Non-Equispaced Fast Fourier Transforms in Turbulence Simulation

NON-EQUISPACED FAST FOURIER TRANSFORMS IN TURBULENCE SIMULATION SEPTEMBER 2017 ADITYA MOHAN KULKARNI, B. E., UNIVERSITY OF PUNE M.S.M.E. UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Prof. Stephen de Bruyn Kops Fourier pseudo-spectral method is one of the approaches used to compute the derivative of a discrete data in Computational Fluid Dynamics. The Fourier transform of the data sampled on equispaced gridpoints is computed using Fast Fourier Transform (FFT), and the Fourier transform of derivative is obtained by multiplying each Fourier coefficient by its corresponding wavenumber and the imaginary number i = √ −1. In a number of turbulent flows like wakes, jets etc., the dynamically important scales of motion are concentrated in some regions, which require a finer gridspacing and other regions may have a coarse grid. Use of non-equispaced grid can potentially lead to reduced memory usage without sacrificing accuracy, which is particularly important as memory throughput is a major limiting factor of the Direct Numerical Simulations (DNS) performed today. The aim of this thesis is to implement the non-equispaced grid in DNS, using the Non-Equispaced Fast Fourier Transform (NFFT)[1] algorithm. In order to be able to achieve the similar accuracy using reduced number of gridpoints, the number of Fourier coefficients needs to be larger than that of the gridpoints. NFFT calculates the Fourier transform by solving a system of linear equations using a variant of conjugate gradient method, which in our case, becomes an under-determined system. A combination of NFFT and an iterative reconstruction algorithm, FOCUSS [2] is used to obtain the accurate Fourier transform of certain test functions, by solving an under-determined system of equations. The combination of NFFT and FOCUSS algorithm is also used to perform a small test case of Direct Numerical Simulation on a grid of 64 points, using Taylor Green initial conditions and the results are found to