It is found that only a small change in either of the undisturbed velocity profiles concerned is required to change them from stable profiles to unstable profiles. The change must be such as to produce a local maximum in the magnitude of the vorticity, or in the case of the pipe, in the magnitude of the vorticity divided by the radius. The actual change in the vorticity (or vorticity/radius) need only be small, but the gradient of the vorticity (or vorticity/radius) must be finite. Viscosity will tend to damp out the distortion in the mean flow that is responsible for the instability, so that if the flow is to become turbulent, non-linear effects must become important before the distortion of the mean flow is reduced to an ineffective level. This requirement leads to the determination of critical Reynolds numbers which depend on the initial (small) distortion of the mean flow and the initial (smaller) amplitude of periodic disturbances. These critical Reynolds numbers are large.

The adequacy of direct one-step chemical kinetics for describing ignition and extinction in initially unmixed gases is studied through the particular case of inviscid axisymmetric stagnation-point flow. Oxidant is assumed to blow from upstream infinity at a non-gaseous reservoir of pure fuel at its boiling (or sublimating) temperature. Before reaching the reservoir the oxidant reacts with gaseous fuel flowing in the opposite direction to form product and release heat. This heat is in part conducted and diffused to the reservoir interface to transform more fuel into the gaseous state and continue the steady-state burning. Second-order Arrhenius kinetics for Lewis-number unity is examined. A critical parameter characterizing the phenomenon is shown to be the first Damkohler similarity group D1, the ratio of a time characterizing the flow to a time characterizing the chemical activity.

For small D1 the reactants convect away heat without releasing the energy stored in their chemical bonds. Regular perturbation about chemically frozen flow establishes this condition as the weak burning limit. For large D1 singular perturbation describes a narrow region of intense chemical activity. For infinite D1 (indefinitely fast rate of reaction) the region is reduced to a surface of discontinuity (the thin-flame kinetics of Burke & Schumann).

For intermediate D1 numerical techniques establish that a solution describing burning of moderate intensity joins the two previously mentioned asymptotic limits. It is suggested that sudden transition of the system between the various branches in this domain of intermediate D1 accounts for the phenomena of ignition and extinction of burning.

The fluid motion associated with puffs (strongly turbulent masses of fluid moving through surroundings with which they mix readily) is considered, for cases in which any buoyancy force acts in the direction of gross motion, the surroundings are unstratified, the internal and external densities are approximately equal, and the motion of the surroundings is directly associated with the motion of the puff. It is assumed that the size of any one such puff is directly proportional to the distance travelled, i.e. that r = z/n.

It is shown that d{ρra(dz/dt)}/dt = CaMag, where t is time, ρ is the density of the surrounding fluid, g is the gravitational acceleration, Ca is a numerical constant, and a = 3 or 2 in the cases of axial or planar mean symmetry. M3 is the mass excess, and M2 is the mass excess per unit length.

Previous experiments with buoyant puffs having zero initial momentum (i.e. experiments with thermals) confirm these equations and give the values C2 [bumpe ] 0·33 and C3 = 0·27. New experiments with non-buoyant puffs having considerable initial momentum also confirm the equations, with the same values for C2 and C3. These results support the view that the turbulence inside thermals is primarily maintained through their mean motion rather than directly by gravitational instability.

Theoretical studies of aerodynamic noise suggest that the sound field of supersonic flows will be dominated by eddy Mach waves. Recent experimental evidence supports this view. In supersonic turbulent boundary layers, and rocket exhaust flows, turbulence occurs in regions of high mean velocity gradient. At low speed, such gradients are known to amplify the sound emitted by turbulence. This paper deals with the corresponding Mach wave problem. The exact equations of sound radiation by turbulence are rearranged in a form where the equivalent sources, derivatives of the turbulence stress tensor, are shown to be dominated by one term. That term is formed from the product of the mean velocity gradient and the rate of change of density. It seems that its resemblance to the dominant source of sound in low speed shear flows is largely fortuitous. In the Mach wave case, the theory is designed to include effects of both temperature gradients and density perturbations, and the approximations of the estimate are of a type that would not be expected to be valid away from the Mach wave condition. The basic theory is used to make an estimate of the sound radiated from supersonic boundary layers, and an approximate equation relating the radiated pressure to the surface pressure is derived. Experimental evidence is then examined to show that the equation is in excellent agreement with observation. The theory is then applied to annular shear flows of the rocket exhaust type. Again an approximate equation relating near and far field pressures is established, and the paper concludes with suggestions for experiments that could check the result.

