bluuebunny/crossref_metadata_2025 huggingface dataset:
An extension of the linearized theory of supersonic flow past quasi-cylindrical bodies, with applications to wing-body interference
Authors: Lock R. C.
Published: March 2006

An extension of the linearized theory of supersonic flow past quasi-cylindrical bodies of almost circular cross-section has been found which enables a direct calculation to be made of the overall forces on wings mounted on such bodies, subject to certain restrictions on the plan-form. The method is applied to two examples: (i) the effect of an arbitrary body distortion on static stability at supersonic speeds; and (ii) the effect of wing-body interference on rectangular wings mounted on a cylindrical body. The drag calculations in the second example are compared with the results of the supersonic area rule, which is found to be in error for moderate values of the ratio of wing chord to body radius, though the discrepancy is not serious from a practical point of view.
Mixing and chemical reaction in the laminar wake of a flat plate
Authors: Cheng S. I. , Kovitz A. A.
Published: March 2006

The initial value problem presented by mixing and chemical reaction in the wake of a flat plate is solved using the boundarylayer approximation. When a cool combustible mixture and its hot combustion products are separated by a finite, perfectly insulating flat plate, the velocity, temperature, and combustible concentration are determined in the vicinity of the trailing edge.

The mixing problem without chemical reaction is solved in terms of a ‘universal solution’ for a given initial temperature ratio and Prandtl number from which the solution for arbitrary temperature ratios can be obtained.

The mixing problem with chemical reaction is solved in terms of a ‘universal solution’ for the first two terms of an assumed series solution for the temperature. In this case the ‘universality’ is with respect to a parameter B characterizing the chemical and hydrodynamic initial conditions.

The axial distance from the trailing edge to the first local temperature maximum is given in terms of the initial conditions and is shown to be greatly shortened by the presence of the viscous wake as compared with non-viscous mixing.
Calculations of unsteady viscous flow past a circular cylinder
Authors: Payne R. B.
Published: March 2006

A numerical solution has been obtained for the starting flow of a viscous fluid past a circular cylinder at Reynolds numbers 40 and 100. The method used is the step-by-step forward integration in time of Helmholtz's vorticity equation. The advantage of working with the vorticity is that calculations can be confined to the region of non-zero vorticity near the cylinder.

The general features of the flow, including the formation of the eddies attached to the rear of the cylinder, have been determined, and the drag has been calculated. At R = 40 the drag on the cylinder decreases with time to a value very near that for the steady flow.
The mean velocity of slightly buoyant and heavy particles in turbulent flow in a pipe
Authors: Binnie A. M. , Phillips O. M.
Published: March 2006

A large number of small spheres of the same size were injected successively into a horizontal pipe conveying water at constant mean velocity, and their times of transit were measured. The mean velocity of the spheres that were either somewhat heavier or lighter than water was less than that of those of neutral density; for those having a terminal velocity in water within ± 1% of the mean velocity of the water in the pipe, the discrepancy was only about 0.1%. The dispersion of the times of transit of the spheres was almost independent of their density. A theory is developed to show how the mean velocity of the spheres depends upon their relative density and size.
Water waves of finite amplitude on a sloping beach
Authors: Carrier G. F. , Greenspan H. P.
Published: March 2006

In this paper, we investigate the behaviour of a wave as it climbs a sloping beach. Explicit solutions of the equations of the non-linear inviscid shallow-water theory are obtained for several physically interesting wave-forms. In particular it is shown that waves can climb a sloping beach without breaking. Formulae for the motions of the instantaneous shoreline as well as the time histories of specific wave-forms are presented.
Finite amplitude cellular convection
Authors: Malkus W. V. R. , Veronis G.
Published: March 2006

When a layer of fluid is heated uniformly from below and cooled from above, a cellular regime of steady convection is set up at values of the Rayleigh number exceeding a critical value. A method is presented here to determine the form and amplitude of this convection. The non-linear equations describing the fields of motion and temperature are expanded in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. We find that there are an infinite number of steady-state finite amplitude solutions (having different horizontal plan-forms) which formally satisfy these equations. A criterion for ‘relative stability’ is deduced which selects as the realized solution that one which has the maximum mean-square temperature gradient. Particular conclusions are that for a large Prandtl number the amplitude of the convection is determined primarily by the distortion of the distribution of mean temperature and only secondarily by the self-distortion of the disturbance, and that when the Prandtl number is less than unity self-distortion plays the dominant role in amplitude determination. The initial heat transport due to convection depends linearly on the Rayleigh number; the heat transport at higher Rayleigh numbers departs only slightly from this linear dependence. Square horizontal plan-forms are preferred to hexagonal plan-forms in ordinary fluids with symmetric boundary conditions. The proposed finite amplitude method is applicable to any model of shear flow or convection with a soluble stability problem.
Characteristics of the equations of motion of a reacting gas
Authors: Broer L. J. F.
Published: March 2006

