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Dec 1973

Volume 16, Issue 12, pp. 2045-2355

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Effect of a Coriolis force on the stability of plane Poiseuille flow

R. Wollkind and R. C. DiPrima

Phys. Fluids 16, 2045 (1973); http://dx.doi.org/10.1063/1.1694263 (7 pages) | Cited 6 times

Online Publication Date: 31 July 2003

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The stability of an exact solution of the Navier‐Stokes equations for the flow between two parallel horizontal plates due to a constant reduced pressure gradient in a system rotating about a vertical axis is studied. For zero rotation the stability problem reduces to the classical Orr‐Sommerfeld equation for the stability of plane Poiseuille flow. The variation of the critical Reynolds number with rotation is determined by an expansion procedure for small values of the dimensionless rotation parameter (Taylor number). The effect of small rotation is destabilizing. By using known results for the stability of the Ekman boundary layer an asymptotic result for the critical Reynolds number is obtained for large values of the Taylor number.

Spatial instability of a jet

Joseph B. Keller, S. I. Rubinow, and Y. O. Tu

Phys. Fluids 16, 2052 (1973); http://dx.doi.org/10.1063/1.1694264 (4 pages) | Cited 91 times

Online Publication Date: 31 July 2003

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The instability of a circular cylindrical jet of liquid in air is studied on the assumption that the wavenumber k of the disturbance is complex while its frequency σ is real. This implies that the disturbance grows with distance along the jet, but that it does not grow with time. The occurence of such disturbances is called spatial instability, in contrast to the temporal instability studied by Rayleigh and others, in which k is real and σ is complex. It is found that there are infinitely many unstable modes for the axially symmetric case and also for each of the asymmetric cases. In the case of high velocity jets, one of these modes for the symmetric case corresponds to the mode Rayleigh found. However, it is not the most rapidly growing mode. Both analytical and numerical solutions of the dispersion equation are given for k as a function of σ and of the dimensionless jet velocity.

Thermal oscillations in convecting fluids

D. T. J. Hurle and E. Jakeman

Phys. Fluids 16, 2056 (1973); http://dx.doi.org/10.1063/1.1694265 (4 pages) | Cited 9 times

Online Publication Date: 31 July 2003

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The occurrence of thermal oscillations in a wedge‐shaped Bénard cell containing water‐methanol mixtures is described and evidence is presented to suggest that the oscillations arise from an instability in the roll‐cell convection produced by the wedge shape. Steady oscillations observed recently in a Benard cell containing water‐ethanol mixtures have been attributed to the Soret effect and this suggestion is critically discussed.

Entrainment by a rising thermal

Girard A. Simons

Phys. Fluids 16, 2060 (1973); http://dx.doi.org/10.1063/1.1694266 (8 pages) | Cited 3 times

Online Publication Date: 31 July 2003

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A theoretical model is proposed to describe the entrainment of ambient air by a rising thermal. The model is appropriate for the first few revolutions of a strongly heated thermal whose motion is dominated by its initial conditions. At sufficiently early times, only the outer edge of the thermal is turbulent. The ambient gas mixes with the thermal gas along the turbulent interface and is convected into the thermal along streamlines. The gas densities and species concentrations on each streamline are obtained by superimposing a turbulent mixing layer on the edge of Hill's spherical vortex. Solutions are obtained in terms of the dimensionless ratio of the mixing length to the thermal radius (l/a ) and the entrainment coefficient is shown to be directly proportional to l/a.

Mean velocity and shear stress distributions in turbulent boundary layers

A. E. Perry and W. H. Schofield

Phys. Fluids 16, 2068 (1973); http://dx.doi.org/10.1063/1.1694267 (7 pages) | Cited 38 times

Online Publication Date: 31 July 2003

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A universal velocity defect law for turbulent boundary layers developing in adverse pressure gradients is proposed. The velocity scale for this law is independent of the wall shear but related instead to the local maximum in the Reynolds stress profile. The proposal and its implications are checked against a very large body of experimental results and a partial connection between the mean velocity field and the shear stress field is established. A comparison between the present theory and the mixing length theory is presented. The experimental data do not appear to support many of the assumptions nor the predictions of the mixing length theory. The distinction between equilibrium and nonequilibrium layers made by Clauser is shown to be unnecessary as all classes of layers appear to conform to the new universal velocity defect law proposed, provided that the maximum shear stress is sufficiently larger than the wall shear stress.

