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Volume 28, Issue 10, October 1985

A predictive Monte Carlo simulation of two‐fluid flow through porous media at finite mobility ratio
View Description Hide DescriptionDiffusion‐limited aggregation simulations of two‐fluid flow through porous media at infinite mobility ratio are extended to finite mobility ratio of driving to driven fluid. Curves for sweep efficiency at breakthrough are generated that come close to data obtained on two‐dimensional sand packs.

Nonlinear mode conversion and anomalous absorption processes during radio‐frequency heating of bumpy torus plasmas
View Description Hide DescriptionThe theory of nonlinear mode conversion is reviewed and applied to the ELMO Bumpy Torus (EBT) [P r o c e e d i n g s o f t h e 5 t h I n t e r n a t i o n a l C o n f e r e n c e o n P l a s m a P h y s i c s a n d C o n t r o l l e d N u c l e a r F u s i o n R e s e a r c h 1 9 7 4 (IAEA, Vienna, 1975), Vol. 2, p. 141] environment in the electron‐cyclotron, upper‐hybrid, lower‐hybrid, and ion‐cyclotron frequency ranges. A variety of parametric excitations in these frequency ranges is studied. Based on cascade saturation mechanisms, the parametric turbulence levels and corresponding anomalous absorption rates of the excited modes are evaluated. The relevance of nonlinear mode conversion for contemporary and future EBT experiments is discussed.

The Stokes drag on a horizontal cylinder falling toward a horizontal plane
View Description Hide DescriptionMeasurements have been made of the Stokes drag on a cylinder of length 2l and diameter 2r falling with its axis horizontal toward a horizontal plane boundary. The distance from the cylinder axis to the plane is h. For l>2.48r and h>3r, the results agree well with the slender body theory of de Mestre and Russel [J. Eng. Math. 9, 81 (1975)]. For l>19r and h<0.5l, the theory of Jeffrey and Onishi [Q. J. Mech. Appl. Math. 3 4, 129 (1981)] for a cylinder of infinite length is in good agreement with the experimental results; this theory also agrees well with the results for cylinders as short as l=0.89r in the region h<2r. Measurements with tori indicate that the theory of Jeffrey and Onishi is not sensitive to curvature of the cylinder centerline in the horizontal plane. For h>2l and for all of the cylinders used (98r≥l≥0.89r) the data confirm the general boundary effect expression of Brenner [J. Fluid Mech. 1 2, 35 (1962)]; and are in better agreement with Brenner’s expression than with the modified form proposed by Williams [J. Fluid Mech. 2 4, 285 (1966)]. For l>5r, the deceleration of the cylinder is observed to be constant when the cylinder velocity is between 0.4 and 0.15 of its value far from the wall, but the deceleration tends to zero as the velocity approaches zero.

Viscous flow through a rotating square channel
View Description Hide DescriptionFully developed flow of an incompressible Newtonian fluid driven by a pressure gradient through a square channel that rotates about an axis perpendicular to the channel roof is analyzed here with the aid of the penalty/Galerkin/finite element method. Coriolis force throws fast‐moving fluid in the channel core in the direction of the cross product of the mean fluid velocity with the channel’s angular velocity. Two vortex cells form when convective inertial force is weak. Asymptotic limits of rectilinear flow and geostrophic plug flow are approached when viscous force or Coriolis force dominates, respectively. A flow structure with an ageostrophic, virtually inviscid core is uncovered when Coriolis and convective inertial forces are both strong. This ageostrophic two‐vortex structure becomes unstable when the strength of convective inertial force increases past a critical value. The two‐vortex family of solutions metamorphoses into a family of four‐vortex solutions at an imperfect bifurcation composed of a pair of turning points.