The equations of motion for a chemically reacting gas in the absence of viscosity and heat conduction are set up. It is shown that the characteristic speed defined by this set of equations is the high-frequency limit of the phase velocity of sound waves as long as the reaction rate is finite. At infinite reaction rate (chemical equilibrium) the characteristics suddenly change to the lowfrequency sound speed. The nature of this transition is discussed in connection with a recent paper of Resler (1957).
Instability in slotted wall tunnels
Authors: King J. L. , Boyle P. , Ogle J. B.
Published: March 2006

A new water tunnel, incorporating a slotted wall working section, was found to suffer from severe vibration. A theoretical explanation is given for this, together with experimental evidence gleaned from this water tunnel and a small wind tunnel. It is shown that the oscillations are hydrodynamic in origin and are associated with the slotted wall design. Consideration is given to methods of elimination or reduction of the oscillations.
The propagation of long waves into a semi-infinite channel in a rotating system
Authors: Crease J.
Published: March 2006

In this paper the propagation of long waves (tides) into a canal is studied. The canal is assumed to be rotating at constant angular velocity and the depth of the fluid is uniform.

The rate of rotation can have a considerable effect on the amplitude of the unattenuated modes in the channel. This is due partly to the modification of the known solution when the rotation is zero, and partly to the fact that even if there is only a single semi-infinite barrier unattenuated waves of a special type (Kelvin waves) may be propagated in the rotating system into what is normally called the ‘shadow’ region behind the barrier (Crease 1956).

The purpose of this theoretical investigation is to seek a partial explanation of the behaviour of tides and storm surges in the North Sea. For this reason two models of this region are discussed. In Example I the model is of two parallel semi-infinite barriers in the path of a plane progressive wave, and in Example II there are two parallel barriers, one semi-infinite and the other infinite.

The ratio of the observed height of the semi-diurnal M2 tide in the North Sea to the height of the incident tide from the Atlantic lies between the results predicted by the two models and is in closer agreement with the model of Example I.
Energy transfer through a dissociated diatomic gas in Couette flow
Authors: Clarke John F.
Published: March 2006

The transfer of energy through a dissociated diatomic gas in Couette flow is considered, taking oxygen as a numerical example. The two extremes of chemical equilibrium flow and chemically frozen flow are dealt with in detail, and it is shown that the surface reaction rate is of prime importance in the latter case. The chemical rate equations in the gas phase are used to estimate the probable chemical state of the gas mixture, this being deduced from the ratio of a characteristic chemical reaction time to a characteristic time for atom diffusion across the layer. The influence of the surface reaction appears to spread outwards through the flow from the wall as gas-phase chemical reaction times decrease. For practical values of the surface reaction rate on a metallic wall, the energy transfer rate may be significantly lower in chemically frozen flow than in chemical equilibrium flow under otherwise similar circumstances.

Similar phenomena to those discussed will arise in the more complicated case of boundary layer flows, so that a treatment of the simpler type of shear layer represented by Couette flow may be of some value in assessing the relative importance of the various parameters.
Two-dimensional planing at high Froude number
Authors: Cumberbatch E.
Published: March 2006

This paper examines the flow characteristics of a body of small slope planing at high Froude number over a water surface. An equation is obtained relating the slope of the planing surface to an integral containing the pressure distribution on the planing surface. The equation is expanded for large Froude number and a solution is obtained by an iteration process. At each stage of the iteration process the integral equation of ordinary thin aerofoil theory is solved. The pressure distribution on the planing surface is derived as a series in inverse powers of the Froude number F, as far as the F−4 term. Computations are performed for the planing of a flat plate, a parabolic surface, and a suitable linear combination of these shapes which results in a flow without a splash at the leading edge.
The mixing of ground water and sea water in permeable subsoils
Authors: Carrier G. F.
Published: March 2006

The subterranean mixing in permeable media of sea water and ground water is studied. The model for this mixing process which was suggested by C. K. Wentworth is adopted, but is soon discarded in favour of a more tractable formulation whose equivalence to the original model is established. The analysis is carried to the point where the determination of the salinity distribution of the ground water in a given subsoil requires only the solution of an elementary linear ordinary differential equation.
On convection cells induced by surface tension
Authors: Pearson J. R. A.
Published: March 2006

A mechanism is proposed by which cellular convective motion of the type observed by H. Bénard, which hitherto has been attributed to the action of buoyancy forces, can also be induced by surface tension forces. Thus when a thin layer of fluid is heated from below, the temperature gradient is such that small variations in the surface temperature lead to surface tractions which cause the fluid to flow and thereby tend to maintain the original temperature variations. A small disturbance analysis, analogous to that carried out by Rayleigh and others for unstable density gradients, leads to a dimensionless number B which expresses the ratio of surface tension to viscous forces, and which must attain a certain minimum critical value for instability to occur. The results obtained are then applied to the original cells described by Bénard, and to the case of drying paint films. It is concluded that surface tension forces are responsible for cellular motion in many such cases where the criteria given in terms of buoyancy forces would not allow of instability.
On the melting of a semi-infinite body of ice placed in a hot stream of air
Authors: Roberts Leonard
Published: March 2006

A simple mathematical model is proposed to describe the steady melting of a body of ice which presents a plane surface transverse to a stream of hot air; the temperature of the air is such that vaporization does not occur.

