work effects are generally rather more important both for gases and liquids. Variations in the properties of the fluid outside the natural convection boundary layer are shown to occur inevitably in this problem and are found to lead to effects comparable with those produced by pressure work. Property variations in the boundary layer itself are also found to be of importance. Such variations, both inside and outside the boundary layer, have been ignored in previous studies. Examples of flows are discussed in which the fluids are either perfect gases or fluids undergoing small changes in temperature and pressure. For both of these fluids, the effects of conditions of either constant temperature or constant entropy in the fluid outside the boundary layer are examined.
Fluid motions induced by a periodic magnetic field in and about a liquid drop
Authors: Sozou C.
Published: March 2006

In this paper we consider the flow field induced by a periodic magnetic field in and about a conducting liquid drop immersed in an incompressible insulating fluid. It is assumed that at infinity the magnetic field varies with time t as cos ωt, where ω is a constant. This magnetic field is associated with a periodic electric field which produces a net electric stress, dependent on the spatial variables, normal to the drop surface. This stress sets up a flow field, in and about the liquid drop, that creates an appropriate viscous stress so that there is stress balance at the drop surface. The flow field is periodic with angular frequency 2ω. For small drop deformations the drop shape at any instant is a spheroid. It is shown that for large ω the amplitude of the velocity field for a conducting drop is approximately independent of ω and for a non-conducting drop it is proportional to ω. The larger velocity amplitude for a non-conducting drop is probably due to the fact that in this case there is no dissipation of electromagnetic energy. The electric stress over the drop surface increases with ω and it is suggested that the drop will burst at large ω unless the amplitude of the applied magnetic field is suitably decreased.
Gasdynamics of a centrifugal machine
Authors: Sakurai Takeo , Matsuda Takuya
Published: March 2006

We consider axisymmetric steady centrifugally driven thermal convection in a compressible fluid in a rapidly rotating circular cylinder. The Boussinesq approximation is not used, because it is not valid for the case of practical interest. We clarify the importance of the effect of the flow-induced volume change of a fluid particle, and propose a widely applicable method of solution.
Motion driven by surface-tension gradients in a tube lining
Authors: Davis Stephen H. , Liu An-Kuo , Sealy George R.
Published: March 2006

A fluid layer that lines the inner surface of a circular tube has motion induced by axial surface-tension gradients. The lubrication equations for the system are analysed and it is found that even for thin layers the motions differ markedly from those in planar layers. The planar case serves as a class of outer solutions. These approximate solutions are modified by a boundary-layer correction where the mean surface tension is important.
Second-order effects in free convection
Authors: Walton I. C.
Published: March 2006

The equations of conservation of momentum, energy and mass together with the equations of state are examined for free convection from a vertical paraboloid. A transformation due to Saville & Churchill is applied to the first- and second-order boundary-layer equations, which are then solved using series about the stagnation point, using asymptotic series far up the body and in between by a method due to Merk. The second-order outer inviscid flow is given in terms of infinite integrals as a solution of Laplace's equation in paraboloidal co-ordinates.

Eight second-order effects are distinguished, depending on longitudinal and transverse curvatures, the displacement flow, heat flux into the boundary layer and the variation of density, viscosity, thermometric conductivity and the coefficient of expansion with temperature. Expressions for the skin friction, heat-transfer coefficient and various flux thicknesses are obtained and a comparison of the second-order effects is made.
Considerations of the stability of certain heterogeneous shear flows including some inflexion-free profiles
Authors: Chimonas George
Published: March 2006

We use some well-known properties of the Taylor-Goldstein equation to generate a set of stably stratified shear flows with known singular neutral-mode solutions. The novel feature of the analysis is that it includes such solutions for flows in which, proceeding upwards from a rigid boundary, the Brunt-Väisälä frequency and the flow shear do not change sign and are monotonically decreasing functions of height. Such profiles are much closer to the conditions met in work on the atmospheric boundary layer than the more frequently used inflected flow profiles.
Reynolds-shear-stress measurements in a compressible boundary layer within a shock-wave-induced adverse pressure gradient
Authors: Rose W. C. , Childs M. E.
Published: March 2006

The results of an experimental investigation of the mean- and fluctuating-flow properties of a compressible turbulent boundary layer in a shock-wave-induced adverse pressure gradient are presented. The turbulent boundary layer developed on the wall of an axially symmetric nozzle and test section whose nominal free-stream Mach number and boundary-layer-thickness Reynolds number were 4 and 105, respectively. The adverse pressure gradient was induced by an externally generated, conical shock wave.

