bluuebunny/crossref_metadata_2025 huggingface dataset:
Unsteady laminar convection in uniformly heated vertical pipes
Authors: Gupta R. K.
Published: March 2006

In this paper an exact solution is presented for the problem of unsteady laminar convective flow under a pressure gradient along a vertical pipe. We have obtained the solution of the problem on the basis of the assumption that the velocity and buoyancy profiles far from the pipe entrance do not change with the height, and the entry lengths have been ignored. The wall of the pipe is heated or cooled uniformly. We have discussed both the cases, when buoyancy forces act together with the pressure gradient or in opposite direction.

In the case when the upflow is heated (or a downflow is cooled) the velocity and thermal boundary layers are formed for sufficiently large Rayleigh numbers. In the second case which has been discussed in detail (when the upflow is cooled or the downflow is heated) we have found the critical value of the Rayleigh number R = R, beyond which the velocity profile and the temperature profile become unsteady and turbulent in all the cases. In the case of the elliptical cylinder R, increases up to 1730 as the ellipticity is increased while in the case of the co-axial pipes this Rayleigh number increases as the gap c between the cylinders is decreased (if c = a/b = 1·2 then R, = 60762, but decreases to 1 when c = 4). Besides this, the time required to reach steady state increases as the Rayleigh number increases in both circular and elliptical pipes; it also increases when the eccentricity is decreased. The cases discussed by Morton (1960 and Dalip Singh (1965 are particular cases of the results derived below.

In this investigation we have dealt with the following ducts: (i) circular tubes, (ii) elliptical tubes and (iii) co-axial tubes. The general solutions for both velocity and temperature fields have been found for the case when the pressure gradient is an arbitrary function of time, with an arbitrary heat source also present. Particular cases when both the parameters are absohte constants have been discussed in detail.

We have made use of finite transforms very frequently; especially for the case of an elliptical tube, a new transform involving Mathieu functions developed by Gupta (1964 has been used. A few new infinite series have been summed with the help of this transform.

Various non-dimensional quantities (for both the cylinders) such as the Nusselt number, volume flux and rate of heat transfer have been found when the pressure gradient and source of heat generation are absolute constants.
The dynamical effect of inertial waves on the gyroscopic motion of a body containing several eccentrically located liquid-filled cylinders
Authors: Scott W. E.
Published: March 2006

A theoretical analysis is made of the effect described in the title. It is shown that both (i) the eigenfrequencies of the inertial waves and (ii) the moment produced by the inertial wave pressure distribution are independent of the location of the off-axis cylinders. Hence, one can use Stewartson's tables computed for a centrally located cylinder to calculate the frequencies and residues for an eccentrically located cylinder.
Flow of dilute polymer solutions about circular cylinders
Authors: Sarpkaya Turgut , Raineyt P. G. , Kell R. E.
Published: March 2006

Experiments on fully attached flows of aqueous solutions of Polyox WSR-301 and pure solvent (tap and distilled water) about circular cylinders were carried out in the range 5 × 104<Re< 3 × 105for the initial purpose of providing definitive experimental information on drag reduction with additives in external flows.

During the course of the investigation, a critically degraded polymer solution was observed to cause, at a particularRe, an unforseen instability in the separation line, similar to that first observed by Brennen (1970 in fully developed cavity flows. This instability, which most easily manifest edit self in the form of an abrupt drag-force jump (here called a drag crisis) from a high subcritical to a low supercritical value, became the central issue in the investigation. Efforts to isolate the causes of this instability led to further experiments with suspensions of nearly neutrally buoyant spherical and non-spherical particles. The measurements have shown that fibres and polymer hasten transition, spherical particles do not appreciably affect the characteristics of the flow and that neither the spherical particles nor fibres give rise to a drag crisis.
Nonlinear Bénard convection with rotation
Authors: Morgan J. C.
Published: March 2006

The equations for nonlinear Bénard convection with rotation for a layer of fluid, thickness d, are derived using the Glansdorff & Prigogine (1964) evolutionary criterion as used by Roberts (1966) in his paper on non-rotational Bénard convection. The parameters of the problem in this case are the Rayleigh number R = αgΔθd/vK, the Taylor number T = 4d4Ω23/v2 and the Prandtl number Pr = v/K, where α is the coefficient of volume expansion, g the acceleration due to gravity, Δθ the temperature difference between the horizontal surfaces, v the kinematic viscosity, K the thermal diffusivity and Ω3 the rotation rate about the vertical direction. The asymptotic solution for two-dimensional cells (rolls) is investigated for large Rayleigh numbers and large Taylor numbers. For rolls the convection equations are found t o be independent of the Prandtl number. However, the solutions depend upon the Prandtl number for another reason. The rotational problem differs from the non-rotational one in that the Rayleigh number and the horizontal wavenumber a of the convection are now functions of the Taylor number. These are taken to be R ∼ ρTα′ and α ∼ ATβ, where α′ and β are positive numbers. Thermal layers develop as R becomes large with ρ or T becoming large. The order in which ρ and T are allowed to increase is important since the horizontal wavenumber a also increases with T and the convection equations can be reduced in this case. A liquid of large Prandtl number such as water has v [Gt ] K. Since R ∼ O (1/vK) and T ∼ O(1/v2), ρ will be greater than T for a given (large) Δθ and Ω3. Similarly, for a liquid of small Prandtl number such as mercury v [Lt ] K, and T is greater than ρ for a given Δθ and Ω3. For rigid-rigid horizontal boundaries with ρ large and then T large the ρ thermal layer has the same structure as for the non-rotating problem. As T → ∞ three types of thermal layers are possible: a linear Ekman layer, a nonlinear Ekman layer and a Blasius-type thermal layer. When the horizontal boundaries are both free the ρ thermal layer is again of the same structure as for non-rotating BBnard convection. As T → ∞ a nonlinear Ekman layer and a Blasius-type thermal layer are possible.

