| 110 | Before the spectra are produced - the data is mapped onto a fixed grid with a resolution equal to the highest level of data. So I looked at the initial velocity field by processing the first frame of the smooth case - but I modified the highest level of data to go from the base grid (level 0) to level 4 - and I used a cube window that was 40x40x40. |
| 111 | |
| 112 | The next figure shows the resulting spectra when only considering the level 0 data, then the level 1 data, and so on out to level 4. (There are 5 levels of data but I need to transfer the data files to a larger cluster to have enough memory to handle the FFT's). Also shown are the Nyquist frequency cutoffs which correspond to the highest wavenumber mode that can be resolved by level's 0, 1, 2, 3, & 4. This wavelength will actually be 2*the cell size (which will correspond with the next coarser level resolution). The spectra continue passed this point but quickly drop off because of incomplete coverage of those wave numbers. Imagine taking a cube and sampling spherical shells... Once the diameter of the shell is larger than the cube width, you will get contributions from only the corners of the cube and not uniformly throughout. So everything on the red line to the right of the first vertical line should be ignored. |
| 113 | |
| 114 | It is good to see that the spectra agree quite well at resolved wave lengths and that with each additional level of data, you extend the spectra to higher wave numbers. Also the rise at the end of the spectra (as well as the smaller one to the left) coincide with the maximum resolvable wavelength which coincides with the cell size of the next coarser grid. Regions that have gradients but are not resolved to the finest level will have a stair-step signal, where each step has 2 points (or 4, or 8 points). This will lead to power with wavelengths corresponding to the nyquist frequency, or twice, four times, etc... On the first frame - the only place where there are gradients that are not completely refined - are at the edges of the colliding flow - where there is a good amount of shear. This is verified below where the initial spike at the nyquist frequency is dominated by solenoidal terms and not compressive terms. |
| 115 | |
| 116 | [[Image(PowerSpectraSmooth0.png, width=400)]] |
| 117 | |
| 118 | |
| 119 | |
| 120 | |
| 121 | ---- |
| 122 | |