A STUDY OF BURGERS MODEL EQUATION WITH QPPLICATION TO THE STATISTICAL THEORY OF TURBULENCE

摘要

This study investigates the statistical properties of some approximate solutions to the following equation and compares them with the statistical properties of real turbulence.nu(x, 0) specifiednIn one case u(x, 0) is assumed to be a Gaussian random function and the statistical behavior of u(x, t) is calculated to several orders in time. It is found that u(x, t) remains normally distributed to the order of the terms calculated although the joint distributions immediately deviate from joint normality. The skewness and flatness factors of derivatives of u(x, t) arc in qualitative agreement with experi¬mental results of real turbulence.nAnother case uses a closed form solution for u(x, t) found by J. D. Cole. A steepest-descents type of approximation leads to a simple expression for u(x, t). Certain restrictions are imposed on u(x, 0) to facilitate calculations. It is found that the probability distributions of u(x, t) become normal after long times. In particular, the flatness factor / decays rapidly from large initial values to a value of 2.9, then rises slightly above the value 3 (appropriate for a normal dis¬tribution), and finally falls to the value 3.nA generalization to the case of three dimensions behaves similarly. It is also found that the energy decreases according to the -5/2 power of the time, as in ex¬perimental turbulence.