When T is large and then ρ made large the differential equations governing the convection are reduced from eighth order to sixth order owing to a becoming large as T → ∞. There are Ekman layers as T → ∞, when the horizontal boundaries are both rigid. The ρ thermal layers now have a different structure from the non-rotating problem for both rigid-rigid and free-free horizontal boundaries. The equation for small amplitude convection near to the marginal case is derived and the solution for free-free horizontal boundaries is obtained.
Selective withdrawal and blocking wave in rotating fluids
Authors: Pao Hsien-Ping , Shih Hsing-Hua , Kao Timothy W.
Published: March 2006

An analysis is made of the axially symmetric flow of a rotating inviscid incompressible fluid into a point sink a t sufficiently low values of the Rossby number. Based on the experimental observations, a theoretical flow model involving a surface of velocity discontinuity which separates the central flowing core from the surrounding stagnant region is proposed. A family of solutions is obtained after posing the problem as one involving a free streamline which is the line of velocity discontinuity in the axial plane. It is found that the flow possesses a minimum flow force as well as a minimum energy flux. Corresponding to such a state, a unique intrinsic Rossby number R′ based on the properties of the flowing core with a value of 1/[npar ]8 is determined. A discussion is made of the flow field development induced by a sudden start of a sink discharge. A theoretical model involving a blocking wave propagating upstream is proposed. The speeds of blocking waves are found to be higher than the maximum group velocity of the infinitesimal waves for R > 0.06. On the other hand, for R < 0.03, the waves are linear and dispersive in nature.
Shear flow instability in a conducting viscous fluid
Authors: Roberts B.
Published: March 2006

The effect of a parallel magnetic field upon the stability of the plane interface between two conducting viscous fluids in uniform relative motion is considered. A parameter reduction, which has not previously been noted, is employed to facilitate the solution of the problem. Neutral stability curves for unrestricted ranges of the governing parameters are found, and the approximate solutions of other authors are examined in this light.
The motion of a cavity in a vertical rotating tube
Authors: Collins R. , Hoath M. T.
Published: March 2006

Experiments are performed to measure the upward velocity U of the cavity resulting from the draining of a vertical liquid-filled tube of radius b, which rotates about its axis at various speeds. With no rotation the cavity moves so that the Froude number U/(gb)½ is constant, where g is the gravitational acceleration. For high angular velocities Ω, however, the cavity appears to adopt a constant value of the Rossby number U/Ωb, implying that gravitational forces are dominaked by centrifugal forces. The cavity velocity finally achieved is found to be the same as the maximum group velocity of infinitesimal waves, so that U/Ωb = 0.52. Approximate theoretical models which satisfactorily describe the development of the motion are constructed.
Generation of magnetic fields by convection
Authors: Busse F. H.
Published: March 2006

The nonlinear hydromagnetic dynamo problem is investigated for the case of convection in a layer of an electrically conducting fluid heated from below. It is shown that two-dimensional convection rolls in conjunction with a longitudinal mean flow are capable of amplifying a magnetic field in the form of a wave propagating in the longitudinal direction. The action of the Lorentz forces causes a reduction of the amplitude of convection with the consequence that the energy of the magnetic field cannot grow beyond an equilibrium value which is determined as a function of the parameters of the problem. The analysis is based on an expansion in powers of the longitudinal wavenumber β of the magnetic field and applies in the case of large values of the magnetic Prandtl number.
On the movement of a spherical particle in vertically oscillating liquid
Authors: Boyadzhiev L.
Published: March 2006

The movement of a heavy spherical particle in vertically oscillating liquid is studied by means of an analog computer. Using the Oseen-Tchen-Houghton force-balance model, it is shown that in the case of asymmetrical oscillations the particle may be made quasi-stationary with a finite amplitude of oscillation. It is suggested that under such conditions a mean movement of the particle in the upward direction is also possible.
Experiments on nonlinear interactions in the transition of a free shear layer
Authors: Miksad Richard W.
Published: March 2006

