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Dec 1965

Volume 8, Issue 12, pp. 2121-2304

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Gas Dynamics with Nonequilibrium Radiative and Collisional Ionization

Joseph H. Clarke and Carlo Ferrari

Phys. Fluids 8, 2121 (1965); http://dx.doi.org/10.1063/1.1761171 (19 pages) | Cited 34 times

Online Publication Date: 9 December 2004

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The dynamics of a gas undergoing nonequilibrium ionization by both radiative (bound‐free) and collisional transitions is formulated. Examination of the theory leads to a number of general deductions. The ionization rates and emission and absorption coefficients are all evaluated explicitly from statistical results available for the monatomic gas considered. This makes the problem concrete; one also sees that the approach for other cross sections, transitions, and gases would be analogous in most or many respects. It is shown that spectral complexities can sometimes be removed by expansion in terms of a large thermal parameter, and there emerges rationally a gas which is approximately gray above the edge frequency. The general theory is applied to the strong normal shock wave structured by the nonequilibrium ionization processes. The nature of this problem is examined, the complete structure is computed, the photoionization in the precursor portion of the shock structure is also determined according to the black‐body emitter model for comparison, and the spectral as well as a directional approximation are assessed.

Radiative Shock Structure

Paul A. Koch

Phys. Fluids 8, 2140 (1965); http://dx.doi.org/10.1063/1.1761172 (8 pages) | Cited 6 times

Online Publication Date: 9 December 2004

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A study is made of plane gas dynamic and transverse magnetogasdynamic shock waves in fully ionized hydrogen at conditions such that radiative processes are significant. The length scale required for a radiative shock wave to be steady is large, and the establishment of a truly steady‐state flow is found only for dimensions relevant to astrophysical cases. In the case of such an established flow the Rankine‐Hugoniot jump equation (i.e., the relationship between the flow variables in the initial and final states) is analyzed. It is shown that of the twelve roots of this equation only two are physical, corresponding to the super‐signal initial and sub‐signal final states. The differential equations which describe the above shock waves are analyzed for cases where the shock is optically thick, i.e., the mean free path for absorption of radiation is much smaller than the characteristic lengths for change of the flow variables. A new criterion is given for the validity of this approximation.

Cylindrical Imploding Shock Waves

John H. Lee and Benedict H. K. Lee

Phys. Fluids 8, 2148 (1965); http://dx.doi.org/10.1063/1.1761173 (5 pages) | Cited 28 times

Online Publication Date: 9 December 2004

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A technique for producing imploding cylindrical shock waves through imploding detonation waves is described. It is found that using the present method of initiation, symmetrical cylindrical imploding detonation waves can be produced in equimolar acetylene‐oxygen mixtures at initial pressures above 150 mm Hg. Due to the effect of area convergence, the detonation wave becomes increasingly over‐driven as it implodes towards the center. In the limit the highly overdriven detonation wave approaches a strong shock wave. Pressure measurements at different radii from the center of symmetry indicate this self‐amplifying characteristic of imploding detonation waves. Close agreement between the experimentally measured detonation pressures and those calculated by the Chester‐Chisnell‐Whitham method is obtained.

Two‐Fluid Model for Shock Wave Structure

K. Toba and J. D. Melnick

Phys. Fluids 8, 2153 (1965); http://dx.doi.org/10.1063/1.1761174 (5 pages) | Cited 1 time

Online Publication Date: 9 December 2004

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A critical study is made of the two‐fluid model of a shock wave which was first introduced by Glansdorff and, also, independently by Rott and Whittenbury, and which has recently been modified by Ziering, Ek, and Koch. It is proposed that a cutoff relative velocity be introduced in the inelastic collision operator which yields various collision frequencies for the transferred quantities. This cutoff velocity is not known initially but is determined by the analysis. Numerical results are given for molecules of rigid sphere, point centers of repulsion, and the Sutherland model. Comparisons with other solutions and experimental data are made.

