Showing papers in "Journal of Fluid Mechanics in 1959"

Small-scale variation of convected quantities like temperature in turbulent fluid Part 1. General discussion and the case of small conductivity

Abstract: When some external agency imposes on a fluid large-scale variations of some dynamically passive, conserved, scalar quantity θ like temperature or concentration of solute, turbulent motion of the fluid generates small-scale variations of θ. This paper describes a theoretical investigation of the form of the spectrum of θ at large wave-numbers, taking into account the two effects of convection with the fluid and molecular diffusion with diffusivity k. Hypotheses of the kind made by Kolmogoroff for the small-scale variations of velocity in a turbulent motion at high Reynolds number are assumed to apply also to small-scale variations of θ.Previous contributions to the problem are reviewed. These have established that the spectrum of θ varies as , the result being given by (4.8). The same result is obtained, using essentially the same approximation about the velocity field, from a different kind of analysis in terms of velocity and θ correlations. Finally, the relation between this work and Townsend's model of the small-scale variations of vorticity in a turbulent fluid is discussed.

Experiments on the flow past a circular cylinder at low Reynolds numbers

Abstract: Part I describes measurements of the drag on circular cylinders, made by observing the bending of quartz fibres, in a stream with the Reynolds number range 0·5-100. Comparisons are made with other experimental values (which cover only the upper part of this range) and with the various theoretical calculations.Part II advances experimental evidence for there being a transition in the mode of the vortex street in the wake of a cylinder at a Reynolds number around 90. Investigations of the nature of this transition and the differences between the flows on either side of it are described. The interpretation that the change is between a vortex street originating in the wake and one originating in the immediate vicinity of the cylinder is suggested.

Turbulent entrainment in stratified flows

Abstract: When a fluid which is lighter than its surroundings is emitted by a source under a sloping roof (or a heavier fluid from a source on a sloping floor), it may flow as a relatively thin turbulent layer. The motion of this layer is governed by the rate at which it entrains the ambient fluid. A theory is presented in which it is assumed that the entrainment is proportional to the velocity of the layer multiplied by an empirical function, E(Ri), of the overall Richardson number for the layer defined by Ri = g(ρa - ρ) h/ρa V2. This theory predicts that in most practical cases the layer will rapidly attain an equilibrium state in which Ri does not vary with distance downstream, and the gravitational force on the layer is just balanced by the drag due to entrainment together with friction on the floor or roof.Two series of laboratory experiments are described from which E(Ri) can be determined. In the first, the spread of a surface jet of fluid lighter than that over which it is flowing is measured; in the second, a study is made of the flow of a heavy liquid down the sloping floor of a channel. These experiments show that E falls off rapidly as Ri increases and is probably negligible when Ri is more than about 0·8.The theoretical and experimental results allow predictions to be made of flow velocities once the rate of supply of density difference is known. An estimate is also given of the uniform velocity which the ambient fluid must possess in order to cause the motion of the layer to be reversed.

The dispersion of marked fluid in turbulent shear flow

Abstract: The analysis used by Taylor (1954) and based on the Reynolds analogy has been extended to describe the diffusion of marked fluid in the turbulent flow in an open channel The coefficient of longitudinal diffusion arising from the combined action of turbulent lateral diffusion and convection by the mean flow is computed to be 5·9uτh, where h is the depth of fluid and uτ the friction velocity This is in agreement with experiments described herein The laterla diffusion coefficient is found by experiment to be 0·23uτh, which is three times larger than the value obtained by the assumption of isotropy The same analysis can be used to describe the longitudinal dispersion of discrete particles, both of zero buoyancy and of finite buoyancy, and comparison is made with observations by Batchelor, Binnie & Phillips (1955) and Binnie & Phillips (1958)

On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres

Abstract: Spatially periodic fundamental solutions of the Stokes equations of motion for a viscous fluid past a periodic array of obstacles are obtained by use of Fourier series. It is made clear that the divergence of the lattice sums pointed out by Burgers may be rescued by taking into account the presence of the mean pressure gradient. As an application of these solutions the force acting on any one of the small spheres forming a periodic array is considered. Cases for three special types of cubic lattice are investigated in detail. It is found that the ratios of the values of this force to that given by the Stokes formula for an isolated sphere are larger than 1 and do not differ so much among these three types provided that the volume concentration of the spheres is the same and small. The method is also applied to the two-dimensional flow past a square array of circular cylinders, and the drag on one of the cylinders is found to agree with that calculated by the use of elliptic functions.

