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Jun 1972

Volume 15, Issue 6, pp. 963-1172

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Existence of Variational Principles for the Navier‐Stokes Equation

Bruce A. Finlayson

Phys. Fluids 15, 963 (1972); http://dx.doi.org/10.1063/1.1694056 (5 pages) | Cited 30 times

Online Publication Date: 1 August 2003

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Fréchet differentials are introduced to decide when a classical variational principle exists for a given nonlinear differential equation. The formalism is applied to the steady‐state Navier‐Stokes equation and the continuity equation, and no variational principle exists unless u × (∇ × u)  =  0 or u⋅ ∇u  =  0. The concept of an adjoint equation is extended to nonlinear equations and a variational principle is derived for the Navier‐Stokes equation and its adjoint.

Use of Streaming Potential Measurements for an Investigation of the Coanda Effect

Jim L. Turpin

Phys. Fluids 15, 968 (1972); http://dx.doi.org/10.1063/1.1694057 (4 pages) | Cited 1 time

Online Publication Date: 1 August 2003

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The streaming potential technique was adapted for an investigation of the Coanda effect. The attachment distances for two‐dimensional submerged, incompressible jets were determined as a function of wall angle and offset. This was done by embedding platinum electrodes in a movable sidewall of a flow model, and determining the electrode pair which exhibited the maximum rms streaming potential fluctuation. Experimentally determined attachment points are presented graphically as a function of wall offset with wall angle as a parameter. It was found that the attachment distance is independent of the nozzle Reynolds number.

Laminar Flow Past a Circle in a Shear Flow

Stanley K. Jordan and Jacob E. Fromm

Phys. Fluids 15, 972 (1972); http://dx.doi.org/10.1063/1.1694058 (5 pages) | Cited 18 times

Online Publication Date: 1 August 2003

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A numerical solution of the equations governing time‐dependent, viscous, incompressible fluid flow past a circle is presented for Reynolds number 400. The free‐stream velocity profile corresponds to the time‐averaged flow of a turbulent jet. The location and strength of the jet are adjusted so that the free stream approximates a linear shear flow in the vicinity of the circle. The asymmetry of the free stream causes a shift of the upstream stagnation streamline (the Pitot‐tube displacement effect) and small biases in the lift and torque exerted upon the circle. This lift bias is suggested as a contributing factor in the observed stability of raindrops in vertical wind tunnels. 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 asymmetry of the free stream manifests itself in terms of fine details within the flow pattern around the circle.

Effects of Couple Stresses on the Stability of Plane Poiseuille Flow

Jai Kumar Jain and Vijay Kumar Stokes

Phys. Fluids 15, 977 (1972); http://dx.doi.org/10.1063/1.1694059 (4 pages)

Online Publication Date: 1 August 2003

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The effects of couple stresses in fluids on the hydrodynamic stability of plane Poiseuille flow have been considered based on the model for fluids proposed by Stokes. Numerical solutions to the problem have been obtained by using an initial‐value technique. It has been shown that the critical Reynolds number first decreases and then increases with the increase in couple stresses. The effects of couple stresses on the different flow parameters, which describe the disturbances, have also been determined.

Statistical Characteristics of Reynolds Stress in a Turbulent Boundary Layer

A. K. Gupta and R. E. Kaplan

Phys. Fluids 15, 981 (1972); http://dx.doi.org/10.1063/1.1694060 (5 pages) | Cited 42 times

Online Publication Date: 1 August 2003

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The first four moments of the Reynolds stress fluctuations uυ(t), and of velocity components u(t) and υ(t) were computed using digital data processing techniques. The results show that over a moderate range of Reynolds number 1900<Reθ<6500 (based on momentum thickness) Reynolds stress fluctuations are intermittent and strongly skewed, while the u and υ fluctuation components separately are not.

“Backward Diffusion” in Turbulent Flow

S. Corrsin

Phys. Fluids 15, 986 (1972); http://dx.doi.org/10.1063/1.1694061 (2 pages) | Cited 4 times

Online Publication Date: 1 August 2003

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For homogeneous, unsheared, stationary turbulence, it is shown that the dispersion of starting position x0 (at t  =  0) for an ensemble of material points which arrive at position x at time t is equal to the dispersion of material points starting at position x, in the same time interval. The question is inspected in both Lagrangian and Eulerian frames.

