Volume 28, Issue 12, December 1985
Index of content:

Instability at the interface between two shearing fluids in a channel
View Description Hide DescriptionThe linear stability of plane Couette flow composed of two immiscible fluids in layers is considered. The fluids have different viscosities and densities. For the case of equal densities, there is a critical Reynolds number above which the interfacial mode of the bounded problem is approximated by that of the unbounded problem for wavelengths that are not short enough to be in the asymptotic short‐wavelength range, as well as for short waves. The full linear analysis reveals unstable situations missed out by the long‐ and short‐wavelength asymptotic analyses, but which have comparable orders of magnitudes for the growth rates. For the case of unequal densities, it is found that the arrangement with the heavier fluid on top can be linearly stable if the viscosity stratification, volume ratio, surface tension, Reynolds number, and Froude number are favorable.

Increased particle confinement observed with the use of an external dc bias field in a spheromak experiment
View Description Hide DescriptionSpheromaks are formed in a mesh flux conserver in the presence of an external dc bias magnetic field. The particle confinement is improved when the spheromak separatrix is put inside the metal mesh by the application of positive bias flux. The spheromaks remain stable to tilt instabilities with ratios of bias‐to‐spheromak flux of up to 47±7%.

Magnetic surface wave instabilities in plasmas
View Description Hide DescriptionTechniques of plasma surface wave theory are introduced to obtain new results for magnetic electron waves at the interface of two plasma regions with distinct properties. The procedure generalizes and simplifies problems involving sharp density jumps often encountered in laser– or beam–plasma interaction. As an example, a new magnetic instability is discussed.

Heat transfer from a cylinder in a time‐dependent cross flow at low Peclet number
View Description Hide DescriptionThe heat transfer from a constant‐temperature cylinder in a uniform, time‐dependent cross flow at low Peclet number is considered. The time dependence is allowed to be strong, so that the velocity fluctuations may be comparable to, or larger than, the mean flow. The first nontrivial term in the Oseen approximation is calculated using matched‐expansion theory, and its physical significance is discussed. As an illustration, the time‐dependent heat transfer is calculated for a steady cross flow with large sinusoidal perturbations.

Transport of sedimenting Brownian particles in a rotating Poiseuille flow
View Description Hide DescriptionGeneralized Taylor dispersion theory is used to analyze the convective and diffusive transport of sedimenting colloidal particles occurring within a Poiseuille flow in a horizontal circular pipe that is being rotated slowly about its symmetry axis. Such rigid‐body rotation serves to keep the particles permanently in suspension despite their non‐neutral buoyancy, thereby preventing deposition of the particles on the cylinder bottom. In the ‘‘large’’ particle limit, where transverse diffusion is small compared with sedimentation, expressions are derived for the mean axial velocity and Taylor dispersivity of the colloidal particles. A novel flow field fractionation (FFF) scheme based thereon is proposed for continuously separating particles of different sizes and/or densities.

High Marangoni number convection in a square cavity
View Description Hide DescriptionThe steady thermocapillary motion in a square cavity with a top free surface in the absence of gravitational forces is considered. The cavity is heated from the side with the vertical boundaries isothermal while the horizontal boundaries are adiabatic. The relative change in the surface tension is very small, i.e., an appropriate capillary number tends to zero, so that the free surface is assumed to remain flat at leading order. A finite‐difference method is employed to compute the flow field. Numerically accurate solutions are obtained for a range of Prandtl numbers and for Reynolds numbers Re as high as 5×10^{4}. Surface deflections are computed as a domain perturbation for small capillary number. In addition, asymptotic methods are used to infer the boundary layer structure in the cavity, in the limit of large values of the Reynolds and Marangoni numbers. For a fixed Prandtl number Pr, it is shown that the Nusselt number, liquid circulation, and maximum vorticity are asymptotic to Re^{1} ^{/} ^{3}, Re^{−} ^{1} ^{/} ^{3}, and Re^{2} ^{/} ^{3}, respectively. These results are in agreement with the computed solutions. The leading‐order solution for the free‐surface deformation is sensitive to the value of Pr. With Pr>1, the depression near the hot corner may exceed the elevation near the cold corner, while a secondary elevation may be induced near the hot corner when Pr<1.

The departure from Darcy flow in natural convection in a vertical porous layer
View Description Hide DescriptionAn analytical and numerical study is reported of steady‐state natural convection in a two‐dimensional porous layer heated from the side. Contrary to previous investigations of the phenomenon, which were all based on the Darcy flow model, a vector generalization of Forchheimer’s one‐dimensional model is used in the present study, which is valid for all values of local Reynolds number based on pore size. A matched boundary layer solution of the type developed by Weber for Darcy flow is developed for the limit of large‐pore Reynolds numbers (the ‘‘non‐Darcy’’ limit). It is shown that the natural convection phenomenon in the non‐Darcy limit is governed by a new dimensionless group, the Rayleigh number for the higher Reynolds number limit, Ra_{∞}. Numerical experiments are reported in the range 1.6×10^{5}≤Ra_{∞}≤1.6×10^{9}, in a porous layer with height/thickness ratio equal to 2, and with a high value of Darcy modified Rayleigh number (Ra=4000). The numerical experiments confirm the flow features and scales anticipated by the matched boundary layer solution for the non‐Darcy limit. The experiments also document the transition from the well‐known Darcy flow to the large‐pore Reynolds‐number limit treated in this paper.

