Showing papers in "Physics of Fluids in 1972"
TL;DR: In this article, a variational principle for the rate of irreversible entropy production in axisymmetric toroidal confinement systems was derived by employing the full Fokker-Planck collision operator, including both like and unlike species collisions.
Abstract: The neoclassical theory of plasma transport in axisymmetric, toroidal confinement systems, is developed by means of a variational principle for the rate of irreversible entropy production. The variational principle derived here employs the full Fokker‐Planck collision operator, including both like and unlike species collisions. Using the variational principle, all the relevant neoclassical transport coefficients are systematically evaluated in the “banana” regime of small collisional frequency, to lowest order in the inverse aspect ratio. These results include both the “diagonal” and “cross” coefficients for the particle fluxes, ion and electron heat flux, and electric current. By combining the transport coefficients with appropriate moments of the drift equation, a closed set of equations which accurately summarize the predictions of neoclassical theory in the banana regime is obtained. The significance of these equations, in particular with regard to recent tokamak experiments, is discussed briefly.
387 citations
TL;DR: In this article, an analytical and numerical study of the nonresonant, modified plasma two-stream instability, which is driven by relative streaming of electrons and ions across a magnetic field B0 is presented.
Abstract: Results of an analytical and numerical study of the nonresonant, modified plasma two‐stream instability, which is driven by relative streaming of electrons and ions across a magnetic field B0 are presented. The instability has characteristic frequency and growth rate comparable to the lower‐hybrid frequency. The linear theory is discussed both in the electrostatic and fully electromagnetic cases, and a detailed numerical study of the dependence of the unstable roots of the dispersion relation for a wide range of plasma parameters is presented. The nonlinear theory includes discussions of (1) quasilinear theory, (2) trapping, which is responsible for nonlinear stabilization, (3) a derivation of a fully nonlinear scaling law which shows how results scale with electron‐ion mass ratio, and (4) the effect of cross‐field vortex‐like motion caused by turbulence induced E × B drifts. One‐and two‐dimensional computer simulations with dense k‐space spectra are presented in support of this theory. The simulations show that the instability can be a very important turbulent heating mechanism that heats the ions (perpendicular to B0) and the electrons (parallel to B0) comparably. The final state has (Ti⊥/mi)1/2 ≈ (Te‖/mi)1/2∼12U, where U is the initial relative drift speed. Applications to experimental situations are discussed.
369 citations
TL;DR: In this article, a wide variety of transverse electromagnetic instabilities in plasmas with different sources and degrees of anisotropy are presented for the electron bi-Maxwellian, electron pinch, and ion-pinch experiments.
Abstract: Theory and simulation experiment are presented for a wide variety of transverse electromagnetic instabilities in plasmas with different sources and degrees of anisotropy. In each of the electron bi‐Maxwellian, electron‐pinch, and ion‐pinch experiments, the bulk response of the system during the initial stages of instability is in good agreement with the predictions of quasilinear theory. Furthermore, the two independent energy constants which derive from the fully nonlinear Vlasov‐Maxwell equations are found to remain constant to very good accuracy, even when the magnetic field energy reaches a substantial fraction of the total system energy. In each simulation experiment it is found that the magnetic energy saturates once the magnetic bounce frequency has increased to a value comparable to the linear growth rate prior to saturation, i.e., when ω¯B∼γ¯k. It is concluded that amplitude limitation for Weibel instabilities is a result of magnetic trapping for a broad range of system parameters. In many experi...
304 citations
TL;DR: In this article, the true critical Rayleigh number for the onset of convective flow of a fluid in a rectangular box of porous material heated from below is found for various box geometries.
Abstract: The true critical Rayleigh number for the onset of a convective flow of a fluid in a rectangular box of porous material heated from below is found for various box geometries. The preferred cellular mode of the motion at Rayleigh numbers just above the critical is determined. In contrast with the established results for the similar problem in a continuous fluid, the roll (a cell with only two nonzero velocity components) is not the only cellular mode and the roll axis direction is such that there is the greatest degree of “squareness” in the cross section of each roll. The invalidity of a frequently used form of Darcy's law and the present form of the energy method for the stability of flows in which fluid crosses the boundaries is discussed.
