Showing papers in "Physics of Fluids in 1967"

Inertial Ranges in Two‐Dimensional Turbulence

Abstract: Two‐dimensional turbulence has both kinetic energy and mean‐square vorticity as inviscid constants of motion. Consequently it admits two formal inertial ranges, E(k)∼e2/3k−5/3 and E(k)∼η2/3k−3, where e is the rate of cascade of kinetic energy per unit mass, η is the rate of cascade of mean‐square vorticity, and the kinetic energy per unit mass is ∫0∞E(k) dk. The −53 range is found to entail backward energy cascade, from higher to lower wavenumbers k, together with zero‐vorticity flow. The −3 range gives an upward vorticity flow and zero‐energy flow. The paradox in these results is resolved by the irreducibly triangular nature of the elementary wavenumber interactions. The formal −3 range gives a nonlocal cascade and consequently must be modified by logarithmic factors. If energy is fed in at a constant rate to a band of wavenumbers ∼ki and the Reynolds number is large, it is conjectured that a quasi‐steady‐state results with a −53 range for k « ki and a −3 range for k » ki, up to the viscous cutoff. The t...

Viscous Sublayer and Adjacent Wall Region in Turbulent Pipe Flow

Abstract: The boundary‐layer research facility utilizing the highly viscous fluid, glycerine, was constructed to permit detailed experimental investigation of the viscous sublayer. At a pipe Reynolds number of 8700 the sublayer thickness corresponding to a nondimensional distance from the wall of yu τ/ν = 5.00 was 0.110 in. Detailed measurements of the streamwise fluctuating velocities were obtained with hot‐film anemometers within the viscous sublayer as well as in the transition region between the linear and logarithmic mean velocity profiles. These data were used to form the space‐time correlation function of the streamwise fluctuating velocities. An eigenfunction decomposition of the streamwise fluctuating velocity into a sum of products of eigenfunctions in the inhomogeneous coordinate direction, with random coefficients dependent on the homogeneous and stationary variables, was obtained from the correlation data. One dominant eigenfunction with a structure nearly identical to the mean velocity in the wall region was found. The dominant large scale structure of the flow in the wall region, obtained with the aid of a mixing length approximation, consisted of randomly distributed counterrotating eddy pairs of elongated streamwise extent.

Instabilities due to Temperature Gradients in Complex Magnetic Field Configurations

Abstract: An integral equation governing an instability due to ion temperature gradients is derived. In the presence of magnetic shear, localized non‐convective normal modes of instability are shown to exist if the relative temperature gradient is larger than that of density, unless the shear is exceedingly strong, i.e., the field shears through a large angle in the distance in which the temperature drops. Quasi‐modes which are less localized in the direction of the gradient can be constructed from these normal modes and a large thermal diffusion may be expected. Conversely the mass diffusion is shown to be rather slow so that it is reasonable to assume that an effective ``divertor'' should keep the actual heat loss quite small.

Toroidal containment of a plasma.

Abstract: The question of plasma containment in a torus is much more complicated than in an open‐ended mirror system. Serious questions arise of the nonexistence of flux surfaces, of noncontained particle drifts, and of nonexistence of self‐consistent equilibria at small gyroradius.

Transport coefficients of partially ionized argon.

Abstract: The complete Chapman‐Enskog‐Burnett expressions for the transport coefficients of multicomponent gas mixtures are applied to the computation of the properties of partially ionized argon. Studies of the rate of convergence of the approximations to the coefficients show that the third approximation to the thermal conductivity and the second to the viscosity are adequate at all degrees of ionization. The ordinary ambipolar diffusion coefficient is given to excellent accuracy by twice the binary ion‐atom diffusion coefficient but, at low ionization, apparently not even the fourth approximation to the electrical conductivity has converged to the true value. The computed electrical and thermal conductivities are compared with experimental measurements.

Simplified Expressions for the Transport Properties of Ionized Monatomic Gases

Abstract: Simplified theoretical expressions for the transport properties of ionized gas mixtures are derived within the framework of the Chapman—Enskog—Burnett method. The properties of equilibrium partially ionized argon are then computed with these expressions and compared with values obtained with the exact theory. Agreement is satisfactory.