Isochoric turbulent mixing of two rapidly reacting chemical species with chemical heat release

César Dopazo and Edward E. O'Brien

Phys. Fluids 16, 2075 (1973); http://dx.doi.org/10.1063/1.1694268 (7 pages) | Cited 9 times

Online Publication Date: 31 July 2003

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The temperature and concentration fields produced by turbulent mixing of two reactants undergoing a one‐step, very rapid, irreversible, exothermic reaction are studied. Temperature and concentration fields decay in two stages: kinetically driven and diffusion controlled. Stochastic solutions are derived for the first stage, the asymptotic state of which displays all the characteristics of species segregration. In particlar, only a weak correlation between the concentration and temperature fields is generated in the first stage. For the special case of equal mass diffusivities and a Lewis number of one the solution to the second stage is obtained in terms of the solution of a nonreacting binary mixing problem. An approximate solution for Lewis numbers different from one is proposed. The temperature field seems to decay with an effective thermal diffusivity determined by the smaller of the molecular mass and thermal diffusivities.

Biasing correction for individual realization of laser anemometer measurements in turbulent flows

D. K. McLaughlin and W. G. Tiederman

Phys. Fluids 16, 2082 (1973); http://dx.doi.org/10.1063/1.1694269 (7 pages) | Cited 145 times

Online Publication Date: 31 July 2003

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In turbulent flow situations the histograms constructed in the individual realization mode of laser anemometry are biased. The biasing occurs because a larger than average volume of fluid, and hence a larger than average number of scattering centers, pass through the probe volume during periods when the velocity is faster than the mean. Similarly, a smaller volume of fluid and a smaller number of scattering particles pass through the probe volume during periods when the velocity is slower than the mean. The proper weighting function needed to correct the biased data is the inverse of the instantaneous velocity vector. However, an analysis using turbulent flow models show that corrections based only upon the streamwise component of the velocity vector are adequate for many flow situations.

Nonlinear stability of a liquid adjacent to a supersonic stream

Ali Hasan Nayfeh and D. T. Mook

Phys. Fluids 16, 2089 (1973); http://dx.doi.org/10.1063/1.1694270 (3 pages) | Cited 1 time

Online Publication Date: 31 July 2003

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A nonlinear second‐order analysis shows that all small disturbances of the interface between a semi‐infinite, inviscid liquid and a uniform supersonic gas stream are unstable, as according to linear theory. Moreover, the growth rates increase with the amplitude of the disturbance.

K‐dependent Brownian diffusion constant in a critical mixture

Kenneth B. Lyons, Richard C. Mockler, and William J. O'Sullivan

Phys. Fluids 16, 2092 (1973); http://dx.doi.org/10.1063/1.1694271 (3 pages) | Cited 3 times

Online Publication Date: 31 July 2003

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The K ‐dependent diffusion constant of 0.3μ Teflon particles in the critical mixture isooctane‐nitroethane has been measured for 0.001 < T − T c < 2 °C. The results are presented along with a simple theory that gives good agreement by considering the surface energy force on the particle due to concentration fluctuations.

Construction of linearized kinetic models for gaseous mixtures and molecular gases

Francis J. McCormack

Phys. Fluids 16, 2095 (1973); http://dx.doi.org/10.1063/1.1694272 (11 pages) | Cited 70 times

Online Publication Date: 31 July 2003

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A simple method of construction of linearized kinetic models is proposed which is based upon the equivalence of moments of the Nth‐order modeled collision operator and the moments of the full collision operator calculated with the Nth‐order approximation to the distribution function. This method is used to construct a third‐order model of the linearized cross‐collision operator for a general multicomponent monatomic gaseous mixture, not restricted to Maxwell molecules. In addition, this method is used to extend the Hanson‐Morse diatomic gas model of the Wang Chang‐Uhlenbeck collision operator to a polyatomic gas. Also presented is a new method of formally extending the Gross‐Jackson modeling procedure to multicomponent monatomic mixtures. It is shown that the method of constructing mixture models based on the equivalence of moments is identical to this new procedure. In addition, the Gross‐Jackson modeling procedure recently developed for mixtures by Boley and Yip is analyzed. It is shown that, under a certain weak restriction, their method is equivalent to the simpler Gross‐Jackson procedure — and leads to the same expression for the Nth‐order model, except for the manner of specification of the constant in the diagonal approximation.

