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Harmonic Balance Method for Turbomachinery Applications

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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.

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\(\mathscr {Q}\) :

Dimensionless passive scalar in the time domain

\(\mathscr {R}\) :

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

t :

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

\(\underline{\underline{A}}\) :

Discrete Fourier expansion matrix

\(\underline{Q}\) :

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

\(\underline{R}\) :

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 \(\mathscr {R}\)

T :

Base period, s

\(\underline{\underline{E}}\) :

Forward DFT matrix

\(\underline{\underline{E}}^{-1}\) :

Backward (inverse) DFT matrix

\(P_{i - j}\) :

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

\(P_l\) :

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

\(\nu \) :

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

\(\rho \) :

Density, kg/m\(^3\)

p :

Pressure, Pa

f :

Base frequency, Hz


Wave amplitudes

\(\phi \) :

Phase shift, s

S :

Sine part

C :

Cosine part

i :

Harmonic index

\(t_j\) :

Discrete time instant


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Correspondence to Gregor Cvijetić .

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Cvijetić, G., Jasak, H. (2019). Harmonic Balance Method for Turbomachinery Applications. In: Nóbrega, J., Jasak, H. (eds) OpenFOAM® . Springer, Cham.

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  • Print ISBN: 978-3-319-60845-7

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