A detailed numerical study has been conducted on the effect of unsteadiness on the dynamics of counterflowing strained diffusion methane/oxygen/nitrogen flames. The modelling included the solution of the unsteady conservation equations of mass, momentum, energy, and species along the stagnation streamline in an opposed jet using detailed descriptions of chemistry and transport. The unsteadiness was introduced by independently imposing sinusoidal variations of the reactant velocity, concentration and temperature at the exits of the nozzles. The results demonstrate that the flame's response is quasi-steady at low frequencies, while at higher frequencies the amplitudes of the induced oscillations are reduced and phase shifted with respect to the imposed signal. At still higher frequencies, the flame no longer responds to the oscillations in the external field. A rigorous physical explanation of the frequency response was provided from first principles by identifying that oscillations imposed at the nozzle exits result in reactant concentration and temperature oscillations at the outer edge of the preflame diffusive zones. The diffusion attenuates the oscillations in this zone in a manner analogous to the velocity attenuation in Stokes’ second problem. The validity of the analogy was confirmed by examining flames over a wide range of frequencies and initial conditions. The current analysis also provided a criterion for the cutoff frequency separating the quasi-steady and transient regimes – information which can be useful in the establishment of laminar flamelet libraries.

Theoretical results based on the methods of Freeman and Garr & Marrone show that the stand-off distance and flow pattern of non-equilibrium dissociating flow of nitrogen over the front part of a blunt body can be correlated in terms of a single reaction rate parameter ω taking account of parameters describing the speed, density, dissociation and temperature of the free stream. The density pattern, which is sensitive to the reaction rate, consists of two distinct regions dominated by the effects of reaction and pressure respectively. The shape and size of these regions depend on Q. Experimental results obtained by optical interferometry in a free-piston shock tunnel confirm the theoretical results. A scale effect consistent with the induction time phenomenon suggested by Shui, Appleton & Keck modifies the theoretical results considerably in the case of small models.

A combined numerical and analytical approach is used to study the low-frequency shock motions observed in shock/turbulent-boundary-layer interactions in the particular case of a shock-reflection configuration. Starting from an exact form of the momentum integral equation and guided by data from large-eddy simulations, a stochastic ordinary differential equation for the reflected-shock-foot low-frequency motions is derived. During the derivation a similarity hypothesis is verified for the streamwise evolution of boundary-layer thickness measures in the interaction zone. In its simplest form, the derived governing equation is mathematically equivalent to that postulated without proof by Plotkin (AIAA J., vol. 13, 1975, p. 1036). In the present contribution, all the terms in the equation are modelled, leading to a closed form of the system, which is then applied to a wide range of input parameters. The resulting map of the most energetic low-frequency motions is presented. It is found that while the mean boundary-layer properties are important in controlling the interaction size, they do not contribute significantly to the dynamics. Moreover, the frequency of the most energetic fluctuations is shown to be a robust feature, in agreement with earlier experimental observations. The model is proved capable of reproducing available low-frequency experimental and numerical wall-pressure spectra. The coupling between the shock and the boundary layer is found to be mathematically equivalent to a first-order low-pass filter. It is argued that the observed low-frequency unsteadiness in such interactions is not necessarily a property of the forcing, either from upstream or downstream of the shock, but an intrinsic property of the coupled system, whose response to white-noise forcing is in excellent agreement with actual spectra.

The linear boundary-layer analysis by Stewartson & Roberts (1963) and by Roberts & Stewartson (1965) for the motion of a viscous fluid inside the spheroidal cavity of a precessing rigid body is extended to include effects due to the nonlinear terms in the boundary-layer equation. The most significant consequence is a differential rotation super-imposed on the constant vorticity flow given by the linear theory. In addition it is shown that a tidal bulge of the cavity forces a fluid motion similar to that caused by the precessional torque. The relevance of both effects for the liquid core of the earth is briefly discussed.

The influence of surface roughness on the slip behaviour of a Newtonian liquid in steady planar shear is investigated using three different approaches, namely Stokes flow calculations, molecular dynamics (MD) simulations and a statistical mechanical model for the friction coefficient between a corrugated wall and the first liquid layer. These approaches are used to probe the behaviour of the slip length as a function of the slope parameter $ka\,{=}\,2 \pi a/\lambda$, where $a$ and $\lambda$ represent the amplitude and wavelength characterizing the periodic corrugation of the bounding surface. The molecular and continuum approaches both confirm a monotonic decay in the slip length with increasing $ka$ but the rate of decay as well as the magnitude of the slip length obtained from the Stokes flow solutions exceed the MD predictions as the wall feature sizes approach the liquid molecular dimensions. In the limit of molecular-scale wall corrugation, a Green–Kubo analysis based on the fluctuation–dissipation theorem accurately reproduces the MD results for the behaviour of the slip length as a function of $a$. In combination, these three approaches provide a detailed picture of the influence of periodic roughness on the slip length which spans multiple length scales ranging from molecular to macroscopic dimensions.

