We have found that the reflected wave that is created by a right-going solitary wave as it travels in a region of slowly changing depth does not satisfy Green's law. The amplitude of the reflected wave is constant along left-going characteristics rather than proportional to the negative fourth root of depth. This new finding allows us to satisfy the mass-flux conservation laws to leading order and establishes that the perturbed Korteweg–de Vries equation is a consistent approximation for the right-going profile.

In order to determine the downstream consequences of the presence of small discrete surface discontinuities situated on otherwise smooth surfaces and subjected to six equilibrium two-dimensional adverse-pressure-gradient turbulent boundary-layer flows, these conditions were first established in a special-purpose wind tunnel. A surface discontinuity is small if it lies within the logarithmic region of the undisturbed boundary layer. Immediately downstream of such discontinuities flow separation ensues. After the subsequent reattachment, measurements were made of the downstream boundary-layer development. Even in strong adverse pressure gradients the local increments of momentum thickness caused by these roughness elements were well predicted by Gaudet & Johnson's zero-pressure-gradient correlation. With highly adverse pressure gradients it was found that these small surface discontinuities have little influence on the flow downstream. The essential outcome of this work is that the incremental drag of small roughness elements depends solely on local wall variables. Thus, when the pressure gradient is strongly adverse and the local skin friction is correspondingly small, the incremental drag of the roughness element becomes similarly small. After reattachment, it has an insignificant effect on the flow downstream of it.

In a three-dimensional simulation higher-order derivative correlations, including skewness and flatness (or kurtosis) factors, are calculated for velocity and passive scalar fields and are compared with structures in the flow. Up to 1283 grid points are used with periodic boundary conditions in all three directions to achieve Rλ to 82.9. The equations are forced to maintain steady-state turbulence and collect statistics. The scalar-derivative flatness is found to increase much faster with Reynolds number than the velocity-derivative flatness, and the velocity- and mixed-derivative skewnesses do not increase with Reynolds number. Separate exponents are found for the various fourth-order velocity-derivative correlations, with the vorticity-flatness exponent the largest. This does not support a major assumption of the lognormal and β models, but is consistent with some aspects of structural models of the small scales. Three-dimensional graphics show strong alignment between the vorticity, rate-of-strain, and scalar-gradient fields. The vorticity is concentrated in tubes with the scalar gradient and the largest principal rate of strain aligned perpendicular to the tubes. Velocity spectra, in Kolmogorov variables, collapse to a single curve and a short $-\frac{5}{3}$ spectral regime is observed.

Unsteady lifting-line theory is developed for a wing of large aspect ratio oscillating at low frequency in inviscid incompressible flow. The wing is assumed to have a rigid chord but a flexible span. Use of the method of matched asymptotic expansions reduces the problem from a singular integral equation to quadrature. The pressure field and airloads, for a prescribed wing shape and motion, are obtained in closed form as expansions in inverse aspect ratio. A rigorous definition of unsteady induced downwash is also obtained. Numerical calculations are presented for an elliptic wing in pitch and heave; compared with numerical lifting-surface theory, computation time is reduced significantly. The present work also identifies and resolves errors in the unsteady lifting-line theory of James (1975), and points out a limitation in that of Van Holten (1975, 1976, 1977).

A one-dimensional inviscid slice model has been used to study numerically the influence of axial microgravity on the breaking of liquid bridges having a volume close to that of gravitationless minimum volume stability limit. Equilibrium shapes and stability limits have been obtained as well as the dependence of the volume of the two drops formed after breaking on both the length and the volume of the liquid bridge. The breaking process has also been studied experimentally. Good agreement has been found between theory and experiment for neutrally buoyant systems.

Bodies that absorb, reflect or generate wave energy are submitted to mean forces. For moving bodies the mean forces in the direction of motion contribute to the drag or propulsion of the body. For flexible and deformable slender bodies swimming in waves at a constant forward velocity U normal to the crests of the waves, the mean rate of working $\overline{W}$ and the mean thrust $\overline{T}$ are evaluated. When the waves are assumed to be not significantly affected by the swimming slender bodies it is found that the Froude efficiency of propulsion for cases without shedding of vorticity is invariably given by U/(U + c), U + c being the phase velocity of the waves with respect to the body. The result remains valid when shedding small amounts of vorticity. $\overline{T}$ is obtained as the result of the radiation stress, and is proportional to $\overline{W}$.

