bluuebunny/crossref_metadata_2025_split huggingface dataset:
On the stability of a laminar incompressible boundary layer over a flexible surface
Authors: Landahl Marten T. 
Published: March 2006

The stability of small two-dimensional travelling-wave disturbances in an incompressible laminar boundary layer over a flexible surface is considered. By first determining the wall admittance required to maintain a wave of given wave-number and phase speed, a characteristic equation is deduced which, in the limit of zero wall flexibility, reduces to that occurring in the ordinary stability theory of Tollmien and Schlichting. The equation obtained represents a slight and probably insignificant improvement upon that given recently by Benjamin (1960). Graphical methods are developed to determine the curve of neutral stability, as well as to identify the various modes of instability classified by Benjamin as ‘Class A’, ‘Class B’, and ‘Kelvin-Helmholtz’ instability, respectively. Also, a method is devised whereby the optimum combination of surface effective mass, wave speed, and damping required to stabilize any given unstable Tollmien-Schlichting wave can be determined by a simple geometrical construction in the complex wall-admittance plane.What is believed to be a complete physical explanation for the influence of an infinite flexible wall on boundary-layer stability is presented. In particular, the effect of damping in the wall is discussed at some length. The seemingly paradoxical result that damping destabilizes class A waves (i.e. waves of the Tollmien-Schlichting type) is explained by considering the related problem of flutter of an infinite panel in incompressible potential flow, for which damping has the same qualitative effect. It is shown that the class A waves are associated with a decrease of the total kinetic and elastic energy of the fluid and the wall, so that any dissipation of energy in the wall will only make the wave amplitude increase to compensate for the lowered energy level. The Kelvin-Helmholtz type of instability will occur when the effective stiffness of the panel is too low to withstand, for all values of the phase speed, the pressure forces induced on the wavy wall.The numerical examples presented show that the increase in the critical Reynolds number that can be achieved with a wall of moderate flexibility is modest, and that some other explanation for the experimentally observed effects of a flexible wall on the friction drag must be considered.
A solution for three-dimensional vortex flows with strong circulation
Authors: Lewellen W. S. 
Published: March 2006

The Navier-Stokes equations for a viscous, incompressible fluid are considered for a steady, axisymmetric flow composed of a strong rotation combined with radial sink flow which exhausts axially inside a finite radius. The equations are reduced to two coupled partial differential equations in terms of the stream function and circulation. The equations contain three dimensionless parameters: the radial Reynolds number, a characteristic ratio of mass flow per unit lenght to circulation, and a characteristic ratio of an axial dimension to a radial dimension. The product of these last two dimensionless parameters is used as a new expansion parameter for generating an asymptotic series solution. To zeroth order in this parameter, the solution for the stream function is a linear distribution between two axial boundary values. First-order correction terms are calculated for a specific example.In discussing these equations the limitations of the exact solutions due to Donaldson & Sullivan (1960) and Long (1961) are noted. These exact solutions are contrasted with the approximate treatment of this type of vortex originated by Einstein & Li (1951) and generalized by Deissler & Perlmutter (1958).
Magnetohydrodynamic flow in a rectangular tube at high Hartmann number
Authors: Williams W. E. 
Published: March 2006

Asymptotic forms, valid for high Hartmann numberM, are obtained for the mean velocity for laminar magnetohydrodynamic flow in a rectangular tube. For a tube with non-conducting walls it is found that, neglecting exponentially damped terms, the mean velocity can be expressed in closed form in terms of tabulated functions. The first three terms of the expansion of the mean velocity in inverse powers ofMare in extremely close agreement with a corresponding expansion obtained by Shercliff (1953) using a boundary-layer method.For perfectly conducting walls the first five terms of an expansion in inverse powers ofMare obtained.
Turbulent free convection above a heated plate inclined at a small angle to the horizontal
Authors: Tritton D. J. 
Published: March 2006

An investigation has been made of the structure of the motion above a heated plate inclined at a small angle (about 10°) to the horizontal. The turbulence is considered in terms of the similarities to and differences from the motion above an exactly horizontal surface. One effect of inclination is, of course, that there is also a mean motion.Accurate data on the mean temperature field and the intensity of the temperature fluctuations have been obtained with platinum resistance thermometers, the signals being processed electronically. More approximate information on the velocity field has been obtained with quartz fibre anemometers. These results have been supplemented qualitatively by simultaneous observations of the temperature and velocity fluctuations and also by smoke experiments.The principal features of the flow inferred from these observations are as follows. The heat transfer and the mean temperature field are not much altered by the inclination, though small, not very systematic, variations may result from the complexities of the velocity field. This supports the view that the mean temperature field is largely governed by the large-scale motions. The temperature fluctuations show a systematic variation with distance from the lower edge and resemble those above a horizontal plate when this distance is large. The largescale motions of the turbulence start close to the lower edge, but the smaller eddies do not attain full intensity until the air has moved some distance up the plate. The mean velocity receives a sizable contribution from a ‘through-flow’ between the side-walls. Superimposed on this are developments that show that the momentum transfer processes are complex and certainly not capable of representation by any simple theory such as an eddy viscosity. On the lower part of the plate there is surprisingly large acceleration, but further up the mixing action of the small eddies has a decelerating effect.
Cavitation of a viscous fluid in narrow passages
Authors: Taylor G. I. 
Published: March 2006

