Showing papers in "Physics of Fluids in 1984"

An electron conductivity model for dense plasmas

Abstract: An electron conductivity model for dense plasmas is described which gives a consistent and complete set of transport coefficients including not only electrical conductivity and thermal conductivity, but also thermoelectric power, and Hall, Nernst, Ettinghausen, and Leduc–Righi coefficients. The model is useful for simulating plasma experiments with strong magnetic fields. The coefficients apply over a wide range of plasma temperature and density and are expressed in a computationally simple form. Different formulas are used for the electron relaxation time in plasma, liquid, and solid phases. Comparisons with recent calculations and available experimental measurement show the model gives results which are sufficiently accurate for many practical applications.

The three‐dimensional flow due to a stretching flat surface

Abstract: An exact similarity solution of the Navier–Stokes equations is found. The solution represents the three‐dimensional fluid motion caused by the stretching of a flat boundary.

Hamiltonian guiding center drift orbit calculation for plasmas of arbitrary cross section

Abstract: A Hamiltonian guiding center drift orbit formalism is developed which permits the efficient calculation of particle trajectories in magnetic field configurations of arbitrary cross section with arbitrary plasma β. The magnetic field is assumed to be a small perturbation from a zero‐order ‘‘equilibrium’’ field possessing magnetic surfaces. The equilibrium field, possessing helical or toroidal symmetry, can be modeled analytically or obtained numerically from equilibrium codes. The formalism is used to study trapped particle precession. Finite banana width corrections to the toroidal precession rate are derived, and the bounce averaged trapped particle motion is expressed in Hamiltonian form. Particle drift‐pumping associated with the ‘‘fishbone’’ oscillation is investigated. A numerical code based on the formalism is used to study particle orbits in circular and bean‐shaped tokamak configurations.

On the scaling of the turbulence energy dissipation rate

Abstract: From an examination of all data to date on the dissipation of turbulent energy in grid turbulence, it is concluded that, for square‐mesh configuration, the ratio of the time scale characteristic of dissipation rate to that characteristic of energy‐containing eddies is a constant independent of Reynolds number, for microscale Reynolds numbers in excess of about 50. Insufficient data available for other grid configurations suggest a possibility that the ratio could assume different numerical values for different configurations. This persistent effect of initial conditions on the time scale ratio is further illustrated by reference to the jet‐grid data of Gad‐el‐Hak and Corrsin.

Statistical analysis of the deviation of the Reynolds stress from its eddy‐viscosity representation

Abstract: An improvement of the eddy‐viscosity representation for Reynolds stress is made from the statistical viewpoint. The Reynolds stress is calculated with the aid of the two‐scale direct‐interaction formalism, and its deviation from the eddy‐viscosity representation is found under general mean flows. This result theoretically elucidates the noncoincidence of the zeros of Reynolds stress and mean strain, which is frequently observed in asymmetric turbulent shear flows.

Electromagnetic ion beam instabilities

Abstract: The linear theory of electromagnetic instabilities driven by an energetic ion beam streaming parallel to a magnetic field in a homogeneous Vlasov plasma is considered. Numerical solutions of the full dispersion equation are presented. At propagation parallel to the magnetic field, there are four distinct instabilities. A sufficiently energetic beam gives rise to two unstable modes with right-hand polarization, one resonant with the beam, the other nonresonant. A beam with sufficiently large T (perpendicular to B)/T (parallel to B) gives rise to the left-hand ion cyclotron anisotropy instability at relatively small beam velocities, and a sufficiently hot beam drives unstable a left-hand beam resonant mode. The parametric dependences of the growth rates for the three high beam velocity instabilities are presented here. In addition, some properties at oblique propagation are examined. It is demonstrated that, as the beam drift velocity is increased, relative maxima in growth rates can arise at harmonics of the ion cyclotron resonance for both right and left elliptically polarized modes.

