bluuebunny/crossref_metadata_2025_split huggingface dataset:
Internal waves of finite amplitude and permanent form
Authors: Benjamin T. Brooke 
Published: March 2006

A theory is derived for the class of long two-dimensional waves, comprising solitary and periodic cnoidal waves, that can propagate with unchanging form in heterogeneous fluids. The treatment is generalized to the extent that the waves are supposed to arise on a horizontal stream of incompressible fluid whose density and velocity are arbitrary functions of height, and the upper surface of the fluid is allowed either to be free or to be fixed in a horizontal plane. Explicit formulae for the wave properties and a general interpretation of the physical conditions for the occurrence of the waves are achieved without need to specify particular physical models; but in a later part of the paper, §4, the results are applied to three examples that have been worked out by other means and so provide checks on the present theory. These general results are also shown to accord nicely with the principle of ‘conjugate-flow pairs’ which was explained by Benjamin (1962b) with reference to swirling flows along cylindrical ducts, but which is known to apply equally well to flow systems of the kind in question here.The theory reveals certain physical peculiarities of a type of flow model often used in theoretical studies of internal-wave phenomena, being specified so as to make the equation for the stream-function linear. In an appendix, some observations are also made regarding the ‘Boussinesq approximation’, which too is often used as a simplifying assumption in this field. It is shown, adding to a recent discussion by Long (1965), that finite internal waves may depend crucially on small effects neglected in this approximation.
Aligned-field hydromagnetic flow past a slender body
Authors: Leibovich S. , Ludford G. S. S. 
Published: March 2006

This is a continuation of our previous work (1965) on the Sears–Resler–Stewartson controversy, in the context of axially symmetric flow. A new approach is presented using boundary-layer arguments, which remove much of the old complexity.For resistive bodies of shape R(x) we uncover a similarity with plane airfoils of shape F(x) = R2(x). As before, Stewartson's slug flows develop fore and aft. For bodies of very high conductivity the Sears–Resler (steady-state) solution turns out to be one possibility. It pertains to bodies (of much higher conductivity than the liquid) into which the initial magnetic field has not diffused.
The entrainment function in turbulent boundary-layer and wall-jet calculations
Authors: Escudier M. P. , Nicoll W. B. 
Published: March 2006

Some of the proposals made by Spalding (1964a) in his ‘Unified theory of friction, heat transfer, and mass transfer’ are examined. The two-component velocity-profile family proposed by Spalding is compared with measured boundary-layer and wall-jet velocity profiles and is shown to be adequate for flows with moderate wake components. The drag law implicit in the velocity-profile family is shown to be in good agreement with experimental data.Recommendations are provided for entrainment functions for both boundary layers and wall jets.Predictions of boundary-layer and wall-jet development, based on the recommended entrainment functions, are presented and compared with experiment. The predictions are in good agreement with experiment except in the vicinity of separation.
Submerged jets in short cylindrical flow vessels
Authors: Mcnaughton K. J. , Sinclair C. G. 
Published: March 2006

Liquid-into-liquid jets in short cylindrical vessels have been investigated under conditions of uniform flow by using an aqueous blue tracer solution in conjunction with tránsparent cylindrical tanks. The vessels had diameters D of 3, 6, 12 and 24in., length-to-diameter ratios L/D of 1, 2 and 3 and inlet diameters d between ¼ and 1 in. Reynolds numbers in the inlet tube, Rei, ranged between 100 and 28,000. Four main types of jet were observed:
\[
\begin{array}{@{}l@{\qquad}l@{\quad}l@{}}
& \hbox{dissipated-laminar jets} & (Re_i < 300\;\hbox{approx}.);\\
& \hbox{fully laminar jets} & (300 < Re_i < 1000\;\hbox{approx}.);\\
& \hbox{semi-turbulent jets} & (1000 < Re_i < 3000\;\hbox{approx}.);\\
\hbox{and} & \hbox{fully turbulent jets} & (Re_i > 3000).
\end{array}
\]The laminar length a of sub-turbulent jets was investigated and correlated with Rei and the geometric parameters by the equation
\[
a/d = 9.97 \times 10^7 Re_i^{-2.46}(D/d)^{-0.48} (L/d)^{0.74}.
\]
Steady free convection in a porous medium heated from below
Authors: Elder J. W. 
Published: March 2006