Measurements of a Precursor Electron Front

J. P. Barach and J. A. Sivinski

Phys. Fluids 8, 2158 (1965); http://dx.doi.org/10.1063/1.1761175 (4 pages) | Cited 3 times

Online Publication Date: 9 December 2004

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An electron current front of a few thousand amperes has been produced in the expansion chamber of an electric shock tube. It precedes the shock and follows the (much faster) precursor breakdown wave. The electron front velocity of up to 2 × 107 cm∕sec is linear with applied voltage gradient and gas conductivity, as measured by double probes. As the operating parameters are varied over a wide range the electron temperature varies from 600 to 8000°K, but the number density in the front remains about 2 × 1013 cm−3. The front heats the gas and increases the speed of the subsequent shock by a factor of about two. This field‐driven wave may exist as a precursor in other shock tubes and must be distinguished from breakdown and hot electron diffusion precursor waves already reported by other authors.

Insulator Ablation in Magnetic Piston Shock Tubes

J. B. Workman

Phys. Fluids 8, 2162 (1965); http://dx.doi.org/10.1063/1.1761176 (7 pages) | Cited 10 times

Online Publication Date: 9 December 2004

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The effect of end wall insulator ablation on the performance of magnetic piston shock tubes has been investigated theoretically. From the analysis of a steady flow model, a simple correlation parameter has been derived that permits a quick determination of the loss in a given experiment. The parameter is a function only of the magnetic drive field, initial test gas density, and the ionization energy of the insulator vapor. The transient heating of the insulator that initiates the vaporization is discussed as is the effect of degassing adsorbed material at the wall surface.

Failure of the Chapman‐Jouguet Theory for Liquid and Solid Explosives

W. C. Davis, B. G. Craig, and J. B. Ramsay

Phys. Fluids 8, 2169 (1965); http://dx.doi.org/10.1063/1.1761177 (14 pages) | Cited 15 times

Online Publication Date: 9 December 2004

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The usual treatment of unsupported detonation, often called the Chapman‐Jouguet theory, is based on four assumptions: (1) the detonation approaches a steady state, (2) the flow is laminar and one‐dimensional, (3) the detonation products approach a state of chemical equilibrium some distance behind the detonation front, and (4) the detonation velocity is the minimum permitted by the conservation conditions. Wood and Fickett proposed experiments to test the validity of the Chapman‐Jouguet ``theory,'' not requiring knowledge or assumptions about the nature of the equation of state of the detonation products, by making variations of the initial state of the explosive. The results of experiments are reported in which the initial state was varied (1) by using mixtures of nitromethane and another liquid which has the same atomic composition as nitromethane, and (2) by using TNT as liquid and as solid. These results show that the Chapman‐Jouguet theory is violated. The calculated pressures are 15 to 20% below the measured pressures. Results of measurements using other explosives are also presented to support the conclusion that the theory fails. No explanation or alternative theory is offered.
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Numerical Calculation of Time‐Dependent Viscous Incompressible Flow of Fluid with Free Surface

Francis H. Harlow and J. Eddie Welch

Phys. Fluids 8, 2182 (1965); http://dx.doi.org/10.1063/1.1761178 (8 pages) | Cited 1627 times

Online Publication Date: 9 December 2004

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A new technique is described for the numerical investigation of the time‐dependent flow of an incompressible fluid, the boundary of which is partially confined and partially free. The full Navier‐Stokes equations are written in finite‐difference form, and the solution is accomplished by finite‐time‐step advancement. The primary dependent variables are the pressure and the velocity components. Also used is a set of marker particles which move with the fluid. The technique is called the marker and cell method. Some examples of the application of this method are presented. All non‐linear effects are completely included, and the transient aspects can be computed for as much elapsed time as desired.

Role of the Interface in the Stability of Stratified Flow down an Inclined Plane

Timothy W. Kao

Phys. Fluids 8, 2190 (1965); http://dx.doi.org/10.1063/1.1761179 (5 pages) | Cited 18 times

Online Publication Date: 9 December 2004

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For the case in which the density of the upper layer is somewhat less than that of the lower layer, a second mode essentially associated with the interface is found to govern the stability of the flow. However, when the upper density becomes much smaller, the free surface mode and the second mode compete for governing the stability. A clarification of the mechanism involved is given. The Reynolds stress transfer of energy at the interface is estimated and is found to be insignificant for long waves.