Distribution Functions in the Statistical Theory of Turbulence

Abstract: A hierarchy of coupled equations for multipoint turbulent velocity distribution functions is derived. These equations resemble the Bogoliubov‐Born‐Green‐Kirkwood‐Yvon equations of kinetic gas theory. The properties of the distribution functions and the equations are discussed and compared with kinetic theory.

Numerical Study of Two Fluid Rayleigh‐Taylor Instability

Abstract: A new computing technique for the time‐dependent calculation of the interaction of two incompressible fluids is used to study the linear and nonlinear phases of Rayleigh‐Taylor instability. The variation of the linear growth rate with Reynolds number is examined for Reynolds numbers < 200 at a ratio of fluid densities of 2 : 1. These findings support Chandrasekhar's predictions of the magnitude of the growth rate and the Reynolds number for which maximum growth rate is attained. A series of inviscid flow calculations, at density ratios between 1.1 : 1 and 10 : 1, is used to explore the effect of density ratio on aspects of the nonlinear Rayleigh‐Taylor instability. It is found that such influences as form drag and Kelvin‐Helmholtz instability decrease rapidly in importance as the density ratio increases.

Couple Stresses in Fluids

Abstract: Applications of couple stress and micropolar theories to the problems of Couette and Poiseuille flows between two parallel plates are discussed and the results are compared.

Similarity and the Turbulent Energy Spectrum

Abstract: Spherically averaged spectral densities of quantities appearing in the turbulent energy equation are assumed to be locally similar: i.e., to depend on external parameters in a similar way when referred to local scales of flux and length. Dynamical equations are given for the influence of production (including anisotropy), viscosity, buoyancy, magnetic field, and elasticity. General solutions of these equations are obtained implicitly, and the leading term in a series expansion explicitly. This similarity is shown to be a generalization of the ideas of Kolmogorov, of the modified Onsager cascade used by Corrsin, and of the formulation of Kovasznay and Shur for the influence of viscosity and buoyancy. In the case of viscosity, the prediction of a spectral cutoff by the Kovasznay formulation is seen to be misleading, and due to the retention of only the leading term in the expansion.

Measurement of Emission and Absorption of Radiation by an Argon Plasma

Abstract: Volumetric radiative loss measurements, correlated with temperature in the range of 10 000 to 26 000°K, have been made on an argon plasma. Pressures of 0.5, 1.0, and 2.0 atm have been used. The 1.0‐atm measurements have been corrected for both absorption and ultraviolet emission and the results agree with those of Emmons in the common temperature range. The 6965 Ar I line has also been studied yielding lineshifts, halfwidths, absorption and emission coefficients. The line shift and halfwidth results are below theoretical predictions. Transition probabilities determined from both emission and absorption studies are found to be in reasonable agreement.

Numerical Solution of the Three-Dimensional Equations of Motion for Laminar Natural Convection

Abstract: A method is presented for numerical finite difference solution of the equations of motion in three dimensions. The complete Navier‐Stokes equations are transformed and expressed in terms of vorticity and a vector potential. The transformed equations are solved using an alternating direction method for the parabolic portion of the problem, and successive over‐relaxation for the elliptic portion. The classical problem of convection in fluid layers bounded by solid walls and heated from below is solved in both two and three dimensions. Comparison with other methods and with prior work in two dimensions shows that the new method presented here has important advantages in speed and accuracy. Apparently, there has been no successful prior work in three dimensions except in cases where one component of the equation of motion is highly simplified.

Breakdown of Eddies and Probability Distributions for Small‐Scale Turbulence

Abstract: A rigorous mathematical description is given for the physical concept of the cascade process of sequential breakdown of turbulent eddies. It is assumed that the probability distribution for the ratio of typical values of turbulent fluctuations averaged over a small volume v to those averaged over a surrounding larger volume V is invariant under the group of space similarity transformations, provided that the length scales of both volumes are much smaller than the external scale of turbulence and much larger than the Kolmogoroff scale. This assumption is in general agreement with the conclusions of Kolmogoroff and Obukhov and with the existing measurements of the power spectrum for the square of the velocity derivative. Some measurements of probability distributions for smallscale turbulent fluctuations are also discussed.