250 citations
TL;DR: In a toroidal plasma with axial symmetry, the three adiabatically invariant actions of a particle are the magnetic moment, the canonical angular momentum and the toroidal flux enclosed by the drift surface.
Abstract: In a toroidal plasma with axial symmetry, the three adiabatically invariant actions of a particle are the magnetic moment, the canonical angular momentum, and the toroidal flux enclosed by the drift surface. Resonant interactions between particles and the normal modes of collective oscillation produce mode growth or decay and random changes in the actions. This random walk is represented by a diffusion equation in action space. Both the diffusion tensor and the growth rate depend upon a coupling coefficient which represents the work done by a normal‐mode field eigenfunction on the current density of an unperturbed particle orbit. The diffusion of the plasma causes adiabatic changes in the electric and magnetic self‐consistent fields. Accordingly, energy is not conserved, but is exchanged with external currents.
247 citations
TL;DR: In this article, a novel state of turbulent plasma characterized by small scale phase-space granulations called "clumps" is proposed, where regions of different phase space density are mixed by the fluctuating electric field.
Abstract: A novel state of turbulent plasma characterized by small scale phase‐space granulations called “clumps” is proposed. Clumps are produced when regions of different phase space density are mixed by the fluctuating electric field. They move along ballistic orbits and drive the turbulent field in a manner similar to that in which thermal fluctuations are driven by particle discreteness. In the coherent wave limit the clumps become the familiar trapped particle eddies of a Bernstein‐Green‐Kruskal mode. The turbulent state can exist in the absence of linear instability although it is more likely to occur in a linearly unstable plasma. The spectrum contains a ballistic portion as well as resonances at the wave (collective) frequencies. The discreteness of the clumps produces collision‐like process. For example, the average distribution function satisfies a Fokker‐Planck equation instead of a quasilinear diffusion equation.
236 citations
TL;DR: Using a two-dimensional Lagrangian hydrodynamic code, this paper showed qualitative agreement with Meshkov's experiments, numerical calculations, and linear analysis are compared, and good agreement was obtained between numerical calculations and linear analyses.
Abstract: Recent experiments by Meshkov have demonstrated that a shock‐accelerated perturbed surface separating two gases of different densities is unstable for shocks traveling from the lighter gas to the heavier gas and vice versa. Using a two‐dimensional Lagrangian hydrodynamic code, the authors show qualitative agreement with Meshkov's experiments. Experiment, numerical calculations, and linear analysis are compared. Good agreement was obtained between numerical calculations and linear analysis.
210 citations
TL;DR: In this article, the effect of two-dimensional roughness elements on boundary-layer transition is investigated, and the authors demonstrate that the behavior can best be understood by considering wave-type disturbances, and that the basic mechanism by which a roughness element induces earlier transition to turbulent flow is by the destabilizing influence of the flow within the recovery zone.
Abstract: An experimental investigation of the effect of two‐dimensional roughness elements on boundary‐layer transition is described. Primary emphasis is given to the nature of disturbances within the recovery zone, i.e., that region in the immediate downstream of the roughness where the mean flow has been distorted by the presence of the roughness. Detailed measurements of mean velocity distributions, of disturbance spectra, and intensity, growth, and decay of disturbances at discrete frequencies were made for a range of unit Reynolds numbers. The measurements demonstrate that the behavior can best be understood by considering wave‐type disturbances, and that the basic mechanism by which a two‐dimensional roughness element induces earlier transition to turbulent flow is by the destabilizing influence of the flow within the recovery zone. Comparison with the behavior expected from stability theory supports this conclusion.
178 citations
TL;DR: In this article, the velocity product, uυ, was sampled in both the turbulent and nonturbulent zones using conditional averaging techniques, leading to an average picture of the velocities in the interfacial bulges.
Abstract: Space‐time correlations with large streamwise separation were obtained in a turbulent boundary layer with a zero‐pressure gradient. The auto and cross correlations of the velocities u and υ with streamwise spatial separation distances up to 20 boundary layer thicknesses revealed a difference in their structure and decay rate. Using conditional averaging techniques, the velocity product, uυ, was sampled in both the turbulent and nonturbulent zones. Further conditional sampling led to an average picture of the velocities in the interfacial bulges. Near the wall the space‐time correlation results are consistent with the idea of retarded fluid being ejected outward from the wall region and influencing the intermittent region.
