Hill & Foda (1998) and Jamali (1998) have presented theoretical and experimental studies of the resonant interaction between a surface wave and two oblique interfacial waves. Despite many similarities between the findings there is one seemingly major difference. Hill & Foda's (1998) analysis indicated that there are only narrow bands of frequency, density ratio and direction angle within which growth is possible. On the other hand, Jamali (1998) predicted and observed wave growth over wide ranges of frequency and direction angle, and for all the density ratios that he investigated. We show that Hill & Foda's (1998) second-order representation of the dynamic interfacial boundary condition is missing a term proportional to the time derivative of the square of the velocity shear across the interface. When this missing term is included in the analysis, the resulting predictions are consistent with the laboratory experiments.

Viewed on a hydrodynamic scale, flames in experiments are often thin so that they may be described as gasdynamic discontinuities separating the dense cold fresh mixture from the light hot burned products. The original model of a flame as a gasdynamic discontinuity was due to Darrieus and to Landau. In addition to the fluid dynamical equations, the model consists of a flame speed relation describing the evolution of the discontinuity surface, and jump conditions across the surface which relate the fluid variables on the two sides of the surface. The Darrieus–Landau model predicts, in contrast to observations, that a uniformly propagating planar flame is absolutely unstable and that the strength of the instability grows with increasing perturbation wavenumber so that there is no high-wavenumber cutoff of the instability. The model was modified by Markstein to exhibit a high-wavenumber cutoff if a phenomenological constant in the model has an appropriate sign. Both models are postulated, rather than derived from first principles, and both ignore the flame structure, which depends on chemical kinetics and transport processes within the flame. At present, there are two models which have been derived, rather than postulated, and which are valid in two non-overlapping regions of parameter space. Sivashinsky derived a generalization of the Darrieus–Landau model which is valid for Lewis numbers (ratio of thermal diffusivity to mass diffusivity of the deficient reaction component) bounded away from unity. Matalon & Matkowsky derived a model valid for Lewis numbers close to unity. Each model has its own advantages and disadvantages. Under appropriate conditions the Matalon–Matkowsky model exhibits a high-wavenumber cutoff of the Darrieus–Landau instability. However, since the Lewis numbers considered lie too close to unity, the Matalon–Matkowsky model does not capture the pulsating instability. The Sivashinsky model does capture the pulsating instability, but does not exhibit its high-wavenumber cutoff. In this paper, we derive a model consisting of a new flame speed relation and new jump conditions, which is valid for arbitrary Lewis numbers. It captures the pulsating instability and exhibits the high-wavenumber cutoff of all instabilities. The flame speed relation includes the effect of short wavelengths, not previously considered, which leads to stabilizing transverse surface diffusion terms.

The stability of a steadily propagating planar premixed flame has been the subject of numerous studies since Darrieus and Landau showed that in their model flames are unstable to perturbations of any wavelength. Moreover, the instability was shown to persist even for very small wavelengths, i.e. there was no high-wavenumber cutoff of the instability. In addition to the Darrieus–Landau instability, which results from thermal expansion, analysis of the diffusional thermal model indicates that premixed flames may exhibit cellular and pulsating instabilities as a consequence of preferential diffusion. However, no previous theory captured all the instabilities including a high-wavenumber cutoff for each. In Class, Matkowsky & Klimenko (2003) a unified theory is proposed which, in appropriate limits and under appropriate assumptions, recovers all the relevant previous theories. It also includes additional new terms, not present in previous theories. In the present paper we consider the stability of a uniformly propagating planar flame as a solution of the unified model. The results are then compared to those based on the models of Darrieus–Landau, Sivashinsky and Matalon–Matkowsky. In particular, it is shown that the unified model is the only model to capture the Darrieus–Landau, cellular and pulsating instabilities including a high-wavenumber cutoff for each.

