### abstract

- Isotropic and anisotropic spectra of passive scalar fluctuations in a turbulent fluid flow with a power law \ensuremath{\propto}${\mathit{k}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\beta}}}$ spectrum are analyzed. The isotropic spectra occur in flows with zero mean external gradient of passive scalar concentration and passive scalar fluctuations can be caused by an external source. On the other hand, in the presence of nonzero mean external gradient of concentration, passive scalar fluctuations are anisotropic and can be excited by ``tangling'' of the mean external gradient of the passive scalar by turbulent fluid flow. The analysis is based on the renormalization procedure in the spirit of Moffatt [J. Fluid Mech. 106, 27 (1981); Rep. Prog. Phys. 46, 621 (1983)]. It is shown that the anisotropic ${\mathit{k}}^{\mathrm{\ensuremath{-}}3}$ spectrum of passive scalar fluctuations is universal, i.e., independent of exponent \ensuremath{\beta} in a turbulent velocity spectrum. In the particular case of the Kolmogorov spectrum (\ensuremath{\beta}=5/3) of turbulent velocity field the derived general spectra recover the known spectra of passive scalar fluctuations \ensuremath{\propto}${\mathit{k}}^{\mathrm{\ensuremath{-}}5/3}$ and \ensuremath{\propto}${\mathit{k}}^{\mathrm{\ensuremath{-}}17/3}$. In addition, the ultimate Prandtl number for large Reynolds numbers is estimated (${\mathrm{Pr}}^{\mathrm{lim}}$\ensuremath{\approxeq}0.792) and is found to be in fairly good agreement with experimental results. \textcopyright{} 1996 The American Physical Society.