155 citations
TL;DR: In this paper, a theory explaining the self-trapping instability (modulational instability) of plasma cyclotron waves is developed and the results are compared with computer experiments using a sheet current model, where the observed growth rate at an initial phase of the instability is in rough agreement with that given by perturbation theory.
Abstract: A theory explaining the self‐trapping instability (modulational instability) of plasma cyclotron waves is developed and the results are compared with computer experiments using a sheet current model. The observed growth rate at an initial phase of the instability is in rough agreement with that given by perturbation theory. However, as the level of the modulation increases, the rate of the growth of the modulation increases due to excitation of additional side bands, and finally, the carrier wave is found to collapse suddenly leading to rapid thermalization. The cause of the sudden collapse of the wave is attributed to the crossing of particles which are accelerated in the direction parallel to the ambient magnetic field by a large pressure gradient force, ∇(B⊥2/2μ0), developed by the instability, where B⊥ is the transverse magnetic field associated with the wave.
149 citations
TL;DR: In this paper, numerical solutions of the equations governing time-dependent, viscous, incompressible fluid flow past a circle are presented for Reynolds numbers 100, 400, and 1000.
Abstract: Numerical solutions of the equations governing time‐dependent, viscous, incompressible fluid flow past a circle are presented for Reynolds numbers 100, 400, and 1000. These solutions show the dramatic rise of the drag coefficient during the development of the Karman vortex street and reveal the oscillatory character of the drag, lift, and torque that are experienced by the circle. Contour plots of the vorticity and stream function are compared with histories of the pressure distribution, drag, lift, torque, and separation angles. These comparisons show how the pressure distribution, drag, lift, and torque on the circle are intimately and logically related to the well‐known flow pattern of the Karman street. A new method is described for implementing the infinity conditions. The use of this technique makes it possible to observe the motion of the upstream stagnation streamline and relate this effect to the lift on the circle. The fact that the drag is larger for the oscillatory wake than the symmetric wake is interpreted as a tendency toward an equilibrium state of maximum energy dissipation. Comparisons are made with experimental results. These comparisons suggest that the present results are a valid description of flow past a circular cylinder for Reynolds numbers in the range from 40 to 400.
TL;DR: In this paper, a theory for the surface profile above a fully developed Poiseuille channel flow is presented, and small disturbances to this flow are examined, and it is shown that if the (channel depth)/(wavelength) ratio is small (shallow waves), and the Reynolds number large enough, these disturbances initially travel at the classical dynamic (Burns) wave speeds.
Abstract: A theory is presented for the surface profile above a fully developed Poiseuille channel flow. Small disturbances to this flow are examined, and it is shown that if the (channel depth)/(wavelength) ratio is small (shallow waves), and the Reynolds number large enough, these disturbances initially travel at the classical dynamic (Burns) wave speeds. However, by introducing appropriate far‐field coordinates it follows that the disturbance eventually travels at a different wave speed—the kinematic wave speed. To confirm this, the dynamic waves are shown to decay by using standard boundary layer techniques. This general result (of decay) agrees with previous one‐dimensional theories. The profile close to the kinematic wave front is examined and shown to satisfy an equation of the form ηT + ηηX + ηXXX = ΔηXX, where η(X, T) is the surface profile. This equation is called the Korteweg‐de Vries‐Burgers equation. The form of the steady solution of this equation exhibits all the characteristics of the undular bore. A bound on Δ agrees with stability requirements found by other authors using different methods.
TL;DR: In this article, the rigorous and general treatment of electrostatic oscillations in a cold plasma of nonuniform density given by Barston can be extended to the case of Alfven waves in an ideal fluid in the presence of a class of inhomogeneous magnetic fields.
Abstract: It has been found that the rigorous and general treatment of electrostatic oscillations in a cold plasma of nonuniform density given by Barston can be extended to the case of Alfven waves in an ideal fluid in the presence of a class of inhomogeneous magnetic fields.