We discuss a series of numerical experiments on the dispersion of neutrally buoyant particles in two-dimensional turbulent flows. The topology of two-dimensional turbulence is parametrized in terms of the relative dominance of deformation or rotation; this leads to a segmentation of the turbulent field into hyperbolic and elliptic domains. We show that some of the characteristic structural domains of two-dimensional turbulent flows, namely coherent structures and circulation cells, generate particle traps and peculiar accelerations which induce several complex properties of the particle dispersion processes at intermediate times. In general, passive particles are progressively pushed from the coherent structures and tend to concentrate in highly hyperbolic regions in the proximity of the isolines of zero vorticity. For large dispersion times, the background turbulent field is a privileged domain of particle richness; there is however a permanent particle exchange between the background field and the energetic circulation cells which surround the coherent structures. At intermediate times, an anomalous dispersion regime may appear, depending upon the relative weight of the different topological domains active in two-dimensional turbulence. The use of appropriate conditional averages allows the basic topology of two-dimensional turbulence to be characterized from a Lagrangian point of view. In particular, an intermediate $t^{\frac{5}{4}}$ anomalous dispersion law is shown to be associated with the action of hyperbolic regions where deformation dominates rotation; the motion of the advected particles in strongly elliptic regions where rotation dominates over deformation is shown to be associated with a $<e1>t<sup>t^{\frac{5}{3}}$ dispersion law. Because neutral particles concentrate on average in hyperbolic regions, the $<e1>t^{\frac{5}{4}}$ dispersion law is quite robust and it can be observed under very general circumstances.

The effect of applied magnetic fields on the deformation of a biocompatible hydrophobic ferrofluid drop suspended in a viscous medium is investigated numerically and compared with experimental data. A numerical formulation for the time-dependent simulation of magnetohydrodynamics of two immiscible non-conducting fluids is used with a volume-of-fluid scheme for fully deformable interfaces. Analytical formulae for ellipsoidal drops and near-spheroidal drops are reviewed and developed for code validation. At low magnetic fields, both the experimental and numerical results follow the asymptotic small deformation theory. The value of interfacial tension is deduced from an optimal fit of a numerically simulated shape with the experimentally obtained drop shape, and appears to be a constant for low applied magnetic fields. At high magnetic fields, on the other hand, experimental measurements deviate from numerical results if a constant interfacial tension is implemented. The difference can be represented as a dependence of apparent interfacial tension on the magnetic field. This idea is investigated computationally by varying the interfacial tension as a function of the applied magnetic field and by comparing the drop shapes with experimental data until a perfect match is found. This estimation method provides a consistent correlation for the variation in interfacial tension at high magnetic fields. A conclusion section provides a discussion of physical effects which may influence the microstructure and contribute to the reported observations.

The flow created by an impulsively started pressure distribution travelling at a constant velocity in a shallow channel is investigated. The restricted Green-Naghdi theory of fluid sheets is used to perform the three-dimensional calculations. The results show remarkable similarity to model tests. In particular, these calculations predict the periodic generation of two-dimensional solitons in front of and travelling faster than the disturbance if the disturbance is large enough. Behind the disturbance a complicated, doubly corrugated set of waves is formed. The computations also predict that periodic creation of solitons is accompanied by a correspondingly periodic oscillation of the wave drag, as well as a dramatic increase in the mean wave drag.

An experimental study of the occurrence of temperature oscillations in molten gallium contained in a rectangular boat and heated from the side is reported. The dependence of the critical temperature difference across the boat for which oscillations set in upon the boat dimensions and upon the strength of a transverse magnetic field is described. The dependence of the frequency of oscillation on these parameters is also reported together with measurements of the variation of the phase of the oscillations over the top surface of the melt. The results are discussed in relation to the theory in the companion paper by Gill (1974).

The stability characteristics of some fluid flows at high Taylor or Gortler numbers are determined using perturbation methods. In particular, the stability characteristics of some fully developed flows between concentric cylinders driven either by a pressure gradient or the motion of the inner cylinder are investigated. The asymptotic structure of short-wavelength disturbances to these flows is obtained and used as a basis for a formal perturbation solution to the corresponding stability problem appropriate to a developing boundary layer. The non-parallel effect of the basic flow on the condition for neutral stability is discussed. The results obtained suggest that the disturbances are concentrated in internal viscous or critical layers well away from the wall and the free stream. The stability of a boundary layer on a concave wall to Görtler vortices that propagate downstream is also considered. These modes are found to be more stable than the usual time-independent modes and they propagate downstream with the speed of the basic flow in the critical layer. Some comparison with previous experimental and theoretical work is given.