Alignment of the eigenvectors for strain-rate tensors and subgrid-scale (SGS) stress tensors in large-eddy simulation (LES) is studied in homogeneous isotropic turbulence. Non-alignment of these two eigenvectors was shown in Tao, Katz & Meneveau (2002). In the present study, the specific term in the decomposition of the SGS stress tensor, which is primarily responsible for causing this non-alignment, is identified using the nonlinear model. The bimodal behaviour of the alignment configuration reported in Tao et al. (2002) was eliminated by reordering the eigenvalues according to the degree of alignment of the corresponding eigenvectors with the vorticity vector. The preferred relative orientation of the eigenvectors was ${\approx}\,42^\circ$. The alignment trends were conditionally sampled based on the relative dominance of strain and vorticity. The effect of the identified term on the alignment was the largest in the region in which the magnitudes of strain and vorticity were comparable and large (flat sheet). The most probable alignment configuration in the flat-sheet region was different from those in the strain-dominated and vorticity-dominated regions. The relative orientation of the eigenvectors was dependent on the degree of resolution for the flat sheet region yielded on the LES mesh. When the alignment was conditionally sampled on the events with the backward scatter of the SGS energy into the grid scale, the interchange of the alignment of the eigenvectors took place. Relevance of the identified term for the generation of turbulence is investigated. It is shown that the identified term makes no contribution to the production of the total SGS energy, but contributes significantly to the generation of the SGS enstrophy. The identified term causes a time-lag in the evolution of the turbulent energy and enstrophy. It is shown that generation of vorticity is markedly attenuated when the magnitude of the identified term is modified, and the original nonlinear model yielded the results which are in the closest agreement with the direct numerical simulation data.

Receptivity of a high-speed boundary layer on a flat plate to acoustic disturbances is investigated using a combined numerical and asymptotic approach. The leading-edge receptivity problem is discussed with emphasis on physical mechanisms associated with scattering and diffraction of acoustic waves. Analytical solutions provide insight into the interplay of these mechanisms as a function of the angle of incidence of external acoustic waves. The theoretical predictions are in good agreement with the wind-tunnel experimental data of Maslov et al. obtained at free-stream Mach number 6. The leading-edge receptivity model is incorporated into the multiple-modes method to account for the inter-modal exchange downstream from the leading edge. This combined modelling resembles basic features of the direct numerical simulation of Ma & Zhong. A comparative analysis of the leading-edge receptivity and the inter-modal exchange associated with non-parallel effects is presented. The theory allows fast evaluation of the receptivity coefficients and clarifies the physics of the receptivity process. The theoretical results may guide further direct numerical simulations and experimental studies of boundary layer receptivity at supersonic and hypersonic speeds.

An experimental investigation has been carried out on a supersonic jet of air issuing from an $M=1.44$ converging–diverging rectangular nozzle of aspect ratio 4. Particle image velocimetry measurements of the flow field along with near field acoustic measurements were made. The effect of injection of a small amount of water ($\sim 5 \%$ of the mass flow rate of the jet) into the shear layer of the jet, on the unsteady flow structure and sound generation were examined. The presence of water droplets in the jet modified the turbulence structure significantly, resulting in axial and normal r.m.s. velocity reductions of about 10% and 30%, respectively, as compared to that of a normal jet. An even larger effect is found on the peak values of the turbulent shear stress with a reduction of up to 40%. The near-field noise levels (OASPL) were found to reduce by about 2–6 dB depending on the location of the injection and the water mass flow rate. Far-field acoustic measurements carried out on a heated $M=0.9$ (jet exit velocity=525 m s$^{-1}$) jet show significant (6 dB) reductions in the OASPL with moderate amounts of water injection (17% of the mass flow rate of the jet) suggesting that the techniques is viable at realistic engine operating conditions.

Shock waves, dead zones and particle-free regions form when a thin surface avalanche of granular material flows around an obstacle or over a change in the bed topography. Understanding and modelling these flows is of considerable practical interest for industrial processes, as well as for the design of defences to protect buildings, structures and people from snow avalanches, debris flows and rockfalls. These flow phenomena also yield useful constitutive information that can be used to improve existing avalanche models. In this paper a simple hydraulic theory, first suggested in the Russian literature, is generalized to model quasi-two-dimensional flows around obstacles. Exact and numerical solutions are then compared with laboratory experiments. These indicate that the theory is adequate to quantitatively describe the formation of normal shocks, oblique shocks, dead zones and granular vacua. Such features are generated by the flow around a pyramidal obstacle, which is typical of some of the defensive structures in use today.

