Volume 15, Issue 1, January 1972
Index of content:

Turbulent Layer in an Ideal Two‐Dimensional Fluid
View Description Hide DescriptionThe problem of the turbulent layer arising in the two‐dimensional ideal fluid is considered. The sharp initial velocity discontinuity was replaced by a row of 100 point vortices whose evolution in time was found by numerical methods. The average velocity profile and velocity fluctuations are given. The picture is very similar to the ordinary turbulent layer in spite of the potential character of the ideal fluid motion outside the discontinuity.

Numerical Solution of the Navier‐Stokes Equations for Flows in the Disk‐Cylinder System
View Description Hide DescriptionA numerical computation of a viscous incompressible fluid confined in a circular cylindrical chamber has been carried out, where the top disk is rotating with a constant angular velocity, and the bottom disk and side wall are held fixed. Using the full Navier‐Stokes equations, steady secondary flows were calculated for various Reynolds numbers up to a maximum value of 400. It is found that, for the Reynolds numbers below 10, the governing equations are essentially linear and the numerical solution agrees very well with the analytic solution. For higher Reynolds numbers, the flow near the boundary is intensified. The detailed flow patterns are compared with previous work of a single disk and two infinite disks. The volume flow rate of the secondary flow due to the centrifugal action and the frictional moment of the disk were also calculated.

High Rayleigh Number Convection in an Enclosure—A Numerical Study
View Description Hide DescriptionSeven finite difference computations for convection in a square cavity at Rayleigh number have been carried out for a variety of dynamical boundary conditions, Rayleigh numbers, and Prandtl numbers. The results indicate that various horizontal dynamical boundary conditions virtually make no difference to the main boundary‐layer flow although free vertical boundaries double the velocity in the boundary layer adjacent to them. The scaled fields in the vertical boundary layers are independent of the Rayleigh number, and to a large extent, independent of the Prandtl number, when the two vertical boundary layers are distinct. The results compare reasonably well with Elder's experiments.

Turbulence Model for Boundary Layers near Walls
View Description Hide DescriptionA 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.

Kinetic Theory Analysis of Temperature Jump in a Polyatomic Gas
View Description Hide DescriptionA kinetic theory analysis of the temperature jump at the wall and the temperature distribution near the wall in polyatomic gases has been made. The collision term in the Boltzmann equation is represented by a classical (BGK‐type) polyatomic model. The problem is linearized by considering small temperature and density variations over their respective mean values, which leads to three simultaneous singular linear integral equations. For the solution of the problem, a direct numerical method is used. The integral equations are reduced to a set of simultaneous linear equations for discrete values of the dependent variables by approximating the integrals with suitable quadrature formulas. This analysis shows that the temperature jump can be expressed as , where denotes the effective number of internal degrees of freedom, and and are the thermal accommodation coefficients for the translational and internal energy transfers, respectively. The parameter is inversely proportional to , the number of collisions to achieve equilibrium between the translational and internal modes of molecular motion. The results show that the temperature jump coefficient is, to good approximation, the same as that for the monatomic case only if . In this case .

Radiative Cooling of Shock‐Heated Air in an Explosively Driven Shock Tube
View Description Hide DescriptionResults are presented of an experimental program to measure the effect of radiative cooling on the enthalpy distribution behind incident shock waves traveling in air. The shock velocity was nominally 16 km/sec and the preshock ambient pressure was varied from 0.4 to 1.6 Torr. Shock‐tube diameters of 4.7 and 9.4 cm were used to investigate the effects of varying optical depths. Radiative cooling rates were determined from spatially resolved measurements of the profile of the line and from absolute measurements of the continuum radiation. The measured enthalpy profiles are in good agreement with the theoretical predictions of Chien and Compton which account for both nongrey and multidimensional aspects of the radiative transport in the shock tube.