Use of the Scattering Kernel Approach in Certain Gas Kinetic Problems

E. C. Whipple

Phys. Fluids 15, 988 (1972); http://dx.doi.org/10.1063/1.1694062 (7 pages) | Cited 13 times

Online Publication Date: 1 August 2003

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The collision gain term of the Boltzmann equation may at times be transformed to a triple integral over the product of the unknown distribution and a scattering kernel in the case of a minor constituent interacting through collisions with a major constituent (scatterer). If isotropy in velocity space may be assumed, the term may be further reduced to an integral over the particle speed. The scattering kernel for hard‐sphere collisions has been used in neutron slowing down and thermalization problems, but this approach does not seem to have been used widely in other gas kinetic applications. The transformation is discussed and cast in a form that is applicable to collision models other than hard‐sphere scattering and to arbitrary velocity distributions of the scatterers. Kernels are obtained for isotropic constant mean‐free‐time scattering, and it is shown that kernels for monoenergetic velocity distributions of the scatterers are useful as well as those for Maxwellian and stationary distributions. Use of the scattering kernel approach is illustrated by finding the steady‐state velocity distribution of excited atoms that are being produced at a steady rate and that are losing kinetic energy via collisions and their excitation energy by radiation. The solution for Maxwellian scatterers is well approximated at high velocities by the solution for stationary scatterers, and at low velocities by the solution for monoenergetic scatterers.

Anisotropic Relaxation in Binary Mixture of Maxwell Molecules

Francis J. McCormack and Anna M. Williams

Phys. Fluids 15, 995 (1972); http://dx.doi.org/10.1063/1.1694063 (4 pages) | Cited 2 times

Online Publication Date: 1 August 2003

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The relaxation of initial temperature difference, stresses and heat fluxes in a binary mixture of Maxwell molecules is studied by solving the two‐temperature 13‐moment equations for the mixture under the assumption of spatial homogeneity. For simplicity, the flow velocities for each gas are assumed to be the same, and the rates at which the stresses and heat fluxes relax are compared with the rate of relaxation of the temperature difference. It is found that, for equal densities and disparate masses, Grad's conjecture on relaxation in “epochs,” or stages, is borne out. For unequal densities, however, it is found that under certain conditions, the heat flux for the gas of larger density relaxes slower than the temperature difference.

Thermal Force on Spherical Particles in a Rarefied Gas

Warren F. Phillips

Phys. Fluids 15, 999 (1972); http://dx.doi.org/10.1063/1.1694064 (5 pages) | Cited 16 times

Online Publication Date: 1 August 2003

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A theoretical analysis is presented for the thermal force exerted on a spherical particle between two flat plates. The case is considered where the Knudsen number KL, based on the plate spacing, is nonzero. The analysis is based on the assumption that the particle radius is much smaller than the plate spacing and the result is valid for all values of KL from zero to infinity. A moment solution to the Boltzmann equation coupled with a two‐stream Chapman‐Enskog distribution function is employed, and the collision integral is evaluated on the basis of Maxwell molecules.

Nearly Free‐Molecular Channel Flow at Finite Pressure Ratio

P. Y. Wang and E. Y. Yu

Phys. Fluids 15, 1004 (1972); http://dx.doi.org/10.1063/1.1694020 (6 pages) | Cited 1 time

Online Publication Date: 1 August 2003

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The problem of nearly free‐molecular flow of a gas from one reservoir to another through a two‐dimensional channel is studied. The width of the channel is much smaller than the length which is of the order of the mean free path of the gas. The ratio of the equilibrium pressures in the reservoirs is finite, whereas the equilibrium temperatures are the same. The fundamental solution of the linear Boltzmann equation is used for the evaluation of total mass flow rate. The result is presented in the form of an asymptotic series, of which the first‐order terms are of the order of α ln α and α, where α is the inverse Knudsen number. Computations shows that the first‐order terms, which represent intermolecular collisions in the counter flows, have a net negative value. The total mass flow rapidly decreases as the length of the channel increases.

Electron‐Beam Investigation of a Hypersonic Shock Wave in Nitrogen

Ronald B. Smith

Phys. Fluids 15, 1010 (1972); http://dx.doi.org/10.1063/1.1694021 (8 pages) | Cited 11 times

Online Publication Date: 1 August 2003

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The electron‐beam fluorescence technique is used to probe a Mach‐26 normal shock wave. A method to determine the distribution of population among the rotational states is developed. The measured number density and rotational temperature profiles are compared with the theoretical results of Brau and Simons. A simple approximate collision model is developed, producing from the experimental results, general information about the rotational equilibration process, and the determination of a rotational excitation cross section.

Slip Coefficients for General Gas‐Surface Interaction

Tomaž Klinc and Ivan Kuščer

Phys. Fluids 15, 1018 (1972); http://dx.doi.org/10.1063/1.1694022 (5 pages) | Cited 18 times

Online Publication Date: 1 August 2003

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The velocity and the temperature slip coefficients are evaluated for general gas‐surface scattering kernels by a variational method. The results are expressed in terms of a set of accommodation coefficients.

