The three-dimensional structure of a moderate-Reynolds-number (≈ 100) plane wake behind a flat plate subjected to periodic spanwise perturbations has been studied both experimentally and numerically. Comparisons between experimental interface visualizations and numerical calculations demonstrate that important features of the development of the three-dimensional evolution can be reproduced by numerical inviscid vortex dynamics.

It is shown that the redistribution, reorientation and stretching of vorticity leads to the formation of counter-rotating pairs of streamwise vortices which superimpose onto the spanwise Kármán-like vortices. These streamwise vortices exhibit lambdashaped structures and are located in the braids connecting consecutive Kármán vortices of opposite sign.

The interaction of the evolving streamwise structure with the spanwise Kármán vortices results in the formation of closed vortex loops. Depending on the orientation of the initial perturbation, the three-dimensional vorticity field of the wake acquires either a symmetric or a non-symmetric configuration. Under the effect of a periodic vertical perturbation, the wake develops a non-symmetric vorticity mode exhibiting a staggered array of closed vortex loops of alternating sign. In contrast, under the effect of a periodic horizontal perturbation, the wake acquires a symmetric vorticity mode with the closed vortex loops of alternating sign aligned in the flow direction.

A numerical investigation of finite-amplitude, non-axisymmetric disturbances, in the form of travelling spiral waves, is made in pipe flow with superimposed solid-body rotation. Rotating pipe flow is found to be supercritically unstable both in the rapid and slow-rotation regimes. Earlier weakly nonlinear calculations, suggesting subcritical instability in the slow-rotation limit, are shown to be in error. Bifurcating neutral waves are calculated for various axial and azimuthal Reynolds numbers and wavenumbers. For fixed axial mean pressure gradient, the axial mean flow induced by these waves gives rise to a significant flux defect, in certain cases as large as 40-50% of the undisturbed mass flux; the possible relevance of this finding to the phenomenon of vortex breakdown is pointed out. In non-rotating pipe flow, no neutral disturbances in the assumed form of spiral waves are found for moderate Reynolds numbers; this indicates that previous conjectures, regarding a possible connection between nonlinear spiral waves in slowly rotating pipe flow and the asymptotic neutral states of Smith & Bodonyi (1982) in non-rotating pipe flow, are not valid.

The behaviour of a turbulent region generated within a linearly-stratified fluid by an external energy source has been studied experimentally. A monoplanar grid that generated small-amplitude oscillations was used as the energy source. The results show that the mixed layer initially grows rapidly, as in an unstratified fluid, but when its physical vertical size becomes rf ∼ (K1/N)½, at a time tf ≈ 4.0 N−1, where N is the buoyancy frequency and K1 is the ‘action’ of the grid, the buoyancy forces become dominant and drastically reduce further vertical growth of the patch. While the patch size remains at rf, a well-defined density interfacial layer is formed at the entrainment interface. An important feature of the interfacial layer is the presence of internal waves, excited by the mixed-layer turbulence. If the grid oscillations are continuously maintained, the interfacial waves break and cause turbulent mixing, thereby increasing the size of the patch beyond rf at a very slow rate. Theoretical estimates are made for the growth characteristics and are compared with the experimental results.

Sound waves can be generated in a colloid by the application of an alternating electric field. In this paper we describe the method for calculating this and the related electro-acoustic phenomenon of electric fields generated by sound waves. As an illustration of the procedure, we obtain formulae for these two effects for a suspension of spherical particles with thin double layers, in a parallel plate geometry.

The unsteady flow of a viscous, incompressible fluid in a channel with a moving indentation in one wall has been studied by numerical solution of the Navier-Stokes equations. The solution was obtained in stream-function-vorticity form using finite differences. Leapfrog time-differencing and the Dufort-Frankel substitution were used in the vorticity transport equation, and the Poisson equation for the stream function was solved by multigrid methods. In order to resolve the boundary-condition difficulties arising from the presence of the moving wall, a time-dependent transformation was applied, complicating the equations but ensuring that the computational domain remained a fixed rectangle.

Downstream of the moving indentation, the flow in the centre of the channel becomes wavy, and eddies are formed between the wave crests/troughs and the walls. Subsequently, certain of these eddies ‘double’, that is a second vortex centre appears upstream of the first. These observations are qualitatively similar to previous experimental findings (Stephanoff et al. 1983, and Pedley & Stephanoff 1985), and quantitative comparisons are also shown to be favourable. Plots of vorticity contours confirm that the wave generation process is essentially inviscid and reveal the vorticity dynamics of eddy doubling, in which viscous diffusion and advection are important at different stages. The maximum magnitude of wall vorticity is found to be much larger than in quasi-steady flow, with possibly important biomedical implications.