The analysis takes into account the convection of heat away from the surface by the water released in melting and the results show that the rate of transfer of heat to the body and thus the rate of melting, is reduced by as much as 46% by this convection.

Simple approximate expressions are obtained for the rate of melting, the thickness of the water layer, and the thickness of the thermal boundary layer in the ice, in terms of a basic parameter S which can be calculated in terms of known quantities. These results are compared with those obtained by a separate Pohlhausen calculation and are found to be in good agreement.

It is also shown that there exists a thermal boundary layer, in the body, of thickness much greater than that of the boundary layer in the air, in which the temperature changes rapidly from its value at the melting surface to its value in the far interior.
Energy content and ionization level in an argon gas jet heated by a high intensity arc
Authors: Cann Gordon L. , Ducati Adriano C.
Published: March 2006

A direct current electric arc has been developed as a heating device for argon gas. Negligible amounts of electrode material are consumed during an operating time of several minutes. Under normal operating conditions 50% to 90% of the input electric power is transferred directly to the gas. The remaining power is absorbed by the water-cooled electrodes. Measurements were made to determine the total gas enthalpy and the proportion of the enthalpy in directed kinetic energy, random particle motion, and ionization energy. From these measurements it is postulated that the gas is initially in a non-equilibrium state on leaving the arc, but approaches equilibrium relatively quickly when confined to a constant diameter jet outside the arc. The gas temperature range in these experiments varies from 5000°K to 15000°K.
On the disturbed motion of a plane vortex sheet
Authors: Miles John W.
Published: March 2006

A formal solution to the initial value problem for a plane vortex sheet in an inviscid fluid is obtained by transform methods. The eigenvalue problem is investigated and the stability criterion determined. This criterion is found to be in agreement with that obtained previously by Landau (1944), Hatanaka (1949), and Pai (1954), all of whom had included spurious eigenvalues in their analyses. It is also established that supersonic disturbances may be unstable; related investigations in hydrodynamic stability have conjectured on this possibility, but the vortex sheet appears to afford the first definite example. Finally, an asymptotic approximation is developed for the displacement of a vortex sheet following a suddenly imposed, spatially periodic velocity.
Influence of diaphragm opening time on shock-tube flows
Authors: White Donald R.
Published: March 2006

Shock waves generated in a shock tube by use of hydrogen or helium as a driver gas and air, nitrogen, oxygen or argon as a driven gas have higher velocities than predicted by simple theory when sufficiently large diaphragm pressure ratios are used. Expected shock-tube performance curves have been constructed using the equilibrium Hugoniot for the driven gas, both for the usual model of shock tube flow, which assumes instantaneous diaphragm removal, and for a suggested model based on a finite rupture time for the diaphragm. Agreement between experiment and the latter model is in general good, and the differences are qualitatively accounted for by the pressure waves expected to result from mixing between driver and driven gases at the contact surface. These waves may be either compression or expansion waves, depending on the relative heat capacities of the two gases. The maximum shock strength observed as a shock goes down the tube was found to occur at a distance from the diaphragm which increases with the shock strength, and the strongest shocks were found to be still accelerating at the end of a 42 ft. long shock tube of 3 1/2 in. square cross-section. Diaphragm breaking time has been measured and found to be consistent with the observations on the shock formation distance.
A modified tangent-gas approximation for two-dimensional steady flow
Authors: Power G. , Smith P.
Published: March 2006

A set of two-dimensional subsonic flows past certain cylinders is obtained using hodograph methods, in which the true pressure-volume relationship is replaced by various straight-line approximations. It is found that the approximation obtained by a least-squares method possibly gives best results. Comparison is made with values obtained by using the von Kármán-Tsien approximation and also with results obtained by the variational approach of Lush & Cherry (1956).
Surface waves on water of non-uniform depth
Authors: Keller Joseph B.
Published: March 2006

Gravity waves occur on the surface of a liquid such as water, and the manner in which they propagate depends upon its depth. Although this dependence is described in principle by the equations of the ‘exact linear theory’ of surface waves, these equations have not been solved except in some special cases. Therefore, oceanographers have been unable to use the theory to describe surface wave propagation in water whose depth varies in a general way. Instead they have employed a simplified geometrical optics theory for this purpose (see, for example, Sverdrup & Munk (1944)). It has been used very successfully, and consequently various attempts, only partially successful, have been made to deduce it from the exact linear theory. It is the purpose of this article to present a derivation which appears to be satisfactory and which also yields corrections to the geometrical optics theory.
Effects of compressibility on a laminar wall jet
Authors: Riley N.
Published: March 2006