The conditions which determine the existence and position of cavitation in the narrow passages of hydrodynamically lubricated bearings have been assumed to be the same as those which produce cavitation bubbles, namely a lowering of pressure below that at which gas separates out of fluid. This assumption enables certain predictions to be made which in some cases are verified, but it does not provide a physical description of the interface between oil and air. Theoretical analysis of the situation seems to be beyond our present capacity, and in none of the experiments so far published has it been possible to measure both the most important relevant data, namely the minimum clearance and the oil flow through it.A method is described here which enables this to be done. It turns out that two physically different kinds of cavitation can occur. One of these is well described by the existing theory and assumption. Surface tension plays no part in it, and in most text books on hydrodynamic lubrication is not even mentioned. The other kind, which is akin to hydrodynamic separation rather than bubble cavitation, depends essentially on surface tension. Both kinds appear clearly in published photographs taken through transparent bearings, but the experimenters do not seem to have distinguished between them.The reason why surface tension, which is only able to supply stresses that are exceedingly small compared with the pressure variation in the fluid itself, may have a large effect on the flow can be understood by considering the flow of a viscous fluid in a tube when blown out by air pressure applied at one end. For any given length of fluid the rate of outflow depends almost entirely on the pressure applied, the surface tension force being negligible; but the amount of fluid left in the tube after the air column has reached the end depends essentially on surface tension.
Observations on oblique shock waves in gaseous detonations
Authors: Edwards D. H. , Jones T. G. , Price B. 
Published: March 2006

An account is given of photographic and pressure observations made on the oblique shock waves occurring in the wake of self-sustaining detonation waves in hydrogen-oxygen mixtures initially at atmospheric pressure. Four explosion tubes were employed, of which three are of circular cross-section with internal diameters of 10, 5 and 1·6 cm and the fourth is a square-section tube of side 1·5 in.On the assumption that the oblique shocks are sufficiently weak to be regarded as Mach waves, the flow Mach number relative to the detonation front is determined; these are found to be substantially higher than the values predicted by deal one-dimensional theory. The measured flow Mach numbers in the rarefaction are then used to calculate the pressure distribution in this region on the basis of the supersonic nozzle model due to Fay (1959, 1962). The predictions of this model are found to disagree with with the observed static pressure profiles. Moreover, the pressure following the initial peak persists at a higher value than the theoretical for distances of the order of 5–10 cm behind the front. This phenomenon implies that the wall boundary-layer pressure remains higher than the C-J value and it is suggested that the pressure difference across the boundary layer can account for the formation of the oblique waves.The supersonic features of the flow can be accounted for by the turbulent-structure hypothesis of White (1961). Some validation of this hypothesis is provided here by the observation of the absence of the oblique shocks in overdriven detonation waves caused by the diminished effects of turbulence. This observation is consistent with the view that the oblique shocks are generated by the pressure difference across the boundary layer near the front as this difference would also be diminished in an over-driven wave.
On free-surface oscillations in a rotating paraboloid
Authors: Miles John W. , Ball F. K. 
Published: March 2006

Lamb's analysis of small-amplitude, shallow-water oscillations in a rotating paraboloid, interpreted by him in the inconsistent context of an approximately plane free surface, is re-interpreted to obtain results that are valid for
$0 \le \omega^2l|2g \; \textless \;1$
(ω = rotational speed, l = latus rectum of paraboloid); no equilibrium is possible for ω2l/2g > 1. It is shown that the frequencies of the dominant modes for the azimuthal wave numbers 0 (axisymmetric motion) and 1 are independent of ω for an observer in a non-rotating reference frame and that the frequencies of all other axisymmetric modes are decreased by rotation (Lamb concluded that they would be increased). An axisymmetric mode of zero frequency, which was over-looked by Lamb, also is found.Exact solutions to the non-linear equations of motion, which reduce to the aforementioned dominant modes for small amplitudes, are determined. The axisymmetric solution is inferred from similarity considerations and is found to contain all harmonics of the fundamental frequency. The finite motion of azimuthal wave-number 1 is a quasi-rigid displacement of the liquid and is found to be simple harmonic except for a second-harmonic component of the free-surface displacement (but the horizontal velocity at a given point remains simple harmonic).
Two-dimensional reattaching jet flows including the effects of curvature on entrainment
Authors: Sawyer R. A. 
Published: March 2006

