A theory is developed to describe the properties of waves on the free surface of a liquid in turbulent motion. The distinction between the interacting wave and turbulent motions is achieved by separating the velocity field uniquely into a surface-induced contribution characteristic of the wave motion and a vorticity-induced contribution associated with the turbulence. The theory is applied to the scattering of gravity waves passing over the surface of deep water which has a turbulent motion of sufficiently low mean square vorticity. Expressions are derived for the directional distribution of the scattered wave (equation 4.19) and for the logarithmic decrement resulting from the scattering (4.20). It is shown that, under typical conditions in open sea, the attenuation from scattering will be greater than that from direct viscous dissipation for wavelengths greater than about 3 m.

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.

A recapitulation is first given of a recent theory of homogeneous turbulence based on the condition that the Fourier amplitudes of the velocity field be as randomly distributed as the dynamical equations permit. This theory involves the average infinitesimal-impulse-response functions of the Fourier amplitudes and employs a new kind of perturbation method which yields what are belived to be exact expansions of third- and higher-order statistical moments of the Fourier amplitudes in terms of second-order moments and these response functions.

In the present paper the theory is applied in lowest approximation (called the direct-interaction approximation) to stationary isotropic turbulence of very high Reynolds number. The characteristic wave-number $k_0 = \epsilon |v^3_0$ and Reynolds number $R_0 = v_0 k^{-1}_0 |v$ where v0 is the r.m.s. velocity in any given direction, ε is the power dissipated per unit mass, and ν is the kinematic viscosity, are introduced. For $R^{\frac {1}{3}}_0 \gg 1$ it is found that the inertial and dissipation ranges extend over wave-numbers k satisfying k0 [Lt ] k [Lt ] R0k0. The time-correlation and average infinitesimal-impulse-response functions of the Fourier amplitudes in these ranges are evaluated. They are found to be asymptotically identical and given by J1(2v0kτ)/v0kτ, where τ is the time interval.

The energy spectrum in these ranges is determined by a non-linear integral equation, involving the time-correlation and response functions, which is suitable for solution by iteration. The solution is of the form E(k)/v0v = (k/kd) $^{- \frac {3} {2}}$f(k/kd), where E(k) is the three-dimensional spectrum function $k_d = R^{\frac {2}{3}}_0 k_0$ is a wave-number characterizing the dissipation range, and f(k/kd) is a universal function. In the inertial range, E(k) = f(0) (ε V0)½$ k^{- \frac {3} {2}}$, asymptotically. Theparameter f(0) can be obtained by quadratures, without solving the integral equation for E(k). Spectral energy transport throughout the inertial and dissipation ranges is found to proceed by a cascade process essentially local in wave-number space; the direct power delivered by all modes below k to all modes above k′ [Gt ] k is of order ε(k/k′)$ ^{ \frac {3} {2}}$ if k and k′ both lie within the inertial range. The mean-square velocity derivatives of all orders are found to be finite. For $R^{\frac {1}{3}}_0 \gg 1$ the skewness factor of the distribution of the nth-order longitudinal velocity derivative is found to have the asymptotic form $A_n R^{-\frac {1}{6}}_0$ where An is a universal constant.

The theory is compared with experiment and is found to be slightly better supported than the Kolmogorov theory. However, it is stressed that extreme caution must be exercised in interpreting the experimental evidence as support for either theory.

An analysis is given of the relations between the Kolmogorov theory, Heisen-berg's heuristic theory, the analytical theories of Heisenberg and Chandrasekhar, the theories of Proudman & Reid and Tatsumi, and the present theory.

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 $C_p(xdC_p|dx)^{\frac {1}{2}} = 0 \cdot 39 (10^{-6}R)^{\frac {1}{10}},$ 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.

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.

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).

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.

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 steady rotational flow of an inviscid fluid in a two-dimensional channel or a circular tube toward a sink is treated. The velocity distribution at infinity is approximated by a cosine curve (which is nearly parabolic) for the two-dimensional case, and is taken as exactly parabolic for the axisymmetric case. The dependence of vorticity on stream-function is assumed to be everywhere the same as it is for streamlines coming from infinity upstream. The resulting linear equations of motion are solved exactly. The solutions show the rather unusual features of separating streamlines and regions of closed flow (corner eddies).

