## 4.4. Harmonic Analysis

A solution to transient periodic flow can be obtained faster using a frequency-based solution method. The ANSYS CFX Harmonic Analysis (HA) is based on the Harmonic Balance (HB) / Time Spectral Method (TSM) (see 228, 229) implemented within a pressure-based solution approach. In this method, the transient flow variation is represented by a Fourier series for a prescribed fundamental frequency, . Typically, harmonics (or modes) are retained in the Fourier series for this approximation. The unsteady period is equally divided into time levels on which the time derivative is evaluated using the spectral approximation, and the system of nonlinear equations coupling all time levels are solved iteratively. For example, if one harmonic is retained () then 3 time levels () are required. If 3 harmonics are retained () then 5 time levels () are required. A pseudo-time marching approach similar to the steady-state solution method is used to solve the time level coupled equations.

In general, complex transient flows containing sharp discontinuities may require a larger number of harmonics to be retained in order to resolve the flow features, while simpler smooth flows require a smaller number of harmonics. Typical flow problems are solved using equal to 1, 3, or 5. The smaller the number of retained harmonics, the more efficient the Harmonic Analysis (HA) method is with respect to the time-marching solution. Therefore, it is essential to use the smallest number of harmonics at which the solution accuracy is acceptable. The efficiency of the solution calculation is also affected by the size of the specified pseudo-time step. Typically, the size of the pseudo-time step is determined as the period of the unsteadiness divided by the intended number of pseudo-time steps per period. A large number of pseudo-time steps per period (this is, a small time step size) makes the solution converge slowly and inefficiently with respect to a time-marching solution. A lower number of pseudo-time steps per period (that is, a large time step size) advances faster towards the converged solution and is more attractive, but may cause numerical stability issues. It has been found that 15 to 30 pseudo-time steps per period typically results in a good balance between calculation efficiency and solution stability.

For details on how HA relates to a transient run with time marching, see Transient versus Harmonic Solution Method in the CFX Reference Guide.

A use case involving HA is described in Blade Flutter using Harmonic Analysis in the CFX-Solver Modeling Guide.

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