An experimental study of transitions from steady bimodal convection to time-dependent forms of convection is described. Using controlled initial conditions for the onset of bimodal convection two mechanisms of instability can be separated from the effects of random noise. The oscillatory instability of bimodal cells introduces standing waves closely resembling those occurring in low Prandtl number convection. The collective instability introduces spoke-pattern convection which is characteristic for turbulent large Prandtl number convection. Both instabilities originate primarily from the momentum advection terms in the equations of motion, as is evident from the strong Prandtl number dependence of the critical Rayleigh number Rt for the onset of oscillations. The results are discussed in relation to previous experiments and recent theoretical work.

The detailed flow fields associated with instability in axisymmetric jets are realized by numerical integration of the time-dependent Navier-Stokes equations.

The mechanism of amplification of small disturbances is shown to be two-dimensional for a thin boundary layer profile and conclusions regarding three-dimensionality which have been inferred from recent models of axisymmetrical systems are clarified.

The computed flow field is shown to be dominated by the large-scale vortex ring structure which has been observed experimentally. Although the wavelength of vortex shedding is found to be slightly variable owing to the randomness of the initial perturbation, the results are shown to agree closely with experiment.

The presence of dissipation in an isothermal atmosphere may cause upwardpropagating small amplitude waves to be reflected downward. For an atmosphere with small dynamic viscosity μ this was demonstrated in Yanowitch (1967b); this will be referred to as case II. Here two problems will be investigated: (i) a thermally conducting atmosphere with small conductivity k (case III) and (ii) a viscous and thermally conducting atmosphere with small k and μ, and a small ratio μ/k, i.e. small Prandtl number (case IV). It will be shown that the validity of the model in case III is questionable. The solution for case IV is determined from the conditions that the average rate of energy dissipation and of entropy increase in a column of fluid be finite, but a radiation condition is required in case III. The solution for case III does not approximate the one for case IV uniformly, and the reflexion coefficient for case IV does not tend to the one for case III as the Prandtl number Pr → 0, but varies periodically with log Pr. Numerical results show that when the Prandtl number is not small the reflexion coefficient can be approximated by the asymptotic value obtained from case II.

Experimental results are presented for two turbulent boundary-layer experiments conducted at a free-stream Mach number of 4 with wall cooling. The first experiment examines a constant-temperature cold-wall boundary layer subjected to adverse and favourable pressure gradients. It is shown that the boundary-layer data display good agreement with Coles’ general composite boundary-layer profile using Van Driest's transformation. Further, the pressuregradient parameter βK found in previous studies to correlate adiabatic highspeed data with low-speed data also correlates the present cooled-wall high-speed data. The second experiment treats the response of a constant-pressure highspeed boundary layer to a near step change in wall temperature. It is found that the growth rate of the thermal boundary layer within the existing turbulent boundary layer varies considerably depending upon the direction of the wall temperature change. For the case of an initially cooled boundary layer flowing onto a wall near the recovery temperature, it is found that δT ∼ x whereas the case of an adiabatic boundary layer flowing onto a cooled wall gives δT ∼ x½. The apparent origin of the thermal boundary layer also changes considerably, which is accounted for by the variation in sublayer thicknesses and growth rates within the sublayer.

A model is developed for the phenomenon of non-Newtonian secretion in tubes. The motivation for this study is the problem of glandular secretion, particularly in the pancreas. Power-law fluids are considered in some detail, as being biologically appropriate. It is found that for a power-law fluid whose exponent is less than unity (the biological case) two types of flow occur. For a sufficiently high secretion pressure, all of the tube is used for secretion, and a nonlinear pressure profile results. Numerical solutions are obtained for the pressure and rate of efflux. When the secretion pressure parameter falls below a certain critical value, the upper end of the tube begins to be choked off, only part of the tube being used for secretion. This phenomenon does not occur for exponents greater than or equal to unity. Physiological implications are considered, and a qualitative discussion given for the case of non-power-law fluids.

