A higher order theory for two-dimensional turbulent boundary-layer flow of a compressible fluid past a plane wall is formulated, for moderately large values of the Reynolds number, by the method of matched asymptotic expansions. The parameters (γ − 1) M2∞ and the molecular Prandtl number are assumed to be of order unity. The analysis deals with the set of Reynolds equations of mean motion (which are underdetermined without an additional set of closure hypotheses) and assumes that the non-dimensional fluctuations in velocity, temperature and density are of order U*, (friction velocity divided by free-stream velocity a t some designation point), while fluctuations in pressure are of order U2*.The first-order results of the present study lead to asymptotic laws for velocity and temperature distributions which correspond to the law of the wall, logarithmic law and defect law, and also to skin friction and heat-transfer laws. It turns out that the first-order defect law depends upon the gradient of entropy and stagnation enthalpy and the law of the wall is independent of viscous dissipation. The second-order terms of the present work (accounting for mean convection due to turbulent mass flux, viscous dissipation in the inner flow and displacement effects in the outer flow) describe the necessary corrections to first-order terms due to low Reynolds number effects. In the overlap region the second-order results, for the law of the wall and the defect law, show bilogarithmic terms along with logarithmic terms.

The transport properties of a diffusive interface with diffusivity ratio $\kappa_S/\kappa_T = {\textstyle\frac{1}{3}}$ have been measured, using salt and sugar as the diffusing components. The flux ratio is constant and equal to (κS/κT)½. The normalized salt flux is related to the density anomaly ratio Rρ = βΔS/αΔT by the power law F*T = 2·59Rρ−12.6 over four decades. Optical measurements show that the vertical gradients of concentration of salt and sugar within the interface are those required if molecular diffusion is to account for the whole flux of each component.

A study is made of the natural convective flow which is induced in an infinite expanse of gas by the presence of a vertical hot flat plate from which hot gas of the same chemical type is being blown. The transpiration rate is assumed to be such that a self-similar boundary-layer type of solution is available. It differs from previous analyses in the following respects. Most important, the density is not assumed to be constant a t any stage in the description of the flow field. Also the form of the induced flow in the outer domain is calculated and proves to be substantially independent of the blowing rate in this case; the induced outer flow is found to be of large lateral extent. Finally, the variable-gas-property problem is carried to second order and solutions are obtained by using an ‘exact’ form of Howarth-Dorodnitsyn variable. The opportunity is taken t o make some comments about the comparison between theory and experiment for finite flat plates without transpiration.

This paper describes an experimental and theoretical study of thermal convection in a sloping porous layer. The saturated layer is bounded by two parallel impermeable planes maintained at different temperatures. Several types of flows were observed: a unicellular movement and a juxtaposition of longitudinal coils or of polyhedral cells.

A theoretical analysis has been made using the standard bases of the linear theory of stability and by taking into account some assumptions suggested by experimental observations. The critical conditions for the transition between unicellular and polycellular flows has been determined. For flow in longitudinal coils or with polyhedral cells the average heat transfer depends mainly on the filtration Rayleigh number and on the slope of the layer.

The experimental study was made in a Rayleigh number range 0–800 and for various slopes (0–90°). For both the transition criterion and the heat transfer, a good fit was observed between the experimental and theoretical results. For maximum slope, i.e. 90°, a correlation which connects the Nusselt number with both the Rayleigh number and the vertical extent of the model is proposed.

In this paper an exact solution is presented for the problem of unsteady laminar convective flow under a pressure gradient along a vertical pipe. We have obtained the solution of the problem on the basis of the assumption that the velocity and buoyancy profiles far from the pipe entrance do not change with the height, and the entry lengths have been ignored. The wall of the pipe is heated or cooled uniformly. We have discussed both the cases, when buoyancy forces act together with the pressure gradient or in opposite direction.

In the case when the upflow is heated (or a downflow is cooled) the velocity and thermal boundary layers are formed for sufficiently large Rayleigh numbers. In the second case which has been discussed in detail (when the upflow is cooled or the downflow is heated) we have found the critical value of the Rayleigh number R = R, beyond which the velocity profile and the temperature profile become unsteady and turbulent in all the cases. In the case of the elliptical cylinder R, increases up to 1730 as the ellipticity is increased while in the case of the co-axial pipes this Rayleigh number increases as the gap c between the cylinders is decreased (if c = a/b = 1·2 then R, = 60762, but decreases to 1 when c = 4). Besides this, the time required to reach steady state increases as the Rayleigh number increases in both circular and elliptical pipes; it also increases when the eccentricity is decreased. The cases discussed by Morton (1960 and Dalip Singh (1965 are particular cases of the results derived below.

In this investigation we have dealt with the following ducts: (i) circular tubes, (ii) elliptical tubes and (iii) co-axial tubes. The general solutions for both velocity and temperature fields have been found for the case when the pressure gradient is an arbitrary function of time, with an arbitrary heat source also present. Particular cases when both the parameters are absohte constants have been discussed in detail.

We have made use of finite transforms very frequently; especially for the case of an elliptical tube, a new transform involving Mathieu functions developed by Gupta (1964 has been used. A few new infinite series have been summed with the help of this transform.

