Volume 14, Issue 3, March 2002
 ARTICLES


Radial response of individual bubbles subjected to shock wave lithotripsy pulses in vitro
View Description Hide DescriptionDirect measurements of individual bubble oscillations in lithotripsy fields have been performed using lightscattering techniques. Studies were performed with bubble clouds in gassy water as well as single levitated bubbles in degassedwater. There is direct evidence that the bubble survives the inertial collapse, rebounding several times before breaking up. Bubble dynamics calculations agree well with the observations, provided that vapor trapping (a reduction in condensation during bubble collapse) is included. Furthermore, the afterbounces are dominated by vapor diffusion, not gas diffusion. Vapor trapping is important in limiting the collapse strength of bubbles, and in sonochemical activity.

Kelvin–Helmholtz instability in a HeleShaw cell
View Description Hide DescriptionA linear stability analysis is presented for the Kelvin–Helmholtz instability in a HeleShaw cell, an analysis based on the Navier–Stokes equation to improve on the previous Euler–Darcy study that Gondret and Rabaud [Phys. Fluids 9, 3267 (1997)] made of their own experiments.

The onset of thermal convection in a horizontal fluid layer heated from below with timedependent heat flux
View Description Hide DescriptionThe onset of convective motion in an initially quiescent, horizontal fluid layer heated from below is analyzed by using propagation theory. When the heat flux is changed suddenly from a constant to a linear ramp with time, cellular convection will set in due to buoyancy forces at a certain time. In this system predictions of the onset time of convective motion are compared with available experimental data and the effect of the aspect ratio is discussed. It is very interesting that propagation theory covers the whole domain of time and the results represent experimental data reasonably well.

Spectral measurements of Rayleigh–Taylor mixing at small Atwood number
View Description Hide DescriptionTemperature measurements have been made in a buoyancy driven mixing layer created by an unstable temperature gradient in water. Experiments were performed with a temperature difference of 5 °C, with a corresponding Atwood number of Two types of analyses were used to determine density fluctuation correlations. The first analysis determined density fluctuation correlations from the mean density profiles, which is sufficient for a twofluid distribution of the density where there was no molecular mixing. The second analysis used measurements taken from a continuous distribution of density, and thus included molecular mixing associated with heat diffusion. The thermocouples used for temperature measurements enabled the collection of large sample sizes and thus a detailed examination of molecular mix fractions, spectral properties of the density fluctuations, and probability distributions of the density through the mixing layer.

Secondorder theory for the deformation of a Newtonian drop in a stationary flow field
View Description Hide DescriptionThe classical perturbative theory for a single drop immersed in a flowing immiscible fluid is revisited, with the wellknown capillary number Ca, ratio of viscous to interfacial stresses, as the expansion parameter. Although the analysis is here limited to the Newtonian case under steadystate conditions, the perturbation method is innovative, as it makes use of rotational invariance to obtain workable tensorial representations of the pressure and velocity fields, and of the drop shape, at any order in Ca. The method is much less cumbersome than the classical one, based on an expansion in spherical harmonics. The analytical secondorder solution thus obtained differs somewhat from previously reported results.

Mean flow precession and temperature probability density functions in turbulent rotating convection
View Description Hide DescriptionLaboratory results for high Rayleigh number convection in a rotating cylinder heated from below are presented. Time series of temperature fluctuations reveal: (a) There is a largescale overturning meridional cell that precesses in the retrograde direction (opposite to the direction of the basic rotation) when viewed from the coordinates attached to the container. This precessing cell is prevalent at low to medium basic rotation rates, but is suppressed when the thermal Rossby number becomes less than about one half. (b) The temperature fluctuations at the centerpoint of the apparatus display a transition from nearly exponential, to Gaussian, then back to toward an exponential probability density function as the Taylor number is raised. The Rayleigh number required for Gaussian statistics is of order As the Rayleigh number increases, the Taylor number interval over which nearly Gaussian statistics are found becomes wider. These findings agree with previous turbulent rotating convection experiments at lower Rayleigh number in which Gaussian statistics were not observed.