Natural convection in a differentially heated corner region
View Description Hide DescriptionThis is a fundamental study of the phenomenon of natural convection in the region formed by a vertical warm wall rising above a cold horizontal wall, or in the region between a cold vertical wall extending downward from a warm horizontal surface. The study consists of scale analysis, numerical simulations, and an asymptotic solution of the low Rayleigh number limit. The scale analysis predicts the persistence of a single cell in the corner region, regardless of Rayleigh number. The cell migrates toward the corner as the Rayleigh number Ra_{ H } increases: the flow rate and the net heat transfer rate vary as Ra^{1/7} _{ H }. The scale analysis is verified qualitatively and quantitatively by means of numerical experiments in the range Ra_{ H } =10^{3}–10^{7}, Pr=0.7–7, H/L=1–4, where Pr is the Prandtl number and H/L is the height/length ratio of the corner region. Additional numerical simulations are presented for cases where one or both walls have uniform heat flux; in these cases, the heat transfer rate shows nearly the same behavior as when the corner walls are both isothermal. The asymptotic solution for the Ra_{ H }→0 limit shows that the flow field is relatively insensitive to whether the wall temperature varies continuously or discontinuously through the corner point.

Stability of flow over axisymmetric bodies with porous suction strips
View Description Hide DescriptionA numerical‐perturbation scheme is used to determine the stability of incompressible flows past axisymmetric bodies with porous strips. Linear triple deck, closed‐form solutions are developed for the basic‐flow quantities, which account for upstream influence. These solutions are linear superpositions of the flow past the body without suction plus the perturbations caused by the suction strips. The flow past the suctionless body is calculated using the Transition Analysis Program System (TAPS). A perturbation scheme is used to determine the increment a _{ i j } in the complex wavenumber at a given location s _{ j } because of the presence of a strip centered at s _{ i }. The end result is a set of influence coefficients that can be used to determine the growth rates and amplification factors for any suction levels without repeating the calculations. The influence coefficients are used to develop a simple linear optimization scheme to determine the number, spacing, and mass flow rate through the strips on an axisymmetric body.

The growth of a shear‐free mixed layer in a linearly stratified fluid
View Description Hide DescriptionAn experimental study is described that deals with the nature of the entrainment zone of a linearly stratified fluid when the turbulence in the mixed layer is generated by an oscillating grid. Measurements include the interfacial‐layer thickness, frequency, and amplitude of the interfacial waves and the buoyancy flux at the entrainment interface. These measurements were compared with a theory proposed by Long [J. Fluid Mech. 8 4, 43 (1978)], part of which (the mixed‐layer deepening law) has already been verified by Folse e t a l. [Phys. Fluids 2 4, 396 (1981)]. After a certain depth of the mixed layer, the entrainment rate shows an abrupt increase, and this may possibly be caused by wall effects as shown in our previous investigations.

The generation of noise in impinging vortex motion past a step
View Description Hide DescriptionThe noise field generated by the motion of one, two, or several rectilinear vortices in an inviscid, irrotational mean flow past a step is considered for low Mach number. The streakline patterns of an impinging vortex are compared with flow visualization studies, and qualitative agreement between the present analytical results and the experiments is achieved. Both acoustic pressure and intensity are strong functions of the vortex motion near the step which, because of the complexity of the geometry, must be computed numerically. Results for the acoustic pressure and intensity are presented for a variety of incident vortex arrays.

Nonlinear lengthening of a triangular acoustic pulse
View Description Hide DescriptionA theoretical and experimental study is reported of the lengthening and attenuation of triangular acoustic pulses of finite amplitude. The pulses studied in the laboratory are the leading portions of individual N waves, which are produced in a conical‐driver shock tube. The facility has been used to produce N waves with nondimensional pressure jumps across the leading front in the range 0.02–0.17. The length of the leading portion of the wave was measured in terms of its transit time at two different locations. Pulse lengthenings were measured in the weak shock wave range and were found to be in agreement with the well‐known theoretical predictions. On the other hand, our measurements of the pulse’s amplitude decay did not compare favorably with the same theory, mainly because of uncertainties associated with the flow field immediately behind the shock front. A new derivation of the theoretical predictions is also presented for lengthening and decay of a triangular pulse that is based on simple physical arguments.