Mean and time-averaged fluctuating-flow data, including the experimental Reynolds shear stresses and experimental turbulent heat-transfer rates, are presented for the boundary layer and external flow, upstream, within and downstream of the pressure gradient. The turbulent mixing properties of the flow were determined experimentally with a hot-wire anemometer. The calibration of the wires and the interpretation of the data are discussed.

From the results of the investigation, it is concluded that the shock-wave/boundary-layer interaction significantly alters the shear-stress characteristics of the boundary layer.
The hydrodynamic interaction of two unequal spheres moving under gravity through quiescent viscous fluid
Authors: Wacholder E. , Sather N. F.
Published: March 2006

The hydrodynamic forces and couples that act on two spherical particles in slow motion through a quiescent fluid are determined as functions of the relative configuration of the particles from the solution of the Stokes equation for the motion of the fluid in the vicinity of the particles. General formulae that relate the translational and rotational velocities of the particles to the ratios of their radii a = a2/a1 and net densities I are obtained, as are asymptotic forms for the velocities in the limiting cases of very large and very small interparticle separation. Relative trajectories of the particles when they move solely under gravity and their own interaction are calculated for several values of I and a. A particularly interesting feature of the results is that, for certain ranges of values of I and a, trajectories of finite length and trajectories having the form of closed periodic orbits may occur.
A laminar boundary layer on a rotating three-dimensional blade
Authors: Miyake Y. , Fujita S.
Published: March 2006

The laminar boundary layer on a rotating thin blade of an axial turbomachine is discussed. A method of deriving the basic equations of the flow for a generalized blade configuration is described, treating a typical blade in practical use as an example. A perturbation technique is applied to obtain a series solution composed of two groups of similar functions. The results give insight into the properties of the boundary layer on helical non-loaded blades in a uniform stream, indicating the influence purely of three-dimensionality of the blade configuration.
Entry flow in a curved pipe
Authors: Singh M. P.
Published: March 2006

This paper deals with the development of the flow in a curved tube near the inlet. The solution is obtained by the method of matched asymptotic expansions. Two inlet conditions are considered: (i) the condition of constant dynamic pressure at the entrance, which may be of practical interest in applications to blood flow in the aorta; and (ii) a uniform entry condition. It is shown that the geometry and the nature of the entry condition appreciably influence the initial development of the flow. The effect of the secondary flow due to the curvature on the wall shear is discussed and it is shown that the cross-over between shear maxima on the inside and the outside of the tube occurs at a downstream distance which is 1·9 times the radius of the tube for entry condition (i) while in the case of entry condition (ii) it is 0·95 times the radius, which is half the distance required in case (i). It is found that the pressure distribution is not significantly influenced by the secondary flow during the initial development of the motion. The analysis, which is developed for steady motion, can be extended to pulsatile flows, which are of greater physiological interest.
Solution of the Boltzmann equation at the upstream and downstream singular points in a shock wave
Authors: Elliott J. P. , Baganoff D.
Published: March 2006

