Derivation of Kolmogorov's Four-Fifth Law-2 — Lesson 4

This lesson covers the concept of structure functions in turbulence, focusing on second and third order correlation functions. It explains how to derive the velocity from the Navier-Strokes equation using the time derivative of the second order correlation function. The lesson also discusses the connection between the second order correlation and the non-linear transfer term. It further explores the assumptions made in the inertia range and how to relate the dissipation rate with the energy term non-linear transfer rate. The lesson concludes with a discussion on higher order structure functions and their relation to the structural theory.

Video Highlights

00:38 - Discussion on the time derivative of the second order correlation function
06:46 - Discussion on the connection between the second order correlation and the non-linear transfer term
12:26 - Discussion on the higher order structure function and its relation to the structural theory
21:59 - Explanation of the log normal distribution
23:45 - Discussion on the deviation from the linear line in the experimental data

Key Takeaways

- The velocity in turbulence can be derived from the Navier-Strokes equation using the time derivative of the second order correlation function.
- The second order correlation is connected with the non-linear transfer term.
- In the inertia range, certain assumptions are made, such as setting the time derivative to 0 and assuming the forcing at large scale.
- The dissipation rate can be connected with the energy term non-linear transfer rate.
- Higher order structure functions can be related to the structural theory, but no close form formula can be derived from the first principle for q not equal to 3.