Statistical properties of large ensembles of exact solutions to Burgers' equation with random initial conditions

The study of Burgers' equation is important to the understanding of the Navier-Stokes equation, as Burgers' equation is so often used in preliminary investigations of turbulence (i.e., statistical closures). In this work, the difficulties in implementing exact solutions to Burgers' equation have been overcome, allowing the study of the statistical properties of Burgers' equation using the exact solution for piecewise linear initial velocity fields. The statistical and spectral properties commonly used in studies of turbulence are presented for three different types of initial conditions. It is found that the energy spectrum quickly evolves to a self-similar functional form which can be simply characterized. A full-spectrum nonlinear similarity transformation based on this characterization is developed. It is found that the probability distribution of the velocity field approaches a bell-shaped distribution, but that the higher moments of this distribution do not rapidly approach those of a Gaussian distribution