A solution of the Boltzmann equation is obtained at the upstream and downstream singular points in a shock wave, for the case of Maxwell molecules. The fluid velocityu, rather than the spatial co-ordinatex, is used as the independent variable, and an equation for ∂f/∂uat a singular point is obtained from the Boltzmann equation by taking the appropriate limit. This equation is solved by using the methods of Grad and of Wang Chang & Uhlenbeck; and it is observed that the two methods are the same, since they involve not only an equivalent system of moment equations but also the same closure relations. Because many quantities are zero at a singular point, the problem becomes sufficiently simple to allow the solution to be carried out to any desired order. At the supersonic singular point, the solution converges very slowly for strong shock waves; but a simple modification to Grad's method provides a rapidly convergent solution. The solution shows that the Navier-Stokes relations, or the first-order Chapman-Enskog results, do not apply unless the shock-wave Mach number is unity, and that they are grossly in error for strong shock waves. The solution confirms the existence of temperature overshoot in a strong shock wave; shows that the critical Mach number in Grad's solution increases monotonically with the order of the solution; provides a simple explanation as to why Grad's closure relations fail and shows how they can be improved; and provides exact boundary values that can be used to guide future numerical solutions of the Boltzmann equation for shock-wave structure.
On the incipient breaking of small scale waves
Authors: Banner M. L. , Phillips O. M.
Published: March 2006

It is shown that the surface wind drift in the ocean substantially reduces the maximum wave height ξxand wave orbital velocity that can be attained before breaking. Ifqis the magnitude of the surface drift at the point where the wave profile crosses the mean water level andcis the wave speed, then\[ \zeta_{\max} = \frac{c^2}{2g}\bigg(1-\frac{q}{c}\bigg)^2. \]Incipient breaking in a steady wave train is characterized by the occurrence of stagnation points at wave crests, but not necessarily by discontinuities in slope. After breaking, there is in the mean flow a stagnation point relative to the wave profile near the crest of the broken wave, on one side of which the water tumbles forward and behind which it recedes more smoothly to the rear. Some simple flow visualization studies indicate the general extent of the wake behind the breaking region.
Molecular models for permeation through thin membranes: the effect of hydrodynamic interaction on permeability
Authors: Malone George H. , Hutchinson Thomas E. , Prager Stephen
Published: March 2006

A variational method involving minimization of the energy dissipation rate which was previously developed for transport in polymer systems is applied here to flow of a continuum solvent through a thin membrane. The membrane is represented by an array of spherical particles undergoing Brownian motion, subject to various interactions with one another and with the motion of the solvent. General upper bounds on the solvent permeability of the membrane are obtained in terms of equilibrium distribution functions, and applications of the method are illustrated for the case where membrane elements are confined to a plane. Calculations which treat all beads equivalently give permeability estimates whose dependence on the number n of beads per unit area of membrane has the form, at low n,
\[
\kappa = (6\pi\eta n)^{-1}(1-\alpha n+\ldots),
\]
where η is the solvent viscosity and α is a constant. More-elaborate trials which allow the drag on a bead to be influenced by the distribution of other beads in the vicinity give the stronger bounding estimate
\[
\kappa = (6\pi\eta n)^{-1}(1+\alpha^{\prime}n\ln n+\ldots).
\]
Comparison with a self-consistent field approach suggests that this logarithmic behaviour is the true first-order correction.
An oscillating-boundary-layer theory for ciliary propulsion
Authors: Brennen C.
Published: March 2006

This paper analyses the locomotion of a finite body propelling itself through a viscous fluid by means of travelling harmonic motions of its surface. The methods are developed with application to the propulsion of ciliated micro-organisms in mind. Provided that the metachronal wavelength (of the surface motions) is much smaller than the overall dimensions of the body, the flow can be divided into an oscillating-boundary-layer flow to which is matched an external complementary Stokes flow. The present paper employs the envelope model of fluid/cili a interaction to construct equations of motion for the oscillating boundary layer. The final solution for the propulsive velocity is obtained by application of the condition of zero total force on the self-propelling body; alternatively, if the organism is held at rest, the thrust it generates can be computed. Various optimum propulsive velocities for self-propelling bodies and optimum thrusts for restrained bodies are analysed in some simple examples. The results are compared with the relatively sparse observations for a number of microorganisms.
The hypersonic laminar boundary layer near sharp compression and expansion corners
Authors: Bloy A. W. , Georgeff M. P.
Published: March 2006

The momentum integral method of Klineberg is shown to provide a good description of the major characteristics of two-dimensional laminar viscous-inviscid interactions a t hypersonic speeds. Surface pressure and heat-transfer-rate measurements were made for sharp compression and expansion corners at Mach 12.2 and are compared with the theoretical predictions. The agreement is found to be good for attached, incipient and fully separated flows.