The values reported by Willmarth & Wooldridge (1962) for the root-mean-square wall pressure and the power spectra of the wall pressure are in error. G. M. Corcos and M. K. Bull pointed out to me, in April 1963, that the area beneath the power spectra of figure 4 does not agree with the values of $\sqrt {(p^2)}| q \infty$ and $\sqrt {(p^2)}| \tau_w$ reported in table 3 and in §4·1. The measurements for the wall pressure at high speed, U∞ = 206ft./sec, were fortunately recorded on magnetic tape. It was therefore possible to check the measurements for that case.

This paper deals with the calculation of the convective heat transfer rate to the end-wall of a shock tube from a monatomic gas heated by a reflected shock. We consider a range of shock strengths for which the equilibrium thermodynamic state is one of appreciable ionization. The resulting boundary-layer problem involves the thermal conductivity and ambipolar diffusion coefficient for a partially ionized monatomic gas. The formulation here is restricted to the case of a catalytic wall and equal temperatures for all species. We ignore the effect of the plasma sheath at the wall. Consideration is given to three limiting cases for which similarity-type solutions of the boundary-layer equations may be found: (1) complete thermodynamic equilibrium behind the reflected shock and within the boundary layer; (2) equilibrium behind the reflected shock, but no gas-phase recombination in the boundary layer; (3) no ionization in either region. Numerical calculations are carried out for argon using estimated values of thermal conductivity and ambipolar diffusion, and compared with shock-tube experiments of Camac & Feinberg (1965). For no ionization, calculations were made with thermal conductivity varying as the ¾ power of the temperature, which fits the estimates of Amdur & Mason (1958) up to 15,000°K. Excellent agreement with experiment is obtained confirming an extrapolation of this power law up to 75,000°K. For ionized cases, based on estimates of Fay (1964), the theory predicts heating rates 20–40% lower than measured values. Some possible reasons for this discrepancy are discussed.

The analysis presented here is a second-order analysis of the resonant interactions triads of waves with wavelengths in the capillary-gravity and pure capillary ranges. The analysis is not a power series perturbation analysis, and one of the objects is the removal of the secularity which arises through power series perturbations. It is further found that the interactions are energy-conserving to the order considered here. Suitable modifications are made to accommodate the inevitable viscous attenuation associated with these small wavelengths. A start is made toward describing more completely the various spectra of random seas in wave-number and frequency regions where these interactions are dynamically significant.

Recently a mathematical model was proposed (Sutera, Maeder & Kestin 1963) to demonstrate that vorticity amplification by stretching was an important mechanism underlying the sensitivity of stagnation-point heat transfer on cylinders to free-stream turbulence. According to the model, vorticity of a scale larger than a certain neutral scale and appropriately oriented can undergo amplification as it is convected towards the boundary layer. Such vorticity, present in the oncoming flow with small intensity, can reach the boundary layer with a greatly magnified intensity and induce substantial three-dimensional effects therein. The mean temperature profile was shown to be much more responsive to these effects than the mean velocity profile and very large increases in the wall-heat-transfer rate were calculated for Prandtl numbers 0·74 and 7·0.

In this work a second, more general, case is treated in which the approaching flow carries vorticity of scale 1·5 times the neutral. By means of iterative procedures applied on an electronic analogue computer, an approximate solution to the full Navier–Stokes equation is generated. The heat-transfer problem is solved simultaneously for Pr = 0·70, 7·0 and 100. It is found that a vorticity input which increases the wall-shear rate by less than 3% is capable of increasing the wall-heat-transfer rate by as much as 40%. The sensitivity of the thermal boundary layer depends on Prandtl number. In the three cases investigated it is greatest for Pr = 7·0 and least for Pr = 100.

Howard's semicircle theorem and a variational principle for the instability of unidirectional flow of an inviscid fluid are applied to the non-uniform vortex sheets discussed in a previous paper (Hocking 1964). Certain results of §5 of that paper are shown to be wrong and the correct results are obtained.

An analysis is made of the two-dimensional flow under gravity of an inviscid non-diffusive stratified fluid into a line sink, involving a velocity discontinuity in the flow field. The fluid above the discontinuity is stagnant and hence is not drawn into the sink. At sufficiently low values of the modified Froude number, this is the only physically possible mode of flow, and is the cause of flow separation in many industrial and natural processes. A proper mathematical solution for flows with a stagnant zone has so far been lacking. This paper presents such a solution, after posing the problem as one involving a free-streamline, which is the line of velocity discontinuity. The solution to be given here is obtained by an inverse method. It is also found herein that the modified Froude number has a value of 0·345 for all separated flows of the kind in question.