Reaction Zone‐Shock Front Coupling in Detonations
View Description Hide DescriptionSome new observations are reported of the delayed effect of chemical reaction on the trajectory of an isolated triple point in a detonation. The implications of these observations relative to the self‐sustenance of finite amplitude transverse waves on a propagating detonation front are developed. A new criterion for the minimum size of a detonation cell is found, based on the requirement that a reaction‐generated compression wave must overtake the triple point within the first half of the cell. This feature and certain other acoustic wave propagation phenomena are incorporated in a new, qualitatively quite detailed model of a detonation cell.

Experimental Study of Strong Transverse Ionizing Shock Waves
View Description Hide DescriptionExperiments are described in which transverse ionizing shock speeds up to 4×10^{8} cm/sec were measured in gaseous hydrogen. Measurements of the magnetic shock structure, shock thickness, and magnetic field jump are described and compared with theoretical predictions. The ion temperature near the device wall was measured by a charge exchange particle detector. A separation distance of about 30 cm between the shock wave and the magnetic piston was measured.

Precursor Ionization Fronts Ahead of Expanding Laser‐Plasmas
View Description Hide DescriptionFast ionization fronts have been experimentally observed to propagate ahead of a laser‐produced plasma expanding into low‐density background gas. The ionizing wave has a velocity of 0.3‐3×10^{8} cm/sec and a thickness of 1‐10 cm both quantities increasing with radius of propagation. Langmuir probe and microwave diagnostics indicate that the wave augments the initial photoionization density by a factor of ; potential fluctuations exist within the wave which can accelerate electrons to energies above the ionization threshold. Equations describing a collisional ionizing electron wave are solved in an approximate manner to yield relations which agree with many features of the measured phenomena.

Magnetohydrodynamics of a Drop
View Description Hide DescriptionThe nature of deformation and drag on a spherical drop of viscous, conducting and incompressible fluid under axisymmetric current distribution in a conducting, viscous, incompressible fluid of infinite extent is considered. The drop fluid and the ambient fluid phases are non‐reacting and non‐diffusive in nature. In low magnetic Reynolds' number approximation, the effects of space charge, either on the total force density or on the distribution of conduction currents are neglected. In the absence of any free induced currents, the drop deformation to oblate or prolate spheroidal shape in Stokes' approximation is primarily determined by the difference in conductivities and permeabilities of two fluid phases. The drag remains the same as that on a spherical drop in a noncurrent carrying fluid because of axisymmetric current distribution. In Oseen's approximation, the drop deformation to oblate spheroidal shape is aided by inertia of flow for large values of drop to ambient fluid density ratio, whereas inertia retards the deformation to prolate spheroidal shape. The drag on the drop increases (or decreases) from its zero‐current carrying ambient fluid counterpart depending on difference in conductivities of two fluid phases. The terminal velocity remains unaffected by currents in Stokes' approximation, but it increases or decreases from its zero‐current carrying ambient fluid counterpart value in Oseen's approximation depending on whether drop has higher or lower conductivity than the ambient fluid. The drag and terminal velocity remains unaffected by the permeability of the drop fluid in these approximations. In higher order, the drop tends to a spherical cap shape since the electromagnetic effects are extremely small in this order.

Magnetohydrodynamic Equilibrium of an Elliptical Plasma Cylinder
View Description Hide DescriptionMagnetohydrodynamic equilibrium of an elliptical plasma cylinder is studied for two different current distributions, with longitudinal current either distributed uniformly over the cross section, or confined to the surface only. Magnetic surfaces are found analytically in the vacuum around the plasma. They are elongated along the major axis, and bound by a separatrix.

Two‐Fluid Theory of the Positive Column of a Gas Discharge
View Description Hide DescriptionThe transition from inertia‐limited flow to mobility‐limited flow of ions to the walls of the positive column of a planar gas discharge is analyzed from a hydrodynamic point of view. The analysis is based on the assumption that the electrons and ions in the positive column behave as inviscid fluids with different temperatures. The resultant formulation is valid for gas pressures from the free‐fall limit to the continuum limit, including the high‐field sheaths at the walls as well as the almost field‐free plasma at the center of the discharge. Numerical results for the characteristic value, wall field, wall potential, and ion energy at the wall are given for many different pressures and central Debye lengths, and it is shown that the results are identical to those of Allis and Rose in the continuum limit and very close to those of Self in the free‐fall limit.