Numerical simulations are used to study homogeneous, forced turbulence in three-dimensional rotating, stably stratified flow in the Boussinesq approximation, where the rotation axis and gravity are both in the zˆ-direction. Energy is injected through a three-dimensional isotropic white-noise forcing localized at small scales. The parameter range studied corresponds to Froude numbers smaller than an O(1) critical value, below which energy is transferred to scales larger than the forcing scales. The values of the ratio N/f range from ≈1/2 to ∞, where N is the Brunt–Väisälä frequency and f is twice the rotation rate. For strongly stratified flows (N/f[Gt ]1), the slow large scales generated by the fast small-scale forcing consist of vertically sheared horizontal flow. Quasi-geostrophic dynamics dominate, at large scales, only when 1/2 [les ] N/f [les ] 2, which is the range where resonant triad interactions cannot occur.

We describe a tensorial generalization of the Navier slip boundary condition and illustrate its use in solving for flows around anisotropic textured surfaces. Tensorial slip can be derived from molecular or microstructural theories or simply postulated as a constitutive relation, subject to certain general constraints on the interfacial mobility. The power of the tensor formalism is to capture complicated effects of surface anisotropy, while preserving a simple fluid domain. This is demonstrated by exact solutions for laminar shear flow and pressure-driven flow between parallel plates of arbitrary and different textures. From such solutions, the effects of rotating a texture follow from simple matrix algebra. Our results may be useful for extracting local slip tensors from global measurements, such as the permeability of a textured channel or the force required to move a patterned surface, in experiments or simulations.

A cloud of solute injected into a pipe or channel is known to spread out by a dispersion process based on cross-sectional diffusion across a velocity shear. The original description of the process is due to Taylor (1953, 1954), and an important subsequent contribution was by Aris (1956), who framed and partially solved equations for the integral moments of the cloud of contaminant. The present work resolves some technical difficulties that occur when Aris’ solution method (separation of variables) is pursued in depth. In particular, it is shown that Aris’ technique has to be modified to give the moments at short and moderate times after the injection of solute into the flow. The paper is concerned with dispersion in those parallel flows for which an associated eigenvalue problem has a discrete spectrum of eigenvalues; fortunately, this case appears to be the rule rather than the exception. Expressions are obtained for the second and third moment about the mean, and the theory is applied to three cases of interest.

With moderate acoustic stimuli, measurements of basilar-membrane vibration (especially, those using a Mössbauer source attached to the membrane) demonstrate:

1. a high degree of asymmetry, in that the response to a pure tone falls extremely sharply above the characteristic frequency, although much more gradually below it;

2. a substantial phase-lag in that response, and one which increases monotonically up to the characteristic frequency;

3. a response to a ‘click’ in the form of a delayed ‘ringing’ oscillation at the characteristic frequency, which persists for around 20 cycles.

The response (iii) demands a travelling-wave model which incorporates an only lightly damped resonance. Admittedly, waveguide systems including resonance are described in classical applied physics. However, a classical waveguide resonance reflects a travelling wave, thus converting it into a standing wave devoid of the substantial phase-lag (ii); and produces a low-frequency cutoff instead of the high-frequency cutoff (i).

By contrast, another general type of travelling-wave system with resonance has become known more recently; initially, in a quite different context (physics of the atmosphere). This is described as critical-layer resonance, or else (because the resonance absorbs energy) critical-layer absorption. It yields a high-frequency cutoff; but, above all, it is characterized by the properties of the energy flow velocity. This falls to zero very steeply as the point of resonance is approached; so that wave energy flow is retarded drastically, giving any light damping which is present an unlimited time in which to dissipate that energy.

Existing mathematical models of cochlear mechanics, whether using one-, two- or three-dimensional representations of cochlear geometry, are analysed from this standpoint. All are found to have been successful (if only light damping is incorporated, as (iii) requires) when and only when they incorporate critical-layer absorption. This resolves the paradox of why certain grossly unrealistic one-dimensional models can give a good prediction of cochlear response; it is because they incorporate the one essential feature of critical-layer absorption.

