OpenFOAM® pp 223-233

# Harmonic Balance Method for Turbomachinery Applications

• Gregor Cvijetić
• Hrvoje Jasak
Chapter

## 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.

## Nomenclature

$$\mathscr {Q}$$

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

$$\omega$$

$$\underline{\underline{A}}$$

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

AB

Wave amplitudes

$$\phi$$

Phase shift, s

## Subscripts

S

Sine part

C

Cosine part

i

Harmonic index

$$t_j$$

Discrete time instant

## References

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