EDQNM MODEL OF A PASSIVE SCALAR WITH A UNIFORM MEAN GRADIENT

Dynamic equations for the scalar autocorrelation and scalar-velocity c ross correlation spectra have been derived for a passive scalar with a uniform mean gradient using the Eddy Damped Quasi Normal Markovian (E DQNM) theory. The presence of a mean gradient in the scalar field make s all correlations involving the scalar axisymmetric with respect to t he axis pointing in the direction of the mean gradient. Equivalently, all scalar spectra will be functions of the wave number k and the cosi ne of the azimuthal angle designated as mu. In spite of this complicat ion, it is shown that the cross correlation vector can be completely c haracterized by a single scalar function Q(k). The scalar autocorrelat ion spectrum, in contrast, has an unknown dependence on mu. However, t his dependency can be expressed as an infinite sum of Legendre polynom ials of mu, as first suggested by Herring [Phys. Fluids 17, 859 (1974) ]. Furthermore, since the scalar field is initially zero, terms beyond the second order of the Legendre expansion are shown to be exactly ze ro. The energy, scalar autocorrelation, and scalar-velocity cross corr elation were solved numerically from the EDQNM equations and compared to results from direct numerical simulations. The results show that th e EDQNM theory is effective in describing single-point and spectral st atistics of a passive scalar in the presence of a mean gradient. (C) 1 996 American Institute of Physics.