The analysis given previously for predicting the average pressure and length of the region of recirculating flow enclosed by a low-speed turbulent jet, issuing parallel to a flat plate, has been modified to take into account the different rates of entrainment by the two edges of the curved jet, the initial mixing region and the pressure forces near reattachment. There is improved correlation between theory and experiment. The analysis has been applied to the flow due to a jet emerging at an angle to a flat plate, and gives good prediction of the length and average pressure of the recirculation region for a particular value of an entrainment-ratio parameter.Curvature has a considerable effect on the rates of entrainment, but a first-order mixing-length theory indicates that this need not necessarily be accompanied by a marked deviation in jet velocity profile from that of a plane jet.
The resistance to a particle of arbitrary shape in translational motion at small Reynolds numbers
Authors: Brenner H. , Cox R. G. 
Published: March 2006

Assuming that the Stokes flow past an arbitrary particle in a uniform stream is known for any three non-coplanar directions of flow, then the force on the body to O(R), for any direction of flow, is given explicitly in terms of these Stokes velocity fields. The Reynolds number (R) based on the maximum particle dimension is assumed small. For bodies with certain types of symmetry it suffices merely to know the Stokes resistance tensor for the body in order to calculate this force. In this case the resulting formula is identical to that of Brenner (1961) and Chester (1962). However, for bodies devoid of such symmetry, their formula is incomplete—there being an additional force at right angles to the uniform stream which remains invariant under a reversal of the flow at infinity. As this additional force is a lift force, it follows that the Brenner-Chester formula furnishes the correct drag on a body of arbitrary shape; moreover, this drag is always reversed to at least O(R) by a reversal of the uniform flow at infinity.Exactly analogous formulae are derived using the classical Oseen equations, and it is shown that although this gives both the correct vector force on bodies with the above types of symmetry and the correct drag on bodies of arbitrary shape, it gives in general an incorrect lift component for completely arbitrary particles.Finally, the singular perturbation result for the force on an arbitrary body is extended to terms of O(R2log R). This higher-order contribution to the force is given explicitly in terms of the Stokes resistance tensor, and has the property of being reversed by a reversal of the flow at infinity, regardless of the geometry of the body.These results are collected in the Summary at the end of the paper.
Calculation of potential flow with separation in a right-angled elbow with unequal branches
Authors: Lichtarowicz A. , Markland E. 
Published: March 2006

A method developed by Roshko (1953a, b) has been applied to the case of flow round a right-angled elbow, in which the fluid separates at the inner corner at a speed which may be higher than the velocity of the separated flow far downstream. Numerical results have been evaluated and are presented for values of the ratio of the channel widths, downstream to upstream of the elbow, equal to 0·1 and 1·0.The elbow may be regarded as one half of a two-dimensional representation of an axisymmetric plate valve, on which experiments indicate considerable variation in performance, the discharge coefficient being found to depend to a marked extent on the length of the valve land.
A steady-state technique for local heat-transfer measurement and its application to the flat plate
Authors: Ginoux J. J. 
Published: March 2006

A steady-state technique of heat-transfer measurement has been developed based on the method of Seban, Emery & Levy (1959) whereby energy is dissipated by the Joule effect in a thin metal sheet on the surface of a model. For the present application, use was made of very thin but mechanically resistant films of metal of very nearly constant thickness, obtained by a simple mirror-silvering technique. The present investigation was prompted by the desire to make very local measurements of heat transfer for application in regions where large variations in convective heat flux and therefore in temperature could be expected.Comparison between theory and experiment has been made in the simple case of a flat plate with constant heat flux for which a rigorous computation could be made based on the theory of Chapman & Rubesin (1949). The model was so conceived that the heat losses were small enough to be neglected. Therefore no corrections, which are often inaccurate, were needed for the experimental results, contrary to what is generally done when using other techniques for heat-transfer measurements. The excellent agreement between theory and experiment gives complete confidence in the method. The theoretical analysis showed that the measurements are simply related to the results that could be obtained in the case of an isothermal surface, because of the constant ratio that exists between the corresponding heat-transfer coefficients.
Mixing-layer flows in a saturated porous medium
Authors: Wooding R. A. 
Published: March 2006

