Karhunen–Loéve expansion of Burgers’ model of turbulence

Abstract

Characteristics of the Karhunen–Loéve expansion of a strongly inhomogeneous random process possessing small viscous length scales and a large outer scale have been investigated in relation to the application of the expansion to turbulent flow fields. Monte Carlo simulations of a randomly forced Burgers’ equation with zero velocity boundary conditions generate the random process numerically and the Karhunen–Loéve (KL) eigenfunctions and the eigenvalue spectra are computed for different Reynolds numbers. The eigenfunctions possess thin viscous boundary layers at the walls and are independent of Reynolds number in the core, where the random process is quasihomogeneous. The eigenfunctions and eigenvalues of the outer, large scale motions obey a principle of Reynolds number similarity. Eigenvalue spectra contain much of the energy in the first few modes, but they are as broad as ordinary trigonometric power spectra. The rate at which the expansion converges to within 90% of the total energy decreases with increasing Reynolds numbers and the expansion of the mean plus the fluctuation converges more rapidly than the expansion of the fluctuation alone.