The motion of bubbles in a vertical temperature gradient

Abstract: It has been observed experimentally that small bubbles in pure liquids can be held stationary or driven downwards by means of a sufficiently strong negative temperature gradient in the vertical direction. This effect is demonstrated to be due to the stresses resulting from the thermal variation of surface tension at the bubble surface. The flow field within and around the bubble is derived, and an expression for the magnitude of the temperature gradient required to hold the bubble stationary is obtained. This expression is verified experimentally.

Two-dimensional convection from heated wires at low Reynolds numbers

Abstract: Measurements of heat transfer from circular wires placed normal to a horizontal airstream have been made in the Reynolds number range 0·01 to 140. The Nusselt number can be related to the Reynolds number and temperature loading by an expression of the form where the values of n, A and B (see table 3) depend on whether the Reynolds number is above or below the value for which a vortex street exists in the wake of the wire. This value of the Reynolds number (R [eDot ] 44) is independent of the intensity and scale of the stream turbulence. The theoretical heat transfer relation based on the Oseen approximation is approached asymptotically as R → 0, provided free convection is negligible.Free convection effects diminish rapidly with increasing Reynolds number so that the orientation of the wire with respect to the vertical has a negligible influence on heat transfer except at very low velocities. For horizontal wires at very low Reynolds numbers, free convection is significant, when, roughly speaking, the Reynolds number is less than the cube root of the Grashof number.

The prediction of separation of the turbulent boundary layer

Abstract: A rapid method for the prediction of flow separation results from an approximate solution of the equations of motion; a single empirical factor is required. The equations are integrated by a modified ‘inner and outer solutions’ technique developed recently for laminar boundary layers, the criterion for separation being obtained as a simple formula applying directly to the separation position. At Reynolds numbers of the order of 106, the criterion is when d2p/dx2 [ges ] 0 and Cp [les ] 4/7; the coefficient 0·39 is replaced by 0·35 when d2p/dx2 < 0.The prediction of the pressure rise to separation is likely to be from 0 to 10% too low, which puts it second in accuracy to those methods, such as Maskell's (1951), which utilize the Ludweig-Tillmann skin friction law. However, the convenience of the method makes the present error acceptable for many applications, while a greater accuracy should be attainable from an improved allowance for the quantity d2p/dx2.The main derivation is for arbitrary pressure distributions, while an extension leads to the pressure distribution which just maintains zero skin friction throughout the region of pressure rise.The concept of a turbulent inner layer with zero wall stress is put forward, and it is deduced that in the neighbourhood of the wall the velocity is proportional to the square root of the distance from the wall.

Shearing flow over a wavy boundary

Abstract: A theoretical study is made of shearing flows bounded by a simple-harmonic wavy surface, the main object being to calculate the normal and tangential stresses on the boundary. The type of flow considered is approximately parallel in the absence of the waves, being exemplified by two-dimensional boundary layers over a plane. Account is taken of viscosity; but, as the Reynolds number is assumed to be large, its effects are seen to be confined within narrow ‘friction layers’, one of which adjoins the wave and another surrounds the ‘critical point’ where the velocity of flow equals the wave velocity. The boundary conditions are made as general as possible by including the three cases where respectively the boundary is rigid, flexible yet still solid, or completely mobile as if it were the interface with a second fluid.The theory is developed on the model of stable laminar flow, although it is proposed that the same theory may usefully be applied also to examples of turbulent flow considered as ‘pseudo-laminar’ with velocity profiles corresponding to the mean-velocity distribution. Use is made of curvilinear co-ordinates which follow the contour of the wave-train. This admits a linearized form of the problem whose validity requires only that the wave amplitude be small in comparison with the wavelength, even when large velocity gradients exist close to the boundary. The analysis is made largely without restriction to particular forms of the velocity profile; but eventually consideration is given to the example of a linear profile and the example of a boundary-layer profile approximated by a quarter-period sinusoid. In § 7 some general methods are set out for the treatment of disturbed boundary-layer proses: these apply with greatest precision to thin boundary layers, but are also useful for the initially very steep but on the whole fairly diffuse profiles which occur in most practical instances of turbulent flow over waves.The phase relationships found between the stresses and the wave elevation are discussed for several examples, and their interest in connexion with problems of wave generation by wind is pointed out. It is shown that in most circumstances the stresses are distributed in much the same way as if the leeward slopes of the waves were sheltered. For instance, the pressure distribution often has a substantial component in phase with the wave slope, just as if a wake were formed behind each wave crest—although of course actual separation effects are outside the scope of the present theory. In this aspect, the analysis amplifies the work of Miles (1957).