The steady two-dimensional seepage flow by gravitational convection, of a fluid of density ρ1surrounded by a fluid of density ρ0(≠ρ1) at rest, leads to a potential problem from which the shape of the interface can be determined. When the two fluids are slightly miscible the interface is replaced by a mixing layer, and it is shown that the first-order Prandtl equations for the flow in the layer possess an exact similarity solution. The profile across the layer is of the same form as the profile of the laminar incompressible half-jet with one fluid at rest, and a formula is obtained for the scale of mixing-layer thickness as a function of distance downstream. Three examples are discussed.(a) Flow of fluid of density ρ1from a horizontal line source. When (ρ1> ρ0) a stagnation point exists above the source, and the fluid ultimately descends in a vertical column of width proportional to the source strength. At the stagnation point, the mixing-layer thickness is finite and is proportional to the square root of the radius of curvature of the interface. At a sufficiently great distance downstream, the thickness increases as the square root of the distance, as in the straight laminar half-jet. These results have been tested experimentally in a Hele-Shaw cell.(b) Symmetrical flow of an ascending column of fluid (ρ1< ρ0) about an obstacle in the form of a finite horizontal strip. The column reforms after passing the obstacle, and the mixing-layer thickness returns to the value corresponding to an unobstructed vertical half-jet. The flow has been produced experimentally.(c) Flow in a lens of fresh water overlying salt water, with inflow due to precipitation, as in a two-dimensional Ghyben-Herzberg lens. Here the potential flow solution is calculated approximately by means of Dupuit-Forchheimer theory. In the steady-state solution the thickness of the mixing layer between fresh and saline water is found to be finite and, as in (a), proportional to the square root of the radius of curvature of the lens.
Rotational stagnation point flow
Authors: Hayes Wallace D. 
Published: March 2006

The constant-density inviscid rotational flow in the neighbourhood of a general stagnation point on a wall is investigated. In all but very special cases, the solution is non-analytic and the vorticity at the wall is infinite; the stagnation streamline is tangent to the wall at the stagnation point; stagnation points of saddlepoint type cannot exist.The boundary-layer equations corresponding to the inviscid solutions studied are presented.
Some perturbation solutions in laminar boundary layer theory Part 2. The energy equation
Authors: Fox Herbert , Libby Paul A. 
Published: March 2006

Solutions for two types of problems involving the energy equation for flows with velocities described by the Blasius solution are presented. The first type arises in flows with arbitrary initial distributions of stagnation enthalpy and with surfaces downstream of the initial station either with constant wall enthalpy or with zero heat transfer. Exact solutions in these cases are obtained for constant ρμ, and Prandtl number of unity; they are given in terms of complete orthogonal sets of functions which can be used to obtain first- and higher-order corrections for the effects of variable ρμ, non-unity Prandtl number, and deviations of the velocity field from that described by the Blasius solution.The second type of problem pertains to flows with power-law descriptions of the wall enthalpy. Again the basic solutions are obtained for Prandtl number of unity and the effect of non-unity Prandtl number is treated as a perturbation.
The dynamics of spheroidal masses of buoyant fluid
Authors: Turner J. S. 
Published: March 2006

It is shown how a simple property of the spherical vortex model can be used to investigate the dynamics of a buoyant, expanding thermal. No details of the vortex motion are required, only the fact that the flow round the buoyant region is potential. The main result is a demonstration that there is a relation between the two constants C and α arising in the dimensional analysis (where in the usual notation w = C (Δr) ½) and r = αz), which have up till now been measured separately and treated as independent. The analysis has been extended to spheroidal thermals by calculating the virtual mass for the appropriate outline, and it has also been generalized to include thermals in which the total buoyancy is increasing with time.Using these risults, and an earlier experimental verification that the mean angles of spread are nearly the same under various conditions of stability, it is suggested that the whole of the mean behaviour of a thermal can be calculated in two nearly independent steps. first, the density difference or Δ as a function of height may be calculated using purely kinematic equations of conservation. Secondly, the velocity is obtained from the local values of Δ and radius r, using a mean value of C, since this has now also been shown to vary little over a wide range of conditions.
Depression of an infinite liquid surface by an incompressible gas jet
Authors: Olmstead W. E. , Raynor S. 
Published: March 2006

The problem of small angle depressions in a liquid surface due to an impinging two-dimensional potential jet is considered. Using conformal mapping methods and finite Hilbert transforms, the problem is formulated as a non-linear singular integral equation. The integral equation is approximated by a set of non-linear algebraic equations which are solved numerically by a method of repeated linear corrections. In addition, an asymptotic solution (for low jet velocity) is derived.From the numerical solutions of the integral equation, the liquid-surface profiles and the free streamlines of the jet are calculated for four cases. These results verify the appearance of lips on the liquid surface which have been observed experimentally by others.