Shock implosion of a small homogeneous pellet
View Description Hide DescriptionA small spherical, or cylindrical, pellet is imploded by an intensive, evenly distributed, short energy pulse. At the surface of the pellet the matter ionizes, its temperature and pressure rapidly rise, and the ablated plasma, by reaction, implodes the inner nucleus of the pellet. The involved structure of the energy absorbing zone is idealized and a sharp deflagration front is considered. With an almost square energy pulse, slightly dropping with time, the solution of the mass, momentum, and energy conservation equations of the compressed matter, is self‐similar. The differential equation of the nondimensional position of the deflagration front, its integral, and the magnitude and shape of the outside energy pulse are derived. The process of ablation is shown to depend solely on the nondimensional velocity of the gas just ahead of the deflagration front, minus the speed of sound, or the ratio of the gas densities across the deflagration front.

High‐resolution measurement of electron velocity distribution functions in a plasma
View Description Hide DescriptionA Thomson scattering diagnostic capable of achieving one‐discharge, high‐resolution spectra through use of a microchannel plate image intensifier coupled to an optical multichannel analyzer has been implemented on a theta‐pinch device. The improved resolution permits measurements of local average electron velocities and current densities. First‐order relativistic corrections and anomalous velocity‐space fluctuations are directly observable. The enhanced fluctuations are attributed to electrons interacting with unstable waves.

Striations of the convective type and feedback in low‐pressure mercury/noble‐gas discharges
View Description Hide DescriptionExperiments indicate that ionization‐diffusion instabilities (striations) of the convective type, present in low‐pressure mercury/noble‐gas lamps, are generated by current fluctuations in the inhomogeneous part at the cathode end of the discharge. Self‐excitation of striations is made possible by feedback via the electrical supply circuit of the lamp. An experimental boundary condition is found at the point of excitation. A simple plasma model, with an extension that takes feedback into account, fully describes the laser‐like phenomenon of mode selection and the influence of the discharge length.

Current and electric field measurements in a double inverse pinch device
View Description Hide DescriptionCurrent and electric field measurements have been made near the magnetic neutral line in a double inverse pinch device. The current in the central region did not show any observable Petschek‐type shock structure, but instead had the appearance of a simple current sheet. A disruption of the induced current system, called the impulsive flux transfer event, was observed. Measurements of the electric field in the central region showed that the space‐charge field, −∇φ, was significant, especially during the current buildup and disruption. When measurements of the total electric field, E=−Ȧ−∇φ, were combined with current density measurements, the conductivity of the plasma in the central region remained sensibly constant throughout the current disruption.

Solitons on a cylindrical pinch with Hall effect
View Description Hide DescriptionIn this paper it is shown by an asymptotic method that the solitons which appear on a cylindrical magnetohydrodynamic (MHD) pinch with axial symmetry persist when the Hall current term is included in the governing equations. On the other hand, the results presented here indicate that, under the linear stability criterion for the cylindrical pinch, nonlinear waves that have an amplitude that does not decay with time can occur.

Kinetic theory of the electromagnetic drift modes driven by pressure gradients
View Description Hide DescriptionKinetic equations for the electromagnetic drift modes are derived and analyzed for the stability of tokamaks in the local approximation. In the dissipationless, hydrodynamic limit, the fifth‐order polynomial dispersion relation previously studied is recovered. The kinetic velocity space integrals in the ion dynamics are shown to modify the five principal modes of oscillation and their polarizations. It is shown that in kinetic stability theory the critical plasma pressure defined in magnetohydrodynamic theory determines a transition from microinstability to macroinstability.