This is an experimental and numerical study of steady free convection in a porous medium, a system dominated by a single non-linear process, the advection of heat. The paper presents results on three topics: (1) a system uniformly heated from below, for which the flow is cellular, as in the analogous Bénard-Rayleigh flows, (ii) the role of end-effects, and (iii) the role of mass discharge. Measurements of heat transfer are used to establish further the validity of the numerical scheme proposed by the author (1966a), while the other flows allow a more extensive study of the numerical scheme under various boundary conditions. The results are very satisfactory even though only moderately non-linear problems can be treated at present.The main new results are as follows. For the Rayleigh-type flow, above a critical Rayleigh number of about 40, the heat transferred across the layer is proportional to the square of the temperature difference across the layer and is independent of the thermal conductivity of the medium or the depth of the layer. This result is modified when the boundary-layer thickness is comparable to the grain size of the medium. The investigation of end-effects reveals variations in horizontal wave-number and a pronounced hysteresis and suggests an alternative explanation of some observations by Malkus (1954).
A simple laboratory model for the oceanic circulation
Authors: Pedlosky J. , Greenspan H. P. 
Published: March 2006

A linear theory is developed for the motion of a viscous, incompressible fluid in a rotating cylinder with a sloping bottom.An analysis of the normal modes of oscillation reveals that the presence of the bottom slope introduces a new set of low frequency inertial oscillations to replace the purely geostrophic modes which are not allowed in this geometry. The new waves possess mean circulation and are the mechanism by which the fluid adjusts to changes in the rotation rate of the container, a process discussed in detail.The steady motion produced in the cylinder when the cylinder's upper surface rotates at a different rate than the bottom surface is studied. It is shown that the presence of the bottom slope inhibits the steady fluid motion in the body of the cylinder and introduces a non-symmetric, high velocity side wall boundary layer.Experimental evidence, presented to validate the theory, reproduces certain important features of the oceanic circulation.
Inviscid hypersonic flow over plane power-law bodies
Authors: Hornung H. G. 
Published: March 2006

Theoretical solutions based on the expansion scheme for largexand largeM∞, as proposed by Freeman (1962), are obtained for the asymptotic inviscid flow over plane bodies of the shapey/d= (x/d)min the range$\frac{2}{3}/\gamma < m < \frac{2}{3}$where blast wave theory applies as a first approximation. In particular, the second-order terms, which are necessary to satisfy the body boundary conditions for the normal velocity are computed. The magnitude of the second-order terms is found to increase from zero at$m=\frac{2}{3}/\gamma $to infinity at$m = \frac{2}{3}.$As a comparison with theory, experiments atM∞= 8·2 were made with two plane power-law bodies in the range$\frac{2}{3}/\gamma < m < \frac{2}{3} $and on a plane parabola with a tangent wedge nose. These consisted of the determination of shock-wave shapes, surface pressure distributions and detailed investigations of the distribution of pitot and static pressure across the shock layer.The experimental results are in good agreement with the theory in the case m =1/2, where the second-order effects are small. At m = 5/8 the region of validity of the theory is limited to much larger distances from the nose of the body and larger Mach numbers. Accordingly, the prediction for the deviation from firstorder theory, although being correct in sign, is too small. Shock-wave shapes on bodies of the same power but of different size are correlated by the similarity theory when scaled with respect to the dimensiond.The experimental results obtained with the wedge-parabola are in very good agreement with a characteristics solution by C. H. Lewis (1965, unpublished).
Three-dimensional boundary layers: a report on Euromech 2
Authors: Horlock J. H. , Norbury J. F. , Cooke J. C. 
Published: March 2006