The effects of compressibility on a radial laminar wall jet are investigated. On the assumption that the coefficient of viscosity is proportional to the temperature, it is shown that a similarity solution for the velocity distribution exists, which is expressible directly in terms of the corresponding solution for an incompressible wall jet. For arbitrary Prandtl number the energy equation is studied in detail and solutions are obtained for a variety of temperature conditions.
A two-dimensional air intake in a sonic stream
Authors: Fraenkel L. E.
Published: March 2006

In this paper transonic small-disturbance theory is applied to a simplified model of the flow near the front of a ducted body. The body is assumed to consist simply of two parallel flat plates which extend from the inlet station to infinity downstream. The velocity far upstream is sonic, and the velocity far downstream in the duct, which is assumed to be known, is slightly subsonic. Air is therefore ‘spilled’ around the intake edges. An analytic solution is found for the resulting flow field up to the ‘limiting Mach wave’, and asymptotic solutions are found for the supersonic flow and for the shock wave far from, and near, the intake edges. The pressure distribution along the outside walls is then known at both ends, and its computation is completed by an empirical procedure. Distributions of pressure along the centre-line and along the inside and outside walls are shown. These results may be used to compute the drag of sharp-edged intakes with a very small frontal area.
Experimental studies of supercavitating flow about simple two-dimensional bodies in a jet
Authors: Silberman Edward
Published: March 2006

A two-dimensional free-jet water tunnel developed at the St Anthony Falls Hydraulic Laboratory of the University of Minnesota is described briefly. Results of experimental measurements on a two-dimensional cup, symmetrical wedges, inclined flat plates, and a circular cylinder in the tunnel are given.

Measured force coefficients at zero cavitation number are in good agreement with theory. Shapes of the cavities were computed for one of the wedges and for one of the plates at zero cavitation number; the observed shapes are also in good agreement with the theory.

For non-zero cavitation numbers, theoretical results for force coefficients were available for comparison in only two cases. For one of these, the cup, agreement between theory and experiment was good up to a cavitation number of about 0.5. For the other, a symmetrical wedge, experimental results were compared with a linear theory with good agreement for cavitation numbers between about 0.1 and 0.3. In the case of the wedge, measured cavity lengths were somewhat shorter than predicted by the linear theory. All other comparisons with theory at non-zero cavitation number had to be made with the theory as developed for infinite fluid. The experimental force coefficients were less than predicted by infinite-fluid theory, but tended to approach the theoretical values as the cavitation number increased. A similar tendency marked the comparison between the experimental data and data taken by others in closed tunnels.
Steady flow through non-uniform gauzes of arbitrary shape
Authors: Elder J.W
Published: March 2006

The steady, two-dimensional flow through an arbitrarily-shaped gauze, of non-uniform properties, placed in a parallel channel is considered for the case in which viscosity can be ignored except in the immediate vicinity of the gauze. The equations are linearized by requiring departures from uniformity both in the flow and in the gauze parameters to be small. Knowledge of any three of the upstream profile, the downstream profile, the shape of the gauze and the gauze parameters, allows the other to be calculated from a linear relation between these four quantities. Particular solutions are given for the production of a uniform shear and the flow through linear and parabolic gauzes. The validity of the solution is verified by experiment. It is shown that the method can also be applied to two-dimensional flow in a diverging channel, axisymmetric flow in a circular pipe and in a circular cone and to flow through multiple gauzes.
The propagation of small disturbances in hydromagnetics
Authors: Ludford G. S. S.
Published: March 2006

This paper is concerned with the propagation of small initial disturbances in a conducting gas under the influence of a uniform external magnetic field.

For a perfect conductor, there are three types of plane waves, each of which depends strongly on the angle at which the magnetic field is crossed. The modifying effects of finite conductivity are determined and, in the case of these waves, this is done uniformly for all angles. A general disturbance may be resolved into two parts, one of which satisfies a fourth-order equation and the other a fifth; for a perfect conductor these reduce to second-and fourth-order equations, respectively.

The free oscillations of the gas are examined when it is contained in a rectangular box, and, in particular, when the external field is very weak or very strong. For vanishingly weak fields the idealization of infinite conductivity proves to be inadequate. Finally, the initial-value problem is discussed.
Ring vortices generated electromagnetically
Authors: Yih Chia-Shun
Published: March 2006

If an electric current of uniform densityj0is passed axially through a stationary fluid between concentric cylinders of radiir1andr2(>r1), the fluid is stable to axisymmetric disturbances only if the damping provided by viscosity and electrical resistivity is sufficiently large. It is shown herein that the fluid may also be stabilized by passing a line currentJalong the axis, sufficient conditions for stability being$J \le -\pi j _0(r^2_2 - r^2_1)$, or$\ge J\; \pi j _0r^2_1$

