Space-time LU-SGS Implicit Time Spectral Method For Unsteady Flow Calculations

Unsteady flow problems are studied widely in theoretical investigation and engineering practice.It is of great importance to develop efficient numerical methods for unsteady flow calculations.The conventional dual-time stepping scheme using the 2^nd order backward difference formula method and explicit pseudo-time marching scheme is usually inefficient due to both the time accuracy and stability requirements.In this dissertation,the time spectral method is employed for time discretization and the implicit pseudo-time stepping scheme is used to solve the time-space coupled equations resulting from time spectral methods.Based on these improvements,an efficient numerical method that can solve periodic as well as general non-periodic unsteady flow problems is developed.The most significant work and the major innovations are as follows:?1?A fast Chebyshev time spectral method is proposed for general non-periodic unsteady flow problems.The existing Fourier time spectral method can only be applied to periodic flow problems.The Chebyshev time spectral method is developed on the basis of approximating flow variables with Chebyshev polynomials.Computational results of a model equation show that the time accuracy of the Chebyshev time spectral method is higher than that of the conventional 2^nd order backward difference fomula method.To achieve solution of equal time accuracy,less time instants are needed in the Chebyshev time spectral method.Hence the computational efficiency is improved.Results of unsteady flow calculations confirm that the Chebyshev time spectral method can be used to efficiently solve general non-periodic unsteady flow problems if the accurate initial flow field is given.?2?An efficient space-time LU-SGS implicit pseudo-time marching scheme is proposed to solve the discretized equations resulting from time spectral methods.The time domain is regarded as another dimension of space when the symmetric Gauss-Seidel sweeps are implemented.Due to the strong implicit temporal coupling,fast convergence is often achieved.Since matrix inversion is avoided,the memory requirement and the CPU time consumption at each iteration are both low.Computational results show that the total CPU time to reach convergence can be greatly saved if the explicit Runge-Kutta scheme is replaced by the space-time LU-SGS implicit scheme.When the periodic flow problems with high frequency or the non-periodic flow problems with small time interval of interest are solved,modifications to enhance stability of the space-time LU-SGS implicit scheme are proposed and verified.?3?A combined frequency search approach is proposed to broaden the search range.In this approach,the frequency is updated using the results of Fourier analysis to the lift coefficient before the gradient based method is applied.Compuational results show that initial values of frequency that are far away from the correct value could be used and the correct value of frequency can be efficiently obtained,such as for the vortex shedding flow befind a stationary circular cylinder.?4?The efficiency of the Fourier time spectral method in solving transonic flows and the shock-free subsonic cases is assessed systematically by analyzing the average error of the computed surface pressure coefficient.The influence of the shock waves on the time accuracy and computational efficiency of the Fourier time spectral method is studied.The results of error analysis show that the Fourier time spectral method is much more efficient than the 2^nd order backward difference folumla method for the shock-free subsonic cases.When shock wave occurs,sufficient Fourier harmonics must be included before the Fourier time spectral method exhibits efficiency advantage.?5?The non-symmetric solution due to the use of odd numbers of intervals in a period is uncovered and studied when the Fourier time spectral method is applied to a symmetric periodic flow problem.Analysis revealed that this phenomenon is not unique to the Fourier time spectral method.The symmetry of numerical solutions is determined by the distribution of the independent instants in a period.To guarantee symmetric solutions,the requirement on the distribution of the independent instants is proposed.The influence of the time resolution and shock waves on the non-symmetric soltions is discussed as well.