A numerical-experimental study of confined flow around rectangular cylinders

Abstract: A previous numerical study by Davis and Moore of vortex shedding from rectangles in infinite domains is extended to include the effects of confining walls. The major changes to the numerical modeling are the addition of a direct solver for the pressure equation and the use of an infinite‐to‐finite mapping downstream of the rectangle. The parameters in the problem are now Reynolds number, rectangle aspect ratio, blockage ratio, and upstream velocity profile. As each of these is varied, the effects upon the forces acting on the rectangle and the structure of the wake are discussed. Streakline plots composed of multishaped passive marker particles provide a clear visualization of the vortices. These plots are compared with smoke‐wire photographs taken from a wind tunnel test. Strouhal numbers obtained both computationally and experimentally are compared for two values of the blockage ratio. Moving recirculation zones which appear between the wake and the walls are discussed.

A collisional drift wave description of plasma edge turbulence

Abstract: Model mode‐coupling equations for the resistive drift wave instability are numerically solved for realistic parameters found in tokamak edge plasmas. The Bohm diffusion is found to result if the parallel wavenumber is chosen to maximize the growth rate for a given value of the perpendicular wavenumber. The saturated turbulence energy has a broad frequency spectrum with a large fluctuation level proportional to κ (=ρs/Ln, the normalized inverse scale length of the density gradient) and a wavenumber spectrum of the two‐dimensional Kolmogorov–Kraichnan type, ∼k−3.

Reduction of Turbulent Skin Friction by Microbubbles

Abstract: Measurements of the effect of microbubbles on a zero pressure gradient turbulent boundary layer generated on the test section wall of a water tunnel are described Microbubbles are created by injecting air through a 05 μm sintered stainless steel plate immediately upstream of a floating element drag balance At the downstream edge of the balance the length Reynolds number is as high as ten million Integrated skin friction reduction of greater than 80% is observed The drag balance results are confirmed by measurements with a surface hot‐film probe For the case in which buoyancy tends to keep the bubbles in the boundary layer, the skin friction data are shown to collapse when plotted against the ratio of air to water volume flow rate The effects of buoyancy on skin friction reduction are also documented

Turbulent structure in the edge plasma of the TEXT tokamak

Abstract: A reversal has been observed in the mean phase velocity of the turbulent fluctuations in the edge plasma of the TEXT tokamak. The observations can be described by a model in which the wave velocity in the lab frame is dominated by a nonuniform Er×B velocity and a gradient driven drift. The measurements also exhibit a localized instability which occurs in the region of maximum velocity shear.

Current drive and helicity injection

Abstract: Magnetic configurations with currents parallel to the magnetic field need means for current drive. The concepts of transport of helicity are found useful for considering methods for current drive. The paper describes a formalism for considering transport of helicity, methods of injection of helicity, and specific examples of arrangements for current drive.

Reconnection rates of magnetic fields including the effects of viscosity

Abstract: The Sweet–Parker and Petschek scalings of the magnetic reconnection rate are modified to include the effect of the viscosity. The modified scalings show that the viscous effect can be important in high‐β plasmas. The theoretical reconnection scalings are compared with numerical simulation results in a tokamak geometry for three different cases: a forced reconnection driven by external coils, the nonlinear m=1 resistive internal kink, and the nonlinear m=2 tearing mode. In the first two cases, the numerical reconnection rate agrees well with the modified Sweet–Parker scaling when the viscosity is sufficiently large. When the viscosity is negligible, a steady state which was assumed in the derivation of the reconnection scalings is not reached and the current sheet in the reconnection layer either remains stable through sloshing motions of the plasma or breaks up to higher m modes. When the current sheet remains stable, a rough comparison with the Sweet–Parker scaling is obtained. In the nonlinear m=2 tearing mode case where the instability is purely resistive, the reconnection occurs on the slower dissipation time scale (ψs∼η). In addition, experimental data of the nonlinear m=1 resistive internal kink in tokamak discharges are analyzed and are found to give reasonable agreement with the modified Sweet–Parker scaling.

Plasma sheath transmission factors for tokamak edge plasmas

Abstract: Simple analytic relations are derived for the floating potential of a solid object in contact with a magnetized plasma as a function of the electron/ion mass ratio, the electron/ion temperature ratio, and the secondary electron emission coefficient. Simple analytic relations are also obtained for the sheath transmission factors, i.e., the particle and energy fluxes to floating and nonfloating surfaces. These analytic formulations, derived from a fluid model treatment of ion flow, are found to give results in close agreement with results derived from more complex analysis based on kinetic theory and numerical models. Tokamak scrape‐off plasmas are often characterized by ion distributions which have finite drift velocities at the point where the ions enter the scrape‐off plasma from the main plasma (via cross‐field diffusion). The Bohm criterion governing the drift velocity of ions as they enter the electrostatic sheath in front of a limiter, divertor plate, or probe is generalized for this case.

Heat transfer by high‐frequency oscillations: A new hydrodynamic technique for achieving large effective thermal conductivities

Abstract: Heat transfer between two fluid reservoirs maintained at different temperatures and connected to each other via a capillary bundle is examined when the fluid within the capillaries is oscillated axially. Very large effective axial heat conduction rates, exceeding those possible with heat pipes by several orders of magnitude, are found to be achievable. A laminar hydrodynamic theory is developed to describe the phenomenon and to show that the enhanced heat conduction is one involving radial heat transfer across very thin Stokes’s boundary layers existing in these flows. Experimental measurements using water as the working fluid show effective thermal diffusivities up to 17 900 times those existing in the absence of oscillations. Since the process involves no net convective mass transport, it offers considerable promise as a means for the rapid removal of heat from radioactive fluids.

Minimum enstrophy vortices

Abstract: Two kinds of minimum enstrophy vortices in two‐dimensional flows are found by variational analysis. Each represents the ideal limit of a selective decay of enstrophy with energy and angular momentum or circulation, respectively, remaining fixed. For each, the vorticity is confined to a disk whose radius is determined by the fixed integrals. The definition of the vortices assures a large degree of stability, but many questions about their formation and destruction have not yet been answered.

Convective thermocapillary instabilities in liquid bridges

Abstract: An axisymmetric liquid bridge is surrounded by a passive gas. A steady shear flow is set up by imposing a temperature gradient along the bridge and driving the motion by thermocapillarity. This dynamic state is susceptible to convective instabilities that lead to propagating hydrothermal waves that feed on the underlying temperature gradients. The convective instabilities of this axisymmetric return‐flow state are presented as functions of the Prandtl number of the liquid and the surface Biot number of the interface. Comparisons are made with the results of Smith and Davis for planar layers and with available experimental data.

Numerical simulation of the inverse cascade in two‐dimensional turbulence

Abstract: A numerical simulation of large‐scale two‐dimensional turbulence fed in a narrow band of wavenumbers is presented. The development of the inverse energy cascade predicted by Kraichnan [Phys. Fluids 10, 1417 (1967)] is observed.

Hamiltonian formulation of reduced magnetohydrodynamics

Abstract: Reduced magnetohydrodynamics (RMHD) is a principal tool for understanding nonlinear processes, including disruptions, in tokamak plasmas. Although analytical studies of RMHD turbulence are useful, the model’s impressive ability to simulate tokamak fluid behavior has been revealed primarily by numerical solution. A new analytical approach, not restricted to turbulent regimes, based on Hamiltonian field theory is described. It is shown that the nonlinear (ideal) RMHD system, in both its high‐beta and low‐beta versions, can be expressed in Hamiltonian form. Thus a Poisson bracket, { , }, is constructed such that each RMHD field quantity ξi evolves according to ξi ={ξi,H}, where H is the total field energy. The new formulation makes RMHD accessible to the methodology of Hamiltonian mechanics; it has lead, in particular, to the recognition of new RMHD invariants and even exact, nonlinear RMHD solutions. A canonical version of the Poisson bracket, which requires the introduction of additional fields, leads to a nonlinear variational principle for time‐dependent RMHD.

