Introduction to the Craya-Herring Basis in Fluid Dynamics — Lesson 1

This lesson covers the Craya-Herring method, a mathematical approach used in fluid dynamics. The lesson delves into the derivation of equations using the Craya-Herring method, explaining the concept of triads and their clockwise and anticlockwise movements. It also discusses the importance of the direction of vectors and the role of unit vectors in the method. The lesson further explores the computation of dot products of unit vectors and the significance of the e1 component. The lesson concludes with examples demonstrating the application of the Craya-Herring method in solving complex problems.

Video Highlights

00:31 - Explanation of the Craya-Herring Basis and its advantages in constructing equations in 2D and 3D.
06:23 - Discussion on the transformation of basis vectors in Fourier space and real space.
09:13 - Explanation of how to calculate the z component of u k in the system.
15:47 - Explanation of how to calculate cross helicity.
20:58 - Explanation of how to calculate pressure in the Craya-Herring basis.
29:50 - Explanation of how to calculate the components of the velocity vector in 2D.

Key Takeaways

- The Craya-Herring method is used to derive the equation of motion for a single triad.
- The concept of wavenumbers, convolution, cross product, and dot product are crucial in understanding the equation of motion.
- The equation of motion is not symmetric, and to make it symmetric, the concept of 'minus k prime' is introduced.
- The vectors k, p, q form a plane due to the condition of vector addition.
- The use of 2D physics simplifies the equation of motion.
- The angles in a triangle play a significant role in understanding the equation of motion.