Theory of semicollisional kinetic Alfvén modes in sheared magnetic fields
View Description Hide DescriptionThe spectra of the semicollisional kinetic Alfvén modes in a sheared slab geometry are investigated, including the effects of finite ion Larmor radius and diamagnetic drift frequencies. The eigenfrequencies of the damped modes are derived analytically via asymptotic analyses. In particular, as one reduces the resistivity, it is found that, because of finite ion Larmor radius effects, the damped mode frequencies asymptotically approach finite real values corresponding to the end points of the kinetic Alfvén continuum.

Electrostatic ion‐cyclotron waves in a nonuniform magnetic field
View Description Hide DescriptionThe properties of electrostatic ion‐cyclotron waves excited in a single‐ended cesium Q machine with a nonuniform magnetic field are described. The electrostatic ion‐cyclotron waves are generated in the usual manner by drawing an electron current to a small exciter disk immersed in the plasma column. The parallel and perpendicular (to B) wavelengths and phase velocities are determined by mapping out two‐dimensional wave phase contours. The wave frequency f depends on the location of the exciter disk in the nonuniform magnetic field, and propagating waves are only observed in the region where f≳f _{ c i }, where f _{ c i } is the local ion‐cyclotron frequency. The parallel phase velocity is in the direction of the electron drift. From measurements of the plasma properties along the axis, it is inferred that the electron drift velocity is not uniform along the entire current channel. The evidence suggests that the waves begin being excited at that axial position where the critical drift velocity is first exceeded, consistent with a current‐driven excitation mechanism.

Forced magnetohydrodynamic turbulence in a uniform external magnetic field
View Description Hide DescriptionTwo‐dimensional dissipative magnetohydrodynamic turbulence is randomly driven at small spatial scales and is studied by numerical simulation in the presence of a strong uniform external magnetic field. A novel behavior is observed which is apparently distinct from the inverse cascade that prevails in the absence of an external magnetic field. The magnetic spectrum becomes dominated by the three longest‐wavelength Alfvén waves in the system allowed by the boundary conditions: those which, in a box size of edge 2π, have wavenumbers (k x ,k y ) =(1,0),(1,1), and (1,−1), where the external magnetic field is in the x direction. At any given instant, one of these three modes dominates the vector potential spectrum, but they do not constitute a resonantly coupled triad. Rather, they are apparently coupled by the smaller‐scale turbulence.

Neoclassical ion transport in rotating axisymmetric plasmas
View Description Hide DescriptionThe neoclassical theory of ion transport in rotating axisymmetric plasmas is formulated. The flow speed is allowed to be of the order of the ion thermal speed. It is shown that the ion distribution function becomes Maxwellian, with temperature uniform on a magnetic surface, and the poloidal flow decays, in a few transit or collision times, in general. A drift kinetic equation is derived which is a simple generalization of the drift kinetic equation for nonrotating plasmas. The radial gradient of the toroidal angular velocity appears as a driving term like the temperature gradient. Both gradients drive the transport of toroidal angular momentum and energy, in general; Onsager relations for the two‐by‐two transport matrix are derived. The off‐diagonal transport coefficients are shown to be zero if the magnetic field has up–down symmetry. A simple expression for the enhancement of the ion thermal conductivity in the banana regime, caused by rotation, is derived. The neoclassical viscosity is shown not to be enhanced by rotation in the banana regime, and to be small in the collisionality parameter in the collisional regime, assuming up–down symmetry. In the collisional regime, the thermal conductivity is shown to be suppressed by the effects of rotation.

Lifetime measurements of electron‐cyclotron‐resonance‐heated electrons
View Description Hide DescriptionThe lifetime of energetic electrons generated via second harmonic electron‐cyclotron‐resonance heating in a symmetric magnetic mirror configuration is investigated experimentally. The decay time is observed to be as much as a factor of 4 shorter than the value calculated for the most rapid classical loss mechanism, Coulomb drag. The nonclassical loss component is shown to be induced by the microwave power available for electron heating. The microwave power required to sustain the temperature and density of the energetic electrons is calculated from the observed lifetimes and stored energy. It is found to be only a few percent of the total injected power which, in this experiment, is equal to or less than 1 kW.