Dispersion of particle pairs and decay of scalar fields in isotropic turbulence

Abstract

The decay of homogeneous scalar fields in isotropic turbulence is addressed by considering the dispersion of particle pairs. The evolution of the variance is studied by considering the dispersion backwards in time from the measurement time to the source time. The decay is shown to depend on the scalar field’s low wave number spectral exponent and a critical exponent of 8/3 is derived. This critical value separates cases where the dominant contribution to the variance at large time comes from the initial large scales from cases where the dominant contribution comes from the initial variance-containing scales. This confirms the results obtained for the particular case of the Kraichnan white noise model by Eyink and Xin and shows that the value of the critical exponent is not an artifact of the white noise model. The evolution of the scalar spectrum and correlation function has also been considered and results have been derived for the backtransfer of variance to low wave numbers, the permanence of large (scalar) eddies, the asymptotic convergence for different initial conditions and the self-similarity of the evolution. In particular it is shown that the form of the backtransfer of variance to low wave numbers previously found in models (i.e., the proportionality of the backtransfer to the fourth power of the wave number) is a direct consequence of the finite variance of particle separations and can be derived without modelling assumptions. Finally the particular case of the “mandoline” source geometry often used in wind tunnel studies is addressed.