Approximate solutions of the Boltzmann equation for shock waves

Norman F. Sather

Phys. Fluids 16, 2106 (1973); http://dx.doi.org/10.1063/1.1694273 (4 pages) | Cited 2 times

Online Publication Date: 31 July 2003

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Several approximate solutions of the Boltzmann equation for shock waves are shown to be applications of the method of weighted residuals. Values of shock thicknesses obtained by these methods are compared, and a systematic procedure for improving the accuracy of the general method is suggested.

Constitutive relations associated with the Mott‐Smith distribution function

M. Nathenson and D. Baganoff

Phys. Fluids 16, 2110 (1973); http://dx.doi.org/10.1063/1.1694274 (6 pages) | Cited 5 times

Online Publication Date: 31 July 2003

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It is shown that the distribution function assumed by Mott‐Smith determines a unique relation between heat flux, stress, and fluid velocity given by q  =  (3/2)τu, i.e., it provides a constitutive relation for heat flux, and it also determines a simple expression for the ratio of third‐order central moments Q  =  〈C x3 〉 / 〈C x C 2〉. These expressions allow the equation of transfer for cx2 to be cast in a form that yields a nonlinear constitutive relation for stress. The results obtained from the Mott‐Smith ansatz are compared with the theory of Baganoff and Nathenson and results from a numerical solution of the Boltzmann equation for shock‐wave structure obtained by Hicks and Yen.

Heat transfer between parallel plates with arbitrary surface accommodation

J. R. Thomas, T. S. Chang, and C. E. Siewert

Phys. Fluids 16, 2116 (1973); http://dx.doi.org/10.1063/1.1694275 (5 pages) | Cited 10 times

Online Publication Date: 31 July 2003

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Elementary solutions of the coupled pair of integrodifferential equations arising from the decomposition of the linearized BGK equation in the kinetic theory of gases are used to solve the problem of heat transfer between parallel plates with arbitrary surface accommodation. A coupled pair of Fredholm equations is derived, and rapidly convergent iterative solutions are constructed. These solutions are then used to obtain accurate values of the heat flux between the plates and the temperature and density profiles, for various values of the accommodation coefficient and inverse Knudsen number. Numerical results for the heat flux are presented and compared to existing variational solutions. Also, explicit results for the temperature and density profiles are given.

Nonsimilar blast waves with transparent gas radiation

Gary G. Erickson and D. B. Olfe

Phys. Fluids 16, 2121 (1973); http://dx.doi.org/10.1063/1.1694276 (11 pages) | Cited 2 times

Online Publication Date: 31 July 2003

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Analytical and numerical calculations are carried out for an initially adiabatic blast wave in which the effect of transparent gas radiation increases as the temperature decreases. After a radiative cooling time is reached, a very dense shell of gas rapidly forms behind the shock front, and a cooling wave slowly progresses through the hot interior gas. The cooling wave reduces the interior pressure, resulting in an attenuation of the shock‐front velocity. For a specific heat ratio γ near unity the shock velocity will remain near the adiabatic value, whereas for γ  =  5/3 the shock velocity will approach the value for a momentum‐conserving shell, as predicted by some authors.

Nonequilibrium structure of argon shock waves

H. F. Nelson

Phys. Fluids 16, 2132 (1973); http://dx.doi.org/10.1063/1.1694277 (11 pages) | Cited 6 times

Online Publication Date: 31 July 2003

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This theoretical study investigates the influence of (1) the nonequilibrium excited state population in the relaxation region and (2) line radiation in the precursor on shock wave structure by considering strong shock waves propagating at Mach 18 and 24 into a theoretical argon‐like gas at a pressure of 1 cm Hg and a temperature of 300 °K. The argon atom is modeled as having two bound states plus a continuum, and the calculations include finite atom‐atom and electron‐atom collisional ionization and excitation rates as well as continuum and line radiation. The electron gas is allowed to be at a different temperature from the atom gas; consequently, three types of of nonequilibrium exist: ionization, excitation, and thermal. In the collisional relaxation region, which serves as a source of radiation to create the precursor, the degrees of ionization and excitation generally lag behind their respective local equilibrium values. By doing so they greatly influence the emission of radiation. The line wing radiation increases the extent of the precursor compared with former studies which considered only continuum radiation. However, elimination of the assumption that the excited states are in equilibrium with the electrons at the electron temperature decreases the magnitude of the precursor compared with results where the assumption has been used because the radiative flux is reduced.

