Research Article
Aeroacoustic properties of supersonic elliptic jets
 KEVIN W. KINZIE, DENNIS K. McLAUGHLIN

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 25 September 1999, pp. 128

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The aerodynamic and acoustic properties of supersonic elliptic and circular jets are experimentally investigated. The jets are perfectly expanded with an exit Mach number of approximately 1.5 and are operated in the Reynolds number range of 25 000 to 50 000. The reduced Reynolds number facilitates the use of conventional hotwire anemometry and a glow discharge excitation technique which preferentially excites the varicose or flapping modes in the jets. In order to simulate the highvelocity and lowdensity effects of heated jets, helium is mixed with the air jets. This allows the largescale structures in the jet shear layer to achieve a high enough convective velocity to radiate noise through the Mach wave emission process.
Experiments in the present work focus on comparisons between the cold and simulated heated jet conditions and on the beneficial aeroacoustic properties of the elliptic jet. When helium is added to the jet, the instability wave phase velocity is found to approach or exceed the ambient sound speed. The radiated noise is also louder and directed at a higher angle from the jet axis. In addition, nearfield hotwire spectra are found to match the farfield acoustic spectra only for the helium/air mixture case. These results demonstrate that there are significant differences between unheated and heated asymmetric jets in the Mach 1.5 speed range, many of which have been found previously for circular jets. The elliptic jet was also found to radiate less noise than the round jet at comparable operating conditions.
Hydrodynamic interactions and collision efficiencies of spherical drops covered with an incompressible surfactant film
 J. BLAWZDZIEWICZ, E. WAJNRYB, MICHAEL LOEWENBERG

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 25 September 1999, pp. 2959

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A theory is developed for the hydrodynamic interactions of surfactantcovered spherical drops in creeping flows. The surfactant is insoluble, and flowinduced changes of surfactant concentration are small, i.e. the film of adsorbed surfactant is incompressible.
For a single surfactantcovered drop in an arbitrary incident flow, the Stokes equations are solved using a decomposition of the flow into surfacesolenoidal and surfaceirrotational components on concentric spherical surfaces. The surfacesolenoidal component is unaffected by surfactant; the surfaceirrotational component satisfies a slipstick boundary condition with slip proportional to the surfactant diffusivity. Pair hydrodynamic interactions of surfactantcovered bubbles are computed from the oneparticle solution using a multiplescattering expansion. Two terms in a lubrication expansion are derived for axisymmetric nearcontact motion.
The pair mobility functions are used to compute collision efficiencies for equalsize surfactantcovered bubbles in linear flows and in Brownian motion. An asymptotic analysis is presented for weak surfactant diffusion and weak van der Waals attraction. In the absence of surfactant diffusion, collision efficiencies for surfactantcovered bubbles are higher than for rigid spheres in straining flow and lower in shear flow. In shear flow, the collision efficiency vanishes for surfactant diffusivities below a critical value if van der Waals attraction is absent.
Thermosolutal bifurcation phenomena in porous enclosures subject to vertical temperature and concentration gradients
 M. MAMOU, P. VASSEUR

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 25 September 1999, pp. 6187

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The Darcy model with the Boussinesq approximations is used to study doublediffusive instability in a horizontal rectangular porous enclosure subject to two sources of buoyancy. The two vertical walls of the cavity are impermeable and adiabatic while Dirichlet or Neumann boundary conditions on temperature and solute are imposed on the horizontal walls. The onset and development of convection are first investigated using the linear and nonlinear perturbation theories. Depending on the governing parameters of the problem, four different regimes are found to exist, namely the stable diffusive, the subcritical convective, the oscillatory and the augmenting direct regimes. The governing parameters are the thermal Rayleigh number, RT, buoyancy ratio, N, Lewis number, Le, normalized porosity of the porous medium, ε, aspect ratio of the enclosure, A, and the thermal and solutal boundary condition type, κ, applied on the horizontal walls. On the basis of the nonlinear perturbation theory and the parallel flow approximation (for slender or shallow enclosures), analytical solutions are derived to predict the flow behaviour. A finite element numerical method is introduced to solve the full governing equations. The results indicate that steady convection can arise at Rayleigh numbers below the supercritical value, indicating the development of subcritical flows. At the vicinity of the threshold of supercritical convection the nonlinear perturbation theory and the parallel flow approximation results are found to agree well with the numerical solution. In the overstable regime, the existence of multiple solutions, for a given set of the governing parameters, is demonstrated. Also, numerical results indicate the possible occurrence of travelling waves in an infinite horizontal enclosure.
Ocean waves and ice sheets
 N. J. BALMFORTH, R. V. CRASTER