Ionizing Potential Waves and High‐Voltage Breakdown Streamers
View Description Hide DescriptionThe structure of ionizing potential waves driven by a strong electric field in a dense gas is discussed. Negative breakdown waves are found to propagate with a velocity proportional to the electric field normal to the wavefront. This causes a curved ionizing potential wavefront to focus down into a filamentary structure, and may provide the reason why breakdown in dense gases propagates in the form of a narrow leader streamer instead of a broad wavefront.

Lorentz Plasma in a Strong Magnetic Field
View Description Hide DescriptionThe electron distribution function has been calculated for a Lorentz plasma in a strong arbitrarily oriented magnetic field and in a weak electric field, temperature gradient, and flow velocity gradient. The magnetic field is strong enough to significantly affect the collision process, but it is not so strong that it completely dominates the plasma. Using the electron distribution function, corrections to the usual thermoelectric transport coefficients and viscosities have been calculated. The corrections are zero, to this order in the magnetic field, for the electrical and thermal conductivities and the thermoelectric coefficients and are typically on the order of 1% or less for the longitudinal, transverse, and Hall viscosities.

Measurement of Plasma Properties from the Cross Spectrum of Thermal Density Fluctuations
View Description Hide DescriptionThe relation between the density cross‐correlation function and the cross spectrum of fluctuations in the electron saturation currents drawn by two Langmuir probes is used to measure basic plasma properties. The measured quantities using this approach are the ambipolar diffusion coefficient and the electron density. It is found that the method has sufficient sensitivity to determine changes in the ionic composition. These measurements are compared with plasma parameters obtained by using conventional probe techniques. Discrepancies are shown to exist and the source of these discrepancies is discussed.

Linearized Variational Analysis of Single‐Species, One‐Dimensional Vlasov Plasmas
View Description Hide DescriptionAnalysis of linearized Vlasov plasmas by means of Hamilton's variational principle is illustrated by application to the initial‐value problem for a neutralized, single‐species plasma in one‐spatial dimension with periodic boundary conditions. For any degree of approximation and for arbitrary equilibrium velocity distributions, the problem is reduced to the solution of a system of ordinary differential equations in time with constant coefficients and time‐dependent driving terms; an exact particular solution has been found. The equations are discussed, including the relation of the equations to Eulerian treatments of the problem, and the use of the equations to obtain numerical results. Numerical examples have been worked out for a Maxwellian equilibrium distribution function, equilibrium distribution functions that are sums of two Maxwellian distributions, and for nonanalytic equilibrium distribution functions that are represented by cubic spline functions. Comparisons are made between numerical results from the variational method and precise solutions of the exact plasma dispersion relation.

Plasma Transport in Toroidal Confinement Systems
View Description Hide DescriptionThe 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.

Sharp‐Boundary Model of a High‐Pressure Tokamak
View Description Hide DescriptionA simple sharp‐boundary model for a high‐pressure tokamak is described. Its relationship to the recent work of Greene, Johnson, and Weimer is demonstrated.

Mode Conversion in an Inhomogeneous Plasma
View Description Hide DescriptionAn expression for the mode conversion efficiency is derived for waves propagating in an inhomogeneous plasma. Special consideration is given to the effects of finite wave vector parallel to a static magnetic field. It is shown that damping results in a reduction of the efficiency in the immediate neighborhood of the conversion layer. This reduction is partially offset by an increase in efficiency with increased density gradient. Efficiences for plasmas found in tokomaks are presented.

Toroidal Equilibrium for a High‐β Theta‐Pinch Plasma
View Description Hide DescriptionThe effect of superimposing a time‐oscillating helical mode on a static straight theta pinch is investigated. The stability obtained for the mode can be used to support the toroidal force which is introduced ad hoc. Conditions for obtaining toroidal equilibrium by this technique are obtained. The theory is within the framework of ideal magnetohydrodynamics.