Similarity of decaying isotropic turbulence with a passive scalar

Abstract

Direct numerical simulations have been carried out for decaying homogeneous isotropic turbulence in a periodic box. Data for both the velocity and passive scalar fields are considered, the latter for several values of the Schmidt number $\hbox{\it Sc}$. The focus is on how the three-dimensional spectra $E(k,t)$ and $E_{\theta} (k,t)$ and the spectral transfer functions $T(k,t)$ and $T_{\theta } (k,t)$ satisfy similarity during decay. The evolution of these four quantities provides qualified support for the equilibrium similarity proposal of George (1992a, b). In particular, this proposal provides a reliable means of calculating the transfer functions, starting with known distributions of $E(k,t)$ and $E_{\theta} (k,t)$. However, at sufficiently large values of the wavenumber $k$, normalizations by Kolmogorov and Batchelor variables yield a better collapse of these quantities than the use of equilibrium similarity The distributions of $E_{\theta} (k,t)$ and $T_{\theta} (k,t)$ do not depend on $\hbox{\it Sc}$, when the latter is in the range $0.7 \,{\leqslant}\, \hbox{\it Sc} \,{\leqslant}\, 7$, irrespective of the normalization adopted. The velocity derivative skewness and mixed velocity–scalar derivative skewness approach constant values as $t$ increases. This is in disagreement with equilibrium similarity but in accord with the observed high-wavenumber collapse of Kolmogorov and Batchelor normalized distributions of $E(k,t)$ and $E_{\theta} (k,t)$.