Some theoretical comparisons between methods based on the Klineberg formulation are made which suggest that the full boundary-layer equations are well described using integral methods that incorporate the energy equation. It is further shown that the properties associated with the stability of the governing differential equations are mathematical properties of the analytical model and should not be associated with any physical characteristics of the boundary layer.

A correlation of hypersonic, cold-wall, incipient separation data is also presented.
Upstream propagation of unsteady disturbances in supersonic boundary layers
Authors: Schneider Wilhelm
Published: March 2006

It is shown that weak harmonic waves can propagate upstream in a supersonic stream along a plane wall over distances that are very large in comparison with the boundary-layer thickness. These waves are characterized by small reduced frequencies, large wavelengths and small phase velocities. Results are presented for the flow along a single plane wall and alsoforthe flow between two plane parallel plates (channel flow).

The progressive-wave solutions are found by asymptotic expansions for small disturbances, large Reynolds numbers of the basic flow, and small reduced frequencies of the unsteady disturbances. It turns out that, as in the corresponding steady theory, four flow regions have to be distinguished: a middle layer which embraces most of the boundary layer; an inner layer near the wall; the outer flow field; and a transition layer between the middle layer and the outer layer. A quasi-steady treatment of the middle, transitional and outer layers is appropriate. Unsteady effects originate in the inner layer.

The relative importance of viscosity and unsteady effects with regard to the waves is characterized by a dimensionless parameterNwhich is the product of certain powers of the reduced frequency and the Reynolds number. ForN→ ∞ and a single wall, Lighthill's steady theory of upstream influence in supersonic boundary layers is recovered as a limiting case of the present theory.
On the stability of a magnetically driven rotating fluid flow
Authors: Richardson A. T.
Published: March 2006

After making the laboratory approximation of small magnetic Reynolds number, the steady, axisymmetric and purely azimuthal velocity profile that in principle can be generated in an incompressible viscous electrically conducting fluid contained in a fixed infinitely long circular cylinder by a magnetic field transverse to the cylinder axis and uniformly rotating with low frequency is subjected to infinitesimal axisymmetric perturbations. The principle of the exchange of stabilities is assumed to hold and the marginal-stability problem becomes a sixth-order eigenvalue problem involving the magnetic Taylor number and the axial wavenumber. An asymptotic analysis, based on the assumption that the magnetic Taylor number is large, and using solutions of the comparison equation d6y/dz6 = zy, is presented in order to obtain first approximations to the neutral-stability curves of the first and second eigenmodes, and compared with the results of direct numerical integration. It is found that at the onset of instability the secondary motions have a multi-cell structure, the motions in the region, near the cylinder wall, of adversely distributed angular momentum driving through weak viscous action the cells in the interior.
Convection of a passive scalar by a quasi-uniform random straining field
Authors: Kraichnan Robert H.
Published: March 2006

The stretching of line elements, surface elements and wave vectors by a random, isotropic, solenoidal velocity field in D dimensions is studied. The rates of growth of line elements and (D – 1)-dimensional surface elements are found to be equal if the statistics are invariant to velocity reversal. The analysis is applied to convection of a sparse distribution of sheets of passive scalar in a random straining field whose correlation scale is large compared with the sheet size. This is Batchelor's (1959) κ−1 spectral regime. Some exact analytical solutions are found when the velocity field varies rapidly in time. These include the dissipation spectrum and a joint probability distribution that describes the simultaneous effect of Stretching and molecular diffusivity κ on the amplitude profile of a sheet. The latter leads to probability distributions of the scalar field and its space derivatives. For a growing κ−1 range at zero κ, these derivatives have essentially lognormal statistics. In the steady-state κ−1 regime at κ > 0, intermittencies measured by moment ratios are much smaller than for lognormal statistics, and they increase less rapidly with the order of the derivative than in the κ = 0 case. The κ > 0 distributions have singularities a t zero amplitude, due to a background of highly diffused sheets. The results do not depend strongly on D. But as D → ∞, temporal fluctuations in the stretching rates become negligible and Batchelor's (1959) constant-strain dissipation spectrum is recovered.
On the development of dunes in erodible channels
Authors: Fredsøe Jørgen
Published: March 2006

