Basic idea of the Harmonic Balance approach

The Harmonic Balance solver in TRACE [12] solves the unsteady RANS equations in the frequency domain. In contrast to the linearised solver (linearTRACE) it is not based on the assumption on small amplitudes. Instead of the unsteady RANS equations,

\begin{equation} \frac{\partial \mathbf q}{\partial t} + \mathbf R(\mathbf q ) = 0 \end{equation}

we solve for the Fourier harmonics of the flow

\begin{equation} \mathbf q(t,\mathbf x) = \Re \left[ \sum_{\omega} \hat {\mathbf q}_\omega (x) e^{\cI \omega t} \right], \end{equation}

for a small set of frequencies \(f = \frac \omega {2\pi}\), specified by the user.

For each angular frequency \(\omega\), the Harmonic Balance equation thus reads

\begin{equation} \cI \omega \hat {\mathbf q}_\omega + \widehat {\mathbf R(\mathbf q)}_\omega = 0. \end{equation}

Here, \(\widehat {\mathbf R(\mathbf q)}_\omega\) denotes the discrete Fourier transform of the nonlinear residual evaluated at an appropriate set of sampling points.

Harmonic Sets

The harmonics are characterised not only by their frequency but by the pair of frequency and interblade phase angle. A harmonic set is given by a base frequency \(f\) and base interblade phase angle, and a finite number of harmonic indices \(k \ge 0\), e.g.

\begin{equation} f = 1000 \mathrm{Hz}, \quad \sigma = 90^\circ,\quad \mathcal{K} = \{0,1,2,3\}, \end{equation}

for a simulation of a periodic disturbance with three higher harmonics.

Sampling Points

Denoting the highest harmonic index of a harmonic set by \(k_{\max}\), TRACE uses

\begin{equation} N = \alpha k_{\max} + 1 \end{equation}

sampling points. By the Nyquist theorem \(\alpha \ge 2\). Note that a period of the highest harmonic is resolved by

\begin{equation} \alpha + 1 \end{equation}

sampling points. In TRACE, this number can be specified by the user. Hence, the lower bound defined by the Nyquist criterion is used if the user specifies a minimum of 3 sampling points per wave length. The default is 5.

Applications

The Harmonic Balance approach may be used for various applications:

Aeroelasticity, i.e. flutter and forced-response analysis. [1]
Tonal noise problems in aeroacoustics.
Deterministic unsteady effects on aerodynamic performance.
Transport of rotational flow asymmetries through an engine component.
Etc.

Solution technique

The Harmonic Balance system is solved analogously to the steady equations by a pseudo-time stepping approach.

Boundary conditions and periodic boundaries

All solid wall boundary conditions as well as leakage inlet/outlet conditions can be used. At periodic boundaries the phase-lag boundary condition is automatically applied.

At inlets 1D and 2D exact non-reflecting boundary conditions should be used. Modal gust disturbances can be prescribed for all harmonics in the frequency and wave number domain, similar to the capabilities of linearTRACE .

Moreover, Riemann boundary conditions can be used for the \(0\)-th harmonic, including the possibility to prescribe a 2D distribution of entry or exit boundary values, e.g. total quantities and flow angles at the inlet.