Growth of turbulent magnetic fields.

Abstract: Weak random initial magnetic field evolution in highly conducting isotropically turbulent fluid, noting exact initial growth expression of magnetic energy spectrum

Diffusion Approximation to Inertial Energy Transfer in Isotropic Turbulence

Abstract: A diffusion approximation is described to the nonlocal inertial energy transfer between wavenumber components in the spectral representation of an isotropic turbulent flow. The approximation yields Kolmogorov's inertial‐range spectrum and is shown to be the local limit of a class of approximations suggested by Kraichnan and Spiegel. A spectrum in the viscous dissipation range is computed with the diffusion approximation and compared to spectra obtained from approximations of Heisenberg and Kovasznay as well as a modification of Obukhov's approximation and a recent prediction of Kraichnan. It agrees closely with the last two. Similarity spectra and associated longitudinal correlation functions are computed for decaying turbulence at infinite Reynolds number. A comparison is made with corresponding results based on the Heisenberg approximation.

Nonlinear Theory of Drift‐Wave Turbulence and Enhanced Diffusion

Abstract: The turbulent plasma state which develops from unstable, current‐driven drift waves is analyzed. In the nonlinear theory, the wave growth predicted by the linear theory is ultimately suppressed by ion damping. Since the phase speeds of the unstable waves are much greater than the ion thermal velocity, the ions cannot absorb wave energy until they become trapped. The amplitude of the turbulent spectrum grows until trapping occurs, and a quasi‐steady state is reached in which the directed electron energy is converted into ion thermal energy at a constant rate. In this state the perturbation of the density gradient due to the turbulence is equal to the mean gradient. Nonlinear limitation due to mode coupling does not suppress the unstable wave growth until much larger density perturbations have developed. Therefore, ion trapping is established as the controlling nonlinear mechanism. In the steady state, the ions diffuse across the magnetic field with a diffusion coefficient D ⊥ ≈ (k ⊥ −2γ)max, where γ is the growth rate predicted by the linear theory and k ⊥ is a perpendicular wavenumber. Although the detailed treatment is for a specific instability, the general conclusions appear to apply to a variety of drift waves.

Turbulence Transport Equations

Abstract: A generalized eddy viscosity function σ, is introduced in order that the Reynolds stress in an incompressible fluid be expressible as a linear combination of the Kronecker and rate‐of‐strain tensors. A transport equation for the eddy viscosity is derived from the general turbulence energy equation, thereby introducing two additional functions, the specific turbulence kinetic energy q, and a scale variable s. To determine the three variables, a transport equation for s is postulated, and a modified Prandtl—Wieghardt relation among the three variables is accepted. The theory is expressed in universal, invariant form, and validity is demonstrated by application to several problems.

Coupled-cluster calculations of nuclear magnetic resonance chemical shifts

Abstract: Theory and implementation of the gauge‐including atomic orbital (GIAO) ansatz for the gauge‐invariant calculation of nuclear magnetic resonance chemical shifts are described for the coupled‐cluster singles and doubles (CCSD) approach. Results for the shielding constants of the hydrides HF, H2O, NH3, and CH4 as well as for a few multiply bonded systems such as CO, N2, and HCN demonstrate the importance of higher‐order correlation corrections, as good agreement with experiment is only obtained at the CCSD level and to some extent at partial fourth‐order many‐body perturbation theory [SDQ‐MBPT(4)] with the latter slightly overestimating correlation effects due to single and double excitations. For relative chemical shifts, GIAO‐CCSD calculations provide in difficult cases (e.g., CO and CF4) more accurate results than previous GIAO‐MBPT(2) calculations. But, it seems that it is often more important to include rovibrational effects (as well as possible molecule–solvent interactions) than higher‐order correlation corrections. Despite that, GIAO‐CCSD proves to be a powerful tool for the accurate calculation of NMR chemical shifts. Its capabilities as well as its limitations are demonstrated in shielding calculations for formaldehyde, diazomethane, and ozone. At least for the latter, the description provided by the CCSD ansatz is not sufficient and even higher excitations need to be considered.