Stability of Frictionally‐Heated Flow

Daniel D. Joseph

Phys. Fluids 8, 2195 (1965); http://dx.doi.org/10.1063/1.1761180 (6 pages) | Cited 24 times

Online Publication Date: 9 December 2004

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Extended results relative to the existence of a critical stress (a finite shear stress or pressure gradient above which fully‐developed steady solutions do not exist) in Couette and Poiseuille motions are reported. The results apply to liquids under general thermal boundary conditions including a conduction‐convection balance at the boundaries. A bound which gives a close a priori estimate of the value of the critical stress is developed. Stability characteristics of frictionally‐heated Couette flow in the inviscid limit are specified. Below the critical stress the steady solutions are double‐valued. The stress parameter first increases, then decreases, with increasing temperature. To the previously reported temperature instability on the second branch can be added a corresponding (inviscid) instability of the motion. On the first branch of the double‐valued solution the flow is stable despite the presence of a single vorticity maximum at the channel center.

Uniform Motion of Vortex System

Franklin Pollock

Phys. Fluids 8, 2201 (1965); http://dx.doi.org/10.1063/1.1761181 (4 pages)

Online Publication Date: 9 December 2004

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The motion of systems of vortices in inviscid fluids that move uniformly without change of structure is considered. The Euler equations are cast in a form that selects out only uniform motion, and the resulting time‐independent equation is examined. It is shown in this case that the requirement that there exist well‐behaved solutions insures that a vortex pair moves with the velocity given by the classical linear theory.

Eddy Formation in an Eccentric Annular Domain

M. Bentwich and C. Elata

Phys. Fluids 8, 2204 (1965); http://dx.doi.org/10.1063/1.1761182 (7 pages) | Cited 4 times

Online Publication Date: 9 December 2004

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The slow motion of a viscous incompressible fluid in the space bounded by two long parallel non‐concentric circular cylindrical solid surfaces is studied. The outer one is stationary while the inner one rotates steadily. Both analysis and experiment show that when the ratio of eccentricity to average gap width is sufficiently large an eddy is formed. In such case part of the fluid rotates around the inner surface. The rest rotates in the opposite direction around a line parallel to the axes. This phenomenon has an interesting analogy in the theory of plates' deflection.

Approximate Solution of the Shear Flow Boundary Layer on a Flat Plate

Leroy Devan

Phys. Fluids 8, 2211 (1965); http://dx.doi.org/10.1063/1.1761183 (5 pages) | Cited 1 time

Online Publication Date: 9 December 2004

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A nonsimilar solution of the incompressible boundary layer over a flat plate in the presence of shear flow is considered. The Dorodnitzyn method of approximate solution, utilizing integral relations, is applied to Crocco forms of the boundary layer equations. Agreement is obtained with the similar solutions of Ting. The computations provide numerical results for the transition between the cases of Li and Murray and of Ting.

Laminar Boundary Layer on a Cone with Uniform Injection

Paul A. Libby

Phys. Fluids 8, 2216 (1965); http://dx.doi.org/10.1063/1.1761184 (3 pages) | Cited 2 times

Online Publication Date: 9 December 2004

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A solution for the laminar boundary layer on a cone with uniform mass transfer is obtained. The velocity field is found for either suction or injection, but the related solution for the energy field is subject to an energy balance at the exposed surface and is therefore valid only for injection. This latter solution is equally applicable to certain species fields as well. The present results along with those presented previously for the two‐dimensional case permit a comparison of the effect of injection on boundary layers over two‐dimensional and conical surfaces.

Self‐Consistent‐Field Approach to Turbulence Theory

J. R. Herring

Phys. Fluids 8, 2219 (1965); http://dx.doi.org/10.1063/1.1761185 (7 pages) | Cited 66 times

Online Publication Date: 9 December 2004

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A self‐consistent‐field type perturbation theory is developed to treat the dynamics of stationary and homogeneous turbulence. The method consists of expanding the full probability distribution function about the product of the exact univariate distributions of all the Fourier modes. The theory is used in second order to find expressions for the turbulent energy spectrum and associated response frequencies. The results for the energy spectrum are identical to a simplified form of the direct‐interaction approximation of Kraichnan and closely resemble the results of the generalized randomphase approximation of Edwards. The relation of the present method to both the above approaches is discussed.