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 25 September 1999, pp. 89124

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A complete analytical study is presented of the reflection and transmission of surface gravity waves incident on icecovered ocean. The ice cover is idealized as a plate of elastic material for which flexural motions are described by the Timoshenko–Mindlin equation. A suitable nondimensionalization extracts parameters useful for the characterization of oceanwave and icesheet interactions, and for scaled laboratory studies. The scattering problem is simplified using Fourier transforms and the Wiener–Hopf technique; the solution is eventually written down in terms of some easily evaluated quadratures. An important feature of this solution is that the physical conditions at the edge of the ice sheet are explicitly built into the analysis, and powerflow theorems provide verification of the results. Asymptotic results for large and small values of the nondimensional parameters are extracted and approximations are given for general parameter values.
The effect of weak shear on finiteamplitude internal solitary waves
 S. R. CLARKE, R. H. J. GRIMSHAW

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 25 September 1999, pp. 125159

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A finiteamplitude longwave equation is derived to describe the effect of weak current shear on internal waves in a uniformly stratified fluid. This effect is manifested through the introduction of a nonlinear term into the amplitude evolution equation, representing a projection of the shear from physical space to amplitude space. For steadily propagating waves the evolution equation reduces to the steady version of the generalized Korteweg–de Vries equation. An analysis of this equation is presented for a wide range of possible shear profiles. The type of waves that occur is found to depend on the number and position of the inflection points of the representation of the shear profile in amplitude space. Up to three possible inflection points for this function are considered, resulting in solitary waves and kinks (dispersionless bores) which can have up to three characteristic lengthscales. The stability of these waves is generally found to decrease as the complexity of the waves increases. These solutions suggest that kinks and solitary waves with multiple lengthscales are only possible for shear profiles (in physical space) with a turning point, while instability is only possible if the shear profile has an inflection point. The unsteady evolution of a periodic initial condition is considered and again the solution is found to depend on the inflection points of the amplitude representation of the shear profile. Two characteristic types of solution occur, the first where the initial condition evolves into a train of rankordered solitary waves, analogous to those generated in the framework of the Korteweg–de Vries equation, and the second where two or more kinks connect regions of constant amplitude. The unsteady solutions demonstrate that finiteamplitude effects can act to halt the critical collapse of solitary waves which occurs in the context of the generalized Korteweg–de Vries equation. The two types of solution are then used to qualititatively relate previously reported observations of shock formation on the internal tide propagating onto the Australian North West Shelf to the observed background current shear.
The concept of blockeffective macrodispersivity and a unified approach for gridscale and plumescaledependent transport
 Y. RUBIN, A. SUN, R. MAXWELL, A. BELLIN

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 25 September 1999, pp. 161180

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We present a new approach for modelling macrodispersivity in spatially variable velocity fields, such as exist in geologically heterogeneous formations. Considering a spectral representation of the velocity, it is recognized that numerical models usually capture lowwavenumber effects, while the largewavenumber effects, associated with subgrid block variability, are suppressed. While this suppression is avoidable if the heterogeneity is captured at minute detail, that goal is impossible to achieve in all but the most trivial cases. Representing the effects of the suppressed variability in the models is made possible using the proposed concept of blockeffective macrodispersivity. A tensor is developed, which we refer to as the blockeffective macrodispersivity tensor, whose terms are functions of the characteristic length scales of heterogeneity, as well as the length scales of the model's homogenized areas, or numerical grid blocks. Closedform expressions are developed for small variability in the logconductivity and unidirectional mean flow, and are tested numerically. The use of the blockeffective macrodispersivities allows conditioning of the velocity field on the measurements on the one hand, while accounting for the effects of unmodelled heterogeneity on the other, in a numerically reasonable setup. It is shown that the effects of the grid scale are similar to those of the plume scale in terms of filtering out the effects of portions of the velocity spectrum. Hence it is easy to expand the concept of the blockeffective dispersivity to account for the scale of the solute body and the porescale dispersion.
On the gravitational displacement of threedimensional fluid droplets from inclined solid surfaces
 P. DIMITRAKOPOULOS, J. J. L. HIGDON

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 25 September 1999, pp. 181209