A theory of dispersion in a porous medium

Abstract: This paper is concerned with the dispersion of a dynamically neutral material quantity in a fluid flowing throuh a porous medium. The medium is regarded as an assemblage of randomly orientated straight pores, and it is assumed that the path of a marked element of the material quantity consists of a sequence of statistically independent steps whose direction and duration vary in some random manner. The probability density function for the displacement of a single marked element is calculated and values for the dispersion of a cloud of marked elements then follow.The case is examined in which the flow satisfies Darcy's law (i.e. the mean velocity is linearly proportional to the mean pressure gradient), and the molecular diffusivity is sufficiently small for the dispersion to be primarily due to the randomness of the streamlines, but it is not assumed that effects of molecular diffusion can be altogether neglected. It is shown that the longitudinal dispersion in the direction of the mean flow may be described asymptotically by an effective diffusivity which is a function of U, l, a, K and T. (U denotes the average velocity, l the pore length, a the pore radius which is shown to be related to the permeability, k the molecular diffusivity, and T the time from the initial instant.) Expressions for the longitudinal diffusivity kl are obtained according to the relative values of l/U, T, t0 = l2/2k and t1 = a2/8k. These are given in §4, equations (4.3), (4.4) and (4.5). Speaking roughly, when t0 [Gt ] T [Gt ] l/U,kl/Ul is alogarithmic function of UT/l and increases with T; when T [Gt ] t0 [Gt ] l/U, which must eventually be the case however small k,kl/Ul is alogarithmic function of Ul/k and independent of T. The theoretical results are compared with experimental data reported in the literature and approximate agreement is obtained when E is put equal to the average diameter of the particles composing the porous medium.The lateral dispersion in the direction perpendicular to that of the mean velocity is found to be governed asymptotically by an effective diffusivity . However, it is pointed out that some of the assumptions, namely that successive steps are statistically independent and that the dispersion of a cloud follows immediately from the statistical properties of the displacement of a single marked element, may not be valid for the lateral dispersion and this result is therefore suspect.Remarks are made in §5 on the dispersion for high values of the Reynolds number Ul/v (v = kinematic viscosity) when Darcy's law is not obeyed, and it is argued that kl/Ul should decrease as Ul/v increases.

Small-scale variation of convected quantities like temperature in turbulent fluid Part 2. The case of large conductivity

Abstract: The analysis reported in Part 1 is extended here to the case in which the conductivity κ is large compared with the viscosity ν, the conduction ‘cut-off’ to the θ-spectrum then being at wave-number (e/κ3)¼. It is shown, with a plausible and consistent hypothesis, that the convective supply of . The consequent form of the theta;-spectrum within this same wave-number range is The way in which conduction influences (and restricts) the effect of convection on the distribution of θ at these wave-numbers beyond the conduction cut-off is discussed.

On the generation of surface waves by shear flows. Part 2

Abstract: 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.

Cellular convection with finite amplitude in a rotating fluid

Abstract: When a rotating layer of fluid is heated uniformly from below and cooled from above, the onset of instability is inhibited by the rotation. The first part of this paper treats the stability problem as it was considered by Chandrasekhar (1953), but with particular emphasis on the physical interpretation of the results. It is shown that the time-dependent (overstable) motions occur because they can reduce the stabilizing effect of rotation. It is also shown that the boundary of a steady convection cell is distorted by the rotation in such a way that the wave length of the cell measured along the distorted boundary is equal to the wavelength of the non-rotating cell. This conservation of cellular wavelength is traced to the constancy of horizontal vorticity in the rotating and non-rotating systems. In the finite-amplitude investigation the analysis, which is pivoted about the linear stability problem, indicates that the fluid can become unstable to finite-amplitude disturbances before it becomes unstable to infinitesimal perturbations. The finite-amplitude motions generate a non-linear vorticity which tends to counteract the vorticity generated by the imposed constraint of rotation. Under experimental conditions the two fluids, mercury and air, which are considered in this paper, will not exhibit this finite amplitude instability. However, a fluid with a sufficiently small Prandtl number will become unstable to finite-amplitude perturbations. The special role of viscosity as an energy releasing mechanism in this problem and in the Orr-Sommerfeld problem suggests that the occurrence of a finite-amplitude instability depends on this dual role of viscosity (i.e. as an energy releasing mechanism as well as the more familiar dissipative mechanism). The relative stability criterion developed by Malkus & Veronis (1958) is used to determine the preferred type of cellular motions which can occur in the fluid. This preferred motion is a function of the Prandtl number and the Taylor number. In the case of air it is shown that overstable square cells become preferred in finite amplitude, even though steady convective motions occur at a lower Rayleigh number.