The second European Mechanics Colloquium, on the subject of three-dimensional boundary layers, was held at Liverpool University from 4 to 7 January 1966 and was attended by thirty-eight people closely associated with current work in this field.The meeting was opened by an introductory review by J. C. Cooke, who successfully sought to provoke discussion by emphasizing areas of apparent agreement and disagreement. The discussions which followed were based on a series of contributions by participants in the Colloquium, and covered the following topics:Laminar boundary layers.Three-dimensional perturbations of two-dimensional turbulent boundary layers.Corner and secondary flows.Boundary layers associated with flow past obstacles.Flow over delta wings.Separation.Flow over rotating surfaces.Heat and mass transfer.Miscellaneous topics.In all these cases, the emphasis was on the three-dimensional nature of the flow, and in this report the discussions are summarized under these headings.
The two-dimensional laminar jet in parallel streaming flow
Authors: Wygnanski I. 
Published: March 2006

Solutions to the problem of a two-dimensional, laminar jet of incompressible fluid issuing into a uniform stream in the direction of the main flow are considered. Two co-ordinate-type expansions are developed. A direct expansion, when suitably transformed, predicts approximately the velocity along the plane of symmetry of the jet for all values of the abscissa, with a maximum error of 7·6% far downstream from the origin. This error is established by comparison with a second, asymptotic expansion valid only at large values of the abscissa. The two expansions are subsequently joined, permitting an approximate determination of a constant which multiplies a third-order term in the asymptotic series and which initially remained unknown even after satisfying all boundary conditions imposed on these series.The decay of velocity excess along the plane of symmetry of the jet is accelerated by the presence of the external stream.
The interaction of a fibre tangle with an airflow
Authors: Taub Paul A. 
Published: March 2006

An analytical model of the interaction of a fibre tangle with an airflow is proposed. This model replaces the discrete fibres by a continuum medium with a non-linear stress-strain law. The governing equations have been examined for one-dimensional unsteady flow configurations and have been found to possess five characteristic directions.A numerical-solution procedure, based upon the method of characteristics, has been outlined and applied to the flow within a dilation chamber. A fibre sample is located at the centre of the chamber, which is alternately pressurized and depressurized.
Transient convection in a porous medium
Authors: Elder J. W. 
Published: March 2006

Laboratory and numerical experiments on non-steady convective flows in a porous medium are reported. The main objective is to note the detailed comparison found between the time-dependent solutions and the time-like development of the iterative solutions of the steady equations originally pointed out by Garabedian (1956) and others.Two flows are chosen for study. The first is the flow which develops when a blob of hot fluid is released at the base of a porous slab. The second is the flow which develops when a portion of the base of a porous slab is suddenly heated. The former flow is very simple and ideally suited for establishing the numerical scheme. The latter flow, however, produces several unexpected features. The gross features of the time development, when the motion is strongly non-linear, show an alternation between periods of slow gradual adjustment and periods of rapid change.
On the oscillatory instability of a differentially heated fluid loop
Authors: Welander Pierre 
Published: March 2006

A theoretical discussion is given of the motion of a fluid contained in a tube forming a closed loop that is heated from below and cooled from above. The fluid is assumed to have uniform temperature over each cross-section, and the heat transfer is assumed proportional to the difference between the local temperatures of the fluid and the tube. The latter temperature is prescribed. The system has one steady solution with warm fluid rising in one branch and cold fluid sinking in the other. This solution may, however, become unstable in an oscillatory manner. A weak instability takes the form of pulsations, the motion being always of one sign, while a strong instability takes the form of oscillations with zero mean motion. These oscillations are irregular and do not repeat themselves even over very long times.These unstable motions are associated with thermal anomalies in the fluid that are advected materially around the loop. The anomalies amplify through the correlated variations in flow rate. A warm pocket of fluid creates maximum flow rate going through the upper part and minimum flow rate going through the lower part of the loop. Accordingly it passes quicker through the heat sink than through the heat source, and the latter becomes more effective. Similarly, the heat sink acts more effectively on a cold pocket of fluid.The curve of neutral stability is worked out as a function of the two parameters of the problem, a non-dimensional gravity and a non-dimensional friction coefficient. The instability has also been studied by direct numerical time integration of the model equations.It is suggested that the mechanism of instability found for this model operates also in more complicated systems, and can explain the pulsative type of motions observed recently in certain convection experiments.
Transition in oscillating boundary layer flows
Authors: Obremski H. J. , Fejer A. A. 
Published: March 2006

