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Volume 31, Issue 12, December 1988

Biological scattering particles for laser Doppler velocimetry
View Description Hide DescriptionThe use of formaldehyde‐killed single‐cell organisms as scattering particles for laser Doppler velocimetry studies in water and salt solutions is discussed. Advantages over traditional scattering particles include ready availability in large quantities, uniformity in size, monodispersity, and the ability to stay suspended in solution for several days. The tracers are colloidal sols for a wide range of densities of aqueous solutions. The microorganisms can be easily stained with a large variety of fluorescent and nonfluorescent dyes before they are used as tracer particles.

Hydrodynamic transport properties of hard‐sphere dispersions. I. Suspensions of freely mobile particles
View Description Hide DescriptionThe hydrodynamic transport properties of hard‐sphere dispersions are calculated for volume fractions (φ) spanning the dilute limit up to the fluid–solid transition at φ=0.49. Particle distributions are generated by a Monte Carlo technique and the hydrodynamic interactions are calculated by Stokesian dynamics simulation. The effects of changing the number of particles in the simulation cell are investigated, and the scaling laws for the finite‐size effects are derived. The effects of using various levels of approximation in computing both the far‐ and near‐field hydrodynamic interactions are also examined. The transport properties associated with freely mobile suspensions—sedimentation velocities, self‐diffusion coefficients, and effective viscosities—are determined here, while the corresponding properties of porous media are determined in a companion paper [Phys. Fluids 3 1, xxxx (1988)]. Comparison of the simulation results is made with both experiment and theory. In particular, the short‐time self‐diffusion coefficients and the suspension viscosities are in excellent agreement with experiment.

Hydrodynamic transport properties of hard‐sphere dispersions. II. Porous media
View Description Hide DescriptionThe hydrodynamic transport properties of hard‐sphere dispersions are calculated for volume fractions (φ) spanning the dilute limit up to the fluid–solid transition at φ=0.49. Particle distributions are generated by a Monte Carlo technique and the hydrodynamic interactions are calculated by Stokesian dynamics simulation. The effects of changing the number of particles in the simulation cell are investigated, and the scaling laws for the finite‐size effects are determined. The effects of using various levels of approximation in computing both the far‐ and near‐field hydrodynamic interactions are also examined. The transport properties associated with porous media—permeabilities and hindered diffusion coefficients—are determined here. The corresponding properties of freely mobile suspensions are determined in a companion paper [Phys. Fluids 3 1, xxxx (1988)].

Parallel flow convection in a tilted two‐dimensional porous layer heated from all sides
View Description Hide DescriptionIn this work natural convection within a two‐dimensional fluid saturated Darcy porous layer is considered. The porous material is in a large aspect ratio rectangle with its sides inclined with respect to the gravity vector. All four faces are exposed to uniform heat fluxes, opposite faces being heated and cooled, respectively. Analytical solutions for the streamfunction and temperature fields in the central region are deduced using a parallel flow assumption. Numerical confirmation of the analytical results is also obtained. For high enough Rayleigh numbers multiple steady states around the rest state are found. These states represent unsymmetrical flows in opposite directions. The critical Rayleigh number for the onset of motion determined from a stability analysis corresponds to that for the existence of unicellular convection using the parallel flow approximation. Linear stability limits of each of the multiple states are also calculated. One of the convective flows is found to be stable to higher Rayleigh numbers than the other.

The sedimentation of nondilute suspensions in inclined settlers
View Description Hide DescriptionThe base‐state convective flows, which are set up when a nondilute sedimenting suspension is placed beneath an inclined wall, are analyzed theoretically using a two‐fluid model. Their hydrodynamic stability and the corresponding spatial growth of small two‐dimensional disturbances at the clear fluid–suspension interface are then determined over the entire range of the governing parameters through numerical solutions of the relevant Orr–Sommerfeld equations. Two mechanisms for the growth of instability waves at the interface are identified. The results demonstrate that the base‐state flow becomes more unstable as inertial effects in the base state become more pronounced and thus, contrary to what has been suggested by earlier investigators, there is no restabilization as the base state approaches the inviscid limit. Increasing the concentration of the suspension is found to have a stabilizing effect on the two‐phase interface, particularly when inertial effects dominate in the base state. Similarly, increasing the angle of inclination enhances the stability of the interface when viscous forces are dominant in the base flow.