The values ofJneeded to stabilize the fluid when the fluid has non-zero viscosity and finite conductivity are calculated for the caser2-r1[Lt ]r1. In this latter case, the ring vortices which exist under conditions of neutral stability are exactly the same as those for flow between rotating cylinders ifJandj0have the same sign, and ifJis not very small compared with πj0r2.
The supersonic flow past a leading edge separation bubble
Authors: Sharp A. W.
Published: March 2006

The supersonic flow field outside a leading edge separation bubble has been computed in detail by a special application of the method of characteristics. Experimental observations on a two-dimensional square-nosed flat plate at M = 1·96 formed the initial data, and base points were established on the plate surface. Computations proceeded outwards and upstream to the sonic line and bow wave, the bow wave entropy gradient being taken into account. The shape of the sonic line was found to be very sensitive to the conditions along the plate surface. Streamlines and iso-Mach lines have been interpolated and are presented. Good agreement was found at the intersection of the network and the bow wave between computed values and those given directly by the shock wave equations. A comparison has been made with interferometric results from France.
Rapid calculations for boundary-layer transfer using wedge solutions and asymptotic expansions
Authors: Merk H. J.
Published: March 2006

Exact transfer calculations for boundary layers with longitudinal pressure gradients are very complicated, but in the literature several approximate methods are known for the rapid calculation of both the wall friction and the heat transfer. A ‘wedge method’ propounded by Meksyn turns out to be one of the most rapid methods, being no less accurate than other approximate methods. A way of refining this method is proposed.

This paper also shows that asymptotic expansions provide convenient relations which are capable of expressing the Nusselt number explicitly in terms of the Prandtl number.

It is shown that, together with the asymptotic expansions, Meksyn's method permits rapid calculation of local heat transfer numbers. Some examples of application are given for elliptical cylinders and spheres for several values of the Prandtl number.
On the drag of a flat plate at zero incidence in almost-free-molecule flow
Authors: Liu V. C.
Published: March 2006

A physical theory is proposed for the skin friction on a flat plate at zero incidence in the transition flow regime, i.e. in the flow of a moderately rarefied gas. The ratio of the molecular mean free path to the characteristic size of the plate is assumed of order unity or larger. A general formula for the perturbation to the well-known friction of the free-molecule theory is given. This perturbation is attributed to the intermolecular collisions which are neglected on the basis of the free-molecule hypothesis. The expected rate of collisions are calculated for rigid spheres, using the classical kinetic theory.

Although this is intended as an approximate theory, the theoretical results check surprisingly well with the limited experimental data that are available. The present theory shows that the ratio of the Reynolds number to the Mach number squared is the governing parameter for determining the intermolecular collision effect on skin friction in the transition flow regime.
Atomic recombination in a hypersonic wind-tunnel nozzle
Authors: Bray K. N. C.
Published: March 2006

The flow of an ideal dissociating gas through a nearly conical nozzle is considered. The equations of one-dimensional motion are solved numerically assuming a simple rate equation together with a number of different values for the rate constant. These calculations suggest that deviations from chemical equilibrium will occur in the nozzle if the rate constant lies within a very wide range of values, and that, once such a deviation has begun, the gas will very rapidly ’freeze’. The dissociation fraction will then remain almost constant if the flow is expanded further, or even if it passes through a constant area section. An approximate method of solution, making use of this property of sudden ’freezing’ of the flow, has been developed and applied to the problem of estimating the deviations from equilibrium under a wide range of conditions. If all the assumptions made in this paper are accepted, then lack of chemical equilibrium may be expected in the working sections of hypersonic wind tunnels and shock tubes. The shape of an optimum nozzle is derived in order to minimize this departure from equilibrium.

It is shown that, while the test section conditions are greatly affected by ’freezing’, the flow behind a normal shock wave is only changed slightly. The heat transfer rate and drag of a blunt body are estimated to be reduced by only about 25% even if complete freezing occurs. However, the shock wave shape is shown to be rather more sensitive to departures from equilibrium.
Energy dissipation in standing waves in rectangular basins
Authors: Keulegan Garbis H.
Published: March 2006

The modulus of decay of standing waves of finite height is derived by assuming that the attenuation of the waves is due to viscous losses in boundary layers close to the solid walls. Dampings are observed in six basins of varying sizes. The basins are duplicated using glass and lucite for the wall materials. With liquids wetting the walls, the losses due to viscosity are slightly increased from causes presumably related to surface tension. With a liquid not wetting the walls (distilled water and lucite), losses from surface activity, of some obscure origin, outweigh many times the losses due to viscosity in the basins of smaller sizes. For moderately large basins, for which surface activity may be neglected, the agreement between the observed and computed rates of decay is found to be satisfactory.
The magnetohydrodynamic flow past a flat plate
Authors: Greenspan H. P. , Carrier G. F.
Published: March 2006