The same efficiency can be realized by two-dimensional bodies oscillating in regular trains of two-dimensional waves. It is also valid for wave-making boats. For three-dimensional cases U/(U + c) represents the upper limit when the outgoing waves are properly beamed. Actuator surfaces with constant loading will be interpreted as vortex wavemakers.

A linear solution modelling the shock-cell structure of an axisymmetric supersonic jet operated at slightly off-design conditions is developed by the method of multiple scales. The model solution takes into account the gradual spatial change of the mean flow in the downstream direction. Turbulence in the mixing layer of the jet has the tendency to smooth out the sharp velocity and density gradients induced by the shocks. To simulate this effect, eddy-viscosity terms are incorporated in the model. Extensive comparisons between the numerical results of the present model and experimental measurements gathered at the NASA Langley Research Center over the Mach number range of |Mj2 − Md2| [les ] 1.0 for underexpanded and overexpanded supersonic jets are carried out. Here Mj is the fully expanded jet Mach number and Md is the design Mach number of the convergent–divergent nozzle. Very favourable agreement is found. This is especially true for the gross features of the shock cells, including the shock-cell spacings and the pressure amplitudes associated with the shocks. The measured data show that the pressure distributions over the first three or four shock cells usually are rich in fine structures. These fine structures are reproduced by the calculated results. Beyond the first few shock cells the model predicts that the shock-cell structure can be represented by a single Fourier mode of the mean flow. This is confirmed by a careful examination of the experimental data. The appropriate turbulent Reynolds number for shock-cell structure calculation is investigated. It is shown that the best choice is the same as the value found to give the best results for jet mean-flow calculation. The present model is used to explain some of the observed characteristics of broadband shock-associated noise.

We consider the motion of two rings of liquids with different viscosities and densities lying between concentric cylinders that rotate with the same angular velocity Ω. Gravity is neglected and interfacial tension is included. We show that rigid motions are globally stable and that the shape of the interface which separates the two fluids is determined by a minimizing problem for a potential [Pscr ] defined as the negative of the sum of the kinetic energies of two rigid motions plus the surface energy of the interface. We show that the stable interface between fluids has a constant radius when heavy fluid is outside and $J = - d^3 [\![\rho]\!]\Omega^2/T$ is larger than four, where d is the mean radius, $[\![\rho]\!] < 0$ the density difference and T the surface tension. When J is negative the heavy fluid is inside and the interface must be corrugated. The potential of flows with heavy fluid outside is smaller, thus relatively more stable, than when light fluid is outside, whenever J is large or for any J when the volume ratio m of heavy to light fluid is greater than one. These results give partial explanation of the stability and shape of rollers of viscous oils rotating in water and the corrugation of the free surface of films coating rotating cylinders.

The classical Heisenberg method of solving the Orr–Sommerfeld equation is modified in such a way that inner and outer expansions are replaced by a uniformly valid successive approximation in which no data on the second derivative of the parallel shear profile are needed. It is shown that this feature enables us to calculate stability characteristics for wider classes of flows with improved accuracy. As a preliminary check for validity of the method, stability of the Blasius flow is calculated and compared with existing methods. It turns out that the method works for high Reynolds numbers, up to about 105, and that the expressions for the eigenfunctions and the eigenvalue condition are much simpler than those found by existing methods.

The parabolic approximation is developed to study the combined refraction/diffraction of weakly nonlinear shallow-water waves. Two methods of approach are used. In the first method Boussinesq equations are used to derive evolution equations for spectral-wave components in a slowly varying two-dimensional domain. The second method modifies the K–P equation (Kadomtsev & Petviashvili 1970) to include varying depth in two dimensions. Comparisons are made between present numerical results, experimental data (Whalin 1971) and previous numerical calculations (Madsen & Warren 1984).