An experimental study is made of nonlinear interactions in a laminar free shear layer. Two disturbances (f1 and f2), excited by sound, amplify and grow independently for small amplitudes. At larger amplitudes the disturbances interact to generate fluctuations of sum and difference frequencies (f2 ± f1). Harmonics and subharmonics of f1 and f2 are also generated and all fluctuations interact to generate additional fluctuations of the form (nf2/m) ± (pf1/q); n, p = 1,2,3,…, m, q = 1,2. Nonlinear mode competition suppresses the growth of f1 or f2, depending on their relative amplitudes, and contributes to finite amplitude equilibration. An upper bound on the modal integral of total u′r.m.s.2 fluctuation energy is found. Fluctuation energy tends to be distributed among all possible frequency components, and its upper bound does not increase as the number of components increases.
Nonlinear gas oscillations in pipes. Part 1. Theory
Authors: Jimenez J.
Published: March 2006

The problem of forced acoustic oscillations in a pipe is studied theoretically. The oscillations are produced by a moving piston in one end of the pipe, while a variety of boundary conditions ranging from a completely closed to a completely open mouth at the other end are considered. All these boundary conditions are modelled by two parameters: a length correction and a reflexion coefficient equivalent to the acoustic impedance.

The linear theory predicts large amplitudes near resonance and nonlinear effects become crucially important. By expanding the equations of motion in a series in the Mach number, both the amplitude and wave form of the oscillation are predicted there.

In both the open- and closed-end cases the need for shock waves in some range of parameters is found. The amplitude of the oscillation is different for the two cases, however, being proportional to the square root of the piston amplitude in the closed-end case and to the cube root for the open end.
Finite amplitude sideways diffusive convection
Authors: Hart J. E.
Published: March 2006

We consider the flow in a differentially heated vertical slot filled with a stably stratified solution. The stability of the flow driven by the differential heating is investigated in the limits of small but finite amplitude disturbances and very large solute Rayleigh number RS = gβ(∂Sa/∂z)D4/KSv. If the Schmidt number H = KT/KS is of order 1, the growth of an initial perturbation at the neutral point is balanced by horizontal advection of solute and heat, and a steady equilibration amplitude is attained. The Nusselt number is independent of all fluid properties and is directly proportional to the Rayleigh number excess ε = (Ra – Rac)/Rac. If H is much greater than |RS|⅙, or if the disturbance wave-number is slightly less than the critical wavenumber, subcritical instabilities are possible. In particular a resonant instability is possible. These theoretical predictions are consistent with previous experimental results and with the laboratory results described in this paper. In the experiments we find that the mixing of the initial sugar gradient is accomplished by convection cells which undergo transitions to larger wavelengths. The breakdown of the interfaces between convection cells is described.
On the hydromagnetic stability of an unsteady Kelvin-Helmholtz flow
Authors: Roberts B.
Published: March 2006

An analysis is made of the stability of an unsteady basic flow of a conducting fluid in the presence of a parallel magnetic field. The particular profile investigated is the classical Kelvin–Helmholtz profile modified by the addition of an oscillatory component. Two cases are considered in detail: that of a perfectly conducting fluid and that of a poorly conducting fluid. The investigation leads, in both cases, to an equation of the Hill type. It is concluded that the magnetic field has a stabilizing influence but is nevertheless unable to suppress the Kelvin–Helmholtz instability in an unsteady (basic) flow.
Numerical experiments on time-dependent rotational Couette flow
Authors: Liu D. C. S. , Chen C. F.
Published: March 2006

The flow induced by impulsively starting the inner cylinder in a Couette flow apparatus is investigated by using a nonlinear analysis. Explicit finite-difference approximations are used to solve the Navier–Stokes equations for axisymmetric flows. Random small perturbations are distributed initially and periodic boundary conditions are applied in the axial direction over a length which, in general, is chosen to be the critical wavelength observed experimentally. Simultaneous occurrence of Taylor vortices is obtained at supercritical Reynolds numbers. The development of streamlines, perturbation velocity components and the kinetic energy of the perturbations is examined in detail. Many salient features of the physical flow are observed in the numerical experiments.
Generalized expressions for secondary vorticity using intrinsic co-ordinates
Authors: Lakshminarayana B. , Horlock J. H.
Published: March 2006

Equations for the development of streamwise (or secondary) vorticity, for stationary and rotating systems, are derived directly from the vector equation for vorticity, using intrinsic co-ordinates. This approach places emphasis on the coupled equations for the vorticity components parallel and normal to the streamline. The equations derived are more general than those hitherto available, being valid for compressible, stratified and viscous flow. Some interpretation is given of the physical mechanism causing secondary flow.
On optimum profiles in Stokes flow
Authors: Pironneau O.
Published: March 2006

In this paper, we obtain the first-order necessary optimality conditions of an optimal control problem for a distributed parameter system with geometric control, namely, the minimum-drag problem in Stokes flow (flow at a very low Reynolds number). We find that the unit-volume body with smallest drag must be such that the magnitude of the normal derivative of the velocity of the fluid is constant on the boundary of the body. In a three-dimensional uniform flow, this condition implies that the body with minimum drag has the shape of a pointed body similar in general shape to a prolate spheroid but with some differences including conical front and rear ends of angle 120°.
On the radiation and scattering of short surface waves. Part 3
Authors: Leppington F. G.
Published: March 2006