Measuring plasma drift velocities in tokamak edge plasmas using probes

Abstract: Plasma drift or rotation is caused at the edge of tokamaks by various mechanisms including flow to surfaces and neutral injection. These drift velocities can be measured using electrostatic probes and a theory is presented providing a method of probe analysis to deduce drift velocity, plasma density, and temperature. Initial probe experiments using this probe analysis yield credible values of drift velocity, but comparison with an independent technique, e.g., spectroscopic Doppler shifts, is sought. OFF

The coupling between scales in shear flows

Abstract: In order to explore the relationship between the large‐ and small‐scale motions in turbulent shear flows, a number of flows have been studied based on short‐time correlation measurements. The shear flows investigated are boundary layers, plane and axisymmetric mixing layers, plane wakes and the far fields of plane and circular jets. The coupling between the scales has been obtained by correlating the low‐frequency component of the u‐velocity signal with a signal that is similar to the envelope of the high‐frequency part of the velocity signal. The coupling is found to be significant in all flows. Phase reversal across the shear region is found to occur in the boundary layers and the mixing layers only. This is interpreted to mean that in boundary layers and mixing layers, the streamwise extent of the large structure over which the small‐scale activity reaches a peak moves at one transverse end of the large structure over which its ensemble averaged u fluctuation is positive to another where it is negative. This phase shift (≂180°) of the location of the peak activity of the small scales with respect to the large structure takes place gradually resulting in a midlayer where the phase difference is about 90°. On the other hand, in the far field of the jets and wakes, all across the layer, the peak activity of the small scales always remains confined to that half of the streamwise extent of the large structure where its ensemble averaged u fluctuation is positive; in the remaining streamwise half of the large structure, the small scales remain dormant.

The formation and expansion of a toroidal drop moving in a viscous fluid

Abstract: The deformation of an initially spherical liquid drop moving under the action of gravity in another fluid with which it is completely miscible is investigated under conditions of small values of the drop Reynolds number. It is found experimentally that such a drop evolves into an open torus which subsequently expands, and this phenomenon is examined theoretically for two limiting drop geometries: (i) a slightly deformed spherical drop, and (ii) a highly expanded, slender open torus. Under the assumptions of zero interfacial tension and creeping flow, the theory provides a qualitative description for the initial stages of the drop evolution [case (i)], but is unable to account for the observed drop expansion during latter stages of deformation [case (ii)]. On the other hand, if small inertial effects are retained in the analysis, the theory predicts that a slender open fluid torus possessing an arbitrary cross‐sectional geometry will expand without change of shape to first order in Reynolds number. Quantitative comparisons of theoretically predicted rates of expansion with experimental measurements suggest the possible existence of a small, time‐dependent interfacial tension across the drop interface.

Nonlinear evolution of the lower‐hybrid drift instability

Abstract: The results of simulations of the lower‐hybrid drift instability in a neutral sheet configuration are described. The simulations use an implicit formulation to relax the usual time step limitations and thus extend previous explicit calculations to weaker gradients, larger mass ratios, and long times compared with the linear growth time. The numerical results give the scaling of the saturation level, heating rates, resistivity, and cross‐field diffusion and a demonstration by comparison with a fluid electron model that dissipation in the lower‐hybrid drift instability is caused by electron kinetic effects.

Microwave absorption and plasma heating due to microwave breakdown in the atmosphere

Abstract: The propagation and absorption of microwaves above the breakdown threshold in the atmosphere with the self‐consistent breakdown plasma are investigated. Typical hydrodynamic calculations show that an ionization front is rapidly formed which moves toward the microwave source and consequently decouples the microwaves from the original ionization region. By focusing the microwaves or using a reflector, ionization can be confined to localized regions where the microwave strength is high enough to cause breakdown even though the incoming microwaves are below threshold. In a strongly collisional atmosphere, it is found that the cutoff plasma density, which roughly equals the collisionless cutoff density multiplied by the collisionality, is much higher than the calculated maximum density. This results in high absorption that increases with microwave power, decreases with atmospheric pressure, and is quite independent of other parameters. In a weakly collisional atmosphere, breakdown easily creates a plasma density higher than the cutoff density and causes reflection. It is found that the reflection decreases with the collisionality of the system and is quite independent of the microwave strength. In general, the microwave field strength in the ionization regions is attenuated to the breakdown value at steady state, and the resulting electron temperature is about 2 eV, independent of the incident microwave flux. Limitations of the calculations are due to the availability of experimental data for the rate coefficients, but comparison to results from recent focused microwave experiments shows excellent agreement.