An experimental investigation of transition phenomena in non-steady boundary layers of sinusoidally oscillating flows was conducted, and the effect of various parameters on the transition process was observed.The manner in which transition occurs appears to be related to a dimensionless grouping designated as the non-steady Reynolds number, (Re)NS= LΔU/2πvν. When this number exceeds a certain critical value, transition begins with turbulent bursts appearing periodically at the frequency of the free stream oscillation; these are preceded by instability waves having the appearance of a Tollmien-Schlichting instability. The transition Reynolds number depends, in this case, only on the amplitude of the free stream oscillations and not on their frequency, at least to the first order. Below a critical value of this parameter, transition occurs at a relatively constant Reynolds number which appears independent of the amplitude and frequency of the oscillation, at least over the range tested.When the bursts appear periodically, their development appears to occur in two succeeding phases: an initial or ‘creative’ phase marked by a sequential appearance of turbulence upstream and rapid development and a latter or ‘convective’ phase marked by turbulent spots having relatively constant leading and trailing edge velocities.The dimensionless amplitude, ΔU/U∞, of the imposed oscillations was varied from 0.014 to 0.29 while the oscillation frequencies extended from 4.5 to 62c/s.Pressure gradients imposed on the flat plate model during the course of the study are given bydCp/dx = − 0.004/ft., −0.081/ft. and 0.045/ft. whereCp= 2p/ρU2∞and x is the distance (ft.) along the plate.
The scalar spectrum in the viscous-convective subrange
Authors: Nye James O. , Brodkey Robert S. 
Published: March 2006

A modified design of a fibre optic light probe was used for the measurement of turbulent concentration fluctuations and the spectra associated with these. The main object was to establish experimentally the nature of the scalar spectrum in the viscous-convective subrange. The existence of a spectral region with a (-1)-power law form supports the uniform straining model proposed by Batchelor (1959) for Schmidt numbers (v/D) much greater than unity. For this range, the data do not agree with the cascading process suggested by Pao (1965).An additional object was to study further the decay of the concentration fluctuations in terms of a measure of the turbulent mixing. More specifically, the limitations of applying the isotropic stationary mixing theory of Corrsin (1957, 1964a) to a shear pipe flow situation and the validity of the earlier data for this case (Lee & Brodkey 1964; Brodkey 1966a, b; and Gegner & Brodkey 1966) were investigated.
Hypersonic weak-interaction similarity solutions for flow past a flat plate
Authors: Bush William B. , Cross Arthur K. 
Published: March 2006

The hypersonic weak-interaction regime for the flow of a viscous, heat-conducting compressible fluid past a flat plate is analysed using the Navier-Stokes equations as a basis. The fluid is assumed to be a perfect gas having constant specific heats, a constant Prandtl number, σ, of order unity, and a viscosity coefficient varying as a power, ω, of the absolute temperature. Limiting forms of solutions are studied for the free-stream Mach number,M, the free-stream Reynolds number (based on the plate length),RL, and the reciprocal of the weak-interaction parameter, (ξ*)−1= [Fscr ](M,RL, ω, σ), greater than order unity.By means of matched asymptotic expansions, it is shown that, for (1 − ω) > 0, the zone between the shock wave and the plate is composed of four distinct regions for which similarity exists. The behaviour of the flow in these four regions is analysed.
An underwater explosive shock gun
Authors: Coombs A. , Thornhill C. K. 
Published: March 2006

A theoretical study is made of a simple design for an underwater shock gun, which consists of a chamber in the form of a hollow circular cone, with a spherical sector of explosive charge fitted into the apex. When the explosive is initiated at the apex, the resulting sector of a spherical blast wave will be diffracted by expansion waves moving inwards after the leading shock has emerged from the rim of the cone.The progress of the expansion wave-fronts is calculated, and the results show a surprising inability of the diffraction process to ‘eat into’ the full-strength sector of spherical blast. It is found to be possible to design such a gun so that it is capable of projecting a high intensity shock-pressure beam over a considerable range, using only a very small explosive charge.
A note on the stability of an infinite fluid heated from below
Authors: Robinson J. L. 
Published: March 2006