A theoretical investigation is presented of the electrohydrostatic stability of a given volume of incompressible dielectric fluid when stressed by the application of a potential difference between bounding conducting fluids. It is assumed that the dielectric fluid is located in a channel of breadth 2 a and height 2h, with h/a [Lt ] 1, whose walls are semi-infinite solid dielectric sheets of thickness 2h. The dielectric fluid may have a volume which differs from that of the channel, so that the presence of menisci at the interfaces between conducting and non-conducting fluids is taken into account. By a suitable method for approximating the electric stress at the interfaces, the electrostatic potential difference across the dielectric is determined as a function of the pressure difference across the interfaces for prescribed values of the discrepancy of the volume of the dielectric from the volume of the channel per unit length, and criteria are obtained for determining the critical electric field which precipitates the instability of the system. The variation of the critical electric field with the dimensionless volume excess 2δ is also found and it is shown that, for δ < −0·5, instability is associated with a symmetric mode of disturbance in which the critical field occurs at the maximum in a plot of potential difference vs. pressure difference. For δ > −0·5, instability arises from an asymmetric disturbance with the critical field occurring at a bifurcation point in the potential difference/pressure difference plane. Bifurcations are shown to occur only when the equilibrium profiles of the interfaces have extrema at the edges of the channel.

The stability of Couette flow of stratified salt solutions is investigated in an apparatus with both the inner and outer cylinders rotating. The ratio of the radius of the inner cylinder to that of the outer cylinder is 0·2. The flow is visualized by means of shadowgraph and dye-trace methods. Compared with homogeneous fluids, the effect of the stabilizing density gradient is to increase the critical speed of the inner cylinder and to decrease the critical wavelength for a given angular speed of the outer cylinder. When the cylinders are rotating in the same direction, in the critical state, the instabilities appear along the inner cylinder in a spiral wave form which is itself not very stable. With counterrotating cylinders, the instabilities appear as regularly spaced vortices which, for the most part, are neither symmetric Taylor vortices nor simple spirals. In addition, these vortices rotate as a whole at a speed generally smaller than that of the inner cylinder. From shadowgraph observations, stability curves are constructed for three density gradients. The critical wavelength and the rotational periods of the vortices are also determined.

Waves of small but finite amplitude in shallow water can occur as periodic wave trains of permanent shape in two known forms, either as Stokes waves for the shorter wavelengths or as cnoidal waves for the longer wavelengths. Calculations are made here of the periodic wave trains of permanent shape which span uniformly the range of increasing wavelength from Stokes waves to cnoidal waves and beyond. The present investigation is concerned with the stability of such permanent waves to periodic disturbances of greater or equal wavelength travelling in the same direction. The waves are found to be stable to infinitesimal and to small but finite disturbances of wavelength greater than the fundamental, the margin of stability decreasing either as the fundamental wave becomes more nonlinear (i.e. contains more harmonics), or as the wavelength of the periodic disturbance becomes large compared with the fundamental wavelength. The decreasing margin of stability is associated with an increasing loss of spatial periodicity of the wave train, to the extent that small but finite disturbances can cause a form of interaction between consecutive crests of the disturbed wave train. In such a case, a small but finite disturbance of wavelength n times the fundamental wavelength converts the wave train into n interacting wave trains. The amplitude of the disturbance subharmonic is then nearly periodic, the time scale being the time taken for repetitions of the pattern of interactions. When the disturbance is of the same wavelength as the permanent wave, the wave is found to be neutrally stable both to infinitesimal and to small but finite disturbances.

A Mach-Zehnder interferometer was used to stud two-dimensional Bénard convection cells. The experiments were performed with distilled water and sea water in the region where density is a linear function of temperature. Two-dimensional convection rolls were formed with Rayleigh numbers as great as 23400. Reversal in the temperature profile was obtained for R/Rc ≥ 3·8, and an overshoot of about 6% was observed at R/Rc = 9·2 and 13·8. This agrees with the values predicted theoretically by Veronis (1966) for stress-free boundaries and Royal (1969) for rigid boundaries. This disagrees with the experimental results of Gille (1967), who reports an overshoot of only 1 ½% at R/Rc = 16. Many of the other results agree with those of other experimenters, such as the relation between the cell height-to-width ratio and Rayleigh number, the relation between the Nusselt number and Rayleigh number, and the value of the critical Rayleigh number.

Two-dimensional waves are incident upon a pair of vertical flat plates intersecting the free surface in a fluid of infinite depth. An asymptotic theory is developed for the resulting wave reflexion and transmission, assuming that the separation between the plates is small. The fluid motion between the plates is a uniform vertical oscillation, matched to the outer wave field by a local flow at the opening beneath the plates. It is shown that the reflexion and transmission coefficients undergo rapid changes, ranging from complete reflexion to complete transmission, in the vicinity of a critical wavenumber where the fluid column between the obstacles is resonant.