Rayleigh–Bénard and interfacial instabilities in two immiscible liquid layers
View Description Hide DescriptionThe linear stability of two immiscible liquid layers heated from below through a rigid perfectly conducting boundary with a free deformable upper surface and a deformable liquid–liquid interface is examined. Three modes of instability are allowed simultaneously in the analysis: surface tension driven at each of the two interfaces, buoyancy driven because of the presence of adverse density gradients in each liquid, and an interfacial mode (the Rayleigh–Taylor instability), related to the density difference and interfacial tension at the interfaces. For purely surface tension driven convection, the presence of a middle interface that can suppress normal deformations makes the two liquid layers more stable than a single layer of the same total depth. The interaction of the buoyancy and interfacial modes leads to overstability when the physical properties of the two liquids are only slightly different from each other. For certain Rayleigh numbers, both stationary and oscillatory modes display positive growth constants over a certain range of wavenumbers. As the Marangoni number is increased, the coupling between the surface tension and buoyancy mechanisms makes the system more unstable but removes the oscillatory eigenmodes. The addition of trace amounts of insoluble surface active agents at the two interfaces has a very strong stabilizing influence by introducing the expected hydrodynamic rigidity to the surfaces. However, their more interesting effect is their ability to change the nature of the most unstable eigenmode from stationary to oscillatory.

Dispersion driven instability in miscible displacement in porous media
View Description Hide DescriptionThe effect of dispersion on the stability of miscible displacement in rectilinear porous media is examined. Following a convection–dispersion (CDE) formalism, the base state of Tan and Homsy [Phys. Fluids 2 9, 3549 (1986)] at conditions of unfavorable mobility contrast is analyzed. Emphasis is placed on the dependence of the dispersion coefficient on flow rate (e.g., mechanical dispersion). It is found that such a dependence induces a destabilizing contribution at short wavelengths. This effect, which is in contrast to the stabilization commonly associated with dispersion, is highly pronounced near the onset of the displacement. It is also near this onset that, for a certain condition, the cutoff wavenumber becomes infinitely large. An analytical expression is derived for this condition and the origin and implications of the instability are discussed. It is also suggested that the present CDE formulation may be inadequate in providing stability criteria for a range of unstable flows.

The dynamics of periodically driven bubble clouds
View Description Hide DescriptionAn averaged two‐fluid model is used to study the motion of a cloud of bubbles. The linearized equations of motion are shown to be a wave equation with both dissipation and dispersion. The fully nonlinear equations are also examined and it is demonstrated that the cutoff frequency of the cloud is equal to the natural frequency of a single bubble. The steady linear response of a periodically driven bubble cloud is then derived. Resonances are seen to arise when the driving frequency is below the cutoff frequency. The inner core of the cloud is shielded by an outer layer when the driving is above the cutoff frequency. The nonlinear dynamics of periodically driven bubble clouds is studied numerically. It is found that the cutoff frequency is crucial in determining whether or not the cloud will behave like a single bubble. Also, under some conditions the cloud is seen to behave like a damped and driven single‐degree‐of‐freedom Hamiltonian system.

The formation of vortex rings
View Description Hide DescriptionVortex rings are usually formed by a brief discharge of fluid from an orifice. In previous investigations, the geometry of the vortex generator has varied greatly from one experiment to another, with important consequences for the ensuing flow. The present work categorizes the generating conditions for vortex rings and classifies the conditions under which a given vortex generator produces either an initially laminar ring, which may or may not undergo instability and transition to turbulence, or an initially turbulent ring. A particularly simple vortex generator was devised and measurements were carried out to provide systematic data over a range of the important dimensionless parameters. The results of this survey are used to construct a transition map that reveals a reasonably well defined boundary separating vortex rings that are turbulent upon formation from those that are not. High‐speed cinephotography of the formation and evolution of turbulent vortex rings suggests a possible connection between the generating conditions and the transition to turbulence.

Influence of streamwise vortices on Tollmien–Schlichting waves
View Description Hide DescriptionAn analysis is presented of the influence of steady quasiperiodic streamwise vortices on Tollmien–Schlichting (TS) waves. The vortices act as parametric exciters for the TS waves. The present analysis is for the subharmonic resonance in which the spanwise wavenumber of the vortices is twice that of the TS wave. Floquet theory is used to derive an eigenvalue problem for the complex streamwise wavenumber, which is then solved using a shooting technique. The results show that there are two components of the solution, one is stabilized and the other is destabilized by the vortices. For the same flow characteristics, calculations were obtained from the theory which Nayfeh [J. Fluid Mech. 1 0 7, 441 (1981)] developed using the method of multiple scales. The results obtained from both approaches are in excellent agreement.