We consider the problem of steady fast flow of a family of Oldroyd fluids into a hole, and show that the field of flow is partitioned into elliptic (subcritical) and hyperbolic (supercritical) regions. We analyse the characteristics and show that the vorticity changes type as in the experiments of Metzner, Uebler & Fong (1969).

The mobility of a sphere moving slowly along the axis of a rotating, viscous, imcompressible fluid has been calculated for zero Reynolds number R and values of the Taylor number T ranging from zero to infinity, using a method of induced forces. For small values of T the mobility has been expanded in a power series in T½; the first seven terms of this series have been evaluated. Very good agreement is found with experimental data, which are only available for T [les ] 0.75.

We compare results of high-resolution direct numerical simulation with equivalent two-point moment closure (the test-field model) for both randomly forced and spin-down problems. Our results indicate that moment closure is an adequate representation of observed spectra only if the random forcing is sufficiently strong to disrupt the dynamical tendency to form intermittent isolated vortices. For strong white-noise forcing near a lower-wavenumber cut-off, theory and simulation are in good agreement except in the dissipation range, with an enstrophy range less steep than the wavenumber to the minus fourth power. If the forcing is weak in amplitude, red noise, and at large wavenumbers, significant errors are made by the closure, particularly in the inverse-cascade range. For spin-down problems at large Reynolds numbers, the closure considerably overestimates enstrophy transfer to small scales, as well as energy transfer to large scales. We finally discuss the possibility that the closure errors are related to intermittency of various types. Intermittency can occur in either the inverse-cascade range (forced equilibrium) or the intermediate scales (spin-down), with isolated concentrations of vorticity forming the associated coherent structures, or it can occur in the dissipation range owing to the nonlinear amplification of variations in the cascade rate (Kraichnan 1967).

Anomalous modes are flows in the Taylor experiment that exist only for sufficiently high Reynolds number R and are always distinct from the primary flow produced by gradually increasing R from small values. They are distinguished from all other secondary modes by having a direction of spiralling of one or both of the end cells such that outward flow is found along the stationary endwall. In this paper we present new observations of these flows and compare them with numerical solutions of the Navier–Stokes equations. A numerical technique for calculating anomalous modes is described and stability curves for 2-, 3-, and 4-cell flows are presented. Streamline plots of the numerical solutions are compared with photographs of the observed flows. The agreement between the calculations and experiments is good. The calculations also confirm certain theoretical predictions made by Benjamin (1978), Benjamin & Mullin (1981) and Hall (1982).

We derive a system of effective equations for wave propagation in a bubbly liquid. Starting from a microscopic description, we obtain the effective equations by using Foldy's approximation in a nonlinear setting. We discuss in detail the range of validity of the effective equations as well as some of their properties.

The standard hyperbolic methods used to solve the compressible Euler equations are not effective in the limit of incompressible flow. The sound waves dominate the system and it becomes poorly conditioned for numerical solution. For steady flow governed by the incompressible Euler equations, artificial compressibility is a technique that removes the troublesome sound waves. It leads to a hyperbolic system of equations that we solve by finite-volume differences centred in space, and explicit multistage time-stepping. The stability of this novel system is analysed, its allowable discontinuities are described, and appropriate far-field and solid-wall boundary conditions are introduced. Results are presented for both two- and three-dimensional flows, including vorticity shed from a delta wing. Whether vorticity is produced or not depends very strongly on the body geometry, the accuracy of the solution method, and the transient discontinuities that evolve in the flow field. The results are analysed for the total-pressure losses in the flow fields, and for the diffusion of the vortex sheets.