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The yield conditions for the gravitational displacement of threedimensional fluid droplets from inclined solid surfaces are studied through a series of numerical computations. The study considers both sessile and pendant droplets and includes interfacial forces with constant surface tension. An extensive study is conducted, covering a wide range of Bond numbers Bd, angles of inclination β and advancing and receding contact angles, θA and θR. This study seeks the optimal shape of the contact line which yields the maximum displacing force (or BT ≡ Bd sin β) for which a droplet can adhere to the surface. The yield conditions BT are presented as functions of (Bd or β, θA, Δθ) where Δθ = θA − θR is the contact angle hysteresis. The solution of the optimization problem provides an upper bound for the yield condition for droplets on inclined solid surfaces. Additional contraints based on experimental observations are considered, and their effect on the yield condition is determined. The numerical solutions are based on the spectral boundary element method, incorporating a novel implementation of Newton's method for the determination of equilibrium free surfaces and an optimization algorithm which is combined with the Newton iteration to solve the nonlinear optimization problem. The numerical results are compared with asymptotic theories (Dussan V. & Chow 1983; Dussan V. 1985) and the useful range of these theories is identified. The normal component of the gravitational force BN ≡ Bd cos β was found to have a weak effect on the displacement of sessile droplets and a strong effect on the displacement of pendant droplets, with qualitatively different results for sessile and pendant droplets.
Stability of nonparabolic flow in a flexible tube
 V. SHANKAR, V. KUMARAN

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 25 September 1999, pp. 211236

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Flows with velocity profiles very different from the parabolic velocity profile can occur in the entrance region of a tube as well as in tubes with converging/diverging crosssections. In this paper, asymptotic and numerical studies are undertaken to analyse the temporal stability of such ‘nonparabolic’ flows in a flexible tube in the limit of high Reynolds numbers. Two specific cases are considered: (i) developing flow in a flexible tube; (ii) flow in a slightly converging flexible tube. Though the mean velocity profile contains both axial and radial components, the flow is assumed to be locally parallel in the stability analysis. The fluid is Newtonian and incompressible, while the flexible wall is modelled as a viscoelastic solid. A high Reynolds number asymptotic analysis shows that the nonparabolic velocity profiles can become unstable in the inviscid limit. This inviscid instability is qualitatively different from that observed in previous studies on the stability of parabolic flow in a flexible tube, and from the instability of developing flow in a rigid tube. The results of the asymptotic analysis are extended numerically to the moderate Reynolds number regime. The numerical results reveal that the developing flow could be unstable at much lower Reynolds numbers than the parabolic flow, and hence this instability can be important in destabilizing the fluid flow through flexible tubes at moderate and high Reynolds number. For flow in a slightly converging tube, even small deviations from the parabolic profile are found to be sufficient for the present instability mechanism to be operative. The dominant nonparallel effects are incorporated using an asymptotic analysis, and this indicates that nonparallel effects do not significantly affect the neutral stability curves. The viscosity of the wall medium is found to have a stabilizing effect on this instability.
A resonant wave theory
 LUNSHIN YAO

 Published online by Cambridge University Press:
 25 September 1999, pp. 237251

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Analysis is used to show that a solution of the Navier–Stokes equations can be computed in terms of wavelike series, which are referred to as waves below. The mean flow is a wave of infinitely long wavelength and period; laminar flows contain only one wave, i.e. the mean flow. With a supercritical instability, there are a mean flow, a dominant wave and its harmonics. Under this scenario, the amplitude of the waves is determined by linear and nonlinear terms. The linear case is the target of flowinstability studies. The nonlinear case involves energy transfer among the waves satisfying resonance conditions so that the wavenumbers are discrete, form a denumerable set, and are homeomorphic to Cantor's set of rational numbers. Since an infinite number of these sets can exist over a finite real interval, nonlinear Navier–Stokes equations have multiple solutions and the initial conditions determine which particular set will be excited. Consequently, the influence of initial conditions can persist forever. This phenomenon has been observed for Couette–Taylor instability, turbulent mixing layers, wakes, jets, pipe flows, etc. This is a commonly known property of chaos.
Directional solidification of a binary alloy into a cellular convective flow: localized morphologies
 Y.J. CHEN, S. H. DAVIS

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 25 September 1999, pp. 253270

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A steady, twodimensional cellular convection modifies the morphological instability of a binary alloy that undergoes directional solidification. When the convection wavelength is far longer than that of the morphological cells, the behaviour of the moving front is described by a slow, spatial–temporal dynamics obtained through a multiplescale analysis. The resulting system has a parametricexcitation structure in space, with complex parameters characterizing the interactions between flow, solute diffusion, and rejection. The convection in general stabilizes twodimensional disturbances, but destabilizes threedimensional disturbances. When the flow is weak, the morphological instability is incommensurate with the flow wavelength, but as the flow gets stronger, the instability becomes quantized and forced to fit into the flow box. At large flow strength the instability is localized, confined in narrow envelopes. In this case the solutions are discrete eigenstates in an unbounded space. Their stability boundaries and asymptotics are obtained by a WKB analysis. The weakly nonlinear interaction is delivered through the Lyapunov–Schmidt method.
The turbulent boundary layer over transverse square cavities
 L. DJENIDI, R. ELAVARASAN, R. A. ANTONIA

