Demonstration of Nonlinear Frequency Domain Methods

Abstract

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

An Extension of the Time-Spectral Method to Overset Solvers

“…!N , such that u = with jk = k (t j ) = e i!ktj (4) serving as the transformation operator. The di↵erentiation operator is applied in the frequency domain, i.e.…”

“…Expanding the temporal variation at each spatial node into a Fourier series transforms the unsteady governing equations into a coupled set of steady equations in integer harmonics that can be tackled with the acceleration techniques a↵orded to steady-state flow solvers. Other similar approaches, such as the Nonlinear Frequency Domain [3,4,5], Reduced Frequency [6], and Time-Spectral [7,8,9] methods, were developed shortly thereafter. Additionally, adjoint-based optimization techniques [10,11] can be applied to provide the ability to perform design optimization without resorting to costly unsteady adjoint methods.…”

Time Spectral Method for Rotorcraft Flow

“…Helicopter flows in forward flight, turbomachinery blades and wind turbine are constantly subjected to unsteady loads. For this class of problems, Gopinath & Jameson 24 have shown that the Time Spectral method is superior in terms of the accuracy and computational efficiency to the typical implicit dual time stepping scheme or the hybrid scheme proposed by Hsu & Jameson. 25 The Time Spectral method takes advantage of the periodic nature of the flow and is simpler than the typical nonlinear frequency domain type solver proposed by Hall et al 26 and McMullen et al [27][28][29] as it does not require the operations of Fourier transforms and inverse Fourier transforms.…”

Computing Stability Derivatives and Their Gradients for Aerodynamic Shape Optimization

“…In an extension of this work, Ekici and Hall [43] applied the technique, known as the harmonic balance technique, to multistage turbomachinery applications where a variety of frequencies may be present. Other spectral methods have been demonstrated by McMullen et al [44,45] (the nonlinear frequency domain (NLFD) method) and Gopinath and Jameson [46] (the time-spectral method). The main differences between the methods are the portions of the solution that are computed in the frequency and time domains.…”