The paper presents new concepts and results for the eddy structure of turbulent convection in a horizontal fluid layer of depth $h$ which lies above a solid base with thickness $h_{b}$. The fluid parameters are the kinematic viscosity $\nu $, the thermal diffusivity $\kappa $, which is taken to be comparable with $\nu$, the density $\rho $, the specific heat $c_{p}$ and the expansion parameter $\beta$. The thermal diffusivity of the solid is $\kappa_{b}$. The results are an extension of the more commonly studied cases, where a constant heat flux or constant temperature is applied at the interface between the fluid and the base. The buoyancy forces induce eddy motions with a typical velocity $w_{\ast} \sim (g \beta F_{\theta} h)^{1/3}$ where $\rho c_{p}F_{\theta}$ is the average heat flux and $F_{\theta}$ the covariance of the fluctuations of the temperature and of the vertical velocity. At moderate Reynolds numbers ($Re=w_{\ast}h/\nu $), say less than about $10^{3}$, an order-of-magnitude analysis shows that for the case of high diffusivity of the base (i.e. $\kappa_{b} \gg \kappa$) elongated ‘plumes’ form at the surface and extend to the top of the fluid layer. When the base diffusivity is low (i.e. $\kappa_{b} \leq \kappa$) the surface cools below the developing ‘plume’ and either the plume breaks up into elongated puffs or, if $\kappa_{b} \ll \kappa$, horizontal pressure gradients form so that only small-scale puffs can form near the surface. At very high Reynolds numbers, approximately greater than $10^{4}$, the surface boundary layer below each puff/plume is highly turbulent with a local logarithmic velocity and temperature profile. An approximate analysis indicates for this case that there is insufficient buoyancy flux from the base, irrespective of its diffusivity, to maintain plumes, because of the high turbulent heat transfer. So puffs dominate high-Reynolds-number thermal convection as numerical simulations and field experiments demonstrate. However, when the surface heat flux is uniform, for example as a result of radiant heat transfer or by forcing with a constant heat flux below a very thin conducting base, plumes are the dominant form of eddy motion, as is commonly observed. In the numerical solutions presented here, where $Re \sim 3 \times 10^{2}$ and the slab thickness $h_{b} = h$, it is shown that the spatial scales of eddy structures in the fluid layer close to the surface become significantly smaller as $\kappa_{b}/\kappa$ is reduced from 100 to 0.1. At the same time in the core of the convective layer the change in the autocorrelation and spatial correlation function indicates that there is a transition from long-duration plumes into shorter-duration and smaller-length-scale elongated puffs. The simulations show that the largest temperature fluctuations near the surface occur when a constant heat flux is applied at the bottom of the fluid layer. The smallest temperature fluctuations are associated with the constant-temperature boundary condition. The finite base diffusivity cases lie in between these limits, with the largest fluctuations occurring when the thermal diffusivity of the base is small. The hypothesis introduced above has been tested qualitatively in a laboratory set-up when the effective diffusivity of the base was varied. The flow structure was observed as it changed from being characterized by nearly steady plumes, into unsteady plumes and finally into puffs when the thickness of the conducting base was first increased and then the diffusivity was decreased.

