The results of measurements of far-field sound emitted from jets are reported. The narrow-band power spectral density of the sound in the far field was measured for three jet diameters, three Mach numbers, and five angular positions. The intensity distribution of mean-square pressure fluctuation in the far field in several wide frequency ranges were also measured. The similarity relations found from the experiments are reported.

It is generally assumed that flows around wall-mounted sharp-edged bluff bodies submerged in thick turbulent boundary layers are essentially independent of the Reynolds number Re, provided that this exceeds some (2–3) × 104. (Re is based on the body height and upstream velocity at that height.) This is a particularization of the general principle of Reynolds-number similarity and it has important implications, most notably that it allows model scale testing in wind tunnels of, for example, atmospheric flows around buildings. A significant part of the literature on wind engineering thus describes work which implicitly rests on the validity of this assumption. This paper presents new wind-tunnel data obtained in the ‘classical’ case of thick fully turbulent boundary-layer flow over a surface-mounted cube, covering an Re range of well over an order of magnitude (that is, a factor of 22). The results are also compared with new field data, providing a further order of magnitude increase in Re. It is demonstrated that if on the one hand the flow around the obstacle does not contain strong concentrated-vortex motions (like the delta-wing-type motions present for a cube oriented at 45° to the oncoming flow), Re effects only appear on fluctuating quantities such as the r.m.s. fluctuating surface pressures. If, on the other hand, the flow is characterized by the presence of such vortex motions, Re effects are significant even on mean-flow quantities such as the mean surface pressures or the mean velocities near the surfaces. It is thus concluded that although, in certain circumstances and for some quantities, the Reynolds-number-independency assumption is valid, there are other important quantities and circumstances for which it is not.

An erodible surface exposed to supercritical flow often devolves into a series of steps that migrate slowly upstream. Each step delineates a headcut with an associated hydraulic jump. These steps can form in a bed of cohesive material which, once eroded, is carried downstream as washload without redeposition. Here the case of purely erosional, one-dimensional periodic, or cyclic steps in cohesive material is considered. The St. Venant shallow-water equations combined with a formulation for sediment erosion are used to construct a complete theory of the erosional case. The solution allows wavelength, wave height, migration speed and bed and water surface profiles to be determined as functions of imposed parameters. The analysis also admits a solution for a solitary step, or single headcut of self-preserving form.

The periodic oscillation of the shock waves in screeching, underexpanded, supersonic jets, issuing from a choked, axisymmetric, nozzle at fully expanded Mach numbers (Mj) of 1.19 and 1.42, is studied experimentally and analytically. The experimental part uses schlieren photography and a new shock detection technique which depends on a recently observed phenomenon of laser light scattering by shock waves. A narrow laser beam is traversed from point to point in the flow field and the appearance of the scattered light is sensed by a photomultiplier tube (PMT). The time-averaged and phase-averaged statistics of the PMT data provide significant insight into the shock motion. It is found that the shocks move the most in the jet core and the least in the shear layer. This is opposite to the intuitive expectation of a larger-amplitude shock motion in the shear layer where organized vortices interact with the shock. The mode of shock motion is the same as that of the emitted screech tone. The instantaneous profiles of the first four shocks over an oscillation cycle were constructed through a detailed phase averaged measurement. Such data show a splitting of each shock (except for the first one) into two weaker ones through a ‘moving staircase-like’ motion. During a cycle of motion the downstream shock progressively fades away while a new shock appears upstream. Spark schlieren photographs demonstrate that a periodic convection of large organized vortices over the shock train results in the above described behaviour. An analytical formulation is constructed to determine the self-excitation of the jet column by the screech sound. The screech waves, while propagating over the jet column, add a periodic pressure fluctuation to the ambient level, which in turn perturbs the pressure distribution inside the jet. The oscillation amplitude of the first shock predicted from this linear analysis shows reasonable agreement with the measured data. Additional reasons for shock oscillation, such as a periodic perturbation of the shock formation mechanism owing to the passage of the organized structures, are also discussed.

