Justification of the harmonic balance method for modeling non-stationary processes in the cooled stage of a gas turbine engine

Аuthors

Popova D. K.^*, Kortikov N. N.

Peter the Great St. Petersburg Polytechnic University, 29, Polytechnicheskaya str., St. Petersburg, 195251, Russia

*e-mail: daria_well96@mail.ru

Abstract

The harmonic balance method (HB) is a promising approach to the numerical solution of non-stationary problems in turbomachinery. The essence of the HB method is to transform the initial time-dependent problem into a system of algebraic equations for the Fourier coefficients. This approach significantly reduces computational costs while maintaining a high level of accuracy for periodic processes. This is especially relevant when modeling steady-state oscillatory modes, such as interaction bet-ween rows of blades or pressure pulsations in cooled elements. In this work, the main focus is on the application of the HB method, implemented in the STAR-CCM+ software package, for the analysis of gas flow in turbine stages.
The non-stationary nature of gas flow in turbomachines is caused by a number of factors, such as subsonic potential interaction, interaction shock waves and wake, as well as temperature unevenness at the exit of the combustion chamber. The circumferential unevenness of the temperature field at the exit from the combustion chamber is one of the most critical factors affecting the thermal load of the turbine elements. This unevenness leads to the fact that that hotter layers of gas are shifted to the surface of the trough of the blade, and cooler layers are shifted to the surface of the back of the blade. This phenomenon has been called “temperature separation”. This redistribution of temperature layers over time creates a stable temperature difference between the bottom and the top of the blade. The presence of such significant temperature gradients leads to local thermal overloads. These overloads can cause thermal defects in the structure, such as cracks.
Despite the widespread use of the harmonic balance method in aerodynamics and acoustics modeling, a review of the literature has shown that its effectiveness in cooling turbine stages under real-world operating conditions remains insufficiently validated. There is a lack of reliable data demonstrating the efficiency and accuracy of this approach in the presence of complex boundary conditions, heat transfer, and strong non-steady disturbances. This study aims to show that there are opportunities for a deeper integration of the HB method into the design of turbine apparatus and their cooling systems.

Keywords:

rotor-stator interaction, combustion chamber, temperature separation, turbine stage, harmonic balance method

References

1. Avgustinovich VG, Shmotin YuN et al. Numerical simulation of unsteady phenomena in gas turbine engines. Moscow: Mashinostroenie; 2005. 536 p. (In Russ.).
2. Sipatov AM. Solving multidisciplinary problems of gas dynamics in the design of aircraft engines. Ekaterinburg: RAN. UrO. 2010. 317 p. (In Russ.).
3. He L. Fourier methods for turbomachinery applications. Progress in Aerospace Sciences. 2010;46(8): 329–341. DOI: 10.1016/j.paerosci.2010.04.001
4. Grigor'ev AV, Yakunin AI, Kuznetsov NB et al. Calculation of non–stationary rotor-stator interaction in the turbine stage by harmonic balance method. Nauchno-tekh-nicheskie vedomosti. SPbGPU. 2013;(1 (166)):183–191. (In Russ.).
5. Hall KC, Thomas JP, Clark WS. Computation of Unsteady Nonlinear Flow sin Cascades Using a Harmonic Balance Technique. AIAA JOURNAL. 2002;40(5): 876–886.
6. Vilmin S, Lorrain E, Hirsch C. The nonlinear harmonic method for rotor-stator interactions applied to thermally perfect gas. 8th International symposium on experimental and computational aerothermodynamics of internal flows. ISAIF8-0066.  (Lyon, 2007). 7 p. 
7. Hall KC, Thomas JP, Ekici K et al. Frequency Domain Techniques for Complex and Nonlinear Flows in Turbomachinery. American Institute of Aeronautics and Ast-ronautics Paper. 2003. 12 p.
8. Junge L, Frey C, Ashcroft G et al. Simulation of indexing and clocking with a new multidimensional time harmonic balance approach. Int. J. Turbomach. Propuls. Power. 2024;9:20.
9. Frey C, Ashcroft G, Kersken H-P et al. Simulation of indexing and clocking with harmonic balance. Int. J. Turbomach. Propuls. Power. 2017;3(1):12. 
10. User Guide STAR-CCM+ 2000.3. CD-adapco. 2020. 10998 p.
11. Gomar A, Bouvy Q, Sicot F et al. Convergence of Fourier - based time methods for turbomachinery wake passing. Journal of computational physics. 2014;(278): 229–256.
12. Tucker PG. Trends in turbomachinery turbulence treatments. Progress in Aerospace Sciences. 2013;63:1–32. DOI: 10.1016/j.paerosci.2013.06.001
13. Marpu RP, Custer CH, Subramanian V et al. Comparison of numerical methods for the prediction of time ave-raged flow quantitiesin cooled multistage turbine. ASME Turbo Expo 2015: Turbine Technical Conference and Exposition GT 2015. (2015, Montreal, Canada). 11 p.
14. Popova DK, Kortikov NN.  Modeling of non-stationary processes in the cooled turbine stage based on the harmonic balance method. XXV SCHOOL-SEMINAR of young scientists and specialists of academician A.I. Leontiev “Problems of gas dynamics and heat and mass transfer”. Collection of materials. (Rybinsk. June 9-13, 2025.) P. 246–247. (In Russ.).
15. Tsereteli AA, El'shin AA. Determination of the temperature of the working fluid in the turbine using a monocrystalline maximum temperature meter. Aviatsionnye dvigateli. 2019;2(3):31–38. (In Russ.).
16. Chernova TA. The influence of non-stationary phenomena on temperature stresses and the life of cooled turbine blades. PhD. thesis. Perm: 2006. 160 p. (In Russ.).
17. Popova DK, Kortikov NN.  Simulation of unsteady heat transfer and temperature separation on the working blades of a gas turbine when changing the position of the combustion chamber injectors. Zhurnal tekhnicheskoi fiziki. 2024;94(10):1639–1645. (In Russ.).
18. Pyatunin KR, Luginina NS, Didenko RA. Testing and adaptation of new approaches to flow modeling in a non-stationary formulation for problems of aerodynamics. Trudy MAI. 2013;(65):21. (In Russ.).
19. Boldyrev YuYa, Rubtsov AO, Kozhukhov YuV et al.  Simulation of unsteady processes in turbomachines based on the nonlinear harmonic NLH method using supercomputers. Supercomputing Days in Russia 2015. Russian Supercomputing Days 2015.  P. 273–279. (In Russ.).
20. Aslanov AR, Kraev VM, Molchanov AM. A model for calculating transients in cryogenic fuel lines of modern aircraft engines. Thermal processes in engineering.  2023;15(4):185–192. (In Russ.).
21. Ivanov IE, Kryukov IA, Larina EV. Effect of relaxation time of turbulent viscosity on modeling of flows in nozzles and jets. Izv. RAN. MZhG. 2014;(5):149–159. (In Russ.).