Transport Coefficients of Ionized Gases

Abstract: The repulsive and attractive screened Coulomb potentials may be used to represent interactions among charged particles in a gas. The classical Chapman‐Enskog collision integrals are calculated for these potentials over a wide range of reduced temperatures, equivalent to a wide range of electron densities and temperatures. At high temperatures the repulsive and attractive collision integrals are equal and agree with the earlier calculations of Kihara and Liboff. At lower temperatures the attractive collision integrals grow larger than the repulsive collision integrals as the temperature falls, and both grow apart from the previous results. It is shown that quantum effects are large in high‐density plasmas at all temperatures and in low‐density plasmas at very high temperatures.

Approximately extensive modifications of the multireference configuration interaction method: A theoretical and practical analysis

Abstract: The extensivity error of configuration interaction (CI) is well understood and unlinked diagram corrections must be applied to get reliable results. Besides the well known a posteriori Davidson‐type corrections, several methods attempt to modify the CI equations a priori to obtain nearly extensive results, while retaining the convenience of working in a configuration space. Such unlinked diagram corrections are particularly important for multireference cases for which coupled‐cluster (CC) calculations, which require a many‐body, integral‐based calculation, are more difficult. Several such multireference methods have been presented recently, ranging from the multireference linearized coupled cluster method (MR‐LCCM), averaged coupled pair functional (MR‐ACPF), through various quasidegenerate variational perturbation theory (QD‐VPT), MR‐coupled electron pair method (MR‐CEPA) to size‐consistent, self‐consistent, selected CI [(SC)2SCI]. We analyze all of these methods theoretically and numerically, paying par...

Quasi‐One‐Dimensional, Nonequilibrium Gas Dynamics of Partially Ionized Two‐Temperature Argon

Abstract: A set of equations governing the quasi‐one‐dimensional flow of a nonequilibrium argon plasma is formulated. The kinetic model includes ionization by atom‐atom and electron‐atom impacts, together with the three‐body recombination counterparts of these processes. The electron gas temperature—a key variable in controlling ionization reaction rates in a noble gas—is not necessarily equal to that of the heavy particles and must in general be found by simultaneous solution of a differential electron energy equation together with the usual conservation equations. Numerical solutions are obtained for the special case of the relaxation zone behind strong normal shocks which give good agreement with shock tube measured ionization times. For these normal‐shock calculations, a local steady‐state approximation to the electron energy equation is found to be useful in view of an insensitivity, demonstrated by the numerical results, to arbitrarily selected initial values of post‐translational shock electron temperature.

Diffusion, Scattering, and Acceleration of Particles by Stochastic Electromagnetic Fields

Abstract: The behavior of charged particles in an electromagnetic field composed of two parts is treated. The first is an unperturbed part, for which the solution of the Vlasov equation is assumed to be known; the second part is a perturbation which is assumed to be random and of small amplitude. Such a ``stochastic'' or ``turbulent'' electromagnetic field leads to diffusion, pitch‐angle scattering, and acceleration. Equations governing these processes may be derived by test‐particle calculations. The problem is first posed in a general way, and solved by a method similar to that used in quasi‐linear theory. The connection of this method with the Fokker‐Planck formulation is discussed. The case which is analyzed in detail is that of relativistic particles moving in a uniform magnetic field under the influence of a steady homogeneous spectrum of electromagnetic fluctuations. Results are given which are valid for any (weak) fluctuation spectrum and for finite test‐particle gyroradii. The physical meaning of the results is discussed, and it is shown that in certain limits there is agreement with earlier, less general analyses.

An extended basis set ab initio study of alkali metal cation-water clusters

Abstract: Ionic clusters comprised of a single alkali metal cation and up to eight water molecules were studied at the Hartree–Fock and correlated levels of theory using the correlation consistent sequence of basis sets. Estimates of the degree of convergence in the computed properties with respect to the complete basis set limit were facilitated by the underlying systematic manner in which the correlation consistent sets approach completeness. In favorable cases, improved property values could be obtained by fitting finite basis set results with a simple analytical expression in order to extrapolate to the complete basis set limit. The sensitivity of structures and binding energies were analyzed with regard to the inclusion of valence and core‐valence correlation recovery at the MP2, MP4, and CCSD(T) levels of theory. The replacement of metal core electrons and the introduction of relativistic contributions via effective core potentials was compared to corresponding all‐electron results.

