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Transient instabilities in turbomachinery

urbomachinery CFD simulations have become a standard in industry, although still presenting quite expensive and time consuming process. In order to alleviate this, a number of tools and methods have been developed, being an approximation or simplification of the ongoing process within turbomachinery. Some of these methods include the steady state Multiple Reference Frame approach (MRF), taking into account the rotation even though a steady state simulation is run. Another more recent method is the Harmonic Balance method, a quasi-steady state method due to a number of time instants being solved via coupled steady state equations. Finally, the approach with least approximations is the transient simulation where a large number of successive time steps are solved, thus obtaining the detailed insight into flow development, wakes propagation, etc. In order for time-accurate simulation to present valid results, a periodic steady state has to be reached: simulating a single period is not enough. It should be made sure that the simulation start instabilities do not affect the solution and that resulting flow features are no longer within the domain, therefore reaching periodic steady state is what makes transient simulation expensive. In certain cases this means simulating 5-10 periods, but if a high level of unsteadiness is present, the number of needed periods can even go up to 50. From the perspective of CPU time consumption, the transient simulation is the most expensive one, followed by Harmonic Balance and then by MRF as the shortest. The focus of this work is on the Harmonic Balance method, which is extensively used for periodic problems, mostly vibrations, acoustics and turbomachinery. Compared to conventional transient methods, the benefit of the Harmonic Balance method is the ability to capture transient flow features at significantly lower CPU time cost, while still being sufficiently accurate. This is presented by performing the comparison with transient approach and steady state MRF method, demonstrating the speed-up of at least an order of magnitude. Depending on the number of harmonics used, the size of the system of equations changes, as well as the accuracy. n number of harmonics yield a system of 2n + 1 coupled equations, where larger number of harmonics will take more time to converge, while offering greater accuracy as higher order effects do not get neglected. As the method is based on the Fourier series expansion, the frequency of the motion should be known in advance, suggesting that problems with imposed periodic motion are the most suitable for Harmonic Balance. In this work, the Harmonic Balance method is implemented in the Finite Volume framework within the open source software OpenFOAM, using a segregated pressure-based algorithm. Turbomachinery start-up and shut-down present a challenging problem for CFD investigation. Furthermore, change of operating points requires special attention as well. Depending on the type of the machine considered, the change of regime can take from several periods up to several dozen periods, making the simulations of such process highly expensive. The change of regime is a transient process during which the flow can change significantly and mass flow through the machine changes according to the newly reached regime. Rotor angular velocity can change as well in this process, making steady state simulations unusable. The modified version of the Harmonic Balance method is deployed here as a quasi-steady method in order to reduce the simulation time and capture the behavior during regime change. Non–periodic process such as start-up or shutdown is made periodic by considering both start-up and shut-down as a complete process. The period then consists of two complementary regime changes, with 2n + 1 coupled simulations throughout the period of start-up and shut-down. Due to two distinctive time-scales of rotor period (inner) and complete startup/ shut-down period (outer), a nested Harmonic Balance structure is deployed. Therefore, the 2n + 1 Harmonic Balance simulations for 2n + 1 time instants are interconnected with additional Harmonic Balance source term for the outer coupling. The validation of the time-spectral Harmonic Balance method is performed on industrially relevant test case of ERCOFTAC centrifugal pump by performing the comparison with other conventional methods (time-accurate, MRF) and with experimental data. Furthermore, the comparison of computational resources is performed in terms of CPU time, showing the Harmonic Balance method approximately 30 times faster than time-accurate simulation. The nested Harmonic Balance is validated using Francis-99 test case for shut-down and start-up processes. Comparison is performed against experimental data for power variation and pressure probes during complete period with good agreement achieved in a fraction of time otherwise needed by the time-accurate simulation.

Harmonic Balance, turbomachinery, nested, Fourier series, spectral, Francis

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