Measurement of electrical conductivity of argon at high pressure

U. Bauder, R. S. Devoto, and D. Mukherjee

Phys. Fluids 16, 2143 (1973); http://dx.doi.org/10.1063/1.1694278 (6 pages) | Cited 12 times

Online Publication Date: 31 July 2003

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Measurements have been carried out of the current, electric field strength, and radial temperature distribution in a 5 mm diam. wall‐stabilized electric arc in argon at pressures from 1‐200 atm. These data have been analyzed to determine the electrical conductivity as a function of temperature and pressure. The results are in reasonable accord with other measurements to 10 atm and with theory.

Electron wave breakdown of helium

Roger N. Blais and R. G. Fowler

Phys. Fluids 16, 2149 (1973); http://dx.doi.org/10.1063/1.1694279 (6 pages) | Cited 6 times

Online Publication Date: 31 July 2003

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A revision of the electrostatic gas breakdown wave apparatus used by Haberstich and Winn has been employed to generate waves for studies of wave speed, and electron temperature and density behind the wave front. The measurements were carried out in helium at applied electrode voltages from 6 to 42 kV of both polarities and over a pressure range of 0.3 to 30.0 Torr. Nuclear data‐handling techniques were adapted to the measurement of the wave speeds to increase the accuracy of the data. Good over‐all agreememt is obtained with the theory of Shelton.

Effect of electrical conductivity and a viscous flow on the stability of a current‐carrying hollow cylinder of fluid

T. H. Date and M. H. Gokhale

Phys. Fluids 16, 2155 (1973); http://dx.doi.org/10.1063/1.1694280 (5 pages)

Online Publication Date: 31 July 2003

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The stability of the axisymmetric modes of a viscous flow of an annular cylindrical layer of fluid carrying an electric current is considered in the presence of an independent parallel electric current along the axis. Even if the electrical conductivity of the fluid is small, sufficiently strong steady‐state flow with a radial velocity gradient can reduce the growth rates so much that the e ‐folding times of the instabilities is larger than the time of flow over a typical length.

Electromagnetic detonation in a magnetic annular shock tube

J. Y. S. Mar, V. Makios, and E. G. Plett

Phys. Fluids 16, 2160 (1973); http://dx.doi.org/10.1063/1.1694281 (7 pages) | Cited 2 times

Online Publication Date: 31 July 2003

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The results of an experimental demonstration of electromagnetic detonation waves are presented. These waves are obtained by discharging a capacitor bank across the annulus of a magnetic annular shock tube filled with hydrogen‐oxygen mixtures. Wave speeds of more than twice the usual steady Chapman‐Jouguet detonation speeds are observed. Experiments with argon in place of the combustible mixture were also conducted to illustrate the basic difference between electromagnetic shock waves and electromagnetic detonations. Theoretically calculated results are included to provide a better understanding of the observed experimental results. It is suggested that techniques similar to those used in these experiments may be used to produce high‐speed detonations for gasdynamic laser applications or to stabilize combustion waves in a supersonic stream.

Laminar electrostatic shock waves generated by an ion beam

H. Ikezi, T. Kamimura, M. Kako, and K. E. Lonngren

Phys. Fluids 16, 2167 (1973); http://dx.doi.org/10.1063/1.1694282 (9 pages) | Cited 27 times

Online Publication Date: 31 July 2003

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Strong laminar electrostatic shock waves have been experimentally observed when an ion beam is injected into a collisionless plasma. The structure of the shock is qualitatively different from one with a trailing wave train. A density depression follows behind the shock front, and no trailing wave train due to wave dispersion is found. A significant amount of ions reflected from and transmitted through the shock front form a precursor. The critical Mach number above which no shock is formed is found to be 1.5. Numerical simulations reported here reproduce the experimental observations very well. An analysis based on the water‐bag model accounts for the observed value of the critical ion‐beam velocity which gives the critical Mach number. It also points out that the reflected ions play an essential role in the persistence of the shock.

Microscopic theory of the transition from ambipolar to free diffusion

Masahiro Numano

Phys. Fluids 16, 2176 (1973); http://dx.doi.org/10.1063/1.1694283 (4 pages) | Cited 2 times

Online Publication Date: 31 July 2003

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The transition from ambipolar to free electron diffusion was investigated by making use of the expression for the diffusion coefficient in terms of the velocity autocorrelation function. It was assumed that the electron mean free path is much smaller than the plasma dimension. The diffusion coefficient was obtained from Vlasov‐Boltzmann type kinetic equations, assuming a BGK model for the terms representing collisions between charged particles and the neutrals. It was found that when the Debye length is much shorter than the plasma dimension, the diffusion is ambipolar and that as the Debye length increases there is transition to free diffusion. The diffusion coefficient obtained was compared with the machine calculation results by Allis and Rose, and a fairly good agreement between them was found. The effect of a magnetic field on the transition of diffusion across the magnetic field was also investigated.