This paper continues the work started in Part 1 (Reznik, Zeitlin & Ben Jelloul 2001) and generalizes it to the case of a stratified environment. Geostrophic adjustment of localized disturbances is considered in the context of the two-layer shallow-water and continuously stratified primitive equations in the vertically bounded and horizontally infinite domain on the $f$-plane. Using multiple-time-scale perturbation expansions in Rossby number $\hbox{\it Ro}$ we show that stratification does not substantially change the adjustment scenario established in Part 1 and any disturbance of well-defined scale is split in a unique way into slow and fast components with characteristic time scales $f_0^{-1}$ and $(f_0 \hbox{\it Ro})^{-1}$ respectively, where $f_0$ is the Coriolis parameter. As in Part 1 we distinguish two basic dynamical regimes: quasi-geostrophic (QG) and frontal geostrophic (FG) with small and large deviations of the isopycnal surfaces, respectively. We show that the dynamics of the FG regime in the two-layer model depends strongly on the ratio of the layer depths. The difference between QG and FG scenarios of adjustment is demonstrated. In the QG case the fast component of the flow essentially does not ‘feel’ the slow one and is rapidly dispersed leaving the slow component to evolve according to the standard QG equation (corrections to this equation are found for times $t\,{\gg}\, (f_0 \hbox{\it Ro})^{-1}$). In the FG case the fast component is a packet of inertial oscillations produced by the initial perturbation. The space-time evolution of the envelope of inertial oscillations obeys a Schrödinger-type modulation equation with coefficients depending on the slow component. In both QG and FG cases we show by direct computations that the fast component does not produce any drag terms in the equations for the slow component; the slow component remains close to the geostrophic balance. However, in the continuously stratified FG regime, as well as in the two-layer regime with the layers of comparable thickness, the splitting is incomplete in the sense that the slow vortical component and the inertial oscillations envelope evolve on the same time scale.

Direct numerical simulations have been carried out for a fully developed turbulent channel flow with a smooth upper wall and a lower wall consisting of square bars separated by a rectangular cavity. A wide range of $w/k$, the cavity width to roughness height ratio, was considered. For $w/k\,{\ge}\,7$, recirculation zones occur immediately upstream and downstream of each element while mean streamlines and spatial distributions of the skin frictional drag indicate that each element is virtually isolated. The maximum form drag occurs at $w/k\,{=}\,7$ and coincides with the minimum skin frictional drag. The dependence on $w/k$ of the Clauser roughness function reflects that of the form drag.

We report results from microgravity experiments on thermocapillary convection in open annuli with outer radius $R_{o}{\,=\,}40\,\hbox{mm}$ and inner radius $R_{i}{\,=\,}20\,\hbox{mm}$ of various aspect ratios $Ar$. The measurements are from more than 230 equilibrated states in the $Ar$–Marangoni-number space. We found time-independent and oscillatory states and report some selected oscillation and Fourier spectra from thermocouple measurements. We measured the critical temperature difference $\uDelta T^{\hspace*{1pt}c}$ for the onset of temperature oscillations in the range $1{\,\le\,}Ar{\,\le\,}8$. We report supercritical oscillation periods and attribute the oscillations in the larger $Ar$ range to hydrothermal waves. This conclusion is supported by the values of the oscillation periods and of the critical Marangoni numbers in that $Ar$ range. The hydrothermal waves exhibit an internal corotating multicellular pattern. For the smaller $Ar$ and near the threshold we report $m$-fold temperature patterns on the free surface with $m$ decreasing for decreasing $Ar$. At $4\uDelta T^{\hspace*{1pt}c}$ these patterns become very irregular. Most of the findings are in accordance with the numerical results reported in Part 2 (Sim et al. (2003)). The experimental $\uDelta T^{\hspace*{1pt}c}$ are higher and the experimental periods $\tau^{c}$ are smaller than the numerical values for Biot number $Bi{\,=\,}0$. However, analysis of the experimental free-surface thermal boundary conditions shows that there was heat input to the free surface. Good agreement with numerical results for $\uDelta T^{\hspace*{1pt}c}$ and $\tau^{\hspace*{1pt}c}$ is obtained with $Bi{\,\ne\,}0$ (heat input).

