Abstract
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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Abbreviations
- \(\mathscr {Q}\) :
-
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
- A, B:
-
Wave amplitudes
- \(\phi \) :
-
Phase shift, s
- S :
-
Sine part
- C :
-
Cosine part
- i :
-
Harmonic index
- \(t_j\) :
-
Discrete time instant
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Cvijetić, G., Jasak, H. (2019). Harmonic Balance Method for Turbomachinery Applications. In: Nóbrega, J., Jasak, H. (eds) OpenFOAM® . Springer, Cham. https://doi.org/10.1007/978-3-319-60846-4_17
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DOI: https://doi.org/10.1007/978-3-319-60846-4_17
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