An infra-red heat-transfer gauge was used in a shock tube for end-wall measurements of the convective heat transfer from argon behind the reflected shock. The thermal conductivity of neutral (un-ionized) argon was measured before the ionization-relaxation time, and was fitted with the power-law temperature dependence 4·2 × 10−5(T/300)0·76±0·03 cal/sec cm°K, where T is measured in °K, and ±0·03 refers to the probable error The free-stream temperature ranged from 20,000 to 75,000°K, corresponding to incident-shock velocities from 3 to 6mm/μsec. At later times, after the free stream established equilibrium ionization, the convective-heat-transfer rate remained the same as the initial rate with neutral argon. Theoretical predictions of Fay & Kemp (1965), assuming equilibrium-boundary-layer conditions, are 20–30% below the experimental values. Also reported in this paper are measurements of the ionization times behind the reflected shock, and these are in agreement with an extrapolation of the Petschek & Byron (1957) measurements behind the incident shock. There is a discussion of the large changes in the gas conditions behind the reflected shock due to the ionization process. The final equilibrium conditions are reached abruptly, as indicated by the continuum-radiation emission which becomes constant immediately after ionization relaxation.

A descriptive experimental study was made in both air and water of the temporally periodic motion that occurs in the vortex whistle and cyclone separator. The motion can be described in terms of an oscillator that derives its energy from hydrodynamic instability of the steady swirling flow and whose frequency is determined by an angular velocity characteristic of this steady flow. The relevant dynamical parameters are the Rossby number and Reynolds number for the steady flow with the addition of the Strouhal number for the time-dependent flow. The results of this study were compared with the vortex breakdown phenomenon over swept-back wings. Breakdown can be described in the same terms as for the other two cases and it appears that all three motions are basically the same.

The asymptotic solution for the incompressible viscous flow past a semi-infinite flat plate constructed by Goldstein (1956, 1960) can be valid only if the solutions of certain ordinary differential equations obey certain constraints (given in Goldstein 1956, 1960). In this paper, we construct the solutions of these equations, show that the necessary constraints are met, and hence establish the validity of the asymptotic solution up to the order considered. The manner in which undetermined constants appear in the solution are discussed.

The instability of a two-dimensional jet with respect to three-dimensional disturbances and that of an axially symmetric jet with respect to azimuthally periodic disturbances are studied, for the inviscid flow of a compressible fluid. In both cases the undisturbed velocity is assumed to be uniform in the jet. It is shown analytically that a two-dimensional jet is unstable under small disturbances, either subsonic or supersonic. There is no upper limit in Mach number, as was found for a plane vortex sheet, above which the flow is completely stable. Numerical calculations for the eigenvalues for both the two-dimensional jet and the axially symmetric jet have been made. The results indicate that the increase of Mach number tends to stabilize the flow. For the two-dimensional jet, the larger the angle between the direction of wave propagation and that of the main flow, the more the flow will be destabilized. For the axially symmetric jet, the flow is more unstable under azimuthally periodic disturbances than under rotationally symmetric ones, at small wave number.

The concept of a law of the wall and a velocity defect law which are related to each other through a common velocity scale and a semilogarithmic velocity profile in the region where they overlap, can be applied successfully to turbulent boundary layers with suction or injection. For turbulent boundary layers at moderate suction rates the velocity scale is proportional to $u_{r}^{2}/{{\nu }_{0}}$. For layers at arbitrary suction or blowing rates the velocity scale, which has been determined empirically, is proportional to (ur + 9v0).

A systematic account is given of the derivation of the dispersion relation for helicon waves in a uniform cylindrical plasma bounded by a vacuum. By retaining finite resistivity in the equations, boundary conditions present no difficulties, since the wave magnetic field is continuous through the plasma-vacuum interface. Two unexpected results are found. First, the wave attenuation remains finite in the limit of vanishing resistivity. This is due to the energy dissipated at the interface by the surface currents required to match the plasma wave field to the vacuum wave field. Zero wave attenuation for zero resistivity is recovered if electron inertia is included. Secondly, it is found that waves with azimuthal numbers m of opposite sign propagate differently, but the sense of polarization at the axis of the cylinder is independent of the sign of m.