Falkner‐Skan Problem in Magnetohydrodynamics

Lewis T. Hildyard

Phys. Fluids 15, 1023 (1972); http://dx.doi.org/10.1063/1.1694023 (5 pages) | Cited 3 times

Online Publication Date: 1 August 2003

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The Falkner‐Skan problem is extended to magnetohydrodynamic flow past a nonconducting body by imposing a magnetic field whose lines of force are parallel to the undisturbed streamlines. The magnetic‐field boundary condition used by Gribben is shown to be inappropriate and results in the form of asymptotic series for large and small values of the magnetic Prandtl number are obtained. Numerical integration is used to establish the validity of the series in each case.

Stability Criterion for the Bluff‐Body Stabilized Electrodeless Arc

R. E. Martin and D. R. Keefer

Phys. Fluids 15, 1028 (1972); http://dx.doi.org/10.1063/1.1694024 (7 pages)

Online Publication Date: 1 August 2003

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It has been found that the electrodeless arc can be stabilized in the presence of flow by placing a bluff body in the entrance stream. A two‐stream boundary layer analysis, analogous to that used in combustion theory, is used to calculate an attachment length, the distance required to heat the entering stream to the arc temperature. The criterion for stability is that the attachment length is less than the length of the recirculation cell formed in the wake of the bluff body. Calculations are made for conditions typical of an atmospheric pressure arc in argon. It is found that the attachment length is approximately a linear function of the ratio of flow velocity to power density.

Onset Point of Ionization

S. S. R. Murty and P. Serou

Phys. Fluids 15, 1035 (1972); http://dx.doi.org/10.1063/1.1694025 (3 pages)

Online Publication Date: 1 August 2003

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It is shown that precursor photoionization is capable of creating electron number densities large enough to virtually eliminate the atom‐atom collision region and the Mach number at which the above phenomenon occurs is calculated for a few initial pressures and temperatures for argon.

Magnetohydrodynamic Turbulence Experiments in a Transverse Magnetic Field

John M. Kellam

Phys. Fluids 15, 1038 (1972); http://dx.doi.org/10.1063/1.1694026 (9 pages) | Cited 1 time

Online Publication Date: 1 August 2003

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Random fluctuations in a transverse magnetic field caused by a helium plasma produced in an arc discharge were measured with magnetic probes. Ensemble averaged power spectra of these fluctuations followed a power law over the frequency range from about 0.2 to 1 MHz, the exponents varying from −11/3 to −19/3. The time average of the fluctuation intensity parallel to the external magnetic field increased nearly linearly with the imposed field up to about 1000 G. Stronger magnetic fields inhibited these fluctuations and retarded the flow of the plasma. Integral and microscales were in the range of 40 to 100 mm and 20 to 40 mm, respectively. The random fluctuations in the magnetic field may have been due partly to random inhomogeneities in the conductivity of the plasma and only partly to vortical turbulence. Magnetic and conductivity probes indicated the presence of substantial electrical conductivity ahead of the luminous plasma. This conductivity was reduced by increasing magnetic field, interpreted as involving an electron flux.

Optical Measurements of the Degree of Turbulence of a Gun Plasma

D. R. Matt and F. R. Scott

Phys. Fluids 15, 1047 (1972); http://dx.doi.org/10.1063/1.1694027 (4 pages) | Cited 6 times

Online Publication Date: 1 August 2003

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Turbulent electric fields of the order of 7 kV/cm were observed in a plasma from a gun. These values were obtained by measuring the ratio of the forbidden companions of the He I 4471 and He I 4921‐Å lines to their corresponding allowed transitions. The electric fields were also observed by examining the shape of the red wing of the He I 4471‐Å and the He II 4686‐Å lines. The measured plasma charge density of 7 × 1012/cm3 was too low to account for the broadening. The plasma turbulence was determined to be 3% of the electron thermal energy. The observed damping was compatible with spatial linear Landau damping for a mean wavelength 30 times the Debye wavelength.

Conductivity of an Anisothermal Magnetoplasma from a Convergent Kinetic Equation

Hon‐Ming Lai

Phys. Fluids 15, 1051 (1972); http://dx.doi.org/10.1063/1.1694028 (5 pages)

Online Publication Date: 1 August 2003

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The convergent classical kinetic equation of Kihara and Aono is used to calculate the collisional part of the ac electrical conductivity of a multispecies, anisothermal magnetoplasma in the long wavelength limit. A two‐component plasma result is derived, and it shows that at Te/Ti ≳ 103, the Coulomb logarithmic term is no longer dominant.