The uniform steady flow of an incompressible, viscous, electrically conducting fluid is distorted by the presence of a symmetrically oriented semi-infinite flat plate. The ambient magnetic field is coincident with the ambient velocity field. The description of the resulting fields depends on the physical co-ordinates measured in units of Reynolds number and on the two parameters ε = ωμν and β = μH2/ρv2. This description of the fields is approximated in three different ways and essentially covers the full range of ε and β. In particular, when β [Gt ] 1, no steady flow which is uniform at large distances from the plate exists.
Development and use of a vane device for boundary-layer measurements
Authors: Burns J. G. , Childs W. H. J. , Nicol A. A. , Ross M. A. S.
Published: March 2006

A hinged vane and a sensitive electrical system for recording the motion of the vane have been developed for the observation of fluctuating y-components of velocity in boundary layers. An approximate theory of the natural oscillations of such vanes is presented and experimentally verified. Using vanes as resonant detectors, meassurements have been made of oscillations injected into the laminar boundary layer on a flat plate in a wind tunnel with 0·3% free-stream turbulence. Points on the neutral Tollmien-Schlichting curve have thereby been obtained which lie close to the theoretical neutral curve.
Torsional oscillations of a plane in a viscous fluid
Authors: Rosenblat S.
Published: March 2006

On the assumption that the rotational oscillations of a rigid plane are small, boundary-layer type solutions of the Navier-Stokes equations are attempted by an expansion of the velocity components in power series of the amplitude. First-and third-order approximations to the transverse velocity are obtained, from which a correction to the moment on a disk of finite radius is found.

The first non-vanishing approximation to the radial-axial flow (a second-order term) is seen to have a steady component and a component with frequency twice that of the plate. The former component appears to persist outside the boundary layer, and at large distances from the plate to have the character of an irrotational stagnation flow. A re-examination not involving the series approximation reveals that although steady radial flow does exist outside the boundary layer, it is a viscous flow and is confined within a secondary layer. The ratio of the thicknesses of the two layers is found to be inversely proportional to the amplitude of the oscillations. These results indicate that a second-order flow in a region where the first-order flow field vanishes should not be accepted without further discussion.
Measurements of intermittency of turbulent motion in a boundary layer
Authors: Sandborn V. A.
Published: March 2006

Previous observations of turbulent motion at large wave-numbers have revealed the existence of an uneven distribution of turbulent energy. The spotty distribution of the turbulent motion at high wave-numbers is here studied experimentally for the turbulent boundary layer. The high wave-number intermittency is observed at all locations through and along the boundary layer from near transition to near separation.

The flatness factors for the longitudinal turbulent component at different wave-numbers are measured to give a quantitative value for the intermittency at particular wave-numbers. Upstream of the separation region the flatness factors are found to depend on wave-number and longitudinal distance, but not on the distance from the wall. It appears that the intermittency develops in the transition region and does not diminish very rapidly with distance downstream. Near separation the flatness factors change radically in distribution near the wall, and are there no longer independent of distance from the wall.
An analysis of the vortex street generated in a viscous fluid
Authors: Schaefer John W. , Eskinazi Salamon
Published: March 2006

An analytic solution for the velocity field of a vortex street generated in a viscous fluid is developed. A method is presented for the determination of the true transverse spacing of vortices. Experimental geometry and velocity data, obtained by hot-wire techniques, are presented.

The experimental results verified the validity of the analytic solution. The vortices of a real viscous vortex street were found to resemble very closely the exponential solution of the Navier-Stokes equations for an isolated axisymmetric rectilinear vortex. Three basic regions of vortex street behaviour were apparent at each Reynolds number investigated-a ‘formation region’ in which the vortex street is developed and large dissipation of vorticity occurs, a ‘stable region’ in which the vortices display a stable periodic laminar regularity, and an ‘unstable region’ in which the street disappears and turbulence develops. Geometry and velocities were determined.
The analysis of temperature fluctuations by pulse-counting techniques
Authors: Townsend A. A.
Published: March 2006

To determine experimentally the mean value of a randomly fluctuating quantity, it may be necessary to measure the average value over a considerable interval of time. This problem arose in a recent study of the temperature fluctuations over a heated horizontal plate, and a system was used that depended on the counting of electrical pulses generated at a rate proportional to the quantity being measured. The advantage of this technique is that mean values may be measured over time intervals of almost unlimited length with little added difficulty for the experimenter. Circuits are described which measure: (a) the mean square of a fluctuating quantity and of its time-derivative, (b) the statistical distribution of the fluctuations, (c) the mean frequency of the fluctuation assuming a particular value, and (d) the mean product of two fluctuating quantities. Over the range of use, the stability and linearity of the calibrations is better than 1%, more than sufficient for work on natural convection. In its present form, the equipment responds uniformly to all frequencies below 100 c/s, but it would not be difficult to extend this range of response to higher frequencies.
A theory of electromagnetically driven shock waves
Authors: Wright J. K. , Black M. C.
Published: March 2006