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 25 September 1999, pp. 271294

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Laserinduced uorescence (LIF) and laser Doppler velocimetry (LDV) are used to explore the structure of a turbulent boundary layer over a wall made up of twodimensional square cavities placed transversely to the flow direction. There is strong evidence of occurrence of outflows of fluid from the cavities as well as inflows into the cavities. These events occur in a pseudorandom manner and are closely associated with the passage of nearwall quasistreamwise vortices. These vortices and the associated lowspeed streaks are similar to those found in a turbulent boundary layer over a smooth wall. It is conjectured that outflows play an important role in maintaining the level of turbulent energy in the layer and enhancing the approach towards selfpreservation. Relative to a smooth wall layer, there is a discernible increase in the magnitudes of all the Reynolds stresses and a smaller streamwise variation of the local skin friction coefficient. A local maximum in the Reynolds shear stress is observed in the shear layers over the cavities.
Linear doublediffusive–inertial instability at the equator
 N. R. EDWARDS, K. J. RICHARDS

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 25 September 1999, pp. 295319

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Motivated by observations of interleaving in the equatorial Pacific, we consider the linear stability of a basic state on an equatorial βplane which is susceptible to both doublediffusive interleaving, driven by a meridional salinity gradient, and inertial instability driven by meridional shear. In a parameter regime compatible with the observations strong interaction can occur between the two processes, indicating that the stability of the system is dependent on the meridional gradients of both salinity and zonal velocity. Meridional shear is found to enhance the interleaving motion even for values of shear well below the cutoff for inertial instability. In the presence of diffusion inertial instability can also be excited by vertical shear, but only if the shear is comparable to the buoyancy frequency. When doublediffusive driving is weak relative to inertial driving the growth can be oscillatory, in which case the mechanism for instability is viscous–diffusive. In this case interleaving layers can slope downwards towards the fresh side of the front in the fingering regime, inhibiting their own growth.
On Faraday resonance of a viscous liquid
 JOHN MILES

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 25 September 1999, pp. 321325

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The tridiagonal determinant for Faraday resonance of a viscous liquid subject to an externally imposed vertical oscillation is expressed in terms of the surfacewave impedance of the liquid and developed as a continued fraction to obtain a systematic sequence of analytical approximations for the threshold acceleration. The impedance is calculated for either a clean or a fully contaminated surface on the assumption that the capillary, gravitational and viscous length scales are small compared with the breadth and depth of the liquid. Limiting approximations for weak and strong viscosity are constructed.
Effect of gravity modulation on the stability of convection in a vertical slot
 WENYAU CHEN, C. F. CHEN

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 25 September 1999, pp. 327344

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The nature of instability occurring in a differentially heated infinite slot under steady gravity depends only on the Prandtl number of the contained Boussinesq fluid. For fluids with Pr < 12.5, the instability is shear dominated and onsets in a steady convection mode; for fluids with Pr > 12.5, the instability is buoyancy dominated and onsets in an oscillatory mode. In this paper, we examine the effect of gravity modulation on the stability characteristics of convection in an infinite slot with both kinds of fluids, in particular, Pr = 1 and Pr = 25. Using the method of Sinha & Wu (1991), we are able to obtain accurate results without excessive numerical integration in the linear stability analysis by Floquet theory. Results show that, for Pr = 1, at a nondimensional oscillation frequency ω = 20, the critical state alternates between the synchronous and subharmonic modes. At higher frequencies, ω > 100, all critical states occur in the synchronous mode. For Pr = 25, with a modulation amplitude ratio of 0.5, resonant interaction occurs in the neighbourhood of ω = 2σc, where σc is the oscillation frequency of the instability at the critical state under steady gravity. This resonant interaction is destabilizing, with the critical Grashof number being reduced by approximately 20% from that at steady gravity. It is due to the presence of a detached subharmonic branch of the marginal stability curve. In frequency ranges where the detached subharmonic branch is absent, the critical state is in the quasiperiodic mode consisting of two waves of different oscillation frequencies whose sum is the forcing frequency. An analysis of the rate of change of the perturbation kinetic energy shows that, for Pr = 1, the instability is shear dominated regardless of the mode of oscillation, synchronous or subharmonic. Similarly, for Pr = 25, the instability is buoyancy dominated whether it is in the quasiperiodic or subharmonic mode. The mode switching is a response to the forcing and is independent of the dominant mechanisms of instability.