Measurements on the Development of Thermal Turbulence in Air between Horizontal Plates

G. E. Willis and J. W. Deardorff

Phys. Fluids 8, 2225 (1965); http://dx.doi.org/10.1063/1.1761187 (5 pages) | Cited 26 times

Online Publication Date: 9 December 2004

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The transition from steady to turbulent thermal convection in air between horizontal plates is inferred from temperature measurements. Spatial and temporal records of air temperature for Rayleigh numbers ranging from 5000 to 1.5 × 106 are presented along with spatial temperature spectra. A region of transition from nearly steady organized convection to irregular turbulent convection was found to exist between Rayleigh numbers of about 6300 and 10 000, although a dominant wavelength continues to exist at much larger Rayleigh numbers. Spatial scales of temperature are noted to extend to both larger and smaller wavelengths with increase in Rayleigh number.

Class of Exact Solutions in Nonlinear Kinetic Theory

Lawrence Sirovich

Phys. Fluids 8, 2230 (1965); http://dx.doi.org/10.1063/1.1761188 (4 pages)

Online Publication Date: 9 December 2004

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A class of exact solutions of the one‐dimensional steady Krooked equation is obtained. The collision frequency of the latter is left arbitrary. The solutions found depict flows widely removed from equilibrium.

Hall Effect in the Two‐Dimensional Flow along an Insulating Plane

H. Hasimoto and G. S. Janowitz

Phys. Fluids 8, 2234 (1965); http://dx.doi.org/10.1063/1.1761189 (6 pages) | Cited 1 time

Online Publication Date: 9 December 2004

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Two‐dimensional flow of a conducting gas interacting weakly with an external inhomogeneous magnetic field is considered. It is proved that for flow over an insulating wall, the total drag force is reduced by the Hall effect, and the total transverse force is zero. General expressions for the Hall current and the induced transversal flow are obtained for the case of flow with small Hall parameter over an infinite insulating plane. Examples are given for the dipole magnetic field. An inversion of the transversal flow, varying with the dipole orientation, is found. This situation also suggests a transversal deflection of the flow behind a magnetized body in the flow of a plasma.

Alfvén Waves in an Incompressible Medium

David P. Hoult

Phys. Fluids 8, 2240 (1965); http://dx.doi.org/10.1063/1.1761190 (5 pages)

Online Publication Date: 9 December 2004

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The theory of Alfvén wave propagation in an incompressible, inviscid, infinitely conducting medium is presented. The theory is presented in terms of canonical equations of motion. Topics included are the generation of arbitrary small amplitude waves, the propagation of finite amplitude waves on a nonuniform magnetic field, and the reflection of finite amplitude waves from an idealized surface of the sun. A nonlinear amplification process associated with reflection is discussed.

Flute Instabilities at Ion Gyrofrequency

Yaakov Shima and T. K. Fowler

Phys. Fluids 8, 2245 (1965); http://dx.doi.org/10.1063/1.1761191 (5 pages) | Cited 30 times

Online Publication Date: 9 December 2004

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Flute modes at the ion gyrofrequency, propagating perpendicular to the magnetic field, are considered. A dispersion relation for inhomogeneous plasmas is obtained from which density thresholds for instabilities are derived. Several plasma distributions are considered. For a loss cone in velocity space, as in mirror machines, the density threshold is given by (ωpi∕ωi) ≈ (R∕rli)¾. In the case of a plasma shell with a radial density proportional to r2 at r = 0, (ωpi∕ωi) ≈ (R∕rli)☒. Here, ωpi is the ion plasma frequency, ωi is the ion gyrofrequency, R is the plasma radius, and rli is the ion gyroradius.

Landau Solution of the Plasma Oscillation Problem

E. C. Taylor

Phys. Fluids 8, 2250 (1965); http://dx.doi.org/10.1063/1.1761192 (5 pages) | Cited 6 times

Online Publication Date: 9 December 2004

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The controversial aspects of Landau's treatment of the initial value problem are discussed. It is demonstrated that any kind of decaying solution other than that of Landau can be obtained for some initial velocity distribution function. Nevertheless, it is shown that several phenomenologically motivated distributions—Maxwellian functions, Lorentzian functions, and smoothly cutoff functions—yield solutions which are dominated by the plasma wave for long periods of time, although the plasma wave may not be present in the asymptotic limit. The distribution functions that are chosen lead to specific examples of each conceivable type of singularity structure for the integrand in the representation of the solution. It is suggested, therefore, that Landau's conclusion that all long‐wavelength disturbances in a plasma produce a damped plasma wave which dominates the ensuing behavior is physically correct.
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Collisionless Damping of Nonlinear Plasma Oscillations

