The oscillating lift and drag on circular cylinders are determined from measurements of the fluctuating pressure on the cylinder surface in the range of Reynolds number from 4 × 103 to just above 105.

The magnitude of the r.m.s. lift coefficient has a maximum of about 0.8 at a Reynolds number of 7 × 104 and falls to about 0.01 at a Reynolds number of 4 × 103. The fluctuating component of the drag was determined for Reynolds numbers greater than 2 × 104 and was found to be an order of magnitude smaller than the lift.

A method for measuring the interfacial structure between a co-current air-water flow using the absorption of light is described. Measurements of the root-meansquared displacement and the frequency spectrum are presented. The use of a Gaussian model to describe the interface is explored.

Induced magnetic fields in shock-tube flows have been measured by previous authors, in order to determine the electrical conductivity of ionized gases. Distortion of the applied magnetic field was taken to be small. In the present work, the full magnetic field solution is developed for idealized boundary conditions. At a magnetic Reynolds number equal to 4 the distortion is found to be considerable, but the field is well represented by a solution correct to third order in magnetic Reynolds number. The axial field in a real shock-tube configuration was computed, and experimentally measured fields were found to agree reasonably with theory.

The power specturm of a passive scalar contaminant undergoing a first-order chemical reaction and isotropic turbulent mixing is deduced for three different spectral ranges: (i) the inertial-convective range; (ii) the viscous-convective and viscous-diffusive ranges for very large Schmidt number; (iii) the inertial-diffusive range for very small Schmidt number. The analysis is restricted to stationary, locally isotropic fields, and to systems so dilute that the heat of reaction has no effect on the reaction rate.

The effects of heat-capacity lag on the flow over slender bodies are examined by means of an extension of Ward's (1949) generalized treatment of the slender-body problem. The results are valid for smooth bodies of arbitrary cross-sectional shape and attitude in the complete Mach number range up to, but not including, hypersonic conditions. Transonic flow can be treated owing to the presence of a dissipative mechanism in the basic differential equation, but the results in this Mach number range are probably of limited practical value.

The results show that cross-wind forces are unaffected to a first approximation, but that drag forces comparable with laminar skin-friction values can arise as a result of the relaxation of the internal degrees of freedom. The magnitude and sign of these effects depend strongly on body shape and free-stream Mach number.

Results are given for the surface pressure coefficient, and the variations of translational and internal mode temperature on and near the body are also found. The influence of these latter effects on heat transfer to the body is discussed.

Recently Kraichnan (1959) has propounded a theory of homogeneous turbulence, based on a novel perturbation method, that leads to closed equations for the velocity covariance. In the present paper, this method is applied to the theory of turbulent diffusion and closed equations are derived for the probability distributions of the positions of marked fluid elements released in a turbulent flow.

Two topics are discussed in detail. The first is the probability distribution, at time t, of the displacement of an element from its initial position. In homogeneous flows, this distribution is found to resemble that for classical diffusion but with a variable coefficient of diffusion which is proportional to $v^2_0 t$ for $t \ll l|v_0$ and which approaches a constant value [eDot ] lv0 for tt [Gt ] l/v0 (l = macroscale, v0 = r.m.s. turbulent velocity).

The second topic treated is the joint probability distribution of the displacements of two fluid elements. Particular attention is focused upon the probability distribution of relative displacement, i.e. Richardson's distance-neighbour function. This is found to be Gaussian for separations r which are large ([Gt ] l). For smaller separations (r [Lt ] l), its behaviour at high Reynolds numbers is found to be quite well expressed in terms of a variable diffusion coefficient K(r,t), as suggested by Richardson (1926). For all but extremely short times, K(r,t) is found to depend only on r and on the form of the inertial range spectrum E(k). On assuming $E(r) \propto v^2_0 l(kl)^{- \frac {3}{2}}$ as results from Kraichnan's approximation (1959), one finds $E(r) \propto v_0 l(r|l)^{ \frac {3}{2}}$. On the basis of similarity arguments of the Kolmogorov type, which give $E(r) \propto v^2_0 l(kl)^{- \frac {5}{2}}$, one finds $E(r) \propto v_0 l(r|l)^{ \frac {4}{3}}$ as, in fact, Richardson originally proposed. The dispersion r2 is proportional to $l^2(v_0 t|l)^4$ on Kraichnan's theory; while $\langle r^2 \rangle \propto l^2 (v_0 t|l)^3$ on the similarity theory. This illustrates that the behaviour of $\langle r^2 \rangle$ is very sensitive to the spectrum.

Incompressible slip flow past a semi-infinite flat plat at zero incidence is treated in terms of the linearized viscous flow equations. A formal solution is obtained using Fourier transforms and the Wiener-Hopf technique. Explicit inversion of the transform is not possible, but asymptotic expansions are discussed. These reveal the inadequacy of boundary-layer theory in predicting the nature of the solution, even at the plate surface. For example, the local shear forces on the plate are significantly different from boundary-layer values, even far downstream, where slip effects are small. The boundary-layer limit is approached as the Reynolds number based on the mean free path or, equivalently, the free-stream Mach number tends to infinity.