Oscillatory thermocapillary convection in open cylindrical annuli heated from the outer wall is investigated numerically. Results at fixed inner/outer radius ratio of 0.5, aspect ratios ($Ar$) of 1, 2.5, 3.33, and 8, zero Biot number, and a Prandtl number of 6.84 are obtained and compared with experiments (Part 1 of this paper). Convection is steady and axisymmetric at sufficiently low values of the Reynolds number (${\it Re}$). Transition to oscillatory states occurs at critical values of $Re$ which depend on $Ar$. With $Ar\,{=}\, 1$, 2.5 and 3.33, we observe 5, 9 and 12 azimuthal wavetrains, respectively, travelling clockwise at the free surface near the critical $Re$. With $Ar \,{=}\, 8$, there are 20 standing waves near the critical $Re$. Experimental results in Part 1 support this finding. A multi-roll structure appears beyond the critical $Re$ in shallow liquid layers with $Ar \,{=}\, 3.33$ and 8. The critical $Re$ and frequency are in qualitative but not in quantitative agreement with the experimental ones. Either heat loss from the free surface or heating from the surroundings to the free surface stabilizes the flow, and the critical $Re$ increases with increasing Biot number while the critical period goes down. The numerical results agree better with the experimental ones if the free surface is assumed to be heated as shown in Part 1. We have also computed supercritical time-dependent states and find that while the non-dimensional frequency increases with increasing $Re$ near the critical region, it approaches an asymptote at supercritical $Re$.

Energy transfers between modes obtained from the proper orthogonal decomposition (POD) of a turbulent flow past a backward-facing step are analysed with the aim of providing guidelines for modelling unresolved modes in truncated POD–Galerkin models. It is observed that energy transfers are local in the POD basis, and that the Fourier-decomposition-based concepts of forward and backward energy cascades are also valid in the POD basis, the net effect being a forward energy cascade. General features of the eddy-viscosity representation of kinetic energy transfers are investigated through a priori tests. It is observed that the ideal eddy-viscosity model should exhibit a cusp behaviour near the cutoff mode.

Capillary-zone electrophoresis (CZE) is an efficient separation method in analytical chemistry. It exploits the difference in electrophoretic migration speeds between charged molecular species in aqueous solution when an external electric field is applied to achieve separation. In most cases the electrophoretic migration of species is also accompanied by a bulk electro-osmotic flow in the capillary due to the presence of a zeta-potential at the capillary wall. Adsorption of charged species at the wall could modify this zeta-potential in a non-uniform manner. This induces axial pressure gradients, so that the flow is no longer uniform over the capillary cross-section. The resulting shear-induced dispersion of the sample is a serious cause of band broadening in CZE particularly for species such as proteins and peptides which adsorb strongly on capillary walls. The problem of the spatio-temporal evolution of the sample concentration is studied in the presence of such wall interactions. An asymptotic theory is developed that is valid provided axial variations have characteristic length scales that are much larger than the capillary radius and temporal variations have a characteristic time scale much larger than the characteristic diffusion time over a capillary radius. These conditions are normally satisfied in CZE, except when the sample is close to the inlet, on account of the capillary length being very much larger than its radius. It is shown that the cross-sectionally averaged sample concentration obeys a one-dimensional partial differential equation. Further, the full three-dimensional concentration field may be calculated once the cross-sectionally averaged concentration field is known. The reduced system is integrated numerically and is shown to lead to predictions consistent with known observations on CZE in the presence of wall interactions.

A theoretical study is made of continuously stratified flow of large depth over topography when small periodic vertical fluctuations are present in the Brunt–Väisälä frequency, the background flow conditions being otherwise uniform. It is known from Phillips (1968) that, owing to nonlinear interactions with such fluctuations, internal gravity waves with vertical wavelength twice that of the background variations become trapped along the vertical, suggesting a waveguide-like behaviour. Using the asymptotic theory of Kantzios & Akylas (1993), we explore the role that this interaction-trapping mechanism plays in the generation of finite-amplitude long-wave disturbances near the hydrostatic limit. As a result of vertical trapping, a resonance phenomenon occurs and the linear hydrostatic response grows unbounded when the flow speed coincides with the long-wave speed of a free propagation mode that is trapped close to the ground. Near this critical flow speed, according to weakly nonlinear analysis, the wave evolution along the streamwise direction is governed by a forced extended Korteweg–de Vries equation, which predicts upstream-propagating solitary waves and bores similar to those obtained in resonant stratified flow of finite depth. The finite-amplitude response is then studied numerically and in some cases features strong upstream influence in the form of vertically trapped solitary waves and bores. On the other hand, incipient wave breaking is often encountered during the evolution of the nonlinear resonant response, and this flow feature, which is beyond the reach of weakly nonlinear theory, arises at topography amplitudes significantly below the critical value for overturning predicted by the classical model of Long (1953) for uniformly stratified steady flow.