The radiation properties of partially immersed three-dimensional bodies, in time-periodic motion, are examined in the short-wave asymptotic limit ε → 0, where ε is a non-dimensional wavelength. The method of matched expansions is used to specify an outer approximation, away from the surface wave region, and an inner approximation where the potential, in the vicinity of the obstacle and free surface, depends only on the local geometry. Finally, the radially spreading surface wave field is estimated by ray-theory arguments. Explicit details are given for the heaving and rolling of a circular dock and for the heaving motion of a hemisphere. Some speculations are made regarding the scattering properties of such obstacles.
On the mixing of a low Reynolds number biological jet with a quiescent outer bathing solution
Authors: Goldgraben J. Richard , Weinbaum Sheldon
Published: March 2006

Existing analyses of the very low Reynolds number Stokes jet, e.g. Birkhoff & Zarantonello (1957) and Förste (1963), have been confined to a single-component fluid satisfying the usual zero-slip boundary conditions at a solid surface. In contrast, the low Reynolds number biological jets which emerge from the pores and channels at the secreting surfaces of numerous human, animal and insect organs entail the movement of water and solute subject to novel boundary conditions that arise from the local osmotic driving forces at the secreting surfaces. These boundary conditions introduce a nonlinear coupling between the fluid momentum and solute conservation equations. This paper first discusses the fundamental dimensionless groups and length scales that characterize these biological jet flows and then examines in detail, using the technique of matched inner and outer expansions, one flow situation important in epithelial membrane transport, namely, the two-dimensional mixing of an inhomogeneous jet with a quiescent outer bathing solution which is bounded by a semi-permeable mem brane in the plane of the exit. The paper concludes with a discussion of other physiologically relevant problems which arise from different orderings of the length scales, and different overall geometrical configurations and flow conditions.
Transverse motion of an elastic sphere in a shear field
Authors: Tam Christopher K. W. , Hyman William A.
Published: March 2006

The forces acting on an elastic particle suspended in a shear field, and moving relative to it, are found for the case in which there are small deformations from an initially spherical shape. The deformation is the result of the viscous stresses acting on the particle. Of principal interest is that there is a component of the force perpendicular to the free-stream direction, so that the particle will migrate across the undisturbed streamlines.
Bubble transport theory with application to the upper ocean
Authors: Garrettson G. A.
Published: March 2006

The formalism of transport theory is adapted to a general description of bubble populations in a moving fluid. The bubble distribution, as a function of position, velocity, radius and time, satisfies a Boltzmann-type transport equation that is derived and then formally solved by the method of characteristics. In order to apply this new analytical tool to the specific problem of gas bubble transport in the upper ocean, an ocean model and a bubble dynamics model must be chosen. For the purpose of illustration, explicit solutions are written, for distributed sources in a stationary ocean with simple expressions for bubble gas diffusion and drag. Calculated results clarify the relations between observed bubble distributions at sea, proposed bubble source mechanisms and known models of single-bubble dynamics.
Flow past a suddenly heated vertical plate
Authors: Brown S. N. , Riley N.
Published: March 2006

An analysis is presented that is appropriate to three distinct phases in the temporal development of the flow past a semi-infinite vertical plate whose temperature is suddenly changed from that of the surrounding fluid. These are the initial stage when a one-dimensional solution describes the flow, then a local solution which describes the early stages of the departure from this, and finally an asymptotic solution which describes the manner in which the final steady state is achieved.
The singularity at the crest of a finite amplitude progressive Stokes wave
Authors: Grant Malcolm A.
Published: March 2006

Expansions have been given in the past for steady Stokes waves at or near a largest wave with a 120° corner. It is shown here that the solution is more complicated than has been assumed: that the corner is not a regular singular point, and that waves of less than maximum amplitude have singularities of a different order.
On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug
Authors: Wygnanski I. J. , Champagne F. H.
Published: March 2006

Conditionally sampled hot-wire measurements were taken in a pipe at Reynolds numbers corresponding to the onset of turbulence. The pipe was smooth and carefully aligned so that turbulent slugs appeared naturally atRe> 5 × 104. Transition could be initiated at lowerReby introducing disturbances into the inlet. For smooth or only slightly disturbed inlets, transition occurs as a result of instabilities in the boundary layer long before the flow becomes fully developed in the pipe. This type of transition gives rise to turbulent slugs which occupy the entire cross-section of the pipe, and they grow in length as they proceed downstream. The leading and trailing ‘fronts’ of a turbulent slug are clearly defined. A unique relation seems to exist between the velocity of the interface and the velocity of the fluid by which relaminarization of turbulent fluid is prevented. The length of slugs is of the same order of magnitude as the length of the pipe, although the lengths of individual slugs differ at the same flow conditions. The structure of the flow in the interior of a slug is identical to that in a fully developed turbulent pipe flow. Near the interfaces, where the mean motion changes from a laminar to a turbulent state, the velocity profiles develop inflexions. The total turbulent intensity near the interfaces is very high and it may reach 15% of the velocity at the centre of the pipe. A turbulent energy balance was made for the flow near the interfaces. All of the terms contributing to the energy balance must vanish identically somewhere on the interface if that portion of the interface does not entrain non-turbulent fluid. It appears that diffusion which also includes pressure transport is the most likely mechanism by which turbulent energy can be transferred to non-turbulent fluid. The dissipation term at the interface is negligible and increases with increasing turbulent energy towards the interior of the slug.