Dynamics of relativistic magnetized blast waves
View Description Hide DescriptionThe dynamics of a relativistic blast wave propagating through a magnetized medium is considered, taking into account possible inhomogeneities of density and magnetic field and additional energy supply. Under the simplifying assumption of a spherically symmetric explosion in a medium with toroidalmagnetic field selfsimilar solutions for the internal dynamics of the flow are derived. In the weakly magnetized case, when the bulk of the flow may be described by the unmagnetized solutions, there is a strongly magnetized sheath near the contact discontinuity (when it exists). Selfsimilar solutions inside the sheath are investigated. In the opposite limit of strongly magnetized upstream plasma new analytical selfsimilar solutions are found. Possible application to the physics of gammaray bursts is discussed.

On the propagation of a twodimensional viscous density current under surface waves
View Description Hide DescriptionThis study aims to develop an asymptotic theory for the slow spreading of a thin layer of viscous immiscible dense liquid on the bottom of a waterway under the combined effects of surface waves and density current. By virtue of the sharply different length and time scales (wave periodic excitation being effective at fast scales, while gravity and streaming currents at slow scales), a multiplescale perturbation analysis is conducted. Evolution equations are deduced for the local and global profile distributions of the dense liquid layer as functions of the slowtime variables. When reflected waves are present, the balance between gravity and streaming will result, on a time scale one order of magnitude longer than the wave period, in an undulating water/liquid interface whose displacement amplitude is much smaller than the thickness of the dense liquid layer. On the global scale, the streaming current can predominate and drive the dense liquid to propagate with a distinct pattern in the direction of the surface waves.

Passive scalar mixing in a planar shear layer with laminar and turbulent inlet conditions
View Description Hide DescriptionThe effect of inlet conditions on the mixing of a passive scalar was investigated in a planar shear layer with inlet boundary layers that were laminar, tripped and naturally turbulent–transitional. Planar laserinduced fluorescence measurements of acetone were used to directly evaluate the shear layer structure, and were processed to determine probability density functions(PDFs) of the mixture fraction. The results agree well with previous studies in aqueous and gaseous systems for laminar inlet conditions. Largescale structures of a nearly homogeneous composition were found, and the structures spanned the mixing layer width giving rise to a nonmarching style PDF. A highspeed boundary layer that differed from the laminar state (produced by tripping a laminar boundary layer, from the naturally turbulent–transitional state at high flow rates, or by tripping the turbulent–transitional condition) gave rise to a hybridstyle PDF that was markedly different from either the laminar nonmarching case, or the marching shape that has been found by other investigators. The hybrid PDF shape had a marching character on the highspeed side of the mixing layer, and a distributed nature not favoring any specific composition on the lowspeed side of the mixing layer. Tripping the lowspeed boundary layer produced no change in the hybrid PDF shape, confirming that the difference observed between the high and lowspeed sides was the result of the shear layer development with turbulent inlet conditions, not an inlet condition effect. In addition, with turbulent and transitional inlet conditions all turbulent passive scalar profiles were found to be selfsimilar and the velocity power spectra displayed a −5/3 slope indicating welldeveloped turbulent conditions prevailed at a relatively low Reynolds number.Secondary structures were observed in images with turbulent–transitional inlet conditions, but not with tripped inlet conditions.

A spectral theory for smallamplitude miscible fingering
View Description Hide DescriptionUsing the selfsimilar symmetry of a diffusing front, we develop a linear spectral theory for miscible fingering at inception that accurately captures the destabilization of localized disturbances (with large transverse wavelengths compared to the front width) by the unsteady front. Our theory predicts a generic selected wavelength for gravity fingering, where η is the transverse to longitudinal dispersion ratio, and an additional factor proportional to the logarithm of the mobility ratio for viscous fingering) at the small time of where is the dispersion coefficient or diffusivity in the flow direction and is the displacement velocity. This wavelength then grows in time and approaches a universal asymptotic wavelength coarsening dynamics of where is the dimensional time, for all smallamplitude miscible fingering phenomena in a slot or in porous media. The exponent in time is due to a unique longwave stabilization mechanism due to transverse convection, which escapes prior quasisteady theory. Explicit and generic scalings are then derived for gravity and viscousmiscible fingering phenomena and are favorably compared to experimental and numerical results on linear coarsening dynamics.

Dynamics of the axisymmetric coreannular flow. II. The less viscous fluid in the core, saw tooth waves
View Description Hide DescriptionThe nonlinear dynamics of the concentric, twophase flow of two immiscible fluids in a circular tube is studied when the viscosity ratio of the fluid in the annulus to that in the core of the tube, μ, is larger than or equal to unity. For these values of the viscosity ratio the perfect coreannular flow(CAF) is linearly unstable and it is necessary to keep the ratio of the thickness of the annulus to the radius of the tube small so that the solutions remain uniformly bounded. The simulations are based on a pseudospectral numerical method while special care has been taken in order to minimize as far as possible the effect of the boundary conditions imposed in the axial direction allowing for multiple waves of different lengths to develop and interact. The time integration originates with the analytical solution for the pressure driven, perfect CAF or the perfect CAF seeded with either the most unstable mode or random disturbances. Quite regular wave patterns are predicted in the first two cases, whereas multiple unstable modes grow and remain even after saturation of the instability in the last case. The resulting waves generally travel in the same direction and faster than the undisturbed interface, except for the case with μ=1 for which they are stationary with respect to it. Depending on parameter values, waves move with the same velocity or interact with each other exchanging their amplitudes or merge and split giving rise to either chaotic or organized solutions. For fluids of equal viscosities and densities (μ=ρ=1) and for a Reynolds number, and an inverse Weber number, both based on the properties of the inner fluid, the tube radius, and the average flow velocity, small amplitude waves are predicted. The increase of μ by almost two orders of magnitude does not affect their amplitudes, but increases their temporal period linearly. Varying W by more than three orders of magnitude increases their amplitudes proportionately, while their period increases with the logarithm of W. Similar to that is the effect of increasing Re. The present analysis confirms and extends results based on long wave expansions, which lead to the Kuramoto–Sivashinsky equation and modifications of it.

Generation of vorticity and turbulent cooling of “hot channels” in gases
View Description Hide Description“Hot channels” (HC), formed by a rapid energy release in a gas, exhibit turbulent mixing and cooling, and their life time can be several orders of magnitude shorter than the molecular heat conduction time scale. Picone and Boris (1983) suggested that turbulent mixing results from the vorticity generation by the baroclinic mechanism during the early, shockwave dominated stage of the dynamics. More recently, alternative scenarios of vorticity generation in the HCs were suggested. This work investigates the vorticity generation and turbulent cooling of the HCs by hydrodynamical simulations in two dimensions. Assuming small perturbations of the cylindrical shape of the energy release region, we follow the evolution of several HCs and determine their cooling time. We identify the details of vorticity generation which results in turbulent flow and fast mixing of the cold ambient gas into the HC. The simulations support, with some modifications, the Picone–Boris scenario. The simulated cooling time scale is in good agreement with experimental results, and the cooling process can be described as turbulent diffusion. The width of the mixed region shows dynamic scaling and compares well to experimental data.

