Numerical study of the return of axisymmetric turbulence to isotropy

Abstract

The spectral method of Orszag & Patterson (1972a, b) is used here to study pressure and velocity fluctuations in axisymmetric, homogeneous, incompressible, decaying turbulence at Reynolds numbers Reλ [lsim ] 40. In real space 323 points are treated. The return to isotropy is simulated for several different sets of anisotropic Gaussian initial conditions. All contributions to the spectral energy balance for the different velocity components are shown as a function of time and wavenumber. The return to isotropy is effected by the pressure-strain correlation. The rate of return is larger at high than at low wavenumbers. The inertial energy transfer tends to create anisotropy at high wavenumbers. This explains the overrelaxation found by Herring (1974). The pressure and the inertial energy transfer are zero initially as the triple correlations are zero for the Gaussian initial values. The two transfer terms are independent of each other but vary with the same characteristic time scale. The pressure-strain correlation becomes small for extremely large anisotropies. This can be explained kinematically. Rotta's (1951) model is approximately valid if the anisotropy is small and if the time scale of the mean flow is much larger than 0·2 Lf/v, which is the time scale of the triple correlations (Lf = integral length scale, v = root-mean-square velocity). The value of Rotta's constant is less dependent upon the Reynolds number if the pressure-strain correlation is scaled by v3/Lf rather than by the dissipation. Lumley & Khajeh-Nouri's (1974) model can be used to account for the influence of large anisotropies. The effect of strain is studied by splitting the total flow field into large- and fine-scale motion. The empirical model of Naot, Shavit & Wolfshtein (1970) has been confirmed in this respect.