Efficient fourier-based algorithms for time-periodic unsteady problems

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. iii Preface This dissertation work proposes two algorithms for the simulation of time-periodic unsteady problems via the solution of Unsteady Reynolds-Averaged Navier-Stokes (URANS) equations. These algorithms use a Fourier representation in time and hence solve for the periodic state directly without resolving transients (which consume most of the resources in a time-accurate scheme). In contrast to conventional Fourier-based techniques which solve the governing equations in frequency space, the new algorithms perform all the calculations in the time domain, and hence require minimal modifications to an existing solver. The complete space-time solution is obtained by iterating in a fifth pseudo-time dimension. Various time-periodic problems such as helicopter rotors, wind turbines, turbomachinery and flapping-wings can be simulated using the Time Spectral method. The algorithm is first validated using pitching airfoil/wing test cases. The method is further extended to turbomachinery problems, and computational results verified by comparison with a time-accurate calculation. The technique can be very memory intensive for large problems, since the solution is computed (and hence stored) simultaneously at all time levels. Often, the blade counts of a turbomachine are rescaled such that a periodic fraction of the annulus can be solved. This approximation enables the solution to obtained at a fraction of the cost of a full-scale time-accurate solution. For a viscous computation over a three-dimensional single-stage rescaled compressor, an order of magnitude savings is achieved. The second algorithm, the reduced-order Harmonic Balance method is applicable only to turbomachinery flows, and offers even larger computational savings than the Time Spectral method. It simulates the true geometry of the turbomachine using only one blade passage per blade row as the computational domain. In each blade row of the turboma-chine, only the dominant frequencies are resolved, namely, combinations of neighbor's blade passing. An appropriate set of frequencies can be chosen by the analyst/designer based on a trade-off between …