Finite‐amplitude solitary waves at the interface between two homogeneous fluids
View Description Hide DescriptionNumerical solutions are presented for finite‐amplitude interfacial waves. Only symmetric waves are calculated. Two cases are considered. In the first case the waves are free‐surface solitary waves propagating on a basic flow with uniform vorticity. Large‐amplitude waves of extreme form are calculated for a range of values of the basic vorticity. In the second case the waves are propagating on the interface between two homogeneous fluids of different densities, which are otherwise at rest. Again large‐amplitude waves of extreme form are calculated for a range of values of the basic density ratio. In particular, in the Boussinesq limit when the density ratio is nearly unity, solitary waves of apparently unlimited amplitude can be found.

The critical Weber number for vortex and jet formation for drops impinging on a liquid pool
View Description Hide DescriptionWhen a liquid drop impacts on a pool, the drop will either coalesce into the host liquid, with the creation of a vortex ring below the surface and little splashing above, or will splash, producing a cavity in the host liquid that collapses inward, producing an upward jet of fluid. It is found that there is a critical Weber number, We_{ c }=U(ρD/T)^{1/2}≂8, below which vortex rings are formed and above which a jet is produced for drops falling into the identical fluid. Here, U is the drop speed at impact, D is the drop diameter, and ρ and T are density and surface tension, respectively. The Weber number criterion is compared with experiments using water and mercury.

The first Landau constant in a viscous free shear layer
View Description Hide DescriptionThe first Landau constant of a free shear layer with parallel velocity profile U=tanh y is calculated numerically according to the amplitude expansion formulation proposed by Herbert [J. Fluid Mech. 1 2 6, 167 (1983)]. The numerical results strongly support Gotoh’s asymptotic expression [J. Phys. Soc. Jpn. 2 4, 1137 (1968)] of the first Landau constant λ_{1} for large Reynolds number R when the amplification factor c _{ i } is smaller than R ^{−} ^{1} ^{/} ^{3}: λ_{1}=−8.177R ^{1} ^{/} ^{3}(1+0.25R c _{ i }). It is also concluded that the third‐order nonlinear term in the Landau equation acts as the stabilizing effect throughout the parameter range where the numerical calculation was performed.

One‐dimensional detonation stability: The spectrum for infinite activation energy
View Description Hide DescriptionThe one‐dimensional stability of detonation waves characterized by one‐step irreversible Arrhenius kinetics is examined. The description of the steady structure in the limit of infinite activation energy is well known, and an examination is made of the linear unsteady perturbation equations in the same limit. The spectrum contains both a single real positive eigenvalue that vanishes in the limit and an infinite number of nonvanishing eigenvalues that define a growth rate which is an increasing function of frequency. Plausible qualitative extrapolation to finite activation energies provides an explanation of experimental results obtained by Alpert and Toong [Astronaut. Acta 1 7, 539 (1972)].

Nonlinear spectral dynamics of a transitioning flow
View Description Hide DescriptionThe nonlinear spectral dynamics of a transitioning flow in the wake of a flat plate is experimentally studied at different downstream positions with a two‐point method. The measurement setup consists of two sensors, which are separated in the downstream direction. The quadratically nonlinear transfer function between the two points is then estimated from the digitized fluctuation data. Such transfer functions permit one to quantify the quadratically nonlinear spectral dynamics occurring between the two sensor points. The method used to estimate the transfer functions and local bicoherency for non‐Gaussian input and output signals, by means of digital spectral analysis techniques, is briefly discussed. The measured quadratic transfer function of the experimental data changes gradually with downstream distance, but its main features are unchanged. The observed appearance of progressively higher harmonics of the fundamental mode and the filling in of the spectral valleys over short downstream distances are, thus, mainly caused by spectral redistribution of energy that is available in the interacting modes and not caused by abrupt changes in the coupling properties. This result is supported by the local bicoherency measurements.

