Skip to main content

Harmonic Balance Method for Turbomachinery Applications

  • Chapter
  • First Online:
OpenFOAM®
  • 4745 Accesses

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

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

AB:

Wave amplitudes

\(\phi \) :

Phase shift, s

S :

Sine part

C :

Cosine part

i :

Harmonic index

\(t_j\) :

Discrete time instant

References

  1. Cvijetic, G., Jasak, H., and Vukcevic, V., “Finite Volume Implementation of the Harmonic Balance Method for Periodic Non–Linear Flows,” 54th AIAA Aerospace Sciences Meeting, 2016, p. 0070.

    Google Scholar 

  2. He, L., “Method of simulating unsteady turbomachinery flows with multiple perturbations,” AIAA Journal, Vol. 30, 11 1992, pp. 2730–2735.

    Article  Google Scholar 

  3. He, L. and Ning, W., “Efficient approach for analysis of unsteady viscous flows in turbomachines,” AIAA Journal, Vol. 36, No. 11, 1998, pp. 2005–2012.

    Article  Google Scholar 

  4. Thomas, J., Custer, C., Dowell, E., and Hall, K., “Unsteady flow computation using a harmonic balance approach implemented about the OVERFLOW 2 flow solver,” 19th AIAA Computational Fluid Dynamics Conference, 2009.

    Google Scholar 

  5. Dufour, G., Sicot, F., Puigt, G., Liauzun, C., and Dugeai, A., “Contrasting the Harmonic Balance and Linearized Methods for Oscillating-Flap Simulations,” AIAA Journal, Vol. 48, No. 4, 2010, pp. 788–797.

    Article  Google Scholar 

  6. Ekici, K. and Hall, K. C., “Harmonic Balance Analysis of Limit Cycle Oscillations in Turbomachinery,” AIAA Journal, Vol. 49, No. 7, 2011, pp. 1478–1487.

    Article  Google Scholar 

  7. Hall, K., Thomas, J., Ekici, K., and Voytovych, D., “Frequency domain techniques for complex and nonlinear flows in turbomachinery,” Vol. 3998, 2003, p. 2003.

    Google Scholar 

  8. Gopinath, A., Van Der Weide, E., Alonso, J., Jameson, A., Ekici, K., and Hall, K., “Three-dimensional unsteady multi-stage turbomachinery simulations using the harmonic balance technique,” 45th AIAA Aerospace Sciences Meeting and Exhibit, Vol. 892, 2007.

    Google Scholar 

  9. Guédeney, T., Gomar, A., and Sicot, F., “Multi-frequential harmonic balance approach for the computation of unsteadiness in multi-stage turbomachines,” AFM, Maison de la Mécanique, 39/41 rue Louis Blanc, 92400 Courbevoie, France (FR), 2013.

    Google Scholar 

  10. Guédeney, T., Gomar, A., Gallard, F., Sicot, F., Dufour, G., and Puigt, G., “Non-uniform time sampling for multiple-frequency harmonic balance computations,” Journal of Computational Physics, Vol. 236, 2013, pp. 317–345.

    Article  Google Scholar 

  11. He, L., “Fourier methods for turbomachinery applications,” Progress in Aerospace Sciences, Vol. 46, No. 8, 2010, pp. 329–341.

    Article  Google Scholar 

  12. Huang, H. and Ekici, K., “Stabilization of High-Dimensional Harmonic Balance Solvers Using Time Spectral Viscosity,” AIAA Journal, Vol. 52, No. 8, 2014, pp. 1784–1794.

    Article  Google Scholar 

  13. Hall, K. C., Thomas, J. P., and Clark, W. S., “Computation of unsteady nonlinear flows in cascades using a harmonic balance technique,” AIAA Journal, Vol. 40, No. 5, 2002, pp. 879–886.

    Article  Google Scholar 

  14. Thomas, J. P., Custer, C. H., Dowell, E. H., Hall, K. C., and Corre, C., “Compact Implementation Strategy for a Harmonic Balance Method Within Implicit Flow Solvers,” AIAA Journal, Vol. 51, No. 6, 2013, pp. 1374–1381.

    Article  Google Scholar 

  15. Sicot, F., Puigt, G., and Montagnac, M., “Block-Jacobi Implicit Algorithms for the Time Spectral Method,” AIAA Journal, Vol. 46, No. 12, 2008, pp. 3080–3089.

    Article  Google Scholar 

  16. Woodgate, M. A. and Badcock, K. J., “Implicit Harmonic Balance Solver for Transonic Flow with Forced Motions,” AIAA Journal, Vol. 47, No. 4, 2009, pp. 893–901.

    Article  Google Scholar 

  17. Su, X. R. and Yuan, X., “Implicit solution of time spectral method for periodic unsteady flows,” International Journal for Numerical Methods in Fluids, Vol. 63, No. 7, 2010, pp. 860–876.

    MathSciNet  MATH  Google Scholar 

  18. Antheaume, S. and Corre, C., “Implicit Time Spectral Method for Periodic Incompressible Flows,” AIAA Journal, Vol. 49, No. 4, 2011, pp. 791–805.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Gregor Cvijetić .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

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

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-60846-4_17

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-60845-7

  • Online ISBN: 978-3-319-60846-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics