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.

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).

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.

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.

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.

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.

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.

Solution of the compressible Rayleigh problem involving an impulsively moved flat plate indicates that transverse velocities can be generated by a purely longitudinal shearing motion. It would be desirable to demonstrate the existence of these waves, and to do so was the motivation for the present investigation. However, the geometry for the Rayleigh problem is prohibitive; and since it was noticed that a similar behaviour should occur for the case of unsteady compressible Couette flow, this was considered for the present investigation. An experimental search for the existence of the transverse velocities was conducted using concentric rotating cylinders to generate the flow. Since these transverse velocities were too small to be measured directly, a temperature gradient was created across the annulus to take advantage of the much greater sensitivity of a hot-wire to temperature fluctuations produced by the fluctuation in the transverse velocity. The periodic temperature fluctuations noted were converted to equivalent transverse velocities. Experimental values obtained agreed qualitatively with the theory. Finally, the role of extraneous effects in the experimental apparatus was considered.

This paper describes an investigation of the turbulent forced plumes generated by steady release of mass, momentum and buoyancy from a source situated in an extensive region of uniform or stably stratified fluid. The treatment, which is an extension of earlier work on buoyant plumes, also brings out the relationship between the jet and the Plume as special cases of forced plumes.

The analysis shows that the behaviour of a forced plume from a source of finite size which delivers buoyancy, mass and momentum can in a uniform environment be related to that from a virtual point source of buoyancy and momentum only, and a treatment is given for the latter type of forced plume. When the environment is weakly stratified it is inferred that forced plumes can be related to point definitely; but when the environment is stably stratified, increasing the release of mass and momentum from a given source of buoyancy has the effect at first of reducing the total height of the plume, and only for very large flux of momentum does the height increase again, without limit. A description is given for the behaviour of vertical jets in a stably stratified environment, and for forced plumes of fluid with negative buoyancy.

By using methods well known in boundary layer theory, the pressure across a contact region is shown to be approximately constant. A partial differential equation for the temperature is then derived. If the ideal-gas flow external to the contact region is known, the temperature profile can be determined. This can then be used to calculate successively the velocity and a better approximation for the pressure of the gas in the contact region. The theory is illustrated by obtaining the temperature, velocity and pressure distributions for a gas in a contact region moving with uniform velocity. The thermal conductivity of the gas is assumed to vary with the temperature τ like k = knτn, where n = 0, 1 or 2. The results are valid for any temperature ratio across the region.

The general theory is also used to determine the motion of a plane shock which is reflected from a plane-conducting wall. The fluid between the reflected shock and the wall is at a higher temperature than that of the wall and a contact region adjacent to the wall results. Expressions for the temperature, velocity and pressure of the fluid are derived, and it is shown that the effect of heat conduction is to decrease the velocity of the reflected shock by an amount which varies as the inverse square root of the time.