Thomas O'Neil

Phys. Fluids 8, 2255 (1965); http://dx.doi.org/10.1063/1.1761193 (8 pages) | Cited 417 times

Online Publication Date: 9 December 2004

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It is well known that the linear theory of collisionless damping breaks down after a time τ ≡ (m∕eϵκ)☒, where κ is the wavenumber and ϵ is the amplitude of the electric field. Jacobi elliptic functions are now used to provide an exact solution of the Vlasov equation for the resonant electrons, and the damping coefficient is generalized to be valid for times greater than t = τ. This generalized damping coefficient reduces to Landau's result when t∕τ ≪ 1; it has an oscillatory behavior when t∕τ is of order unity, and it phase mixes to zero as t∕τ approaches infinity. The above results are all shown to have simple physical interpretations.

Computer Experiments on the Randomization of Electrons in a Collisionless Plasma

P. Burger, D. A. Dunn, and A. S. Halsted

Phys. Fluids 8, 2263 (1965); http://dx.doi.org/10.1063/1.1761194 (10 pages) | Cited 10 times

Online Publication Date: 9 December 2004

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The large‐signal, time‐dependent behavior of a one‐dimensional, collisionless plasma is simulated on a computer by calculating the trajectories of a large number of charge sheets in a one‐dimensional space. The equations of motion of the charge sheets are calculated self‐consistently with the electric fields that the charges themselves set up in the diode. A plasma is formed in the diode by the generation of electron and ion sheets at a constant rate in time and space. The velocity distribution, charge distribution, and potential distribution of the simulated plasma are investigated. Electrons and ions are generated with prescribed initial velocities. Because of the heavy mass of the ions, the potential in the middle of the diode becomes positive with respect to the wall. The potential does not settle down to a dc state, however, but exhibits fluctuations at the electron plasma frequency. It is shown that these rf fluctuations randomize the velocities of the electrons. A clear similarity is shown between the effects of these rf fluctuations and elastic collisions.

Current‐Driven Instabilities in Configurations with Sheared Magnetic Fields

B. Coppi

Phys. Fluids 8, 2273 (1965); http://dx.doi.org/10.1063/1.1761195 (8 pages) | Cited 70 times

Online Publication Date: 9 December 2004

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A set of instabilities which are fastest in the weakly collisional regime, being slowed down by increasing electron‐ion collisions, is presented. They take place in configurations having very low pressures, sheared magnetic field, and current flowing along it. These instabilities are driven by the transverse gradient of the longitudinal current, are connected with the effects of electron inertia, and characterized by two types of mode. Modes of one type are electrostatic and localized, having eddies of relatively short scale length, and exist for small and finite wavelengths. The other modes are nonelectrostatic, nonlocalized, and result from one which is singular in the hydromagnetic limit. These exist only for wavelengths of the order of the width of the plasma column, in which limit they are faster than the electrostatic type of modes. The relationship between these instabilities and the known ``tearing'' resistive mode is studied. The growth rate of this is, in fact, increased by ion‐electron collisions and corresponds to transfer of magnetic energy into plasma kinetic energy by decoupling the motion of particles and lines of force. A brief discussion of experimental results offering indication for the modes discussed above is finally given.

Excitation of Hydromagnetic Waves by a Gyrating Proton Stream

Jacob Neufeld and Harvel Wright

Phys. Fluids 8, 2281 (1965); http://dx.doi.org/10.1063/1.1761196 (7 pages) | Cited 3 times

Online Publication Date: 9 December 2004

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It is shown that hydromagnetic waves excited by a stream of protons gyrating around the lines of a static magnetic field (helical proton beam) result from superluminous and counterstreaming instabilities. Physical conditions under which hydromagnetic P and B instabilities can occur are investigated with particular reference to the intensities of the helical proton beam and the rate of growth of the excited waves. It is found that in some instances the qualitative behavior of instabilities is different for different beam intensities. When the beam is very tenuous, the hydromagnetic B instability has a lower rate of growth than the P instability. However, for larger beam intensities the B instability has the larger rate of growth. Therefore it represents the dominant instability. The applicability of the results of the investigation to the analysis of low frequency pulsations of the terrestrial magnetic field is discussed.
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