An examination was made of the laminar travelling waves that disturbed the glassy surface in an open water channel, fitted with a trumpet entry and inclined downwards over the range 1 to $2 \frac{3}{4}^{\circ}$. The Reynolds number at their first appearance was observed, and measurements were obtained of their velocities and lengths, the latter being highly irregular. Comparison was made with the theoretical values after corrections had been introduced to allow for the higher velocities of flow near the walls due to the increased depth caused by surface tension. At greater discharges the onset of turbulence occurred in a random manner, leading to the production of intermittent bores moving more slowly than the laminar stream. Transition was found to have usually begun at a Reynolds number R = 2500, R being defined as the discharge per unit width divided by the kinematic viscosity. Except close to the inlet, the channel was continuously occupied by turbulent water when R was in the range 4000 to 4500.

This paper gives the extension of the approximate theory developed in Part 1 (Whitham 1957) to three-dimensional problems. The basic equations are derived in §1, using the original assumption of a functional relation between the strength of the shock wave at any point and the area of the ray tube. An analogy with steady supersonic flow is found. For the diffraction of a plane shock wave by an obstacle, the equations and boundary conditions are exactly the same as those for steady supersonic potential flow past that obstacle, with a special choice of the density-speed relation. The successive positions of the shock wave are the equipotential surfaces of the supersonic flow. The 'shock-shocks’ introduced in Part 1, i.e. discontinuities in the slope and Mach number of the shock wave, correspond to the steady oblique shock waves in the supersonic flow problem. They arise when Mach reflexion occurs.

In §2 the theory is applied in detail to the diffraction of a plane shock wave by a cone. Then, in §3, a small perturbation theory is applied to the two typical problems of (i) diffraction by a slender axi-symmetrical body of general shape, and (ii) the stability of a plane shock. Many further applications would be possible and some brief comments on these are made in §4.

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.

The jet-stream in a rotating fluid is treated as a thermal boundary layer, but viscous effects are omitted from the first approximation. A theoretical justification for this treatment is presented, and a particular solution of the resulting equations is found. This solution is shown to give a reasonable picture of the flow in the neighbourhood of the stream far from solid boundaries.

A method has been developed for calculating the viscous effects of the hypersonic flow on the fore part of various bodies. This method takes into account the finite leading-edge thickness of the body and the detached shock wave. The calculated pressure distributions agree with all experimental results for both air and helium over a wide range of Mach numbers and Reynolds numbers. The calculation predicts skin frictions of the order of twice those predicted by ordinary boundary-layer theory.

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.

A pulsating sphere, which performs a sequence of virtually impulsive changes in its radius with time, is completely surrounded by an inviscid, incompressible fluid whose velocity field is generally rotational. This paper indicates how it is possible, by means of Helmholtz's theorem, to relate the corresponding vorticity and velocity fields immediately before and after such expansions or contractions.

The method is then applied to the case of a spherical mass of fluid initially in uniform rotation in which a spherical core undergoes a single sudden expansion, followed after a short interval by an equally rapid contraction back to the original radius. An interesting meridional flow is thereby induced, which tends to decrease the angular velocity of rotation of the fluid near the poles at the outer surface, relative to that of the equatorial fluid. It is perhaps significant that this is in qualitative agreement with the variation of angular velocities observed at the surface of the sun.

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.

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.

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.

The one-dimensional theory for the interaction between a normal shock wave and a discontinuous reduction in a area is applied to the flow in a two-dimensional channel. It is shown that, for a given overall area ratio, a channel shape consisting of two equal discontinuous reductions in area produces a larger gain in shock strength than a channel having a single discontinuity. The gain in shock strength continues to increase with the number of steps, until the ideal limiting case of an infinite number of vanishingly small changes in area is reached.

A drum camera was used to measure, in a shock tube, the increase in speed of normal shock waves passing through two-dimensional converging channels of different wall shape. It was found that a single discontinuous change produced a gain in shock strength which was less than half that given by the one-dimensional analysis. However, the gain increased as the area change was made more gradual and for long smooth nozzles the value for the ideal case was nearly attained.