During the last few years, many experimental devices have been built in which strong shock waves are generated in gases by electromagnetic forces on currentcarrying gas particles. The general theory of these devices is discussed, taking external circuit inductance into account. It is shown that a shock wave of constant speed is finally attained. This shock wave is travelling at 90% of its final speed when the circuit inductance has increased to 3·0 times its initial value.
The pressure distribution on dihedral wings at supersonic speed
Authors: Ting Lu
Published: March 2006

For wings with supersonic edges and with arbitrary dihedral, twist, camber and thickness distribution, the pressure distribution on the wing exterior to and along the two Mach lines emanating from the vertex of the wing is equal to the corresponding pressure distribution for a planar wing. The problem is to find the pressure distribution inside the two Mach lines. In the present paper, the unknown pressure distribution is approximated by an elementary function of the two surface variables. The (as yet undetermined) constants in the function are then found by the conditions: (i) that the function takes on the corresponding planar values along the two Mach lines, (ii) that it fulfils certain generalized integral relationships (Ting 1959), and (iii) that it satisfies the averaging property of solutions of the wave equation to be developed in this paper. The generalized integral relationship relates the integral of the pressure distribution along the line of intersection of a Mach plane with the wing to the integral along the same line of the prescribed normal velocity. The averaging property relates the pressure distribution along the line of intersection of the surface of the dihedral wing to that on a planar wing.
Non-equilibrium flow over a wavy wall
Authors: Vincenti Walter G.
Published: March 2006

A small-disturbance solution is obtained for the steady two-dimensional flow over a sinusoidal wall of an inviscid gas in vibrational or chemical non-equilibrium. The results are based on a single, linear, third-order partial differential equation, which plays the same role here as does the Prandtl–Glauert equation in equilibrium flow. The solution is valid throughout the range from subsonic to supersonic speeds and for all values of the rate parameter from equilibrium to frozen flow (in both of which limits it reduces to Ackert's classical solution of the Prandtl–Glauert equation). The results illustrate in simple fashion some of the properties of non-equilibrium flow, such as the occurrence of pressure drag at subsonic speeds and the absence of the discontinuous phenomena that characterize the Prandtl–Glauert theory when the flow changes from subsonic to supersonic.
Pressure and velocity measurements on detonation waves in hydrogen-oxygen mixtures
Authors: Edwards D. H. , Williams G. T. , Breeze J. C.
Published: March 2006

Measurements are described of the static pressures and velocities of detonation waves in hydrogen-oxygen mixtures, together with the pressures arising on their normal reflexion at the closed end of the explosion tube. Two explosion tubes, of diameter 10 and 1·6 cm, were employed to study the diameter effect on the wave pressures. The experimental results are compared with calculated values of the wave properties for a range of hydrogen-oxygen mixtures initially at atmospheric pressure. In the 10 cm tube the static pressures and velocities are found to agree well with theory for mixtures with hydrogen content in the approximate range 50-75%; the evidence from pressure profiles and wave velocities indicates that mixtures outside this range may not be able to support ideal C–J waves. Detonation waves in all the mixtures studied in the 1·6 cm tube are found to be subideal. A possible explanation, in terms of energy loss to the tube wall, of the discrepancy between experiment and theory is discussed. Spinning occurs in mixtures near the limits of detonation in the smaller tube; the measured frequencies are found to be in reasonable agreement with the values predicted by the theories of Manson (1947) and Fay (1952).
Continuum equations in the dynamics of rarefied gases
Authors: Krook Max
Published: March 2006

A procedure is given for translating boundary-value problems of gas dynamics from microscopic form into approximately equivalent continuum form. The continuum formulations involve state-variables that are either half-space moments, or complete moments of the molecular distribution functions. Moment equations derived from the kinetic equations are reduced to a determinate set by representing the distribution functions as sums of ‘modified Maxwellian functions based on various characteristic temperatures and velocities’. The particular choice of such a representation depends on the Knudsen number and on the nature of the microscopic boundary conditions.
On the generation of surface waves by shear flows. Part 2
Authors: Miles John W.
Published: March 2006

A previous analysis for the generation of surface waves by a parallel shear flow (Miles 1957 a) is extended by: (a) presenting results based on a more accurate solution of the differential equation; (b) imposing the boundary condition at the surface wave, rather than at the mean surface; and (c) including the dominant viscous term in the complete Orr-Sommerfeld equation. The modification (a) yields an energy transfer somewhat smaller than that predicted previously but of the same order of magnitude as, and in rather better agreement with, observation, while (b) has no effect and (c) only a small effect for gravity waves. The analysis is based on the equations of motion in intrinsic co-ordinates (rather than the usual Orr-Sommerfeld equation) and may be of interest in other problems of hydrodynamic stability.
On the generation of surface waves by shear flows Part 3. Kelvin-Helmholtz instability
Authors: Miles John W.
Published: March 2006