Large coherent structures over vegetation canopies are responsible for a substantial part of the turbulent transfer of momentum, heat and mass between the canopy and the atmosphere. As forested landscapes are often fragmented, edge regions may be of importance in turbulent transfer. The development of coherent structures from the leading edge of a forest is investigated here for the first time. For this purpose, the turbulent flow over a clearing–forest pattern is simulated using the Advanced Regional Prediction System (ARPS). In previous studies the code has been modified so as to simulate turbulent flows at very fine scale (0.1h, where h is the mean canopy height) within and above heterogeneous vegetation canopies, using a large-eddy simulation (LES) approach. Validations have also been performed over homogeneous forest canopies and over a simple forest–clearing–forest pattern, against field and wind-tunnel measurements. Here, a schematic picture of the development of coherent eddies downstream from the leading edge of a forest is extracted from the mean vorticity components, the Q-criterion field, the cross-correlation of the wind velocity components and the length and separation length scales of coherent structures, determined by using a wavelet transform. This schematic picture shows strong similarities with the development of coherent structures observed in a mixing layer, with four different regions: (i) close to the edge, Kelvin–Helmholtz instabilities develop when a strong wind gust hits the canopy; (ii) these instabilities roll over to form transverse vortices from around 3h downstream from the edge, characterized by a length scale close to the depth of the internal boundary layer that develops from the canopy edge; (iii) secondary instabilities destabilize these rollers and increase the vertical and streamwise vorticity components from around 6h, and two counter-rotating streamwise vortices appear; (iv) at about 9h the initial rollers have become complex three-dimensional coherent structures, with spatially constant mean length and separation length scales. These four stages of development occur closer to the edge with increasing canopy density. While this average picture of the development of coherent structures is similar to that observed in a mixing layer, the analysis of instantaneous fields shows that coherent structures behind the leading edge appear as resulting from the ‘branching’ of tubes localized in regions of low pressure, where their cores are characterized by high values of enstrophy and Q-criterion.

Measurements of the characteristics of small-amplitude shape oscillations of drops immersed in a host liquid have been carried out by acoustical means. The resonance frequencies of the first few modes have been measured, as well as the damping constant for the fundamental mode, as functions of the drop radius and viscosities of the two liquids. A qualitative photographic study during steady oscillations has revealed a simple internal fluid-particle flow field with no circulation. The theory available at the present time has been found to provide results which are in general agreement with experimental findings for low-viscosity liquids.

The refraction of progressive internal waves on sloping bottoms is treated for the case of constant Brunt—Väisälä frequency. In two dimensions simple, explicit expressions for the changing wavelengths and amplitudes are found. For small slopes, the solutions reduce to simple propagating waves at infinity.

The singularity along a characteristic is shown to be removable, though the solutions are now inhomogeneous waves. The viscous boundary layers of the wedge geometry are briefly considered with the inviscid solutions remaining as interior solutions.

A theory valid for small slopes is obtained for three-dimensional waves. The waves are refracted in the usual manner, turning parallel to the beach in shallow water.

In Part 1 of this work (Barenblatt 1993) a non-universal scaling law (depending on the Reynolds number) for the mean velocity distribution in fully developed turbulent shear flow was proposed, together with the corresponding skin friction law. The universal logarithmic law was also discussed and it was shown that it can be understood, in fact, as an asymptotic branch of the envelope of the curves corresponding to the scaling law.

Here in Part 2 the comparisons with experimental data are presented in detail. The whole set of classic Nikuradze (1932) data, concerning both velocity distribution and skin friction, was chosen for comparison. The instructive coincidence of predictions with experimental data suggests the conclusion that the influence of molecular viscosity within the main body of fully developed turbulent shear flows remains essential, even at very large Reynolds numbers. Meanwhile, some incompleteness of the experimental data presented in the work of Nikuradze (1932) is noticed, namely the lack of data in the range of parameters where the difference between scaling law and universal logarithmic law predictions should be the largest.

Experimental and computational results are presented on an aerofoil undergoing pitch oscillations in ground effect, that is, close to a solid boundary. The time-averaged thrust is found to increase monotonically as the mean position of the aerofoil approaches the boundary while the propulsive efficiency stays relatively constant, showing that ground effect can enhance thrust at little extra cost for a pitching aerofoil. Vortices shed into the wake form pairs rather than vortex streets, so that in the mean a momentum jet is formed that angles away from the boundary. The time-averaged lift production is found to have two distinct regimes. When the pitching aerofoil is between 0.4 and 1 chord lengths from the ground, the lift force pulls the aerofoil towards the ground. In contrast, for wall proximities between 0.25 and 0.4 chord lengths, the lift force pushes the aerofoil away from the ground. Between these two regimes there is a stable equilibrium point where the time-averaged lift is zero and thrust is enhanced by approximately 40 %.