At any point in a physical system, the high-frequency limit of energy flow velocity is the slope of the graph of frequency against wavenumberIn any travelling wave, the wavenumber is the rate of change of phase with distance; for example, it is 2π/λ in a sine wave of length λ. at that point. In the cochlea, this is a good approximation at frequencies above about 1 kHz; and, even at much lower frequencies, remains good for wavenumbers above about 0·2 mm−1 (which excludes only a relatively unimportant region near the base).

Frequency of vibration at any point can vary with wavenumber either because stiffness or inertia varies with wavenumber. However, we find that models incorporating a wavenumber-dependent membrane stiffness must be abandoned because they fail to give critical-layer absorption; this is why their predictions (when realistically light damping is used) have been unsuccessful. Similarly, models neglecting the inertia of the cochlear partition must be rejected.

One-dimensional modelling becomes physically unrealistic for wavenumbers above about 0.7 mni-1, and the error increases with wavenumber. The main trouble is that a one-dimensional theory makes the effective inertia ‘flatten out’ to its limiting value (inertia of the cochlear partition alone) too rapidly as wavenumber increases. Fortunately, a two-dimensional, or even a three-dimensional, model can readily be used to calculate a more realistic, and significantly more gradual, ‘flattening out’ of this inertia. All of the models give a fair representation of the experimental data, because they all predict critical-layer absorption. However, the more realistic two- or three-dimensional models must be preferred. These retard the wave energy flow still more, thus facilitating its absorption by even a very modest level of damping. The paper indicates many other features of these models.

The analysis described above is preceded by a discussion of waves generated a t the oval window. They necessarily include:

1. the already-mentioned travelling wave, or ‘slow wave’, in which the speed of energy flow falls from around 100 m s−l at the base to zero at the position of resonance;

2. a pure sound wave, or ‘fast wave’, travelling a t 1400 m s-1, with reflection at the apex which makes i t into a standing wave. Half of the rate of working of the stapes footplate against the oval window is communicated as an energy flow a t this much higher speed down the scala vestibuli, across the cochlear partition and back up the scala tympani to the round window, whence it becomes part of the slow general apical progress of the travelling wave; a progress which, as described above, comes to a halt altogether just in front of the position of resonance.

Mathematical detail is avoided in the discussion of cochlear energy flow in the main part of the paper (§ 1–10), but a variety of relevant mathematical analysis is given in appendices A–E. These include, also, new comments about the functions of the tunnel of Corti (appendix A) and the helicotrema (appendix C).

When deep-water surface gravity waves traverse an area with a curved or otherwise variable current, the current can act analogously to an optical lens, to focus wave action into a caustic region. In this region, waves of surprisingly large size, alternatively called freak, rogue, or giant waves are produced. We show how this mechanism produces freak waves at random locations when ocean swell traverses an area of random current. When the current has a constant (possibly zero) mean with small random fluctuations, we show that the probability distribution for the formation of a freak wave is universal, that is, it does not depend on the statistics of the current, but only on a single distance scale parameter, provided that this parameter is finite and non-zero. Our numerical simulations show excellent agreement with the theory, even for current standard deviation as large as 1.0 m s−1. Since many of these results are derived for arbitrary dispersion relations with certain general properties, they include as a special case previously published work on caustics in geometrical optics.

The flow field of an ‘axisymmetric’ vortex breakdown has been mapped using a laser-Doppler anemometer. The interior of the recirculation zone is dominated by energetic, non-axisymmetric, low frequency periodic fluctuations. Spectra for a number of points inside this zone, as well as time-averaged swirl and axial velocity profiles both inside and outside the recirculation zone, have been obtained. The time-averaged streamlines in the interior show an unexpected two-celled structure attributed to the action of the fluctuations. Although the present experiment deals with one particular breakdown, flow-visualization studies indicate that the case examined is typical of the ‘axisymmetric’ form of breakdown over a range of flow conditions.

The screech noise generation process from supersonic underexpanded jets, issuing from a sonic nozzle at pressure ratios of 2.4 and 3.3 (fully expanded Mach number, Mj=1.19 and 1.42), was investigated experimentally. The extremely detailed data provide a fresh, new look at the screech generation mechanism. Spark schlieren visualization at different phases of the screech cycle clearly shows the convection of the organized turbulent structures over a train of shock waves. The potential pressure field (hydrodynamic fluctuations) associated with the organized structures is fairly intense and extends outside the shear layer. The time evolution of the near-field pressure fluctuations was obtained from phase-averaged microphone measurements. Phase-matched combined views of schlieren photographs and pressure fluctuations show the sound generation process. The individual compression and rarefaction parts of the sound waves are found to be generated from similar hydrodynamic fluctuations. A partial interference between the upstream-propagating sound waves and the downstream-propagating hydrodynamic waves is found to be present along the jet boundary. The partial interference manifests itself as a standing wave in the root-mean-square pressure fluctuation data. The standing wavelength is found to be close to, but somewhat different from, the shock spacing. An outcome of the interference is a curious ‘pause and go’ motion of the sound waves along the jet periphery. Interestingly, a length scale identical to the standing wavelength is found to be present inside the jet shear layer. The coherent fluctuations and the convective velocity of the organized vortices are found to be modulated periodically, and the periodicity is found to match with the standing wavelength distance rather than the shock spacing. The reason for the appearance of this additional length scale, different from the shock spacing, could not be explained. Nevertheless, it is demonstrated that an exact screech frequency formula can be derived from the simple standing wave relationship. The exact relationship shows that the correct spacing between the sources, for a point source model similar to that of Powell (1953), should be a standing wavelength (not the shock spacing).

