B. G. Nickel

Small-scale structure of the Taylor–Green vortex

The dynamics of both the inviscid and viscous Taylor–Green (TG) three-dimensional vortex flows are investigated. This flow is perhaps the simplest system in which one can study the generation of small scales by three-dimensional vortex stretching and the resulting turbulence. The problem is studied by both direct spectral numerical solution of the Navier–Stokes equations (with up to 256 3 modes) and by power-series analysis in time. The inviscid dynamics are strongly influenced by symmetries which confine the flow to an impermeable box with stress-free boundaries. There is an early stage during which the flow is strongly anisotropic with well-organized (laminar) small-scale excitation in the form of vortex sheets located near the walls of this box. The flow is smooth but has complex-space singularities within a distance

Nonasymptotic critical behavior from field theory at d=3. II. The ordered-phase case

\hat{\delta}(t)

Addendum-erratum to: ``Nonasymptotic critical behavior from field theory at d=3. II. The ordered-phase case. Phys. Rev. B35, 3585 (1987)

of real (physical) space which give rise to an exponential tail in the energy spectrum. It is found that

Perturbation theory for a polymer chain with excluded volume interaction

\hat{\delta}(t)

Stiff chain model—functional integral approach

decreases exponentially in time to the limit of our resolution. Indirect evidence is presented that more violent vortex stretching takes place at later times, possibly leading to a real singularity (

Expansion of a polymer chain with excluded volume interaction

\hat{\delta}(t) = 0

On the singularity structure of the 2D Ising model susceptibility

) at a finite time. These direct integration results are consistent with new temporal power-series results that extend the Morf, Orszag & Frisch (1980) analysis from order t 44 to order t 80 . Still, convincing evidence for or against the existence of a real singularity will require even more sophisticated analysis. The viscous dynamics (decay) have been studied for Reynolds numbers R (based on an integral scale) up to 3000 and beyond the time t max at which the maximum energy dissipation is achieved. Early-time, high- R dynamics are essentially inviscid and laminar. The inviscidly formed vortex sheets are observed to roll up and are then subject to instabilities accompanied by reconnection processes which make the flow increasingly chaotic (turbulent) with extended high-vorticity patches appearing away from the impermeable walls. Near t max the small scales of the flow are nearly isotropic provided that R [gsim ] 1000. Various features characteristic of fully developed turbulence are observed near t max when R = 3000 and R λ = 110: a k − n inertial range in the energy spectrum is obtained with n ≈ 1.6–2.2 (in contrast with a much steeper spectrum at earlier times); th energy dissipation has considerable spatial intermittency; its spectrum has a k −1+μ inertial range with the codimension μ ≈ 0.3−0.7. Skewness and flatness results are also presented.

High-Temperature Series for Scalar-Field Lattice Models: Generation and Analysis

We present the first detailed calculations in the ordered phase using the massive φ4 field theory directly at d=3. It is shown that an adapted expansion allows the renormalization functions of the symmetric theory to be kept unchanged. Extending results in a previous paper [C. Bagnuls and C. Bervillier, Phys. Rev. B 32, 7209 (1985)] (noted I), we obtain, for Ising-type systems, nonasymptotic functions of temperature for the spontaneous magnetization, the susceptibility, and the specific heat along the critical isochore, which include all the quantitative universal characteristics of critical behavior in the real preasymptotic critical domain Dpreas. All universal leading- and first-correction amplitude combinations (including the new one RBcr-) are accurately estimated and are compared with previous theoretical and experimental estimates. We also show that the functions are well adapted to a suitable comparison with experiment and we describe how the adjustable parameters, limited to only three (the same as in I), enter in nonasymptotic critical behavior. Together with I, this work provides experimentalists with an efficient and coherent method which will facilitate the experimental test of the renormalization-group predictions.

Addendum to `On the singularity structure of the 2D Ising model susceptibility'

This note is intended to emphasize the existence of estimated Feynman integrals in three dimensions for the free energy of the O(1) scalar theory up to five loops which may be useful for other work. We also correct some misprints of the published paper.

Evaluation of simple Feynman graphs

We present a simple derivation of the mean square end‐to‐end distance 〈R2〉 of a linear flexible chain as a perturbation series in the dimensionless excluded volume parameter zd. Our results, to orders six and four in space dimension d=3 and 2, respectively, are 〈R2〉= Ll[1+ (4)/(3)  z3−2.075 385 396 z23+6.296 879 676 z33 GFIX−25.057 250 72 z43+116.134 785 z53 GFIX−594.716 63 z63+⋅⋅⋅],  d=3, 〈R2〉= Ll[1+ 1/2 z2−0.121 545 25 z22+0.026 631 36 z32 GFIX−0.132 236 03 z42+⋅⋅⋅],  d=2, where z3=(3/2πl)3/2 wL1/2 and z2=wL/πl with L the contour length of the chain, l the effective or Kuhn segment length, and wl2 the effective binary cluster integral for a pair of segments. Our method uses in an essential way Laplace transforms with respect to the contour length L; the resulting graphical expansion, when combined with the field theoretical methods, is far simpler than that in the conventional cluster expansion approach. Furthermore, we prove that 〈R2〉 is free of ln L terms to all orders in the perturbation theory in bo...