TL;DR: In this article, the initiation, development, and propagation of thermonuclear reaction waves in a solid density deuterium-tritium plasma are presented and the qualitative behavior of the physical variables is discussed and solutions of the equations are obtained by employing finite difference numerical techniques.
Abstract: The initiation, development, and propagation of thermonuclear reaction waves in a solid density deuterium‐tritium plasma are presented. Physical effects due to thermonuclear reactions, heat conduction, electron‐ion equilibration, bremsstrahlung, and fluid dynamics are contained in the analysis. The qualitative behavior of the physical variables is discussed and solutions of the equations are obtained by employing finite difference numerical techniques. Due to plasma properties and the form of the fusion reaction cross section, important differences exist between fusion and chemical detonation waves. Threshold ignition energy is found for simple geometries and the complete history of the development of the wave is studied. The structure of a fully developed fusion reaction wave is presented. Finally, the thermonuclear energy yield from small D‐T pellets is presented.
TL;DR: In this paper, the stability of a simple viscoelastic fluid heated from below is investigated and it is shown that under certain conditions the fluid is overstable; namely, an oscillating cell structure will be created before the classical (Benard) steady secondary flow instability appears.
Abstract: The stability problem for a plane layer of a general viscoelastic (simple) fluid heated from below is investigated. The nature of the problem suggests that linear viscoelasticity assumptions are sufficient to fully describe the phenomena. It is shown that under certain conditions the fluid is overstable; namely, an oscillating cell structure will be created before the classical (Benard) steady secondary‐flow instability appears. Stability criteria for the oscillatory modes have been found as well as wavenumber and oscillation periods for both rigid and free boundaries. The theoretical results have been applied to a Maxwell fluid and to some real viscoelastic solutions. The numerical results for the latter suggest that although oscillation of Benard cells is theoretically possible, very high‐temperature gradients or high gravitational fields would be required before the oscillating cells could be observed in common polymer solutions of moderate viscosity.
TL;DR: In this paper, it was shown that a strong shear in the interior of a magnetically confined Q-machine plasma column by means of an externally controlled nonuniform radial electric field is produced in the velocity shear layer.
Abstract: Strong shear in the plasma rotation is produced in the interior of a magnetically confined Q‐machine plasma column by means of an externally controlled nonuniform radial electric field. Coherent low‐frequency oscillations are observed localized in the velocity shear layer, which contains up to seven ion gyroradii. The observed oscillations are shown to result from the transverse Kelvin‐Helmholtz instability by the good agreement with finite‐Larmor‐radius fluid theory for the mode frequencies, radial dependences of the potential and density fluctuations, and the mode structure at large magnetic field and small rotation. At large electric fields the low‐frequency Kelvin‐Helmholtz modes are completely suppressed. Simultaneously, there appear oscillations with ω>ωc, whose properties are described by the fluid equations when extended to arbitrary rotation and wave frequencies.
TL;DR: In this article, a turbulence model is proposed for the prediction of boundary-layer flows near walls, and four cases of self-similar flow are predicted with the model and found to compare favorably with the relevant experimental data.
Abstract: A turbulence model is proposed for the prediction of boundary‐layer flows near walls. Two differential equations are solved: one for the kinetic energy of turbulence, and one for its length scale. The local effective viscosity in the flow is taken as proportional to the product of the length and the square root of the energy. The constants appearing in the equations are determined by reference to experimental data. Four cases of self‐similar flow are predicted with the model and found to compare favorably with the relevant experimental data. Satisfactory predictions for more general flows are also reported.
TL;DR: In this article, an extension of the analysis of Nield is made to more completely characterize the onset of convection in an infinite horizontal porous medium stratified by temperature and concentration.
Abstract: An extension of the analysis of Nield is made to more completely characterize the onset of convection in an infinite horizontal porous medium stratified by temperature and concentration. Comparisons are made with thermohaline convection in Newtonian fluids. Major differences lie in the concentration Rayleigh number dependence of the wavenumber at the marginal state of overstability and the dependence of the horizontal wavenumber in the “salt finger” region of stationary convection on both temperature and concentration Rayleigh numbers. Suggestions of geological applications and laboratory verification using a Hele‐Shaw cell are presented.
