With the aid of a hot-wire anemometer, the mean velocity distribution of the incompressible, turbulent, plane wall-jet has been examined in some detail. As previously reported, this fully developed boundary layer belongs to the class of self-preserving flows. Over the entire range of experimental conditions, a single velocity scale $(\overline {U}_m)$ and a single length scale (δ) seem to correlate all the velocity data. Further, it is shown theoretically that δ must vary as x, and $\overline {U}_m$ must vary as xa over this range. The exponent a has been empirically determined as -0.555. Values of the Reynolds shear stress, which are proportional to $\overline {u} \overline {v}$, the Boussinesq exchange coefficient ε, and the shear stress at the wall have been evaluated. The Reynolds number based on the maximum velocity and the thickness of the boundary layer varied from 22,000 to 106,000.

Small perturbations of a parallel shear flow U(y) in an inviscid, incompressible fluid of variable density ρ0(y) are considered. It is deduced that dynamic instability of statically stable flows ($\rho ^{\prime}_0 (y)\; \textless \; 0$) cannot be other than exponential, in consequence of which it suffices to consider spatially periodic, travelling waves. The general solution of the resulting differential equation is considered in some detail, with special emphasis on the Reynolds stress that transfers energy from the mean flow to the travelling wave. It is proved (as originally conjectured by G. I. Taylor) that sufficient conditions for stability are $U^{\prime}(y) \not= 0$ and $J(y)\; \textgreater \frac {1} {4}$ throughout the flow, where $J(y) = -g \rho^{\prime}_0(y)|\rho (y)U^{\prime 2}(y)$ is the local Richardson number. It also is pointed out that the kinetic energy of a normal mode in an ideal fluid may be infinite if $0 \; \textless \; J(y_c) \; \textless \; \frac {1}{4}$, where $U(y_c)$ is the wave speed.

The theorem X established by Miles in the preceding paper is here given a simpler and more general proof. Some further theoretical results concerning the stability of heterogeneous shear flows are also presented, in particular a demonstration that the complex wave velocity of any unstable mode must lie in a certain semicircle.

The general equations of motion are developed for a compressible, inviscid flow in which a non-uniform distribution of heat transfer is applied to the fluid or a non-uniform generation of heat per unit volume occurs. In general, vorticity can be created if the cross-products of the temperature and entropy gradients are finite. If the temperature gradients in the flow are small (first-order), then a non-uniform heat addition across the stream will produce a second-order change in vorticity. For this type of flow, solutions are obtained for the variations of velocity and density that occur in a two-dimensional plane flow and an axially symmetric three-dimensional flow. A simple expression is also obtained for the streamline displacement caused by the non-uniform addition of heat.

The subject of this note is the behaviour of three-dimensional small disturbances to plane parallel flows, which have a variation in the direction normal to the plane of mean flow, in relation to two-dimensional disturbances which vary in the plane of mean flow only. It was pointed out by Squire (1933) that, in linearized theory, the disturbancewhichisneutrallystable at the criticalReynolds number R, is two-dimensional in form. More recently interest has turned to the question as to which kind of disturbance is most rapidly amplified at a given Reynolds number above the critical. Jungclaus (1957) pointed out that for certain values of R and of the resolved wavelength in the plane of mean flow, three-dimensional disturbances may be more unstable than plane ones. Recently, Watson (1960) has shown further that a two-dimensional disturbance is the one most rapidly amplified in a certain range of R starting from the critical. In thi8 note we take a slightly different view of the problem which enables us to define specifically the upper end of this range of R, when it exists.

The common assumption that the energy of waves on a non-uniform current U is propagated with a velocity (U + c) where cg is the group-velocity, and that no further interaction takes place, is shown in this paper to be incorrect. In fact the current does additional work on the waves at a rate γijSij where γij is the symmetric rate-of-strain tensor associated with the current, and Sij is the radiation stress tensor introduced earlier (Longuet-Higgins & Stewart 1960).

In the present paper we first obtain an asymptotic solution for the combined velocity potential in the simple case (1) when the non-uniform current U is in the direction of wave propagation and the horizontal variation of U is compensated by a vertical upwelling from below. The change in wave amplitude is shown to be such as would be found by inclusion of the radiation stress term.

In a second example (2) the current on the x-axis is assumed to be as in (1), but the horizontal variation in U is compensated by a small horizontal inflow from the sides. It is found that in that case the wave amplitude is also affected by the horizontal advection of wave energy from the sides.