Bruun's (tabulated) universal function for the response of a hot-wire anemometer is examined in detail to see if the function has some unique relationship with King's law. As a result of the analysis, an approximate expression is derived and proposed for the response over a wide range of wind speeds. Although the expression is like King's law with an additional correction term and is thus quite simple in form, it agrees with Bruun's universal function (and also with the derivative, i.e. the velocity sensitivity) to within 3% over the range of wind speeds from 2 to 120 m/s. There is little doubt that the proposed response equation may serve as a helpful guide in calibrating a hot-wire probe, in constructing a simple linearizer circuit of high accuracy, and in linearization by means of a computer.

A new approach to contaminant diffusion in homogeneous turbulence is proposed. This approach is based on solving for the Lagrangian trajectories of many particles taking into account the interaction among their velocities. The velocity field at a given instant is composed of many ‘eddies’ distributed randomly and uniformly in space. The velocity of each eddy is proportional to the cube root of its size. In this way the calculated Eulerian correlation function between any two points is consistent with observations. The present model is used to calculate concentration fluctuations, concentration averages and intermittency as functions of location and time. Results were found to be in accordance with experimental measurements. Probability distributions as functions of time and location are also calculated.

For a long cylinder floating on the sea surface, incident sea waves with a narrow frequency band excite body oscillations of short and long periods. Depending on the stiffness of the mooring system, the body displacement of the long-period motion can be comparable with, or even greater than that of the short-period oscillations. By combining the asymptotic methods of multiple scales and inner and outer expansions, we describe an essentially analytical theory for slow sway of both small and large amplitudes. Besides showing results for various quasi-steady and transient incident waves for a rectangular cylinder, we examine the effect of the gap between the keel of the body and the sea bottom. It is found in particular that a small gap can enhance moderate resonance by blocking the flow due to long waves and increase the apparent mass of the cylinder. Real-fluid effects are not included.

We present results of the calculation of superharmonic normal mode perturbations to the exact nonlinear deep-water capilary wave solution of Crapper (1957). By using the method of Longuet-Higgins (1978a), we are able for the first time to consider all waveheights up to and including the maximum for two-dimensional perturbations. We find agreement with the recent asymptotic analysis of Hogan, Gruman & Stiassnie (1988). Superharmonic instabilities are found at various waveheights less than the maximum.

Laminar flow through slots is investigated using a flow-visualization technique and the numerical solution of the Navier-Stokes equations for steady flow. In the flow situation studied here, the fluid enters an upper channel blocked at the rear end and leaves through a lower channel blocked at the front end. The two channels are interconnected by one, two and three slots. The flow-visualization technique effectively brings out the various features of the flow through slot(s). The ratio of the slot width to the channel height w/h is varied between 0.5 to 4.0 and the Reynolds number Re, based on the velocity at the entry to the channel and the height of the channel, is varied between 300 and 2000. Both w/h and Re influence the flow in general and the extent of the regions of recirculating flow in particular. The Reynolds number at which the vortex shedding begins depends on w/h. Computations are carried out using the computer code 2/E/FIX of Pun & Spalding (1977). The computed flow patterns closely resemble the observed patterns at various Reynolds numbers investigated except around the Reynolds number where the vortex shedding begins.

Time-dependent mean velocities and dispersion coefficients are evaluated for a general two-dimensional laminar flow. A Lagrangian method is adopted by which a Brownian particle is traced in an artificially restructured velocity field. Asymptotic expressions for short, medium and long periods of time are obtained for Couette flow, plane Poiseuille flow and open-channel flow over an inclined flat surface. A new formula is suggested by which the Taylor dispersion coefficient can be evaluated from purely kinematical considerations. Within an error of less than one percent, over the entire time domain and for various flow fields, a very simple analytical expression is derived for the time-dependent dispersion coefficient \[ \tilde{D}(\tau) = D + D^T\left(1-\frac{1-{\rm e}^{-\alpha\tau}}{a\tau}\right), \] where D is the molecular diffusion coefficient, DT denotes the Taylor dispersion coefficient, τ stands for the non-dimensional time π2Dt/Y/, Y is the distance between walls and a = (N + 1)2 is an integer which is determined by the number of symmetry planes N that the flow field possesses. For Couette and open-channel flow there are no planes of symmetry and a = 1; for Poiseuille flow there is one plane of symmetry and a = 4.