Kinetic theory for plane flows of a dense gas of identical, rough, inelastic, circular disks
View Description Hide DescriptionGrad’s method of moments is employed to derive balance laws and constitutive relations for plane flows of a dense gas consisting of identical, rough, inelastic, circular disks. Two temperatures are involved; these are proportional to the kinetic energies associated with fluctuations in translational velocity and spin, respectively. When the single particle velocity distribution function is assumed to be close to a two‐temperature Maxwellian, two distinct theories are obtained. One applies when the particles are almost smooth and the collisions between them are nearly elastic; the other applies to nearly elastic particles that, in a collision, almost reverse the relative velocity of their points of contact. I both cases energy is nearly conserved in collisions.

The thick, turbulent boundary layer on a cylinder: Mean and fluctuating velocities
View Description Hide DescriptionThe mean and fluctuating velocities in a turbulent boundary layer on a cylinder have been experimentally characterized for the case where the boundary layer is thick compared to the radius of transverse curvature. The mean velocity measurements suggest a mixed scaling for the ‘‘log law of the wall’’ using the wall coordinate y U _{τ}/ν and the ratio of the local boundary layer thickness to the radius of the cylinder δ/a. A relation for the slope and intercept of the log law of the wall as functions of δ/a based on empirical results and simple analysis is presented. Measurements of the Reynolds stress for δ/a of order 10 show that the Reynolds stress drops off much more quickly with distance from the wall than for a turbulent boundary layer on a flat plate. Both the Reynolds stress data and the turbulent intensity in the mean flow direction data are functions of the inverse radial distance from the center of the cylinder.

Laminar‐turbulent transition in a slowly pulsating pipe flow
View Description Hide DescriptionTransitional pulsating flow in a pipe is investigated experimentally in the frequency region where its behavior is assumed to be quasisteady. Instantaneous velocity measurements were performed at several radial locations at the exit plane of the pipe. The output of flush‐mounted hot wires at two upstream positions was also recorded simultaneously. The results indicate that the quasisteady assumption is valid in general. The flow behavior is the function primarily of the instantaneous Reynolds number, although the laminarization process and retransition to turbulence are qualitatively different. The laminarization at subcritical instantaneous Reynolds numbers in a time‐dependent flow is a gradual process, similar to the observed turbulence decay in flow geometries where Re is varied spatially. The retransition to turbulence occurs by means of the generation of turbulent slugs at various locations along the pipe. These slugs are convected downstream and coalesce, eventually resulting in fully developed turbulent flow in the whole pipe at higher instantaneous Reynolds numbers.

The possibility of a resonance mechanism in the developing two‐dimensional jet
View Description Hide DescriptionAn experimental investigation of a developing planar turbulent jet discharging into a quiescent environment was undertaken. The measurements now reported suggest that the pressure field associated with the large‐scale flow structures occurring downstream excite the nascent jet shear layers near the jet exit. Such structures are characterized by the formation of an antisymmetric structural array near the end of the potential core and extending far into the similarity region. The results suggest that coupling or feedback between downstream coherent structures and the initial region may be important in the development of the flow.

Flow field effects on nucleation in a reacting mixing layer
View Description Hide DescriptionChemical nucleation has been studied numerically in a stagnation point mixing layer in which reactants in two counter‐flowing streams form a condensable monomer. The response of the subsequent nucleation kinetics to the velocity gradient in the flow is described in terms of a Damkohler number. Two limiting cases have been established. First, if the Damkohler number for monomer production is small, i.e., the rate of monomer production is slow, then the nucleation of particles can be strongly affected by the flow field in a manner which is equivalent to the effect of supersaturation in a uniform vapor. Second, if the Damkohler numbers for cluster growth are small (because of a small accommodation factor for monomer–cluster interactions), the concentrations of clusters do not achieve equilibrium levels. This can result in the suppression of particle formation over a critical range of Damkohler numbers. In this case the behavior of the nucleation kinetics is analogous to the transient phase of nucleation in a uniform vapor.

The evolution of axisymmetric vortex systems in liquid He II
View Description Hide DescriptionThe main features of the evolution of the normal and superfluid vorticity distributions (for high line density) in liquid helium are investigated for simple systems in the case of axial symmetry. The variation of initially Gaussian and top‐hat vorticity distributions for different ratios of the two components is analyzed. The ratio of the obtained characteristic time scales of the variation of the vorticity on the centerline and the spreading of the vorticity distribution depends essentially on the initial conditions.