Various non-dimensional quantities (for both the cylinders) such as the Nusselt number, volume flux and rate of heat transfer have been found when the pressure gradient and source of heat generation are absolute constants.

This paper investigates the Bénard problem in a binary mixture of dilute gases in which an imposed vertical temperature gradient induces a concentration gradient owing to the thermal diffusion effect. The transfer equations are derived by first-order perturbation theory which leads to instability criteria. Numerical results indicate that instability will set in only as stationary convection. This is distinctly different from the cases of liquids and concentrated gases, in which the thermal diffusion (or Soret) effect gives rise to oscillatory instability. It is disclosed in the study that the destabilization of the dilute gas-mixture layer is enhanced by an increase in the thermal diffusion ratio and/or the molecular weight ratio of the species.

This paper discusses the theory of the reflexion and scattering by an irregular coastline of a Poincaré-type wave on a rotating ocean. It is assumed that the coast is straight except for small deviations from the rectilinear form, and that these deviations may be regarded as a random function of position along the coast. The rigorous theory of energy-transfer processes in random media is applied to determine the power flux from the incident Poincard wave into the scattered Kelvin wave, which propagates in a unique direction along the coast, and into Poincard ocean wave noise. The relative efficiencies of generation of these waves is examined in some detail, and studied in particular for varying ranges of values of certain non-dimensional parameters characterizing the coastal configuration. Detailed estimates are given for a shoreline whose irregularities are specified by a Gaussian spectrum of Fourier components, and the results extra-polated in the concluding section of the paper to give a general qualitative discussion of the effects of an arbitrary coastline on an incident wave.

Flow separation can be observed (1) at the leading edge of a spilling breaker or ‘white-cap’, (2) at the lower edge of a tidal bore or hydraulic jump and (3) upstream of an obstacle abutting a steady free-surface flow. At the point of flow separation there is a discontinuity in the slope of the free surface. The flow upstream of this point is relatively smooth; the flow downstream of the discontinuity is turbulent.

In this note, a local solution for the flow in the neighbourhood of the discontinuity is derived. The turbulence is represented by a constant eddy viscosity N, and the tangential stress across the interface between the laminar and turbulent zones is expressed in terms of a drag coefficient C. It is shown that the inclinations of the free surface of the two sides of the discontinuity depend on C only, and are independent of N and g. As C increases from zero to large values, so the inclination of the free surface in the turbulent zone increases from 10° 54′ to 30°. In the laminar zone the inclination of the free surface simultaneously decreases from 10° 54′ to 0°, the densities in the two zones being assumed equal.

Owing to the possible entrainment of air at the separation point, the effective density ρ′ in the turbulent zone may be less than the density ρ in the laminar zone. When these densities are allowed to be different it is found that the possible flows are of two distinct types. Flows of the first type, called ‘quasi-static’, are contiguous to a state of rest. Flows of the second type, called ‘dynamic’, are contiguous with the frictional flows described above, for which ρ′ = ρ At a given positive value of C there exists generally only one quasi-static solution. There is also just one dynamic solution provided ρ′/ρ > 0·50012. On the other hand, if ρ′/ρ < 0·5 there may be either two or no dynamic flows, depending on the value of C; and when 0·5 < ρ′/ρ > 0·50012 there may be three such flows.

The inclination of the free surface is studied as a function of C and ρ′/ρ.

For flow of a viscous fluid in a divergent channel of small angle it is shown that small disturbances to the basic Jeffery-Hamel flow may grow, according to nonlinear theory, to produce a secondary (supercritical) flow, in which the main flow winds from side to side in the channel and vortices form, with the whole pattern moving slowly downstream.

A theoretical analysis is made of the effect described in the title. It is shown that both (i) the eigenfrequencies of the inertial waves and (ii) the moment produced by the inertial wave pressure distribution are independent of the location of the off-axis cylinders. Hence, one can use Stewartson's tables computed for a centrally located cylinder to calculate the frequencies and residues for an eccentrically located cylinder.

The work of Dean and that of McConalogue & Srivastava on the steady motion of an incompressible fluid through a curved tube of circular cross-section is extended through the entire range of Reynolds numbers for which the flow is laminar. The coupled nonlinear system of partial differential equations which defines the motion is solved numerically by finite differences. Computer calculations are described and physical implications are discussed.

Experiments on fully attached flows of aqueous solutions of Polyox WSR-301 and pure solvent (tap and distilled water) about circular cylinders were carried out in the range 5 × 104 < Re < 3 × 105 for the initial purpose of providing definitive experimental information on drag reduction with additives in external flows.

During the course of the investigation, a critically degraded polymer solution was observed to cause, at a particular Re, an unforseen instability in the separation line, similar to that first observed by Brennen (1970 in fully developed cavity flows. This instability, which most easily manifest edit self in the form of an abrupt drag-force jump (here called a drag crisis) from a high subcritical to a low supercritical value, became the central issue in the investigation. Efforts to isolate the causes of this instability led to further experiments with suspensions of nearly neutrally buoyant spherical and non-spherical particles. The measurements have shown that fibres and polymer hasten transition, spherical particles do not appreciably affect the characteristics of the flow and that neither the spherical particles nor fibres give rise to a drag crisis.