TL;DR: In this paper, a theory of cyclotron instability was developed on the basis of computer simulation experiments, which holds that at certain turbulence levels, electron cross-field diffusion which supresses the electron gyroresonances is created by turbulent wave-particle interactions in a plasma beam after a period of quasi-linear exponential development of turbulence.
Abstract: A theory of plasma beam cyclotron instability is developed on the basis of computer simulation experiments. The theory holds that at certain turbulence levels, electron cross-field diffusion which supresses the electron gyroresonances is created by turbulent wave-particle interactions in a plasma beam after a period of quasi-linear exponential development of turbulence. The stabilizing effect of Landau ion damping is noted. The behavior of cold and hot ions is discussed.
TL;DR: In this paper, an analytical solution is obtained for the steady laminar flow of an incompressible Newtonian fluid in an axisymmetric conduit with irregular surface where the spread of the surface roughness is large compared with the mean radius of the conduit.
Abstract: To understand the abnormal flow conditions caused by the presence of stenoses in arteries, an analytical solution is obtained for the steady laminar flow of an incompressible Newtonian fluid in an axisymmetric conduit with irregular surface where the spread of the surface roughness is large compared with the mean radius of the conduit. Numerical results are presented for the streamlines, the distributions of velocity, vorticity, and pressure, the energy dissipation, and the separation and reattachment points for conduits with sinusoidal wall variations. The analysis is also applicable to a locally constricted conduit, provided the separation zone does not extend into the straight wall portion of the conduit.
TL;DR: In this article, the KortewegdeVries equation describing nonlinear ion-acoustic waves in a plasma with finite ion temperature is derived and the temperature dependences of soliton width and speed are obtained.
Abstract: The Korteweg‐deVries equation describing nonlinear ion‐acoustic waves in a plasma with finite ion temperature is derived and the temperature dependences of soliton width and speed are obtained.
TL;DR: In this paper, the anomalous high frequency resistivity in a plasma driven by a large electric field oscillating near the electron plasma frequency is investigated, and a simple nonlinear theory gives results in reasonable agreement with simulations.
Abstract: In one‐ and two‐dimensional computer simulations an anomalous high‐frequency resistivity in a plasma driven by a large electric field oscillating near the electron plasma frequency is investigated. The large field excites the oscillating two‐stream and the ion‐acoustic decay instabilities in agreement with the linear theory. When the ion and electron fluctuations saturate, a strong anomalous heating of the plasma sets in. This strong heating is due to an efficient coupling of the externally imposed large electric field to the plasma by ion fluctuations. The anomalous collision frequency and the saturation fluctuation amplitudes are determined as a function of the external field amplitude and frequency, and the electron‐ion mass ratio. A simple nonlinear theory gives results in reasonable agreement with simulations.
TL;DR: In this article, the time-dependent Navier-Stokes equations are solved by the finite element method, where a steady-state solution is assumed when the timedependent solution becomes convergent.
Abstract: Occurrences of viscous fluid flows in arbitrary internal passages are numerous, an analytical solution of the governing Navier‐Stokes equations cannot be obtained. Even a numerical approach faces difficulties arising from the nonlinearity and complexity of the geometry involved. To remedy these difficulties, the time‐dependent Navier‐Stokes equations are solved by the finite element method wherein a steady‐state solution is assumed when the time‐dependent solution becomes convergent. A family of locally constricted channels was considered in the computations, and in each case, the shear stress at the wall was found to be sharply increased at and near the region of constriction. Computations were carried out to Reynolds numbers when a separation eddy was established. The numerical scheme used seems to be fairly stable.
TL;DR: In this article, the time asymptotic distribution functions corresponding to adiabatic and sudden excitation of an electrostatic wave are calculated and compared and used to calculate the nonlinear response of the plasma, and Poisson's equation is used to find a nonlinear dispersion relation.
Abstract: The time asymptotic distribution functions corresponding to adiabatic and sudden excitation of an electrostatic wave are calculated. These distributions are compared and used to calculate the nonlinear response of the plasma, and Poisson's equation is used to find a nonlinear dispersion relation.