Boundary Layer over a Thin Needle

Abstract: An equation governing the motion of an incompressible fluid flowing axially over a thin paraboloid of revolution is derived. The asymptotic behaviors and an approximate solution are discussed, and numerical solutions computed. From the numerical solutions, an extrapolation towards vanishing boundary thickness is made. It is found that for progressively thin needles, the displacement thickness and drag per unit length diminish very slowly, but eventually become zero as the needle vanishes.

Note on Decay of Homogeneous Turbulence

Abstract: The assumption of self‐similarity and the existence of an exact invariant are combined to predict the decay rate of homogeneous turbulence.

Anomalous skin effect in a plasma.

Abstract: The theory of the skin effect in a plasma is developed for the case in which the relation between current and electric field is not local.

Traveling‐Wave Bulk Electroconvection Induced across a Temperature Gradient

Abstract: If a temperature gradient is imposed on a slightly conducting liquid, a gradient in natural electrical conductivity generally results. It is shown that if the liquid is then subjected to a wave of electric field traveling perpendicular to the temperature and conductivity gradients, charges are induced in the liquid bulk. These charges relax to form a traveling wave which interacts with the imposed field to pump the liquid. The sign of the conductivity gradient determines whether the liquid is pumped in the same direction or a direction opposite to that of the traveling wave. Equations are given for the velocity profile in plane flow, showing the effect of fluid properties as well as of the frequency, wavelength, and potential of the traveling wave. Experiments support the significance of the theory. Observations of a type of bulk Rayleigh‐Taylor instability are discussed.

Unified Shock Profile in a Plasma

Abstract: Previously [P.N.Hu, Phys. Fluids 9, 89 (1966)], a universal monotonic profile was found for a weak plasma shock propagating perpendicular to a magnetic field using an elaborate set of moment equations which included electron and ion viscosity, coupled electron and ion heat conduction, thermal diffusion, and resistivity, together with distinct Hall terms for the resistivity, stress, and heat flow. Inclusion of an additional term related to electron inertia now gives a one‐parameter family of profiles which spans the entire range from collision dominated monotonic profiles to damped oscillatory profiles and the limiting collisionless case of purely oscillatory and solitary wave profiles.

Dynamical Properties of Truncated Wiener‐Hermite Expansions

Abstract: The use of truncated Wiener‐Hermite functional expansions as a basis for the theory of turbulence is critically examined. An account is given of the application of such expansions to Burgers' model equation. The nature of Wiener‐Hermite expansions is such that certain important consistency requirements, such as realizability, follow immediately. In order to gain insight into other aspects of the predicted turbulence dynamics, a simpler model problem in which there are only three interacting modes is first studied. It is shown that Wiener‐Hermite closures do not faithfully represent the dynamics for this latter model. Then, the analysis is extended to Burgers' equation. It is shown that Wiener‐Hermite closures do not preserve fundamental properties of the exact dynamics in equilibrium. Inviscid equipartition solutions do not survive in the closures. In addition, it is shown that these closures do not treat the effects of large scales on small scales properly. Numerical calculations be extended to the corresponding theory of Navier‐Stokes turbulence. We conclude that truncated Wiener‐Hermite expansions are unsuitable for the theory of high Reynolds‐number turbulence.

Temperature and Density Measurements in Free Jets and Shock Waves

Abstract: The fluorescence stimulated by a 17.5‐kV electron beam probe was used to obtain measurements of rotational temperature and gas density in supersonic nitrogen jets expanding from room temperature. The parameter P0d (where P0 is the stagnation pressure in Torricelli and d is the orifice exit diameter in millimeters) was varied from 15 to 480 Torr‐mm. Density measurements were made using an interference filter‐photomultiplier combination. The experimental density data follow the axial isentropic density distribution in regions of rotational nonequilibrium. Rotational temperature measurements were obtained from rotational spectra of the 0–0 band of the N2+ first negative system. The experimentally determined rotational temperatures, which initially follow the axial isentropic temperature distribution in a free jet, depart from the isentropic curve at low temperatures and freeze at a constant value, indicating a substantial degree of nonequilibrium in the expansion. A shock holder was inserted in the jet and a...