The purpose of this paper is to study the basic principle of fish propulsion. As a simplified model, the two-dimensional potential flow over a waving plate of finite chord is treated. The solid plate, assumed to be flexible and thin, is capable of performing the motion which consists of a progressing wave of given wavelength and phase velocity along the chord, the envelope of the wave train being an arbitrary function of the distance from the leading edge. The problem is solved by applying the general theory for oscillating deformable airfoils. The thrust, power required, and the energy imparted to the wake are calculated, and the propulsive efficiency is also evaluated. As a numerical example, the waving motion with linearly varying amplitude is carried out in detail. Finally, the basic mechanism of swimming is elucidated by applying the principle of action and reaction.

This paper presents experimental investigations of the diffraction of plane strong shocks by several cones, a cylinder, and a sphere. The diffraction pattern, in particular the loci of Mach triple points and the shape of the diffracted shocks are compared with theoretical results obtained from a diffraction theory proposed by Whitham (1957, 1958, 1959). The agreement between theory and experiment is shown to be good. Also given are extensive numerical results supplementing Whitham's papers, and theortical considerations applying Whitham's theory to very blunt bodies.

Measurements of the amount of fluid left behind when a viscous liquid is blown from an open-ended tube are described.

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.

A long bubble of a fluid of negligible viscosity is moving steadily in a tube filled with liquid of viscosity μ at small Reynolds number, the interfacial tension being σ. The angle of contact at the wall is zero. Two related problems are treated here.

In the first the tube radius r is so small that gravitational effects are negligible, and theory shows that the speed U of the bubble exceeds the average speed of the fluid in the tube by an amount UW, where $W \simeq 1\cdot 29(3 \mu U|\sigma)^{\frac {2}{3}}\;\;\; as\;\;\; \mu U|\sigma$ (This result is in error by no more than 10% provided $\mu U |\sigma \; \textless \;5 \times 10^{-3}\rightarrow 0$). The pressure drop, P, across such a bubble is given by $P \simeq 3\cdot 58(3\mu U|\sigma)^{\frac {2}{3}}\sigma|r \; \; \;as\; \; \; \mu U|\sigma \rightarrow 0$ and W is uniquely determined by conditions near the leading meniscus. The interface near the rear meniscus has a wave-like appearance. This provides a partial theory of the indicator bubble commonly used to measure liquid flowrates in capillaries. A similar theory is applicable to the two-dimensional motion round a meniscus between two parallel plates. Experimental results given here for the value of W agree well neither with theory nor with previous experiments by other workers. No explanation is given for the discrepancies.

In the second problem the tube is wider, vertical, and sealed at one end. The bubble now moves under the effect of gravity, but it is shown that it will not rise at all if $\rho gr^2| \sigma \; \textless \; 0 \cdot 842,$ where ρ is the difference in density between the fluids inside and outside the bubble. If $0 \cdot 842 \; \textless \; 1 \cdot 04,$ then $\rho gr^2| \sigma - 0 \cdot842 \simeq 1 \cdot 25 (\mu U|\sigma)^{\frac {2}{9}} + 2 \cdot 24(\mu U|\sigma)^{\frac {1}{3}},$ accurate to within 10%. Experiments are adduced in support of these results, though there is disagreement with previous work.

Measurements on a large circular cylinder in a pressurized wind tunnel at Reynolds numbers from 106 to 107 reveal a high Reynolds number transition in which the drag coefficient increases from its low supercritical value to a value 0.7 at R = 3.5 × 106 and then becomes constant. Also, for R > 3.5 × 106, definite vortex shedding occurs, with Strouhal number 0.27.

The Stokes flow problem is considered here for the case in which an axially symmetric body is uniformly rotating about its axis of symmetry. Analytic solutions are presented for the heretofore unsolved cases of a spindle, a torus, a lens, and various special configurations of a lens. Formulas are derived for the angular velocity of the flow field and for the couple experienced by the body in each case.