Mixed laminar and turbulent flows were observed far downstream for\[ 2000 < Re < 2700 \]when a large disturbance was introduced into the inlet. The flow in the vicinity of the inlet, however, was turbulent at much lowerRe. The turbulent regions which are convected downstream at a velocity which is slightly smaller than the average velocity in the pipe we shall henceforth call puffs. The leading front of a puff does not have a clearly defined interface and the trailing front is clearly defined only in the vicinity of the centre-line. The length and structure of the puff is independent of the character of the obstruction which created it, provided that the latter is big enough to produce turbulent flow at the inlet. The puff will be discussed in more detail later.
Upwelling of a stratified fluid in a rotating annulus: steady state. Part 2. Numerical solutions
Authors: Allen J. S.
Published: March 2006

Numerical solutions of finite-difference approximations to the Navier–Stokes equations have been obtained for the axisymmetric motion of a Boussinesq liquid in a rigidly bounded rotating annulus. For most of the cases studied, a temperature difference is maintained between the top and bottom surfaces such that essentially a basic stable density stratification is imposed on the fluid. The side walls are thermally insulated and the motion is driven by a differential rotation of the top surface. Approximate steady-state solutions are obtained for various values of the Rossby number ε and the stratification parameter S = N2/Ω2, where N is the Brunt–Väisälä frequency and Ω is the rotational frequency. The changes in the flow field with the variation of these parameters is studied. Particular attention is given to an investigation of the meridional, or up welling, circulation and its dependence on the stratification parameter. The effects on the flow of different boundary conditions, such as an applied stress driving, specified temperature at the side walls and an applied heat flux at the top and bottom surfaces, are also investigated.
Resonant surface waves
Authors: Ockendon J. R. , Ockendon H.
Published: March 2006

Small amplitude forced horizontal or vertical oscillations of a container of liquid with a free surface may give rise to motions in the liquid on a scale much greater than the forcing amplitude. Three such situations are analysed and, in those cases where the response is still small compared with the dimensions of the container, explicit asymptotic solutions for the liquid motion are obtained.
Aerodynamic noise emission from turbulent shear layers
Authors: Pao S. P.
Published: March 2006

The Phillips (1960) convected wave equation is employed in this paper to study aerodynamic noise emission processes in subsonic and supersonic shear layers. The wave equation in three spatial dimensions is first reduced to an ordinary differential equation by Fourier transformation, then solved via the WKBJ method. Three typical solutions are required for discussions in this paper. The current results are different from the classical conclusions. The effects of refraction, convection, Mach-number dependence and temperature dependence of turbulent noise emission are analysed in the light of solutions to the Phillips equation. Owing to the inherent restrictions of the WKBJ transformation, the results of the present paper should be applied to wave radiation from shear layers whose thickness is no less than approximately one quarter of a wavelength. Such a condition is satisfied for turbulent round jets with an exit velocity greater than 0·6 times the ambient speed of sound.
Viscous–gravity spreading of an oil slick
Authors: Buckmaster J.
Published: March 2006

A two-dimensional oil slick is examined as it spreads on water under the influence of gravitational and viscous forces. An exact analytical and numerical description of the flow is obtained in a certain rational asymptotic limit. The central result is an expression for the size of the slick as a function of time. This expression contains no free parameters to be empirically determined, and is in substantial agreement with experiment.
Nonlinear streaming effects associated with oscillating cylinders
Authors: Bertelsen A. , Svardal A. , Tjøtta S.
Published: March 2006

This paper deals with nonlinear streaming effects associated with oscillatory motion in a viscous fluid. A previous theory by Holtsmarket al.(1954) for the streaming near a circular cylinder in an incompressible fluid of infinite extent is reconsidered and used to obtain new numerical results, which are compared with earlier observations. The regime of validity of this theory is considered. The condition to be satisfied by the Reynolds number is found to be less stringent than was previously supposed.

The more recent theory by Wang (1968) based on the outer–inner expansion technique is discussed and corrected with the Stokes drift.

