The application of a simple discrete velocity model to low Mach number Couette and Rayleigh flow is investigated. In the model, the molecular velocities are restricted to a finite set and in this study only eight equal speed velocities are allowed. The Boltzmann equation is reduced by this approximation to a set of coupled differential equations which can be solved in closed form. The fluid velocity and shear stress in Couette flow are in approximate accord with those of Wang Chang & Uhlenbeck (1954) and of Lees (1959) over the complete range of Knudsen number. Similarly, the Rayleigh flow solution is remarkably like those found by other investigators using moment methods.

This paper concerns an investigation of turbulence in the density stratified shear flow of a specially designed wind tunnel in which the density gradient is created by differential heating of the air. The first three sections of the paper consist of a description of the apparatus and of the mean temperature and velocity gradients in the tunnel, together with a discussion of a method of measuring low wind speeds based on the periodic shedding of vortices by a circular cylinder. In the remaining sections details of the experimentally determined structure of the turbulence of the flow and of its eddy conductivity and viscosity are presented and their dependence on the over-all gradient form of Richardson number, $[g \partial T | \partial z]|[T(\partial U| \partial z)^2]$, considered.

The steady separated flow past a circular cylinder was investigated experimentally. By artificially stabilizing the steady wake, this system was studied up to Reynolds numbers R considerably larger than any previously attained, thus providing a much clearer insight into the asymptotic character of such flows at high Reynolds numbers. Some of the experimental results were unexpected. It was found that the pressure coefficient at the rear of the cylinder remained unchanged for 25 [les ] R [les ] 177, that the circulation velocity within the wake approached a non-zero limit as the Reynolds number increased, and that the wake length increased in direct proportion to the Reynolds number.

The Rayleigh stability equation of inviscid linearized stability theory was integrated numerically for amplified disturbances of the hyperbolic-tangent velocity profile. The evaluation of the eigenvalues and eigenfunctions is followed by a discussion of the streamline pattern of the disturbed flow. Here no qualitative distinction is found between an amplified and the neutral disturbance. But considering the vorticity distribution of the disturbed flow it is shown that in the case of amplified disturbances two concentrations of vorticity occur within a disturbance wavelength, while in the neutral case only one maximum of vorticity exists. The results are discussed with respect to the instability mechanism of free boundary-layer flow.

Further consideration is given to the stability of the flow of an idealized elasticoviscous liquid contained in the narrow channel between two rotating coaxial cylinders. The work of Part 1 (Thomas & Walters 1964) is extended to include highly elastic liquids. To facilitate this, use is made of the orthogonal functions used by Reid (1958) in his discussion of the associated Dean-type stability problem. It is shown that the critical Taylor number Tc decreases steadily as the amount of elasticity in the liquid increases, until a transition is reached after which the roots of the determinantal equation which determines the Taylor number T as a function of the wave-number ε become complex. It is concluded that the principle of exchange of stabilities may not hold for highly elastic liquids.

In this paper flow-visualization techniques are used to study the flow instability which occurs at the orifice in the free convection open thermosyphon. The influence of the shape of the orifice on the measured heat transfer of the system is also studied under both laminar and turbulent conditions. The dominant mode of penetration of the hot stream by the cold fluid entering at the base of the reservoir is found to instigate a mixing region at the orifice. In turbulent flow this spreads into and eventually fills the tube as the Rayleigh number is increased. In such circumstances a sharp-edged orifice appears to give better overall heat transfer than a rounded orifice.

The constant-density inviscid rotational flow in the neighbourhood of a general stagnation point on a wall is investigated. In all but very special cases, the solution is non-analytic and the vorticity at the wall is infinite; the stagnation streamline is tangent to the wall at the stagnation point; stagnation points of saddlepoint type cannot exist.

The boundary-layer equations corresponding to the inviscid solutions studied are presented.

A weak expansion wave propagating in a relaxing gas is discussed with particular reference to the ‘near-equilibrium’ and ‘near-frozen’ regions. The concept of bulk viscosity is used in conjunction with Burger's equation in the near-equilibrium region. The asymptotic equilibrium simple wave is modified by diffusive regions in the neighbourhood of the first and last rays. It is shown that in the case of a weak expansion wave, Chu's asymptotic solution of the acoustic equation describes the wave-form for a finite time interval before convection effects become noticeable. In the near-frozen region a characteristic perturbation method is used to describe the flow near the wave-front.