An experimental flow with zero skin friction throughout its region of pressure rise

Abstract: A flow has been produced having effectively zero skin friction throughout its region of pressure rise, which extended for a distance of 3 ft. No fundamental difficulty was encountered in establishing the flow and it had, moreover, a good margin of stability. The dynamic head in the zero skin friction boundary layer was found to be linear at the wall (i.e. u ∞ y½), as predicted theoretically in the previous paper (Stratford 1959).The flow appears to achieve any specified pressure rise in the shortest possible distance and with probably the least possible dissipation of energy for a given initial boundary layer. Thus an aerofoil which could utilize it immediately after transition from laminar flow would be expected to have a very low drag. A design pressure distribution (besides having the usual safety margin against stall) should have a slightly more gradual start to the pressure rise than in the present experiment, as small errors close to the discontinuity can cause difficulty.

The rise of a gas bubble in a viscous liquid

Abstract: The rise of a gas bubble in a viscous liquid at high Reynolds number is investigated, it being shown that in this case the irrotational solution for the flow past the bubble gives a uniform approximation to the velocity field. The drag force experienced by the bubble is calculated on this hypothesis and the drag coefficent is found to be 32/R, where R is the Reynolds number (based on diameter) of the bubbles rising motion. This result is shown to be in fair agreement with experiment.The theory is extended to non-spherical bubbles and the relation of the resulting theory, which enables both bubble shape and velocity of rise to be predicted, to experiment is discussed.Finally, an inviscid model of the spherical cap bubble involving separated flow is considered.

Temperature fluctuations over a heated horizontal surface

Abstract: The previous work of Thomas (1956) on the turbulent convection over a single, heated, horizontal surface has been extended using improved methods of analysis of the temperature fluctuations, and it has been possible to measure the distributions of mean temperature, the mean squares of the temperature fluctuation, the temperature gradient and the rate-of-change of temperature, and the statistical distributions of these quantities. These measurements were made for three different values of the convective heat flux, and the results are consistent with the dimensional consequences of assuming that the convection near the surface is independent of the distant boundaries and determined by the heat flux and the viscosity and conductivity of the fluid.The most striking feature of the observations is that the fluctuations of temperature, temperature gradient, and rate-of-change of temperature, all show periods of activity, characterized by large fluctuations, alternating with periods of quiescence with comparatively small ones. Both the proportion and frequency of occurrence of the active periods decrease with increasing distance from the surface and they probably occur when rising columns of hot air pass through the point of observation. The quiescent periods occur when the point of observation lies outside the columns, and analysis of the statistical distributions of the various fluctuations shows that, during these periods, they are nearly independent of height. It is concluded that the quiescent fluctuations are typical of the turbulent convection far from the surface while the active fluctuations are the manifesta- tion of the convective processes arising near the rigid boundary. These processes may be described as the detachment of columns of hot air from the edge of the conduction layer and the erosion of these rising columns by contact with the surrounding air which is in vigorous turbulent motion. Since the variations of the intensities with height is dominated by the contributions of the active periods, it is not surprising that no agreement is found with the predictions of the similarity theory which assumes the convection to be independent of the conduction layer at a sufficient distance from it. The Malkus theory of turbulence, which empha- sizes dependence on the conduction layer, is in qualitative agreement with this inferred mechanism of the convection and is in quantitative agreement with the observed distribution of mean temperature. A brief discussion is given of the effect of a horizontal shearing motion on the convection and of the relation of these measurements to measurements of the temperature distribution in the earth's boundary layer with upward flux of total heat.

Atomic recombination in a hypersonic wind-tunnel nozzle

Abstract: 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.

On the stability of a spinning top containing liquid

Abstract: The stability of a heavy top, containing a cylindrical cavity partly full of liquid, for small displacements from the sleeping position is studied. It is shown theoretically that instability can occur when any one of the periods of free oscillation of the liquid, which are doubly infinite in number, is sufficiently near to the period of nutation of the empty top. In experiments carried out by Prof. Ward, only the two principal instabilities could be distinguished.

Energy dissipation in standing waves in rectangular basins

Abstract: 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.