Dilute suspension of high inertia particles in the turbulent flow near the wall
View Description Hide DescriptionA steady flow of a very dilute suspension of solid particles in the turbulent flow near the wall is considered. The particle phase is described by the system of mean field mass, momentum, and turbulent Reynolds stress conservation equations incorporating the closure approximation for the thirdorder correlation of particle fluctuatingvelocity; this closure approximation, as well as the mean field equations, were derived earlier from the PDF (probability density function) kinetic equation governing the particle transport in the turbulent flow. In the turbulent flow near the wall, particles can be characterized as “high inertia” or “low inertia” by small or large values, respectively, of a single nondimensional parameter: the normalized particle relaxation time combining the wall frictionvelocity and physical parameters of the fluid and particle phase. Making use of the technique of matching asymptotic expansions, the distributions are found of the normal and shear turbulent stress, volume fraction, and mean velocity in the particle phase of the suspension of high inertia particles.

Nonlinear stability analysis of viscoelastic Taylor–Couette flow in the presence of viscous heating
View Description Hide DescriptionRecently, based on a linear stability analysis we demonstrated the existence of a new thermoelastic mode of instability in the viscoelastic Taylor–Couette flow [AlMubaiyedh et al., Phys. Fluids 11, 3217 (1999); J. Rheol. 44, 1121 (2000)]. In this work, we use direct timedependent simulations to examine the nonlinear evolution of finite amplitude disturbances arising as a result of this new mode of instability in the postcritical regime of purely elastic (i.e., Re=0), nonisothermal Taylor–Couette flow. Based on these simulations, it is shown that over a wide range of parameter space that includes the experimental conditions of White and Muller [Phys. Rev. Lett. 84, 5130 (2000)], the primary bifurcation is supercritical and leads to a stationary and axisymmetric toroidal flow pattern. Moreover, the onset time associated with the evolution of finite amplitude disturbances to the final state is comparable to the thermal diffusion time. These simulations are consistent with the experimental findings.

Velocity field statistics in homogeneous steady turbulence obtained using a highresolution direct numerical simulation
View Description Hide DescriptionVelocity field statistics in the inertial to dissipation range of threedimensional homogeneous steady turbulent flow are studied using a highresolution DNS with up to grid points. The range of the Taylor microscale Reynolds number is between 38 and 460. Isotropy at the small scales of motion is well satisfied from half the integral scale (L) down to the Kolmogorov scale (η). The Kolmogorov constant is 1.64±0.04, which is close to experimentally determined values. The third order moment of the longitudinal velocity difference scales as the separation distance r, and its coefficient is close to 4/5. A clear inertial range is observed for moments of the velocity difference up to the tenth order, between 2λ≈100η and where λ is the Taylor microscale. The scaling exponents are measured directly from the structure functions; the transverse scaling exponents are smaller than the longitudinal exponents when the order is greater than four. The crossover length of the longitudinal velocity structure function increases with the order and approaches 2λ, while that of the transverse function remains approximately constant at λ. The crossover length and importance of the Taylor microscale are discussed.

Solitary waves on inclined films: Flow structure and binary interactions
View Description Hide DescriptionThe downstream evolution of disturbances, introduced at the inlet of a liquid film flowing along an inclined plane wall, is studied numerically by solving the full, timedependent Navier–Stokes equation. Computational results are validated against the predictions of spatial linear stability analysis and against detailed data of the entire evolution process. The structure of the flow field below the waves is analyzed, and the results are used to test assumptions frequently invoked in the theoretical study of filmflow by longwave equations. An interesting prediction is that solitary waves exhibit strongly nonparabolic velocity profiles in front of the main hump, including a slim region of backflow. The computational scheme is subsequently used to study solitary wave interactions. It is predicted that coalescence (the inelastic collision of two humps) is not inevitable but occurs only when the waves differ appreciably in height. Waves of similar size repel monotonically, whereas for intermediate differences in height a strong oscillatory interaction between the two humps is predicted. Encouraging qualitative agreement with the limited experimental information available is noted.