The viscometer investigations of Donnelly & Ozima (1962) on the stability of flow between rotating cylinders in the presence of a magnetic field have been extended by making careful torque measurements in the subcritical range using improved mercury. The results show that a small difference in stability can be observed with conducting cylinders as compared with insulating cylinders. The results also suggest that a heretofore unnoticed instability begins well before the instability studied earlier. This relatively weak instability accounts for some of the anomalous effective viscosity observed by Donnelly & Ozima.

The problem of small angle depressions in a liquid surface due to an impinging two-dimensional potential jet is considered. Using conformal mapping methods and finite Hilbert transforms, the problem is formulated as a non-linear singular integral equation. The integral equation is approximated by a set of non-linear algebraic equations which are solved numerically by a method of repeated linear corrections. In addition, an asymptotic solution (for low jet velocity) is derived.

From the numerical solutions of the integral equation, the liquid-surface profiles and the free streamlines of the jet are calculated for four cases. These results verify the appearance of lips on the liquid surface which have been observed experimentally by others.

The steady two-dimensional seepage flow by gravitational convection, of a fluid of density ρ1 surrounded by a fluid of density ρ0(≠ρ1) at rest, leads to a potential problem from which the shape of the interface can be determined. When the two fluids are slightly miscible the interface is replaced by a mixing layer, and it is shown that the first-order Prandtl equations for the flow in the layer possess an exact similarity solution. The profile across the layer is of the same form as the profile of the laminar incompressible half-jet with one fluid at rest, and a formula is obtained for the scale of mixing-layer thickness as a function of distance downstream. Three examples are discussed.

(a) Flow of fluid of density ρ1 from a horizontal line source. When (ρ1 > ρ0) a stagnation point exists above the source, and the fluid ultimately descends in a vertical column of width proportional to the source strength. At the stagnation point, the mixing-layer thickness is finite and is proportional to the square root of the radius of curvature of the interface. At a sufficiently great distance downstream, the thickness increases as the square root of the distance, as in the straight laminar half-jet. These results have been tested experimentally in a Hele-Shaw cell.

(b) Symmetrical flow of an ascending column of fluid (ρ1 < ρ0) about an obstacle in the form of a finite horizontal strip. The column reforms after passing the obstacle, and the mixing-layer thickness returns to the value corresponding to an unobstructed vertical half-jet. The flow has been produced experimentally.

(c) Flow in a lens of fresh water overlying salt water, with inflow due to precipitation, as in a two-dimensional Ghyben-Herzberg lens. Here the potential flow solution is calculated approximately by means of Dupuit-Forchheimer theory. In the steady-state solution the thickness of the mixing layer between fresh and saline water is found to be finite and, as in (a), proportional to the square root of the radius of curvature of the lens.

The uniqueness condition that was utilized by the author (Concus 1962) is considered. The condition, which excludes certain fluid depths, is shown to be physically unacceptable because it is essentially impossible to satisfy in practice. The resulting in validation of the perturbation method is discussed, and a revision is presented, which invokes the presence of viscosity and allows retention of the previously obtained solutions. The revision may also be applied to the work of other authors who utilized the same method to solve other standing-wave problems.

The travelling-wave plasma converter is analyzed allowing for the compressible nature of the fluid. The study is restricted to conditions for which the magnetic Reynolds number is small. Steady and time-varying flow conditions are investigated using small perturbation theory. Relations for optimum conversion effectiveness are established and the effects of variable electrical conductivity are investigated. Conditions for electromechanical resonance are derived and it is shown that the conversion process introduces a damping effect.

This report is an extension of an earlier note in which a simple method of estimating the distribution of vibrational temperature along a hypersonic nozzle was described. Results were presented for hyperbolic, axisymmetric nozzles with reservoir conditions 1000[les ]p0[les ] 4000 p.s.i.a., 1000[les ] T0 [les ] 3000°K. The problem was subsequently programmed for the Ferranti Mercury computer at the University of London computing centre, and the results of these computations are given here. The vibrational temperatures are compared with those of the previous simple method. The distributions of pressure and temperature through the nozzle are also given and a simple method of estimating the vibrational temperature is described.