This paper reports an analytical and numerical study of the natural convection in a horizontal porous layer filled with a binary fluid. A uniform heat flux is applied to the horizontal walls while the vertical walls are impermeable and adiabatic. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the horizontal walls (double-diffusive convection, $a\,{=}\,0$) or by temperature gradients (Soret effects, $a\,{=}\,1$). The governing parameters for the problem are the thermal Rayleigh number, $R_T$, the Lewis number, $Le$, the solutal Rayleigh number, $R_S$, the aspect ratio of the cavity, $A$, the normalized porosity of the porous medium, $\varepsilon$, and the constant $a$. The onset of convection in the layer is studied using a linear stability analysis. The thresholds for finite-amplitude, oscillatory and monotonic convection instabilities are determined in terms of the governing parameters. For convection in an infinite layer, an analytical solution of the steady form of the governing equations is obtained by assuming parallel flow in the core of the cavity. The critical Rayleigh numbers for the onset of supercritical, $R_{\hbox{\scriptsize\it TC}}^{\hbox{\scriptsize\it sup}}$, or subcritical, $R_{\hbox{\scriptsize\it TC}}^{\hbox{\scriptsize\it sub}}$, convection are predicted by the present theory. A linear stability analysis of the parallel flow pattern is conducted in order to predict the thresholds for Hopf bifurcation. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical prediction and the numerical simulations.

A term-by-term wavelet decomposition of the equation for turbulence kinetic energy in turbulent channel flow is used to provide a dual space-scale description of the production and flux of energy. Wavelet filtering, analogous to that used in large-eddy simulation, is performed on the nonlinear term that constitutes the energy flux. Meneveau's term, $\pi_{{sg}}^{(m)}[{\bm i}]$ is used to represent forward scatter and backscatter. This term is highly intermittent, much more so than the equivalent terms for production at the same scale. Virtually all of $\pi_{{sg}}^{(m)}[{\bm i}]$ appears in only two components that involve subgrid flux of streamwise momentum in the wall-normal and spanwise directions. An equivalent term that is the wavelet transform of the pressure-gradient term is shown to be several orders of magnitude smaller, consistent with its neglect in current subgrid modelling techniques. However, the mean-square pressure-gradient fluctuations (that reach a maximum in the range of wavenumbers in which the velocity spectra exhibit a −5/3 slope) are responsible for the significant spatial intermittency observed in the energy flux.

This paper investigates the heat propagation process in a gas from concentrated energy sources with deposition times, $t'_d$, of the order of the characteristic acoustic time, $t'_a$, across the region where the temperature will be increased by a factor of order unity. Heat propagation takes place by two different mechanisms that act separately in two different neatly defined spatial regions of comparable size. Around the source, we find a conductive region of very high temperature where the spatial pressure variations are negligible. The edge of the resulting strongly heated low-density region appears as a contact surface that acts as a piston for the outer flow, where the pressure disturbances, of order of the ambient pressure in the distinguished regime $t'_d \,{\sim}\, t'_a$ considered here, generate a shock wave that heats up the outer gas as it propagates outwards. The mass and energy balances for the conductive region provide a differential equation linking its pressure with the velocity of its bounding contact surface, which is used, together with the jump conditions across the shock, when integrating the Euler equations for the outer compressible flow. Solutions for the heating history are obtained for point, line and planar sources for different values of the ratio $t'_d/t'_a$, including weak sources with $t'_d \,{\gg}\, t'_a$ and very intense sources with $t'_d \,{\ll}\, t'_a$. The solution determines in particular the temperature profile emerging as the pressure perturbations become negligible for times much larger than the acoustic time.