The Kelvin-Helmholtz model for the formation of surface waves at the interface between two fluids in relative motion is generalized for parallel shear flows. It is assumed that phase changes across the flow are negligible and hence that the aerodynamic pressure on the wave is in phase with its displacement (rather than its slope). A variational formulation is established and leads to the determination of appropriately weighted means for the velocity profiles. The principal application is to flow of a light inviscid fluid over a viscous liquid; it is shown that the principle of exchange of stabilities is applicable to such a configuration, and a critical wind speed in satisfactory agreement with observation is predicted for an air-oil interface. The results also are applied to an air-water interface and lead to the conclusion that Kelvin-Helmholtz instability of such an interface is unlikely at commonly observed wind speeds. A more general formulation of the Kelvin-Helmholtz boundary-value problem and variational principle, allowing for variations in both velocity and density, is given in two appendices.
The symmetric state of a rotating fluid differentially heated in the horizonal
Authors: Robinson Allan R.
Published: March 2006

The motion of a fluid inside a rotating annulus of square cross-section, whose dimensions are small compared with the distance from the axis of rotation, is considered. The rigid side walls are held at different constant temperatures and the fluid motions that occur are strongly influenced by Coriolis accelerations. A detailed study is made of the azimuthally independent state, a Hadley cell, in the limit of small thermal Rossby number. It is convenient to employ a boundary layer type analysis, essentially with respect to the Taylor number and all the imposed boundary conditions are rigorously satisfied.

An entirely geostrophic thermal wind is found to obtain over the main body of the fluid. The circulation in the plane of the annular cross-section is entirely confined within narrow boundary layers and consists of a superposition of three cellular motions: a cell occupying the cross-section and two additional cells confined to the side-wall boundary layers. These motions are intimately related to the rotational constraint. The temperature distribution and its relation to the conductive and convective processes are determined.
The time-dependent magnetohydrodynamic flow past a flat plate
Authors: Carrier G. F. , Greenspan H. P.
Published: March 2006

Two time-dependent magnetohydrodynamic flow problems are discussed. In Part I we consider the situation in which a semi-infinite flat plate is moved impulsively in its own plane into an electrically conducting viscous fluid. The ambient magnetic field has the same direction as the motion of the plate; it is found that when $\mu H^2_0| \rho U^2_0 \; \textless \; 1$, the flow pattern approaches asymptotically the steady flow found earlier (Greenspan & Carrier 1959). When $\mu H^2_0| \rho U^2_0 \; \textgreater \; 1$, the asymptotic state is one in which the fluid accompanies the plate in a rigid body motion as was anticipated in the earlier work.

In Part II, an infinite plate is moved impulsively in its own plane in the presence of an ambient magnetic field which is perpendicular to the plane of the plate. It is shown that the problem is not uniquely set until one specifies what three-dimensional problem reduces in the limit to the two-dimensional problem so defined. The answers in the conceptually acceptable limit case investigated here (the plate being a pipe of very large radius) have an asymmetry which at first sight is unexpected.
Forced small-amplitude water waves: a comparison of theory and experiment
Authors: Ursell F. , Dean R. G. , Yu Y. S.
Published: March 2006

This paper describes an attempt to verify experimentally the wavemaker theory for a piston-type wavemaker. The theory is based upon the usual assumptions of classical hydrodynamics, i.e. that the fluid is inviscid, of uniform density, that motion starts from rest, and that non-linear terms are neglected. If the water depth, wavelength, wave period, and wavemaker stroke (of a harmonically oscillating wavemaker) are known, then the wavemaker theory predicts the wave motion everywhere, and in particular the wave height a few depths away from the wavemaker.

The experiments were conducted in a 100 ft. wave channel, and the wave-height envelope was measured with a combination hook-and-point gauge. A plane beach (sloping 1:15) to absorb the wave energy was located at the far end of the channel. The amplitude-reflexion coefficient was usually less than 10%. Unless this reflexion effect is corrected for, it imposes one of the most serious limitations upon experimental accuracy. In the analysis of the present set of measurements, the reflexion effect is taken into account.

The first series of tests was concerned with verifying the wavemaker theory for waves of small steepness (0.002 ≤ H/L ≤ 0.03). For this range of wave steepnesses, the measured wave heights were found to be on the average 3.4% below the height predicted by theory. The experimental error, as measured by the scatter about aline 3.4% below the theory, was of the order of 3%. The systematic deviation of 3.4% is believed to be partly due to finite-amplitude effects and possibly to imperfections in the wavemaker motion.

The second series of tests was concerned with determining the effects of finite amplitude. For therange of wave steepnesses 0.045 ≤ H/L ≤ 0.048, themeasured wave heights were found to be on the average 10% below the heightspredictedfrom the small-amplitude theory. The experimental error was again of the order of 3%.

It is considered that these measurements confirm the validity of the small-amplitude wave theory. No confirmation of this accuracy has hitherto been given for forced motions.