The three-dimensional problem of blunt-body impact onto the free surface of an ideal incompressible liquid is considered within the Wagner theory. The theory is formally valid during an initial stage of the impact. The problem has been extensively studied in both two-dimensional and axisymmetric cases. However, there are no exact truly three-dimensional solutions of the problem even within the Wagner theory. At present, three-dimensional effects in impact problems are mainly handled approximately by using a sequence of two-dimensional solutions and/or aspect-ratio correction factor. In this paper we present exact analytical rather than approximate solutions to the three-dimensional Wagner problem. The solutions are obtained by the inverse method. In this method the body velocity and the projection on the horizontal plane of the contact line between the liquid free surface and the surface of the entering body are assumed to be given at any time instant. The shape of the impacting body is determined from the Wagner condition. It is proved that an elliptic paraboloid entering calm water at a constant velocity has an elliptic contact line with the free surface. Most of the results are presented for elliptic contact lines, for which analytical solutions of the inverse Wagner problem are available. The results obtained can be helpful in testing other numerical approaches and studying the influence of three-dimensional effects on the liquid flow and the hydrodynamic loads.

An experimental investigation of the two-dimensional incompressible mixing layer was carried out. The measurements provide new information on the development of the mean and turbulent fields towards a self-preserving state and on the higher-order statistical characteristics of the turbulent field. The relevance of initial conditions to the development of the flow is discussed in the light of both present and previous data. Measurements of spectra, probability densities and moments to eighth order of all three velocity-component fluctuations at various transverse positions across the flow were carried out using an on-line digital data acquisition system. The probability density distributions of the derivative and the squared derivative of the longitudinal and lateral velocity fluctuations were also determined. Direct measurements of moments to eighth order of the velocity derivatives were attempted and are discussed in the light of the simultaneously measured histograms. The problems in obtaining higher-order statistical data are considered in some detail. Estimates of the integral time scale of many of the higher-order statistics are presented. The high wave-number structure was found to be locally anisotropic according to both spectral and turbulent velocity-gradient moment requirements. Higher-order spectra to fourth order of the longitudinal velocity fluctuations were measured and are discussed. Finally the lognormality of the squared longitudinal and lateral velocity-derivative fluctuations was investigated and the universal lognormal constant μ was evaluated.

An experimental study has been made of the concept of controlling boundary-layer transition by superimposing in the flow Tollmien–Schlichting waves that are of equal amplitude and antiphased to the disturbances that grow and lead to transition. The cases that have been considered are transition arising from a single-frequency two-dimensional disturbance and transition arising from a nonlinear interaction between two waves of different frequency. A feedback system for controlling transition has also been studied. In each case, both hot-wire surveys and flow visualization have shown that it is possible to delay transition but that the flow cannot be restored completely to its undisturbed state. This appears to be a consequence of interactions between the very weak three-dimensional background disturbances in the flow and the primary two-dimensional waves. The implications of these findings in an implementation of the concept are discussed.

Dissipative heating is produced by irreversible processes, such as viscous or ohmic heating, in a convecting fluid; its importance depends on the ratio d/HT of the depth of the convecting region to the temperature scale height. Integrating the entropy equation for steady flow yields an upper bound to the total rate of dissipative heating in a convecting layer. For liquids there is a regime in which the ratio of dissipative heating to the convected heat flux is approximately equal to c(d/HT), where the constant c is independent of the Rayleigh number. This result is confirmed by numerical experiments using the Boussinesq approximation, which is valid only if d/HT is small. For deep layers the dissipative heating rate may be much greater than the convected heat flux. If the earth's magnetic field is maintained by a convectively driven dynamo, ohmic losses are limited to 5% of the convected flux emerging from the core. In the earth's mantle viscous heating may be important locally beneath ridges and behind island arcs.

Using numerical schemes, this paper demonstrates how viscous resistance to volume changes modifies the simplest shock wave solutions presented in Part 1. For an initial condition chosen to form a step-function shock, viscous resistance causes the shock to disperse into a rank-ordered wavetrain of solitary waves. Large obstructions in flux produce large-amplitude, slow-moving wavetrains while smaller shocks shed small-amplitude waves. While the viscous resistance term is initially important over a narrow boundary layer, information about obstructions in the flux can propagate over many compaction lengths through the formation of non-zero wavelength porosity waves. For large-amplitude shocks, information can actually propagate backwards relative to the matrix. The physics of dispersion is discussed and a physical argument is presented to parameterize the amplitude of the wavetrain as a function of the amplitude of the predicted shock. This quantitative relationship between the prediction of shocks and the development of solitary waves also holds when mass transfer between solid and liquid is included. Melting causes solitary waves to decrease in amplitude but the process is reversible and freezing can cause small perturbations in the fluid flux to amplify into large-amplitude waves. These model problems show that the equations governing volume changes of the matrix are inherently time dependent. Perturbations to steady-state solutions propagate as nonlinear waves and these problems demonstrate several initial conditions that do not relax to steady state. If these equations describe processes such as magma migration in the Earth, then these processes should be inherently episodic in space and time.

