Axisymmetric homogeneous turbulence: a comparison of direct spectral simulations with the direct-interaction approximation

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Abstract
The return to isotropy of homogeneous axisymmetric turbulence at Reλ ≈ 28 is studied by means of the direct spectral simulation (DSS) technique of Orszag & Patterson (1972) and the direct-interaction approximation (DIA) of Kraichnan (1959) as implemented by Herring (1974). The results of the two methods are compared for different initial degrees of anisotropy. The general agreement between the methods is good. Most of the discrepancies can be attributed to present technical limitations in implementing both schemes. The DSS has been found to be superior for strong anisotropies, because the numerical method used for solving the DIA equations is limited in its angular resolution. For small anisotropies the angular anisotropy becomes less important and the DIA results are accurate; in this case the DIA seems to be superior as it is free from the statistical uncertainties inherent in the DSS method. With respect to a return-to-isotropy study these statistical errors are large, in particular for small anisotropies. The physical interpretation of the angular energy distribution is discussed also. The numerical results and theoretical considerations for the DIA equations show that one should retain angular moments at least up to the fourth in order to obtain accurate values of the Rotta constant at moderate anisotropies.