Global instabilities in countercurrent jets
View Description Hide DescriptionSelfexcited instabilities in supersonic, countercurrent (CC) jets are demonstrated based on numerical simulations, and it is argued that they can be used to explain unresolved discrepancies between laboratory and theoreticalanalysis. Selfexcitation is based on upstream feedback mechanisms acting on the subsonic outer jet regions, and associated with underexpanded initial conditions or partial confinement effects introduced by a collar surrounding the CC jet shear layer. Recognition of these instabilities provides new insights on the role played by the collar in the laboratory CC jet systems, suggesting that practical approaches to the active control of CC jets might be based on suitable direct excitation of the shear layer within the region of influence of the collar.

Laminar flow in a porous channel with large wall suction and a weakly oscillatory pressure
View Description Hide DescriptionThe laminar oscillatory flow inside a rectangular channel with wall suction is considered here. The scope is limited to large suction imposed uniformly along the permeable walls. Inside the channel, the onset of small amplitude pressure disturbances produces an oscillatory field that we wish to investigate. Based on the normalized pressurewave amplitude, the conservation equations are linearized and split into leadingorder (steady) and firstorder (timedependent) equations. The firstorder set is subdivided into an acoustic, pressuredriven, wave equation, and a vortical, boundarydriven, viscousequation. For longitudinal pressure oscillations, both equations are written to the order of the wall suction Mach number. The resulting equations are then solved in an exact fashion. The novelty lies in the vortical response that reduces to a Weber equation following a Liouville–Green transformation. The emerging rotational solution is expressible in terms of confluent hypergeometric functions of the suction Reynolds number, Strouhal number, and spatial coordinates. The total solution is then constructed and found to coincide with the numerical solution of the linearized momentum equation. The oscillatory velocity exhibits similar characteristics to the exact Stokes profile for oscillations inside a long channel with hard walls. In particular, a thin rotational layer is observed in addition to the small velocity overshoot near the wall. Both depth and overshoot are nowhere near their values obtained by switching from mass extraction to mass addition. In contrast to former studies involving injection, the socalled acoustic boundary layer is found to depreciate when suction is increased or when viscosity is reduced. This response is similar to that of the Stokes layer over hard walls. Overall, the effect of increasing frequency is that of compressing the rotational layer near the wall.

Nonlinear regime of a multimode Richtmyer–Meshkov instability: A simplified perturbation theory
View Description Hide DescriptionIn this paper we present a drastic simplification of the perturbation method for the Richtmyer–Meshkov instability developed by Zhang and Sohn [Phys. Fluids 9, 1106 (1997)]. This theory is devoted to the calculus of the growth rate of the perturbation of the interface in the weakly nonlinear stage. In the standard approach, expansions appear to be power series in time. We build accurate approximations by retaining only the terms with the highest power in time. This simplifies and accelerates the solution. Highorder expressions are then easily reachable. Furthermore, computations for multimode interfaces become tractable. The accuracy of this approach is checked against twodimensional numerical simulations. The selection mode process is studied and the phase between modes is shown to be as important as the wave number or the amplitude. Inferences for the intermediate nonlinear regime are also proposed. In particular, a class of homothetic configurations is inferred; its validity is verified with numerical simulations even as vortexstructures appear at the interface.

Polymer stress statistics in the nearwall turbulent flow of a dragreducing solution
View Description Hide DescriptionThe direct numerical simulation of the turbulent flow of a dilute polymer solution in a plane channel at lowReynolds number has been performed in order to investigate the reduction in frictiondrag. The polymer solution has been represented as a continuum fluid whose constitutive equations have been derived on the basis of a modified FENEP dumbbell model. The mean polymer dynamics in the turbulent flow have been studied through statistical moments of the configuration tensor. The analysis of the results obtained suggests that polymers can be effective in terms of drag reduction only if their relaxation time is comparable to the characteristic time of their convection in the normaltothewall direction within nearwall turbulent structures. The energy budget of the normal components of the Reynolds stresstensor suggests that elongated polymers inhibit turbulence regeneration by opposing pressure redistribution from streamwise to crossflow velocity fluctuations.