A theory is developed for the incompressible flow of a fluid with high electrical conductivity about thin cylinders (airfoils) in non-uniform motion. A uniform magnetic field is applied parallel to the free stream and solutions are obtained subject to the restriction of small perturbations. The effects of viscosity are included, for the most part, only through the use of the Kutta condition, where applicable, for lifting airfoils. The validity and range of applicability of the infinite-conductivity assumption are determined on the basis of an order-of-magnitude analysis; the general character of the flow is discussed at length.

The flow-field for infinite conductivity is changed from the non-magnetic case only through the new transport speed of vorticity; the forces on the airfoil are changed due to surface currents. For the case of the Alfvén speed less than the free-stream speed, the airfoil lift and pitching moment are given in integral form for general unsteady-airfoil motion and are given in closed form for harmonic ocsillations. The forces at moderate frequencies may be larger than in the corresponding non-magnetic case. The response to a unit-step change in the downwash is studied and the asymptotic form of the lift is obtained for small and large time.

For the case of the Alfvén speed greater than the free-stream speed, vorticity and current are shed from both the leading and trailing edges. Therefore the extension of the usual Kutta condition is not obvious. It is shown that if finite viscosity and / or conductivity tend to remove the trailing-edge singularity, the flow is unstable and no steady flow can be obtained.

1. The diffusion terms in the mean velocity and temperature equations of turbulent flow are analysed to decide when variations of fluid properties can produce appreciable errors.

2. A theoretical demonstration is given that in the mean-flow continuity equation for a gas the error in assuming constant density is small if the flow is turbulent, even when the temperature variations are large.

3. Separate discussion is given of the case of local heat sources in turbulence, as large errors can occur there.

The effects of density variation and body force on the stability of a heterogeneous horizontal shear layer are investigated. The density is assumed to decrease exponentially with height, and the body force is assumed to be derivable from a potential; the velocity distribution in the shear layer is taken to be U(y) = tanh y. The method of small disturbances is employed to obtain a family of neutral stability curves depending on the choice of the Richardson number. It is demonstrated, furthermore, that the value of the critical Richardson number depends on the magnitude of the non-dimensional density gradient.

Let D0 be the Stokes drag on an arbitrary body moving parallel to a principal axis of resistance, with velocity U, through an unbounded fluid. The Oseen drag, D, experienced by this same body moving with equal velocity and identical orientation through the unbounded fluid is then given by the expression $\frac {D}{D_0} = 1 + \frac {D_0}{16 \pi \mu c U} R + O(R^2)$ where c is any characteristic particle dimension and $R = cU \rho|\mu$ is the particle Reynolds number. An analogous expression is given for the case where the motion is not parallel axis. Finally, an expression is given for the Oseen resistance of an arbitrary falling parallel to a principal axis of resistance along the axis of a cylindrical tube of finite radius.

In turbulent flow past rigid boundaries, there can be distinguished regions close to the wall in which the local rates of energy production and dissipation are so large that aspects of the turbulent motion concerned with these processes are determined almost solely by the distribution of shear stress within the region and are independent of conditions outside it. These regions are here called equilibrium layers because of the equilibrium existing between local rates of energy production and dissipation. Three kinds of equilibrium layer have been studied experimentally, the constant-stress layer, the transpiration layer and the zero-stress layer, but there are other possible forms. One that is of importance in the theory of self-preserving flow in boundary layers and in diffusers is the ‘linear-stress’ layer in which the stress increases linearly with distance from the wall. The properties of these various equilibrium layers are considered and the distributions of mean velocity are derived from the equation for the turbulent kinetic energy and certain assumptions of flow similarity.

The theory of self-preserving wall flow, usually expressed as a combination of the law of the wall and the defect law, assumes compatibility between the outer flow and the equilibrium layer, and the course of development depends on the kind of equilibrium layer. Earlier work by the author, which assumed the defect law, is only valid if the whole of the equilibrium layer is a constant-stress layer and this is not true in strong adverse pressure gradients. A consistent theory is developed for these flows by assuming a ‘linear-stress’ layer, and the solutions show the relation between flows of finite stress and of zero stress and provide a plausible explanation of the phenomenon of downstream instability observed by Clauser. Self-preserving flow in wedges is treated on similar lines.

A solution is given for a viscous vortex in an infinite liquid. Similarity arguments lead to a reduction of the equations of motion to a set of ordinary differential equations. These are integrated numerically. A uniform feature is the constant circulation K outside the vortex core, which is also a viscous boundary layer. The circulation decreases monotonically towards the axis. The axial velocity profiles and the radial velocity profiles have several characteristic shapes, depending on the value of the non-dimensional momentum transfer M. The solution has a singular point on the axis of the vortex. The radius of the core increases linearly with distance along the axis from the singularity, and, at a given distance, is proportional to the coefficient of viscosity and inversely proportional to K.

