Stability Theory in Fluid Dynamics — Lesson 1

This lesson covers the concept of stability theory in fluid dynamics, focusing on the transition from laminar to turbulent flows. It explains the significance of the critical Reynolds number, which indicates the point at which flow becomes turbulent. The lesson introduces the concept of perturbations in parallel flows and how these disturbances, if small, can be analyzed using linear stability analysis. It also discusses the use of Fourier analysis and the Squire's theorem to simplify the momentum equation into a linear, fourth-order ordinary differential equation known as the Orr-Sommerfeld equation. By solving this equation and finding the eigenvalues, one can determine the critical Reynolds number.

Video Highlights

01:35 - Discussion on the significane of the Reynolds number in determining turbulence.
03:19 - Discussion on the concept of small perturbation in linear stability analysis.
17:41 - Introduction to the concept of Fourier analysis or normal mode analysis.
28:13 - Explanation of the square theorem and its implications.
42:38 - Explanation of how to eliminate pressure between equations using the square transformation.
47:54 - Discussion on the Orr Sommerfeld equation, a linear and 4th order ordinary differential equation.

Key Takeaways

- The critical Reynolds number indicates the point at which flow becomes turbulent.
- Small perturbations in parallel flows can be analyzed using linear stability analysis.
- Fourier analysis and the Squire's theorem can be used to simplify the momentum equation into the Orr-Sommerfeld equation.
- Solving the Orr-Sommerfeld equation and finding the eigenvalues helps determine the critical Reynolds number.