The behaviour of weak homogeneous turbulence subjected to a uniform distortion is studied by means of linearized equations that include the effect of inertial interaction between the mean flow and the turbulent fluctuations and of viscous dissipation. Three particular cases of distortion are investigated in varying degrees of detail; those of uniform rotation, uniform shear, and uniform irrotational distortion. For the first two, the total energy associated with the turbulence is found to decay, but for the third it is found in general to increase without limit. A general solution in terms of the spectrum functions for the restricted case of irrotational distortions is given, particular consideration being paid to the asymptotic limits approached as the distortions (not necessarily the rates of distortion) become large. These show that for an arbitrary initial turbulent spectrum the asymptotic rates of growth are functions of the precise geometrical nature of the distortion, of an integral parameter of the initial spectrum function, of a Reynolds number based on the rate of distortion and the length scale of the initial turbulence, and of the contraction. A particular example, that of initially isotropic turbulence in its final period of decay subjected to an axisymmetric distortion, is worked out in detail and te solution (described in terms of mean turbulent intensities) displayed graphically for several values of the Reynolds number.

In the laboratory experiments by Fultz & Riehl (1957) and by Hide (1958) on heated rotating liquids contained in the annulus between two cylinders, it has been observed that a strongly marked jet stream appears on the free surface of the liquid under certain conditions of rotating and heating. This jet stream meanders around the annulus in a regular wave-like pattern alternately approaching the outer and inner cylindrical boundaries. The present paper puts forward an analytical theory for this jet, or Rossby regime, in the course of which exact solutions are presented of certain fundamental non-linear partial differential equations. The assumption is made that the flow is geostrophic at the first approximation and that the heat transfer across the stream lines of geostrophic flow (that is, the isobars) is due to molecular conduction. From a calculation of the heat flow it appears that this leads to values of the heat transfer which are too small, so that the ageostrophic terms must be of importance in the actual heat transfer; never-theless, the exact solution obtained here probably reveals the mechanism of the change from one wave pattern to another and certainly provides an explanation for the observed upper limit to the number of waves in a given geometrical configuration, as discussed by Hide. It has been established that the mean zonal flow and mean zonal temperature field are dependent upon the amplitude function of a finite amplitude wave solution. In this exact solution it is found that the amplitude and phase functions of the wave patterns are themselves interdependent and that the shape of the wave depends on the quantity of heat and angular momentum being transferred. It is shown that the wave pattern consisting of an integral number m lobes or petals can exist only in a restricted range of the Rossby number S—this is well-known from the experimental work of Fultz and Hide.

If an electric current of uniform density j0 is passed axially through a stationary fluid between concentric cylinders of radii r1 and r2 (> 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 current J along 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 of J needed to stabilize the fluid when the fluid has non-zero viscosity and finite conductivity are calculated for the case r2-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 if J and j0 have the same sign, and if J is not very small compared with πj0r2.

In this paper we discuss the structure of those oblique shocks which occur in plane hydromagnetic flow. We find that not all pairs of states satisfying the shock conditions can be linked by a one-dimensional flow of a viscous, electrically conducting fluid, while other pairs can be linked in more than one way. In the latter case a particular transition is singled out when further information is given concerning the three-dimensional problem of which the plane motion is an idealization.

As a primary step towards understanding the flow of a partially ionized gas in a magnetic field, we have studied both theoretically and experimentally the problem in which the gas flow is one-dimensional. This simplification permits a detailed calculation of the flow field and a quantitative comparison of the theory with observations made in a shock tube.

An ionized gas is composed of three species: electrons, ions and neutral particles. To take complete account of all the phenomena occurring when the high-velocity gas interacts with the magnetic field, the motion of all three species must be considered. When this is done, it is found that the electrical conductivity of the gas is a tensor dependent on both the magnitude and geometry of the magnetic field. However, when the collision frequency for the electrons is greater than their cyclotron frequency in a magnetic field, the gas may be treated as a continuum with a scalar conductivity. For such gas states, all of the observed effects for two experimental geometries in which the gas current forms closed loops in the magnetic field can be explained with a simple theory. The interaction produces a flow completely analogous to pipe flow with friction and no heat transfer, where the wall friction force is replaced by the magnetic body force which can choke the flow if the body forces are larger than a certain minimum value.