From cases (1) and (2) the general law of interaction between short waves and non-uniform currents is inferred. This is then applied to a third example (3) when waves encounter a current with vertical axis of shear, at an oblique angle. The change in wave amplitude is shown to differ somewhat from the previously accepted value.

The conclusion that non-linear interactions affect the amplification of the waves has some bearing on the theoretical efficiency of hydraulic and pneumatic breakwaters.

Sphere drag has been measured in a low-density supersonic wind tunnel by a simple displacement technique. A conical nozzle of fixed geometry, operated at a constant supply temperature of about 300 °K was used. Test section Mach numbers ranged between 3.8 and 4.3 depending on supply pressure. The Reynolds number and Knudsen number range, based on free-stream conditions and sphere diameter, were 50 < Re∞ < 1000 and 0·106 < Kn∞ < 0·006, respectively. This range was achieved by varying sphere size and supply pressure. The drag coefficient was found to increase from a value of CD = 1·17 at Re∞ = 1000 to CD = 1·73 at Re∞ = 50.

The motion of drops of liquid through an immiscible field liquid has been observed and drag coefficients have been measured at Reynolds numbers from 0.01 to 1500. Several kinds of motion have been distinguished and it has been noticed that a change in the motion is always preceded by an increase in the drag coefficient and is usually followed by a return to the drag coefficient for solid spheres. When the drop is deformed, the drag coefficient and the form of the motion is not determined solely by the Reynolds number. The effect of the walls of a containing vessel on the motion is computed for low Reynolds numbers.

Simple ideas of dimensional analysis and of limiting cases are used to elucidate the stability characteristics of a steady basic parallel flow of a viscous incompressible fluid. The principal result is that the stability characteristics of a smoothly varying velocity profile for wave disturbances of small wave-number can be found by use of a discontinuous velocity profile. The boundary conditions for a disturbance at a discontinuity of the basic flow are derived, and are used to find the stability characteristics of broken-line representations of the hal-jet and jet. These findings are in agreement with previous ones.

Scorer (1950) has described an extension of the method of stationary phase. This method is used to derive an approximation to the wave pattern at large distances from the ship in a simple manner. The approximation is valid in the vicinity of the critical angle and the nature of the disturbance in this region is readily seen. The divergent and transverse wave systems in a critical region are shown, together with their variation of amplitude and phase.

This paper is a study of the flow of a viscous incompressible fluid in the immediate neighbourhood of a saddle point of attachment, near which the external flow is irrotational with components {ax, by, −(a + b)z}, where a > 0 > b. It is shown that the flow is of a boundary-layer character, and that part of the boundary-layer flow is reversed when b/a < − 0·4294.

On the assumption that such flows are physically plausible, the problem may be solved for all values of b/a [ges ] −1. Even in the limiting case b/a = −1, an effect of the boundary layer is everywhere to draw fluid towards the wall, so that vorticity is still convected towards the wall.

Numerical solutions have been computed, and some of the results are presented in the tables and diagrams.

By expanding the dependent variables in series of orthogonal functions on the one hand, and expanding the coefficients of these functions in power series of a parameter η on the other hand, a solution has been obtained for the system of non-linear equations of cellular convection. The expansion parameter η is chosen in such a way as to make it remain less than 1 for all finite values of the Rayleigh number. The solution so obtained is found to be valid for a large range of the imposed temperature difference, and converges rapidly. This solution provides a quantitative theory for the convective heat transport as a function of the temperature difference in the range of laminar flow.

The solution also reveals that when the actual Rayleigh number is greater than twice the critical Rayleigh number, a layer of isothermal (adiabatic lapserate in a gas medium) mean temperature develops in the middle of the fluid layer. The thickness of this layer increases as the actual Rayleigh number increases, and at the same time the temperature gradient increases in the boundary layer so that an increase in the heat transport is accomplished.

The solution reveals further that the large temperture gradients are concentrated in the region where the cold descending current approaches the lower boundary and where the warm ascending current approaches the upper boundary. It is also shown that these ascending and descending currents spread out in mushroom-like patterns, a feature characteristic of the convection of isolated hot bubbles, but one which never has been considered as the form for finite cellular convection. Recent optical observations indicate that this is the most common form of the temperature field.

The heat transport given by this solution fits a power law of exponent 1.24, which is very close to the observed power law of exponent 1.25 for laminar flow.