Flow Equations in Fourier Space — Lesson 2

This lesson covers the concept of Flow Equations in Fourier Space, focusing on the incompressible Navie-Stokes. The lesson explains how to convert real space equations into Fourier space using the Fourier series. It discusses the basic idea of commuting the sum with d/dt and how the Fourier mode changes with time. The lesson also elaborates on the terms of the equation, including the non-linear term, pressure gradient term, force in Fourier space, and the viscous term. It further explains how to compute the pressure from the velocity field and the external force. The lesson concludes by highlighting the advantages of solving flow patterns at different scales using the spectral method or Fourier transform.

Video Highlights

00:30 - Conversion of the equation into Fourier space
02:42 - Explanation of the terms in the equation
05:34 - Explanation of the Fourier transform
11:38 - Discussion on the equation for pressure
13:14 - Explanation of the finite difference method
14:26 - Explanation of the tensorial equation and the incompressible term

Key Takeaways

- The Fourier series is used to convert real space equations into Fourier space.
- The Fourier mode changes with time.
- The equation's terms include the non-linear term, pressure gradient term, force in Fourier space, and the viscous term.
- Pressure is computed from the velocity field and the external force.
- The spectral method or Fourier transform allows for solving flow patterns at different scales, providing a multiscale feature.