Mathematical Analysis of Turbulence I

Michael Jolly
Indiana University

This lecture is meant to provide an introduction to several classical turbulence theories.
It is aimed at a general audience of applied mathematicians. All terms below will be defined. The main elements are:

1. 3D statistical theory

We present the main features of Kolmogorov's theory for 3D fully developed turbulence
Certain hypotheses are recalled from which further laws are deduced without direct use of the equations of motion. After defining the 2nd order structure function and the energy spectrum, we show how the 2/3-law for the former is consistent with a 5/3 power in the latter. We then state the energy dissipation law and provide a heuristic derivation.

2. Using the 3D Navier-Stokes equations (NSE)

We present a sketch of Frisch's proof of Kolmogorov's 4/5-law for 3rd order structure functions. The Taylor length scale is introduced in order to make several hypotheses consistent.

3. Phenomenological theory

We recall Richardson's cascade mechanism for the transfer of energy to smaller length scales, and Obukov's derivation of the 5/3 spectrum.

4. Summary of 3D laws and connections

5. Functional form of the NSE, weak solutions, Reynolds number vs. Grashof number

6. Comparison with Kraichnan theory of 2D turbulence

7. Power law implications (if time permits)

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