The argument of the dispersion function is complex and numerical results were obtained using a computer. The method of programming is described, and results are given applicable to propagation in metals at low temperatures, or in a typical gas discharge plasma for the m = 0 and m = ± 1 modes.

An example of the amplitude of the wave fields as a function of radius is given for the axisymmetric mode, and of amplitude and phase for the m = ± 1 modes.

A mathematical analysis of the behaviour of small disturbances to Poiseuille flow in a circular pipe can be tackled in two different ways. In the first of these, the disturbance velocity is expressed as a combination of terms, each of which represents a disturbance of a fixed wavelength that decays with time. In the second approach, each term which contributes to the disturbance velocity represents a disturbance of a fixed frequency that decays with downstream distance. The latter method is given more prominence in this paper, since, in experiments in which disturbances are introduced into the flow in a controlled manner, a constant frequency-generating device is commonly used.

In the main, it is assumed not only that the Reynolds number of the Poiseuille flow is large, but that the frequency, f, of the disturbance is large compared with the frequency, v/a2, where a is the radius of the pipe and v the kinematic viscosity of the fluid. The mathematical problem is then of the singular perturbation type, and the disturbances with the smallest damping rates are confined to thin layers. A simple, but crude, analysis shows among other things that the radius a t which the disturbance velocity is a maximum is roughly that at which the velocity of the Poiseuille flow is equal to the frequency, f, times the disturbance wavelength. Eigenfunctions are found precisely for the two limiting cases in which, as fa2/v tends to infinity, the disturbance becomes confined to a thin layer situated (a) near the centre of the pipe, and (b) near the wall. The eigenfunctions are presented graphically in such a way that immediate comparison can be made with some of Leite's experimental results. Good agreement is found. A real disturbance, however, is made up of several modes, each of which is damped at a different rate with increasing downstream distance. Possible changes in the form and apparent damping rate of a disturbance are discussed in terms of particular case.

Next, an asymptotic procedure is carried out, which proves to give a good approximation to the eigenvalues and eigenfunctions over a wide range of conditions. From the eigenvalue equation so obtained it is possible to calculate the wave-speed and damping rate for each mode as a function of the non-dimensional frequency, fa2/v, and the Reynolds number. For simplicity, the calculations are carried out for the case in which the Reynolds number is infinite, so that the eigenvalues depend only on fa2/v. For each mode it is found that the damping rate is an increasing function of the frequency for high frequencies, but the frequency is decreased the damping rate approaches a limiting value. These limiting values can be quite small for the Reynolds numbers, R, at which experiments are normally carried out, and for some low-frequency disturbances the distance required for the disturbance amplitude to be reduced by half is as much as R/1000 diameters.

The problem of the stability and growth of disturbances in a fluid with time-dependent heating is investigated. The analysis is restricted to the case when the temperature gradient is large in a layer which is narrow by comparison with the overall depth of the fluid. An approximate method of solution is presented. Results of computations are also presented which illustrate the method of solution and the essential features of the problem.

A mono-energetic electron plasma is trapped in a one-dimensional parabolic potential well. In the undisturbed state a suitable background charge provides space-charge neutrality. It is shown that stationary disturbances of infinitesimal amplitude can exist in the plasma for certain critical values of the parameter $\Lambda \,=\,{4\bar{n}{{e}^{2}}}/{m{{\omega }^{2}}}\;\$, where $\bar{n}\$ is the mean electron density and ω/2π is the frequency of oscillation of the electrons in the well. The first few critical values are Λ = 0, 4·12, 8·2.

The boundary conditions at the end of the plasma are non-linear. As a result stationary disturbances of finite amplitude in a given mode, say the rth, require that Λ shall exceed Λr. Further it can be shown that disturbances of small amplitude in the rth mode are unstable when Λ exceeds Λr. This applies even for Λr = 0; in this case there exist nearby an unstable even and an unstable odd mode.

It seems likely that these results can be extended to all cases in which the potential well is symmetrical.

A model equation is constructed whose analysis reveals the same features, including stable hexagonal cells, as analysis of genuine approximate equations for the thermal convection problem. Taking advantage of the relative simplicity of the calculations an attempt is made to clarify certain procedures customarily used in non-linear stability theory; in particular, the basis for the usual formal expansions and the appropriateness of neglecting terms of fourth and higher order are discussed. It is demonstrated that a whole class of equations leads to hexagonal cells, thereby giving confidence that results on convection cells found elsewhere in idealized situations will remain valid when more realistic situations are studied.