TL;DR: In this paper, a new kind of mechanism which induces a flow around a solid body in a slightly rarefied gas is proposed, and the behavior of gas around a sphere with a constant temperature which is placed in an infinite expanse of gas at rest with a uniform temperature gradient is investigated.
Abstract: A new kind of mechanism which induces a flow around a solid body in a slightly rarefied gas is proposed In order to demonstrate the flow induced by this mechanism the behavior of gas around a sphere with a constant temperature which is placed in an infinite expanse of gas at rest with a uniform temperature gradient is investigated on the basis of the asymptotic theory for a slightly rarefied gas A flow with magnitude of the order of the Knudsen number squared is induced from the hotter to the colder region The sphere is subject to a force in the direction of the given temperature gradient
TL;DR: In this article, the growth rate of magnetohydrodynamic instabilities is analyzed in the tokomak approximation and numerically without approximation for the Lundquist field and the constant pitch magnetic field is shown to be unstable to quasi-kinks, quasi-interchanges and pure interchanges.
Abstract: In order to obtain growth rates of magnetohydrodynamic instabilities, the equation of motion of a diffuse linear pinch is solved analytically in the tokomak approximation and numerically without approximation. The growth rates of kinks are calculated for the Lundquist field. The constant‐pitch magnetic field is shown to be unstable to quasi‐kinks, quasi‐interchanges, and pure interchanges which dominate, respectively, at increasing values of the negative pressure gradient. Analytical expressions and numerical values for the growth rates of these modes are given. Interchanges in a sheared magnetic field are investigated numerically and checked by means of perturbation theory.
TL;DR: The stability of a two-component fluid layer subjected to a temperature gradient has been studied, and the associated thermal diffusion separation has been found to exert a large influence even when the separations are small.
Abstract: The stability of a two‐component fluid layer subjected to a temperature gradient has been studied, and the associated thermal diffusion separation has been found to exert a large influence even when the separations are small. The most unexpected and perhaps important result is that an instability has been found which can give rise to convection currents even though the density gradient is not adverse. Thus, a system heated from above can become unstable even when the fluid is less dense at the top of the system provided the more dense substance rises to the upper plate. Many measurements of the Soret coefficient could be subject to this instabilitity.
TL;DR: In this article, a more realistic description of the gross stability properties in high-β plasmas than that given by ideal magnetohydrodynamics was presented, and a model was presented which provided a more accurate representation of the stability properties.
Abstract: A model is presented which provides a more realistic description of the gross stability properties in high‐β plasmas than that given by ideal magnetohydrodynamics.
TL;DR: In this paper, ideal magnetohydrodynamic equilibria in current-carrying diffuse toroidal pinches of circular cross section (tokamaks) were studied numerically to determine possible limitations on β = 8πp/B2.
Abstract: Ideal magnetohydrodynamic equilibria in current‐carrying diffuse toroidal pinches of circular cross section (tokamaks) are studied numerically to determine possible limitations on β = 8πp/B2. Equilibrium considerations alone give no limitations even for β∼1 or βp≃βq2A2∼A2, but the stability requirement q>1 for magnetohydrodynamic kink modes does limit β. The best case found has β∼0.1 with q>1, A = 3, and βp∼2.
TL;DR: In this paper, the authors studied the condensation of ethanol by homogeneous nucleation of liquid droplets in a supersonic nozzle with continuous static pressure measurements on the nozzle centerline and light scattering measurements.
Abstract: Condensation of ethanol by homogeneous nucleation of liquid droplets has been studied in a supersonic nozzle. Continuous static pressure measurements on the nozzle centerline and light scattering measurements permit the condensation process to be resolved in detail for comparison with theory. Nucleation rates found in the present work are in general agreement with previous results obtained in diffusion and expansion cloud chambers. It is seen that the predictions of the classical theory of homogeneous nucleation or that of the statistical mechanical theory with the assumptions of Dunning roughly agree with the experimental results. Various terms entering different versions of nucleation theory are given quantitatively. The pressure variations in the flow found beyond the onset point of condensation are used to study droplet growth processes. It is shown that a simple kinetic growth law with a constant mass condensation coefficient describes the condensation process well.