The effect of wall suction on the organized motion of a tubulent boundary layer is examined experimentally both in a wind tunnel and in a water tunnel. In the windtunnel boundary layer, which developed over a slighly heated surface, temperature fluctuations were simultaneously obtained at several points, aligned in either the x (streamwise) or y (normal to the wall) direction. The temperature traces reveal the existence of two spatially coherent events, characterized either by a sudden decrease (cooling) or by a sudden increase (heating) of temperature. Estimates are presented for the average convection velocity, and average frequency of these events. The average convection velocity of ‘coolings’ is about 15% larger than that of ‘heatings’, the velocity of both events exhibiting an important local maximum in the buffer region. Near the wall, the convection velocity of both events is increased slightly by suction while their average frequency is reduced by suction. Away from the wall, the average inclination of ‘coolings’ and ‘heatings’ is about 40° without suction; suction does not alter the inclination of ‘coolings’ but increases that of ‘heatings’ to about 50°. Visualizations in the water tunnel indicate that suction increases the stability and the longitudinal coherence of low-speed streaks. They also show that suction reduces the average frequency of dye ejections into the outer layer.

A two-component laser-Doppler velocimeter was used to measure simultaneously velocity components parallel and normal to the wall in two fully developed, wellmixed, low-concentration (1-2 p.p.m.) drag-reducing channel flows and one turbulent channel flow of water. The mean velocity profiles, root-mean-square velocity profiles and the distributions of the ūv turbulent correlation confirm that the additives modify the buffer region of the flow. The principal influence of the additives is to damp velocity fluctuations normal to the wall in the buffer region.

The structural results show that the average time between bursts increased for the drag-reducing flows. When compared to a water flow at the same wall shear stress, this increase in the timescale was equal to the increase in the average streak spacing. Conditionally averaged velocity signals of y+ = 30 centred on the leading edge of a burst, as well as those centred on the trailing edge, have the same general characteristics in all three flows indicating that the basic structure of the fundamental momentum transport event is the same in these drag-reducing flows. However, it was clearly shown that the lower-threshold Reynolds-stress-producing motions were damped while the higher-threshold motions were not damped. In the buffer region of the drag-reducing flows this yields a larger mean velocity gradient with damped fluctuations normal to the wall and increased fluctuations in the streamwise direction. It is hypothesized that some strong turbulent motions are required to maintain extended polymer molecules, which produce a solution with properties that can damp lower threshold turbulence and thereby reduce viscous drag.

Smoke-wire flow visualization and hot-wire anemometry have been used to study near and far wakes of two-dimensional bluff bodies. For the case of a circular cylinder at 70 < Re < 2000, a very rapid (exponential) decay of velocity fluctuations at the Kármán-vortex-street frequency is observed. Beyond this region of decay, larger-scale (lower wavenumber) structure can be seen. In the far wake (beyond one hundred diameters) a broad band of frequencies is selectively amplified and then damped, the centre of the band shifting to lower frequencies as downstream distance is increased.

The far-wake structure does not depend directly on the scale or frequency of Kármán vortices shed from the cylinder; i.e. it does not result from amalgamation of shed vortices. The growth of this structure is due to hydrodynamic instability of the developing mean wake profile. Under certain conditions amalgamation can take place, but is purely incidental, and is not the driving mechanism responsible for the growth of larger-scale structure. Similar large structure is observed downstream of porous flat plates (Re ≈ 6000), which do not initially shed Kármán-type vortices into the wake.

Measured prominent frequencies in the far cylinder wake are in good agreement with those estimated by two-dimensional locally parallel inviscid linear stability theory, when streamwise growth of wake width is taken into account. Finally, three-dimensionality in the far wake of a circular cylinder is briefly discussed and a mechanism for its development is suggested based on a secondary parametric instability of the subharmonic type.

The motion of a spherical droplet in unbounded exterior phase in the presence of a stagnant cap of adsorbed insoluble surfactants has been recently presented by Sadhal & Johnson (1983). The present study considers the axisymmetric motion of a gas bubble at the exit of a circular orifice in the presence of a similar stagnant cap.

The solution procedure utilizes the boundary integral representation in order to obtain the drag correction factor for a bubble translating away from the orifice in otherwise quiescent fluid, and for a fixed bubble exposed to Sampson's flow towards the orifice. It is demonstrated that the presence of the confining orifice boundaries substantially increases the drag acting on the bubble, and the solution approaches the exact result of Sadhal and Johnson as the distance between the bubble and the orifice is increased. Furthermore, it is shown that for a fixed amount of surfactants on the bubble surface the cap angle increases with distance from the orifice due to the diminishing hydrodynamic interaction. Hence, the quasi-steady terminal velocity of a bubble rising from an orifice is reduced by the viscous boundary interaction, and by the growing immobile cap size.