Closure of the Reynolds stress and scalar flux equations
View Description Hide DescriptionA second‐order, single‐point closure model for calculating the transport of momentum and passive scalar quantities in turbulent flows is described. Of the unknown terms that appear in the Reynolds stress and scalar flux balance equations, it is those which involve the fluctuating pressure that exert a dominant influence in the majority of turbulent flows. A closure approximation (linear in the Reynolds stress) has been formulated for the velocity‐pressure gradient correlation appearing in the Reynolds stress equation. When this is used in conjunction with previous proposals for the other unknown terms in the stress equation, the proposed model closely simulates most of the data on high Reynolds number homogeneous turbulent flows. For the fluctuating scalar‐pressure gradient correlation appearing in the scalar flux equation, an approximation has been devised that satisfies the linear transformation properties of the exact equation. Additional characteristics of the fluctuating scalar field are obtained from the solution of modeled balance equations for the scalar variance and its ‘‘dissipation’’ rate. The resulting complete scalar field model is capable of reproducing measured data in decaying scalar grid turbulence and strongly sheared, nearly homogeneous flow in the presence of a mean scalar gradient. In addition, applications to the thermal mixing layer developing downstream from a partially heated grid and to a slightly heated plane jet issuing into stagnant surrounds result in calculated profiles in close agreement with those measured.

Flow visualization of Dean vortices in a curved channel with 40 to 1 aspect ratio
View Description Hide DescriptionResults from a flow visualization study of Dean vortex flow are presented. These were obtained over a range of Dean numbers from 40 to 220 using a transparent channel with mild curvature, an aspect ratio of 40 to 1, and an inner to outer radius ratio of 0.979. Observations and photographs show evidence of pairs of counter‐rotating Dean vortices indicated by mushroom‐shaped smoke patterns for Dean numbers greater than 64 and angular positions at least 85° from the start of curvature. Photographs showing nonsymmetric Dean vortices with rocking motion are presented and believed to be evidence of a twisting mode of oscillations. Dean vortices with oscillations mostly in the radial direction are also observed, which are believed to strongly depend on the small amplitude disturbances that trigger initial vortex development. Photographic evidence of small secondary vortex pairs, and vortices with simultaneous radial and spanwise oscillations are also given along with a domain map showing the experimental conditions for different types of vortex behavior.

Thermophoresis of a spherical particle in a rarefied gas of a transition regime
View Description Hide DescriptionA theoretical study is made of thermophoresis of a solid sphere in a rarefied gas in which a uniform temperature gradient and a uniform velocity at infinity exist. The analysis is carried out on the basis of the linearized Bhatnager–Gross–Krook (BGK) equation, from which simultaneous integral equations for the density, flow velocity, and temperature are derived. These equations are solved numerically over a wide range of Knudsen numbers covering the area from the slip flow to the nearly free molecular flow. A formula for the variation of the thermophoretic force acting on the sphere versus the Knudsen number is obtained for any value of thermal conductivity of the sphere when there is no imposed flow at infinity. The thermophoretic velocity of a suspended sphere in a gas is also calculated. The flow patterns as well as the distributions of temperature are shown.

Shock wave derivatives
View Description Hide DescriptionA general theory is developed for obtaining the tangential and normal derivatives of thermodynamic and kinematic properties just downstream of a curved, unsteady shock wave. The flow upstream of the shock need not be steady or uniform and the gas need not be thermally or calorically perfect. Because of the complexity of the results, explicit formulas are provided for the above derivatives when the flow is steady and two‐dimensional or axisymmetric, and the gas is perfect.

Nearly incompressible magnetohydrodynamics at low Mach number
View Description Hide DescriptionThe dynamics of a compressible magnetofluid plasma with a polytropic equation of state are considered in the limit of low plasma frame acoustic Mach number. The relationship between the equations describing the low Mach number flow and the equations of idealized incompressible magnetohydrodynamics is investigated using a multiple time scale asymptotic expansion procedure, which is justified by appealing to several rigorous theorems concerning both hydrodynamics and magnetohydrodynamics. When appropriate assumptions are adopted concerning the degree of departure from incompressibility, the lowest‐order behavior is that of incompressible magnetohydrodynamics, associated with order Mach number‐squared ‘‘pseudosound’’ density fluctuations. The first corrections to incompressible flow take the form of magnetoacoustic fluctuations, with associated pressure fluctuations at the same order as the pseudosound pressure. Resumming the asymptotic series gives rise to a simple set of equations that describes ‘‘nearly incompressible magnetohydrodynamics.’’ The theory provides a justification for the turbulent density spectrum theory of Montgomery, Brown, and Matthaeus [J. Geophys. Res. 9 2, 282 (1987)] and clarifies several issues pertaining to Alfvén wave turbulence in the solar wind. The nearly incompressible description may also be useful in other theoretical contexts, particularly in extensions of incompressible magnetohydrodynamic turbulence theory, since it is expected to be valid for finite times (until possible shock structures form) when the global Mach number is sufficiently small.