An experimental investigation has been made of the processes occurring during the natural transition from laminar to turbulent flow of natural convection flow of water adjacent to a flat vertical surface where the surface heat flux is uniform. Measurements of both the velocity and temperature fields were made over wide ranges of the heat flux and at various downstream locations. Of principal interest were the definitions of the boundaries of the transition regime and their determination at several values of the surface heat flux. The interaction of the velocity and temperature fields during transition was measured. Our results show that transition events are not correlated in terms of the Grashof number G*. The form G*/xn, where n is of order ½ was found to give satisfactory correlations. Measurements of the frequency and growth rate of disturbances indicate the primacy of the velocity field during transition and show that the growth of turbulence in the temperature field lags behind that in the velocity field. The study of the turbulence growth, in terms of intermittency factors in both the velocity and temperature fields, resulted in unambiguous criteria for the boundaries of the transition regime. Our results suggest a kinetic energy flux parameter E and a single value closely correlates both our measurements of the onset of transition as well as those from all past studies known to us, for both different fluids and heating conditions.

An analysis of two-dimensional steady flow of an incompressible, viscous, electrically conducting fluid past an infinite vertical porous plate is carried out under the following assumptions: (i) that the suction velocity normal to the plate is constant, (ii) that the plate temperature is constant, (iii) that the difference between the temperatures of the plate and the free stream is moderately large, causing free convection currents, (iv) that the transversely applied magnetic field and magnetic Reynolds number are very small and hence the induced magnetic field is negligible.

Approximate solutions to the coupled nonlinear equations governing the steady velocity and temperature are derived. They are shown graphically. During the course of discussion, the effects of positive and negative G (the Grashof number: G > 0 implies cooling of the plate, G < 0 heating of the plate), of P (the Prandtl number), of positive and negative E (the Eckert number) and of M (the magnetic field parameter) are presented quantitatively.

Using a test-particle Monte Carlo method the general features of a gas expanding from a circular nozzle into a vacuum were investigated. Calculations were performed by considering the flow to be uniform and axial at the exit plane of the nozzle with an exit Mach number of 3 and a throat Reynolds number of 25. Results are presented for the density, mean flow velocity and kinetic temperature of the gas. Near the axis of the jet the calculated density distributions were found to agree well with those given by Robertson & Willis. Results are also presented for regions far from the jet's axis where no other solution is available.

The strength (initial circulation) and spacing of vortices in the wake of a circular cylinder have been obtained for conditions under which the body undergoes lateral vibrations. The vibrations of the cylinder were at all times synchronized with those in the wake, thereby suppressing the natural Strouhal frequency in favour of a common synchronized or ‘locked-in’ frequency for the body-wake system. All experiments were performed at a Reynolds number of 144 or 190. An inverse relation between the initial circulation K and the length lF of the vortex formation region was obtained for cylinder oscillations of up to 50% of a diameter, at vibration frequencies both above and below the Strouhal shedding frequency. The initial circulation K of the vortices was increased by as much as 65%, at lF = 1·6 diameters, from the stationary-cylinder value of K corresponding to lF = 3·2d. An increase in the rate A of vorticity generation of 80% from the stationary-cylinder wake value was obtained with the cylinder vibrating at 30% of a diameter and 110% of the Strouhal frequency. Both flow-visualization and hot-wire results show that the lateral spacing of the vortex street decreases as the vibration amplitude of the cylinder is increased, but that the longitudinal vortex spacing is independent of changes in amplitude. The longitudinal spacing, however, varies inversely with the vibration frequency. The street approaches a single line of vortices of alternating sign as the amplitude of vibration approaches values near a full cylinder diameter, and secondary vortex formation at these large amplitudes is associated with the vanishing lateral spacing of the street. Observation of the wake has elucidated the mechanism of vortex formation; the entrainment processes in the formation region have been observed at small intervals over a cycle of the cylinder's motion.