Finally, a discussion is given to indicate that intense vortices above a plate, like the confined experimental vortex, or above the ground, like the atmospheric tornado and dust whirl, will not resemble the theoretical vortex except, possibly, far above the plate.

The flow about a spinning sphere moving in a viscous fluid is calculated for small values of the Reynolds number. With this solution the force and torque on the sphere are computed. It is found that in addition to the drag force determined by Stokes, the sphere experiences a force FL orthogonal to its direction of motion. This force is given by ${\bf F}_L = \pi a^3 \rho \Omega \times {\bf V}[1 + O(R)]$ .

Here a is the radius of the sphere, Ω is its angular velocity, V is its velocity, ρ is the fluid density and R is the Reynolds number, $R = \rho \mu ^{-1} Va$. For small values of R, the transverse force is independent of the viscosity μ. This force is in such a direction as to account for the curving of a pitched baseball, the long range of a spinning golf ball, etc. It is used as a basis for the discussion of the flow of a suspension of spheres through a tube.

The calculation involves the Stokes and Oseen expansions. A representation of solutions of the Oseen equations in terms of two scalar functions is also presented.

The flow of an inviscid, incompressible electrical conducting fluid in a channel of constant rectangular cross-section is considered, when the flow enters a region which contains a magnetic field transverse to the flow and electrodes on opposite sides of the channel. This geometry is typical of a d.c. induction pump or magnetohydrodynamic generator. The conducting fluid external to the magnetic field acts as a shunt and produces a non-uniform electric potential field and hence a non-uniform Lorenz force on the fluid, and causes the fluid velocity profile to be distorted. These effects are calculated theoretically for small magnetic Reynolds number and small magnetic interaction parameter. It is found that the velocity at the centre-line of the channel is retarded and at the walls the velocity is accelerated. The fractional change of velocity at the wall is equal to approximately 0·44 times a modified magnetic interaction parameter.

In this paper a study of the energy-transfer processes associated with the motion of a viscous, heat-conducting fluid is begun. The class of motions considered are unsteady, two-dimensional, vortical flows. After developing simplified equations of motion and energy appropriate to this type of flow in the low Mach-number limit, general solutions of the momentum equations are presented.

The concept of a line impulse of angular momentum is introduced as an example of this class of motions for which a solution of the energy field is obtainable in closed form. The solution for the line impulse can be viewed as a combination of velocity, pressure, and temperature waves concurrently radiating from the origin of the impulse and decaying with time. Particular examples of the development of the energy field of the impulse in both liquids and gases are presented for selected values of Prandtl number. The energy-transfer processes are discussed in some detail, and the resulting differences in the energy fields for liquid gases are emphasized.

The force on a sphere moving through an inviscid, conducting fluid in the presence of a uniform magnetic field B0 is calculated for the low-conductivity case where the hydrodynamic motion deviates only slightly from potential flow. The magnetic Reynolds number is assumed small. The force on the sphere is found to consist of both a drag and a deflective component which tends to orient its motion parallel to a magnetic field line; if the sphere's velocity is V, the force may be written $\bf {R} = -AB^2_0\bf {V} + \bf C(V.B_0)B_0$ where the coefficients A and C depend on the conductivities of both sphere and fluid. The coefficients are evaluated by calculating the Joule dissipation for particular orientations of V relative to B0. In one case the force is also calculated directly from the perturbed pressure distribution in the fluid. In an analogous way, a spinning sphere in a conducting fluid experiences both resistive and gyroscopic torques.

The effect of turbulence on an applied magnetic field is considered in the case when the magnetic Reynolds number Rm is large compared with unity but small compared with the ordinary Reynolds number R of the turbulence. When the applied field is sufficiently weak, it is argued that its effect on the velocity field is negligible. The equation for the field is then linear and its spectrum may be obtained throughout the equlibrium range of wave-numbers. It appears that the spectrum increases as $k^{\frac {1}{3}}$ up to a wave-number kc marking the threshold of conduction effects, and falls off as $k^{-\frac {11}{3}}$ beyond kc. The net effect of the turbulence is expressed in terms of an eddy conductivity equal to $R_m^{- \frac {5}{2}}$ times the electrical conductivity of the fluid. The effect of magnetic forces when these are not negligible is also tentatively considered.

The incompressible laminar boundary layer on a semi-infinite flat plate is considered, when the main stream has uniform shear. A solution is obtained for the first two terms of an asymptotic solution for small viscosity. It is shown that one of the principal effects of free-stream vorticity is to introduce a modified pressure field outside the boundary-layer region.

The shedding of vorticity from the edges of a plate, which is accelerated normal to itself from rest in still air, is investigated experimentally by means of a new flow-visualization technique using spark-lighted shadowgraphs and heated air, benzene or other vapours to introduce sufficient changes in density along the surface of discontinuity.

The photographs show small-scale undulations and the formation of a number of centres of vorticity along the well known big-scale vortex sheets which roll up along their edges.