Conductivity of a Magnetoplasma from a Quantum Mechanical Convergent Kinetic Equation

Agnar Pytte and Hon‐Ming Lai

Phys. Fluids 15, 1056 (1972); http://dx.doi.org/10.1063/1.1694029 (3 pages) | Cited 3 times

Online Publication Date: 1 August 2003

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The quantum‐mechanical version of Baldwin's convergent kinetic equation is used to calculate the collisional part of the ac electrical conductivity tensor of a multispecies magnetoplasma to second order in the wave vector. Both dominant (Coulomb logarithm) and subdominant terms are calculated exactly to first order in the plasma parameter. Full account is taken of the heavy ion motion.

Microscopic Theory of Conductivity of a Weakly Ionized Plasma

W. R. Chappell and R. H. Williams

Phys. Fluids 15, 1059 (1972); http://dx.doi.org/10.1063/1.1694030 (4 pages) | Cited 1 time

Online Publication Date: 1 August 2003

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Through the use of the microscopic density functions in μ space the conductivity for a weakly ionized plasma can be expressed in terms of a function describing single particle diffusion in the plasma. This diffusion function or transition probability has been shown to obey a linear Boltzmann equation. As a result the approximate solution of the conductivity problem can be accomplished either by modeling the collision term in the kinetic equation or by a stochastic modeling of the diffusion function itself. The two approaches are seen to be completely equivalent.
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Quasilinear Diffusion of an Axisymmetric Toroidal Plasma

Allan N. Kaufman

Phys. Fluids 15, 1063 (1972); http://dx.doi.org/10.1063/1.1694031 (7 pages) | Cited 132 times

Online Publication Date: 1 August 2003

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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.

Near‐Steady Oblique Shock Waves in a Collisionless Plasma

Lynn M. Olson

Phys. Fluids 15, 1070 (1972); http://dx.doi.org/10.1063/1.1694032 (12 pages) | Cited 2 times

Online Publication Date: 1 August 2003

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Waves of small amplitude ϵ which travel at an oblique angle θ to a magnetic field in a cold collisionless plasma and whose head approaches steadiness are studied. The relevant double limit
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is treated in terms of appropriate sets of single limits. Near‐steady waves which can lead from one equilibrium to another are shown to be possible only at propagation speeds corresponding to Alfvén Mach number 1 or cosθ. In the first case, either the magnetic pressure at the head of the wave decreases monotonically from its initial value and the wave spreads linearly with time, or else the head of the wave consists of a near‐periodic train of cnoidal waves, the first crest of which approaches a soliton. For near‐normal shocks, the magnetic pressure in the near‐periodic wave train always exceeds its initial equilibrium value. But for M cotθ>1, where M  =  (m+/m−)1/2−(m−/m+)1/2, it always falls short of that value. A transition with precursor wave is shown to be impossible for a cold collisionless plasma. At an Alfvén Mach number cosθ, the only near‐steady solution consists of a main wave front followed by an oscillatory tail; on the length scale considered, the electric field transverse to the propagation direction oscillates without bound at the tail.

Ion and Electron Pressure Effects on Magnetosonic Shock Formation

R. J. Mason

Phys. Fluids 15, 1082 (1972); http://dx.doi.org/10.1063/1.1694033 (8 pages) | Cited 18 times

Online Publication Date: 1 August 2003

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Piston generated magnetosonic shocks have been investigated with a hybrid particle‐in‐cell simulation scheme that retains electron inertial effects and pressure but neglects resistivity. At low piston Mach numbers, Mp ≤ 1.0, the c/ωpe scale wave‐train shocks of Auer, Hurwitz, and Kilb are reproduced. For Mp  =  1.7 post‐frontally flat shocks with strong precursors are obtained. For Mp ≥ 2.4 pure piston reflected flow is observed. All these flows are predominantly laminar. Electron pressure effects are found to scale as (1 +βe0)/Mp2, βe0  =  Pe0/(B02/8π). Finite ion pressure smooths the post‐frontal flow through ion Landau damping. Comparisons are made with results from other recent simulations, and with observations from laboratory experiments at Culham and Garching.

Growth Rates of Instabilities of a Diffuse Linear Pinch

J. P. Goedbloed and H. J. L. Hagebeuk

Phys. Fluids 15, 1090 (1972); http://dx.doi.org/10.1063/1.1694034 (12 pages) | Cited 60 times

Online Publication Date: 1 August 2003

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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.

Vlasov‐Fluid Model for Studying Gross Stability of High‐β Plasmas

J. P. Freidberg

Phys. Fluids 15, 1102 (1972); http://dx.doi.org/10.1063/1.1694035 (7 pages) | Cited 82 times

Online Publication Date: 1 August 2003

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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.
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