A dispersion relation is derived for plane space-charge waves of finite amplitude in a plasma containing two oppositely charged streams of particles. The relation may be expressed simply in parametric form; it can also be quite well approximated, in two different ranges of amplitude, by expressions for the wavelength in terms of the maximum variation of the electrostatic potential.

The physics of shock-waves with vibrational relaxation regions is recapitulated, and it is shown that exact methods of analysis can be developed from the classical Rayleigh-line equations by treating the real gas as an ideal gas with heat transfer. By using these methods to analyse experimental records of density distributions in relaxation regions, a large number of local values of the relaxation frequency, rather than a single over-all value, may be obtained from each shock-wave record.

It is shown to be probable that a large proportion of the drag exerted by a water surface on the wind is in the form of wave drag. As a result the usual relation between the wind profile and the surface stress and roughness length are modified. In particular, close to the surface the relation between the transport of momentum and that of heat and water vapour are different from that obtaining over a rough solid surface.

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 flatness factor F of the signal transmitted through a band-pass filter has been measured for the turbulence in a free shear layer and for a squared Gaussian noise. They both show flatness factor increasing with centre frequency fc. In the turbulence, band-passed signals look intermittent and have larger F than the full signal, but in the squared noise, band-passed signals all have smaller F than the full signal although they look more intermittent.

It is shown analytically that the derivative of a smoothed, squared Gaussian noise may have flatness factor either greater or less than the undifferentiated signal.

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 transient effects generated when a shock wave is suddenly disturbed by a field of cellular vortices have been studied. Both the pressure disturbance on the shock and the local shock velocity are found to be strong functions of the cell geometry. Disturbances are resolved into transient components and sinusoidal components of constant amplitude. The transients are found to die out as t−3/2, t being the interaction time, except for one particular case of the cell geometry for which they diminish as t−1/2. Furthermore, the analysis indicates that the initial magnitude of the transient components may be quite appreciable in comparison with the sinusoidal component. The theory is extended to treat the convection through the shock of a single column of vortex cells.

The validity of various solutions for the vibrational relaxation region in shock-waves, and of the assumptions on which they are based, has been assesed by comparison with an exact solution obtained by numerical integration of the relaxation equation, and also by use of the Rayleigh-line equations. Estimates of errors in the values of the relaxation frequency, determined by means of these solutions, are given.

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.

Three-dimensional periodic oscillations in the shear flow region between two parallel streams are considered up to that second order of the oscillation amplitude. It is shown that, as an integral part of the oscillation, there is a mean secondary flow in the nature of a longitudinal vortex. Despite the dissimilarity in the profile of the basic flows, several of the principal features of the calculated results can be compared with those observed for the Blasius flow by Schubauer and Klebanoff & Tidstrom at the National Bureau of Standards.

In an earlier contribution (Francis 1956), a report was made of the speed of small solid bodies floating in the upper layers of a free surface stream. The bodies were cylinders of several diameters floating with their axes vertical and ballasted to the same draft. All the floats travelled at the same speed in the long, wide stream which was of constant depth and was strongly sheared, and this speed was the mean speed of the upper layers of the stream. It was pointed out that this result is unexpected, since the solid boundaries of the floats prevent turbulent interchange there, so that the floats are not locked to the stream in the same way as the corresponding volume of turbulent water would be locked. Thus the floats would be expected to travel a little faster than the stream, large diameter floats travelling faster than small floats.

Several interesting features of the flow field in free-molecule flow through an orifice are discussed. An estimate is then made of the deviation of the mass flow $\dot{m}$ through the orifice from its limiting free-molecule value $\dot{m}$ for small departures from the limit. Using an iteration method proposed by Willis, it is shown that this deviation is of the first order in ε, the inverse Knudsen number, defined as the ratio of the radius of the hole to the mean free path in the gas at upstream infinity. An estimate of the coefficient is obtained making some reasonable assumptions about the three-dimensional nature of the flow, and the value so derived, giving $\dot{m}=\dot{m}(1+0.25\epsi)$, shows fair agreement with the measurements of Liepmann. It appears that ‘nearly’ free-molecular conditions prevail up to ε ∼ 1.0.