Modeling the Reynolds Stress Spectra in Turbulent Channel Flow

The Reynolds stress spectra in turbulent channel flow are studied by numerical simulations combined with spectral analyses of the Orr Sommerfeld/Squire (OS/SQ) system of equations. The simulations vary in Re_τ from 180 to 5200. The OS/SQ equations are assumed to have stochastically-forced source terms representing the nonlinear terms in the full dynamical equations for ?² and ω_y. Their solutions are written in terms of eigenfunction expansions leading to expressions for the v and ω_y spectra. These expressions contain the known eigenfunctions, the corresponding eigenvalues, and the yet unknown statistics of the forcing term for each mode with possible cross correlations between modes. The modeling proceeds by attempting to represent the known simulation results for the spectra by an optimal choice of the covariance matrix of the eigenmode forcing amplitudes. The modes chosen for the expansion are ranked in importance by the real part of the eigenvalues so that those with smaller real parts (slower decaying modes) are ranked higher. Once the v and ω_y spectra and their crossspectrum are determined the remaining Reynolds stresses may be computed. We investigate in particular the contributions to the stresses associated with highly elongated streamwise structures.