Secondary flows in non-circular ducts are accompanied by a longitudinal component of vorticity. The equation of motion defining this component in a turbulent flow is composed of three terms giving the rates of production, diffusion and convection. Since the expression for production is the second derivative of Reynolds strees components, longitudinal vorticity cannot exist in laminar flow. For turbulent flow in a square duct the Reynolds stress tensor is examined in detail. Symmetry requirements alone provide relationships showing that the production is zero along all lines of symmetry. General characteristics of flow in circular pipes are sufficient to indicate where the production must be greatest. Experimental measurements verify this result and define the point density of production, diffusion and convection of vorticity. Data also indicate that the basic pattern of secondary flow is independent of Reynolds number, but that with increasing values of Reynolds number the flows penetrate the corners and approach the walls. A similar experimental investigation of a rectangular duct shows that the corner bisectors separate independent secondary flow circulation zones. Production of vorticity is again associated with the region near the bisector. However, there is some evidence that the secondary flow pattern is not so complex as inferred from the distortion of the main longitudinal flow.

A general theory of unsteady cavitating flow past hydrofoils and other obstacles is given for the case of cavities of finite length L. If the circulation Γ, the cavity volume V and L are known as functions of time, the theory yields explicit formulae for the velocity over the wetted surface and for the cavitation number σ. The theory is based on the approximation that the cavity is bounded by stream-lines, and so is valid only for slow rates of change of L, V and Γ. The possibility of allowing for the presence of a vortex sheet behind the cavity is discussed. The theory is extended to the case of a cascade of hydrofoils behind which extend growing cavities.

Two examples of the theory are discussed, namely the unsteady flow past a symmetrical wedge, and the unsteady flow past a flat plate hydrofoil cavitating from the leading edge.

Magnetohydrodynamic squeeze films are investigated theoretically and experimentally. The theory of magnetohydrodynamic lubrication as applied to squeeze films is extended to include fluid-inertia effects and buoyant forces. Excellent agreement is obtained between theory and experiment.

Large-amplitude atmospheric flows past mountain ridges are investigated. The flows are assumed to be steady and two-dimensional. Diffusive and viscous effects are neglected but static compressibility is taken into account.

The larger part of the investigation is devoted to the study of waves in the lee of mountain ridges. The major contribution consists in the treatment of the large-amplitude motion. The flows are governed by an equation which turns out to be linear for certain upstream conditions. These conditions impose some restrictions on the wind profile and stratification of the entropy and specific energy far upstream. However, flow patterns representing realistic upstream conditions have been obtained.

A comparison between a compressible flow and an incompressible flow with equivalent upstream conditions is included.

The flow in the noise-producing region of a circular jet is found to be dominated by a group of large eddies, containing nearly a quarter of the turbulent shear stress in the quasi-plane region of the shear layer: their contribution to the shear stress decreases as the effects of axisymmetry become noticeable at more than about two diameters downstream of the nozzle. These large eddies appear to be almost entirely responsible for the irrotational fluctuations near the nozzle, which, for this and other reasons, are larger relative to the reference dynamic pressure than in other shear flows. As a consequence of this, the convection velocity near the high- and low-velocity edges of the flow is biased towards the mean velocity in the high-intensity region. The dominance of the large eddies therefore explains the measurements of near-field pressure fluctuations by Franklin & Foxwell (1958), and of convection velocity by Davies, Barratt & Fisher (1963) and the present authors. The strength of these large eddies, compared with those in the boundary layer or wake, is remarkable.

The large eddies appear to be mixing-jets similar to those found by Grant (1958) in the wake, but with their projection in the (y, z)-plane inclined at about 45° to the y (radial) axis instead of lying along the y-axis as in the wake.

It is suggested that the augmentation of these large eddies by artificial means could be used to increase the mixing rate and permit the reduction of jet noise by means of acceptably short ejector shrouds.

The medium-scale motion is found to be far from isotropic in scales, although the two scales associated with a given vorticity component are more nearly equal. This phenomenon is also noticeable in the wake.

It is found that the departure from self-preservation, which starts when the shear layer thickness is no longer small compared with the nozzle radius, does not grossly affect the region of high turbulence intensity and maximum noise production until this region itself is no longer small compared with the radius. The maximum shear stress seven diameters downstream of the exit is still 70% of its value near the exit, and the non-dimensional mean velocity gradient is practically unchanged.