Kinetic Energy in Fourier Space — Lesson 3

This lesson covers the third part of Fourier Space, focusing on Kinetic Energy. It explains the concept of velocity field and how kinetic energy is constructed from it. The lesson further delves into the concept of Fourier space energy of a Fourier mode, also known as modal kinetic energy. It discusses the calculation of total kinetic energy and introduces Parseval’s theorem. The lesson also explains how energy changes with time for every k mode and the concept of energy supply rate. It further discusses the role of viscosity in energy loss and the concept of shell spectrum. The lesson concludes with the explanation of energy injection rate and dissipation rate.

Video Highlights

00:37 - Discussion on Fourier space energy of a Fourier mode
02:52 - Explanation of the concept of complex conjugate and equation for kinetic energy
06:48 - Derivation of the non-linear term in the equation for kinetic energy
11:45 - Discussion on the concept of shell spectrum and its importance in Kolmogorov theory
13:52 - Discussion on the role of pressure in energy transfer

Key Takeaways

- Kinetic energy is constructed from the velocity field in Fourier Space.
- The Fourier space energy of a Fourier mode is referred to as modal kinetic energy.
- The total kinetic energy is calculated by summing over all ks.
- Parseval’s theorem establishes an important relation used in simulations.
- Energy changes with time for every k mode.
- Energy supply rate and the role of viscosity in energy loss are key concepts in understanding energy dynamics.
- Shell spectrum is a concept used to simplify the understanding of energy in isotropic systems.
- Energy injection rate and dissipation rate are crucial in understanding energy interactions in fluid flows.