Two-dimensional laminar flow of an incompressible viscous fluid through a channel with a sudden expansion is investigated. A continuation method is applied to study the bifurcation structure of the discretized governing equations. The stability of the different solution branches is determined by an Arnoldi-based iterative method for calculating the most unstable eigenmodes of the linearized equations for the perturbation quantities. The bifurcation picture is extended by computing additional solution branches and bifurcation points. The behaviour of the critical eigenvalues in the neighbourhood of these bifurcation points is studied. Limiting cases for the geometrical and flow parameters are considered and numerical results are compared with analytical solutions for these cases.

When a horizontal layer of fluid is heated from below and cooled from above with the mean temperature and physical parameters of the fluid constant, the two-dimensional roll is known to be the stable solution near the critical Rayleigh number. In this study, with the mean temperature changing steadily at a rate η, the Rayleigh number and the velocity and temperature fields governed by the Boussinesq equations are expanded in two parameters: η, and the amplitude ε. Hexagons are shown to be the stable solution near the critical Rayleigh number. The direction of the motion depends upon the sign of η. A finite amplitude instability is possible with an associated hysteresis in the heat flux as the critical Rayleigh number is approached from below or from above.

This paper presents results obtained from a numerical solution to a stream function–vorticity formulation of the Navier–Stokes equations for the flow around a circular cylinder in planar oscillating flow at small Keulegan–Carpenter numbers (KC) in the subcritical Reynolds number (Re) range. The equations are solved by finite-difference methods. For very small KC ([les ] 1), the numerical results coincide with analytical solutions. As KC is increased, the incipient separation and instability leading to an asymmetrical flow with vortex shedding are predicted. Computed flow fields at small KC values are compared to flow visualizations, and good agreement is found for moderate β-values (≈ 250). The well-documented flow regimes with the transverse vortex street, single-, double- and three-pair shedding, are predicted by the model. Although the flow is not fully resolved for the highest Re values, comparisons of calculated drag and inertia coefficients with experimental data for three different values of the frequency parameter β in the range 196–1035 for 0 < KC < 26 show good agreement.

Large-eddy simulation is used to investigate fully developed turbulent flow in a neutral channel wherein the lower wall is sinusoidal. The numerical results are compared with experimental observations for wave slopes ranging from 0 to 0.628. Particular emphasis is placed on the separated flow induced by a large-amplitude wave. A detailed comparison with the data of Buckles, Hanratty & Adrian (1984) shows generally good agreement. Large-eddy simulation surface pressures are integrated to calculate form drag as a function of wave slope. Drag is found to increase quadratically with slope for small-amplitude waves, with a somewhat slower increase for larger amplitudes. However, comparison with experimental measurements is confounded by uncertainties with the values reported in the literature. An interesting feature characteristic of all wavy-surface simulations is an increase in transverse velocity fluctuations on the wave upslope. Although the precise mechanism responsible is not known, analysis shows it to be associated with temporally persistent vortex-like structures localized near the surface. The magnitude of the fluctuation increase appears to scale quadratically with slope for small-amplitude waves, in contrast to the streamwise fluctuations, which increase linearly.

Rayleigh-Bénard convection in the presence of a plane Couette flow is investigated by numerical computations. From earlier work it is well known that longitudinal rolls are preferred at the onset of convection and that at Prandtl numbers of the order unity or less these rolls become unstable with respect to the wavy instability which introduces wavy distortions perpendicular to the axis of the rolls. In the present analysis the three-dimensional flows arising from these distortions are studied and their stability is considered. A main result is the subcritical existence of three-dimensional flows at Rayleigh numbers far below the critical value for onset of convection.

This paper reports an experimental investigation of pressure and force distributions on a sharp-nosed circular cylinder inclined to a uniform low-speed air flow under conditions of laminar separation of the boundary layer. The main concern is with the out-of-plane force (i.e. the side force if the body is at incidence). The experimental model consisted of an extensively pressure-tapped cylinder to which four different noses were fitted. The results show that there is an oscillatory distribution of out-of-plane force along the cylinder for most of the inclination range 0-90°. The amplitude of this distribution is strongly affected by nose shape in conditions where the out-of-plane force extends onto the nose. At very high angles of inclination the oscillatory distribution disappears and is replaced by a vortex pattern like that found on an infinite yawed cylinder. The general nature of the out-of-plane force is found to be consistent with the impulsively started flow analogy. Unsteadiness in the flow was found to cause a serious reduction in many of the time-averaged values. The unsteadiness is ascribed to the switching of the flow pattern due to free-stream turbulence. Measurements of the time histories of certain pressures enabled values of the force in the unswitched state to be calculated. The Reynolds number was found to have an important influence at inclinations above 55°. However, it was also found that the range of Reynolds numbers over which this effect occurs can depend on the scale of the model.