A two-dimensional stability analysis of the flow in a straight alluvial channel has been carried out, using the vorticity transport equation. In the analysis an attempt has been made to account for the influence of gravity on bed-load transport, and this turned out to change the stability quite significantly.

In the case of instability, the further growth of the dunes has been investigated using a second-order approximation, This nonlinear theory explains the experimental fact that the dunes very soon become asymmetric.
An analysis of flow through sudden enlargements in pipes
Authors: Teyssandiert R. G. , Wilson M. P.
Published: March 2006

An integral analysis of the type used to predict the flow of co-flowing jets has been applied to the problem of a sudden enlargement in a pipe (Borda–Carnot expansion). This technique successfully predicts all the overall flow parameters of interest (e.g. reattachment lengths, pressure profile, etc.). The analysis indicates that the downstream conditions (up to reattachment) are insensitive to wall shear and the point of minimum pressure does not coincide with the location of the maximum return-flow velocity.
The meniscus on the outside of a small circular cylinder
Authors: James David F.
Published: March 2006

The method of matched asymptotic expansions is used to solve the differential equation describing the shape of a meniscus on the outside of a circular cylinder. Since the perturbation quantity is proportional to the cylinder radius, the solution is valid basically for small oylinders. The predicted meniscus height is compared with numerical data to determine the accuracy of the two-term result; the third term is found but does not improve the estimate.
Finite-amplitude neutral disturbances in plane Poiseuille flow
Authors: George W. D. , Hellums J. D. , Martin B.
Published: March 2006

Finite-amplitude disturbances in plane Poiseuille flow are studied by a method involving Fourier expansion with numerical solution of the resulting partial differential equations in the coefficient functions. A number of solutions are developed which extend to relatively long times so that asymptotic stability or instability can be established with a degree of confidence. The amplitude for neutral stability is established for a fixed wavenumber for two values of the Reynolds number. Details of the neutral velocity fluctuation are presented. These and earlier results are expressed in terms of the asymptotic amplitude and compared with results obtained by prior workers. The results indicate that the expansion methods used by prior workers may be valid only for amplitudes considerably smaller than 0·01.
Nonlinear interactions among standing surface and internal gravity waves
Authors: Joyce Terrence M.
Published: March 2006

A laboratory study has been undertaken to measure the energy transfer from two surface waves to one internal gravity wave in a nonlinear, resonant interaction. The interacting waves form triads for which
\[
\sigma_{1s} - \sigma_{2s} \pm\sigma_1 = 0\quad {\rm and}\quad \kappa_{1s} - \kappa_2s} \pm \kappa_I = 0;
\]
σj and κj being the frequency and wavenumber of the jth wave. Unlike previously published results involving single triplets of interacting waves, all waves here considered are standing waves. For both a diffuse, two-layer density field and a linearly increasing density with depth, the growth to steady state of a resonant internal wave is observed while two deep water surface eigen-modes are simultaneously forced by a paddle. Internal-wave amplitudes, phases and initial growth rates are compared with theoretical results derived assuming an arbitrary Boussinesq stratification, viscous dissipation and slight detuning of the internal wave. Inclusion of viscous dissipation and slight detuning permit predictions of steady-state amplitudes and phases as well as initial growth rates. Satisfactory agreement is found between predicted and measured amplitudes and phases. Results also suggest that the internal wave in a resonant triad can act as a catalyst, permitting appreciable energy transfer among surface waves.