The problem of the stability of a fluid with time-dependent heating has been investigated by Morton (1957), Lick (1965) and Foster (1965). Morton and Lick assumed that the rate of change of the temperature profile is small compared with the growth rate of the disturbances (quasi-static assumption). This assumption is invalid near the onset of instability (as defined by ∂/∂t = 0), and Foster has therefore used an initial-value approach.In this paper the range of validity of the quasi-static assumption is discussed, and results of a time-scaled analysis and calculations based on this are compared with the work of Foster; the agreement is found to be good. We restrict our attention to a semi-infinite fluid initially at a constant temperature; at time t = 0 a temperature difference ΔT is applied at the (lower) horizontal boundary (case (A) of Foster).
The thermohaline Rayleigh-Jeffreys problem
Authors: Nield D. A. 
Published: March 2006

The onset of convection induced by thermal and solute concentration gradients, in a horizontal layer of a viscous fluid, is studied by means of linear stability analysis. A Fourier series method is used to obtain the eigenvalue equation, which involves a thermal Rayleigh numberRand an analogous solute Rayleigh numberS, for a general set of boundary conditions. Numerical solutions are obtained for selected cases. Both oscillatory and monotonic instability are considered, but only the latter is treated in detail. The former can occur when a strongly stabilizing solvent gradient is opposed by a destablizing thermal gradient. When the same boundary equations are required to be satisfied by the temperature and concentration perturbations, the monotonic stability boundary curve in the (R, S)-plane is a straight line. Otherwise this curve is concave towards the origin. For certain combinations of boundary conditions the critical value ofRdoes not depend onS(for some range ofS) or vice versa. This situation pertains when the critical horizontal wave-number is zero.A general discussion of the possibility and significance of convection at ‘zero’ wave-number (single convection cell) is presented in an appendix.
Internal waves of permanent form in fluids of great depth
Authors: Benjamin T. Brooke 
Published: March 2006

This paper presents a general theoretical treatment of a new class of long stationary waves with finite amplitude. As the property in common amongst physical systems capable of manifesting these waves, the density of the (incompressible) fluid varies only within a layer whose thickness h is much smaller than the total depth, and it is h rather than the total depth that must be considered as the fundamental scale against which wave amplitude and length are to be measured. Internal-wave motions supported by the oceanic thermocline appear to be the most promising field of practical application for the theory, although applications to atmospheric studies are also a possibility.The waves in question differ in important respects from long waves of more familiar kinds, and in § 1 their character is discussed on the basis of a comparison with solitary-wave and cnoidal-wave theories on customary lines, such as apply to internal waves in fluids of limited depth. A summary of some simple experiments is included at the end of § 1. Then, in § 2, the comparatively easy example of two-fluid systems is examined, again to illustrate principles and to prepare the way for the main analysis in § 3. This proceeds to a second stage of approximation in powers of wave amplitude, and its leading result is an equation (3·51) determining, for arbitrary specifications of the density distribution, the form of the streamlines in the layer of heterogeneous fluid. Periodic solutions of this equation are obtained, and their properties are discussed with regard to the interpretation of internal bores and wave-resistance phenomena. Solutions representing solitary waves are then obtained in the form
\[
f(x) = a\lambda^2/(x^2+\lambda^2),
\]
where xis the horizontal co-ordinate and where aΛ = O(h2). (The latter relation between wave amplitude and length scale contrasts with the customary one, aΛ2 = O(h3)). The main analysis is developed with particular reference to systems in which the heterogeneous layer lies on a rigid horizontal bottom below an infinite expanse of homogeneous fluid; but in § 4 ways are given to apply the results to various other systems, including ones in which the heterogeneous layer is uppermost and is bounded by a free surface. Finally, in §5, three specific examples are treated: the density variation with depth is taken, respectively, to have a discontinuous, an exponential and a ‘tanh’ profile.