Thin coating flows inside a rotating circular cylinder are investigated when the axis of rotation is perpendicular to the direction of gravity. Attention is restricted to flows of power-law fluids having negligible inertia. Four distinct steady-state liquid-film profiles are found to be possible. Two of the cases correspond to a continuous coating, i.e. films that cover the entire inner surface of the cylinder. The other two cases involve partial films covering a limited portion of the cylinder surface. Of the two continuous films, one is the expected configuration involving a coating that gradually changes in thickness as one moves around the cylinder, the film being thicker on the ascending portion of the cylinder and thinner on the descending portion. The second continuous-film configuration has regions on the rising side of cylinder where a rapid change in depth is possible. This case also has the potential to have recirculating zones where a portion of the fluid is trapped in either one or two eddies at fixed locations on the rising side of the cylinder. Of the two partial films, one corresponds to a weakly deformed puddle at the bottom of the cylinder and is the appropriate solution at small rotation rates. The second partial film is a film which coats a portion of the ascending side of the cylinder, the extent of which depends on the film volume.

In this paper experimentally studied characteristics of various frequency components of incoherent velocity fluctuations in the near wake behind a thin normal plate immersed in a uniform flow are described. Measurements were made at a position 8 plate heights downstream of the plate where the wake had a marked periodicity, so that coherent vortices shed from the plate are expected to have a small dispersion in streamwise spacing, transverse location, strength and shape. Shearing stress associated with the incoherent fluctuations is mainly contributed by components with frequencies around half of the vortex-shedding frequency fs on one side of the wake. The ½fs frequency component appears to be caused by the spanwise locations of ribs, which are connected to the coherent vortices, being different from vortex to vortex. A probable spanwise arrangement of the ribs is suggested.

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

This paper is an account of a summer programme for the study of the ideas and models of turbulent flows, using the results of direct numerical stimulations of the Navier-Stokes equations. These results had been obtained on the computers and stored as accessible databases at the Center for Turbulence Research (CTR) of NASA Ames Research Center and Stanford University. At this first summer programme, some 32 visiting researchers joined those at the CTR to test hypotheses and models in five aspects of turbulence research: turbulence decomposition, bifurcation and chaos; two-point closure (or k-space) modelling; structure of turbulent boundary layers; Reynolds-stress modelling; scalar transport and reacting flows.

A number of new results emerged including: computation of space and space-time correlations in isotropic turbulence can be related to each other and modelled in terms of the advection of small scales by large-scale motion; the wall layer in turbulent boundary layers is dominated by shear layers which protrude into the outer layers, and have long lifetimes; some aspects of the ejection mechanism for these layers can be described in terms of the two-dimensional finite-amplitude Navier-Stokes solutions; a self-similar form of the two-point, cross-correlation data of the turbulence in boundary layers (when normalized by the r.m.s. value at the furthest point from the wall) shows how both the blocking of eddies by the wall and straining by the mean shear control the lengthscales; the intercomponent transfer (pressure-strain) is highly localized in space, usually in regions of concentrated vorticity; conditioned pressure gradients are linear in the conditioning of velocity and independent of vorticity in homogeneous shear flow; some features of coherent structures in the boundary layer are similar to experimental measurements of structures in mixing-layers, jets and wakes.

The availability of comprehensive velocity and pressure data certainly helps the investigation of concepts and models. But a striking feature of the summer programme was the diversity of interpretation of the same computed velocity fields. There are few signs of any convergence in turbulence research! But with new computational facilities the divergent approaches can at least be related to each other.

This paper deals with the introduction of a nonlinear term into Darcy's equation to describe inertial effects in a porous medium. The method chosen is the numerical resolution of flow equations at a pore scale. The medium is modelled by cylinders of either equal or unequal diameters arranged in a regular pattern with a square or triangular base. For a given flow through this medium the pressure drop is evaluated numerically.

The Navier-Stokes equations are discretized by the mixed finite-element method. The numerical solution is based on operator-splitting methods whose purpose is to separate the difficulties due to the nonlinear operator in the equation of motion and the necessity of taking into account the continuity equation. The associated Stokes problems are solved by a mixed formulation proposed by Glowinski & Pironneau.

For Reynolds numbers lower than 1, the relationship between the global pressure gradient and the filtration velocity is linear as predicted by Darcy's law. For higher values of the Reynolds number the pressure drop is influenced by inertial effects which can be interpreted by the addition of a quadratic term in Darcy's law.

On the one hand this study confirms the presence of a nonlinear term in the motion equation as experimentally predicted by several authors, and on the other hand analyses the fluid behaviour in simple media. In addition to the detailed numerical solutions, an estimation of the hydrodynamical constants in the Forchheimer equation is given in terms of porosity and the geometrical characteristics of the models studied.