The case of an incompressible fluid enclosed between two coaxial cylinders, one of which is oscillating, is considered in detail. New theoretical and experimental results are given for various values of the parameters involved (Reynolds number, amplitude and cylinder radii).
Steady separated flow in a linearly decelerated free stream
Authors: Leal L. G.
Published: March 2006

Numerical methods are used to investigate the separated flow over a finite flat plate when the flow at large distances is given by the stream function ψ∞ = −xy and the plate is situated on the x axis from -1 to 1. The range of nominal Reynolds number is 10–800. Reduced-mesh calculations are used for fine resolution of the flow field in the immediate vicinity of the separation point. Streamlines, equi-vorticity lines, and shear stress and pressure gradient at the plate surface illustrate the overall structure of the flow. In each case the streamwise pressure gradient is less than that for undisturbed potential flow and the position of separation is consequently downstream of that predicted by classical boundary-layer theory. The boundary-layer structure in the vicinity of the separation point shows a direct transition between the regular upstream behaviour and Dean's (1950) solution right at separation with no sign whatever of intermediate singular behaviour of the type predicted by Goldstein (1948). The implications of these results for the structure of high Reynolds number, steady, laminar flow are discussed.
The structure of internal intermittency in turbulent flows at large Reynolds number: experiments on scale similarity
Authors: Atta C. W. Van , Yeh T. T.
Published: March 2006

Some of the statistical characteristics of the breakdown coefficient, defined as the ratio of averages over different spatial regions of positive variables characterizing the fine structure and internal intermittency in high Reynolds number turbulence, have been investigated using experimental data for the streamwise velocity derivative ∂u/∂tmeasured in an atmospheric boundary layer. The assumptions and predictions of the hypothesis of scale similarity developed by Novikov and by Gurvich & Yaglom do not adequately describe or predict the statistical characteristics of the breakdown coefficientqr,lof the square of the streamwise velocity derivative. Systematic variations in the measured probability densities and consistent variations in the measured moments show that the assumption that the probability density of the breakdown coefficient is a function only of the scale ratio is not satisfied. The small positive correlation between adjoint values ofqr,land measurements of higher moments indicate that the assumption that the probability densities for adjoint values ofqr,lare statistically independent is also not satisfied. The moments ofqr,ldo not have the simple power-law character that is a consequence of scale similarity.

As the scale ratiol/rchanges, the probability density ofqr,levolves from a sharply peaked, highly negatively skewed density for large values of the scale ratio to a very symmetrical distribution when the scale ratio is equal to two, and then to a highly positively skewed density as the scale ratio approaches one. There is a considerable effect of heterogeneity on the values of the higher moments, and a small but measurable effect on the mean value. The moments are roughly symmetrical functions of the displacement of the shorter segment from the centre of the larger one, with a minimum value when the shorter segment is centrally located within the larger one.
On a stagnation condition for combining or branching inviscid flows
Authors: Sarpkaya Turgut
Published: March 2006

A unique stagnation condition is derived as a guide for analysing systems of combined or branched flows through the use of the free-streamline theory and the Schwarz–Christoffel transformation. Moreover, the physical significance of such a condition is illustrated through its application to the analysis of the penetration and deflexion of jets normal to a bounded stream.
Non-parallel flow corrections for the stability of shear flows
Authors: Ling Chi-Hai , Reynolds W. C.
Published: March 2006

The proper corrections for non-parallel flow to the eigenvalues for small disturbances on a nearly parallel shear flow have been determined through a perturbation about the parallel flow solutions. The resulting shifts in the neutral stability curves have been calculated for the Blasius boundary layer, for the two-dimensional jet, and for the two-dimensional flat-plate wake.
Periodic waves in shallow water
Authors: Bryant P. J.
Published: March 2006

An investigation is made into the evolution, from a sinusoidal initial wave train, of long periodic waves of small but finite amplitude propagating in one direction over water in a uniform channel. The spatially periodic surface displacement is expanded in a Fourier series with time-dependent coefficients. Equations for the Fourier coefficients are derived from three sources, namely the Korteweg–de Vries equation, the regularized long-wave equation proposed by Benjamin, Bona & Mahony (1972) and the relevant nonlinear boundary-value problem for Laplace's equation. Solutions are found by analytical and by numerical methods, and the three models of the system are compared. The surface displacement is found to take the form of an almost linear superposition of wave trains of the same wavelength as the initial wave train.
Pressure fields over hypersonic wing-bodies at moderate incidence
Authors: Malmuth N.
Published: March 2006

Delta wings with conically subsonic cones-bodies mounted on their compressive side are analysed in the hypersonic small disturbance regime. The weakly three-dimensional conditions associated with slender parabolic Mach cones are used to validate a linearized rotational approximation of the flow field. A combined integral–series representation is obtained for the pressure distribution between the wing-body and shock wave for arbitrary body cross-sections, and is specialized to give analytical formulae for arbitrary-order polynomial transversal contours. Numerical results are presented for wedge, parabolic and higher order cross-sections illustrating the dominant character of the cross-flow stagnation singularity associated with sharp wing-body intersections having a finite slope discontinuity. It is shown that the pressure has a logarithmic infinity at this secondary leading edge, as in corresponding Prandtl–Glauert irrotational flows. The relation of this finding to Lighthill's theorem on cross-stream vorticity is discussed. Other features of the pressure field are considered with particular emphasis on their relationship to a recently derived area rule for such configurations, and possibilities for favourable interference.
The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1
Authors: Segur Harvey
Published: March 2006