Bénard convection with time‐periodic heating

Abstract: A thin liquid layer, which is heated from below, has its lower boundary modulated sinusoidally in time with amplitude δ. Weakly‐nonlinear stability theory shows that the modulation produces a range of stable hexagons near the critical Rayleigh number. For small δ the range is O(δ4) in size and decreases with modulation frequency. These hexagons bifurcate subcritically and correspond to downflow at cell centers.

A symmetric regularized‐long‐wave equation

Abstract: A symmetric version of the regularized‐long‐wave equation is shown to describe weakly nonlinear ion acoustic and space‐charge waves. The equation possesses hyperbolic secant squared solitary waves and has four known invariants. Numerical solutions are compared with previous results on the regularized‐long‐wave equation.

Flow development in a porous channel and tube

Abstract: The spatial development of the inlet velocity profile in a uniformly porous channel and tube is investigated. For tubes which are very long compared with their radius, it is shown that above a critical Reynolds number of 2.3 the inlet velocity profile does not necessarily decay into the fully developed, similarity profile for an infinite tube. Rather, the structure of the flow throughout the entire tube is influenced by the inlet profile. This loss of validity of the similarity solution is due to the fact that the tube is of finite length and the inlet profile is not of the similarity form. The actual length of the tube is, however, unimportant. Analogous results hold for the flow in a porous channel, with a critical Reynolds number of approximately 6.

Transport associated with the collisionless detrapping/retrapping orbits in a nonaxisymmetric torus

Abstract: The differences among the various scalings in the transport calculations associated with the collisionless detrapping/retrapping orbits in a nonaxisymmetric torus are clarified. A simple random walk argument indicates that these differences are mainly due to an implicit assumption made about the retrapping rate of the toroidally trapped particles.

The stability of two-dimensional linear flows

Abstract: A theoretical investigation is made of the linear stability of a viscous incompressible fluid undergoing a steady, unbounded two‐dimensional flow in which the velocity field is a linear function of position. Such flows are approximately generated by a four‐roll mill device which has many experimental applications, and can be characterized completely by a single parameter λ which ranges from λ=0 for simple shear flow to λ=1 for pure extensional flow. The linearized velocity disturbance equations are analyzed for an arbitrary spatially periodic initial disturbance to give the asymptotic behavior of the disturbance at large time for 0≤λ≤1. In addition, a complete analytical solution of the vorticity disturbance equation is obtained for the case λ=1. It is found that unbounded flows with 0<λ≤1 are unconditionally unstable. An instability criterion relating the initial disturbance wave vector α to the steady flow strain rate E, kinematic viscosity ν, and the parameter λ is obtained. This criterion shows that for all admissible values of E and ν, a wave vector α may be found which corresponds to disturbances that grow exponentially in time. The growth of these disturbances is accompanied by a growth of vorticity oriented along the principal axis of extensional strain in the case λ=1. The results of this investigation also confirm the established fact that simple shear flow (λ=0) is stable to all infinitesimal spatially periodic disturbances.

Nonlinear reduced fluid equations for toroidal plasmas

Abstract: Nonlinear reduced fluid equations are derived for studying resistive instabilities in large‐aspect‐ ratio, low‐beta toroidal plasmas. An ordering is developed in which plasma compressibility as well as the poloidal curvature are retained. The nonlinear equations can be linearized and used to reproduce the Mercier criterion in the large‐aspect‐ratio, low‐beta limit. A second set of reduced equations is derived from the Braginskii fluid equations. These equations, which are very similar to the reduced magnetohydrodynamic equations, contain diamagnetic effects as well as parallel transport associated with magnetic fluctuations. Both sets of equations conserve energy exactly.

The stability of an evaporating liquid surface

Abstract: A linearized stability analysis is carried out for an evaporating liquid surface with a view of understanding some observations with highly superheated liquids. The analytical results of this study depend on the unperturbed temperature near the liquid surface. The absence of this data renders a comparison with experiment impossible. However, on the basis of several different assumptions for this temperature distribution, instabilities of the interface of a rapidly evaporating liquid are found for a range of wavenumbers of the surface wave perturbation. At large evaporating mass flow rates the instability is very strong with growth times of a millisecond or less. A discussion of the physical mechanism leading to the instability is given.