This paper considers the dispersion of a cloud of passive contaminant released from an instantaneous source in the steady two-dimensional laminar flow near the forward stagnation point on a bluff body. The body is replaced by its tangent plane y = 0 with x measuring distance along the plane. Far away from y = 0 the flow is irrotational with velocity potential ½l(x2 – y2), where l is a positive constant. When the boundary layer is ignored the equation governing the distribution of concentration can be solved exactly. Consequences of this solution are that for large times the centre of mass moves parallel to the body at a speed proportional to exp (lt) while the cloud spreads out along the body symmetrically about the centre of mass with the magnitude of the spread also proportional to exp (lt). However, this solution is unrealistic because most of the contaminant is confined to a layer adjoining the body of thickness of order (k/l)½, where k is the molecular diffusivity, and this layer normally lies within the boundary layer, which is of thickness of order (v/l)½, where v is the kinematic viscosity. An approximate analysis, based on ideas similar to those supporting the Pohlhausen method in boundary-layer theory, suggests that when the boundary layer is taken into account the conclusions above remain true provided that exp (lt) is replaced by exp (βlt), where β is a constant depending on v/k. Calculations give values of β ranging from 0·73 when v/k = 0·5 to 0·10 when v/k = 103.

The stability of natural convective flow in a porous medium heated both uniformly and non-uniformly from below is studied in order to determine the possibility of oscillatory and other unsteady flows, and to explore the conditions under which they may occur. The results of the numerical work are directly comparable with experiments using a Hele Shaw cell and also, in the uniformly heated case, with the results of Combarnous & Le Fur (1969) and Caltagirone, Cloupeau & Combarnous (1971). It is shown that for the uniformly heated problem there exist, in certain cases, two distinct possible modes of flow, one of which is fluctuating, the other being steady. However in the non-uniformly heated case the boundary conditions force the solution into a unique mode of flow which is regularly oscillatory when there is considerable non-uniformity in the heat input at the lower boundary, provided that the Rayleigh number is sufficiently high.

In this paper an exact formulation of the strength of the disturbance overtaking a shock is presented. The similarity solution is used to find the five interaction terms at all points of the flow. This work confirms that when the strength of the overtaking disturbance is known the CCW approximation may be modified to become an exact theory.

We observed a gradual change in the structure of a shock wave passing through a long tube of bubbly liquid, which we attribute to the motion of the bubbles relative to the liquid. We show that the effect of the motion on the structure of a shock wave is like that of thermal relaxation on gasdynamic shock waves: the pertinent relaxation time is the time viscous forces in the fluid take to alter the velocity of a bubble to that of the fluid. Our theory predicts certain changes in the speed of the shock wave and in its structure. We could not verify the prediction as to wave speed: in dilute mixtures it is too small to be measured. But we report experiments on the structure of the wave, which support our theoretical conclusion that the observed changes are due to the relative motion.

A layer of viscous incompressible fluid is confined between two horizontal plates which rotate rapidly in their own plane with a constant angular velocity. A hemisphere has its plane face joined to the lower plate and when a uniform flow is forced past such an obstacle, a Taylor column bounded by thin detached vertical shear layers forms. The linear theory for this problem, wherein the Rossby number ε is set equal to zero on the assumption that the flow is slow, is examined in detail. The nonlinear modifications of the shear layers are then investigated for the case when ε ∼ E½, where E is the Ekman number. In particular, it is shown that provided that the Rossby number is large enough separation occurs in the free shear layers. The extension of the theory to flow past arbitrary spheroids is indicated.

The two-dimensional steady flow of an incompressible viscous fluid past a circular cylinder, placed symmetrically in a simple shear field, has been studied for both the stationary and the freely rotating case by solving numerically the Navier-Stokes equations for values of the Reynolds number R in the range \[ 0.047 \leqslant R \leqslant 70. \] At R = 0·047, the present results are in substantial agreement with the analytic small-R perturbation solution given by Robertson & Acrivos (1970). Inertia effects were found, however, to play a significant role even at R = 1, and hence the calculated flow pattern for R ≥ 1 differs significantly from that of the creeping-flow solution. Specifically, for the freely rotating case, the region of closed streamlines decreases rapidly in extent with increasing R, two symmetrically placed wakes are formed on either side of the cylinder and the dimensionless rotational speed of the freely suspended cylinder decreases as R-½. In fact, for a value of R as low as 70, many of the gross features of the flow are surprisingly similar to those described by an inviscid solution except for the difference in the two sets of pressure profiles and the fact that the numerical results do not as yet exhibit the expected flow separation on the surface of the cylinder.