The method of solution of the Korteweg–de Vries equation outlined by Gardneret al.(1967) is exploited to solve the equation. A convergent series representation of the solution is obtained, and previously known aspects of the solution are related to this general form. Asymptotic properties of the solution, valid for large time, are examined. Several simple methods of obtaining approximate asymptotic results are considered.
Flow through non-uniform gauze screens
Authors: Livesey J. L. , Laws E. M.
Published: March 2006

An experimental investigation into flow through shaped gauze screens in two-dimensional and axisymmetric situations has shown that there is disagreement between measured velocity profiles and those computed from the method developed by Elder (1958). This disagreement has been shown to be attributable to the retention of a term of second order in the basic linearization and comparisons between experimental and theoretical profiles omitting this term are in good agreement.
Helical turbulence and absolute equilibrium
Authors: Kraichnan Robert H.
Published: March 2006

The interaction of two pure helical (circularly polarized) velocity waves according to the incompressible Navier–Stokes equation produces modulation products of mixed helicity. In general, the interaction of waves of opposite helicity is stronger than that of waves with the same helicity. The inference is that strong net helicity depresses overall turbulent energy transfer. The conservation laws strongly inhibit energy transfer from higher to lower wavenumbers, when the helicity is large. The absolute equilibrium spectra of velocity and helicity for an inviscid flow system truncated at an upper wavenumber k2 are
\[
U(k) = 2\alpha/(\alpha^2-\beta^2k^2),\quad Q(k) = 2\beta k^2/(\alpha^2-\beta^2k^2),
\]
where the velocity variance and helicity/unit volume are ∫U(k)d3k and ∫Q(k)d3k, respectively. The temperature parameters α and β are constrained by α > 0 and |βk2| < α. There are no analogues of the negative-temperature equilibrium states known for two-dimensional inviscid flow. It is argued that the inertial-range energy cascade in isotropic turbulence driven by helical input should not differ asymptotically from that of non-helical turbulence. The absolute equilibrium distributions suggest that, in contrast to the analogous two-dimensional situation, statistically steady helical input at middle wavenumbers should not produce a significant downward cascade of energy to lower wavenumbers.
A note on boundary-layer growth at an axisymmetric rear stagnation point
Authors: Howarth J. A.
Published: March 2006

This paper presents the analysis and numerical solution for the axisymmetric analogue of the problem considered by Proudman & Johnson (1962) and Robins & Howarth (1972).
On the coexistence of laminar and turbulent flow in a narrow triangular duct
Authors: Bandopadhayay P. C. , Hinwood J. B.
Published: March 2006

The possible coexistence of laminar and turbulent flow in a triangular duct had been a point of controversy among several authors. Since the nature of the flow-laminar or turbulent-has a major influence on the head loss and heattransfer characteristics of the flow, resolution of this controversy is of practical as well as theoretical value.
A note on a breakdown of the multiplicative composition of inner and outer expansions
Authors: Schneider Wilhelm
Published: March 2006

It is shown that the multiplicative rule of the method of matched asymptotic expansions fails under certain conditions. The velocity distribution in front of an elliptic airfoil is given as an example. The reason for the breakdown is explained by inspecting the usual formal justification of multiplicative composition.
The initial development of a submerged laminar round jet
Authors: Abramovich S. , Solan A.
Published: March 2006

Experimental results are presented for the speed of travel of the spherical vortex at the front of a suddenly switched on, submerged, laminar jet in the Reynolds number range 80 < Re < 500. The results show that the speed of advance of the front is approximately one half of the speed of a fluid element on the axis of the steady jet. The experimental data are well correlated by an approximate model of the jet–vortex interaction in which the vortex is treated as a liquid drop with averaged properties. An auxiliary result is a new correlation for the axial variation of the velocity of a steady jet.
Second-harmonic resonance in the interaction of an air stream with capillary–gravity waves
Authors: Nayfeh Ali Hasan
Published: March 2006

The method of multiple scales is used to derive equations governing the temporal and spatial variation of the amplitudes and phases of in viscid capillary–gravity travelling waves in the case of second-harmonic resonance (Wilton's ripples), but including the effects of: (i) near resonance, (ii) liquid depth, and (iii) pressure perturbations exerted by an external subsonic gas on the liquid–gas interface. The spatial form of the equations shows that, below a critical gas velocity, energy is transferred between the fundamental and its first harmonic in keeping with the energy conservation law. However, the amplitude of the first harmonic decreases with increasing gas velocity. Above the critical gas velocity, the displacement of the gas–liquid interface grows monotonically with distance. It is found that pure amplitude-modulated waves are possible only at perfect resonance. Pure phase-modulated, near-resonant waves are periodic, as the resonance forces a readjustment of the phases to produce perfect resonance. The effectiveness of the resonance in rippling the interface increases as the liquid depth decreases.
Tsunami generation
Authors: Braddock R. D. , Driessche P. Van Den , Peady G. W.
Published: March 2006

