Calculate Spectra in Irregular Geometry - Method Derivation

00:00 The Helmholtz Decomposition
01:19 Boundary Conditions for the Helmholtz potentials
09:22 Statement of the boundary value problems for the Helmholtz potentials
11:25 Relating the Helmholtz potentials to kinetic energy
16:20 Eliminating the cross-product term from the total kinetic energy
20:25 Discussion of spectra and projections onto known basis functions
23:33 Motivating new basis functions from appropriate Sturm-Liouville problems
26:36 Writing the Helmholtz potentials as the superposition of Dirichlet and Neumann modes
28:24 Attributing each mode to specific length scales through the eigenvalues
30:30 Expressing kinetic energy in terms of the Dirichlet and Neumann modes
32:03 Showing orthogonality and relating the kinetic energy to the spectral coefficients
43:01 Discussing Parseval's equality and the kinetic energy density
45:33 Using Green's functions to resolve impacts of inhomogeneous boundary conditions
53:08 A clever trick - writing Green's functions in terms of the eigenmodes
56:44 The spectral coefficients for the divergent part of the velocity field
1:00:58 Calculating the spectral coefficients for the rotational part of the velocity field
1:03:35 Summary of our theory for spectra in irregular geometry
1:11:32 What we will do in the next video

Friends don't let friends use Fourier Series. If you're working with complex geometry and you need to calculate spectra of a velocity field, you can window your data, convolve it with a smooth compact support function, and then use Fourier analysis. Or, you can work with the velocity field data as is! In this video, I derive the mathematical framework for calculating spectra using a Helmholtz decomposition and basis functions generated from suitable Sturm-Liouville problems.

Background music :
Fluidscape by Kevin MacLeod is licensed under a Creative Commons Attribution 4.0 license. creativecommons.org/licenses/by/4.0/

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