Nonlinear Wave Expansion for Turbulence in the Burgers' Model of a Fluid

Abstract

A new scheme of expansion of the turbulent velocity field is proposed and applied to the Burgers' one‐dimensional model turbulence. The velocity field of the Burgers' turbulence is expanded into a series of nonlinear waves of sawtooth profile, each of which is an asymptotic solution of the Burgers' equation for extremely large Reynolds numbers R . The effect of the interaction between waves of different wavenumbers k is estimated approximately, and the possible form of the energy distribution of nonlinear waves G(k∥ is determined. The energy spectrum E(k) , the velocity correlation, and the energy decay law are calculated numerically adopting a particular distribution of G(k) . The spectrum has the asymptotic tail E(k) ∝k −2 for k → ∞ reflecting the discontinuous structure of the turbulent field at infinite R . The energy decays in time inversely proportional to the square of the time. Theoretical results are compared with results due to the numerical experiment of Jeng and others, and the agreement appears to be excellent.