OpenFOAM® pp 223-233 | Cite as

Harmonic Balance Method for Turbomachinery Applications

  • Gregor CvijetićEmail author
  • Hrvoje Jasak


The Harmonic Balance Method for nonlinear periodic flows is presented in this paper. Assuming a temporally periodic flow, a Fourier transformation is deployed in order to formulate a transient problem as a multiple quasi-steady-state problem. A solution of the obtained equations yields flow fields at discrete instants of time throughout a representative harmonic period, while still capturing the transient effect. The method is implemented in foam-extend, a community-driven fork of OpenFOAM\(^{\textregistered }\) and developed for multi-frequential use in turbomachinery applications. For validation, a 2D turbomachinery test case is used. Pump head, efficiency, and torque obtained with Harmonic Balance will be compared to a transient and steady-state simulation. Furthermore, pressure contours on rotor blades will be compared. And finally, in order to present the method’s efficiency along with its accuracy, a CPU time comparison will also be presented.


\(\mathscr {Q}\)

Dimensionless passive scalar in the time domain

\(\mathscr {R}\)

Convection–diffusion transport operator for a passive scalar in the time domain


Time, s

\({\mathbf {u}}\)

Velocity field, m/s

\(\gamma \)

Diffusion coefficient, m\(^2\)/s

\(S_{\mathscr {Q}}\)

Source terms for a passive scalar, 1/s

\(\omega \)

Base radian frequency, rad/s


Discrete Fourier expansion matrix


Vector of Fourier harmonics for \(\mathscr {Q}\)


Vector of Fourier harmonics for \(\mathscr {R}\)

\(\underline{\mathscr {Q}}\)

Vector of discrete time instant values for \(\mathscr {Q}\)

\(\underline{\mathscr {R}}\)

Vector of discrete time instant values for \(\mathscr {R}\)


Base period, s


Forward DFT matrix


Backward (inverse) DFT matrix

\(P_{i - j}\)

Coupling coefficient for \(t_i\) and \(t_j\) time instants


Coupling coefficient equivalent to \(P_{i - j}\)

\(\nu \)

Kinematic viscosity, m\(^2\)/s

\(\rho \)

Density, kg/m\(^3\)


Pressure, Pa


Base frequency, Hz


Wave amplitudes

\(\phi \)

Phase shift, s



Sine part


Cosine part


Harmonic index


Discrete time instant


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of EnergyPower Engineering and EnvironmentZagrebCroatia

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