On the generation of surface waves by shear flows Part 3. Kelvin-Helmholtz instability

Abstract: 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.

An unsteady turbulent boundary layer

Abstract: A low speed parallel flow, whose velocity fluctuates sinusoidally in magnitude about a constant mean, has been produced in a boundary layer wind tunnel. Hot-wire measuring techniques have been developed to permit an investigation of the turbulent boundary layer developing on a flat plate with this free stream condition. The response of the layer at a Reynolds number $R_{\delta *} = (\overline {U}_\infty \delta^*)|v = 3\cdot 6 \times 10^3$ was measured for free stream fluctuation amplitudes up to 34% of the mean velocity and frequencies ranging from 0 to 48 cycles/sec. Here $\delta^* = \int ^\infty _0 \left(1 - \frac {\overline{U}} {\overline{U}_\infty} \right)dy$ is the boundary displacement thickness, $\overline {U}(x, y)$ ( x , y ) the mean velocity in the boundary layer, $\overline {U}_\infty$ the mean velocity in the free stream, and v the kinematic viscosity. Measurements were made of the mean velocity, the amplitudes of in- and out-of-phase components of the first harmonic of the periodic fluctuations, and the intensity of higher harmonics and turbulence. It was found that non-linear effects, even at the largest fluctuation amplitudes, were so small that they were obscured by experimental errors.

The magnetohydrodynamic flow past a flat plate

Abstract: 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 e = ωμν and β = μH2/ρv2. This description of the fields is approximated in three different ways and essentially covers the full range of e and β. In particular, when β [Gt ] 1, no steady flow which is uniform at large distances from the plate exists.

An analysis of the vortex street generated in a viscous fluid

Abstract: 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.

Steady flow through non-uniform gauzes of arbitrary shape

Abstract: 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.

Rapid calculations for boundary-layer transfer using wedge solutions and asymptotic expansions

Abstract: 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.

The slow motion of two or more spheres through a viscous fluid

Abstract: Expressions are derived for the velocity of two spheres, moving slowly under external forces through a viscous fluid, as a function of their separation and radii. They compare favourably with the available experimental data. A discussion of the interactions of three particles and some general comments on the settling of a swarm of spheres are also included.

An experimental investigation of the stability of Poiseuille flow

Abstract: An axially symmetric laminar flow of air was established in a long smooth pipe. This flow was steady up to Reynolds numbers of about 20,000, the capacity of the system. Small, nearly axially, symmetric disturbances were superimposed by longitudinally oscillating a thin sleeve adjacent to the inner wall of the pipe. Hot-wire anemometer measurements consisting of radial and longitudinal traverses were made downstream of the sleeve. These measurements indicated that within the Reynolds number range investigated (up to 13,000), the flow is stable to small disturbances. In general, the radial distribution of disturbance amplitudes was not independent of distance downstream; while the disturbances, as generated, exhibited imperfect axial symmetry, the non-symmetric part decayed more rapidly than the symmetric part. Results were interpreted in such a way that rates of propagation and rates of decay of the disturbances could be compared with those given by a recent theoretical stability analysis. It was found that the rates of decay are predicted fairly satisfactorily by the theory; however, the rates of propagation are not. In addition, it was found that transition to turbulent flow occurs whenever the amplitude of the disturbance exceeds a threshold value which decreases with increasing Reynolds number. Due to the departures from axial symmetry in the amplitude of the disturbance, it was not possible to obtain a quantitative measure of the threshold. A mathematical idealization of the disturbances, believed to be more akin to experimental perturbations than the classical model used in small-perturbation analyses, is proposed.

Torsional oscillations of a plane in a viscous fluid

Abstract: 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.

Theory of thin airfoils in fluids of high electrical conductivity

Abstract: Steady, plane flow of incompressible fluid past thin cylindrical obstacles is treated with two different orientations of the undisturbed, uniform magnetic field; namely, parallel and perpendicular, respectively, to the undisturbed, uniform stream. In the first case, the flow of an infinitely conducting fluid is shown to be irrotational and current-free except for surface curents at the walls of the obstacles. With large but finite conductivity the surface currents are replaced by thin boundary layers of large current density.In the second case, for infinite conductivity the flow field is made up of an irrotational current-free part and a system of waves involving currents and vorticity extending out from the body. For large, finite conductivity these waves attenuate exponentially with distance from the body.In both cases the forces on sinusoidal walls and on airfoils are calculated. In the second case positive drag occurs.