KARHUNEN-LOEVE EXPANSION OF THE DERIVATIVE OF AN INHOMOGENEOUS PROCESS

The properties of the Karhunen-Loeve (KL) expansion of the derivative u(x)(x) of an inhomogeneous random process possessing viscous boundary -layer behavior are studied in relation to questions of efficient repr esentation for numerical Galerkan schemes for computational simulation of turbulence. Eigenfunctions and eigenvalue spectra are calculated f or the randomly forced one-dimensional Burgers' model of turbulence. C onvergence of the expansion of u(x) is much slower than convergence of the expansion of u(x), and direct expansion of u(x) is not significan tly more efficient than differentiating the expansion of u. The ordere d eigenvalue spectrum of u(x) is proportional to the square of the ord er parameter times the eigenvalue spectrum of u. The underlying cause of slow convergence is the earlier onset of locally sinusoidal behavio r of the KL eigenfunctions when the expansion is performed over the en tire domain of the solution.