Instantaneous electron energy distribution function in ion waves

D. Grésillon and P. L. Galison

Phys. Fluids 16, 2180 (1973); http://dx.doi.org/10.1063/1.1694284 (4 pages) | Cited 15 times

Online Publication Date: 31 July 2003

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The assumptions involved in the use of Boltzmann's law to describe the electron density variations in ion acoustic waves are discussed. The need for electron collisions to fill the regions of maximum potential with low‐energy electrons is pointed out. A direct method for measuring the electron energy distribution at any time during the period of an ion wave is presented. Experimental results show the existence of electrons with energy small enough for their motion to be constrained to the wave potential well. For higher energies, the distribution stays approximately the same. The method provides a direct measure of plasma potential, which is related to density. The Boltzmann law is still seen to be accurate for relative fluctuations of the order of up to 20%.

Collision terms and density correlations for strongly correlated plasmas

Mitsuhiro Nambu

Phys. Fluids 16, 2184 (1973); http://dx.doi.org/10.1063/1.1694285 (6 pages)

Online Publication Date: 31 July 2003

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New collision terms for stable plasmas are formally derived within the framework that the ternary correlation function be expressed in terms of the products of the single‐particle and pair‐correlation functions. The collision terms obtained are nonlinear integral equations involving both single particle and pair correlation functions. It is investigated as to how the conventional Balescu‐Lenard collision terms are modified through pair‐correlation functions. As a consequence of such a modification of effective interactions, the decay of pair correlations becomes oscillatory. The effective Debye length for strongly correlated plasmas is introduced, and the sum rule for strongly correlated plasma is also investigated.

Interaction of force‐free fields with external circuits

A. Kadish and J. W. A. Zwart

Phys. Fluids 16, 2190 (1973); http://dx.doi.org/10.1063/1.1694286 (9 pages) | Cited 4 times

Online Publication Date: 31 July 2003

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The interaction of a perfectly conducting force‐free magnetic field with external circuits during the adiabatic compression of a perfectly conducting column is studied in a slab geometry. It is shown that the circuit equations of a cylindrical screw pinch discharge together with an initially specified magnetic field determine a unique motion of the plasma, provided the E × B vector is initially inward. It is found that after a finite time, the model breaks down. Estimates for this time, which correspond to the self‐crowbarring time of the plasma, are obtained. Exact nonlinear solutions are obtained for special initial discharges.

Equilibrium and stability of mirror‐confined nonneutral E layers

Ronald C. Davidson, Adam T. Drobot, and C. A. Kapetanakos

Phys. Fluids 16, 2199 (1973); http://dx.doi.org/10.1063/1.1694287 (12 pages) | Cited 20 times

Online Publication Date: 31 July 2003

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The steady‐state Vlasov‐Maxwell equations are used to study the equilibrium properties of a nonneutral E layer confined both axially and radially, by a static external mirror field, mirror field, B0ext(x). Equilibrium properties are calculated for the electron distribution function fe0 (H , P θ) in which all electrons have the same total energy (H) and the same canonical angular momentum (P θ), i.e., fe0 (H , P θ)  =  N0δ(H − H0)δ(Pθ − P0), where N 0, H 0, and P 0 are positive constants. For a low‐density E layer, the electrostatic potential energy of an electron, − e ϕ0(r , z ), is small in comparison with H 0. Neglecting terms of order ∣e ϕ0(r , z)/ H 0∣, a closed zero‐order expression for the r ‐ z boundary of the E layer is obtained. Iterating, the equilibrium electrostatic potential ϕ0(r , z) is then computed to lowest order, together with O [e ϕ0(r , z) / H0] corrections to the boundary of the E layer. Computer simulation experiments are used to investigate electrostatic stability properties for the case of azimuthally symmetric perturbations (∂/∂θ  =  0). The code allows only r and z spatial variations but self‐consistently follows the three velocity components of 6264 macroparticles. When the electrons are initially loaded close to the equilibrium state described by fe0(H , Pθ)  =  N 0δ(H − H 0)δ(P θ − P 0), the E layer is observed to achieve a stable quasisteady configuration in about 15 ωp e−1 with negligible particle loss out the mirrors.
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