The problems of tsunami generation are treated by standard integral-transform and modified stationary-phase methods to yield asymptotic approximations to the surface disturbances. The effects of asymmetry are considered in a one-dimensional ocean. Series representations are used to produce sets of normal-mode oscillations in a two-dimensional ocean, and the magnitudes of the wave front and wave train are discussed in relation to the asymmetry of the generating region.
The stability of plane Couette flow with viscous heating
Authors: Sukanek Peter C. , Goldstein Charles A. , Laurence Robert L.
Published: March 2006

An investigation of the stability of plane Couette flow with viscous heating of a Navier–Stokes–Pourier fluid with an exponential dependence of viscosity upon temperature is presented. Using classical small perturbation theory, the stability of the flow can be described by a sixth-order set of coupled ordinary differential equations. Using Galerkin's method, these equations are reduced to an algebraic eigenvalue problem. An eigenvalue with a negative real part means that the flow is unstable.

Neutral stability curves are determined at Brinkman numbers of 15, 19, 25, 30,40,80 and 600 for Prandtl numbers of 1, s and 50. A Brinkman number of 19 corresponds approximately to the maximum shear stress which can be applied to the system.

The results indicate that four different modes of instability occur: one termed an inviscid mode, arising from an inflexion point in the primary flow; a viscous mode, due to the stratification of viscosity in the flow field and an associated diffusive mechanism; a coupling mode, resulting from the convective and viscous dissipation terms in the energy equation; and finally a purely thermal mode.
A generalized slender-body theory for fish-like forms
Authors: Newman J. N. , Wu T. Y.
Published: March 2006

A consistent slender-body approximation is developed for the flow past a fish- like body with arbitrary combinations of body thickness and low-aspect-ratio fin appendages, but with the fins confined to the plane of symmetry of the body. Attention is focused on the interaction of the fin lifting surfaces with the body thickness, and especially on the dynamics of the vortex sheets shed from the fin trailing edges. This vorticity is convected by the (non-lifting) flow past the stretched-straight body, and departs significantly from the purely longitudinal orientation of conventional lifting-surface theory. Explicit results are given for axisymmetric bodies having fins with abrupt trailing edges, and calculations of the total lift force are presented for bodies with symmetric and asymmetric fin configurations, moving with a constant angle of attack.
A linearized potential flow theory for airfoils with spoilers
Authors: Brown G. P. , Parkinson G. V.
Published: March 2006

Linearized two-dimensional potential flow theory is applied to an airfoil with an upper surface spoiler. The spoiler wake is modelled as a cavity of empirically given constant pressure, and a sequence of conformal transformations maps the linearized physical plane, with a slit on the real axis representing the airfoil plus cavity, onto the upper half of the plane exterior to the unit circle. The complex acceleration potential is used, and its real part is specified on the real axis, repre- senting the cavity boundary, while its imaginary part is specified on the unit semicircle, representing the wetted surface of the airfoil and spoiler. Solutions are found for both the steady-state lift and the transient lift after spoiler actuation for airfoils of arbitrary camber, thickness and incidence, with and without a simple flap, and with spoilers of arbitrary position, height and angle. The empirical cavity pressure is arbitrary for the steady-state solution, but is assumed to have the free-stream value for the transient solutions. Comparisons are made with the results of wind-tunnel experiments, and, for the steady-state solutions, with predictions of an earlier theory. The agreement of the present theoretical predictions with the experimental results is generally good, and is in most cases somewhat better than that of the earlier theory.
Surface relief accompanying natural convection in liquid pools heated from below
Authors: Kayser William V. , Berg John C.
Published: March 2006

Free-surface deformations accompanying natural convection in shallow liquid pools heated from below by a straight wire are studied. Two-dimensional flow ensues as density gradients are set up in the bulk liquid and surface tension gradients are set up in the interface. Surface profiles are measured experimentally using a reflecting schlieren technique. Profiles concave above the rising warm current are produced in the shallower pools and convex profiles in the deeper pools. The profile for intermediate depths is a hybrid with a shallow crest rising from a shallow trough. A mathematical model of the system is developed and yields numerical results in qualitative agreement with experiment. The import- ance of various fluid properties and system parameters on the surface profile is examined using further calculations based on the model.
Time-dependent shear flows of a suspension of particles with weak Brownian rotations
Authors: Hinch E. J. , Leal L. G.
Published: March 2006

A dilute suspension of rigid axisymmetric particles in a time-dependent shear flow is considered in circumstances where the shear flow alignment of the particles dominates small but not unimportant Brownian disorientations. Three cases are studied: stress relaxation on the cessation of a steady shear, the sudden application of a steady shear from a state of rest and the sudden change from one steady shear rate to another. The rheology exhibits effects on two basic time scales provided that the particle aspect ratio r is not extreme: oscillatory features with a frequency proportional to the shear rate γ which are due to the rotation of the particles about their Jeffery orbits, and an exponentially fading memory due to Brownian diffusion with a characteristic time praportional to the inverse diffusion rate D−1. When the particle aspect ratio r becomes large, the oscillation frequency is reduced to γ/r while the diffusion rate is enhanced to Dr2 for some features and to Dr4 for others.