Modeling Two-Point Spatial Statistics in Turbulent Channel Flow

Two-point spatial statistics 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 to have stochastically-forced source terms representing the nonlinear terms in the full dynamical equations for ▽²v and ω_y. Using eigenfunction expansions we derive the two-point correlations for v , ω_y , and their cross-correlation explicitly in terms of their corresponding nonlinear source correlations. These expressions contain the known eigenfunctions, the corresponding eigenvalues, and the 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 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 a smaller magnitude of the real part (slower decaying modes) are ranked higher. Once the nonlinear source statistics are determined the remaining two-point statistics involving u, w, and uv may be computed.