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Non-Equispaced Fast Fourier Transforms in Turbulence Simulation

Abstract
Fourier pseudo-spectral method on equispaced grid is one of the approaches in turbulence simulation, to compute derivative of discrete data, using fast Fourier Transform (FFT) and gives low dispersion and dissipation errors. In many turbulent flows the dynamically important scales of motion are concentrated in certain regions which requires a coarser grid for higher accuracy. A coarser grid in other regions minimizes the memory requirement. This requires the use of Non-equispaced Fast Fourier Transform (NFFT) to compute the Fourier transform, by solving a system of linear equations. To achieve similar accuracy, the NFFT needs to return more Fourier coefficients than the number of non-equispaced gridpoints, making it an under-determined system. The minimum L2 norm solution of the system is refined using an iterative reconstruction algorithm, FOCUSS. The NFFT and FOCUSS algorithms yield accurate results with smaller test case of a Direct Numerical Simulation on a grid of 64 gridpoints in each dimension, using Taylor green initial condition. The computational speed for this case was found to be unacceptably slow and few methods to improve the performance have been discussed. The approach of NFFT and FOCUSS was tested on a line extracted from 3-dimensional turbulent flow field. Fourier transform of the extracted line, sampled on 1024 non-equispaced gridpoints, computed for 2048 coefficients and the corresponding numerical derivative are found to be inaccurate. It can be observed that the NFFT and FOCUSS approach works for sparse Fourier transform, but not for turbulent fields having a wideband Fourier transform.
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