Nonlinear waves in superposed magnetic fluids
View Description Hide DescriptionAn asymptotic weakly nonlinear wave propagation in Rayleigh–Taylor magnetic fluid flows is investigated. The waves are found to be unstable against modulation. The stability analysis reveals that there exist two regions of instabilities. The magnetic field has a stabilizing influence in one region and an opposite effect in the second region. The nonlinear cutoff wavenumber which separates the region of stability from that of instability is also obtained. The magnetic field has a destabilizing effect on the cutoff wavenumber.

Plasma kinetic theory in action‐angle variables
View Description Hide DescriptionAn appropriate canonical perturbation theory to correctly deal with general electromagnetic field perturbation is developed and is used to set up plasma kinetic theory in action‐angle variables. A variety of test problems are solved to show the unifying power of the method. Basic linear, quaislinear, and nonlinear equations, which can serve as the starting point for a whole range of plasma problems, are derived.

The dispersion functional for multidimensional equilibria
View Description Hide DescriptionNumerical study of the linear stability of plasmas is very difficult when one or more of the plasma species is collisionless and the equilibrium is multidimensional, that is, characterized by two or more nonignorable spatial coordinates. The problem arises, for example, in evaluating kinetic stabilizing effects on the internal tilting mode (an n=1 ballooning mode) in field‐reversed configurations. In this paper, the Laplace transform of the perturbation distribution function for a collisionless species is derived for all classes of phase‐space trajectories and used to construct the dispersion functional for multidimensional equilibria. The kinetic part of the dispersion functional is expressed in terms of the Laplace transform of autocorrelation functions with respect to a certain delay time. It is shown how to obtain the same result formally by using Liouville eigenfunctions. For the case of the Vlasov‐fluid model, the dispersion functional is transformed in a way that is particularly appropriate for computation of the kinetic stability of field‐reversed configurations to the internal tilting mode.

The long‐time evolution approximation for a quasi‐one‐dimensional plasma system
View Description Hide DescriptionThe long‐time evolution of a plasma through a series of equilibrium states for the case of the field‐reversed configuration (FRC) is considered. A formulation of the transport model in magnetic flux variables is given for the approximate geometry where the magnetic field lines are straight. Thus a complicated two‐dimensional elliptic differential equation amenable only to numerical solution is avoided. Radial force balance is enforced pointwise while axial force balance is enforced only globally. The equations formulated in this manner are relatively simple; some of their salient features are discussed. Although a particular plasma–magnetic field configuration is considered, the type of analytical method presented is more general and applies to other coupled initial‐boundary value problems. The absence of complicated geometry and flux surface averaging involved for other systems makes the essential aspects of the transformations employed transparent.

Physical mechanisms for hot‐electron stabilization of low‐frequency interchange modes
View Description Hide DescriptionThe stabilization of low‐frequency background and hot‐electron interchange modes is studied by using a relatively simple multi‐fluid model. Physical pictures for the stabilization mechanisms such as the ‘‘charge‐uncovering’’ effect are given and compared with the stabilization of interchange modes by the finite Larmor radius (FLR) effect.

Nearly perpendicular wave propagation at the fundamental electron‐cyclotron resonance
View Description Hide DescriptionWaves propagating nearly perpendicular to the equilibrium magnetic field across the fundamental cyclotron resonance layer are studied by a boundary layer analysis for a weakly relativistic, inhomogeneous Vlasov plasma. The plasma is assumed to be perpendicularly stratified. It is found that the wave energy associated with the ordinary mode transmitted through the layer is independent of the relativistic corrections and is given by a geometrical optics formula. It is also found that there is no reflected energy associated with this mode when it is incident from the high‐field side. These results are the same as the nonrelativistic case with purely perpendicular propagation. Relativistic effects produce a significant reduction of the reflection coefficient for low‐field side incidence from the nonrelativistic value. Correspondingly, for this mode there is a considerable increase in the absorption rate for sufficiently high, but moderate, electron density and temperature.

Linear and nonlinear evolution of double‐humped ion distributions in strong unmagnetized shock structures
View Description Hide DescriptionIon distributions in shock structures are determined from the Fokker–Planck equation. If the shock Mack number is high, and the electron temperature T _{ e } in the shock structure is sufficiently higher than the ion temperature T _{1} far ahead of the shock wave, the ion distribution has a double‐humped shape, and is unstable in the sense of a linear Landau analysis. Characteristic times in which the wave–ion interaction, the wave–wave interaction, and the ion trapping modify the ion distribution and/or the Landau‐amplified electric field to a certain extent are estimated. If T _{ e }/T _{1} and the shock Mach number are sufficiently high, the wave–ion interaction has a strong influence on the evolution of the ion distribution, and competes with the ion–ion collisions in the shock structure.