The classical lubrication-theory model of steady peristaltic transport of periodic sinusoidal waves in infinite-length tubes (Shapiro et al. 1969) is generalized to arbitrary wave shape and wavenumber in tubes of finite length. Whereas the classical model is steady in a frame of reference moving with the peristaltic waves, peristaltic transport in a finite-length tube is inherently non-steady. It may be shown, however, that pumping performance is independent of tube length if there exists an integral number of peristaltic waves in the tube. Three particularly interesting characteristics of non-steady peristalsis are described: (i) fluctuations in pressure and shear stress arise due to a non-integral number of waves in the finite-length tube; (ii) retrograde motion of fluid particles during peristaltic transport (reflux) has inherently different behaviour with single peristaltic waves as compared to multiple ‘train waves’, and (iii) finite tube length, the number of peristaltic waves and the degree of tube occlusion affect global pumping performance. We find that, whereas significant increases in pressure and shear stress result from the tube-to-wave length ratio being non-integral, global pumping performance is only slightly degraded by the existence of a non-integral number of waves in the tube during peristaltic transport. Furthermore, the extent of retrograde motion of fluid particles is much greater with single waves than with train waves. These results suggest that in the design and analysis of peristaltic pumps attention should be paid to the unsteady effects of finite tube length and to the differences between single and multiple peristaltic waves.

Linear stability and weak nonlinear theories are used to investigate analytically the Coriolis effect on three-dimensional gravity-driven convection in a rotating porous layer heated from below. Major differences as well as similarities with the corresponding problem in pure fluids (non-porous domains) are particularly highlighted. As such, it is found that, in contrast to the problem in pure fluids, overstable convection in porous media is not limited to a particular domain of Prandtl number values (in pure fluids the necessary condition is Pr<1). Moreover, it is also established that in the porous-media problem the critical wavenumber in the plane containing the streamlines for stationary convection is not identical to the critical wavenumber associated with convection without rotation, and is therefore not independent of rotation, a result which is quite distinct from the corresponding pure-fluids problem. Nevertheless it is evident that in porous media, just as in the case of pure fluids subject to rotation and heated from below, the viscosity at high rotation rates has a destabilizing effect on the onset of stationary convection, i.e. the higher the viscosity the less stable the fluid. Finite-amplitude results obtained by using a weak nonlinear analysis provide differential equations for the amplitude, corresponding to both stationary and overstable convection. These amplitude equations permit one to identify from the post-transient conditions that the fluid is subject to a pitchfork bifurcation in the stationary convection case and to a Hopf bifurcation associated with the overstable convection. Heat transfer results were evaluated from the amplitude solution and are presented in terms of Nusselt number for both stationary and overstable convection. They show that rotation has in general a retarding effect on convective heat transfer, except for a narrow region of small values of the parameter containing the Prandtl number where rotation enhances the heat transfer associated with overstable convection.

When a viscous liquid is displaced by a long air bubble in a capillary, it leaves behind a wetting liquid film. A lubrication analysis by Bretherton (1961), which assumes a mobile surface, underpredicts the film thickness at low bubble speeds. In this investigation, the Marangoni effect of small amounts of impurities is shown to be capable of explaining this discrepancy. We carry out an asymptotic analysis for different convective, diffusive and kinetic timescales and show that, if transport in the film is mass-transfer limited such that a bulk concentration gradient exists in the film, the film thickness increases by a maximum factor of 4 2/3; over Bretherton's mobile result at low bubble speeds. Moreover, at large bubble speeds, Bretherton's mobile prediction is approached for all ranges of timescales. For intermediate bubble speeds, the film thickness varies with respect to the bubble speed with an exponent smaller than 2/3 of the mobile theory. These results are favourably compared to literature data on film thickness.