Turbulent and laminar pulsating flows in a straight smooth pipe are compared at identical frequencies and Reynolds numbers. Most measurements were made at a mean Reynolds number of 4000, but the influence of Re was checked for 2900 < Re < 7500. The period of forcing ranged from 0.5 to 5 s, with corresponding change in the non-dimensional frequency parameter α = R √(ω/ν) from 4.5 to 15. The amplitude of the imposed oscillations did not exceed 35% of the mean in order to avoid flow reversal or relaminarization. Velocities at the exit plane of the pipe and pressure drop along the pipe were measured simultaneously; velocity measurements were made with arrays of normal hot wires. The introduction of the periodic surging had no significant effect on the time-averaged quantities, regardless of the flow regime (i.e. in both laminar and turbulent flows). The time-dependent components at the forcing frequency, represented by a radial distribution of amplitudes and phases, are qualitatively different in laminar and turbulent flows. The ensemble-averaged turbulent quantities may also be represented by an amplitude and a phase; however, the non-harmonic content of these intensities increases with increasing amplitude of the imposed oscillations. A normalization procedure is proposed which relates phase-locked turbulent flow parameters in unsteady flow to similar time-averaged quantities. An integral momentum equation in a time-dependent flow requires that a triad of forces (pressure, inertia and shear) will be in equilibrium at any instant of time. All the terms in the force-balance equation were measured independently, providing a good check of data. The analysis of the experimental results suggests that turbulence adjusts rather slowly to the local mean-flow conditions. A simple eddy-viscosity model described by a complex function can account for ‘memory’ of turbulence and explain the different phase distribution in laminar and turbulent flows.

Numerical predictions are presented for the centrifugally driven free convection in a sealed rotating cavity and for the buoyancy-affected flow through a cavity with an inner cylindrical source and an outer cylindrical sink. Results for a sealed cavity filled with a high-viscosity silicone oil are in good agreement with previously published experimental measurements of the mean Nusselt number. When the heat transfer is conduction-dominated the results away from the cylindrical surface agree with Dorfman's (1968) similarity solution, but as convection becomes important they depart from this solution. In an air-filled cavity, for both the free convection and radial outflow cases, the results away from the cylindrical surface are generally in reasonable agreement with Chew's (1982) similarity solution, although property variations and radial heat conduction do cause some departure from this solution. The extent of the region in which the heat transfer was influenced by the presence of the cylindrical surface, and the Nusselt number distribution in this region are shown to be sensitive to the thermal boundary conditions imposed on this surface.

Previous contributions made by physical models to the understanding of cochlear mechanics suggested that a new cochlear model should be constructed. This paper illustrates the results obtained with a rectilinear, three-chamber model. The model was geometrically scaled 50:1 and contained the constituent elements of the cochlear cross-section including the basilar membrane, Reissner's membrane, the tectorial membrane and the organ of Corti. The basilar membrane was stretched crosswise in order to simulate real basilar membrane anisotropy. Two kinds (rigid and elastic) of tectorial membranes were used. The ductus and the sulcus were made visible and the model was also provided with displacement transducers to measure the axial and cross components of the oscillating fluid motion in the scala media. The adoption of a highly flexible membrane, simulating Reissner's membrane, made it possible to vary the viscosity of the scala media compared to that of the other two scalae. The reasons why the simplifications of the previous models were partially rejected and the criteria adopted to assure dynamic similitude between the model and the real cochlea are described in the paper. The results of tests carried out to determine the partial distribution of the amplitude maximum, the phase velocity along the axis of the model and the dispersion curves are shown. The same tests were repeated with partially filled scala vestibuli. Lastly a typical nonlinear feature, that is a continuous flow in the scala media, is described.

The Stokes force on a fluid droplet is obtained when the droplet is placed in an unbounded fluid medium and motion ensues due to an arbitrary interfacial-tension gradient on the droplet surface. The force, derived here for a spherical droplet, is proportional to the integral of the interfacial-tension gradient over the droplet surface. It may be calculated without solving the complete governing equations from a knowledge of this integral and the ratio of the viscosity of the droplet phase to that of the continuous phase, as shown in the principal result displayed in (29).

When the interfacial-tension gradients are caused by temperature or concentration variations, the result for the force may be further specialized when convective transport effects are negligible. In this case, it is possible to express the force in terms of the gradient of the undisturbed temperature (or concentration) field evaluated at the location of the droplet centre in a form analogous to Faxén's force law.