Split-Domain Harmonic Balance Solutions to Burger's Equation for Large-Amplitude Disturbances

Abstract

A new split-domain harmonic balance approach is presented. The split-domain approach combines the conventional multidomain harmonic balance approach with a split-operator technique in a unique way to solve periodic unsteady ow problems ef ciently. The new technique is applied to Burger's equation to obtain solutions for two large-amplitude periodic boundary conditions-a single-frequency sine wave and a simulated wake function. Solutions containing strong moving discontinuities are obtained with Fourier series containing up to 48 frequencies for various grid densities. The split-domain harmonic balance solutions are compared with conventional time-accurate solutions. The differences between the two are found to be asymptotic with respect to the number of Fourier frequencies included. In addition, the harmonic balance approach was found to be sensitive to grid density. Nomenclaturea = Fourier cosine coef cient Q a = amplitude of unsteady input disturbance b = Fourier sine coef cient Q F = discrete Fourier transform operator O F = harmonic balance ux term vector f = frequency of unsteady input disturbance (1/s) O S = harmonic balance source term vector U = vector of time-sampled dependent variables O U = vector of Fourier coef cients u = scalar dependent variable in Burger's equation°= component of frequency domain ux vector 1¿ = pseudotime numerical integration step size ! = frequency of unsteady input disturbance (rad/s) Subscripts i = grid point n = Fourier frequency number t = differentiation with respect to time x = differentiation with respect to space ¿ = differentiation with respect to pseudotime Superscripts n = time level T = vector transpose ¡1 = operator inverse

Adaptive harmonic balance method for nonlinear time-periodic flows

“…(1) can be transformed into a split-domain harmonic balance form [12]. To simplify the discussion, this transformation is presented for a homogeneous scalar conservation equation of the form…”

“…Numerical stability considerations require a reduced time step when solving Eq. (13) for large N [9,12]. To improve the stability characteristics of the numerical solution, Eq.…”

“…Considerable computational savings can be gained by transforming Eq. (14a) to the time domain [8,10,12]. Given the vector of Fourier coefficientsn n 1 , and a discrete Fourier transform (DFT) operator, F, that produces positive-frequency Fourier coefficients from a set of real numbers, define n such that…”

“…To achieve shorter computation times for this class of problem, CFD techniques have been developed that take advantage of the time-periodic nature of the flow. These include the time-linearization technique [1][2][3][4], the time-averaged nonlinear harmonic technique [5][6][7], and the harmonic balance technique [8][9][10][11][12][13].…”

An Extension of the Time-Spectral Method to Overset Solvers

“…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. Frequency-adaptive methods [12,13,14] can o↵er even greater e ciency by refining the the number of modes resolved at each grid point to the frequency content in its solution.…”