The approach of axisymmetric, homogeneous turbulence to isotropy using the direct interaction approximation is investigated. The turbulence is characterized by two parameters, the spectral variance transverse and parallel to the direction of axisymmetry. The dependence of these energy components on wave vector orientation is developed into a spherical harmonic expansion, and only low order terms are examined in detail. In terms of this characterization of the theory, the general qualitative nature of the relaxation to isotropy is discussed and numerical results for the energy spectrum and transfer functions are presented. It is shown that the simplest characterization of the theory leads to an almost linear relaxation to isotropy. The numerical results at moderate Reynolds numbers are compared to the phenomenological theory of Rotta [J. C. Rotta, Z. Physik 129, 547 (1951)]. A simple analytic estimate of the Rotta relaxation rate is also presented.

The character of transition from laminar to chaotic Rayleigh–Bénard convection in a fluid layer bounded by free-slip walls is studied numerically in two and three space dimensions. While the behaviour of finite-mode, limited-spatial-resolution dynamical systems may indicate the existence of two-dimensional chaotic solutions, we find that, this chaos is a product of inadequate spatial resolution. It is shown that as the order of a finite-mode model increases from three (the Lorenz model) to the full Boussinesq system, the degree of chaos increases irregularly at first and then abruptly decreases; no strong chaos is observed with sufficiently high resolution.In high-Prandtl-number σ two-dimensional Boussinesq convection, it is found that there are finite critical Rayleigh numbers Ra for the onset of single- and two-frequency oscillatory motion, Ra [gsim ] 60 Rac and Ra [gsim ] 290 Rac respectively, for σ = 6.8. These critical Rayleigh numbers are much higher than those at which three-dimensional convection achieves multifrequency oscillatory states. However, in two dimensions no additional complicating fluctuations are found, and the system seems to revert to periodic, single-frequency convection at high Rayleigh number, e.g. when Ra [gsim ] 800Rac at σ = 6.8.In three dimensions with σ = 10 and aspect ratio 1/√2, single-frequency convection begins at Ra ≈ 40Rac and two-frequency convection starts at Ra ≈ 50Rac. The onset of chaos seems coincident with the appearance of a third discrete frequency when Ra [gsim ] 65Rac. This three-dimensional transition process may be consistent with the scenario of Ruelle, Takens & Newhouse (1978).As Ra increases through the chaotic regime, various measures of chaos show an increasing degree of small-scale structure, horizontal mixing and other characteristics of thermal turbulence. While the three-dimensional energy in these flows is still quite small, it is evidently sufficient to overcome the strong dynamical constraints imposed by two dimensions.Gollub & Benson (1980) found experimentally that frequency modulation of lower boundary temperature Ra(t) = Ra(0) [1 + ε sin ωt] induces chaotic behaviour in a quasi-periodic flow close to transition. We investigate numerically the effects of finite modulation of Ra on the flow far below natural transition (R = 50Rac). By choosing ε = 0.1 and the Rayleigh-number oscillation frequency ω incommensurate with the frequencies of the quasi-periodic motion, transition to chaos is induced early. This result also seems consistent with the Ruelle et al. scenario and leads to the conjecture that periodic modulation of the Rayleigh number of the above form in a two-frequency flow may provide the third frequency necessary for chaotic flow.For moderate Prandtl number, σ = 1, our results show that two-dimensional flow seems free of oscillation, while three-dimensional flow is vigorously turbulent for Ra [gsim ] 70Rac.

The decay of two-dimensional, homogeneous, isotropic, incompressible turbulence is investigated both by means of numerical simulation (in spectral as well as in grid-point form), and theoretically by use of the direct-interaction approximation and the test-field model. The calculations cover the range of Reynolds numbers 50 ≤ RL ≤ 100. Comparison of spectral methods with finite-difference methods shows that one of the former with a given resolution is equivalent in accuracy to one of the latter with twice the resolution. The numerical simulations at the larger Reynolds numbers suggest that earlier reported simulations cannot be used in testing inertial-range theories. However, the large-scale features of the flow field appear to be remarkably independent of Reynolds number.The direct-interaction approximation is in satisfactory agreement with simulations in the energy-containing range, but grossly underestimates enstrophy transfer at high wavenumbers. The latter failing is traced to an inability to distinguish between convection and intrinsic distortion of small parcels of fluid. The test-field model on the other hand appears to be in excellent agreement with simulations at all wavenumbers, and for all Reynolds numbers investigated.

We examine results of direct numerical simulations (DNS) of homogeneous turbulence in the presence of stable stratification. We focus on the effects of stratification on eddy diffusion, and the distribution of pairs of particles released in the flow. DNS results are presented over a range of stratification, and at Reynolds numbers compatible with aliased free spectral results for a resolution of 128 mesh points. We compare results for particle dispersion to simple analytic theories such as that proposed by Csanady (1964) and Pearson et al. (1983) by adapting the basic Langevin model to decaying turbulence at low Reynolds numbers. Stable stratification is found to arrest both single particle displacements and pair separation in the direction of stratification, but it leaves these quantities nearly unaltered in the transverse direction. With respect to the dynamics of stratified flows, we find that regions of strong viscous dissipation are intermittently spaced, and are associated with large horizontal vorticity, consistent with recent experimental results by Fincham et al. (1994).