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This paper demonstrates the accuracy of the nonlinear frequency domain method in applications to unsteady flow calculations. The basis of the method is a pseudospectral approach to recast a nonlinear unsteady system of equations in the temporal domain into a stationary system in the frequency domain. The nonlinear frequency domain method, in principle, provides the rapid convergence of a spectral method with increasing numbers of modes, and, in this sense, it is an optimal scheme for time-periodic problems. In practice it can also be effectively used as a reduced order method in which users deliberately choose not to resolve temporal modes in the solution. A variable-time-period method has been proposed such that the nonlinear frequency domain method can be applied to problems where the time period of the unsteadiness is either known or unknown a priori. To validate the latter case, results from this method have been compared with experimental results of vortex shedding in low Reynolds number flows past cylinders. Validation of the first case utilizes experimental data of a pitching airfoil in transonic flow. These comparisons demonstrate the efficiency of the nonlinear frequency domain method in representing complex nonlinear flow field physics with a limited number of temporal modes. = spatial dimension y = nondimensionalized distance from wall = ratio of specific heats ij = Kronecker delta = heat transfer coefficient = absolute viscosity = density = shear stress tensor = pseudotime

This paper presents a technique used to accelerate the convergence of unsteady calculations of time-periodic flows to a periodic steady state. The basis of the procedure is the use of the discrete Fourier transform in time, and is similar to the harmonic balance procedure that has been pursued by Hall et. al. The technique is amenable to parallel processing, and convergence acceleration techniques such as multi-grid and implicit residual averaging. The computational efficiency of this method is compared with dual time stepping algorithms. Sample calculations are provided, and a comparison between solutions with varying temporal resolution is presented. The results show that the computational efficiency of the harmonic balance technique is largely a function of the temporal resolution. Initial experiments confirm the promise of the harmonic balance method to achieve significant reductions in computational cost. IntroductionThe calculation of periodic unsteady flows in turbomachinery continues to present a severe challenge to Computational Fluid Dynamics (CFD). The unsteadiness stems mainly from the relative motion of the rotating blade fields, and has a fundamental period which depends on the rate of rotation and the number of blade passages. Currently, a popular approach to the computation of this problem is to introduce a fully implicit A-stable time discretization of the flow equations requiring the solution of a set of coupled nonlinear equations at each physical time step. These equations are typically solved with an explicit inner iteration by introducing a pseudo-time variable and marching to a steady state. Converged solutions attained at the end of the inner iterations represent a solution of the implicit equations at the end of the physical time step.1-4 A variety of techniques such as variable local pseudo-time steps, implicit residual averaging and multigrid are used to accelerate the convergence of the inner iterations. The efficiency of this dual time stepping method depends on the effectiveness of these acceleration techniques in restricting the number of inner iterations. If the physical time step can be increased in comparison to an explicit scheme by a factor larger than the number of inner iterations, the implicit scheme will be more efficient than the explicit scheme. In practice it has been found that for simple geometries and flows, on the order of 24-36 implicit time steps are sufficient to resolve the features * Graduate Student, Student Member AIAA † T.V. of one period in the oscillation of the flow. Typically one restricts the number of inner iterations to the order of 30-80. At a minimum, capturing one period requires 1000 inner iterations. As the number of periods needed to obtain convergence increases the calculation can become prohibitively expensive. This is particularly true in multistage turbomachinery flows where convergence to a periodic steady state can be slow since it requires the physical propagation of disturbances back and forth between the front and the back of the turbom...

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This paper presents an adjoint method for the optimum shape design of unsteady flows. The goal is to develop a set of discrete unsteady adjoint equations and the corresponding boundary condition for the non-linear frequency domain method. First, this paper presents the complete formulation of the time dependent optimal design problem. Second, we present the non-linear frequency domain adjoint equations for three-dimensional flows. Third, we present results that demonstrate the application of the theory to a three-dimensional wing.