Using the same experimental geometries, the gas state is then adjusted so that the electrical conductivity is a tensor. Outstanding among the observed effects are ion slip, where the ions and neutrals travel through the field at different velocities, and Hall currents, generated by the drift of the charged particles across magnetic field lines. The observed effects again agree with the predicted values.

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.

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.

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.

The stability of infinitesimal axially symmetric disturbances in fully developed pipe flow is examined anew. The classical eigenvalue problem is treated in part by asymptotic methods and leads to an algebraic relation between the eigenvalue c, the disturbance wavelength 2π/α, and the Reynolds number. Examination of the limiting cases of this relation reveals the existence of two families of characteristic numbers, the value of which tends to unity and to zero as the Reynolds number increases without bounds. For the latter, a more accurate solution is required and given. It is found that all eigenvalues yield stable solutions and that for a given wave number and Reynolds number only a finite number of eigenvalues exists.

The limitations of the analysis are discussed in the light of a recent experimental study of the same problem.

The equation of motion is derived for the steady flow under gravity of a liquid in a uniform channel, including the effects of the friction of the bed and of the eddy viscosity for lateral mixing. The equation is solved to find the distribution of mean velocity across the channel when the cross-section is rectangular, triangular or trapezoidal. Numerical values are also given to indicate the extent to which eddy viscosity may affect the lateral distribution of mean velocity.

This paper deals with the propagation of a sound pulse into a gas which initially has solid-body rotation and constant temperature, the initial pressure and density increasing outwards like ex, where x is the square of a certain dimensionless radial co-ordinate. The perturbations are due to a source-like disturbance on the axis of symmetry, which begins to act at time t = 0: most attention is paid to source strengths which vary in time like a Dirac pulse or a step function, but the following remarks apply generally.

Immediately behind the wave front the perturbation velocity and temperature decay like e-½x, while the (absolute) perturbation pressure and density grow like e½x (the relative pressure and density increments, which are referred to local conditions in the undisturbed state, then also decay like e-½x). The rotation also introduces oscillations in flows which, with the same disturbance at the origin and no rotation, would vary monotonically with time at a given point.

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.

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.

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.

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 $ n^{- \frac {5} {3}}$ (n being wave-number) at the low wave-number end of the equilibrium range, but there has been some confusion about the wave-number marking the upper end of the range of validity of this relation. The existence of a conduction ‘cut-off’ near $ n = \isin{\sol}k {^3}) {^\frac {1} {4}}$ as put forward by Obukhoff and Corrsin is shown to hold only when ν [Lt ] κ, and that near n = (ε/νκ2) $ ^{\frac {1} {4}}$ put forward by Batchelor is shown to apply only when ν [Gt ] κ. In the case ν [Lt ] κ, the remaining problem is to determine the form of the spectrum of θ beyond the conduction cut-off; this is done in Part 2. In the case ν [Gt ] κ, the conduction cut-off occurs at wave-numbers much higher than (ε/ν3)$ ^{\frac {1} {4}}$, which is where the energy spectrum is cut off by viscosity, and where the spectrum of θ ceases to vary as $ n^{- \frac {5} {3}}$.

The form of the spectrum of θ in this latter case is determined over the range n > (ε/ν3)$ ^{\frac {1} {4}}$ by analysing the effect of the velocity field, regarded as effectively a persistent uniform straining motion for these small-scale variations of θ, and of molecular diffusion on a single Fourier component of θ. The wave-number of this sinusoidal variation of θ is changed (and generally increased in magnitude) by the straining motion and the amplitude is diminished by diffusion. By supposing that the level of the spectrum of θ is kept steady at wave-numbers near (ε/ν3)$ ^{\frac {1} {4}}$ by some mechanism of transfer from lower wave-numbers, the linearity of the equation for θ then allows the determination of the spectrum for n > (ε/ν3) $ ^{\frac {1} {4}}$, 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.

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 (ε/κ3)¼. It is shown, with a plausible and consistent hypothesis, that the convective supply of $\overline {\theta^2}$-stuff to Fourier components of θ with wave-numbers n in the range (ε/κ3)¼ [Lt ] n [Gt ] (ε/ν3)¼ is due primarily to motion on a length scale of order n-1 acting on a uniform gradient of θ of magnitude $[(\overline {\nabla \theta)^2}]^{\frac {1}{2}}$. The consequent form of the theta;-spectrum within this same wave-number range is $\Gamma (n) = \frac {1}{3}C \chi \epsilon ^{\frac {2}{3}} k^{-3}n^ {-\frac {17} {3}}.$

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.