A STUDY OF BURGERS' MODEL EQUATION WITH APPLICATION TO THE STATISTICAL THEORY OF TURBULENCE,
Abstract
Investigation is made of the statistical properties of some approximate solutions to Burger's equation for free turbulence where there is no mean motion and compares them with the statistical properties of real turbulence; u(x, O) specified. In one case u(x, O) 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) are in qualitative agreement with experimental results of real turbulence. Another case uses a closed form solution for u(x, t). A steepest-descents type of approximation leads to a simple expression for u(x, t). Certain restrictions are imposed on u(x, O) 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 distribution), and finally falls to the value 3. A 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 tim , as in experimental turbulence. (Author)
Document Details
- Document Type
- Technical Report
- Publication Date
- Dec 01, 1963
- Accession Number
- AD0426321
Entities
People
- David W. Moomaw
Organizations
- University of Michigan